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RIN
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PETRIKUS OLUR
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TUEBOR
SCIENTIA
OF THE
QUÆRIS-PENINSULAM AMŒNAMA
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Received in ExchanGE
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caranda durdodaustaunti
668 Mathematics for Practical Mon, a
Common Place Book of Pure and Mixed Mathe-
matics, by O. Gregory, with engravings, thick
8vo, cloth, 1,50.
Phila. 1854
:
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MATHEMATICS
རྗ་
FOR PRACTICAL MEN:
A COMMON-PLACE BOOK
}
BEING
PRINCIPLES, THEOREMS, RULES, AND TABLES,
IN VARIOUS DEPARTMENTS
OF
OF
PURE AND MIXED MATHEMATICS,
WITH THEIR APPLICATION;
TO THE PURSUITS OF SURVEYORS, ARCHITECTS, MECHANICS,
AND CIVIL ENGINEERS.
↓
ESPECIALLY
WITH NUMEROUS ENGRAVINGS.
OLINTHUS GREGORY, LL.D., F.R.A.S.,
Corresponding Associate of the Academy of Dijon ; Ilonorary Member of the Literary and Philosophical Society of
New York; of the New York Historical Society; of the Literary and Philosophical, and the Antiquarian
BY
Societies of Newcastle-upon-Tyne; of the Cambridge Philosophical Society of the Institution
of Civil Engineers, &c. &c., and Professor of Mathematics in the Royal Military Academy,
FIRST AMERICAN FROM THE SECOND LONDON EDITION,
CORRECTED AND IMPROVED.
Only let men awake, and fix their eyes, one while on the nature of things, another while
on the application of them to the use and service of mankind."-Lord Bacon.
PHILADELPHIA:
E. L. CAREY AND A. HART, CHESNUT ST.
AND SOLD BY ALL THE PRINCIPAL BOOKSELLERS IN THE UNITED STATES.
1834.
in
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PRESIDENT;
AND TO THE VARIOUS
THOMAS TELFORD, Esq. F.R.S.E.
OF THE
-
OFFICERS AND MEMBERS
OF
/
INSTITUTION OF CIVIL ENGINEERS;
}
THIS COMPENDIUM
ENDIUM
¿
BY THEIR
MATHEMATICS FOR PRACTICAL MEN,
IS MOST RESPECTFULLY
DEDICATED,
FAITHFUL AND OBLIGED SERVANT
THE AUTHOR.
I
#
PREFACE.
THE work now presented to the public had its origin in a desire
which I felt to draw up an Essay on the principles and applications
of the mechanical sciences, for the use of the younger members of the
Institution of Civil Engineers. The eminent individuals who are
deservedly regarded as the main pillars of that useful Institution,
stand in need of no such instructions as are in my power to impart :
but it seemed expedient to prepare an Essay, comprised within
moderate limits, which might furnish scientific instruction for the
many young men of ardour and enterprise who have of late years
devoted themselves to the interesting and important profession, of
whose members that Institution is principally constituted. My first
design was to compose a paper which might be read at one or two
of the meetings of that Society; but, as often happens in such cases,
the embryo thought has grown, during meditation, from an essay to a
book: and what was first meant to be a very compendious selection
of principles and rules, has, in its execution, assumed the appearance
of a systematic analysis of principles, theorems, rules, and tables.
Indeed, the circumstances in which the inhabitants of this country
are now placed, with regard to the love and acquisition of knowledge,
impelled me, almost unconsciously, to such an extension of my
original plan, as sprung from a desire to contribute to the instruction of
that numerous class, the practical mechanics of this country. Besides
the early disadvantages under. which many of them have laboured,
there is another which results from the activity of their pursuits.
Unable, therefore, to go through the details of an extensive systematic
course, they must, for the most part, be satisfied with imperfect views
of theorics and principles, and take much upon trust: an evil, how-
ever, which the establishment of Societies, and the composition of
treatises, with an express view to their benefit, will probably soon
diminish.
LORD BROUGHAM, in his "Practical Observations upon the Education
of the People," remarks that " a most essential service will be rendered
A 2
V
vi
PREFACE.
to the cause of knowledge, by him who shall devote his time to the
composition of elementary treatises on the Mathematics, sufficiently
clear, and yet sufficiently compendious, to exemplify the method of
reasoning employed in that science, and to impart an accurate know-
ledge of the most useful fundamental propositions, with their appli-
cation to practical purposes; and treatises upon Natural Philosophy,
which may teach the great principles of physics, and their practical
application, to readers who have but a general knowledge of mathe-
matics, or who are even wholly ignorant of the science beyond the
common rules of arithmetic." And again, "He who shall prepare a
treatise simply and concisely unfolding the doctrines of Algebra,
Geometry, and Mechanics, and adding examples calculated to strike
the imagination of their connexion with other branches of knowledge,
and with the arts of common life, may fairly claim a large share in
that rich harvest of discovery and invention which must be reaped by
the thousands of ingenious and active men, thus enabled to bend their
faculties towards objects at once useful and sublime."
I do not attempt to persuade myself that the present volume will
be thought adequately to supply the desiderata to which the passages
advert; yet I could not but be gratified, after full two-thirds of it were
written, to find that the views which guided me in its execution ac-
corded so far with the judgment of an individual, distinguished as
Lord Brougham was, in early life, for the elegance and profundity
of his mathematical researches.
With a view to the elementary instruction of those who have not
previously studied mathematics, I have commenced with brief, but, I
hope, perspicuous, treatises on Arithmetic and Algebra; a competent
acquaintance with both of these being necessary to ensure that ac-
curacy in computation which every practical man ought to attain,
and that ready comprehension of scientific theorems and formulæ
which becomes the key to the stores of higher knowledge. As no
man sharpens his tool or his weapon, merely that it may be sharp, but
that it may be the fitter for use; so no thoughtful man learns arith-
metic and algebra for the mere sake of knowing those branches of
science, but that he may employ them; and these being possessed as
valuable pre-requisites, the course of an author is thereby facilitated:
for then, while he endeavours to express even common matters so
that the learned shall not be disgusted, he may so express the more
abstract and difficult that the comparatively ignorant (and the mere
knowledge of arithmetic and algebra is, in our times, comparative
ignorance) may practically understand and apply them.
Guth
1
vii
"O'
After the first 103 pages, the remaining matter is synoptical. The
general topics of geometry, L metry, conic sections, curves,
perspective, mensuration, statics, dynamics, hydrostatics, hydrody-
namics, and pneumatics, are thus treated. The definitions and prin-
ciples are exhibited in an orderly series; but investigations and
demonstrations are only sparingly introduced. This portion of the
work is akin in its nature to a syllabus of a Course of Lectures on the
departments of science which it treats; with this difference, however,
occasioned by the leading object of the publication, that popular
illustrations are more frequently introduced, practical applications
incessantly borne in mind, and such tables as seemed best calculated
to save the labour of architects, mechanics, and civil engineers,
inserted under their appropriate heads. Of these latter, several have
been collected from former treatises, &c., but not a few have been
either computed or contributed expressly for this Common-place Book.
In a work like this, it would be absurd to pretend to originality.
The plan, arrangement, and execution, are my own; but the materials
have long been regarded, and rightly, as common property. It has
been my aim to reduce them into the smallest possible space, con-
sistently with my general object; but, wherever I have found the
work in this respect prepared to my hands, I have transcribed it into
the following pages, with the usual references to the sources from
whence it was taken. They who are conversant with the best writers
on subjects of mixed mathematics and natural philosophy, will know
that Smeaton, Robison, Playfair, Young, Du Buat, Leslie, Hachette,
Bland, Tredgold, &c. are authors who ought to be consulted, in the
preparation of a volume like this. I hope it will appear that I have
duly, yet, at the same time, honourably, availed myself of the ad-
vantages which they supply. I have, also, made such selections
from my own earlier publications as were obviously suitable to my
present purpose; but not so copiously, I trust, as to diminish the
utility of those volumes, or to make me an unfair borrower even from
myself.
PREFACE.
Besides our junior Civil Engineers, and the numerous Practical
Mechanics who are anxious to store their minds with scientific facts
and principles; there are others to whom, I flatter myself, the follow-
ing pages will be found useful. Teachers of mathematics and those
departments of natural philosophy which are introduced into our
more respectable seminaries, may probably find this volume to occupy
a convenient intermediate station between the merely popular ex-
hibitions of the truths of mechanics, hydrostatics, &c., and the larger
ľ
viii
PREFACE.
FX
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i
•
treatises, in which the whole chain of inquiry and demonstration is
carefully presented, link by link, and the successive portions firmly
connected upon irrefragable principles. While students who have
recently terminated a scientific course, whether in our universities
or other institutions, public or private, may, I would fain believe, find
in this Common-place Book an abridged repository of the most
valuable principles and theorems, and of hints for their applications
to practical purposes.
The only performances with which I am acquainted, that bear any
direct analogy to this, are Martin's Young Student's Memorial Book,
Jones's Synopsis Palmariorum Matheseos, and Brunton's Compen-
dium of Mechanics; the latter of which I had not seen until the
present volume was nearly completed. The first and last mentioned
of these are neat and meritorious productions; but restricted in their
utility by the narrow space into which they are compressed. The
other, written by the father of the late Sir William Jones, is a truly
elegant introduction to the principles of Mathematics, considering the
time in which it was written (1706); but as it is altogether theoretical,
and is, moreover, now becoming exceedingly scarce, it by no means
supersedes the necessity, for such I have been induced to regard it,
of a Compendium like that which I now offer to the public.
In its execution I have aimed at no higher reputation than that of
being perspicuous, correct, and useful; and if I shall be so fortunate
as to have succeeded in those points, I shall be perfectly satisfied.
OLINTHUS Gregory.
Royal Military Academy,
Woolwich, October 1st, 1825.
!
In this new edition I have corrected a few errors which had escaped
my notice in the former impression. I have also made a few such
additions and improvements as the lapse of time and the progress of
discovery rendered desirable; and such as will, I hope, give the work
new claims on public approbation.
July 1st, 1833.
*
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C
Definitions and notation
Addition of whole numbers
Subtraction of whole numbers
Multiplication of whole numbers
Division of whole numbers
Proof of the first four rules
Tables of weights and measures
French and English weights and measures compared
Vulgar fractions
Sha
CONTENTS..
-
ARITHMETIC.
Sand
A
1
Addition, subtraction, multiplication, and division of ditto
Decimal fractions
↓
Powers and roots, square, cube, and higher roots
Proportion, rule of three, compound proportion, &c.
Properties of numbers
Numerical problem on the reduction of ratios
ALGEBRA.
Definitions and notation
Addition, subtraction, multiplication, division
Involution of monomials and polynomials.
Evolution
1
wh
1
Simple equations, extermination, &c.
Quadratic equations
Equations in general
Reduction of decimals
Addition, subtraction, multiplication, and division of ditto.
Complex fractions used in the arts and commerce
Addition, subtraction, &c. of ditto
Duodecimals
Progressions, arithmetical and geometrical
Principles of logarithms
Computation and use of formula
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1
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1
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1
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K
W
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Surds, reduction, addition, subtraction, multiplication, division,
involution, evolution
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AN
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PAGE
1
4
5
6
K
10
12
14
21
23
26
28
29
32
33
35
38
39
44
50
53
88529
60
67
70
22263
72
78
86
91
94
99
- 101
2
ix
X
CONTENTS.
Definitions
Angles, right lines, and their rectangles
Triangles
da
GEOMETRY.
Quadrilaterals and polygons
Circles, and inscribed and circumscribed figures
Planes and solids
Practical geometry
H
TRIGONOMETRY.
Definitions
CONIC SECTIONS.
Definitions, properties of the ellipse
Properties of the hyperbola and parabola
General application to architecture
Plane trigonometry
Determination of the heights and distances of objects
by approximate and mechanical methods
to chain bridges
Professor Farish's isometrical perspective
MENSURATION.
Mensuration of superficies.
Mensuration of solids
Approximate rules for both surfaces and solids
CURVES USEFUL TO ARCHITECTS, &c.
MECHANICS.
STATICS.
I
Parallelogram of forces, and applications
Centre of gravity
1
1
of structures
Equilibrium of piers
Pressure of earth against walls
1
(
I
Conchoid, cissoid, cycloid, quadratrix, catenary
Table of relations, strains, &c. in catenarian curves, applicable
1
1 1
B
1
1
1
1
1
1
1 1
11 I
PAGE
- 104
- 105
S
- 106
- 109
- 112
- 119
- 123
]
Ta
- 133
143
- 150
- 155
- 161
- 167
- 168
- 179
· 181
Váno
Mechanical powers, lever, wheel and axle, pulley, inclined plane,
wedge, screw
General application of the principles of statics to the equilibrium
- 195
- 196
- 198
- 206
- 209
ܝ
205
ཀ
214
222
222
224
CONTENTS.
xi
Equilibrium of polygons, centering roofs, &c. -
Stability of arches
Models:-Smart's mathematical chain-bridge
Definitions
Uniform motions
Motion uniformly accelerated
-
DYNAMICS.
on pulleys, inclined planes, &c.
Portable pendulum
Gridiron and other compensation pendulums
Table of lengths and vibrations of pendulums
Principles of rotation
Central forces, steam-engine governors, fly-wheels
Percussion or collision
Principles of chronometers, escapements, &c.
Select mechanical experiments
•
HYDROSTATICS.
C
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HYDRODYNAMICS.
1
Motions about a centre or axis
Pendulum, simple and compound, centres of oscillation, percus-
sion, and gyration -
1
PNEUMATICS.
1
Equilibrium of air and elastic fluids.
1
PAGE
- 226
- 230
- 232
1
Definitions
Pressure of non-elastic fluids
Illustrations and applications: hydrostatic paradox and bellows,
Bramah's press, embankments, strength of pipes of oak or
iron
Contrivances to measure the velocity of running waters
Effects of the old London bridge on the tides, &c.
Watermills: undershot, overshot, breast, Barker's
- 235
- 236
- 236
- 238
- 243
244
249
- 253
- 255
- 259
- 261
- 265
- 269
- 278
Floating bodies, buoyancy, Farey's self-acting flood gate
Specific gravities, Coates's hydrostatic balance, tables of specific
gravities and weights of various substances
284
- 285
-
Rules for weights of leaden pipes, and rims of cast-iron fly-
wheels
-
Definitions: motion and effluence of fluids
Motion of water in conduit pipes and open canals, over weirs,
&c., with various tables
G
Table of rise of water occasioned by piers of bridges and other
contractions
ہے
286
291
-
293
300
302
304
- 316
- 317
- 321
325
330
xii
CONTENTS.
PAGE
346
Pumps, sucking, lifting, forcing, fire-engine, Clark's quicksilver
pump, Archimedes's screw, spiral pump, hydraulic ram, &c. 333
Wind and windmills, Smeaton, Coulomb, &c.
Steam and steam-engines; Savery, Newcomen and Cawley,
Watt, Woolf, Oliver Evans, Fenton and Co., Nuncarrow,
&c.
.i
Useful tables and remarks on steam-engines, rail-roads, canals,
and turnpike roads
'
ACTIVE AND PASSIVE STRENGTH.
Active strength, or animal energy, as of men, horses, &c.
Schulze's experiments, Coulomb's, Bevan's, Morisot's, Regnier's,
Hachette's, Tredgold's, &c.
Passive strength, modulus of elasticity, &c.
Cohesive strength, modulus of cohesion, results of Leslie, Duha-
mel, Rennie, Bevan, &c.
SUPPLEMENTARY TABLES.
Of useful factors connected with the circle
Of circles; from which knowing the diameters, the areas, cir-
cumferences, and sides of equal squares are found
Of relations of the arc, abscissa, ordinate, and subnormal, in the
catenary
!
A
PLATES.
(To be placed at the end.)
1. Isometrical Perspective.
2. Steam-engines, Evans's, &c.
3. Fenton and Co.'s engine.
B
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med
353
- 386
402
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373
384
406
413
415
424
6
A
COMMON-PLACE BOOK,
&c. &c.
CHAPTER I.
ARITHMETIC.
SECTION I.-Definitions and Notation.
ARITHMETIC is the science of numbers.
We give the name of number to the assemblage of many
units, or of many parts of an assumed unit; unit being the
quantity which, among all those of the same kind, forms a
whole, which may be regarded as the base or element. Thus,
when I speak of one house, one guinea, I speak of units, of
which the first is the thing called a house, the second that
called a guinea. But when I say four houses, ten guineas,
three quarters of a guinea, I speak of numbers, of which the
first is the unit house repeated four times; the second is the
unit guinea repeated ten times; the third is the fourth part
of the unit guinea repeated three times.
In every particular classification of numbers, the unit is a
measure taken arbitrarily, or established by usage and con-
vention.
Numbers formed by the repetition of an unbroken unit are
called whole numbers, or integers, as seven miles, thirty shil-
lings: those which are formed by the assemblage of many parts
of a unit are called fractional numbers, or simply fractions;
as two-thirds of a yard, three-eighths of a mile.
When the unit is restricted to a certain thing in particular,
as one man, one horse, one pound, the collection of many of
those units is called a concrete number, as ten men, twenty
horses, fifty pounds. But if the unit does not denote any parti-
cular thing, and is expressed simply by one, numbers which are
constituted of such units are denominated discrete or abstract,
as five, ten, thirty. Hence, it is evident that abstract numbers
B
1
2
ARITHMETIC: NOTATION.
J
1
can only be compared with their unit, as concrete numbers are
compared with, or measured by, theirs; but that it is not possible
to compare an abstract with a concrete number, or a concrete
number of one kind with a concrete number of another; for
there can exist no measurable relations but between quantities
of the same kind.
The series of numbers is indefinite; but only the first nine
of them are expressed by different characters, called figures:
thus,
Names. one, two, three, four, five, six, seven, eight, nine.
Figures. 1, 2, 3, 4, 5, 6, 7, 8, 9.
Besides these, another character is employed, namely 0,
called the cipher or zero; which has no particular value of
itself, but by its position is made to change the value of any
significant figures with which it is connected.
In the system of numeration now generally adopted, and
borrowed from the Indians, an infinitude of words and cha-
racters is avoided, by a simple yet most ingenious expedient,
which is this:-every figure placed to the left of another
assumes ten times the value that it would have if it occupied
the place of the latter.
Thus, to express the number that is the sum of 9 and 1, or
ten units, called ten, we place a 1 to the left of a 0, thus 10.
So again the sum of 10 and 1, or eleven, is represented by 11;
the sum of 11 and 1, or of 10 and 2, called twelve, is repre-
sented by 12; and so on for thirteen, fourteen, fifteen, &c.
denoted respectively by 13, 14, 15, &c., the figure 1 being all
along equivalent to ten, because it occupies the second rank.
In like manner, twenty, twenty-one, twenty-two, &c. are
represented by 20, 21, 22, because the 2 in the second rank is
equivalent to twice ten, or twenty. And thus we may proceed
with respect to the numbers that fall between twenty and three
tens or thirty 30, four tens or forty 40, five tens or fifty 50,
six tens or sixty 60, seven tens or seventy 70, eight tens or
eighty 80, ninc tens or ninety 90. After 9 are added to the 90
(ninety) numbers can no longer be expressed by two figures,
but require a third rank to the left hand of the second.
The figure that occupies the third rank, or of hundredths, is
expressed by the word hundred. Thus 369, is read three
hundred and sixty-nine; 428, is read four hundred and twenty-
eight; 837, eight hundred and thirty-seven and so on for all
numbers that can be represented by three figures.
But if the number be so large that more than three figures.
are required to express it, then it is customary to divide it into
periods of three figures each, reckoning from the right hand
towards the left, and to distinguish each by a peculiar name.
-
ARITHMETIC: NOTATION.
3
The second period is called that of thousands, the third that
of millions, the fourth that of milliards or billions,* the fifth
that of trillions, and so on; the terms units, tens, and hundreds,
being successively applied to the first, second, and third ranks
of figures from the right towards the left, in each of these
periods.
Thus, 1111, is read one thousand one hundred and eleven.
23456, twenty-three thousands, four hundred and fifty-six.
421835, four hundred and twenty-one thousands, eight hun-
dred and thirty-five.
732846915, seven hundred and thirty-two millions, eight
hundred and forty-six thousands, nine hundred and fifteen.
The manner of estimating and expressing numbers we have
here described is conformable to what is denominated the deci-
mal notation. But, besides this, there are other kinds invented
by philosophers, and others indeed in common use; as the
duodecimal, in which every superior name contains twelve
units of its next inferior name; and the sexagesimal, in which
sixty of an inferior name are equivalent to one of its next supe-
rior. The former of these is employed in the measurement and
computation of artificer's work; the latter in the division of a
circle, and of an hour in time.
To the head of notation we may also refer the explanation
of the principal symbols or characters employed to express
operations or results in computation. Thus,
The sign + (plus) belongs to addition, and indicates that the
numbers between which it is placed are to be added together.
Thus, 5+7 expresses the sum of 5 and 7, or that 5 and 7 are to
be added together.
The sign (minus) indicates that the number which is
placed after it is to be subtracted from that which precedes it.
So, 9.- 3 denotes that 3 is to be taken from 9.
The sign denotes difference, and is placed between two
quantities when it is not immediately evident which of them is
the greater.
The sign (into), for multiplication, indicates the product
of two numbers between which it is placed. Thus 8 x 5 denotes
8 times 5, or 40.
The sign (by), for division, indicates that the number
which precedes it is to be divided by that which follows it;
and the quotient that results from this operation is often
* It has been customary in England to give the name of billions to millions of
millions, of trillions to millions of millions of millions, and so on: but the method
hore given of dividing numbers into periods of three figures instead of six, is uni-
versal on the continent; and, as it seems more simple and uniform than the othor,
I have adopted it.
4
ARITHMETIC NOTATION.
represented by placing the first number over the second with
a small bar between them. Thus, 15-8 denotes that 15 is to
be divided by 8, and the quotient is expressed thus.
The sign, two equal and parallel lines placed horizontally,
is that of equality. Thus, 2+3+4=9, means that the sum of
2, 3, and 4, is equal to 9.
Inequality is represented by two lines so drawn as to form
an angle, and placed between two numbers, so that the angular
point turns towards the least. Thus, 74, and A > B, indi-
cate that 7 is greater than 4, and the quantity represented by
A greater than the quantity represented by B: on the other
hand, 3 < 5 and C D indicate that 3 is less than 5, and C
less than D.
Colons and double colons are placed between quantities to
denote their proportionality. So, 3:59:15, signifies that
3 are to 5 as 9 to 15, or g
3 19/5.
The extraction of roots is indicated by the sign ✔,
figure occasionally placed over it to express the
root. Thus✔ 4 signifies the square root of 4, 27 the cube
root of 27, ✔ 16 the fourth or biquadrate root of 16; and so on.
These characters find their most frequent use in algebra and
the higher departments of mathematics; but may, without
hesitation, be employed whenever they secure brevity without
a sacrifice of perspicuity.
with a
degree of the
SECTION II.-Addition of Whole Numbers.
ADDITION is the rule by which two or more numbers are
collected into one aggregate or sum.
Suppose it were required to find the sum of the numbers
3731, 349, 12487, and 54.
It is evident that if we computed
separately the sums of the units, of the tens, of the hundreds,
of the thousands, &c. their combined results would still amount
to the same.
We should thus have 15 thousands + 14 hundreds
+20 tens 21 units, or 15000+1400+200+21; operating
again upon these, in like manner, rank by rank, we should have
10 thousands + 6 thousands + 6 hundreds + 2 tens + 1, or
16621, which is the sum required.
But the calculation is more commodiously effected by this
RULE.
Place the given numbers under each other, so that units
stand under units, tens under tens, hundreds under hun-
dreds, &c.
Add up all the figures in the column of units, and observe
for every ten in its amount to carry one to the place of tens in
ARITHMETIC: SUBTRACTION OF WHOLE NUMBERS.
5
:
the second column, putting the overplus figure in the first
column.
Proceed in the same manner with the second column, then
with the third, and so on till all the columns be added up; the
figures thus obtained in the several amounts indicate, according
to the rules of notation, the sum required.
Note. Whether the addition be conducted upwards or
downwards, the result will be the same; but the operation is
most frequently conducted by adding upwards.
57
762
5389
97615
Example. Taking the same numbers as before,
and disposing them as the rule directs, we have
4+7+9+1=21, of which we put down the 1 in
the place of units, and carry the 2 to the tens:
then 2+5 +8 + 4 + 3=22, of which we put down
the left hand 2 in the place of tens, and carry the
other to the hundreds: then 2+4+3+7
16, of
which the 6 is put in the place of hundreds, and the
1 carried to the thousands. This progress continued will give
the same sum as before.
103823
Other Examples.
77756
3388
9763
90257
6475
9830
2764
5937
25006
181164
?
RULE.
10376786
789632
1589
73
11168080
3731
349
12487
54
16621
SECTION III.-Subtraction of Whole Numbers.
SUBTRACTION is the rule by which one number is taken from
another, so as to show the difference or excess.
The number to be subtracted or taken away is called the
subtrahend; the number from which it is to be taken, the
minuend: the quantity resulting, the remainder.
Write the minuend and the subtrahend in two separate lines,
units under units, tens under tens, and so on.
Beginning at the place of units, take each figure in the
subtrahend from its corresponding figure in the minuend,
and write the difference under those figures in the same rank
or place.
But if the figure in the subtrahend be greater than its cor-
B 2
6 ARITHMETIC: MULTIPLICATION OF WHOLE NUMBERS.
I
responding figure in the minuend, add ten to the latter, and
then take the figure in the subtrahend from the sum, putting
down the remainder, as before; and in this case add 1 to the
next figure to the left in the subtrahend, to compensate for
the ten borrowed in the preceding place.
Thus proceed till all the figures are subtracted.
Note. It is customary to place the minuend above the sub-
trahend; but this is not absolutely necessary. Indeed, it is
often convenient in computation to find the difference between
a number and a greater that naturally stands beneath it; it is
therefore, expedient to practise the operation in both ways, so
that it may, however it occur, be performed without hesi-
tation.
Example:
pdate
i
Minuend
Subtrahend
Remainder
26565874
9853642
1.6712232
}
Here the five figures on the right of the subtrahend are each
less than the corresponding figures in the minuend, and may
therefore be taken from them, one by one. But the sixth
figure, viz. 8, cannot be taken from the 5 above it. Yet, as a
unit in the seventh place is equivalent to 10 in the sixth, this
unit borrowed (for such is the technical word here employed)
makes the 5 become 15. Then 8 taken from 15 leaves 7,
which is put down; and 1 is added to the 9 in the 7th place
of the subtrahend, to compensate or balance the 1 which was
borrowed from the 7th place in the minuend. Recourse
must be had to a like process whenever a figure in the sub-
trahend exceeds the corresponding one in the minuend.
Other Examples.
Take 21498 Take 45624
From 76262 From 80200
From 8217 From 44444
Take 3456 Take 3456
Remains 4761 Remains 40988 Remains 54764 Remains 34576
SECTION IV.-Multiplication of Whole Numbers.
MULTIPLICATION of whole numbers is a rule by which we
find what a given number will amount to when it is repeated
as many times as are represented by another number.
* This definition, though not the most scientific that might be given, is placed
here, because others depend implicitly, if not explicitly, on proportion, and there-
fore cannot logically be introduced thus early in the course.
ARITHMETIC: MULTIPLICATION OF WHOLE NUMBERS.
7
The number to be multiplied, or repeated, is called the multi-
plicand, and may be either an abstract or a concrete number.
The number to be multiplied by is called the multiplier, and
must be an abstract number, because it simply denotes the
number of times the multiplicand is to be repeated.
Both multiplicand and multiplier are called factors.
The number that results from the multiplication is called the
product.
Before any operation can be performed in multiplication, the
learner must commit to memory the following table of products,
from 2 times 2 to 12 times 12.
times 2

2
468
36912
4 8 12 16
6
5 10 15 20
7
8
10
3
11
14
12
12 18 24
16
4 5
9 18 27 36
20 30 40
21 28
24
22 33
24 36
32
44
48
10 12
15 18
20
6
30
25 30
35
45
36
24 28
42
40 48
7
14
50 60
21
42
49
35 40
56
54 63
70
8| 9 10 11 12
16 18
20
22
24
24
1
32 36
48
56
64
72
27
80
45
54
63
30 33
81
40
50
44
72 80
55
60 6
70 77
88
36
48
55 66 77 88
60 72 84 96
It is very advantageous in practice to have this table carried
on, at least intellectually, to 20 times 20. All the products to
this extent are easily remembered.
60
72
84
96
90 99 108
90 100 110 120
99 110 121 132
108 120 132 144
The learner will perceive that in this table 7 times 5 is equal
to 5 times 7, or 7 × 5 35 5 X 7. In like manner that 8
X 3 24 3 X 8, 4 x 11 = 44 11 × 4, and so of other
products. This is often made a subject of formal proof, as well
as that 3 × 5 × 8 3 X 8 X 5 = 5 × 3 × 8 = 5 × 8 × 3,
But to attempt the demonstration of things so nearly
axiomatical as these is quite unnecessary.
;
1
8 ARITHMETIC MULTIPLICATION OF WHOLE NUMBERS.
Previously to exhibiting the rules, let us take a simple
example, and multiply 4827 by 8. Here placing
the numbers as in the margin, and multiplying in
their order 7 units by 8, 2 tens by 8, 8 hundreds by
8, 4 thousands by 8, the several products are 56
units, 16 tens, 64 hundreds, 32 thousands: these
placed in their several ranks, according to the rules
of notation, and then added up, give for the sum of
the whole, or for the product of 4827 multiplied by 8.
the number 38616.
1
4827
8
56
16
64
32
38616
The same example may be worked thus:
8 X
7
8 X 20
8 X 800
8 4000
Multiply 4827
By
8
Products 38616
56
160
6400
32000
f
38616
CASE I. To multiply a number, consisting of several figures,
by a number not exceeding 12.
Multiply each figure of the multiplicand by the multiplier,
beginning at the units; write under each figure the units of the
product, and carry on the tens to be added as units to the pro-
duct following.
}
Examples.
218043
9
which is evidently
the same in effect
as before.
1962387
440052
11
1
4840572
8765400
12
105184800
CASE II. To perform multiplication when cach factor
exceeds 12.
Place the factors under each other (usually the smallest at
bottom), and so that units stand under units, tens under tens,
and so on.
Multiply the multiplicand by the figure which stands in the
unit's place of the multiplicr, and dispose the product so that
its unit's place shall stand under the unit of the multiplicand;
then multiply successively by the figure in the place of tens,
hundreds, &c. of the multiplier, and place the first figure of
cach product under that figure of the multiplier which gave the
said product.
The sum of these products will be the product required.
ARITHMETIC MULTIPLICATION OF WHOLE NUMBERS. 9
Multiply 8214356 by 132.
8214356 x2=
8214356 × 3 tens =
8214356 × 1 hundred
8214356 × 132=
Multiply 821436
by 672576
4928616
5750052
4107180
1642872
Example.
Other Examples.
5750052
4928616
Multiplicand 8214356
Multiplier
132
Product 552478139136
Multiply 8214356
by
11
And this product 90357916
by
12
16428712
}
Product as before 1084294992
24643068
8214356
Product 3450743782950
1084294992
Multiply 8210075
by 420306
Note.-Multiplication may frequently be shortened by sepa-
rating the multiplier into its component parts or factors, and
multiplying by them in succession. Thus, since 132 times any
number are equal to 12 times 11 times that number, the first
example may be performed in this manner:
49260450
24630225
16420150
32840300
Here one line of
multiplication,
and one of addi-
tion, are saved.
So, again, the multiplier of the second example, viz. 672576,
divides into three numbers, 600000, 72000, and 576; where,
}
10
ARITHMETIC: DIVISION OF WHOLE NUMBERS.
1
omitting the cipher, we have 72 = 12 × 6, and 576 = 8 × 72.
Hence the operation may be performed thus:-
Multiplicand 821436
Multiply by 6
in the 6th place
4928616
59143392 for 72 thousands.
473147136 for 576 units.
552478139136: three lines saved.
Previous product × 12
Second product x8
Same product as before .
Other modes of contraction will appear as we proceed.
SECTION V.-Division of Whole Numbers.
DIVISION is a rule by which we determine how often one
number is contained in another. Or, it is a rule by which,
when we know a product of one of the factors which produced.
it, we can find the other.
The number to be divided is called the dividend.
That by which it is divided, the divisor.
That which results from the division, the quotient, when
division and multiplication are regarded as reciprocal ope-
rations.
The dividend is equivalent to the product..
The divisor
The quotient
multiplier
multiplicand.
RULE.
Draw a curved line both on the right and left of the dividend,
and place the divisor on the left.
Find the number of times the divisor is contained in as many
of the left hand figures of the dividend as are just necessary,
and place that number on the right.
Multiply the divisor by that number, and place the product
under the above mentioned figures of the dividend.
Subtract the said product from that part of the dividend under
which it stands, and bring down the next figure of the dividend.
to the right of the remainder.
i
Divide the remainder thus increased, as before; and if at
any time it be found less than the divisor, put a cipher in the
quotient, bring down the next figure of the dividend, and con-
tinue the process till the whole is finished: the quotient figures
thus arranged will be that required.
ARITHMETIC: DIVISION OF WHOLE NUMBERS.
11
Sara an
}
Dividend.
Divisor 324)743256(2294 Quotient.
648
135
71
644
639
Remain.
71)29754(419
284
pap
66
64
952
648
21
16
3045
2916
1296
1296
56
56
8)38616(4827
32
#
Example.
Divide 743256 by 324.
5
77
i
131
407
393
I
146
131
:
131)135076(1031-15
Divisor 324
Quotient 2294
Divide 38616 by 8.
5 Remain.
15 Remain.
Note. When the divisor does not exceed 12, the operation
may readily be performed in a single line; as will appear very
evident if the following example be compared with the two
methods of working the first example in multiplication.
1296
2916
648
Proof 743256
Dividend 38616
Divisor 8
Quotient 4827
648
In these two ex-
amples the num-
bers which re-
main are placed
over their TC-
spective divisors,
and attached to
the quotients.-—
The meaning of
this will be ex-
plained when we
treat of frac-
tions.
Herc 8 in 38 go 4 times and 6 over ; these
carried as 6 tens to the next 6, make 66:
8 in 66 go 8 times and 2 over; these car-
ried as 2 tens to the next figure 1 make 21;
and so of the rest.
12
ARITHMETIC: DIVISION OF WHOLE NUMBERS.
J
In division, also, upon the same
principle as in multiplication, the
labour may often be abridged by
taking component parts of the
divisor. Thus, in the first example,
the divisor is equal to 4 times 81, or
4 times 9 times 9. Hence the di-
vidend may be divided by 4, 9, and
9, successively, as in the margin,
and the result will be the same as
before.
120675 Answer.
Sadd
5821 00
4
Divide 743256
by 4
23284
this quotient 185814
by 9
Since 25 is a fourth part of 100, and 125 the 8th part of 1000,
it will be easy to multiply or to divide by either of these num-
bers in a single line—thus,
"
and this
by
Quotient
To multiply 4827 by 25, put | To multiply 6218 by 125, put
two ciphers on the right, 3 ciphers, which is equivalent.
which is equivalent to multi- to multiplying by 1000; then
plying by 100; and divide divide by 8.
by 4.
4)482700
8)6218000
20646
9
2294
To divide 582100 by 25, strike | To divide 4567000 by 125,
off two figures on the right strike off three figures on the
hand, which is equivalent to right hand, which is equiva-
dividing by 100; then multiply lent to dividing by 1000; then
by 4.
multiply by 8.
4567 000
8
777250 Answer.
36536
Magi
Proof of the first four rules of Arithmetic.
Simple as these four rules are, it is not unusual to commit
errors in working them: it is, therefore, useful to possess
modes of proof.
1. Now, addition may be proved by adding downwards, as
•
·
ARITHMETIC: PROOF OF THE FOUR RULES.
well as upwards, and observing whe-
ther the two sums agree: or, by
dividing the numbers, to be added
into two portions, finding the sum
of each, and then the sum of those
two separate amounts. Thus, in the
margin, the sum of the four num-
bers is 7355, the sum of the two
upper ones 5857, of the two lower
ones 1498, and their sum is 7355,
the same as before.
What
2758
3099
469
1029
7355
2758
3099
469.
1029
5857
1498
13
7355
2. The proof of subtraction is effected by adding the re-
mainder to the subtrahend; if their sum agrees with the
minuend the work is right, otherwise not.
3. Multiplication and division reciprocally prove each
other.
There is also another proof for multiplication known tech-
nically by the phrase casting out the nines. Add together
the numbers from left to right in the multiplicand, dropping
9 whenever the sum excceds 9, and carry on the remainder,
dropping the nines as often as the amount is beyond them;
and note the last remainder. Do the same with the multiplier
and with the product; then if the product of the first two
remainders is equal to the last remainder, this is regarded as
a test that the work is right. Thus, taking the second ex-
ample in multiplication, the figures in the multiplicand
amount to 6 above two nines, those in the multiplier to 6
above three nincs, those in the product to 0 above six nines;
the product 6 x 6 of the two first excesses is 36, or 0 above
four nines: the coincidence of the two O's is the proof. It is
plain, however, that the proof will be precisely the same so
long as the figures in the product be the same, whatever be
their order: the proof, therefore, though ingenious, is de-
fective.*
A similar proof applies to division.
the multi-
* The correctness of this proof, with the exception above specified, may be shown
algebraically, thus-put A and N the number of nines in the multiplicand
and multiplier respectively, m and n their excesses; then, 9 M + m =
plicand, 9 N+n the multiplier, and the product of those factors will be 81
M N+ 9 M n + 9 Nm + mn; but the three first terms are cach a precise num-
ber of nines; because one of the factors in each is so; these, therefore, being neg-
lected, there remains mn to be divided by nine; but m n is the product of the two
former excessos; therefore the truth of the method is evident.
Q. E. D.
+
C
14
ARITHMETIC: WEIGHTS AND MEASURES.
丢
​WEIGHTS AND MEASURES,
Agreeably to the Act of Uniformity, which took effect
1st January, 1826.
The term Measure is the most comprehensive of the two,
and it is distinguishable into six kinds, viz:-
3. Solidity, or Capacity.
Measures of 4. Force of Gravity, or what is commonly called
Weight.
5. Angles.
6. Time.
The several denominations of these measures have reference
to certain standards, which are entirely arbitrary, and conse-
quently vary among different nations. In this kingdom,
The standard of
1. Length.
2. Surface.
69
The standards of Angular Measure, and of Time, are the
same in all European, and most other, countries.
12
3 Feet
5
40
8
1
Inches
Yards
Poles
Length is a Yard.
Surface is a Square Yard, the 4 of an
4840
Acre.
S Solidity is a Cubic Yard.
Capacity is a Gallon.
Weight is a Pound.
Furlongs
Miles
1. MEASURE OF LENGTH.
1 Foot.
= 1 Yard.
;
1 Rod or Pole.
1 Furlong.
1 Mile.
=1 Degree of a Great Circle of the Earth.
The imperial standard yard, when compared with a pendu-
lum, vibrating seconds of mean time in the latitude of London,
in a vacuum, at the level of the sea, is in the proportion of 36
inches to 39·1393.
Since the passing of this act, however, some very elaborate
and scientific experiments of Mr. Francis Baily have shown
that errors of sufficient moment to be taken into the account
in an inquiry of this kind render the above proportion inac-
curate. We do not, in fact, yet know the length of a seconds
ARITHMETIC: WEIGHTS AND MEASURES.
15
pendulum at London, vibrating in the circumstances pro-
posed.
The following standard yards, made with great accuracy, give
the annexed results:
General Lambton's scale, used in India
Sir Geo. Shuckburgh's scale
General Ray's scale
Royal Society's Standard
Ramsden's bar
Its copy, at Marischal College, Aberdeen
•
A Nail
An inch is the smallest lineal measure to which a name is
given; but subdivisions are used for many purposes. Among
mechanics the inch is commonly divided into eighths. By the
officers of the revenue, and by scientific persons, it is divided
into tenths, hundredths, &c. Formerly it was made to consist
of 12 parts, called lines, but these have properly fallen into
disuse.
Yard
Ell
Particular Measures of Length.
2 Inches
Quarter= 4 Nails
Hand
Fathom
Link
Chain
{, potent
4 Quarters
5 Quarters
4 Inches
6
Feet
7 Inc., 92
hdths.
= 100 Links
•
640
144 Square Inches
9 Square Feet
30 Square Yards
40 Perches
4 Roods
Acres
•
Used for measuring cloth of all
kinds.
2. MEASURE OF SURFACE.
Used for the height of horses.
Used in measuring depths.
Used in Land Measure to facili-
tate computation of the content,
10 square chains being equal to
an acre.
Inches.
35.99934
35.99998
36.00088
36.00135
36.00249
36.00244
1728 Cubic Inches
27 Cubic Feet
K
1 Square Foot.
= 1 Square Yard.
1 Perch, or Rod.
1 Rood.
1 Acre.
= 1 Square Mile.
3. MEASURES OF SOLIDITY AND CAPACITY.
Division I-Solidity.
1 Cubic Foot.
1 Cubic Yard.
16
ARITHMETIC: WEIGHTS AND MEASURES.
>
Division II.
Imperial Measure of Capacity for all liquids, and for all
dry goods, except such as are comprised in the Third Divi-
sion:
4 Gills
2 Pints
4 Quarts
2 Gall.
8 Gall.
8 Bush.
5 Qrs.
1 Pint
1 Quart
1 Gallon
1 Peck
1 Bushel
1 Quarter=
1 Load
34 Cubic Inches, nearly.
691
277
5542
2 Gallons 1 Peck
8 Gallons
1 Bushel
1 Sack
3 Bushels
12 Sacks
1 Chald.
2218
:
The four last denominations are used for dry goods only.
For liquids several denominations have been heretofore
adopted, viz. -For Beer, the Firkin of 9 gallons, the Kilder-
kin of 18, the Barrel of 36, the Hogshead of 54, and the Butt
of 108 gallons. These will probably continue to be used in
practice. For Wine and Spirits, there are, the Anker,
Runlet, Tierce, Hogshead, Puncheon, Pipe, Butt, and Tun;
but these may be considered rather as the names of the casks
in which such commodities are imported, than as expressing
any definite number of gallons. It is the practice to guage all
such vessels, and to charge them according to their actual
content.
101 Cubic Feet, nearly.
514
Flour is sold, nominally, by measure, but actually by weight,
reckoned at 7lb. avoirdupois to a gallon.
Division III.
Imperial Measure of Capacity for coals, culm, lime, fish,
potatoes, fruit, and other goods, commonly sold by heaped
measure:
Bushel
94 inches.
Half-bushel 15 inches.
704
2815
2
4
58 S
Cubic Inches, nearly.
Cubic Fect, nearly.
The goods are to be heaped up in the form of a cone, to a
height above the rim of the measure of at least of its depth.
The outside diameter of measures used for heaped goods are to
be at least double the depth; consequently not less than the
following dimensions:-
Peck
Gallon
Half-gallon 73 inches.
12 inches.
9 inches
ARITHMETIC: WEIGHTS AND MEASURES.
17
The Imperial Measures described in the second and third
divisions were established by act 5 Geo. IV., c. 74. Before
that time there were four different measures of capacity used in
England:-1. For wine, spirits, cider, oils, milk, &c. ; this was
one-sixth less than the Imperial Measure. 2. For malt liquor;
this was part greater than the Imperial Measure. 3. For
corn, and all other dry goods not heaped; this was part less
than the Imperial Measure. 4. For coals, which did not differ
sensibly from the Imperial Measure.
1
៩ ខ
1
33
The Imperial Gallon contains exactly 10lbs. avoirdupois of
pure water, at the temperature of 62° on Fahrenheit's ther-
mometer, the barometer being at 30 inches; consequently the
pint will hold 14lb., and the bushel 80lbs.
4. MEASURE OF WEIGHT.
Division I.—Avoirdupois Weight.
2711 Grains
16 Drams
16 Ounces
28 Pounds
4 Quarters
20 Cwt.
8. Pounds
14 Pounds
2 Stone
6½ Tod
2 Weys
12 Sacks
1 Dram
1 Ounce
1 Pound (lb.)
1 Quarter (qr.)
1 Hundred-weight (cwt.)
1 Ton.
This weight is used in almost all commercial transactions,
and in the common dealings of life.
Particular weights belonging to this division:-
cwt. qr. lb.
Used for Meat.
1 Stone
1 Stone
2711 Grains.
4371/ 2
7000
1 Tod
1 Wey
1 Sack
1 Last
24 Grains
20 Pennyweights.
12 Ounces
0 0 14
0 1 0
1 2 14
3 1 0
39 O 0
1
Division II.-Troy Weight.
= 1 Pennyweight
1 Ounce
1 Pound
Used in the Wool Trade.
24 Grains.
480
5760
These are the denominations of Troy Weight when used
for weighing gold, silver, and precious stones (except dia-
monds). But Troy Weight is also used by apothecaries in
compounding medicines, and by them the ounce is divided into
8 drams, and the dram into 3 scruples, so that the latter is equal
to 20 grains.
For scientific purposes the grain only is used; and sets of
C 2
18
ARITHMETIC: WEIGHTS AND MEASURES.
weights are constructed in decimal progression, from 10,000
grains downwards to of a grain.
By comparing the number of grains in the Avoirdupois and
Troy pound and ounce respectively, it appears that the Troy
pound is less than the Avoirdupois, in the proportion of 14 to
17 nearly; but the Troy ounce is greater than the Avoirdupois,
in the proportion of 79 to 72 nearly.
The Troy pound is equal to the weight of 22.815 cubic inches
of distilled water, weighed in air, temperature 62° Fahrenheit,
barometer at 30 inches.
1
100
1 lb. Avoirdupois
1 oz.
1 dr.
0 1 3
The carat, used for weighing diamonds, is 3 grains. The
term, however, when used to express the fineness of gold, has a
relative meaning only. Every mass of alloyed gold is supposed
to be divided into 24 equal parts; thus the standard for coin is
22 carats fine, that is, it consists of 22 parts of pure gold, and 2
parts of alloy. What is called the new standard, used for
watch-cases, &c. is 18 carats fine.
oz. dwts. grs.
14 11 15 Troy.
0 18 5/1/
5. ANGULAR MEASURE; OR, DIVISIONS OF THE CIRCLE.
60 Seconds
= 1 Minute.
60 Minutes
30 Degrees
60 Seconds
60 Minutes
24 Hours
7 Days
28 Days
1 Degree.
1 Sign.
90 Degrees
= 1 Quadrant.
360 Degrees, or 12 Signs 1 Circumference.
Formerly the subdivisions were carried on by sixties; thus,
the second was divided into 60 thirds, the third into 60 fourths,
&c. At present the second is more generally divided de-
cimally into 10ths, 100ths, &c. The degree is frequently so
divided.
28, 29, 30, or 31 Days
12 Calendar Months
365 Days
366 Days
365 Days
365 D. 5 H. 48 M. 45 S.
!
=
6. MEASURE OF TIME.
1 Minute.
1 Hour.
1 Day.
1 Week.
1 Lunar Month.
1 Calendar Month.
1 Year.
1 Common Year.
1 Leap Year.
= 1 Julian Year.
1 Solar Year.
ARITHMETIC: WEIGHTS AND MEASURES.
19
In 400 years, 97 are leap years, and 303 common.
The same remark, as in the case of Angular Measure, applies
to the mode of subdividing the second of time.
COMPARISON OF MEASURES
The old ale gallon contained 282 cubic inches.
The old wine gallon contained 231 cubic inches.
The old Winchester bushel contained 2150 cubic inches.
The imperial gallon contains 277-274 cubic inches.
The corn bushel, eight times the above.
Hence, with respect to Ale, Wine, and Corn, it will be ex-
pedient to possess a
TABLE OF FACTORS,
For converting old measures into new, and the contrary.
By Decimals.
By vulgar Fractions
nearly.
Winc Ale
Mea- Mea-
sure.
sure.

Corn Wine Ale
Measure. Measure. Measure.
To convert old
measures to new.
To convert new
measures to old
32
3 1
N. B. For reducing the prices, these numbers must all be reversed.
•96943 •83311
Royal
Medium
1-03153 1·20032. ⚫98324
Wove antique
Double elephant
Atlas
Columbier
Elephant
Imperial
Super-royal
SIZES OF DRAWING-PAPER
x 2 ft. 7
4 ft. 4
3 ft. 4
× 2 ft. 2
2 ft. 9 × 2 ft. 2
1·01704
1 Aum of hock contains
1 Barrel, imperial measure
anchovies
Coru
Mca-
sure.
2 ft.
2 ft.
2 ft. 5
2 ft. 3
2 ft. 0
1 ft. 10
soap
herrings
salmon or eels
31
32
94 × 1 ft. 11
93
32 × 1 ft. 101
× 1 ft. 91
× 1 ft. 7
x 1 ft. 7
X 1 ft. 6
MISCELLANEOUS INFORMATION.
oper
30 pounds
256 pounds
32 gallons
42 gallons
5
6/1
5
36 gallons
9981-864 cubic inches
60
59
988
5
60
20
ARITHMETIC: WEIGHTS AND MEASURES.
1 Bushel of coal
flour
1 Butt of Sherry
1 Chaldron of coals, with
ingrain
1 Chaldron of coals, without}
1 Chaldron of coals at Newcastle is
1 Clove of wool
1 Firkin of butter
99818.64 cubic inches
53 cwt.
[By an act of parliament in 1831, coals within 25 miles of the General Post
Office, London, must be sold by weight.]
soap
soap
1 Fodder of lead, at Stockton
at Newcastle
at London
1 Gross
1 Great gross
1 Hand
1 Hogshead of claret
tent
1 Hundred of salt
1 Keg of sturgeon
1 Last of salt
gunpowder
beer
pot-ash
cod-fish
herrings
meal
soap
pitch and tar
flax
feathers
wool
1 Pack of wool
1 Palm
1 Pipe of Madeira
Cape Madeira
Teneriffe
Bucellas
Barcelona
Vidonia
Mountain
Port
88 pounds
56 pounds
130 gallons
104809.572 cubic inches
•
▸
7 pounds
56 pounds
64 pounds
8 gallons
22 cwt.
21 cwt.
19 cwt.
12 dozen
12 gross
4 inches
58 gallons
63 gallons
7 lasts
4 or 8 gallons
18 barrels
24 barrels
12 barrels
12 barrels
12 barrels
12 barrels
12 barrels
12 barrels
12 barrels
17 cwt.
17 cwt.
4368 pounds
240 pounds
3 inches
110 gallons
110 gallons
120 gallons
140 gallons
120 gallons
120 gallons
120 gallons
138 gallons
1
ARITHMETIC: WEIGHTS AND MEASURES.
21
1 Pipe of Lisbon
1 Pole, Woodland
Plantation.
Cheshire
1 Sack of wool
1 Seam of glass
1 Span
1 Stone of meat
fish
•
•
•
(horseman's weight)
glass
wool
1 Tun of vegetable oil
animal oil
1 Tod of wool
1 Wey of cheese, in Suffolk
in Essex
1 Wey of wool
140 gallons
18 feet
21 feet
24 feet
364 pounds
124 pounds
9 inches
8 pounds
8 pounds
14 pounds
5 pounds
14 pounds
ENGLISH.
1 Inch (1-36th of a yard)
1 Foot (1-3d of a yard)
Yard imperial
Fathom (2 yards)
Polo, or perch (5 1-2 yards)
Furlong (220 yards)
Mile (1760 yards)
236 gallons
252 gallons
28 pounds
256 pounds
336 pounds
182 pounds
DIGGING.
24 Cubic feet of sand, or 18 cubic feet of earth, or 17 cubic
feet of clay, make 1 ton.
1 Yard cube of solid gravel or earth contains 18 heaped
bushels before digging, and 27 heaped bushels when dug.
27 Heaped bushels make 1 load.
FRENCH AND ENGLISH WEIGHTS AND
MEASURES COMPARED.
0
The following is a comparative Table of the Weights and
Measures of England and France, which were published by
the Royal and Central Society of Agriculture of Paris, in the
Annuary for 1829, and founded on a Report, made by Mr.
Mathieu, to the Royal Academy of Sciences of France, on
the bill passed the 17th of May, 1824, relative to the Weights
and Measures termed 66
Imperial," which are now used in
Great Britain.
MEASURES OF LENGTH.
FRENCH.
2.539954 centimetres
3.0479449 decimetres
0·91438348 metre
1-82876696 metro
5.02911 metres
201·16437 metres
1609-3149 metres
5
22
ARITHMETIC: WEIGHTS AND MEASURES.
FRENCH.
1 Millemetre
1 Centimetre
1 Decimetre
1 Metre
Myriamctre.
ENGLISH.
1 Yard square
1 Rod (square perch)
1 Rood (1210 yards square)
1 Acre (4840 yards square)
FRENCH.
1 Metre square
1 Are.
1 Hectare
ENGLISHI.
1 Pint (1-8th of a gallon)
1 Quart (1-4th of a gallon)
1 Gallon imperial
1 Peck (2 gallons)
1 Bushel (8 gallons)
1 Sack (3 Bushels)
1 Quarter (8 bushels)
1 Chaldron (12 Sacks)
FRENCH.
1 Litre
1 Decalitre.
1 Hectolitre
FRENCH.
1 Gramme
ENGLISH AVOIRDUPOIS.
1 Drachm (1-16th of an ounce)
1 Ounce (1-16th of a pound)
1 Pound avoirdupois imperial
•
SQUARE MEASURES.
•
ENGLISH TROY.
1 Grain (1-24th of a pennyweight)
1 Pennyweight (1-20th of an ounce)
1 Ounce (1-12th of a pound troy)
1 Pound troy imperial.
1 Hundred-weight (112 pounds)
1 Ton (20 hundred-weight) .
1 Kilogramme
ENGLISH.
0.03937 inch
0.393708 inch
3.937079 inches
·
39.37079 inches
3-2808992 feet
1·093633 yard
6.2138 miles
SOLID MEASURES.
FRENCH.
0.836097 metre square
25.291939 metres square
10.116775 are
0-404671 hectares
ENGLISH.
1-196033 yard square
0.098845 rood
2-473614 acres
FRENCH.
0.567932 litre
1.135864 litre
4.54345794 litres
9-0869159 litres
36-347664 litros
1.09043 hectolitre
2.907813 hectolitres
13.08516 hectolitres
ENGLISH.
1.760773 pint
0-2200967 gallon
2.2009667 gallons
22-009667 gallons
WEIGHTS.
FRENCH.
0.06477 gramme
1.55456 gramme
31-0913 grammes
0.3730956 kilogramme
FRENCH.
1-7712 gramme
28-3384 grammes
0-4534148 kilogramme
50-78246 kilogrammes
1015.649 kilogrammes
ENGLISH.
15-438 grains troy
0.643 pennyweight
0.03216 ounce troy
2-68027 pounds troy
2-20548 pounds avoirdupois.
ARITHMETIC: FRACTIONS.
23
i
SECTION VI.-Vulgar Fractions.
The fractions of which we have already spoken in section
the 1st, are usually denominated Vulgar Fractions, to distin-
guish them from another kind, hereafter to be mentioned, called
decimal fractions.
A fraction is an expression for a part of a unit, or integer,
when it represents a whole of any kind. Thus, if a pound
sterling be the unit, then a shilling will be the twentieth part
of that unit, and four pence will be four-twelfths of that
twentieth part. These represented according to the usual
notation of Vulgar Fractions, will be and of respec-
tively.
20
12
The lower number of a fraction thus represented (denoting
the number of parts into which the integer is supposed to be
divided) is called the denominator; and the upper figure (which
indicates the number of those parts expressed by the fraction)
the numerator. Thus, in the fractions, 8, 7 and 15 are
denominators, 5 and 8 numerators.
Vulgar fractions are divided into proper, improper, mixed,
simple, compound, and complex.
Proper fractions have their numerators less than their de-
nominators, as,, &c.
numbers as
Improper fractions have their numerators equal to, or greater
than, their denominators, as 4, 12, &c.
Mixed fractions, or numbers, are those compounded of whole
numbers and fractions, as 7, 123, &c.
Simple fractions are expressions for parts of given units, as
4 5. &c.
91 69
Compound fractions are expressions for the parts of given
fractions, as off, of, &c.
3
Complex fractions have either one or both terms mixed
1
20
2
3
Q
6%
52 12 65
24' 143' 123'
&c.
Any number which will divide two or more numbers with-
out remainder is called their common measure.
Reduction of Vulgar Fractions.
This consists principally in changing them into a more
commodious form for the operations of addition, subtraction,
&c.
Case 1.-To reduce fractions to their lowest terms:
Divide the numerator and denominator of a fraction by
24
ARITHMETIC FRACTIONS.
any number that will divide them both, without a remainder ;
the quotient again, if possible, by any other number and so
on, till 1 is the greatest divisor.
Thus, 1478
spectively are the divisors.
, where 5, 3, 7, 7, re-
2205
4)
294
441
Or, 14702, by dividing at once by 735.
2205
Note. This number 735 is called the greatest common
measure of the terms of the fraction: it is found thus-Divide
the greater of the two numbers by the less; the last divisor by
the last remainder, and so on till nothing remains: the last
divisor is the greatest common measure required.*
Case 2. To reduce an improper fraction to its equivalent
whole or mixed number.
97
86
98
147
11
Divide the numerator by the denominator, and the quotient
will be the answer: as is evident from the nature of division.
Ex.-Let 25 and 540 be reduced to their equivalent whole
or mixed numbers.
43
274
43)957(2211 Answer.
86
1 4
2
274)5480(20 Answer.
548
3
0
* The following theorems are useful for abbreviating Vulgar Fractions:
THEOREMS.
1. If any number terminates on the right hand with a cipher, or a digit divisible
by 2, the whole is divisible by 2: for the one which remains in the second place is
10; but 2 measures 10; therefore the whole is divisible by 2.
2. If any number terminates on the right hand with a cipher or 5, the whole is
divisible by 5; for every unit which remains in the second place is 10; but 5 mea-
sures every multiple of 10; therefore the whole is divisible by 5.
3. If the two right hand figures of any number are divisible by 4, the whole is
divisible by 4: for every unit which remains in the third place is 100; but 4
measures every multiple of 100; therefore the whole is divisible by 4.
4. If the three right-hand figures of any number are divisible by 8, the whole is
divisible by 8: for every unit which remains in the fourth place is 1000; but 8
measures every multiple of 1000; therefore the whole is divisible by 8.
5. If the sum of the digits constituting any number be divisible by 3 or 9, the
whole is divisible by 3 or 9.
6. If the sum of the digits constituting any number be divisible by 6, and the
right-hand digit by 2, the whole is divisible by 6: for by the data it is divisible both
by 2 and 3.
7. If the sum of the 1st, 3d, 5th, &c. digits constituting any number be equal
to that of the 2d, 4th, 6th, &c. that number is divisible by 11: for if a, b, c, d, e,
}
ARITHMETIC FRACTIONS.
25
!
Case 3.-To reduce a mixed number to its equivalent im-
proper fraction; or a whole number to an equivalent fraction
having any assigned denominator.
This is, evidently, the reverse of Case 2; therefore multiply
the whole number by the denominator of the fraction, and add
the numerator (if there be one) to obtain the numerator of the
fraction required.
Ex. Reduce 2211 to an improper fraction, and 20 to a
fraction whose denominator shall be 274.
M
(22 × 43) + 11 957 new numerator, and 957 the first
fraction.
43
20 × 274 5480 new numerator, and 5480 the second
fraction.
274
Case 4.-To reduce a compound fraction to an equivalent
simple one.
Multiply all the numerators together for the numerator, and
all the denominators together for the denominator, of the simple
fraction required.
If part of the compound fraction be a mixed or a whole num-
ber, reduce the former to an improper fraction, and make the
latter a fraction by placing 1 under the numerator.
When like factors are found in the numerators and denomi-
nators, cancel them both.
Ex.-Reduce of of of 7 of 8 to a simple fraction.
2 2 5
3
I X 5 X 4 20
2 X 3 X 5 X 7 X 8 2 X 5 X 8
3 X 4 X 7 X 9 X 11 4 X 9 X 11
1 X 9 X 11 99
Here the 3 and 7 common to numerator and denominator are
first cancelled; then the fraction is divided by 2; and then by
2 again.
Ex.-Reduce three farthings to the fraction of a pound
sterling.
A farthing is the fourth of a penny, a penny the twelfth of a
shilling, and a shilling the twentieth of a pound.
운
​ΤΣ
Therefore of 12 of 20
Ex.-Simplify the complex fraction
Here, reducing the mixed numbers to improper fractions, we
have multiplying by 3, to get quit of the denominator of
3
B
3
760
(8)
(7)
(6)
a,
a, + b,
b, + c,
where the odd terms are to the even.
D
(5)
c, + d,
1 X 5 X 8
2 X 9 X 11
1 the answer.
320
m, n, be the digits, constituting any number, its digits, when multiplied by 11, will
become
323/3/13
22
(4)
d, + e,
43
5
(3)
e, + m,
(2)
m + n
(1)
n;
26
ARITHMETIC: FRACTIONS.
8
5
the upper fraction, we have 72: multiplying by 5, to get quit
of the denominator of the lower fraction, we have 49; dividing
both terms of this fraction by 8, there results for the simple
fraction required.
Case 5. To reduce fractions of different denominators to
equivalent fractions having a common denominator.
Multiply each numerator into all the denominators except its
own, for new numerators; and all the denominators together
for a common denominator.
6
Ex.-Reduce 2, %, and, to equivalent fractions having a
common denominator.
2 X 7 X 9
6 × 3 × 9
5 × 3 × 7
3 X 7 X 9
J
K
126
162
the numerators.
105
189, the common denominator.
Hence the fractions are 126 162 105
breviated.
189 189 189
Hence, also, it appears that exceed, and that ex-
ceed 2.
3
B
or
5
Ex.-Reduce 4 of a penny, and of a shilling, each to the
fraction of a pound'; and then reduce the two to fractions having
a common denominator.
K
42
6 31
A
1200
1
30
20
2
of a penny = 4 of 11/2
of of
2 of a shilling of
2 1
Hence of a shilling are 10 times as much as of a penny.
3
20
GO
3
Note. Other methods of reduction will occur to the student
after tolerable practice, and still more after the principles of
algebra are acquired.
5 4
6 31 639
35, when ab-
3 of a pound.
1
300
10 of a pound.
300
Addition and Subtraction of Fractions.
RULE. If the fractions have a common denominator, add or
subtract the numerators, and place the sum or difference as a
new numerator over the common denominator.
If the fractions have not a common denominator, they must
be reduced to that state before the operation is performed.
In addition of mixed numbers, it is usually best to take the
sum of the integers, and that of the fractions, separately; and
then their sum, for the result required.
ARITHMETIC FRACTIONS.
27
1
1
12
1. Find the sum
5, and 3
4
of 2
2+ + 2 = 56 +82 +83=172-211.
5
3
32 72
6 0 6
84 84
3
84
84
4
12
2. Take of a shilling from of a pound sterling.
of a shilling of of a pound = 18
Also of a pound
=
4
Hence
480
Examples.
3 1
20
40
480*
1 1 11 pence.
240
3. Find the difference between 125 and 83.
125-8377 — 43-385-258
-
3
80
3. Multiply £2 13s.
£2 13s. 4d. =2 +
Examples
3
1. Multiply by 2, and divide & by 5.
小
​2 × 2 == 24
£4 15s. ÷ 31
117 =£1 8s. 6d.
440
4.
2. Multiply 22 by 2, and divide & by
3
8
8
de 2, and
223/2 × 23/03
11, ans.
Multiplication and Division of Fractions.
RULE 1. To multiply a fraction by a whole number, mul-
tiply the numerator by that number, and retain the denomi-
11, and ÷ 5
g
न
127
30
nator.
2. To divide a fraction by a whole number, multiply the de-
nominator by that number, and retain the numerator.
3. To multiply two or more fractions is the same as to take
a fraction of a fraction; and is, therefore, effected by taking
the product of the numerators for a new numerator, and of
the denominators for a new denominator.
new denominator. (The product
is evidently smaller than either factor when each is less than
unity.)
4. To divide one fraction by another, invert the divisor,
and proceed as in multiplication. (The quotient is always
greater than the dividend when the divisor is less than
unity.)
3
480°
19
小
​1 8
480
6
7X5
430.
Cla
22
48.0
6
35, ans.
1 × 555
X
4d. by 31, and divide £4 15s. by 31.
+ 4 of 1 2 =, and
1 3
20
g
× 31
12
20
G
28
1 × 13/14 = 500
X
£9
6s. 8d.
3
3
10
43/3 ÷ 3/13/1
-
3
19
10 × 130
ΤΟ
10
可
​57
4 0
28
ARITHMETIC,: FRACTIONS.
Note. In the multiplication of
mixed numbers, it is often less
laborious to perform the multipli-
cation of each part separately, and
collect their sum, as in the margin,
than to reduce the mixed numbers
to improper fractions, and reduce
their product back again to a mixed
number.
!
Thus, also, ·1
•01
·001
·0001
·7
SECTION VII.Decimal Fractions.
43
•125
7.3
12.85
57.217
is the same as
45 X 7
45 x 1 ten
3223232 X3
45
17 x
&c. &c. &c.
The embarrassment and loss of time occasioned by the com-
putation of quantities expressed in vulgar or ordinary fractions,
have inspired the idea of fixing the denominator so as to know
what it is without actually expressing it. Hence originate two dis-
positions of numbers, decimal fractions and complex numbers.
Of the latter, such, for example, as when we express lineal
measures in yards, in feet (or thirds of a yard), and inches (or
twelfths of a foot), we shall treat after a few pages. We shall
now treat of the former.
1
10
1
100
1
Decimal fractions, or substantively, decimals, are fractions
expressed as whole numbers, but whose values decrease from
the place of units progressively to the right hand in the same
decuple or tenfold proportion as the common scale of whole
numbers increase to the left. They are usually separated from
the integers by a dot placed between the upper part of the
figures. Thus, 227 expressed according to the decimal notation
is 22.7.
1000
1
1000 0
7
10
43
100
1 2 5
1000
Multiply 453
By 17/3/
315
45.
73
10
42.85
100
23/0
57.217
1000
•
14
2
.30
•
123
Product 808
1
ARITHMETIC: DECIMALS.
29
The value of a decimal fraction is not altered by ciphers on
the right hand for 500, or 50, is in value the same as, or
•5, that is 1.
2
When decimals terminate after a certain number of figures,
they are called finite, as 125
9 5 8
11, 958
1000
Kateg
237
250
When one or more figures in the decimal become repeated,
it is called a repeating or circulating decimal; as 333333, &c.
1,66666, &c.
•428571428571, &c.
and many
•
16
2
39
others.
Rules for the management of this latter kind of decimals are
given by several authors; but, in general, it is more simple and
commodious to perform the requisite operations by means of
the equivalent vulgar fractions, from which circulating decimals
are educed.
2)1.0
Reduction of Decimals
Reduction of Decimals is a rule by which the known parts
of given integers are converted into equivalent decimals, and
vice versa.
Example.
1. Reduce,,,, to equivalent decimals.
4)3.00
Case 1.-To reduce a given vulgar fraction to an equivalent
decimal.
4)7.0000
4)1.7500
Annex ciphers to the numerator, divide by the denominator,
and the quotient will be the decimal required
•5 decimal =
4375 decimal
1 2 5
1000
1-12-20
76
64
;
·
3235-789
75 decimal
7)
8)11.000
8) 1.375000
4/0
C.
•171875 decimal = 11.
64
6
D 2
30
ARITHMETIC DECIMALS.
2. Reduce and to equivalent decimals.
27 11/1 3
3)4.000000
7) 11.0000000
9)1.333333
9) 1.5714285714285
•1746031746031,
27
•148148, &c.
4.
27
decimal
decimal
These two are evidently circulating decimals, in the former
of which the figures 148 become indefinitely repeated, in the
latter the figures 174603.
3. Reduce 14s. 6d.
First 14s. 6d.
Then 22
7.25
10
40
399
daddy
4. Reduce 44 to its equivalent decimal.
57)44-000000(77192, &c, decimal 44.
410
399
110
57
530
513
63
170
114
to the decimal of a pound.
16 + 1 of 2 = 20 + 10
1 28 + 10 = 28·
29
14
--
20
20
40
40
=725, the decimal required.
3
56
Note. The above fraction is
1, of which the two
denominators are both prime numbers (that is, divisible by no
other number than unity), the entire equivalent decimal is a
circulator of 18 places, i. e. one less than the last prime ...
771929824561403508, 7719, &c. over again ad infinitum.*
* There are many curious properties of fractions
whose denominators are prime numbers, one of
which may be here shown in reference to frac-
tions having the denominator 7. The circulating
figures of the equivalent decimals are precisely the
same, for 4, 3, &c. and in the same order: the cir-
culate merely commences at a different place for
cach numerator.
·
1/11/
63
Pale che steret
O
&c.
•
∙14285714, &c.
•28571428, &c.
•42857142, &c.
•57142857, &c.
71428571, &c.
•85714285, &c.
ARITHMETIC: DECIMALS.
31
Case 2. Any decimal being given to find its equivalent
vulgar fraction; or to express its value by integers of lower de-
nominations.
When the equivalent vulgar fraction is required, place under
the decimal as a denominator a unit with as many ciphers as
there are figures in the proposed decimal; and let the fraction
so constituted be reduced to its lowest terms.
Or, if the value of the decimal be required in lower deno-
minations, multiply the given decimal by the value of its in-
teger in the next inferior order; and point off, from right to
left, as many figures of the product as there were places in the
given decimal.
I
Multiply the decimal last pointed off by the value of its
integer, in the next inferior order, pointing off the same number
of decimals as before and thus continue the process to the
lowest integer, or until the decimals cut off become all ciphers;
then will the several numbers on the left of the separating
points, together with the remaining decimal, if any, express the
required value of the given decimal.
Examples
1. Find the vulgar fractions equivalent to 25 and ·375.
•25 =
12050 ; and .375 1000
§3, answers.
1
375
2. Find the value in shillings, &c. of 528125 of a £.
.528125
20
10.562500
12
6.7500=63
3. Find the value of 74375 of an acre.
•74375
4
•
2.97500
40
Ans. 10s. 62d.
39.000
Ans. 2 roods 39 perches.
32
ARITHMETIC: DECIMALS.
Addition and Subtraction of Decimals.
These are performed precisely as in whole numbers, the
numbers being so arranged that units stand under units, tens
under tens, &c. or, which amounts to the same thing, that the
decimal points stand under one another.
Thus,
Add
together
43.7
39.1
437
421.75
Sum 465.5692
3933
1311
32.8165
•0027
11.
Multiplication and Division of Decimals.
Here, again, the operations are performed as in integers.
Then, in multiplication, let the product contain as many decimal
places as there are in both multiplier and multiplicand, ciphers
being prefixed, if necessary, to make that number; and, in
division, point off as many decimals in the quotient as the num-
ber in the dividend (including the ciphers supplied, if there be
any) exceeds that in the divisor
170.867
Examples.
1. Multiply 43·7 by 3·91, and 2·4542 by 0053.
2.4542
·0053 Here one cipher
is prefixed to
73626 make the requi-
site number of
decimals in the
122710
Here 4 3.7 × 3·91
X
3 9 1
100
170-867
437
To
1 7 0 8 6 7
1000
From 2486.173
Take
Remains 2471·60511
Proof 2486·17300
10009
as in the decimal
operation.
14.56789
1
01300726 product.
ARITHMETIC: DECIMALS.
33
2. Divide 172.8 by 144, and 192 by 5-423.
•144)172.8(1200' quotient.
144
288
288
..00
In the first of these examples the
two ciphers brought down, together
with the decimal 8, make the num-
ber of decimals in the dividend the
same as in the divisor, therefore the
quotient is entirely of integers. In
the second example, 3, the decimal
places in the divisor, taken from 8,
the decimal places in the dividend
(including those brought down),
leave 5 for the decimal places in the
quotient.
5.423)192.000 (35.40475
16269
1
20
29310
27115
21950
21692
1
21
25800
21692
41080
37961
31190
27115
4075
SECTION VIII.-Complex Fractions used in the Arts and
Commerce.
1
In the arts and in commerce, it is customary to assume a
series of units having a constant relation to each other, so that
the units of one denomination become fractions of another.
One farthing, for example, is of a penny, 1 penny of a
shilling, 1 shilling of a pound, or of a guinea. One
lineal inch, again, is of a foot, 1 foot of a yard, and so
on, according to the relations expressed in the tables at the
end of the fifth section. The arithmetical operations on
complex numbers of these kinds are usually effected by
simpler rules than those which apply to vulgar fractions
generally; of which it will therefore be proper here to specify
a few.
12
3
T2
Reduction.
Here we have two general cases:
Case 1.-When the numbers are to be reduced from a higher
denomination to a lower.
34
ARITHMETIC : COMPLEX NUMBERS.
1. Multiply the number in the higher denomination by as
many of the next lower as make an integer, or one, in that
higher, and set down the product.
2. To this product add the number, if any, which was in this
lower denomination before; and multiply the sum by as many
of the next lower denomination as make an integer in the pre-
sent one.
3. Proceed in the same manner through all the denominations
to the lowest, and the number last found will be the value of
all the numbers which were in the higher denominations taken
together.
Case 2. When the numbers are to be reduced from a lower
denomination to a higher.
1. Divide the given number by as many of that denomi-
nation as make one of the next higher, and set down what
remains.
2. Divide the quotient by as many of this as make one of the
next higher denomination, and set down what remains in like
manner as before.
3. Proceed in the same manner through all the denominations
to the highest; and the quotient last found, together with the
several remainders, if any, will be of the same value as the first
number proposed.
The method of proof is to work the question back again.
Examples.
1. Reduce £14 to shillings, pence and farthings; and 24316
farthings into pounds, &c.
14
20
280 shillings
12
3360 pence
4.
13440 farthings
I
24316
÷4.
6079 pence
-12
506 7
20
£25 6s. 7d.
ARITHMETIC: COMPLEX NUMBERS.
35
"
1
2. Reduce 22 Ac. 3 R. 24 P. into perches and 52187
perches into acres.
22 3 24
4.
1
91 roods
40
1
3664 perches
£ s.
Add 368 10
257 10
88 11
41
33 10 0
12 13 5
8 8
8
Sum 769 4 2
Addition and Subtraction.
Rule.-Write, one under the other, the parts which have
the same denomination, and operate successively on each of
them, beginning with the smallest. If any sum surpass the
number of units necessary to form one or more units of the
next superior order, put down the excess, and carry on the
other. Proceed similarly with regard to what is borrowed in
subtraction.
£ S.
d.
From 16 12 8 3
Take 10 11
6/1
Rem. 6 1 21
HIPK
23 10
d.
5
52187
40
1304 27
÷÷÷÷4
Ac. 326 0 R. 27 P.
Examples.
lb. oz. dwt. gr.
Add 14 6 12 13
17 5 3 12
15 0 9 16
2
7 15 20
13 2 10 19
4 1 5 21
£ S. d.
From 21 13 4
Take 18 9
83
Rem. 3 3 81
Į
lb. oz. dut. gr.
Add 10 8 11 17
42 5 16 12
12 2 14 18
51 6 0 22
24 9 17 17
29 4 18 22
Sum 66 11 18 5 Sum 171 2 0 12
16. oz. dwt. gr.
From 18 9 10 8
Take
9 10 15 20
Rem.
8 10 14 12
36
ARITHMETIC: COMPLEX NUMBERS.
Multiplication and Division.
1. In Compound Multiplication, place the multiplier under
the lowest denomination of the multiplicand.-Multiply the
number in the lowest denomination by the multiplier, and find
how many integers of the next higher denomination are con-
tained in the product, and write down what remains.-Carry
the integers, thus found, to the product of the next higher de-
nomination, with which proceed as before; and so on, through
all the denominations to the highest; and this product, together
with the several remainders, taken as one number, will be the
whole amount required.
If the multiplier exceed 12, multiply successively by its
component parts; as in the following examples.
2. In Compound Division, place the divisor and dividend
as in simple division.-Begin at the left hand or highest de-
nomination of the dividend, which divide by the divisor, and
write down the quotient.-If there be any remainder after this
division, find how many integers of the next lower denomi-
nation it is equal to, and add them to the number, if any,
which stands in that denomination.-Divide this number so
found, by the divisor, and write the quotient under its proper
denomination.-Proceed in the same manner through all the
denominations to the lowest, and the whole quotient, thus found,
will be the answer required.
Examples.
£ s. d.
a. r. p.
1. Multiply 4 17 61 by 441, and 3 2 14 by 531.
10
9 X 7 X 7 = 441
1
£ S. d.
4 17 61
9
43 17 101
77
307 4 112
7
Ans. £2150 14 101
35 3 20
10
358 3 0 for 100
5
1793 3 0 for 500
107 2 20 3 times 10
3 2 14 1 top line.
Ans. 1904 3 34
B
ARITHMETIC: COMPLEX NUMBERS.
37
1
4
£
S.
d.
2 Divide 521 18 6 by 432
432 12 × 12 × 3.
Therefore, by short division :
£521 18 6
12
3 12 5 × a farthing.
÷÷÷3
Quotient £1 4 12 × % of a farthing
89
20
432)1798(4
1728
By long division:
£ s.
d. £
S. d.
432)521 18 6(1 4 14 × of a farthing.
432
70
12
432)846(1
432
414
4
432)1656(3
1296
43 9 10
12
360
1
6
432
380=30=80
៖៖
6
5
6
i
1
7
E
$38
ARITHMETIC: DUODECIMALS.
Fractions whose denominators are 12, 144, 1728, &c. are
called duodecimals; and the division and sub-division of the
integer are understood without being expressed, as in decimals.
The method of operating by this class of fractions is principally
in use among artificers, in computing the contents of work, of
which the dimensions were taken in feet, inches, and twelfths of
an inch.
RULE.-Set down the two dimensions to be multiplied to-
gether, one under the other, so that feet shall stand under feet,
inches under inches, &c. Multiply each term in the multipli-
cand, beginning at the lowest, by the fect in the multiplier, and
set the result of each immediately under its corresponding term,
observing to carry 1 for every 12, from the inches to the feet.
In like manner, multiply all the multiplicand by the inches of
the multiplier, and then by the twelfth parts, setting the result
of each term one place removed to the right hand when the
multiplier is inches, and two places when the parts become the
multiplier. The sum of these partial products will be the an-
swer required.
{
Or, instead of multiplying by the inches, &c. take such parts
of the multiplicand as these are of a foot.
36
1
Duodecimais.
Examples.
1. Multiply 4 f. 7 inc. into 8 f. 4 inc.
or,
4 f. 7 i.
8
4
8
6 4
38 2 4
· 4 = }}
4 f. 7 i.
8
36
1
8
6 /
38 2
Here the 2 which stands in the second place does not denote
square inches, but rectangles of an inch broad and a foot long,
which are to be added to the square inches in the third place, so
that (2 × 12) × 4 28 are the square inches, and the product
is 38 square feet, 28 square inches
ARITHMETIC
3.9
DUODECIMALS.
1
¿
2. Multiply 35 f. 4 inc. into 12 f. 3 inc.
35 4
6
or,
12
3
4
424
434 3 11 0 0
6
0
8 10 1
1
1st) 2d
2
11 9 6 0
{
LO
Here, again, the product is 435 square feet, + (3 × 12) +
11 inches, or 434 square feet, 47 square inches. And this man-
ner of estimating the inches must be observed in all cases
where two dimensions in feet and inches are thus multiplied
together
SECTION IX.—Powers and Roots.
A power is a quantity produced by multiplying any given
number, called the root or radix, a certain number of times con-
tinually by itself. The operation of thus raising powers is
called involution.
5
3—3 is the root, or 1st power of 3;
3.x 3=32=9, is the 2d power, or square of 3
3 × 3 × 3 = 3³=27, is the 3d power, or cube of 3
3 × 3 × 3 × 3 = 34-81, do. 4th power or biquadrate of 3
&c. &c. &c.
Table of the first Nine Powers of the first Nine Numbers.
8
9
1
4
3d 4th 5th 6th
11
Jord
81
8
3 9 27 81 243
4 16 64 256 1024
16
1
32
1
64
424 6
3-1 8 10 11
3 of 3 11 9
729
7th
1
128
2187
35 4
12
434 3 11

4096 16384
78125
8th
1
256
9th
25 125 | 625 3125 15625
6 36 216 1296 7776 46656 279936
1679616 10077696
77
49 343 2401 16807117649 823543
5764801 40353607
64 512 4096 32768 262144 2097152 16777216 134217728
729 6561 59049 531441| 4782969 |43046721 387420489,
1
512
19683
6561
65536 262144
390625 1953125
40
ARITHMETIC: SQUARE ROOT.
2
2 8
2/3
9
So again, x-square of ;
cube of ;X
4 × 23/0
14, biquadrate of 2; and so of others. Where it is evident,
that while the powers of integers become successively larger
and larger, the powers of pure or proper fractions become suc-
cessively smaller and smaller.
Evolution.
8
27
Evolution, or the extraction of roots, is the reverse of in-
volution.
Any power of a given number may be found exactly; but
we cannot, conversely, find every root of a given number
exactly. Thus, we know the square root of 4 exactly, being
2; but we cannot assign exactly the cube root of 4. So again,
though we know the cube root of 8, viz. 2, we cannot exactly
assign the square root of 8. But of 64 we can assign both the
square root and the cube root, the former being 8, the latter 4.
By means of decimals we can in all cases approximate the
root to any proposed degree of exactness.
T
Those roots which only approximate are called surd roots, or
surds, or irrational numbers; as √2, 5, 9, &c., while
№ W/
those which can be found exactly are called rational; as 9
=3, 125=5, / 16=2.
4
A
To extract the square root.
RULE. Divide the given number into periods of two figures.
each, by setting a point over the place of units, another over
the place of hundreds, and so on over every second figure, both
to the left hand in integers, and to the right hand in decimals.
Find the greatest square in the first period on the left hand,
and set its root on the right hand of the given number, after
the manner of a quotient figure in Division.
Subtract the square thus found from the said period, and to
the remainder annex the two figures of the next following
period, for a dividend.
Double the root above mentioned for a divisor, and find how
often it is contained in the said dividend, exclusive of its right-
hand figure; and set that quotient figure both in the quotient
and divisor.
Multiply the whole augmented divisor by this last quotient
ARITHMETIC: SQUARE ROOT.
41
figure, and subtract the product from the said dividend,
bringing down to the next period of the given number for a
new dividend.
Repeat the same process, viz. find another new divisor, by
doubling all the figures now found in the root; from which,
and the last dividend, find the next figure of the root as
before; and so on through all the periods, to the last.*
Note. The best way of doubling the root, to form the
new divisors, is by adding the last figure always to the last
divisor, as appears in the following Examples. Also, after
the figures belonging to the given number are all exhausted,
the operation may be continued into decimals at pleasure,
by adding any number of periods of ciphers, two in each
period.
Examples.
1. Find the square root of 17.3056.
81
1
826
17.3056(4.16 the root in which the
number of decimal places is
the same as the number of
decimal periods into which
81 the given number was di-
vided.
130
16
4956
4956
*The reason for separating the figures of the dividend into periods or portions
of two places each, is, that the square of any single figure never consists of more
than two places; the square of a number of two figures of not more than four
places, and so on. So that there will be as many figures in the root as the given
number contains periods so divided or parted off.
And the reason of the several steps in the operation appears from the algebraic
form of the square of any number of terms, whether two or three, or more. Thus,
302 +2.30.5+52 or generally (a + b)²= a² + 2 a b + b² = a² +
2 a + b) b, the square of two terms; where it appears that a is the first term of
the root, and b the second term; also a the first divisor, and the new divisor is 2 a
+ b, or double the first term increased by the second. And hence the manner of
extraction is as in the rule.
E 2
42
ARITHMETIC EVOLUTION.
}
1
!
L
2. Find the square root of 2, to six decimals.
2(1.414213 root.
\
}
I
}
1
{
{
ļ
5
12
24
4
281
1
28282
124
2824 | 11900
4
11296
4
F
282841
|
1285
1
5
100
96
3. Find the square root of 2.
416666666, &c.
1
12904
2828423 10075900
8485269
4
|
1
2
400
281
|
0.416666(0.64549, &c.
36
3
|
60400
56564
566
496
383600
282841
1590631
7066.
6425
64166
51616
129089 1255066
1.161801
93265
I
J
1
!
1
1
1
ARITHMETIC: EVOLUTION.
43
De
Note. In cases where the square roots of all the integers
up to 1000 are tabulated, such an example as the above may
be done more easily by a little reduction. Thus ✔ 12
='645497, &c.
5
7.7459667
60
12
6
√(152 • 12) = √104 =
1
12
*
Cube and higher roots.
The rules usually given in books of arithmetic for the cube
and higher roots, are very tedious in practice: on which
account it is advisable to work either by means of approxi-
mating rules, or by means of logarithms. The latter is, gene-
rally speaking, the best method. We shall merely present
here Dr. Hutton's approximating rule for the cube root, which
may sometimes be serviceable when logarithmic tables are not
at hand.
·
RULE. By trials take the nearest rational cube to the given
number, whether it be greater or less, and call it the assumed
cube.
Then say, by the Rule of Three, as the sum of the given
number and double the assumed cube, is to the sum of the
assumed cube and double the given number, so is the root of the
assumed cube, to the root required, nearly. Or, as the first sum
is to the difference of the given and assumed cube, so is the as-
sumed root, to the difference of the roots, nearly.
Again, by using, in like manner, the cube of the root last
found as a new assumed cube, another root will be obtained
still nearer. And so on as far as we please; using always the
cube of the last found root, for the assumed cube.
Example.
To find the cube root of 21035·8.
Here we soon find that the root lies between 20 and 30, and
then between 27 and 28. Taking therefore 27, its cube is
19683, which is the assumed cube. "Then
44
ARITHMETIC CUBE ROOT, &c.
19683
2
39366
21035.8
21035.8
42071.6
19683
As 604018 : 61754·6::27 : 27·6047
27
2
1
4322822
1235092
459338
36525
60401-8) 1667374-2(27.6047 the root nearly.
Again, assuming 27.6
and working as before,
the root will be found
27.60491.
284
42
}
J
SECTION X.-Proportion.
Two magnitudes may be compared under two different
points of view, that is to say, either by inquiring what is the
excess of one above the other, or how often one is contained
in the other. The result of this comparison is obtained by
subtraction in the first case, by division in the second. The
ratio of two numbers is indicated by the quotient resulting
from dividing one by the other. Thus 3 may be regarded as
the ratio of 12 to 4, since 12 or 3 is the quotient of the numbers
12 and 4.
矍
​The first of two numbers constituting a ratio is called the
antecedent, the second the consequent.
The difference of two numbers is not changed by adding
one and the same number to each, or by subtracting the same
number from each.
Thus 12 5 = (12 + 2 − (5 + 2) = 14 — 7
(12 — 2)
(5 2) — 10 — 3.
In like manner, a ratio is not changed by either multiplying
both its terms, or dividing both its terms by the same
number.
!
1
42
༣
14
2
Thus 14
(4 ⋅ 3 =) 43 = ( 4 ÷ } = ) }·
.
Since surds enter arithmetical calculations by means of
1
·
i
45
ARITHMETIC PROPORTION.
their approximate values, a sufficiently precise idea may be ob-
√3 1.73205
tained of their ratio: thus,
This ratio is often
√2 1.41421
commensurable even with respect to surds: as
√(12÷3) √4 2
✓(3÷3) 1
1
✓12
√3
Equality of differences, or equidifference, is a term used to
indicate that the difference between two numbers is the same as
the difference between other two, or other two. Such, for
example, as 12 — 9 8—5—7 — 4.
Equality of ratios, or proportion, is similarly employed to
denote that the ratio of two numbers is the same
as that
between two others. Thus 20 and 10, 14 and 7, have 2
for the measure of the ratio: we have therefore a proportion
between 20 and 10, 14 and 7, which is thus expressed, 20: 10
:: 14: 7; and thus read 20 are to 10 as 14 are to 7. The same
proportion may also be represented thus, 20 14. Though,
by whatever notation it be represented, it is best to read or
enumerate it as above. It is true, however, that in all cases
when two fractions are equal, the numerator of one of them is to
its denominator, as the numerator of the other to its denominator.
In a proportion, as 20: 10 :: 14: 7, the second and third
terms are called the means, the first and fourth the extremes.
When the two means are equal, the proportion is said to be
continued. Thus 3:6 :: 6 : 12 are in continued proportion.
This is usually expressed thus 3: 6:12; and the second
term is called the mean proportional.
In the case of equidifference, as 1297-4, the sum of
the extremes (12 and 4) is equal to that of the means (9 and 7).
In like manner in a proportion, as 20: 10 :: 14: 7, the product
of the extremes (20 and 7) is equal to that of the means (10 and
14). The converse of this likewise obtains, that if 20 × 7
10 × 14, then 20: 10: 14: 7.
Hence, 1. If there be four numbers, 5, 3, 15, 9, such that the
products 5 × 9 and 3 × 15 are found equal, we may infer the
equality of their ratios, or the proportion 15, or 5: 3 ::
15: 9. So that a proportion may always be constituted with
the factors of two equal products.
可
​2. If the means are equal, their product becomes a square;
therefore the mean proportional between two numbers is equal
to the square root of their product. Thus, between 4 and 9 the
mean proportional is (4 x 9) = 6.
3. If a proportion contain an unknown term, such, for
example, as 53 15 the unknown quantity; since 5 times
:
5
8
46
ARITHMETIC PROPORTION.
the unknown quantity must be equal to 3 × 15 or 45, that
quantity itself is equal to 455 or 9. Generally one of the
extremes is equal to the product of the means divided by the
other extreme; and one of the means is cqual to the product of
the extremes divided by the other mean.
4. We may, without affecting the correctness of a proportion,
subject the several terms which compose it to all the changes
which can be made, while the product of the extremes remains
equal to that of the means. Thus, for 5: 3 :: 15:9, which
gives 5 × 9 = 3 × 15, we may
I. Change the places of the means without changing those
of the extremes, or change the places of the extremes with-
out changing those of the means: this is denoted by the term
alternando.
Thus 5: 3::15:9
become 5: 15: 3:9
9: 3:15:5
9:15:: 3:5
or
or
II. Put the extremes in the places of the means ; this is
called invertendo.
3:59: 15
III. Multiply or divide the two antecedents or the two con-
sequents by the same number.
14
7
It also appears, with regard to proportions, that the sum or
the difference of the antecedents is to that of the consequents, as
either antecedent is to its consequent.
And, that the sum of the antecedents is to their difference, as
the sum of the consequents is to their difference.
5+15
159
and
Thus
5+15 5 15
3+9 3 9
3+ 9
If there be a series of equal ratios represented by = 10
5
6+10+14-30 60
3+5+7+15
&c.
62
30
Therefore, in a series of equal ratios, the sum of the antecedents
is to the sum of the consequents, as any one antecedent is to its
consequent.
If there be two proportions, as 30: 15 :: 6 : 3, and 2: 3 :: 4:6,
then multiplying them term by term we shall have 30 × 2:
15 × 36 × 43 × 6, which is evidently a proportion, be-
cause 30 × 2 × 3 × 6 15 × 3 × 6 × 4
15
5
na
33898358
30 we shall have
Sat
Hence
1 0
5
1080.
Thus, also, any powers of quantities in proportion arc in
proportion; and conversely of the roots.
2:36:9
22: 32:: 62: 92
2:36:9
2:3 3: : 6
9
23: 33:: 63: 93
ARITHMETIC: RULE OF THREE.
47
Rule of Three.
When the elements of a problem will form a proportion of
which the unknown quantity is the last term, a simple calcula-
tion will determine it, and the problem is said to belong to the
Golden Rule, or Rule of Three. The operation is regulated
by the foregoing principles of proportion.
Of the three given numbers, two are called the terms of
supposition, and the other the term of demand. Now if the
term of demand be greater or less than the other term of the
same kind, and the question require the term sought to be re-
spectively greater or less than the other, the question belongs to
the Rule of Three direct: otherwise it belongs to the Rule of
Three inverse.
Matt
For the Rule of Three direct we have this
RULE. Write the three given terms in the following order,
viz. let that which implies or asks the demand be put in the
third place, and the other of the same kind in the first then
will the remaining term, which is similar to the fourth or re-
quired one, occupy the second place. Having thus disposed
the numbers, called stating the question, reduce the first and
third terms to one and the same denomination; and if the
second term be a compound one, reduce it to the lowest name
mentioned. Multiply the second and third terms together,
and divide the product by the first, and the quotient will be
the answer, in the same denomination to which you reduced the
second term.
When the second term is a compound one, and the third a
composite number, it is generally better to multiply the second
term, without any previous reduction, by the component parts
of the third, as in compound multiplication, after which divide
the compound product by the first term, or, by its factors. (Here
the first and third terms are homogeneous, in a given ratio, the
second and fourth in the same.)
For the Inverse Rule.
State the question and reduce the terms in the direct rule:
then multiply the first and second terms together and
divide the product by the third, and the quotient will be the
answer.
48
ARITHMETIC: RULE OF THREE.
1
1. If 3 gallons of brandy
cost 19s., what will 126 gal-
lons cost at the same rate?
1
gal. s. gal.
3:19:126: ?
I
19
(
1134
126
29
27
20
3) 2394 (798
21
24
24
1
1
Examples in the Direct Rule.
÷3
397. 18s. Ans.
98
÷7
{
3. If 21 yards of cloth cost
247. 10s. what will 160 yards
cost?
£
S.
Here, 21:24 10:: 160:?
X
4
}
392 0
X 10
0
X4
3920 O
•
1
1306 13 4
£
4×4×10=160 Here, 1
20
2. How much brandy may
be bought for 397. 18s. ; at
the rate of 3 gallons for 19
shillings?
£186 13s. 4d. Ans.
s. gal. £ S.
19: 3:39 18: ?
20
798s.
3
gal.
19) 2394 (126 Ans.
16
22
49
38
T
114
144
4. If selling by cloth at
17. 2s. per yard, 10 per cent.
is gained, what would be gain-
ed if it had been sold at 17. 5s.
per yard?
[
S.
£ $.
2:110::1 5:?
22
20
25
X 110
2)2750
11)1375
Amount £125
Deduct 100
Gain per cent. £25
ARITHMETIC :
49
RULE OF THREE.
5. What is the simple inte-
rest of 5601. for 5 years, at 4
per cent. per annum ?
£ £
I
f
£
Here, 100: 4 :: 560
4
100) 2240
£22.4
20
80
Interest for one year £22 8s.
y. £ S.
y.
Then 122 8::5
5
£112 0 Answer.
7. If a person travel 300
miles in 10 days of 12 hours
each, in how many days of 16
hours cach may he travel 600
miles?
|
First, if the days were of
the same length, it would be,
by direct proportion,
can
6. If 100 workmen
finish a piece of work in 12
days, how many men work-
ing equally hard would have
finished it in 3 days?
This example is manifestly
in the Rule of Three inverse.
d.
พ.
h.
Hence, 12: 100 :: 3 : ?
12
A distinct rule is usually given for the working of problems
in Compound Proportion; but they may generally be solved
with greater mental facility by means of separate statings.
Thus :
3 ) 1200
Answer 400 workmen.
8. If a family of 9 persons
spend £480 in 8 months, how
much will serve a family (liv-
ing upon the same scale) of 24
persons 16 months?
p. £ p.
First, as 9: 480::24:
= £1280.
K
480 X 24
9
m. d.
m.
As 300: 10 :: 600 : 20 days.
But these would be days of
12 hours each, instead of 16,
of which fewer will be re-
quired. Hence, by inversc
proportion,
h. d.
As 12: 20:16:
So that the answer is 15 days.
12 × 20
16
15.
Note. The Rule of Three receives its application in ques-
tions of Interest, Discount, Fellowship, Barter, &c.
F
11.
But this would only be the
expense for 8 months. Hence,
again,
£
112. £
As 8:1280:: 16:2560, the
expense of the 24 persons for
16 months.
Į
50
}
PROPERTIES OF NUMBERS.
Properties of Numbers.
To render these intelligible to the student, we shall here col-
lect a few definitions.
1. An unit, or unity, is the representation of any thing con-
sidered individually, without regard to the parts of which it is
composed.
2. An integer is either a unit or an assemblage of units;
and a fraction is any port or parts of a unit.
3. A multiple of any number is that which contains it some
exact number of times.
4. One number is said to measure another, when it divides
it without leaving any remainder.
5. And if a number exactly divides two, or more numbers, it is
then called their common measure.
6. An even number is that which can be halved, or divided
into two equal parts.
7. An odd number is that which cannot be halved, or which
differs from an even number by unity.
8. A prime number is that which can only be measured by
1, or unity.
9. One number is said to be prime to another when unity is
the only number by which they can both be measured.
10. A composite number is that which can be measured by
some number greater than unity.
2
3
11. A perfect number, is that which is equal to the sum of
all its aliquot parts: thus 6=&+ 8 +8.
+을​+
Prop. 1.-The sum or difference of any two even numbers
is an even number.
2. The sum or difference of any two odd numbers is even ;
but the sum of three odd numbers is odd.
but
3. The sum of any even number of odd numbers is even ;
the sum of any odd number of odd numbers is odd.
4. The sum or difference of an even and an odd number is
odd.
5. The product of an even or an odd number, or of two even
numbers, is even.
6. An odd number cannot be divided by an even number,
without a remainder.
7. Any power of an even number is even.
8. The product of any two odd numbers is an odd number.
9. The product of any number of odd numbers is odd ; and
every power of an odd number is odd.
10. If an odd number divides an even number, it will also
divide the half of it.
PROPERTIES OF NUMBERS.
51
11. If a number consist of many parts, and each of those parts
have a common divisor d, then will the whole number, taken
collectively, be divisible by d.
12. Neither the sum nor the difference of two fractions,
which are in their lowest terms, and of which the denominator
of the one contains a factor not common to the other, can be
equal to an integer number.
13. If a square number be either multiplied or divided by a
square, the product or quotient is a square; and conversely,
if a square number be either multiplied or divided by a
number that is not a square, the product or quotient is not a
square.
14. The product arising from two different prime numbers
cannot be a square number.
15. The product of no two different numbers prime to each
other can make a square, unless each of those numbers be a
square.
16. The square root of an integer number, that is not a com-
plete square, can neither be expressed by an integer nor by any
rational fraction.
17. The cube root of an integer that is not a complete
cube cannot be expressed by either an integer or a rational
fraction.
18. Every prime number greater than 2, is of one of the
forms 4 n + 1, or 4 n
- 1.
19. Every prime number greater than 3, is of one of the
forms 6 n + 1, or 6 n - 1.
20. No algebraical formula can contain prime numbers
only.
M
21. The number of prime numbers is unlimited.
22. The first twenty prime numbers are 1, 2, 3, 5, 7, 11, 13,
17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, and 67.
23. A square number cannot terminate with an odd number
of ciphers.
24. If a square number terminate with a 4, the last figure
but one (towards the right hand) will be an even number.
25. If a square number terminate with 5, it will terminate
with 25.
•
26. If a square number terminate with an odd digit, the last
figure but one will be even; and if it terminate with any even
digit, except 4, the last figure but one will be odd.
27. No square number can terminate with two equal digits,
except two ciphers or two fours.
28. No number whose last, or right hand, digit is 2, 3, 7, or
8, is a square number.
52
PROPERTIES OF NUMBERS.
29. If a cube number be divisible by 7, it is also divisible by
the cube of 7.
30. The difference between any integral cube and its root is
always divisible by 6.
31. Neither the sum nor the difference of two cubes can be a
cube.
32. A cube number may end with any of the natural num-
bers 1, 2, 3, 4, 5, 6, 7, 8, 9, or 0.
33. If any series of numbers, beginning from 1, be in con-
tinued geometrical proportion, the 3d, 5th, 7th, &c. will be
squares; the 4th, 7th, 10th, &c. cubes; and the 7th, of course,
both a square and a cube.
34. All the powers of any number that end with either 5 or
6, will end with 5 or 6, respectively.
35. Any power, n, of the natural numbers, 1, 2, 3, 4, 5, 6,
&c. has as many orders of differences as there are units in the
common exponent of all the numbers; and the last of those
differences is a constant quantity, and equal to the continual
product 1 × 2 × 3 × 4 ×
×n, continued till the last
factor, or the number of factors be n, the exponent of the powers.
Thus,
The 1st powers 1, 2, 3, 4, 5, &c. have but one order of
differences 1 1 1 1 &c. and that difference is 1.
The 2d powers 1, 4, 9, 16, 25, &c. have two orders of
differences 3 5 7 9
222
of which the last is constantly 2
1 x 2.
The 3d powers 1, 8, 27, 64, 125, &c. have three orders of
differences 7 19 37 61
12 18 24
6 6
of which the last is 6
1
5
•
•
2
5
1 X 2 X 3.
24
In like manner, the 4th, or last, differences of the 4th powers,
are each
1 × 2 × 3 × 4; and the 5th, or last
differences of the 5th powers, are cach 125 = 1 × 2 × 3 ×
4 X 5.
36. If unity be divided into any two unequal parts, the sum
of one of those parts added to the square of the other, is equal to
the sum of the other part added to the square of that. Thus,
of the two parts and,+()² = ÷ + (3)² = 2; so, again,
of the parts and 3, + (3) } + (3)² = 23·
4.
5
25
3
5
19
25
For the demonstrations of these and a variety of other pro-
perties of numbers, those who wish to pursue this curious line
+
A
}
1
PROPERTIES OF .
NUMBERS.
53
}
of inquiry may consult Legendre "Sur la Theorie des Nom-
bres," the "Disquisitiones Arithmetica" of Gauss, or Barlow's
"Elementary Investigation of the Theory of Numbers."
Also, for the highly interesting properties of Circulating
Decimals, and their connexion with prime numbers, consult
the curious works of the late Mr. H. Goodwyn, entitled "A
First Centenary," and "A Table of the Circles arising from the
Division of a Unit by all the Integers from 1 to 1024."
NUMBERSA useful Numerical Problem, to reduce a given fraction or
a given ratio, to the least terms; and as near as may be
of the same value.
RULE 1.-Let A, B, be the two numbers. Divide the latter
B by the former A, and you will have 1 for A; and some num-
ber and a fraction annexed, for B, call this C. Place these in
the first step.
}
Then subtract the fractional parts from the denominator, and
what remains put after C + 1, with a negative sign.
Then
throw away the denominator, and place 1 and that last number
in the second step. This is the foundation of all the rest.
If the fractional parts in both be nearly equal, add these two
steps together; if not, multiply the less by such a number
as will make the fractional parts, in both, nearly equal, and
then add. And this multiplier is found by dividing the greater
fraction by the less, so far as to get an integer quotient. When
you add the steps together, you must subtract the fractional
parts from one another, because they have contrary signs.
The process is to be continued on, the same way, adding the
last step, or its multiple, to a foregoing step, viz. to that which
has the least fraction.
Note. The ratios thus found will be alternately greater and
less than the true one, but continually approaching nearer
and nearer. And that is the nearest in small numbers, which
precedes far larger numbers: and the excess or defect of any
one is visible in the operation.
9
54
DETERMINATION OF RATIOS.
૭
3
Example 1.
To find the ratio of 10000 to 7854, in small numbers.
A
B.
1
4
5
7
8
9
10
11
1
12
1
3
4
5
9
14
210
219
233
452
904
13 1137
3+1416, 2d ratio.
4—0730, 3d ratio.
7+0686, 4th ratio.
11-0044, 5th ratio.
·0044 ) ⋅0686 ( 15
165-0660
14 4548
15| 5000
1
0+7854
1-2146, 1st ratio.
2146) 7854 (3)
3-6438
172+0026, 6th ratio.
183-0018, 7th ratio.
355+0008, 8th ratio.
·0008 ) ·0018 ( 2
710+0016
893-0002, 9th ratio.
·0002) ·0008 (4)
3572-0008
t
3927±0000, 10th ratio.
Explanation.
The ratio of 10000 to 7854 is the same as 1 to 0 + ·7854 or
1 to 1 2146; here I and 1 is the first ratio. But 2146 be-
ing less than 7854, divide the latter by the former, and you get
3 in the quotient, then multiply 1 and 1-2146 by 3, pro-
duces 3 and 3—6438 as in the 3d step. This third step added
to the first step produces 4 and 3 for the integers, and subtract-
ing the fractional parts, leaves 1416. So the 4th step is 4 and
In this
3 + ·1416; and the integers 4 and 3 is the 2d ratio.
manner it is continued to the end; and the several ratios ap-
proximating nearer and nearer are 1, 3, 4, 4, 11, 172, 183 2 3 5 3 2
1137 and 5000 Here is the nearest in small numbers, the
defect being only ro
5 9 14
233 452
57
893
3027
TT
44
T
1
55
DETERMINATION OF RATIOS.
{
(
E
1
Example 2.
To find the ratio of 268.8 to 282 in the least numbers.
1
1 Q
2
3,
4
5
6
7
1
2688) 2820 (1,132=2—3888.
2688
Į
19
20
1
1.
132) 2556 (19
40
41
61
8 183
9 224
132
48 ) 132 ( 2
12) 36 ( 3
1+0132, first ratio.
2-2556
19+2508
21- 48, 2d ratio.
96,
36, 3d ratio.
12, 4th ratio.
36
42-
F
43+
64-
2556
192-
235
"
5th ratio.
20 4 1 6 1
41
So the several ratios are 1, 1, 33, 84, 333,
fect or excess is plain by inspection, e. g.
the truth only parts; and 21, but 48 such parts.
43
3 6
And the de-
differs from
RULE 2.-Divide the greater number by the less, and the
divisor by the remainder, and the last divisor by the last re-
mainder, and so on until 0 remains. Then
1 divided by the first quotient, gives the first ratio.
And the terms of the first ratio multiplied by the second
quotient, and 1 added to the denominator, give the second
ratio.
And in general the terms of any ratio, multiplied by the next
quotient, and the terms of the foregoing ratio added, give the
next succeeding ratio.
F
!
56
DETERMINATION OF RATIOS.
Į
|
Į
Let the numbers be 10000 and 31416, or the ratio 10000
10000) 31416 (3
31416
30000
t
Example 3.
1
1416) 10000 (7
9912
1
22 × 16+3
113× 11+ 7
355 × 11+22
Then 1st or least ratio.
1x7
3X7+1
7×16+1
2688 ) 2820 (1
2688
or
1
or
88 ) 1416 (16
88
or
77
22
113
355
1250
3927
536
528
132) 2688 (20
264
8) 88 (11.
88
Example 4.
The ratio of 268.8 to 282 is required.
0
=2d ratio.
3d ratio.
4th ratio.
48 ) 132 (2
96
36) 48 (1
36
12) 36 (3
36
0
.
1
1
DETERMINATION OF RATIOS.
57
1
1
Then
= first ratio.
20
21
1 X 20
1x 20+1
20 × 2+1
21x2+
41 X1+20
43×1+21
1
61 × 3+41
64×3+43
or
or
or
or
41
43
1
61
64
224
235
F
J
1
2d ratio.
3d ratio.
4th ratio.
5th ratio.
1
t
1
"
}
}
58
ALGEBRA DEFINITIONS.
1
$
CHAPTRE II.
ALGEBRA.
1
SECTION I.-Definitions and Notation.
Algebra is the science of the computation of magnitudes in
general, as arithmetic is the particular science of the computa-
tion of numbers.
Every figure or arithmetical character has a determinate
and individual value; the figure 5, for example, represents
always one and the same number, namely, the collection of
5 units, of an order depending upon the position and use of
the figure itself. Algebraical characters, on the contrary,
must be, in general, independent of all particular significa-
tion, and proper to represent all sorts of numbers or quan-
tities, according to the nature of the questions to which we
apply them. They should, moreover, be simple and casy to
trace, so as to fatigue neither the attention nor the memory.
These advantages are obtained by employing the letters of
the alphabet, a, b, c, &c. to represent any kinds of magnitudes
which becomes the subjects of mathematical research. The
consequence is that when we have resolved by a single alge-
braical computation all the problems of the same kind proposed
in the utmost generality of which they are susceptible; the
application of the investigation to all particular cases requires
no more than arithmetical operations.
1
It is usual, though by no means absolutely necessary, to re-
present quantities that are known by the commencing letters
of the alphabet, as a, b, c, d, &c. and those that are unknown
by the concluding letters, w, x, Y, Z. But it is often conve-
nient, especially as it assists the memory, to represent any quan-
tity, whether known or unknown, which enters an investiga-
tion, by its initial letter, as sum by s, product by p, density
by d, velocity by v, time by t, and so of others.
Now, if s denotes the sum of four numbers represented by a,
b, c, and d, then, adopting the other symbols explained at
the beginning of arithmetic, we should express this algebrai-
cally by writing s=a+b+c+d.
Kad
ALGEBRA: NOTATION.
59
If the four quantities be all equal, or s = a + a + a + a,
this evidently reduces to s = 4 x a, or simply s = 4 a, drop-
ping the sign of multiplication, which is here understood.
The figure 4 is named the coefficient. In the quantities 3 a, 5
a, 7 a, na; 3, 5, 7, and n, are respectively the coefficients.
The continual product of three or more quantities is expressed
either by interposing the sign of multiplication, as a xbx c
× d; or by interposing dots, which have the same signification,
as a.b.c.d; or, lastly, by placing the letters in juxtaposition,
as a b c d.
When the quantities are equal, their continued multiplication
produces powers, as a a, a a a, a a a a, &c. which are usually
represented, instead of repeating the letters, by placing a figure
a little above the single letter, to expound or tell how many
equal factors are multiplied together; this figure is called the
exponent. Thus, instead of a a, a a a, a a a a, we put a², a³,
a¹, the figures 2, 3, 4, being the exponents.
Since roots are the reverse of powers, they are expressed by
exponents, which are the reciprocals of those that express the
corresponding powers. Thus the square root of a is represented
cither by a, or by a; the cube root of a + b, either by
yu + b, or by (a + b); the fourth root of a + b — c,
either by (a + b-c), or by (a + b — c)³.
-
We give the name term to any quantity separated from
another by the sign + or
A monomial has one term; a
binomial has two terms, as a + b, a c 4 ab; when the
second term of a binomial is it is frequently called a resi-
dual. A trinomial has three terms, as a + b + c, a d 4 ab
+5bc. A quadrinomial has four, as a+b+c-d. A
multinomial, or polynomial, has many terms.
>
A g
The signs and, which in arithmetic simply indicate
the operations of addition and subtraction, are employed.
more extensively in algebra, to denote, besides addition and
subtraction, any two operations or any two states which are
as opposed in their nature as addition and subtraction are.
And if, in an algebraical process, the sign is prefixed to a
quantity to mark that it exists in a certain state, position,
direction, &c., then, whenever the sign occurs in connex-
ion with such quantity, it must indicate precisely the con-
trary state, position, &c. and no intermediate one.
This is a
matter of pure convention, and not of metaphysical reasoning.
Other characters might have been contrived to denote this
opposition; but they would be superfluous, because the cha
60
ALGEBRA NOTATION.
>
racters
and, though originally restricted to denote
addition and subtraction, may safely be extended to other
purposes.
Thus if + a(added above
signifies any
to the right,
thing
forwards,
If+a sig-
nifies any
assigned
Increase,
Money due,
Gravity,
Motion upwards,
a sig-
nifies
subtracted below,
to the left,
backwards.
- a signi- (Decrease,
fies a cor-
respond-
ing
And so on in every kind of contrariety. And two such
quantities connected together in any case destroy each other's
effect, or are equal to nothing, as + a a = 0. Thus, if a
man has but 107. and at the same time owes 107. he is worth
nothing. And, if a vessel which would, otherwise, sail six
miles an hour, be carried back six miles an hour by a current,
it makes no advance.
Levity,
Money owing,
Motion downward.
Like signs are either all positive (+), or all negative (—).
And unlike are when some are positive and others negative.
If there be no sign before a quantity, the sign + is under-
stood..
An equation is when two sets of quantities which make an
equal aggregate are placed with the sign of equality ( = ) be-
tween them:
Thus, 3 a, +4 α, + 7 a,
5 a³ b,
KB)
As 12 + 5 20 - 3, or x + y a + b = cd.
The quantities placed on both sides the sign of equality are
called respectively the members of the equation.
SECTION II.-Addition and Subtraction.
1. Properly speaking, there is not in algebra either addi-
tion or subtraction, but a reduction, namely, the algebraic
operation, by which several terms are, when it is possible,
combined into one term. This, however, can only be effected
upon quantities that differ in their coefficients and their
signs, while they are formed of the same letters and the same
exponents.
3ab, 48 ab, are evidently reducible.
@b, }
ALGEBRA ADDITION AND SUBTRACTION.
In the first set, the incorporation gives (3+4+7) a=14 a,
in the second (5—3+8) a ³b=10 a ³b.
2. Generally, taking the similar terms the reduction affects
their coefficients, which are to be added when their signs are
alike, subtracted when they are different: in the first case,
give to the result the common sign; in the second, the sign
of the quantity having the greatest coefficient.
3. When quantities are presented promiscuously, it is best to
classify them, previously to the incorporation.
Thus, 3 a², 3 bc, + 2 c², +
4 d, + 7 a², +5 b c, + a², — 2 c²,
- 4 bc, when arranged become
as in the margin, and their sum
is readily obtained, as in the fourth
line.
Thus,
become
1
Result
4 a b —
(2 a b —
3 b c
6 b c)
4 ab
3bc
2 a b + 6 b c
2 a b + 3 b c
4. If it were required to subtract a residual, as b
c, from
a single term, as a; it is evident that the required difference
would not be changed if a quantity c were added to both. We
should then have to take b c+c, or b, from a + c, that is,
we should, have a + c b, for the difference sought, in which,
as is manifest, the signs of the letters b and c, which were to be
subtracted, have become changed.
Hence, to subtract a polynomial, change all the signs, and
reduce by incorporating the coefficients, where that is pos-
sible.
And
become
3 a³ +
7 a²
Q3
Result
K
5 b c 2 c²
3 b c + 4 d
4 b c +
2 c²
Kad
11 a² 2 b c + 4 d
da
61
4 ab- 3 c² + b c
-(ab
c²-2bc)
3 c³ + b c
4 a b
-ab +
c³ + 2 b c
3 ab + 2 c²+3 b c
Mag
5. In addition and subtraction of algebraic fractions, the
quantities must be reduced to a common denominator, and
occasionally undergo other reductions similar to those in
vulgar fractions in arithmetic and then the sum of the dif-
ference of the numerators may be placed over the common
denominator, as required.
10
G
62
ALGEBRA ADDITION AND SUBTRACTION.
Thus, 2+
+
Also,
And,
b2
c²
+
a b c a b c
a
ō
с
d
And,
a c + c x
a
S
bc
a + x
d
+
ad b c
+
b d
b
a c
+
с
a b
+
a² b2
a b c
b d
a² + b² + c² + a² b² + a² c² + b² c²
a b c
с
d
+
a-x
C
a d + d x
+
c d
b + x
h X
And, b-x
b + x
(b² + 2 bx+x²)—(b?
b2 x2
a d - b c
b d
a d + b c
b d
ab. ac
+ +
с b
a² c²
a b c
+
a c + c x
c d
a (c-
b2 c²
a b c
a d
2 b x + x²)
*
b c
a
dc
d) + x (c + d)
c d
d x
4 b x
b2
M
x2
a²
a b c
SECTION III.-Multiplication.
1. To multiply monomials, multiply their coefficients, add
together the exponents which affect the same letters (ascrib-
ing the exponent I to quantities which have none), then write
in order the coefficients and letters thus obtained.
:
2. To find the product of two polynomials, multiply each
term of the one into all those of the other, following the rule
given for monomials and observe to take each partial product
negatively when its factors have contrary signs, and positively
when they have the same signs. Or, briefly, observe that like
signs give +, unlike signs.
3. To multiply algebraic fractions, take the product of the
numerators for the new numerator, and that of the denomina-
tors for the new denominator.
Note. The general rule for the signs may be rendered
evident from the following definition: multiplication is the
ALGEBRA: MULTIPLICATION.
63
finding a magnitude which has to the multiplicand the pro-
portion of the. multiplier to unity. Hence, the multiplier
must be an abstract number, and, if a simple term, can have
neither nor prefixed to its notation. Now 1st, + a ×
+ m +ma, for the quality of a cannot be altered by in-
creasing or diminishing its value in any proportion; therefore
the product is of the quality plus, and m a by the definition is
the product of a and m. Secondly, a x + m =
m a,
for the same reasons as before, mutatis mutandis. Thirdly,
+ax m has no meaning; for m must be an abstract
number, therefore here we can have no proof. But + a x
(m n) = m a n a, n being less than m; for a taken as
often as there are units in m is = ma by the first case; but a
was to have been taken only as often as there are units in
n; therefore a has been taken too often by the units in n;
consequently a taken n times or n a, must be subtracted;
and of course m a na is the true product. Fourthly,
a × (228 n) ma + na. For a x m =
(by case 2); but this, as above, is too great by
na; there-
fore m'a with n a subtracted from it is the true product;
but this, by the rule of subtraction, is ma + na.
m
m a
T
2.
63+1
3.
1. 4 a b × 5 c d
4.
p
Sa + b
a + b
M
8 a³ b³ x 4 a5 b
32 at b¹.
a² + a b
J
Multiply 2 a + b c
By 2 a
b
Product 4 a³
Examples.
4.5.ab.cd
a b + b³
a² + 2 a b + bº
4 a² + 2 a b c
M
2 a b c
M
•
262
c + 2 b³
b³ c³
5.
4. a². a5. b³. b
daddy
20 a b c d.
A
4 a be
b³ c² + 2 b³ c
+ 4 a b² + 2 b³ c
+ 4 b³ c
Sa² + 2 ab+b³
a + b
a³ + 2 a³ b + a b³
K
32 a²+5
4 64
4 64
+ a² b + 2 a b² + bs
a³ + 3 a² b + 3 a b² + 63
64
ALGEBRA DIVISION.
1
7.
8.
a + b
C
2 x
a
2 a b c
X
S
X
}
α
d
3 a b
с
b
X
6.
Sa+b
га b
T
a² + a b
αγ
a²
(a + b) (a
c x d
3 ac
26
ped
b²
- b³
b) a³
18 a² b c x
2 a b c
M
c d
b³
[
SECTION IV.—Division.
9 a x
1
1
I a x.
2 a b c
Note. From the above examples (4, 5, and 6) we may
learn-
1. That the square of the sum of two quantities is equal to
the sum of the squares of the two quantities together with twice
their product.
X
I
2. That the product of the sum and difference of two quan-
tities is equal to the difference of their squares.
3. That the cube of the binomial a + b, is a³ + 3 a³ b+
3 a b² + b³.
S
1. To divide one monomial by another, suppress the letters
that are common to both, subtract the exponents which affect
the same letters, and divide the coefficients one by another.
2. To divide a polynomial by a monomial, divide each term
of the polynomial by the monomial according to the rule 1, and
connect the results by their proper signs.
3. To divide two polynomials one by the other, arrange them.
with respect to the same letter, then divide the first terms one
by the other, and thence will result one term of the quotient;
multiply the divisor by this, and subtract the product from the
dividend proceed with the remainder in the same manner.
Observe in the partial divisions the same rules for the determi-
nation of the signs as in multiplication.
ALGEBRA DIVISION.
65
4. To divide algebraic fractions, invert the terms of the
divisor, and proceed as in multiplication.
1. 12 a³ b³ c ÷ 3 a b
2. 15 a³ b5 ÷ 5 a² b³
bá
3. 6 x² + 12 x y
2 x + 4 y
9 x y z
3 x
4. Divide x³
X Z)x3
X3
J
—
5. Divide as
a
- b) a³ —
a5
Į
Ja
↓
Examples
1
12 a³ -162
3
3 x² z + 3 x z³
x² Z
2 x² z + 3 x z³
2 x² z + 2 x z²
D
Here the second term of
the dividend is brought
down to stand over the
corresponding term in
the last product.
3 x³ z + 3 x z²
S
15 a³ - 2 b5
5
9 x y z ÷ 3 x =
3 y z.
જ્ઞ
X Z3
X Z³
*
a4b
at b as bo
a³ bs
a³ b³
z³ by x
Z³(23
11
J
1 62 - 1 c
2
*
z3
23
Sp
b5 by a
b.
b³ (aª + a³ b + aª b³ + a b³ + b quotient.
a¹ b
· a bз
G 2
a³ bз
as f s
— 4 a³ b c.
3 a b³.
6 x³
3x
Na p
Z.
2 x z + z² quotient.
аза
+
pptatt
a b 4
a b 4
12 x y
3 x
65
b5
[
66
ALGEBRA DIVISION.
11
6. Divide 1 by 1
1 — x)1
7.
+
x2
8.
}
2 x2
a³ + x³
2 x
pak
уч
X
X2
XA
p
1-
a x + a²
X04
b4
b4
X
XC
Add
X.
X
a + x
(1 + x + x² + 203 + x4 +
1
X
x2
X2
X2
II
10. Divide 10 a³ + 11 a
15 b c² 5 b³ c by 3 a b +
11. Divide x² + y² +
X3
2 b x + b²
Xth
x² + b²
J
6º)
X
x (x + b) (x b)
x (x³
9. Divide 96 6 a¹ by 6-3 a.
+ 2 a³.
X3
X03
2 x³
a³ + x³
ba
x² + bx
X b
1
S
204
Badland
X04
X4
X
a + x
x¹
(x
205
+8
Pap
b 19 a b c
5 a³ 5 bc.
ენ
Y² by x + y +
X2
XC5
64
b)⁹
x+
b2
x
Quot. 16 + 8 a + 4 a³
X
XC
2 x² (a + x)
(α³ + x³) x
x_b
x(x+b)
15 aº c + 3 a b³ +
Quot. 2a + b 3 c.
J K
y³
Quot. xy
1
f
TL
ALGEBRA INVOLUTION.
67
SECTION V.-Involution.
1.—To involve or raise monomials to any proposed power.
Involve the coefficient to the power required, for a new co-
efficient. Multiply the index of each letter by the index of the
required power. Place each product over its respective letter,
and prefix the coefficient found as above: the result will be the
power required.
All the powers of an affirmative quantity will be + ; of a
negalive quantity, the even powers, as the 2d, 4th, 6th, &c.
will be; the odd powers, as the 3d, 5th, 7th, &c. will
be
To involve fractions, apply these rules to both numerator and
denominator.
Examples.
1. Find the fourth power of 2 a³.
2×2×2×2=16, new coefficient.
2
10
2×4=8, new exponent. Hence 16 as the answer.
2. The fifth
of
power 3 y² is 243 y¹º.
3. The fourth power of — 4 x³ is 256 x¹².
2 z 64 26
4. The sixth power of is
12
3 y2 729 y12
2. To involve polynomials.
Multiply the given quantity into itself as many times, want-
ing one, as there are units in the index of the required power,
and the last product will be the power required.
Cube xz and 2 x
x + z
x ± z
x² + x z
±x z + z²
x²º ± 2 x z + z²º squares.
x + z
x³± 2x² z+x z²
Example.
3 z.
土 ​x³ z + 2 x z² + z ³
x³±3x² + 3x zº+z³ cubes.
2 x
2 x
M
4 x² - 6 x Z
3 Z
3 z
T
8 x³
4x³ 12 x z +9 z³
3 z
2 x
6 x z +9 zº
24 x² z+18x z²
12 x³ z+36 x zª.
8 x³ — 36 x³ z +54 x x²
Men
27 23
27 z3
68
ALGEBRA INVOLUTION.
?
The operation required by the preceding rules, however
simple in its nature, becomes tedious when even a binomial
is raised to a high power. In such cases it is usual to employ
Sir Isaac Newton's Rule for involving a Binomial.
1. To find the terms without the coefficients. The index
of the first, or leading quantity, begins with that of the given
power, and decreases continually by 1, in every term to the
last; and in the following quantity the indices of the terms are
0, 1, 2, 3, 4, &c.
2. To find the unciæ or coefficient.-The first is always 1,
and the second is the index of the power and in general, if
the coefficient of any term be multiplied by the index of the
leading quantity, and the product be divided by the number of
terms to that place, it will give the coefficient of the term next
following.*
Note. The whole number of terms will be one more than
the index of the given power; and when both terms of the root
are +, all the terms of the power will be +; but if the second
term be, all the odd terms will be +, and the even terms
11
!
(a+b)"
2
n
Examples.
1. Let a + x be involved to the fifth power.
pak
3
The terms without the coefficients will be
1, 5,
*This rule, expressed in general terms, is as follows:
Katal
121
2
2
að, a¹ x, α³ x³, ɑ² x³, α x¹, x5,
a³, а³
and the coefficients will be
5 X 4 10 × 3 10 × 2 5 × 1
9
-"
2
3
4
5
And therefore the fifth power is
a5+5 a¹x +10 a³ x²+10 a² x³ +5 a x1+x.
an-3b3, &c.
=an + n. an -16 +2.
an-3 b³, &c.
A
12
(a - b)n =an n. an −1 b + n.
71
2
1
an-262 +n.
an-262_n.
22 + 1
2
71-
Sp
2
1
3
The same theorem applied to fractional exponents, and with a slight modifica-
tion, servos for the extraction of roots in infinito sorics; as will be shown a little
farther on.
1
ALGEBRA BINOMIAL RULE.
69
Here we have, for the sake of perspicuity, exhibited sepa-
rately, the manner of obtaining the several terms and their re-
spective coefficients. But in practice the separation of the two
operations is inconvenient. The best way to obtain the coeffi-
cients is to perform the division first, upon either the requisite
coefficient or exponent (one or other of which may always be
divided without a remainder), and to multiply the quotient into
the other. Thus, the result may be obtained at once in a single
line, nearly as rapidly as it can be written down.
x² + 7 x 6 z + 21 x5 z³ + 35 x1 ±³ + 35 x³ z¹
x z° + =7.
2. (x + y)7
+ 21 x² z5 + 7
3. (x — z)³
56 x
8 x7 z + 28 xº z² . 56 x5 z³ + 70 x¹ z¹
7
8
8 x z² + z³.
X8
528 x³ z6
For Trinomials and Quadrinomials.-Let two of the terms
be regarded as one, and the remaining term or terms as the
other; and proceed as above.
11
M
Example.
Involve x + y z to the fourth power.
z as
x01 +
2)¹,
Let x be regarded as one term of the binomial, and y
the other then will (x + y - 2)4 {x + (y − z)}^
4 x³ ( y − z) + 6 x³ (y z)² + 4 x (y · -2)² + (y
where the powers of y≈ being expanded by the same rule,
and multiplied into their respective factors, we shall at length
have x + 4x³ 4x³ ≈ + 6x³ y² 12 x³ y z + 6 x³ z³ +
4 x ys 12 x yº z + 12 x y z³. 4 x z³ + y¹
4 y³ ≈ + 6
Y³ z³ — 4 y z³, the fourth power required.
Y
Had (x + y) and z been taken for the two terms of the
binomial, the result would have been the same.
Note. The rule for the involution of multinomials is too
complex for this place.
▬▬▬▬▬▬▬
Maddy
70
r
Į
SECTION VI.—Evolution.
1. To find the roots of monomials.-Extract the correspond-
ing root of the coefficient for the new coefficient: then mul-
tiply the index of the letter or letters by the index of the root,
the result will be the exponents of the letter or letters to be
placed after the coefficient for the root required.
ALGEBRA EVOLUTION.
3.
Examples.
1. Find the fourth root of 81 a4 z8.
First 8193, new coefficient.
4
Then 4 x = 1, exponent of a, and 8 x = 2, expo-
1
nent of z.
Hence 3 a z² is the root required.
3 18 23
2729
2. §/(32 α5 x1º) = 5/32 × a³× 3
a5 ×x10×
3 X 1/1/0
13/27 XZ
9X3
2. To find the square root of a polynomial.-Proceed as
in the extraction of the square root, in arithmetic.
138 × x
×x10×13—2 а x³
2 x
3z3.
2 a³ +2 a x) 4 a³ x + 6 a² x²
4 a² x³
2
4 a³ x +
Examples.
1. Extract the square root of aª + 4 a³ x + 6 a² x² + 4 a x³
+24
a² + 4 a³ x + 6 a²x² + 4 a x³ + x¹ (α² + 2 a x + x²
a₁
[root req.
2 a² + 4 a x + x²) 2 a² x² + 4 α x³ +x¹
a
2 a²x² + 4 α x³ + x^
*
Tag
*
I
*
ALGEBRA : EVOLUTION.
71
2. Extract the square root of x4
x 1
X4
X04
G
St
2 x² x)
2 x³ + 23x²
ენ
X6
• 2 x8 + 23x²
2 x3 + X2
2 x² - 2 x + 1) 12/1
(x²)®×3=3x¹)
XC6
*
Xo
1. Find the cube root of x6
6 x + 1.
1 x²
god da
1/1/1 x 2
x²
2
*
· 6 x + 15 x¹
*
[
3. To find the roots of powers in general.—If they be not
the roots of high powers that are required, the following rule
may be employed:
Find the root of the first term, and place it in the quotient.
-Subtract its power, and bring down the second term for a
dividend.—Involve the root, last found, to the next lowest
power, and multiply it by the index of the given power for a
divisor. Divide the dividend by the divisor, and the quotient
will be the next term of the root.-Involve the whole root,
and subtract and divide as before; and so on till the whole is
finished.
2
+
2018 2018
*
6 25
6x5 + 12 x4
*
3 x4)3 24
6 x5 + 15 x
+16 (2³ — x + 1 root.
Examples.
+
2 x³ + 232 x² 1 x + 1/6 •
1
16
1
16
*
6 x5 + 15 xª 20 x³ 15 x³
Spa Ange
20 x³ +15 x³ — 6 x+1)x² — 2 x
[+ 1 root req.
8x8
20 x 15 x²-6x + 1
*
*
*
*
72
ALGEBRA: EVOLUTION.
1
2. Find the 4th root of 16 a¹
216 a x² + 81 xª.
16 a4
16 a¹
8 a³ × 4 = 32 a³)
16 a4
1
*
M
96 a³ x + 216 a²x²
96 as x
96 a³ x + 216 a² x²
*
96 a³ x + 216 a² x² — 216 a x³ + 81 x¹
!
- 216 a x³ +81 x¹ (2 α
[— 3x, root sought.
Note. In the higher roots proceed thus:
For the biquadrate, extract the
sixth root
eighth root,
ninth root,
square root of the square root.
cube root of the square root.
sq. rt. of the sq. rt. of the sq. rt.
the cube root of the cube root.
Examples, however, of such high roots seldom occur in any
practical inquiries.
*
SECTION VII.-Surds.
A Surd, or irrational quantity, is a quantity under a radical
sign or fractional index, the root of which cannot be exactly ob-
tained. (See ARITH. Sect. 9. Evolution.)
Surds, as well as other quantitics, may be considered as
either simple or compound, the first being monomials, as 3,
}
a
5
, a b³, the others polynomials, as √3 + √5, ŵa + √b
√c³, &c.
3
Rational quantities may be expressed in the form of surds,
and the operation, when effected, often diminishes subsequent
labour.
Reduction.
1. To reduce surds into their simplest expressions.
1. If the surd be not fractional, but consist of integers or in-
tegral factors under the radical sign:
Divide the given power by the greatest power, denoted by
the index, contained therein, that measures it without re-
mainder; let the quotient be affected by the radical sign, and
ALGEBRA : SURDS.
73
have the root of the divisor prefixed as a coefficient, or connected
by the sign X.
1. ✓75:
2. 3/448
3.
/176
4. √(8x³
√(2 x
4
5. 108 x³ y¹
4
3 x y № 4y.
6. №³/ (56 x³ y + 8 x³)
(7 y + 1) = 2 x
Examples.
√(25 × 3) = √25 × √3
13/(64 × 7) =✔/64 × ŵ/7
~/ (16x11)= /16 × 4/112/11.
12 x³ y) = √4 x² (2 x — 3 y) = √ 4x² ×
2 x 3 y).
3 y)
x³ y³ × 4 y) = √27 x³ y³ × № 4 y
kattapa
1. √/{} = √({} . 3)
5
2. √/} = √(} . §)
5
5
3.
6.
× 150 a x³
16
81
1
J = √ CD
X 18
લ
2. If the surd be fractional, it may be reduced to an equiva-
lent integral one, thus:
Multiply the numerator of the fraction under the radical
sign, by that power of its denominator whose exponent is
1 less than the exponent of the surd. Take the denominator
from under the radical sign, and divide the coefficient (whether
unity, number, or letter) by it, for a new coefficient to stand
before the surd so reduced.
√n.
25
4. 3/4= (4.28) = 3/108
125
2 a
5.
2 a
5 x
5 x
Note. This reduction saves the labour of actually dividing
by an approximated root; and will often enable the student
to value any surd expressions by means of a table of roots of
integers.
J &
1
pred
8
9
5 x
•
(27
18.
2
9
•
25 x²
D
Examples.
1
√3 = √3 × √3 = √3.
1
√28 = √25 × √5 =} √ 5.
5
25
ค
n
•
3/8 x³ (7 y + 1) 13/8 x³ X
(7 y + 1).
)
25 x2
50 a x³.
8
9.9
2 9
9
H
112
5/3.
3
4 13/7.
× √ n
11
115 × 100
50 a x²
125 x3
•
8
729
18
3
1
125 x³
8
729
100.
1
74
ALGEBRA SURDS.
3. If the denominator of the fraction be a binomial or re-
sidual, of which one or both terms are irrational and roots of
squares:
Then, multiply this fraction by another which shall have
its numerator and denominator alike, and each to contain the
same two quantities as the denominator of the given expression,
but connected with a different sign.
Note. By means of this rule, since any fraction whose nu-
merator and denominator are the same, is equal to unity, the
quantity to be reduced assumes a new appearance without
changing its value; while the expression becomes freed from
the surds in the denominator, because the product of the sum
and difference of two quantities is equal to the difference of
their squares.
8
1.
/3
√/5 — 、
4 (√5 + √3).
2.
√5
3.
16
4.
K
pagtatagalpatt
3
√5 + √2
√2.
√20 √/12
√/5 + √/3
260
+
Va b
/5+/3
8
• (a b).
8
√5 −√
D
Examples.
3
√/5
√5 + √
√/5 + √/2 * √/5 −√//2
√20
√5+ √3 /5.
√5 + √3' √/5-√//3
/3
√5+ √3
√5 + √3
√12
Vab
4/5 + 3
4/3
ναι
Kaladka ming
√5 −√/3
√/5
√5 + √3
√60 8 2 15.
2100
8 (5+ √3)
2
4/5—4/3
4/5-4/3
3
3 (1/5/2)
3
(5%
Ma
✔/100—2、/60+/36
5 3
5-
/5-
vab
1
1
3
52 353
32 — 3)
n
21
-1
3
Note 2.-Upon the same general principle any binomial or
residual surd, as AB may be rendered rational by
taking A ⇓ (A→B) + ŵ (Aª¬³ B³) ‡ ~/ (A "—^ B³)
+ &c. for a multiplier : where the upper signs must be taken
with the upper, the lower with the lower, and the series con-
tinued to n terms.
Thus, the expression
ab³, multiplied by ✔a +
Ya⁰ b² + 4 us be + /bº, gives the rational product a"-b".
6
8
ALGEBRA SURDS.
75
Malayalam, ka
2. To reduce surds having different exponents to equiva-
lent ones that have a common exponent.-Involve the powers
reciprocally, according to each other's exponent, for new powers:
and let the product of the exponents be the common exponent.
Nole. Hence, rational quantities may be reduced to the form
of any assigned root; and roots with rational coefficients may
be so reduced as to be brought entirely under the radical sign.
2. a2 and b³, become
2
3. 3 and 23, become
and /4.
Examples.
1
1. a and b, become an
1.m
πι
11. 131305
2
a
or a
}
or a
m
1 ก
m n and m n
b
3ª and 2
38 and 23,
3
and
✓8=√36=√3 × √6 = √ √6
302
or
Addition.
6
3
1 2
b
2 2
[
7
or bm
2
or o.
4. (a+b)³, and (a—b)³, become ⁄ (a+b)¹ and W/(a — b)³.
12
12
m n
5. The rational quantity a becomes a¹, a®, /a³, or
am.
6. 4 a 5 b, becomes (4 a)³ × ŵ5 b, № 64 a³ × 5 b, or
320 a³ b.
33 and 2/2 or 2/27
These and other obvious reductions, which will at once sug-
gest themselves, being effected, the operations of addition, sub-
traction, &c. are so easily performed upon such surd quantities as
usually occur, that it will merely suffice to present a few exam-
ples without detailing rules.
Ex. 1. √8+√/18=√(4.2) +√/(9.2)=2 √2+3 √2=5√2.
8
2. Add together √54, √}, and ✔✅ 27.
✓54=√(9.6)=√✅✔✅/9× √6=3√6]
M
(3+}+})√6=3,7 √6.
---
The sum of these is
8
4
√27=√17+3)=√ H= 3√6
3. 27 ux+ √3 aª x= √(9 a¹. 3 x)+(a². 3 x)=3 a³√3
x+a/3x=(3 a²+a) 3 x.
√✅✅
4. 8 Y a³ b+ Ÿ aº b= (8 va³×ŵb)+(Waª×ŵb)=8 ab
+a³ ✔b=(8 a+a³) vb.
76
ALGEBRA: SURDS.
Ex. 1. 2√✓/50—√✓/18=2√(25.2)—✓✓(9.2)=2.5√✓/2—3√✓/2—
(10-3)/2=7√2.
2. √3—√5=√(§•§)—√(57•})=√}}−√}}=} √15 —
27
1
27
4
9
15=√15.
3. 13/2/ .
13/9
3 ½ = 1/ (2·3) — ✔ (3½·2)=√1;— 3/18
蛋 ​18-418=18.
2
3.9
64
2
V
4. 250 a³ x — ✅ 16 a³ x=✔ (125 a³.2 x) — (8 a³. 2 x)
= 5 a 3/2 x—2 a 3/2 x=3 a 3/2 x.
Subtraction.
10. am xa
5. √45 s¹ x— √20 s² x³= √(9 s¹. 5 x) — √(4 s² x². 5 x)=
(3 s³
2 s x) √5x.
Jarak
12
a
c²
c²
··(9) * - () * - ; × (-_-) — ~ × (-1) = "=" (1) ".
W
6.
X
X
a
с
a
а с
а с
a
+
·12
Multiplication.
Ex. 1. 18x54=5 (18.4)=5 (4.2.9) 5 (8.9) =
5.2 3/9-10 9.
7
X & TO
2. & √ 1 × √ √6=}•£ √(1·7%) =¿ √ √ %=
2 8
√(6)=√100-220 350
7
ठ ठ
8
1
7 5
35
1
35.
8
3
2
2
3
3
× a ¹³ — a³ + ½ — a¹³ï + îî_1.
12 12
3. a³ x a
α
a
XX
4. (x+z)² ×(x+z)* =(x+z)
*=(x+z)² + *=(x+z)..
5. (x+ √y)×(x — √y)=x³-y.
6. (x+ √y)*×(x−√/Y)*=(x²—y)³.
1
m
a
11. √—a× √-
12. ✓ a × √
✔ab x 1
9
1
am x
an
m-In
1
8
27
1
7. z n x 2m = zn
X zm
*
6
8. dx ab= d³ xa³ b³=a* b² d³.
9. ✔ab√3.X √a+ √b−√3= √(aª—b+ √3.)
am-n
12
Z m n
=
Jordan)
-1
√α√−1 × √α√—1
- b
ναν 1 × √ √
Vab.
Parametra
-C.
1
ALGEBRA SURDS.
77
1
L
الله
a
Ex. 1.1000÷24-41000=2/250=
2(125.2)=102.
2. & 3 ÷ ÷ 3 3/3=4·5 (2·4)=25 / 8 =
5 2 3/
133/
7
4.
7 2
14
25 8.3
24 V (3·3) = 24 (4·3)=25·23=213.
/
3/3:
8
27
14
14
9
2
4
2
3. x³÷x³
1
X
T
Jo
26
a2 x
1
a
m
("")
a x2
x
1
I
-
1
3 3 -x ³÷1÷÷÷x³.
1
4. zz
5. (x-xd-b+d √/b)÷(x — √b)=x+ √b-d.
ас a d
α.
a cad
6.
✓ (α³ x
a² x − a x³) ÷ 2 b √ (a — x) =
26
26
Z"
}
TA
Ja d
✓ (a±b) = √ata
2
2
3
Division.
m-n
m n
土
​= (cd) √ ax.
Ex. 1. (z a³)²=z ·z ·a¹¹¹=‡ a³=‡ ✔a².
+
3
3
1
√
27
8
1
1
1
16
2. (} })³· } · J · √✓ (} ·↓· }) ==
27 √(6·2)=27 √(76•2)=37·4 √2=8B √2.
3. (3+ √5)²={(3+ √5) (3+ √5)}=14+6 √5.
4. (ab)³a³ — 3 a² √b+з a b
108
2
3
3
bb.
F
Involution.
Evolution.
Ex. 1. √10³= √1000= √(100.10)= ✓✔100 × √10
=10 √10.
4
2. 4/81 a¹ y° ≈= √(81 a¹ y¹·y° =)=9 a³ y³ ¹/y³ z.
3. (a³ 4 a ✓b+b²) = a -2b, the operation being
performed as in the arithmetical extraction of the square
root.
X
Note.—The square root of å binomial or residual a±b, or
even of a trinomial or quadrinomial, may often be convenient-
ly extracted thus :-Take d = √(a³-b³); then
a
d
This is evident: for, if Ja+d±
2
2
d
Jabe squared, it will give a + √(a² — d²) or a + b, as
2
12
H 2
78
ALGEBRA SURDS.
it ought: and, in like manner, the square of
a d
A
√(a³
2
Ex. 1. Find the square root of 3+2 √2.
Here a=3, b=2, √2, d= √(9-8)=1,
And
d
·9
is a
6+ √8
2
a + d
2
Conseq.
Šting my fat
+
t
a
1
dº), or a
J
3 + 1
2
- ს.
2
√ ½ +√3/= √2+ √1=1+ √2.
2. Find the square root of 6 — 2 √5.
Here a=6, b=2 √5, d=√(36 -20)=√16=4,
d 16+4
ja + d
And
2
2
10
√12 - √32/3 = √5—1.
号
​2
+
4
Co
3. Find the square root of 6+ √8 −√12 ·√24.
Here a=6+ √⁄8,b= √12+ √24, d= √(6+ √8)³
−(√12+ √24)= √(44+128362
=√(4436+12812 √8) = √8.
12.24)
a
a + d 6+2/8
2
2
√/8 3.
3+8, and
2
Ja + d
2
d
1
૫
But (Ex. 1), (3+2 √2)=√(3+ √8)=1+ √2.
Therefore the root required is 1+ 2√3.*
I
{
SECTION VIII.-Simple Equations.
An algebraic equation is an expression by which two quanti-
tics called members (whether simple or compound), are indi-
cated to be equal to each other, by means of the sign of equality
-placed between them.
C
In equations consisting of known and unknown quantities,
when the unknown quantity is expressed by a simple power,
as x, x³, x³, &c. they are called simple equations, generally;
*For the cube and higher roots of binomials, &c. the reader may consult the
treatises on Algebra by Maclaurin, Emerson, Lacroix, Bonnycastle, J. R. Young,
and Hine.
1
ALGEBRA SIMPLE EQUATIONS.
79
。
and particularly, simple or pure quadratics, cubics, &c. accord-
ing to the exponent of the unknown quantity. But when the
unknown quantity appears in two or more different powers in
the same equation, it is named an adfected equation. Thus
x² = a + 15, is a simple quadratic equation: x² + ax = b,
an adfected quadratic.
It is the former class of equations that we shall first con-
sider.
G
The reduction of an equation consists in so managing its
terms, that, at the end of the process, the unknown quantity
may stand alone, and in its first power, on one side of the
sign and known quantities, whether denoted by letters, or
figures, on the other. Thus, what was previously unknown is
now affirmed to be equal to the aggregate of the terms in the
second number of the equation.
"
"In general the unknown quantity is disengaged from
the known ones, by performing upon both members the RE-
VERSE OPERATIONS,"* to those indicated by the equation, what-
ever they may be. Thus,
If any known quantity be added to the unknown quantity,
let it be subtracted from both members or sides of the
cquation.
If any such quantity be subtracted, let it be added.t
If the unknown quantity have a multiplier, let the equation be
divided by it.
If it be divided by any quantity, let that become the multi-
plier.
*
Ca
If any power of the unknown quantity be given, take the
corresponding root.
If any root of it be known, find the corresponding power.
If the unknown quantity be found in the terms of a propor-
tion (Arith. Sect. 10), let the respective products of the means
and extremes constitute an equation; and then apply the gene-
ral principle, as above.
* This simple direction, comprehending the seven or eight particular rules for
the reduction of equations given by most writers on algebra, from the time of
Newton down to the present day, is due to Dr. Hutton. It is obviously founded
upon the mathematical axiom, that equal operations performed upon equal things
produce equal results.
†These two operations constitute what is usually denominated transposition.
80
ALGEBRA SIMPLE EQUATIONS.
1
{
Then, by sub-
Examples.
1. Given x 3 + 5
9, to find x.
First, by adding we have x + 5 = 9 + 3 = 12.
3 to both sides,
x + 5 — 5
}
12
tracting 5.
Otherwise in appearance only, not in effect,
Są
1
Tada je p
}
bow
60
By transposing the 3, and changing its sign, x+5=93.
By transposing the, 5, and changing its sign, x = 9 + 3 --
5
7.
2. Given 3 x + 5 20, to find x.
=
=
5.
First, by transposing the 5, 3 x 205 15
by dividing by 3, x= 1-55
3. Given
+ d = 3 b
2 c to find x.
3
J
a
I
First, transposing d, 36-2 c + d.
a
Then, multiplying by a, x=3 ab 2 ac+ ad.
4. Given (3 x + 4) + 2 6, to find x.
First transposing the 2, (3 x + 4) 6 2
4
60
= 3
Then, cubing, 3 x + 4 48 64.
Then, transposing the 4, 3 x 64
Lastly, dividing by 3, x0 = 20.
5. Given 4 a x 5 b 3 d x + 4 c, to find x.
3 d x
4 a x
First, transposing 5 b and 3 d
That is, by collecting the
5b+ 4 c.
J
x,
coefficients,
(4 a
Therefore, by dividing by 4 a 3 d, x
Jakk
A
Gigant
30 x + 24 x
} x = 3, to find x.
}
ह
6. Given x + 1/2 x
Multiplying by 120
4 × 5 × 6, we have
That is, collecting the coefficients, 34 x
Therefore, dividing by 34, x = 300
7. Given & x: a5b: 3 c, to find x.
Mult. means and extremes, 2 c x
3
5 a b,
Dividing by c, x = 5 ab÷ 2 c =
20 a b
9 c
}
5, or x =
padd
1 8 0
17
= 7.
4 a
p
!
4.
5 b + 4 c.
3 d) x
J
56 + 4 c
3 ď
20 x = 360.
360.
1010.
1
ALGEBRA SIMPLE EQUATIONS.
81
I
8. Given a + x √ a³ + x √ (4 b² + x²), to find x.
First, by squaring, we have, a
+ 2 a x + x² = a² +
x √(4 b² + x².)
1
Then striking out a from both sides, 2 a x + x³
x √(4 b² + x²)
dividing by x, 2 a + x =
squaring, 4 a² + 4 a x + x²
striking out x, and 2
transposing 4 a³ Š
Opp
dividing by 4 a, x =
9. Given c x a c =
First, dividing by
4 α x =
a
4 62
¿
-
√(4 b² + x²)
4 b² + x²
- 4 a²
――
4 a
transposing the 1, c1, or
4 62
4 a²
b+ √ x
xa, we have
a
S
C.
62
inverting and transposing the fractions,
multiplying by b, (xa)
squaring both sides, x
transposing a, x = a +
10. Given 13
✓ c
√c - 1
1
Pa
a
a, to find x.
c =
h
8
√(4 + y²)
Extermination.
JAPA
A.
1.
√(x — a)
12
2 √c + 1
&
√(x — a)
√(x − a)
b
62
2/c+1
√13 + 3 x, to find x.
**
√3 x =
11. Given y + √4 + y²
to find y.
Ans. y = 3√3.
12. Given (x + 1) + (x + 3) = (x + 4) + 16, to find x.
1
= }
Ans. x = 41.
Ans. 11.
Ans. x
13. Given: √(x-1) :: 3 : 1, to find x.
1
14. Given (b + 2) = (a + x²) to find x.
2
1
Ans. x =
+1
1
1
√c — 1
↓
2
12.
When two or more unknown quantities occur in the con-
sideration of an algebraical problem, they are determinable by
a series of given independent equations. In order, however,
82
ALGEBRA : EXTERMINATION.
1
that specific and finite solutions may be obtained, this condition
must be observed, that there be given as many independent
equations as there are unknown quantities. For, if the
number of independent equations be fewer than the unknown
quantities, the question proposed will be susceptible of an
indefinite number of solutions:* while, on the other hand, a
greater number of independent equations than of unknown
quantities, indicates the impossibility or the absurdity of the
thing attempted.
Where two unknown quantities are to be determined from
two independent equations, one or other of the following rules
may be employed.
1. Find the value of one of the unknown letters in each of
the given equations; make those two values equal to one
another in a third equation, and from thence deduce the value
of the other unknown letter. This substituted for it in either
of the former equations, will lead to the determination of the
first unknown quantity.
2. Find the value of either of the unknown quantities in one
of the equations, and substitute this value for it in the other
equation so will the other unknown quantity become known,
and then the first, as before.
3. Or, after due reduction when requisite, multiply the first
equation by the coefficient of one of the unknown quantities in
the second equation, and the second equation by the coefficient
of the same unknown quantity in the first equation: then the
addition or subtraction of the resulting equations (according as
the signs of the unknown quantity whose coefficients are now
made equal, are unlike or like) will exterminate that unknown
quantity, and lead to the determination of the other by former
rules.
Noles. The third rule is usually the most commodious and
expeditious in practice.
The same precepts may be applied, mutatis mutandis, to
cquations comprising three, four, or more unknown quantities:
and they often serve to depress equations, or reduce them from
a higher to a lower degree.
*This, though generally true, has one striking exception, namely, in the case
of equations constituted partly of rational quantitics and partly of quadratic surds;
where two unknown quantities are determinable by one equation, four unknown
quantities by two equations; and so on.
Thus, if x +√y= a + √o
and z
Vo= с √d
Then xa, y = b, z = c, v = d.
Jokeralateral
ALGEBRA EXTERMINATION.
83
1. Given 4 x³ + 3 y
and y.
1st equa. x by 3, gives 12 x + 9 y =123.
2d equa. × by 4, gives 12 x³
16 Y
The difference of these, 25 y =
Then, from equa. 2d, 3 x³
Whence dividing by 3, x³
48.
whence y
3.
75,
12 + 4y 12 + 12 = 24.
8, or x = 2.
Ex. 2. Given x + y + z =
x + 3y + 4 z = 134.
Examples.
41, and 3 x³
Ex. 3. Given x + y
x, y, and z.
1. x +
y +
2. x + 2y + 3 ≈
3. x + 3y + 4 z
Ans. x =
=
c b' — b c'
a b' — ba'
4. 1st equa. taken from 2d, gives y + 2 z = 52
y +
29
5. 2d equa. taken from 3d,
6. 5th equa. taken from 4th,
7. 6th equa. taken from 5th,
8. 5th equa. taken from 1st,
53, x + 2y + 3 z =
1. x + y
2.x + z =
3.y + ==
Sp
and y
a
b
C
22
4. 1st. + 2d + 3d, gives 2 x + 2y +
5. Half equa. 4th gives x + y + z
6. 3d equa. taken from 5th, gives x
7. 2d equa. taken from 5th,
Y
8. 1st equa. taken from 5th,
The
a c'
ab'
Y
53
105
134
4 y
a, x + z = b, y + z
+
Z
BOHLOFLOT 10
12, to find x
· c a'
ba'
z=23
|| || ||
105, and
6
24.
2 z = a + b + c.
C.
a + b b +
b
a +
Ex. 4. Given a x + by = c, and a x + b'y
c, and a x + b'y - c', to find
x and y.
c, to find
$/£5-£9}~
C.
α
11 6 + C.
3 a + b + c.
•
}
}
84
Ex. 5. Given a x + by + cz
a" x + b" y + c" z = d" to find x, y and z.
Ans.x
y
X
ALGEBRA EXTERMINATION.
x and y.
d b' c" d c' b" + c d' b
daddy
c d' b" —b d' c''+b c' d' — c b'd"
b'c" —a c' b'+c a'b' —b a'c''+b c'a'' —c b'a'
a
1
a d' c" ·a c'd"+ca'd"
a b' c''—a c'b'+c a' b"
a b' d'
a b'c"
S
Ex. 6. Given x (x + y + z)
y (x + y + z)
z (x + y + z)
Ex. 8. Given
Ex. 7. Given (x + y)
·a d'b''+d a' b'—b a'd''+b d' a"—d b' a"
a c'b''+c a' b'—b a' c''+b c' a'—c b'a'"
x +
Į x + \ y + ÷
y +
HKH HRO KO
d, a' x + b' y + c' z = d'
KATA,
X
= 60, and (x + y) = = 23, to find
Ans. x = 10,y
y
x
2.
= 18
27
to find x, y, and z.
36
Ans. x = 2, y = 3,'z
N N N
da'c'+dc' a" ·c d'a'
ba'c''+b c'a'—c b'a'
62
47
38
Ans. x = 24, y
≈
=
x + / y + z = =
4.
to find x, y, and z.
60, z = 120.
Solution of General Problems.
K
A general algebraic problem is that in which all the quanti-
ties concerned, both known and unknown, are expressed by let-
ters, or other general characters. Not only such problems as
have their conditions proposed in general terms are here im-
plied; every particular numerical problem may be made general,
by substituting letters for the known quantities concerned in it;
when this is done, the problem which was originally proposed in
a particular form becomes general.
In solving a problem algebraically some letter of the alphabet
must be substituted for an unknown quantity. And if there be
more unknown quantities than one, the second, third, &c. must
either be expressed by means of their dependence upon the first
and one or other of the data conjointly, or by so many distinct
letters. Thus, so many separate equations will be obtained, the
resolution of which, by some of the foregoing rules, will lead to
the determination of the quantities required.
1
}
+
ALGEBRA GENERAL PROBLEMS.
85
Examples.
1. Given the sum of two magnitudes, and the difference of
their squares, to find those magnitudes separately.
Let the given sum be denoted by s, the difference of the
squares by D; and let the two magnitudes be represented by x
and y respectively.
Then, the first condition of the problem expressed algebraic-
ally is x + y = s
And the second is x²
2
y³ = D
Equa. 2 divided by equa. 1, gives x
=
D
S
Equa. 1 added to equa. 3, gives 2 x =
S² + D
28
Equa. 4 divided by 2, gives x =
چوری
28
and y
Suppose, again, s =
25 +5
then x =
and 2.
X
dag med
..
Y
Equa. 2 × by y, gives x
Substituting this value
of x for it in equa. 1
15
X
p
Dividing by q, y²
q
Y
q y
q y² = p.
Equa. 5 taken from equa. 1, gives y s
2 s
12,
To apply this general solution to a particular example, sup-
pose the sum to be 6, and the difference of the squares 12.
Then s 6 and D =
S² + D
2 s
D
and x =
36 12 48
12
12
36- 12 24
12
12
5, D = 5:
3, and y
25 - 5
10
10
Ex. 2. Given the product of two numbers, and their quotient,
to find the numbers.
4.
2.
D
N
1
S
+ s =
s² + D
S
S² + D
28
2.
Let the given product be represented by p, the quotient by q;
and the required numbers by x and y, as before.
Then we have, 1. xy=p
ი2.
D
1
13
I
f
86
Extracting the square root, y=
Then, by substitution, x = q y = q
ALGEBRA : GENERAL PROBLEMS.
'p q⋅
Suppose the product were 50 and the quotient 2.
✓ 25 = 5, and x
50
Then y
J
'
q
Ans. x 1
1
½ s + ½ √ 2 s
2
2
✓100
= 10.
Again, suppose the product 36, and the quotient 24.
136
Then y
√
21
P
B
I
I
12
q
81
9.
Ex. 3. Given the sum (s) of two numbers, and the sum of
their squares s, to find those numbers.
s², and
A
p q²
q
√ p q
Ex. 4. The sum and product of two numbers are equal, and
if to either sum or product the sum of the squares be added, the
result will be 12. What are the numbers?
✓ 16 = 4, and x = ✔ · p q =
Ans. each = 2.
1
Ex. 5. The square of the greater of two numbers multi-
plied into the less, produces 75; and the square of the less.
multiplied into the greater produces 45. What are the
numbers?
Ex. 6. A man has six sons whose successive ages differ by
four years, and the eldest is thrice as old as the youngest.
Required their several ages?—Ans. 10, 14, 18, 22, 26, and 30
years.
f
SECTION IX.-Quadratic Equations.
When, after due reduction, equations assume the general
form a x² + в x + c 0; then dividing by A, the coefficient
of the first term, there results a² + x + 0, or, making
n
с
A
A
we have x² + p x + q
"
Y 11/1/1 S ½ √2 s — s³.
p
0......(1)
A
an equation which may represent all those of the second degree,
p and q being known numbers positive or negative.
1
ALGEBRA QUADRATICS.
87
2
Let a be a number or quantity which when substituted for x
renders x²+p x+q=0; then a²+pa+q=0, or q=-a-pa.
Consequently x² + p x + q, is the same thing as x²
a² + p
px-pa, or as (x+a) (x − a) + p (x—a), or, lastly, as (x—a)
(x + a +p).
The inquiry, then, is reduced to this, viz. to find all the
values of x which shall render the product of the above two
factors equal to nothing. This will evidently be the case when
either of the factors is = 0; but in no other case. Hence, we
have a
a,=0, and x+a+p=0, or x-a, and x
And hence we may conclude—
a — p.
pagkaka
1. That every equation of the second degree whose condi-
tions are satisfied by one value a of x, admits also of another
valuc
-p.
These values are called the roots of the quad-
ratic equation.
a
2. The sum of the two roots a and-a-p is—p; their pro-
duct is - a² a p, which as appears above is=q.
above is q. So that the
coefficient, p, of the second term is the sum of the roots with
a contrary sign; the known term, q, is their product.
3. It is easy to constitute a quadratic equation whose roots
shall be any given quantities b and d. It is evidently -
x³
(b+d) x+b d=0.
4. The determination of the roots of the proposed equation
(1) is equivalent to the finding two numbers whose sum is-p,
and product q.
X
5. If the roots b and dare equal, then the factors x b and
d are equal; and x²+p x+q is the square of one of them.
I kn
To solve a quadratic equation of the form
2
+p x+q=0, let
it be considered that the square of x + p is a trinomial, x²+
px+p³, of which the first two terms agree with the first two
terms of the given equation, or with the first member of that
equation when q is transposed.
That is, with x²+px=— 9.
Let then be added, we have, a+pa+ } p² = {p² — q
of which the first member is a complete square.
* If it bo affirmed that the given equation admits of another value of a, besides
the above, 6 for instance, it may he proved as before that —
b
e b must be of the num-
ber of the factors of a2+x+, or of (x − a) (x+a+p). But a
a and
x+a+p being prime to each other, or having no common factor, their product
cannot have any other factor then thoy. Consequently 6 must either be equal to a
or to-a-p; and the number of roots is restricted to two.
88
ALGEBRA : QUADRATICS.
2
Its root is + p=± √(} p³ — q)
2
and consequently xp± √ (4 p² — q)
otherwise, from number 2 above, we have x + x'
}
and x x' = q
2
Taking 4 times the second of these equations from the square
of the first, there remains a³ - 2 x x'+x'²=p³ — 4 q
Whence, by taking the root, x-x'= √(p² - 4 q)
Half this added to half equa. 1, gives xp +
} √ ( p² — 4 q) - ½ p + √(‡ p³ - 9)
ž p
And the same taken from half equa. 1, gives x'
} √ ( p³ — 4 q) ž p √(4 p² — q) which two values of
a evidently agree with the preceding.
1. x²+p x=Y
2. x²-p x=q
3. x²+px=
4. x³-p x===
It would be easy to analyze the several cases which may
arise, according to the different signs and different values, of p
and q.
But these need not here be traced. It is evident that
whether there be given
զ
զ
N
L
Magi
htt
The general method of solution is by completing the square,
that is, adding the square of p, to both members of the equa-
tion, and then extracting the root.
P
It may farther be observed that all equations may be solved
as quadratics, by completing the square, in which there are
two terms involving the unknown quantity or any function
of it, and the index of one double that of the other. Thus,
q
y 2
n
X
2
x6 ± p
X3
¶, x²n ±p xn I, x³±p x¹
I, (x² + px + 9)³
± (x²+p x+q)=r, (x²¹ — x²)² + (x³n — x)=q, &c., are of the
same form as quadratics, and admit of a like determination of
the unknown quantity. Many equations, also, in which more
than one unknown quantity are involved, may be reduced to
lower dimensions, by completing the square and reducing; such,
for example, as (x²+y³)³±p(≈³+y³)
XB
土
​9, and
px
Y
so on.
Note. In some cases a quadratic equation may be conve-
niently solved without dividing by the coefficients of the
square, and thus without introducing fractions. To solve the
general equation a x² + bx = c, for example, multiply the
whole by 4 a, whence 4 a²x² + 4 a b x = 4 a c, adding b³ to
complete the square, 4 a2 x2 + 4 a b x+b²=4 a c+b²
taking the square root, 2 ax + b = ± √(4.a c+b³) ;
2
ALGEBRA: QUADRATICS.
89
whence x =
general theorem.
2. Given-
= 45
Examples.
1. Given x²
8x+10= 19, to find x.
transposing the 10, x 8 x 19—10—9
completing the squ., x³-8x+16=9+16=25
extracting the root, x -4=±5
consequently x=4± 5-9 or 1.
49
10 14
9
Fb ± √(4 a c+b³)
2 a
JA, A
X
2 x
X3
ppt
J
S
12
4. Given " 2 a x²
J
22
9
22 x³
9
9
multiplying by x², 10 x — 14+2 x =
transposing, 22 x 12 x
x³
dividing by 22, 2° -4x=-1},
5
complet. squ. x — $ 4 x + ( 7 )² =
2 721-17=387,
extract. root, x 1221713
+ 11,
transposing, x = 23 or 1.
5
6
1
21
6
11
that is, x + 1 = 5 or 9
hence x = 4 or 10.
P
AJ
which will serve for a
to find the values of x.
3. Given
+2 x+4 √x²+2 x+1=44, to find x.
adding 1, we have (x² + 2 x+1) + 4 √(x² + 2 x + 1)
complet. squ. (x² + 2 x + 1) + 4 √(x² + 2 x + 1)+4
extract. root, √(x² +2 x+1)+2 = ± 7
transposing the 2 √(x² + 2 x + 1) = ± 7 − 2 = 5 01
14,
6 3
IT'
c, to find x
71
complet. squ. "—2 α x² + a²=c+aº
n
extract. root, x² a = ± √(c+a³)
ต
2
transposing, x = a ± √(c+a²)
al2
consequently, x = (a ± √c+a³)"
}
36
1
I 2
90
ALGEBRA : QUADRATICS,
5. Given
·|·
2.2
4 x
at
32
Y
5 y³
complet. squ. in equa. 1,
20
Extracting root + 2
Y
у
M
y or
x=2 y or =—
6 Y.
Substituting the former value of x in the 2d equa. it becomes
2 y―y=2, or y=2; whence x=4.
Again, substituting the 2d value of x, in equa. 2, it becomes
6 Y
2. and x
7 y
2; whence y
+
6. Given x² y² — 5 — 4 x y, and ½ x y
x and y.
o&o
"
2
2
x²
equa. 1, by transposition, becomes x2 y24 xy=5
completing the square, xy-4xy+4=9
extracting the root, xy-2-±3
J
7. Given
2 :
1 ✔/25= w/ 5.
ह्र
(ix:3
12, and x
T
X3
K
Whence x y=5 or
1.
3
Substituting the first of these values for x y in equa. 2, it be-
comes y³½: whence y=1 and x=5.
Substituting the 2d value in the same equation, it becomes
whence y
25, and x — —
- 1
✓/ / /
4)3 + 203
1)
X2
P
6
y = 2, to find x and y.
X
y²+4+4=16.
Y
4: whence 2 or 6, and
X
у
351
4 25 x⁹
}
to find x.
5
Ans. x =
1 2
T
y³, to find
± 3, or ± √39
11
S. A man travelled 105 miles at a uniform rate, and then
found that if he had not travelled so fast by two miles an hour,
he would have been six hours longer in performing the same
journey. How many miles did he travel per hour?
Ans. 7 miles per hour.
Į
9. Find two such numbers that the sum, product, and dif-
ference of their squares may be equal.
Àns. ½ + ½ √5, and 3+ 5.
2
2
10. A waterman who can row eleven miles an hour with the
tide, and two miles an hour against it, rows five miles up a river
and back again in three hours: now, supposing the tide to run
uniformly the same way during these three hours, it is required
to find its velocity?
Ans. 4 miles per hour.
ALGEBRA EQUATIONS.
SECTION X.-Equations in General.
A
a, x =
x =
a=0,x
X
Equations in general may be prepared or constituted by the
multiplication of factors, as we have shown in quadratics. Thus,
suppose the values of the unknown quantity x in any equation
were to be expressed by a, b, c, d, &c. that is, let x =
d, &c. disjunctively, then will x —
C, X
C 0, x
d 0, &c. be the simple radical equations
of which those of the higher orders are composed. Then, as the
product of any two of these gives a quadratic equation; so the
product of any three of them, as (x — a) (x - b) (x - c) = 0,
will give a cubic equation, or one of three dimensions. And
the product of four of them will constitute a biquadratic equa-
tion, or one of four dimensions; and so on. Therefore, in ge-
neral, the highest dimension of the unknown quantity x is
equal to the number of simple equations that are multiplied
together to produce it.
a
་
When any equation equivalent to this biquadratic (x
a)
(x — b) (x — c) (xd) = 0 is proposed to be resolved, the
whole difficulty consists in finding the simple equations x-
0, x - b = 0, x
d =0, by whose multiplica-
0, x
tion it is produced; for each of these simple equations gives one
of the values of x, and one solution of the proposed equation.
For, if any of the values of a deduced from those simple equa-
tions be substituted in the proposed equation in place of x, then
all the terms of that equation will vanish, and the whole be
found equal to nothing. Because when it is supposed that x = a,
or x b, or x = c, or x = d, then the product (x - a) (x — b)
(x —c) (xd) vanishes, because one of the factors is equal to
nothing. There are, therefore, four suppositions that give
(x — a) (x — b) (x — c) (x — d) 0, according to the pro-
posed equation; that is, there are four roots of the pro-
posed equation. And after the same manner any other equa-
tion admits of as many solutions as there are simple equa-
tions multiplied by one another that produce it, or as many as
there are units in the highest dimensions of the unknown quan-
tity in the proposed equation.
с
M
91
0
b,
b=0,
p
Sta
- But as there are no other quantities whatsoever besides.
these four (a, b, c, d,) that, substituted in the proposed product
in the place of x, will make that product vanish: therefore,
the equation (x — a) (x —b) (x — c) (x — d) O, cannot
possibly have more than these four roots, and cannot admit
1
92
ALGEBRA EQUATIONS.
1
M
of more solutions than four. If we substitute in that product
a quantity neither equal to a, nor b, nor c, nor d, which suppose
e, then since neither, e
a, e
b, e
c, nor e d, is equal to
nothing; their product cannot be equal to nothing, but must be
some real product and therefore, there is no supposition bc-
sides one of the aforesaid four, that gives a just value of x ac-
cording to the proposed equation. So that it can have no more
than these four roots. And after the same manner it appears,
that no equation can have more roots than it contains dimen-
sions of the unknown quantity.
To make all this still plainer by an example in numbers,
suppose the equation to be resolved to be x4. 10 x³ + 35 x²
50 x + 24 = 0, and that we discover that this equation is the
same with the product of (x — 1) (x — 2) (x — 3) (x — 4),
then we certainly infer that the four values of x are 1, 2, 3, 4;
seeing any of these numbers, placed for x, makes that product,
and consequently x 10 2335 x 50 x + 24, equal to
nothing, according to the proposed equation. And it is certain
that there can be no other values of x besides these four; for
when we substitute any other number for x in those factors
x−1,x -2, x 3, 4, none of them vanish, and therefore
their product cannot be equal to nothing, according to the equa-
tion.
4
M
http
A variety of rules, some of them very ingenious, for the so-
lution of equations, may be found in the best writers on Alge-
bra ;* but we shall simply exhibit the easy rule of Trial-and-
Error, as it is given by Dr. Hutton in the 1st volume of his
"Course of Mathematics." ""
RULE.
K
"1. Find, by trial, two numbers as near the true root as pos-
sible, and substitute them in the given equation instead of the
unknown quantity; marking the errors which arise from each
of them.
"2. Multiply the difference of the two numbers, found by
trial, by the least error, and divide the product by the differ-
ence of the errors, when they are alike, but by their sum when
they are unlike. Or say, as the difference or sum of the errors
is to the difference of the two numbers, so is the least error to
the correction of its supposed number.
* See the treatises of Lacroix, Bonnycastle, Wood, J. R. Young, &c.
ALGEBRA EQUATIONS.
93
葛
​"3. Add the quotient, last found, to the number belonging to
the least error, when that number is too little, but subtract it
when too great, and the result will give the true root nearly.
"4. Take this root and the nearest of the two former, or any
other that may be found nearer; and, by proceeding in like man-
ner, a root will be had still nearer than before; and so on to any
degree of exactness required.
t
"Note. It is best to employ always two assumed numbers
that shall differ from each other only by unity in the last figure
on the right; because then the difference, or multiplier, is 1."
Example.
To find the root of the cubic equation x3 + x² + x
the value of x in it.
Here it is soon found that xl Again, suppose 4.2, and 4.3,
lies between 4 and 5. Assume, and repeat the work as follows:
therefore, these two numbers,
and the operation will be as fol-
lows:
1st Sup.
4
16
64
84
16.
28
ش روایتی به
sums
errors
2d Sup. 1st Sup.
5 4.2
25 17.64
125 74.088
155 95.928
the sum of which is 71.
Then as 71: 1 :: 16: 225.
Hence = 4.225 nearly.
+55-4072
x
X3
X3
sums
100, or
errors
+2.297
the sum of which is 6.369.
As 6.369: 1 :: 2.297 : 0.036
This taken from
·
4.300
leaves x nearly
2d Sup.
4.3
18.49
79.507
102.297
4.264
14
94
ALGEBRA EQUATIONS.
}
Again, suppose 4.264 and 4.265, and work as follows:
4.264
18.181696
77.526752
99.972448
-0.027552
XC
X3
X3
sums
errors
4.265
18.190225
77.581310
•
100.036535
the sum of which is 064087
Then as 064087: 001 :: 027552 : 0·0004299
To this adding
4.264
+0.036535
↓
gives a very nearly
When one of the roots of an equation has been thus found,
then take for a dividend the given equation with the known
term transposed to the unknown side of the equation; and for
a divisor take x minus the root just determined: the quotient
will be equal to nothing, and will be a new equation de-
pressed a degree lower than the former. From this a new
value of x may be found and so on, till the equation is re-
duced to a quadratic, of which the roots may be found by the
proper rules.
:
4.2644299
SECTION XI.-Progression.
When a series of terms proceed according to an assignable
order, either from less to greater or from greater to less, by con-
tinual equal differences or by successive equal products or quo-
tients, they are said to form a progression.
If the quantities proceed by successive equi-differences they
are said to be in Arithmetical Progression. But if they pro-
ceed in the same continued proportion, or by equal multiplica-
tions or divisions, they are said to be in Geometrical Pro-
gression.
Konta
If the terms of a progression successively increase, it is called
an ascending progression: if they successively decrease, it is
called a descending progression.
Thus, 1, 3, 5, 7, 9, &c. form an ascending arithmetical
24, 22, 20, 18, 16, &c. form a descending arithmetical
1, 3, 9, 27, 81, &c. form an ascending geometrical
4, 2, 1, 4, 1, &c. form a descending geometrical
Progression
ALGEBRA: PROGRESSION.
95
1
Arithmetical Progression.
1. Let a be the first term of an arithmetical progression
d the common difference of the terms
z the last term
n the number of terms
s the sum of all the terms.
Then a, a+d, a +2 d, a + 3 d, &c. is an ascending progression.
and a, ad, a 2 d, a 3 d, &c. a descending progression.
Hence, in an ascending progression, a + (n − 1) d, is the last
p
term;
in a descending progression, a-- (n 1) d, is the last
term.
2. Let a series be a + (a + d) + (a + 2 d) + (a+3 d).
The same inverted, (a + 3 d) + (a + 2 d) + (a + d) + a.
The sum of the two, (2 a+ 3 d) + (2a + 3 d) + (2a + 3 d)
+ (2a + 3 d)
2 s.
That is, (2 a + 3 d.) × 4, in this case (a + a + 3 d) n = 2 s.
Consequently, s ½ (a + a = 3 d) n, or= (a + z) n, since
ž
z is here a + 3 d. The same would be obtained if the pro-
gression were descending; and let the number of terms be what
it may.
=
3. From the equations z=a + (n 1) d, s ½ n (a + z),
and s½n (a+a+(n − 1) d), we may readily deduce the fol-
lowing theorems applicable to ascending series. When the se-
ries is descending, either the signs of the terms affected with d
must be changed, or a must be taken for z; and vice versa.
(1.) a = z — n d + d = √(— 2 sd + z 2 + d z + { d³) +
1
28
S
¿ d = — + 4 d — in d=
N
(2.) d
K
a
1 2 s
(3.) ≈ = a + n d - d
(5.) n
N
Databas
Add
22
Z2
28
a + z
S
d.
12
ጎ
a d + 1 d³)
a³
a
2.
a
p
2 z n
No
+ ½ n d − z ď
2 $
½ d
N
√(2 s d + a³
(4.) s=&n (a+z) = (a + ½ n d − d) n = (z—} n d + ½ d) n
z³ — a³ + d (z + a)
-
2 d
ping
1.
ጎ
28
a.
2 s− 2 na
No
N
96
ALGEBRA PROGRESSION.
Hậ
1
Examples.
1. Required the sum of 20 terms of the progression, 1, 3, 5,
7, 9, &c.
Here a 1, d=2, n
2, n = 20; which is substituted in the
theorem s = (a + ½ n d d) n, transform it to s
(1 201) 20
+
2
20 x 20
400, the sum required.
pop
Note. In any other case the sum of a series of odd numbers
beginning with unity, would be = n², the square of the number
of terms.
Here n =
2. The first term of an arithmetical progression is 5, the last
term 41, the sum 299. Required the number of the terms, and
the common difference.
and d
Here d
n
Here z =
the last term,
and d
Z
2 s
a + z
Z
Ma
p
a
ጎ 1
Magya
α
There are 8 equidifferent numbers: the least is 4, the greatest
32. What are the numbers ?
1
a 32 4
4, the common difference.
N
1
Whence 4, 8, 12, 16, 20, 24, 28, 32, are the numbers.
2 s
n
41
598
46
4. The first term of an arithmetical progression is 3, the num-
ber of terms 50, the sum of the progression 2600. Required
the last term and the common difference.
a
Ind
12
Cate
101
- 5
pat
5200
50
49
13, the number of terms,
3
3, the common difference.
1
P
3 104
1
1
- 3 = 101,
mą kalną
2, the common difference.
-
5. The sum of six numbers in arithmetical progression is 48;
and if the common difference d be multiplied into the less ex-
treme, the product equals the number of terms. Required those
terms.
Ans. 3, 5, 7, 9, 11, and 13.
*
Geometrical Progression.
1. Let a be the first term of a geometrical series;
r the common ratio; z the last term;
n the number; and s the sum of the terms.
-1
Then a, r à, r² a, p³ ɑ, p²-¹ ɑ, is a geometrical progression,
which will be ascending or descending, according as r is an in-
teger or a fraction.
2. Let the prog, a+r a+r² a+r³ a+r¹ a=s, be × by r, it
becomes r atr² atp³ atr¹ a+p5 a=rs
8
=rs
S.
The diff. of these is, а + доба
But r a is the last term of the original progression multi-
plied by r, or in general terms -1 a xr, that is, pa. Conse-
quently r¹ a
S
Whence s =
2012
2°
1
A similar method will
ever be the value of n.
(3.) s
Z por
Jr7~1
(2.) ≈=α p²-1.
J01-1
(4.) r =
A
1
becomes transformed to s =
1
ALGEBRA: PROGRESSION.
gr
a = r S
γη α-α
S
s
Ꮽ
3. Now from these values of z and s the following theorems
may be deduced.
Z (r — 1) s
(1.) a =
2012
1
a(r"—1)
- 1
Stage)
a
r s
20th
r
İ
lead to a like expression for s, what-
If be a fraction, the expression.
доп
M
2
a (1
1 -
Ma
1
a, the sum of the series.
s + a s (2012
J+12
r z
Jr
(2) 1.
=strz — 7 S.
Z
And, if the logarithm of
that of r=R: then,
a
↑
r
α.
2071)
K
pot ~1)
1
- 1
97
PA,
N, that of
a
sr-sta
a
M, and
}
+
98
ALGEBRA PROGRESSION.
N
(5.) n ==+ 1 =
R
R
Also, if when r is a fraction, n is infinite, then is 7"-0, and
the expression for s becomes.
a
(6.) s
mation of infinite series.]
1 - p
j - 1
g
M
(type"""matt
1 -
Job
Examples.
1. The least of ten terms in geometrical progression is 1, the
ratio 2. Required the greatest term, and the sum.
Here= a pn-¹=1 × 2º 512, the greatest term ;
r z
α
1022
and s
=1023, the sum.
[This expression is often of use in the sum-
1
1
2. Find the sum of 12 terms of the progression 1, 3, 3, 27,
1 &c.
1
Jon
1
Here s
the sum.
r
J
Consequently, s =
Jak batt
Here theor. 6, that is s =
the sum required.
1
1
53 144 1
1
114
3
1 p
3. Find the sum of the series 1, 1, 4, §, &c. carried to infinity.
a
1
becomes s
2,
265720
177147'
1 — µ³
1
3 6
99
1800 ÷ 700
=
4. Find the vulgar fraction equivalent to the circulating de-
cimal 36363636.
Here a=1, r—— x, and s=
1
36
This decimal, expressed in the form of a series, is, +
10000+10030000+&c. where a=3%, and r=0.
36
6
36
100
a
1
a
Ma
119
C
742
sought.
5. Find the sum of the descending infinite series 1 x +
x³+xª, &c.
X2
the fraction
Stat
1
the sum req.
до 1 + x
And, by way of proof, it will be found that if 1 be divided
by 1+x, the quotient will be the above series.
6. Of four numbers in geometrical progression the product
of the two least is 8, and of the two greatest 128. What are the
numbers ?
Ans. 2, 4, 8, and 16.
ALGEBRA : LOGARITHMS.
99
SECTION XII.-Logarithms.
The logarithm of any given number is the index of such a
power of some other number, as is equal to the given one.
Let us suppose that the number r greater than unity, is the
base of a system of logarithms, and let there be given to it the
variable exponent p, in such manner that the expression rº shall
represent all possible numbers, by attributing successively dif-
ferent values to the exponent p. It is manifest,
1. That the logarithm of unity will be always zero or nothing,
whatever be the base r: for, in general, rº=1
2. That the logarithm of the base r will be 1, since r is the
same thing as r¹.
3. That all numbers above 1 will have positive numbers for
their logarithms. Thus, supposing 10, then the number
r =
10000 or 10¹ has for its logarithm the positive number 4.
4. All fractions, or numbers below unity, have negative num-
bers for their logarithms. Thus, if r=10, then 1 or 0001
or 10-4 has 4 for its logarithm.
5. Assuming two numbers N and N', to which correspond
respectively the two logarithms p and p', to the same base or
root r we have N=2, and N'=', and consequently N+N'
=2¹² ×?•P' = p² + P'. Whence it appears that in every system of
logarithms, the logarithm p+p' of a product N N', composed of
two factors, is equal to the sum of the logarithms of those fac-
=pP+r'.
tors.
N
8. We have also λ AP
6. If we have any numbers A, B, C, D, how many soever,
we may prove in a similar manner, that (using the initial λ to
denote the logarithm) we shall have 2 (A. B . C . D) = 2 A +
λ в + λ C + λ D.
7. If A = B = C= D, we shall have 2 (AXA XA× a), or λ aª
L
11
λ ^ + λ ^ + λ ^ +λ a 4 λA: and in general λA = ηλ Α.
Thus it appears that the logarithm of any integral positive
power, n, of any number A, is equal to n times the logarithm
of A.
n
1
10000
22
2 A (n and p being positive in-
7
p
tegers). For, let AK, and consequently A" λ K. From
κ.
n
the equation AK, we have by raising the
power p, A"
K", and, of consequence, nλ A
whole to the
pax, or by
N
division A
p
λπ πλα
N
M
S
100
ALGEBRA LOGARITHMS.
F
•
*M
A
9. From the same principle it follows, that 2
B
10. Farther, 2 a
A
For, let-q, and consequently A=BXQ: we shall then have
B
λ λ =λ B + λa; whence λ q=λ a—2 B.
Q
So that the logarithm
of the quotient is equal to the logarithm of the dividend minus.
that of the divisor; or the logarithm of a fraction is equal to
the logarithm of the numerator made less by that of the deno-
minator.
11. Again, λ A
n
ηλΑ.
2 ^" = 21λA"= 0 nλA: which is no other than
Λ
λι
ηλA,
P
n
= 0
gle
n
0 — 2
Wedd
N
p
·λ A =
For A
2 A. For A
n
-끔
​p
n
I
n
-λ a.
1
λ A
M
A
λ B.
グレ
​T
For A¯ = 1÷A
1A": whence
results 2 A
p
12. Suppose there be two systems of logarithms whose roots
or bases are r and s. Let any number N have p for its loga-
rithm in the first system, and q for its logarithm in the second :
p
: therefore
N
27
we shall have N=7” and N=s"; which gives rº=s, and s =
Therefore, taking the logarithms for the system r, we shall have
2s=22r; or, if in the system r we have 2 r=1, then λ s=
p
I
q
p
1
or q
px
Thus, knowing the logarithm p of any
λο
λς
number N, for the system whose base is r, we may obtain the
logarithm q of the same number for the system s, by multiply-
ing p by a fraction whose numcrator is unity and denominator
the logarithm of s taken in the system r.
13. In the system of logarithms first constructed by Baron
Napier, the great inventor, r = 2*718281828459, &c. and the
exponents are usually denominated Napierian or Hyperbolic
logarithms; the latter name being given because of the
relation between these logarithms and the lines and asymptotic
spaces in the equilateral hyperbola: so that in this system n
is always the hyperbolic logarithm of (2-71828, &c.)". But
in the system constructed by Mr. Briggs (corresponding with
the spaces in a hyperbola whose asymptotes make an angle
of 25° 44′ 25″ 28″), called common or Briggean logarithms,
10; so that the common logarithm of any number is the
index of that power of 10 which is equal to the said number.
p
ALGEBRA LOGARITHMS.
101
and 9023
10
,
Thus, if 50=10
; then is 1.69897
the common logarithm of 50, and 3.955351 the common loga-
rithm of 9023.
•69897
14. The rules for the management and application of loga-
rithms being given in the best collections of logarithmic tables,
are here omitted. The tables published in England by Dr. Hut-
ton, those published in France by Callet, and those recently
published by Professor Babbage, may be recommended as the
most correct and best fitted for scientific use. Mr. Galbraith's
tables are correct and valuable.
1
12
If the reader wish for neatly arranged tables in small com-
pass, for the practical purposes of a man of business, I would
recommend those of Mr. Woollgar, given in the Mechanics'
Magazine, those recently published by Mr. J. R. Young, and
those given by Mr. Carr, in his valuable " Synopsis of Prac-
tical Philosophy."
3.955351
SECTION XIII-Computation of Formula.
Since the comprehension, and the numerical computation of
formulæ expressed algebraically, are of the utmost consequence
to practical men, enabling them to avail themselves advanta-
geously of the theoretical results of men of science, as well as
to express in scientific language the results of their own experi-
mental or other researches; it has appeared expedient to present
brief treatises of Arithmetic and Algebra. The thorough un-
derstanding of these two initiatory departments of science will
serve essentially in the application of all that follows in the pre-
sent volume; and that application may probably be facilitated
by a few examples, as below :-
G
Ex. 1. Let a=5, b=12, c=13, and s=a+b+c; then what
is the numerical value of the expression,
A
ad
-
2
✓ [ ½ s (3 s − a) (3 s — b) (ž s -c)], which denotes the area
of the triangle whose sides are 5, 12, and 13?
Here s=a+b+c=5+12+13=30; ½ s=15;
A S
a=15—5—10; å s -b=15 12=3; è s
=15 13=2.
Consequently, by substituting the numerical values of the
several quantities between the parentheses for them, we shall
have
padm
pag
Jag
с
✓ (15 × 10 × 3 × 2)=√/900=30, the value required.
Ex. 2. Suppose g = 321, 16: required the value of
4g to, an expression denoting the space in feet which a heavy
15
K 2
102
NUMERICAL COMPUTATION OF FORMULA.
body would fall vertically from quiescence in six seconds, in
the latitude of London.
Here ½ g t²=167½½ × 6º — 96 ½ × 6=579 feet.
12
Ex. 3. Given D=6, d=4, h 12, ₪ ≈ 3·141593; required
the value of л h (D²+D d +ď), a theorem for the solid content
→ 12
of a conic frustrum whose diameters of the two ends are D, d,
and height h.
Here D*=36, D d=6 × 4—24, d³
2618 nearly.
2
Hence (D²+D d+d²)=2618 (36+24+16) 12
2618 × 76 × 12—3∙141593 × 76=238.761068.
12
Ex. 4. Let a=1, h=25, g=193 inches: what is the value
of 2 a gh? This being the expression for the cubic inches
of water discharged in a second, from an orifice whose area is a,
and depth below the upper surface of water in the vessel or re-
servoir, h, both in inches.
Here 2 ag h=2 √✓✓(25 × 193)=10 ✔✅✅/193 10 X 13.89244
138.9244 cubic inches.
Ex. 5. Suppose the velocity of the wind to be known in
miles per hour; required short approximative expressions for
the yards per minute, and for the feet per second.
First 176060—8—29}=30 nearly.
16,
Also 5280÷(60x60)=5280-844-1 nearly.
If, therefore, n denote the number of miles per hour :
30
2
30 ʼn will express the yards per minute; and 1½ n, the feet
per second.
π
12
These are approximative results to render them correct,
where complete accuracy is required, subtract from each result
its 45th part, or the fifth part of its ninth part.
Thus suppose the wind blows at the rate of 20 miles per
hour:
2
3
Ba
Then 30 n=30 × 20 600 yards per minute, or more cor-
rectly 600-900=600-13=586 yards.
Also 14 n= =30 feet per second.
or, correctly, 30
=30
138=
Conversely, of the feet per second will indicate the miles
per hour, correct within the 45th part, which is to be added to
obtain the true result.
30
=29} feet.
3
Ex. 6. To find a theorem by means of which it may be as-
certained when a general law exists, and what that law is.
Suppose, for example, it were required to determine the
law which prevailed between the resistances of bodies moving
in the air and other resisting media, and the velocities with
which they move. Let v, v, denote any two velocities, and R,r
M
1
L
gives xx log.
1
COMPUTATION OF FORMULE.
the corresponding resistances experienced by a body moving
with those velocities: we wish to ascertain what power of v it
is to which R is proportional. Let x denote the index or ex-
ponent of the power; then will v* : v2 :: R: r, if a law subsist.
v²
XC
r
Consequently () =
R
พ
X
за
Div. the consequents by the antecedents, we have 1: (~~) ::1:
1: //
บ
R
This, expressed
or x
V
log. r
log. v
L
*
log.-;
R
+
103
log. R
- log. v
Hence, the quotient of the differences of the logs. of the re-
sistances, divided by the difference of the corresponding velo-
cities, will express the exponent
required.
This theorem is of very frequent application in reference to
the motion of cannon balls, of barges on canals, of carriages on
rail roads, &c. and may indeed be applied to the planetary
motions.
folded padded
A
logarithmically,
}
1
104
1
}
7
PLANE GEOMETRY.
CHAPTER III.
PRINCIPLES OF GEOMETRY.
Definitions.
1. GEOMETRY is a department of science, by means of which
we demonstrate the properties, affections, and measures of all
sorts of magnitude.
2. Magnitude is a continued quantity, or any thing that is
extended; as a line, surface, or solid.
3. A point is that which has no parts: i. e. neither length,
breadth, nor thickness.
4. A line is a length without breadth or thickness.
Cor. The extremes of a line are points.
5. A right line is that which lies evenly, or in the same di-
rection, between two points. A curve line continually changes
its direction.
Þ
Cor. Hence there can only be one species of right lines, but
there is infinite variety in the species of curves.
6. An angle is the inclination of two lines to
one another, meeting in a point, called the an-
gular point. When it is formed by two right
lines, it is a plane angle, as A; if by curve lines,
it is a curvilinear angle.
7. A right angle is that which is made by
one right line A в falling upon another c n,
and making the angles on each side equal,
A B C = A B D
ABD; So that A B does not incline
more to one side than another: A B is called a
perpendicular. All other angles are called
oblique angles.
8. An obtuse angle is greater than a right
angle, as R.
A
C B
R
S
D
9. An acute angle is less than a right angle, as s.
10. Contiguous angles are those made by one line falling
upon another, and joining to one another, as R, S.
1

PLANE GEOMETRY.
105
11. Vertical or opposite angles, are those made
on contrary sides of two lines intersecting one
another, as A, B.
12. A surface has only length and breadth. The extremes
or limits of a surface are lines.
13. A plane is that surface which lies perfectly even be-
tween its extremes; or in which, right lines any way drawn
coincide.
A
14. A solid is a magnitude extended every way, or which
has length, breadth, and depth.
The terms or extremes of a solid are surfaces.
15. The square of a right line is the space included by four
right lines equal to it, set perpendicular to one another.
16. The rectangle of two lines is the space included by four
lines equal to them, set perpendicular to one another, the oppo-
site ones being equal.
2. If two right lines, A B, C D, cut one
another, the opposite angles E and G will be
equal.
SEC. I.-Of Angles and Right Lines, and their Rectangles.
Prop. 1. If to any point c in a right line a B,
several other right lines DC, E c are drawn on
the same side; all the angles formed at the
point o, taken together, are equal to two right
angles, ACD + D C E + EC B two right
angles.
A
A
4. If a right line o &, intersects two parallels
A D, F H; the alternate angles, A B E, and B E H
will be equal.
B
Cor. 1. All the angles made about one point in a plane, be-
ing taken together, are equal to four right angles.
Cor. 2. If all the angles at o, on one side of the line A B,
are found to be equal to two right angles: then A o B is a
straight line.
EL
F
G
V
C
G
H
E
D
3. A right line, в H, which is perpendicular to one of two
parallels, is perpendicular to the other.
E B
A B C D
B
*
J

106
PLANE GEOMETRY: ANGLES, RECTANGLES, &c.
Cor. 1. The external angle c B D, is equal to the internal
angle on the same side B E H.
Cor. 2. The two internal angles on the same side are equal
to two right angles D B E B E H = two right angles.
+
5. Right lines parallel to the same right line, are parallel
to one another.
6. If a right line A c be divided into two
parts, A B, B´C: the square of the whole line is
equal to the squares of both the parts, and
twice the rectangle of the parts, a c² A B³ +
2
B C² + 2 AB. B C.
Jada P
G
D
A
H I

E
B
F
7. The square of the difference of two lines A C, B c, is equal
to the sum of their squares, wanting twice their rectangle,
A Cº + B C³ -2 AC. B C.
A B³
8. The rectangle of the sum and difference of two lines is
equal to the difference of their squares.
9. The square of the sum, together with the square of the
difference of two lines, is equal to twice the sum of their
squares.
SECTION II-Of Triangles.
Definitions.
1. A triangle is a plane figure bounded by three right lines,
called the sides of the triangle.
2. An equilateral triangle is one which has three equal
sides.
3. An equiangular triangle is one which has three equal
angles; and two triangles are said to be equiangular, when
the angles in the one are respectively equal to those in the
other.
4. An isosceles triangle has two sides equal.
5. A right-angled triangle is that which has a right angle.
The side opposite to the right angle is called the hypothenuse.
6. An oblique triangle is one having oblique angles.
7. An obtuse angled triangle has one obtuse angle.
8. An acute angled triangle has three acute angles.
9. A scalene triangle has three unequal sides.
10. Similar triangles are those whose angles are respective-
ly equal, each to each. And homologous sides are those lying
between equal angles.
1. The base of a triangle, is the side on which a perpen-
PLANE GEOMETRY: TRIANGLES.
107
dicular is drawn from the opposite angle called the vertex;
the two sides, proceeding from the vertex, are called the legs.
Prop. 1. In any triangle A B C, if one side
B c be produced or drawn out; the external an-
gle A c d will be equal to the two internal oppo-
site angles A B.
P

A
Δ
B
C
D
2. In any triangle, the sum of the three angles is equal to two
right angles.
Cor. 1. If two angles in one triangle be equal to two angles
in another the third will also be equal to the third.
6. If two triangles A B C, ab c, have
two sides, and the included angle equal in
each; these triangles, and their corres-
ponding parts, shall be equal.
Cor. 2. If one angle of a triangle be a right angle, the sum
of the other two will be equal to a right angle.
3. The angles at the base of an isosceles triangle, are
equal.
r
Cor. 1. An equilateral triangle is also equiangular; and the
contrary.
Cor. 2. The line which is perpendicular to the base of an
isosceles triangle, bisects it and the verticle angle.
4. In any triangle, the greatest side is opposite to the greatest.
angle, and the least to the least.
5. In any triangle A B C, the sum of any
two sides BA, A C, is greater than the
third в c, and their difference is less than
the third side.
B
a
13
11. If a line p E be drawn parallel to one side
Be, of a triangle; the segments of the other
sides will be proportional ; a d:DB:: AE: EC.
13
E
A
D
C
7. If two triangles A B C and a be, have two angles and an
included side equal, each to each; the remaining parts shall
be equal, and the whole triangles equal.
8. If two triangles have all their sides respectively equal :
all the angles will be equal, and the wholes equal.
9. Triangles of equal bases and heights are equal.
10. Triangles of the same height, are in proportion to one
another as their bases.
A
F
C
D
+
Cor. 1. If the segments be proportional, a n : n B :: AE : E 0 ;
then the line D is parallel to the side B Č.
Cor. 2. If several lines be drawn parallel to one side of a
triangle, all the segments will be proportional.
/
1
108
PLANE GEOMETRY: TRIANGLES.
1
Cor. 3. A line drawn parallel to any side of a triangle, cuts
off a triangle similar to the whole.
A

12. In similar triangles the ho-
mologous sides are proportional;
A BACDED F.
B
··
1
----་་
0
13. Like triangles are in the duplicate ratio, or as the squares
of their homologous side.
B
14. In a right angled triangle в A C,
if a perpendicular be let fall from the
right angle upon the hypothenuse, it will
divide it into two triangles similar to one
another and to the whole, A B D, A D C.
Cor. 1. The rectangle of the hypothenuse and either seg-
ment is equal to the square of the adjoining side.
D
C
15. The distance A o of the right angle, from the middle
of the hypothenuse is equal to half the hypothenuse.
25. If D be any point in the base of a
scalene triangle, A B C : then is A B². DC +
A C². B D A D².B C + B C. B D. DC.
D
E
16. In a right-angled triangle, the square of the hypothenuse
is equal to the sum of the squares of the two sides.
17. If the square of one side of a triangle be equal to the sum
of the squares of the other two sides; then the angle compre-
hended by them is a right angle.
18. If an angle A, of a triangle в a c, be bi-
sected by a right line a D, which cuts the
base ; the segments of the base will be propor-
tional to the adjoining sides of the triangle;
B
A
BDD C::AB: A C,
B
D
19. If the sides be as the segments of the base, the line A D
bisects the angle a.
Alerts N
20. Three lines drawn from the three angles of a triangle to
the middle of the opposite sides, all meet in one point.
21. Three perpendicular lines erected on the middle of the
three sides of any triangle, all meet in one point.
22. The point of intersection of the three perpendiculars,
will be equally distant from the three angles: or, it will be the
centre of the circumscribing circle.
D
F

A
23. Three perpendiculars drawn from the three angles of a
triangle, upon the opposite sides, all meet in one point.
24. Three lines bisecting the three angles of a triangle, all
mcet in one point.
E
A

с
PLANE GEOMETRY: QUADrangles and POLYGONS: 109
SECTION III.-Of Quadrilaterals and Polygons.
Definitions.
1. A quadrangle or quadrilateral, is a plane figure bounded
by four right lines.
A
G
2. A parallelogram is a quadrangle whose opposite sides.
are parallel, as A G B H. The line A B drawn to the opposite
corners is called the diameter or diagonal. And if two lines
be drawn parallel to the two sides,
through any point of the diagonal; they
divide it into several others, and then
C, D are called parallelograms about the
diameter and E, F the complements: and H
the figure EDF a gnomon.
3. A rectangle is a parallelogram whose sides are perpen-
dicular to one another.

7. A trapezoid is a quadrangle, having only
two sides parallel.
E
1
F
4. A square is a rectangle of four equal sides and four equal
angles.
5. A rhombus is a parallelogram,
whose sides are equal, and angles ob-
lique.
6. A rhomboid is a parallelogram, whose sides are unequal
and angles oblique.
8. A trapezium is a quadrangle, that has no
two sides parallel.
9. A polygon is a plane figure enclosed
by many right lines. If all the sides and
angles are equal, it is called a regular poly-
gon, and denominated according to the num-
ber of sides or angles, as a pentagon 5, a
hexagon 6, a heptagon 7, &c.
D.
B
W



B
16
L
110 PLANE GEOMETRY: QUADRILATERALS AND POLYGONS.
10. The diagonal of a quadrangle or polygon is a line
drawn between any two opposite corners of the figure, as
A B.
11. The height of a figure is a line drawn from the top per-
pendicular to the base, or opposite side on which it stands.
12. Like or similar figures, are those whose several angles
are equal to one another, and the sides about the equal angles.
proportional.
13. Homologous sides of two like figures are those between
two angles, respectively equal.
14. The perimeter or circumference of a figure, is the com-
pass of it, or sum of all the lines that enclose it.
15. The internal angles of a figure are those
on the inside, made by the lines that bound
the figure, A D C B.
16. The external angle of a figure is the
angle made by one side of a figure, and the
adjoining side drawn out, as в A F.
3. Any line в c passing through the mid-
dle of the diagonal of a parallelogram P,
divides the arca into two equal parts.
4.
ין
5. In any parallelogram A B D C, the com-
plements c 1, and I в are equal.
A
B
f
A
E
A
H
D
Prop. 1. In any parallelogram the opposite sides and angles
are equal; and the diagonal divides it into two equal
triangles.
2. The diagonals of a parallelogram intersect each other in
the middle point of both.
A
B
P
C
B

C

19
B
4. Any right line в c drawn through the middle point P of
the diagonal of a parallelogram, is bisected in that point;
BPP C. (See preceding fig.)
G
10
F
D
P
C
E
6. Parallelograms of equal bases and heights are equal.
7. A parallelogram is double a triangle of the same, or an
equal base and height.
8. Parallelograms of the same height are to one another as
their bases.
·

PLANE GEOMETRY: QUADRILATERALS AND POLYGONS.
111
9. Parallelograms of equal bases are as their heights.
10. Parallelograms are to one another, as their bases and
heights.
11. In any parallelogram the sum of the squares of the
diagonals is equal to the sum of the squares of all the four
sides.
12. The sum of the four internal angles of any quadrilateral
figure, is equal to four right angles.
13. If two angles of a quadrangle be right angles, the sum
of the other two amounts to two right angles.
14. The sum of all t e internal angles of a polygon is equal
to twice as many right angles, abating four, as the polygon has
sides.
15. Hence all right-lined figures of the same number of sides,
have the sum of all the internal angles equal.
16. The sum of the external angles of any polygon is equal
to four right angles.
17. All right-lined figures have the sum of their external
angles equal.
B
S
D
E
18. In two similar figures a c, PR; if two lines в E, Q T, be
drawn after a like manner, as suppose,
to make the angle C B E = R Q T ; then
these lines have the same proportion, as
any two homologous sides of the figure;
VİZ. B E : Q T :: B C : Q R :: A B : P Q :: A D : P S.
BL
19. All similar figures are to one another as the squares of
their homologous sides.
20. Any figure described on the hy-
pothenuse of a right-angled triangle, is
equal to two similar figures described
the same way upon the two sides : B F C
ALCA G B.
1)
B
21. Any regular figure A B C D E, is equal
to a triangle whose base is the perimeter
A B C D E A; and height, the perpendicular
or, drawn from the centre, perpendicular to
one side.
1
A
E
(
P
L
APR
ין
R
C
D
22. Only three sorts of regular figures can fill up a plane
surface, that is, the whole space round an assumed point; and
these are six triangles, four squares, and three hexagons.
:


1
112
1
PLANE GEOMETRY: CIRCLES, &c.
SECTION IV. Of the Circle, and Inscribed and Circum
scribed Figures.
Definitions.

1. A circle is a plane figure described by
a right line moving about a fixed point, as
A c about c or it is a figure bounded by one
line equidistant from a fixed point.
5. The diameter is a line drawn through
the centre, from one side to the other,
A D.
2. The centre of a circle is the fixed point about which the
line moves, c.
3. The radius is the line that describes the circle, c a.
Cor. All the radii of a circle are equal.
8. An arch is any part of the circum-
ference A B.
4. The circumference is the line described by the extreme
end of the moving line, A B D a.
Ꮐ
band
A
O
A
F
B
D
C
-C
10
D)
6. A semicircle is half the circle, cut off by the diameter, as
A B D.
7. A quadrant, or quarter of a circle, is the part between two
radii perpendicular to one another, as C D E.
C
B

B
DE F, or DAB F.
11. A cord, a right line drawn through the circle, as D F.
D

F
E
9. A sector is a part bounded by two radii, and the arch be-
tween them, a C B.
10. A segment is a part cut off by a right line, or cord,
PLANE GEOMETRY
113
CIRCLES, &c.
12. Angle at the centre is that whose angular point is at the
centre A C B. (See the last figure.)

13. Angle at the circumference is when
the angular point is in the circumference,
as B A D.
B
C
14. Angle in a segment, is the angle made by two lines
drawn from some point of the arch of that segment to the ends
of the base; as B C D is an angle in the segment B C D.
A
15. Angle upon a segment, is the angle made in the oppo-
site segment, whose sides stand upon the base of the first; as
BA D, which stands upon the segment B C D.
16. A tangent is a line touching a circle, which produced,
does not cut it, as G A F. (Fig. to def. 5.)
17. Circles are said to touch one another, which meet, but do
not cut one another.
18. Similar arches, or similar sectors, are those bounded
by radii that make the same angle.
19. Similar segments are those which contain similar tri-
angles, alike placed.
20. A figure is said to be inscribed in a circle, or a circle
circumscribed about a figure, when all the angular points of
the figure are in the circumference of the circle.
21. A circle is said to be inscribed in a figure, or a figure
circumscribed about a circle, when the circle touches all the
sides of the figure.
Prop. 1. The radius c R, bisects any cord at
right angles, which passes not through the
centre, as a B.
22. One figure is inscribed in another, when all the angles
of the inscribed figure are in the sides of the other.

1
B
R
Cor. 1. If a line bisects a cord at right angles, it passes
through the centre of the circle.
Cor. 2. The radius that bisccts the cord also bisects the
arch.
2. In a circle cqual cords are equally distant from the
centre.
3. If several lines be drawn through a circle, the greatest is
the diameter, and those that are nearer the centre are greater
than those that are farther off.
L 2
ľ
114
PLANE GEOMETRY
CIRCLES, &c.
4. If from any point three equal right lines can be drawn to
the circumference; that point is the centre.
5. No circle can cut another in more than two points
6. There can only two equal lines be drawn from any exte-
rior point r, to the circumference of a circle.
7. In any circle, if several radii be drawn mak-
ing equal angles, the arches and sectors compre-
hended thereby will be equal, if ACB=BCD: then,
arch A B=arch BD; and sector A C B B CD.
12. In a circle the angle
at the centre is double the
angle at the circumference,
standing upon the same arch;
BD C 2 BAC.
P
1
A
8. In the same or equal circles, the arches, and also the sec-
tors, are proportional to the angles intercepted by the radii.
9. The circumferences of circles are to one another as their
diameters.
10. A right line, perpendicular to the diameter of a circle,
at the extreme point, touches the circle in that point, and lies
wholly without the circle.
F
11. If two circles touch one another either inwardly or out-
wardly, the line passing through their centres shall also pass
through the point of contact.
D
Λ
C
15. The angle A B C in a semicircle is a
right angle.
F
13. All angles in the same segment of a
circle are equal, D A C=D B C, and D G C—D H C.
B
A
D
D
Α
C
B
1

D
F
0
D

B
G
14. If the extremities of two equal arches D A, B c, be joined
by right lines, DC, AB; they will be parallel.
A

H

F
G
16. The angle A B G, in a greater segment A B F G, 1s less than
a right angle; and the angle A B F, in a.less segment A в F, is
greater than a right angle.
PLANE GEOMETRY CIRCLES, &c.
115
17. If two lines cut-
ting a circle, intersect G
one another in a; and H
there be made at the cen-
tre ECF = BAD;
A
19. The angle a = LBH D
+ HDG, when A is within; or
BHD HD G, when a is
without the circle.
A
G
B
A
E
4.
F
D
A
20. In a circle, the angle made at the
point of contact between the tangent and
any chord, is equal to the angle in the al-
ternate segment; E C F = E B C, and E C A
=EG C.
THE
A
;
Then arch BD + GH = 2 E F, if A is within the circle; or
arch B D G H = 2 E F, if A is without.
18. If from a point without, two lines touch a circle: the
angle made by them is equal to the angle at the centre, standing
on half the difference, of these two parts of the circumference.
24. In a circle if the diameter A D be drawn,
and from the ends of the cords A B, A c, per-
pendiculars be drawn upon the diameter;
the squares of the chords will be as the seg-
ments of the diameter; A E AFA B2 :
A C².
A E A F
B
pa
25. If two circles touch one another in P,
and the line P DE be drawn through their
centres; and any line P A B is drawn through
that point to cut the circles, that line will
be divided in proportion to the diameters;
PAPB PD PE.
D
H
C
A
P
A
B
II
G
B
D
C
D
B


A
C
21. A tangent to the middle point of an arch, is parallel to
the chord of it.
B
E
22. If from any point в in a semicircle,
a perpendicular в D be let fall upon the
diameter, it will be a mean proportional
between the segments of the diameter
A
ADD BD BD C.
23. The chord is a mean proportional between the adjoining
segment and the diameter, from the similarity of the triangles:
that is, A D AB::AB: AC; and CD:CB.: C B C A.
A
E
D
I
F
بت
G
F
D
C
B
E
G


B
E
מ!



D

116
PLANE GEOMETRY CIRCLES, &c.
26. If through any point r in the diameter
of a circle, any chord cr D be drawn, the
rectangle of the segments of the chord is
equal to the rectangle of the segments of the
diameter; c F. F D — A F. F B = also G F . E

FE.
27. If through any point r out of the circle
in the diameter в A produced, any line F C D
be drawn through the circle: the rectangle of
the whole line and the external part is equal
to the rectangle of the whole line passing
through the centre, and the external part;
DF. F C =
=AF. F B.
Τ
F
K
ľ D
A
B
←
D
H
O
D
G
B
28. Let H r be a tangent at ; then the rectangle c r. F D =
square of the tangent F H.
29. If from the same point r, two tangents be drawn to the
circle, they will be equal; r H = F I.
с
30. If a line P F c be drawn perpendicular to the diameter
A D of a circle, and any
line drawn from a to cut
the circle and the per-
pendicular; then the rec-
tangle of the distances of a
the sections from a, will
be equal to the rectan-
gle of the diameter and
the distance of the perpendicular from A; ABXAC = APX AD.
Also, a в × A C — A K².
d
31. In a circle EDF whose centre is c, and radius c E, if
the points в A, be so placed in the
diameter produced, that c B, C E,
CA, be in continual proportion, then
two lines û D, A D drawn from these
points to any point in the circumfer-
ence of the circle will always be in
the given ratio of в E, to A E.
32. In a circle, if a perpendicular D B be
let fall from any point D, upon the diame-
ter c 1, and the tangent Do drawn from D,
then a B, a c, a o, will be continually propor-
tional.
n
P
B
B
0
r
D
E
B C
D
F
C
E
D P
BA
G

p
I



P

PLANE GEOMETRY
117
CIRCLES, &c.

33. If a triangle B DF be inscribed in a
circle, and a perpendicular D P let fall from D
on the opposite side B F, and the diameter
DA drawn; then, as the perpendicular is to
one side including the angle D, so is the
other side to the diameter of the circle ;
B
DPDB:: DF: D A.
34. The rectangle of the sides of an inscribed triangle is equal
to the rectangle of the diameter, and the perpendicular on the
third side; B D. D F=A D. D p.
35. If a triangle B A C be inscribed in a
circle, and the angle A bisected by the
right line A E D, then as one side to the
segment of the bisccting line, within the
triangle, so the whole bisecting line to
the other side; A B A E:: AD: AC; and
A B . A C=B E . E C+A E³.
36. If a quadrilateral A B C D be in-
scribed in a circle, the sum of two oppo-
site angles is equal to two right angles;
ADC+ABC=two right angles.
B
E B
A
43. If an equilateral triangle A B C be in-
scribed in a circle; the square of the side
thereof is equal to three times the square of
the radius: A B³—3 A D³.
A
17
E
D
с
B
C
37. If a quadrangle be inscribed in a circle, the rectangle of
the diagonals is equal to the sum of the rectangles of the oppo-
site sides.
P
38. A circle is equal to a triangle whose base is the circum-
ference of the circle; and height, its radius.
A
D
39. The area of a circle is equal to the rectangle of half the
circumference and half the diameter.
D
40. Circles (that is, their areas) are to one another as the
squares of their diameters, or as the squares of the radii, or as
the squares of the circumferences.
F
с
41. Similar polygons inscribed in circles, are to one another
as the circles wherein they are inscribed.
42. A circle is to any circumscribed rectilineal figure, as the
circle's periphery to the periphery of the figure.
E
Ꭰ
A
F



118
PLANE GEOMETRY CIRCLES, &c.
44. A square inscribed in a circle, is equal to twice the
square of the radius.
45. The side of a regular hexagon inscribed
in a circle, is equal to the radius of the circle;
B E B C.
46. If two chords in a circle mutually inter-
sect at right angles, the sum of the squares of
the segments of the chords is equal to the
square of the diameter of the circle. A p³ +
PB² + P C² + P D²= diam.*
RT is the arithmetical mean,
RS is the geometrical mean,
RV is the harmonical mean.
48. If the arcs P Q, Q R, RS, &c. be
equal, and there be drawn the chords
P Q, P R, P S, &c. then it will be P Q : P R ::
PRP QPS:: PS: PR+PT::PT:
rs + P v, &c.
1
47. If the diameter P Q be divided into two
parts at any point R, and if R s be drawn perpen-
dicular to PQ; also R T applied equal to the
radius, and T R produced to the circumference P
at v then, between the two segments P R,
R Q,
Q
E
P
F
A
R
B
D
G
S
B

IC
P
C
T S
R
=
K
V
π

C

Q

V
W
I
PLANE GEOMETRY: CIRCLES, &c,
119
1
49. The centre of a circle being o, and
P a point in the radius, or in the radius.
produced; if the circumference be divided
into as many equal parts A B, B C, C D, &c.
as there are units in 2 n, and lines be
drawn from P to all the points of division;
then shall the continual product of all the
alternate lines, viz. P A X PC X PE &c.
be доог when P is within the
circle, or
pn when P is
xn
without the circle; and the product.
of the rest of the lines, viz. P B X P D
XPF,
&c. pn+xn: where r =
A o the radius, and x = o p the dis-
P
tance of p from the centre.
P
*
Ka
P
50. A circle may thus be divided into any
number of parts that shall be equal to one
another both in area and perimeter. Divide
the diameter QR into the same number of
equal parts at the points s, T, v, &c.; then
r,
on one side of the diameter describe semi-
circles on the diameters Q s, QT, Q v, and on
the other side of it describe semicircles on
R V, R T, R S; so shall the parts, 1 7, 3 5,
5 3, 7 1, be all equal, both in area and peri-
meter.
A
B
K
B
K
P
1
с
I
S
3
SECTION V.-Of Planes and Solids.
7
C
O
5
T
LO
O
H
5
H
D
D

7
V
3
E
G
E
F

R
Definitions.
1. The common section of two planes, is the line in which
they meet, or cut cach other.
2. A line is perpendicular to a plane, when it is perpendi-
cular to every line in that plane which meets it.
3. One plane is perpendicular to another, when every line.
of the one, which is perpendicular to the line of their common
section, is perpendicular to the other.
120
SOLID GEOMETRY.
*
4. The inclination of one plane to another, or the angle they
form between them, is the angle contained by two lines, drawn
from any point in the common section, and at right angles to the
same, one of these lines in each plane.
5. Parallel planes are such as being produced ever so far
both ways, will never meet, or which are everywhere at an
equal perpendicular distance.
6. A solid angle is that which is made by three or more plane
angles, meeting each other in the same point.
7. Similar solids, contained by plane figures, are such as
have all their solid angles equal, each to each, and are bounded
by the same number of similar planes, alike placed.
8. A prism is a solid whose ends are parallel, equal, and like
plane figures, and its sides, connecting those ends, are parallelo-
grams.
9. A prism takes particular names according to the figure of
its base or ends, whether triangular, square, rectangular, penta-
gonal, hexagonal, &c.
10. A right or upright prism, is that which has the planes
of the sides perpendicular to the planes of the ends or base.
11. A parallelopiped, or a parallelopipedon,
is a prism bounded by six parallelograms, every
opposite two of which are equal, alike, and
parallel.
12. A rectangular parallelopipedon is that whose bound-
ing planes are all rectangles, which are perpendicular to each
other

13. A cube is a square prism, being bounded
by six equal square sides or faces, which are per-
pendicular to each other.
Gove
14. A cylinder is a round prism having circles for
its ends; and is conceived to be formed by the rota-
tion of a right line about the circumferences of two
equal and parallel circles, always parallel to the axis.
15. The axis of a cylinder is the right line joining the cen-
tres of the two parallel circles about which the figure is de-
scribed.
SOLID GEOMETRY.
121
16. A Pyramid is a solid whose base is any right-
lined plane figure, and its sides triangles, having all
their vertices meeting together in a point above the
base, called the vertex of the pyramid.
17. Pyramids, like prisms, take particular names from the
figure of their base.
18. A cone is a round pyramid having a circular
base, and is conceived to be generated by the rota-
tion of a right line about the circumference of a circle,
one end of which is fixed at a point above the plane of
that circle.
19. The axis of a cone is the right line, joining the vertex
or fixed point, and the centre of the circle about which the
figure is described.
K
20. Similar cones and cylinders, are such as have their al-
titudes and the diameters of their bases proportional.
21. A sphere is a solid bounded by one curve surface, which
is every where equally distant from a certain point within,
called the centre. It is conceived to be generated by the rota-
tion of a semicircle about its diameter, which remains fixed.
22. The axis of a sphere is the right line about which the
semicircle revolves, and the centre is the same as that of the
revolving semicircle.
23. The diameter of a sphere is any right line passing
through the centre, and terminated both ways by the surface.
24. The altitude of a solid is the perpendicular drawn from
the vertex to the opposite side or base.
Prop. 1. If any prism be cut by a plane parallel to its base,
the section will be equal and like to the base.
2. If a cylinder be cut by a planc parallel to its base, the
section will be a circle, equal to the base.
3. All prisms and cylinders, of equal bases and altitudes,
are equal to each other.
4. Rectangular parallelopipedons, of equal altitudes, are to
each other as their bases.
5. Rectangular parallelopipedons, of equal bases, are to each
other as their altitudes.
6. Because, prisms and cylinders are as their altitudes, when
their bases are equal: and, as their bases when their altitudes
are equal. Therefore, universally, when neither are equal,
they are to one another as the product of their bases and alti-
tudes hence, also, these products are the proper numeral
measures of their quantities or magnitudes.
M
122
SOLID GEOMETRY.
7. Similar prisms and cylinders are to each other as the cubes
of their altitudes, or of any like linear dimensions.
8. In any pyramid a section parallel to the base is similar to
the base; and these two planes are to each other as the squares
of their distances from the vertex.
9. In a cone, any section parallel to the base is a circle; and
this section is to the base as the squares of their distances from
the vertex.
10. All pyramids and cones, of equal bases and altitudes,
are equal to one another.
11. Every pyramid is a third part of a prism of the same
base and altitude.
12. If a sphere be cut by a plane, the section will be a
circle.
13. Every sphere is two-thirds of its circumscribing cy-
linder.
14. A cone hemisphere, and cylinder of the same base and
altitude, are to each other as the numbers 1, 2, 3.
15. All spheres are to each other as the cubes of their diame-
ters; all these being like parts of their circumscribing cylinders.
16. None but three sorts of regular plane figures joined
together can make a solid angle and these are, 3, 4, or 5
triangles, 3 squares, and three pentagons.
And therefore there can only be five regular bodies, the py-
ramid, cube, octaedron, dodecaedron, and icosaedron.
17. No other but only one sort of the five regular bodies,
joined at their angles, can completely fill a solid space; viz.
eight cubes.
18. A sphere is to any circumscribing
solid B F, (all whose planes touch the
sphere); as the surface of the sphere to the
surface of the solid.
B
P
D
J

F
E
19. All bodies circumscribing the same sphere, are to one
another as their surfaces.
20. The sphere is the greatest or most capacious of all
bodies of equal surface.
+
PRACTICAL GEOMETRY.
123
SECTION VI.-Practical Geometry.
It is not intended in this place to present a complete collec-
tion of Geometrical Problems, but merely a selection of the most
useful, epecially in reference to the employments of mechanics
and engineers.
The instruments well known to be used in geometrical con-
structions, are the scale and compasses, the semicircular or the
circular protractor, the sector, and a parallel ruler. To these
a few other useful instruments may be added, which we shall
describe as we proceed; speaking first of the Triangle and
Ruler.
These are, as their names indicate, a triangle, that is to say,
an isosceles right-angled triangle, and a ruler, both made of
well seasoned wood, or of ivory, ebony, or metal. Each side
A B, A c, of the triangle, about the right angle A, being 3, 4, 6,
or 8 inches, according to the magnitude of the figures, in
whose construction it is
likely to be employed.
About the middle of the
triangle there should be a
circular orifice, as shown
A

R
D
C
in the figures; and if a
t
scale of equal parts be
placed along each of the three sides, all the better. The ruler
may be from 12 to 18 inches in length; and it also may, use-
fully, have a scale along one of its sides. The conjoined appli-
cation of these instruments is of great utility; as will soon
appear.
a
ď
E
PROB. I. To bisect a given line.
Let a b be the line proposed. Lay the longest side в c of
the triangle so as to coincide with a b, and so that its angle B
shall coincide with the point a; and along the side B A of the
triangle draw a line a d. Then slide the
base B C of the triangle along the line
a b, until c coincides with b, and draw
in coincidence with the side c A, the line
b d intersecting the former in d. Next
bring the ruler to coincide with a b,and
in contact with it lay one of the legs в A of the triangle; then
slide the triangle along the ruler, until the other leg A c passes
through the point d: draw along a c, so posited, the line di;
it will be perpendicular to a b, and will bisect it in i, the point
required.
b
P)

124
PRACTICAL GEOMETRY.
PROB. 2. Through a given point, c, to draw a line parallel to
a given line A B.
F
E
C
LX
A
B

A
KOMA, WAV
Make a Sunday
E C
me me den an yu qe me .
A
B
A B.
Place one of the sides of the triangle in contact with the line
Lay the ruler against one of the other sides of the tri-
angle; and keeping it steady, slide along the triangle until
the same side which had been made to coincide with part of
the line A B touches the point c: then, along that side, draw
through a the line EF; it will be parallel to A в as required.
B
PROB. 3. To bisect a given angle: then to bisect its half;
and so on
C
---
D
F
1
E
B

P
Let B AP be the proposed angle. Through any point в
draw в E parallel to A P (by the former problem). Upon в E
set off, with the compasses, from the scale at the edge of the
ruler, B C = B A join a c; it will bisect the angle B A P.
Again, set off, upon в E, from C, C D C A: join a d; it
will bisect CA P, or quadrisect B A P.
Again, set off, upon в E, DEDA: join EA; So shall
E AP be of BAP and so on.
ठ्ठ
PROB. 4. To erect a perpendicular at any given point c, in
a given line a B.
1st Method. Apply one of the legs, E F, of the triangle,
upon the line AB. Lay the side of the ruler H1, against the hypo-
thenuse, EG, of the triangle, and, keeping it steady, slide
the triangle upwards until the side r & touches the point c.
PRACTICAL GEOMETRY.
125
A
Then draw C D in contact with that leg, and it will be the per
pendicular required.


D
C F
a p
em a man at mak
- vna pakar m
-
II
B
A
A
D
C
E
G
2d Method. Apply the hypothenuse, & E, of the triangle to
the line AB. Lay the edge of the ruler H I, against the leg
G E. Keep it steady, and turn the triangle so that the other
leg FE may be laid against the ruler. Then slide the trian-
gle upwards until the hypothenuse touches the point c: then
in coincidence with it draw c D, and it will be the perpendicu-
lar required.
Note.-After similar methods may a perpendicular be let fall
from a point D above a line A в upon it.
H
C
3d Method, by a ruler and compasses only: as suppose it were
required to cut the end of a plank square. Let A B C D he the
plank, of which the end BD is required to be squared. The
edge A B being quite straight,
open the compasses to any con-
venient distance, and place the
point of one leg at B, and the
other at any point as F. Keep one
leg at F, and turn the other round
till it touches the edge A B at E;
keep them firm, and apply the
straight edge to E F, as the figure
shows; keep the leg still at F, and turn them over into the posi-
tion FG, & being close to the straight edge, and make a mark at G.
Now, if the straight edge be applied to & and B, and G B be
drawn, it will be square to the edge A B.
Note. In this construction it is evident that F is the centre,
and E G the diameter of a semicircle that passes through B; con-
quently в is a right angle.
F
E B

D
FD
B
18
M 2
126
PRACTICAL GEOMETRY.
PROB. 5. To divide a given line A B into any proposed num-
ber of equal parts.
C
1st Method. Draw any other line a c, forming any angle with
the given line AB on which set off as many of
any equal parts, A D, D E, E F, r c, as the line A B
is to be divided into. Join BC; parallel to which
draw the other lines FG, EH, DI: then these
will divide A B in the manner as required.
A
From the centres A and c, with any one ra-
dius, describe the arcs DE, F G. Then, with
radius DE, and centre F, describe an arc cutting
F G in G. Through & draw the line A &; and
it will form the angle required.
PROB. 7. To find the centre of a circle.
2d Method, without drawing parallel lines.
line which is to be divided into n equal parts.
extremity a draw any right line A C,
upon which set off n + 1 equal parts,
the point D being at the termination of
the (n+1)th part. Join D B and pro-
duce it until the prolongation вE=B D.
Let F
be the termination of the
(n-1)th part. Join F E, and the right
line of junction will cut the given line
A B in the point r, such that B P= A B; and of course distances
equal to в P set off upon в A, will divide it, as required.*
N
Draw any chord A B, and bisect it perpendicu-
larly with the line c p. Then bisect CD in
0, the
centre required.
D
GRA
Let A B be the
Through one
D)
C
C
A IHG B
E
A
PROB. 6. At a given point a, in a given line à в, to make an
angle equal to a given angle c.
Р
A
F

1
D
C
E
E
B

0
])
FR
13
* The truth of this method is easily demonstrated. Through r the intermediate
point of division, on a c, between r and ¤, draw 1 B. Then, because » в = }} ¢
and » 1 = 1 F, in is parallel to F r. Consequently, BP: DA:: IF : 1 A : : 1 : 1,
by construction.
PRACTICAL GEOMETRY.
127
PROB. 8. To describe the circumference of a circle through
three given points, A, B, C.

From the middle point в draw chords в A,
в c, to the other two points, and bisect these
chords perpendicularly by lines meeting in o,
which will be the centre. Then from the
centre o, at the distance of any of the points,
as o A, describe a circle, and it will pass
through the two other points B, C, as required.
A
WHummut.co
PROB. 9. On a given chord à в to describe an arc of a circle
that shall contain any number of degrees; performing the ope-
ration without compasses, and without finding the centre of the
circle.
Jum
с
Place two rulers, forming an angle A c B, equal to the supple-
ment of half the given number of degrees, and fix them in c.
Place two pins at the extremities of the given chord, and hold
a pencil in c;
then move the
edges of this in-
strument against
the pins, and the
my
הנות
A
B
R
tra
B
pencil will describe the arc required.
Suppose it is required to describe an arc of 50 degrees on the
given chord A B; subtract 25 degrees (which is half the given
angle) from 180, and the difference, 155 degrees, will be the sup-
plement. Then form an angle a C B of 155° with the two rulers,
and proceed as has been shown above.
V
с

PROB. 10. To describe mechanically the circumference of a
circle, through three given points, A, B, C, when the centre is
inaccessible; or the circle too large to be described with com-
passes.
Place two rulers, M N, R s, crossways, touching the three
points A B C. Fix them in v by a pin, and by a transverse piece
T.
M $
Hold a pencil in A, and describe the
arc в A C, by moving the angle R A N, so
as to keep the outside edges of the rulers
against the pins BC. Remove the instru-
ment RV N, and on the are described
mark two points, D, E, so that their dis-
tance shall be equal to the length B C.
Apply the edges of the instrument
against D E, and with a pencil in a de-
scribe the arc в c, which will complete
the circumference of the circle required.

2 MM UL
128
PRACTICAL GEOMETRY.
A
Otherwise. Let an axle of 12 or 15 inches long carry two
unequal wheels A and B, of which one, A, shall be fixed, while
the other, B, shall be susceptible of motion along the axle, and
then placed at any assigned distance, A, B upon the paper or
plane, on which the circle is to be described. Then will a and
в be analogous to the ends of a conic frustrum, the vertex of
the complete cone being the centre of the circle which will be
described by the rim, or edge,
of the wheel î, as it rolls upon
the proposed plane. Then, it
will be, as the diameter of
the wheel A, is to the differ-
ence of the diameters of A and B, so is the radius of the circle
proposed to be described by A, to the distance A в, at which
the two wheels must be asunder, measured upon the plane on
which the circle is to be described.
B
A

G
The wheel B will evidently describe, simultaneously, another
circle, whose radius will be less than that of the former by a B.
PROB. 11. To divide any given angle A B C into three equal
parts.
From B, with any radius, describe
the circle ACDA. Bisect the angle ABC
by в E, and produce a в to D. On the
edge of a ruler mark off the length
of the radius A B. Lay the ruler on
D, and move it till one of the marks
on the edge intersects B E, and the
other the arc A c in G. Set off the
distance c & from & to F: and draw
the lines в F, and B G, they will tri-
sect the angle a B c.

A
B
F
A
E
W
ZANOHO
с
Otherwise, by means of Mr. R. Christie's ingenious instru-
ment for the mechanical trisection of an angle.
PRACTICAL GEOMETRY.
129

B
G
mapapalaglagi japan mat
Ad
E
2.
1.
6
h
E
&
2
D
•
...
This instrument may be made either of wood or metal. Fig.
1 represents it applied to, and trisecting the angle H C в, and
fig. 2 represents it shut up. The pieces H C, E C, F C, and
GC, are all of the same length, and moveable on the joint c.
The joints a, b, c, and d, are all equally distant from c. The
connecting pieces a e, e b, b h, hc, c i, and i d, are all equal;
and the pieces ef, fh, h g, and g i, are equal to each other, but
longer than the preceding pieces. Two sockets, ƒ and g, fit,
and move up or down on the pieces E c and r c.
The pieces are
all connected by pivots at the joints, represented by the small
letters a, b, c, d, &c., and the connecting pieces fit in between
the other when the apparatus is shut.
In applying this instrument it is only necessary to lay the
centre c on the vertex of the given angle, c G, on one of the
sides forming it, and to move H c till it coincides with the other;
then each of the angles, H C E, E C F, and F C G, will be a third
of the angle H C G. For it is manifest, that the angle н с E can-
not be increased without increasing the angle a e b, and that
a e b cannot be increased without diminishing the angle bef
and the distance fb. But because be is equal to b h,fe to fh,
and ƒ b common to the two triangles ƒe b and ƒ h b; the an-
gle f h b must be always equal to the angle fe b, and conse-
quently b he to a e b; therefore н CE must in all positions of
the apparatus, continue equal to E C F. In the same manner it
might be shown that the angles E C F and F C G will always con-
tinue equal. Hence the angle H C B has been trisected by the
straight lines E C and F c. If the instrument had been applied
to the angle D C B, it would have taken the position represented
by the dotted lines.
Jetta
Note. It is evident that instruments may be made on the
same principle to divide an angle into any other number of equal
parts.
130
PRACTICAL GEOMETRY.
1
PROB. 12. To cut off from a given line A B, supposed to be
very short, any proportional part.
3
12'
Suppose, for example, it were required to find the 12, 12,
&c. of the line A в in the first figure below. From the ends a
and û draw A D D C perpendicular to A B.
From A to D set off
any opening of the compasses 12 times, and the same from B
to c. Through the divisions 1, 2, 3, &c. draw lines 1 f, 2 g,
&c. parallel to a B. Draw the diagonal a c, and 1 d will be
the of A B; 2 c,, and so on. The same method is applica-
ble to any other part of a given line.
12
A
2
109
1
2
3
4
Lo
A
5
6
d
7
8
9
10
11
12 D
B
4- 20
g
A
G
D) 12
RC
E
H
B
2
F
=
PROB 13. To make a diagonal scale, say of feet, inches, and
tenths of an inch.
Draw an indefinite line A B, on which set off from A to в the
given length for one foot, any required number of times. From
the divisions A, C, H, в, draw A D, C E, &c. perpendicular to a b.
On A D and B F set off any length ten times; through these di-
visions draw lines parallel to a B.
Divide A c and D E into 12
equal parts, each of which will be one inch.
Draw the lines ▲ 1,
G 2, &c. and they will form the scale 1equired; viz. each of the
larger divisions from E to 1, 1 to 2, &c. will represent a foot;
cach of the twelve divisions between D and E an inch; and the
several perpendiculars parallel to R C in the triangle E CR, 7,
1,7%, &c. of an inch.
1
3
Note. If the scale be meant to represent feet, or any other
unit, and tenths and hundredths, then D E must be divided into
ten instead of twelve equal parts.
PROB. 14. Given the side of a regular polygon of any number
of sides, to find the radius of the circle in which it may be in-
scribed.
K


Multiply the given side of the polygon by the number which
stands opposite the given number of sides in the column enti-
tled radius of circum. circle; the product will be the radius
required.
:
PRACTICAL GEOMETRY.
131
Thus, suppose the polygon was to be an octagon, and each
side 12, then 1.3065628 × 12=15.6687536 would be the radius
sought. Take 15.67 as a radius from a diagonal scale, describe
a circle, and from the same scale, taking off 12, it may be applied
as the side of an octagon in that circle.
PROB. 15. Given the radius of a circle to find the side of any
regular polygon (sides not exceeding 12) inscribed in it.
Multiply the given radius by the number in the column en-
titled factors for sides, standing opposite the number of the
proposed polygon; the product is the side required.
Thus, suppose the radius of the circle to be 5, then
5×1·732051=8.66025, will be the side of the inscribed equila-
teral triangle.
No. of
sides.
3 LO
4
5
6
7
9
10
11
12
Names.
TABLE OF POLYGONS.

Trigon
Tetragon, or Square
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Undecagon
Dodecagon
Multipliers
for areas.
Radius of
circum.cir.
0.4330127
0.5773503
1-732051
1.0000000
0.7071068
1.414214
1.7204774
0.8506508 1.175570
2.5980762 1.0000000 1.000000
3.6339124 1.1523824 0.867767
4.8284271 1.3065628 0-765367
6.1818242 1.4619022 0.684040
7.6942088 1.6180340 0.618034
9.3656399 1.7747324 0.563465
11.1961524 1.9318517 0.517638
PROB. 16. To reduce a rectilinear figure of 6, 7, or more sides,
to a triangle of equal area.
This is a very useful problem, as it saves much labour in com-
putation.
Factors
for sides.
4
Suppose A B C D E F G to be the proposed space to be reduced
to a triangle. Lay a parallel ruler from A to c, and move
it until it pass through B, marking the point 1 in which it
cuts A G continued. Then lay the ruler
through 1 and D, and move it until it pass
through c, and mark the point 2 where it
Next lay the ruler through 2
and F, move it up till it pass through D,
marking the point 3 where it cuts A G
continued. Again, lay the ruler through
3 and F, move it up until it pass through E,
the point of intersection with G A produced.
cuts A G.
G

1
A
2
B
D
F
E
and mark 4,
Lastly, draw
132
PRACTICAL GEOMETRY.
the right line 4 F; so shall the triangle 4 r & be equal in area
to the irregular polygon A B C D E F G.
Here в 1 is parallel to A c; so that if c 1 were drawn, the
triangle a 1 c would be equal to AC B and by the mechanical
process this reduction is effected. In like manner, the other
triangles are referred, one by one, to equal triangles, having
their bases on G A or its prolongation. Hence the principle of
the reduction is obvious.
PROB. 17. To reduce a simple rectilinear figure to a similar
a
one upon either a smaller or a larger scale.
Pitch upon a point p any where
about the given figure A B C D E,
either within it, or without it, or
in one side or angle; but near the
middle is best. From that point p
draw lines through all the angles;
upon one of which take p a to r A
in the proposed proportion of the
scales, or linear dimensions; then draw a b parallel to A в, b c
to в c, &c.; so shall a b c d e be the reduced figures sought, either
greater or smaller than the original. (Hutton's Mens.)
a
Otherwise to reduce a Figure by a Scale.-Measure all the
sides and diagonals of the figure, as A B C D E, by a scale; and
lay down the same measures respectively from another scale, in
the proportion required.
To reduce a Map, Design, or Figure, by Squares.-Divide
the original into a number of little squares, and divide a fresh
paper, of the dimensions required, into the same number of other
squares, either greater or smaller, as required. This done, in
every square of the second figure, draw what is found in the cor-
responding square of the first or original figure.
The cross lines forming these squares may be drawn with a
pencil, and rubbed out again after the work is finished. But a
more ready and convenient way, especially when such reduc-
tions are often wanted, would be to keep always at hand frames
of squares ready made, of several sizes; for by only just laying
them down upon the papers, the corresponding parts may be
readily copied. These frames may be made of four stiff or in-
flexible bars, strung across with horse hairs, or fine catgut.
g
When figures are rather complex, the reduction to a different
scale will be best accomplished by means of such an instrument
as Professor Wallace's Eidograph, or by means of a Panto-
graph, an instrument which is now considerably improved by
simply changing the place of the fulcrum. See the Mechanics'
Oracle, part II. page 33.

:
A
B
a
b
E
с
d
C
D
d
STAND
PLANE TRIGONOMETRY.
133
CHAPTER IV.
TRIGONOMETRY.
SECTION I-Plane Trigonometry.
1. Plane Trigonometry is that branch of mathematics by
which we learn how to determine or compute three of the six
parts of a plane, or rectilinear triangle, from the other three,
when that is possible.
The determination of the mutual relation of the sines, tan-
gents, secants, &c. of the sums, differences, multiples, &c. of arcs
or angles; or the investigation of the connected formulæ, is, also,
usually classed under plane trigonometry.
2. Let A C B be a rectilinear angle: if about c as a centre, with
any radius, c A, a circle be described, intersecting c A, C в, in
A, B, the arc A B is called the measure of the angle A C B. (See
the next figure.)
a,
ww
3. The circumference of a circle is supposed to be divided
or to be divisible into 360 equal parts, called degrees; each de-
gree into 60 equal parts, called minutes; each of these into 60
equal parts, called seconds; and so on to the minutest possible
subdivisions. Of these, the first is indicated by a small circle,
the second by a single accent, the third by a double accent, &c.
Thus, 47° 18' 34" 45", denotes 47 degrees, 18 minutes, 34
seconds, and 45 thirds. So many degrees, minutes, seconds, &c.
as are contained in any arc, of so many degrees, minutes, seconds,
&c. is the angle of which that arc is the measure said to be.
Thus, since a quadrant, or quarter of a circle, contains 90 de-
grees, and a quadrantal arc is the measure of a right angle, a
right angle is said to be one of 90 degrees.
4. The complement of an arc is its difference from a quad-
rant; and the complement of an angle is its difference from a
right angle.
5. The supplement of an arc is its difference from a semicir-
cle, and the supplement of an angle is its difference from two
right angles.
6. The sine of an arc is a perpendicular let fall from one ex-
tremity upon a diameter passing through the other.
;
19
N
134
PLANE TRIGONOMETRY.
>
1
7. The versed sine of an arc is that part of the diameter
which is intercepted between the foot of the sine and the arc.
8. The tangent of an arc is a right line which touches it in
one extremity, and is limited by a right line drawn from the
centre of the circle through the other extremity.
9. The secant of an arc is a sloping line which thus limits
the tangent.
10. These are also, by way of accommodation, said to be the
sine, tangent, &c. of the angle measured by the aforesaid arc,
to its determinate radius.
11. The cosine of an arc or angle, is the sine of the comple-
ment of that arc or angle: the cotangent of an arc or angle is
the tangent of the complement of that arc or angle. The co-
versed sine and co-secant are defined similarly.
To exemplify these definitions by the annexed diagram let
A B be an assumed arc of a circle described with the radius A c,
and let ▲ E be a quadrantal arc; let в D be demitted perpen-
B
dicularly from the extremity в upon
the diameter a A'; parallel to it let
A T be drawn, and limited by c r: let
G B and E м be drawn parallel to a a',
the latter being limited by or or cr
produced. Then B E is the comple-
ment of B A, and angle в C E the com-
plement of angle B C A; BE A' is the
supplement of B A, and angle B C A' the supplement of в CA;
BD is the sine, D A the versed sine, Ar the tangent, c r the
secant, & B the cosine, a r the coversed sine, E M the cotangent,
and c м the cosecant, of the arc A E, or, by convention, of the
angle A C B.
Note.-These terms are indicated by obvious contractions :
Thus, for sine of the arc A B we use
ditto
ditto
ditto
ditto
ditto
tangent
secant
versed sine
cosine .
•
cotangent
cosecant
coversed sine
ditto
ditto
A'
T
D'
B/
E
G
E
с
sin A B,
tan A B,
sec A B,
versin A B,
B
D.
T
COS A B,
cot A B,
cosec A B,
coversin A B.
A
F
M

PLANE TRIGONOMETRY.
135
Corollaries from the above Definitions.
12. (A.) Of any arc less than a quadrant, the arc is less than
its corresponding tangent: and of any arc whatever, the chord
is less than the arc, and the sine less than the chord.
(B.) The sine B D of an arc A B, is half the chord B F of the
double arc B A F.
(c.) An arc and its supplement have the same sine, tangent,
and secant. (The two latter, however, are affected by different
signs, or —, according as they appertain to marks less or
greater than a quadrant.)
+
(D.) When the arc is evanescent, the sine, tangent, and versed
sine, are evanescent also, and the secant becomes equal to the
radius, being its minimum limit. As the arc increases from
this state, the sines, tangents, secants, and versed sines increase ;
thus they continue till the arc becomes equal to a quadrant a E,
and then the sine is in its maximum state, being equal to radius,
thence called the sine total; the versed sine is also then equal
to the radius: and the secant and tangent becoming incapable
of mutually limiting each other, are regarded as infinite.
(E.) The versed sine of an arc, together with its cosine, are
equal to the radius. Thus, A D+B G=A D+DC=A C. (This is
not restricted to arcs less than a quadrant.)
(F.) The radius, tangent, and secant, constitute a right-angled
triangle c A T. The cosine, sine, and radius, constitute another
right-angled triangle C D B, similar to the former. So, again,
the cotangent, radius, and cosecant, constitute a third right-
angled triangle, м E c, similar to both the preceding. Hence,
when the sine and radius are known, the cosine is determined
by the property of the right-angled triangle.
The same may be said of the determination of the secant, from
the tangent and radius, &c. &c. &c.
(G.) Further, since CD:DB:: C A A T, we see that the tan-
gent is a fourth proportional to the cosine, sine and radius.
Also, O D C B::CA: CT; that is, the secant is a third pro-
portional to the cosine and radius.
Again, C GG BCE: EM; that is, the cotangent is a fourth
proportional to the sine, cosine, and radius.
And B DBC::CE: CM; that is, the cosecant is a third pro-
portional to the sine and radius.
136
PLANE TRIGONOMETRY.
(H.) Thus, employing the usual abbreviations, we should
have
2
1. cos=
✔(radº — sin³).
3. sec (rad³ + tan³).
=
rad X sin
5. tan
COS
7. sec
rad2
COS
8. cosec=
1. cos= √(1
V(1−sin®).
3. sec= √(1 + tan³).
sin
5. tan=
COS
1
sin
rad2
cot
These, when unity is regarded as the radius of the circle,
become
1
cot
2. tan= √(sec³ -
rad³).
4. cosec= √(rad³+cot³).
rad 2
rad X cos
sin
tan
8. cosec_ rad²
sin
6. cot.
·
2. tan= ✓✓(sec² 1).
4. cosec =√(1+cot³).
7. sec=
6. cot-
COS
sin
1
tan
Ad
1
COS
13. From these, and other properties, and theorems, mathe-
maticians have computed the lengths of the sines, tangents,
secants, and versed sines, to an assumed radius, that corres-
pond to arcs from one second of a degree, through all the
gradations of magnitude, up to a quadrant, or 90°. The results
of the computations are arranged in tables called Trigonome-
trical Tables for use. The arrangement is generally appro-
priated to two distinct kinds of these artificial numbers,
classed in their regular order upon pages that face each other.
On the left hand pages are placed the sines, tangents, secants,
&c. adapted at least to every degree, and minute, in the quad-
rant, computed to the radius 1, and expressed decimally.
On the right hand pages are placed in succession the corres-
ponding logarithms of the numbers that denote the several
sines, tangents, &c. on the respective opposite pages. Only,
that the necessity of using negative indices in the logarithms
may be precluded, they are supposed to be the logarithms of
sines, tangents, secants, &c. computed to the radius 10000000000.
The numbers thus computed and placed on the successive right
hand pages are called logarithmic sines, tangents, &c. The
numbers of which these are the logarithms, and which are
arranged on the left hand pages, are called natural sines, tan-
gents, &c.
The tables of Hutton, Galbraith, Ursin, and
Young, will serve well for the usual purposes: if very accurate
computations occur, in which the sines, tangents, &c. are re-
quired to seconds, the tables of Taylor and Bagay may be ad-
vantageously consulted.
PLANE TRIGONOMÈTRY.
137
II.-General Properties and Mutual Relations.
1. The chord of any arc is a mean proportional between the
versed sine of that arc and the diameter of the circle.
2. As radius, to the cosine of any arc; so is twice the sine of
that arc, to the sine of double the arc.
3. The secant of any arc is equal to the sum of its tangent,
and the tangent of half its complement.
4. The sum of the tangent and secant of any arc, is equal to
the tangent of an arc exceeding that by half its complement.
Or, the sum of the tangent and secant of an arc is equal to the
tangent of 45° plus half the arc.
5. The chord of 60° is equal to the radius of the circle; the
versed sine and cosine of 60° are each equal to half the radius,
and the secant of 60° is equal to double the radius.
6. The tangent of 45° is equal to the radius.
7. The square of the sine of half any arc or angle is equal to
a rectangle under half the radius and the versed sine of the
whole; and the square of its cosine, equal to a rectangle under
half the radius and the versed sine of the supplement of the
whole arc or angle.
8. The rectangle under the radius and the sine of the sum or
the difference of two arcs is equal to the sum of the difference
of the rectangles under their alternate sines and cosines.
9. The rectangle under the radius and the cosine of the sum
or the difference of two arcs, is equal to the difference or the sum
of the rectangles under their respective cosines and sines.
10. As the difference or sum of the square of the radius and
the rectangle under the tangents of two arcs, is to the square of
the radius; so is the sum or difference of their tangents, to the
tangent of the sum or difference of the arcs.
11. As the sum of the sincs of two unequal arcs, is to their
difference; so is the tangent of half the sum of those two arcs
to the tangent of half their difference.
12. Of any thrce equidifferent arcs, it will be as radius, to the
cosine of their common difference, so is the sine of the mean
arc, to half the sum of the sines of the extremes; and, as radius
to the sine of the common difference, so is the cosine of the
mean arc to half the difference of the sines of the two extremes.
(A.) If the sine of the mean of three equidifferent arcs.
+
1
}
1
1
N 2
138
PLANE TRIGONOMETRY.
dius being unity) be multiplied into twice the cosine of the com-
mon difference, and the sine of either extreme be deducted from
the product, the remainder will be the sine of the other ex-
treme.
K
(B.) The sine of any arc above 60°, is equal to the sine of ano-
ther arc as much below 60°, together with the sine of its excess
above 60°.
J
Remark. From this latter proposition, the sines below 60°
being known, those of arcs above 60° are determinable by addi-
tion only.
13. In any right-angled triangle, the hypothenuse is to one
of the legs, as the radius to the sine of the angle opposite to that
leg; and one of the legs is to the other as the radius to the tan-
gent of the angle opposite to the latter.
14. In any plane triangle, as one of the sides is to another, so
is the sine of the angle opposite to the former to the sine of the
angle opposite to the latter.
15. In any plane triangle it will be, as the sum of the sides
about the vertical angle is to their difference, so is the tangent
of half the sum of the angles at the base, to the tangent of half
their difference.
16. In any plane triangle it will be, as the cosine of the dif-
ference of the angles at the base, is to the cosine of half their
sum, so is the sum of the sides about the vertical angles to the
third side. Also, as the sine of half the difference of the angles
at the base, is to the sine of half their sum, so is the difference
of the sides about the vertical angle to the third side, or base.*
17. In any plane triangle it will be, as the base, to the sum
of the two other sides, so is the difference of those sides to the
difference of the segments of the base made by a perpendicular
let fall from the vertical angle.
18. In any plane triangle it will be, as twice the rectangle
under any two sides, is to the difference of the sum of the
squares of those two sides and the square of the base, so is
the radius to the cosine of the angle contained by the two
sides.
gles.
Cor. When unity is assumed as radius, then if A c, A B,
в c, are the sides of a triangle, this prop. gives cos. c =
A c²+B C². A B2
and similar expressions for the other an-
S
2 CB.CA
* These propositions were first given by Thacker in his Mathematical Miscel-
luny, published in 1743; their practical utility has been recently shown by Pro-
fessor Wallace, in the Edinburgh Philosophical Transactions.
PLANE TRIGONOMETRY.
139
19. As the sum of the tangents of any two unequal angles is
to their difference, so is the sine of the sum of those angles to
the sine of their difference.
"
F
20. As the sine of the difference of any two unequal angles
is to the difference of their sines, so is the sum of those sines to
the sine of the sum of the angles.
These and other propositions are the foundation of various for-
mulæ, for which the reader who wishes to pursue the inquiry
may consult the best treatises on Trigonometry.
III.-Solution of the Cases of Plane Triangles.
Although the three sides and three angles of a plane triangle,
when combined three and three, constitute twenty varieties, yet
they furnish only three distinct cases in which separate rules
are required.
CASE I.
When a side and an angle are two of the given parts.
The solution may be effected by prop. 14 of the preceding
section, wherein it is affirmed that the sides of plane triangles
are respectively proportional to the sines of their opposite an-
gles.
V
In practice, if a side be required, begin the proportion with
a sine, and say,
As the sine of the given angle,
To its opposite side;
So is the sine of either of the other angles,
To its opposite side.
If an angle be required, begin the proportion with a side,
and say,
As one of the given sides,
Is to the sine of its opposite angle;
So is the other given side,
To the sine of its opposite angle.
The third angle becomes known by taking the sum of the two
former from 180º.
Note 1.—Since sines are lines, there can be no impropriety in
comparing them with the sides of triangles; and the rule is bet-
ter remembered by young mathematicians than when the sines
and sides are compared each to each.
140
PLANE TRIGONOMETRY.
-
Note 2.-It is usually, though not always, best to work the
proportions in trigonometry by means of the logarithms, taking
the logarithm of the first term from the sum of the logarithms
of the second and third, to obtain the logarithm of the fourth
Or, adding the arithmetical complement of the loga-
rithm of the first term to the logarithms of the other two, to
obtain that of the fourth.
When two sides and the included angle are given,
The solution may be effected by means of props. 15 and 16
of the preceding section.
Thus: take the given angle from 180°, the remainder will be
the sum of the other two angles.
CASE II.
Then say,-As the sum of the given sides,
Is to their difference;
So is the tangent of half the sum of
the remaining angles,
To the tangent of half their difference.
Then, secondly, say,-As the cosine of half the said difference,
Is to the cosine of half the sum of the angles;
So is the sum of the given sides
To the third, or required side.
Or, As the sine of half the diff. of the angles,
Is to the sine of half their sum;
So is the difference of the given sides,
To the third side.
Example. In the triangle A B C are given
A C 450, B C = 540, and the included angle
c =80°; to find the third side, and the two re-
maining angles.
Hence, BC+AC
BC- AC
To
So is tan (A + B)
So tan (A — B)
T
Here в C+AC=990, B c — A c=90, 180°— c=100°
=A+B.
990
90
50°
Log.
Log. =
Log.
A
6° 11' Log.
B
2.9956352
1.9542425
10.0761865
9.0347938
PLANE TRIGONOMETRY.
141
64
}
B)
Cos (A
Cos
(A + B)
So is B C+AC
6° 11' Log.=9.9974660
50° Log.=9.8080675
990 Log.=2·9956352
640.08 Log.=2·8062367
Also
(A+B) +
(A — B)
B) = 56° 11′ =^; and ½ (A + B)
B.
½ (A — B) = 43° 49′
2
Here, much time will be saved in the work by taking
cos (A+B) from the tables, at the same time with tan (A+B);
and cos ½ (A — B) as soon as tan ½ (A —— B) is found. Observe,
also, that the log. of в C+A c is the same in the second opera-
tion as in the first. Thus the tables need only be opened in
five places for both operations.
To A B
Another Solution to Case II.
Supposing c to be the given angle, and c A, c в, the given
sides; then the third side may be found by this theorem,
viz.
2
2
A в= √(A c²+в c³ 2 A C. C B. Cos c).
Thus, taking a c=450, в c=540, c=80°, its cos ·1736482
A B= √(450°+540°-2.450. 540 x 1736482)
•
= √[90° (5°+62-2.5.6 x 1736482)]
= 9050-58118-90 × 7·112=640.08, as before.
p
CASE III.
10. When the three sides of a plane triangle are given, to
find the angles.
1st Method. Assume the longest of the three sides as base,
then say, conformably with prop. 16,
As the base,
To the sum of the two other sides;
So is the difference of those sides,
Gl
*
To the difference of the segments of the base.
Half the base added to the said difference gives the greater
segment, and made less by it gives the less; and thus, by
means of the perpendicular from the vertical angle, divides the
original triangle into two, cach of which falls under the first
case.
2d Method. Find any one of the angles by means of prop.
18 of the preceding section; and the remaining angles either
by a repetition of the same rule, or by the relation of sides to
the sines of their opposite angles.
20
I
*
142
PLANE TRIGONOMETRY.
Thus, cos c=
and cos a
J
N
A C²+B C²
2 AC
2
BA²+A C²
2
•
A B³
B C
2 AB. AC
G
;
B C²
COS B:
Right-angled Plane Triangles.
1. Right-angled triangles may, as well as others, be solved by
means of the rule to the respective case under which any spe-
cified example falls and it will then be found, since a right an-
gle is always one of the data, that the rule usually becomes
simplified in its application.
Ga
A B²+B C
2. When two of the sides are given, the third may be
found by means of the property in Plane Geom. Triangles,
prop. 16.
2 A B.B C
Hypoth. (base + perp.)
Base
(hyp. perp.)
Perp.
(hyp.+perp.). (hyp. - perp.)
✓(hyp."— basc")=(hyp.+base).. (hyp.- base.)
3. There is another method for right angled triangles, known
by the phrase making any side radius; which is this.
A C
Is to the given side;
So is the name of the required side,
To the required side.”
Is to radius;
So is the other given side,
To the name of that side,
拳
​"To find a side.-Call any one of the sides radius, and write
upon it the word radius; observe whether the other sides be-
come sines, tangents, or secants, and write those words upon
them accordingly. Call the word written upon each side the
name of each side
then say,
As the name of the given side,
which determines the opposite angle."
"To find an angle.-Call either of the given sides radius,
and write upon it the word radius; observe whether the other
sides becomes sines, tangents, or sccants, and write those words
on them accordingly. Call the word written upon each side the
name of that side. Then say,
I
As the side made radius,
1
4. When the numbers which measure the sides of the tri-
angle are either under 12, or resolvable into factors which are
each less than 12, the solution may be obtained, conformably
}
1
1
1.
HEIGHTS AND DISTANCES.
with this rule, easier without logarithms than with them.
For,
2.
с
Let A B C be a right angled triangle, in which A в, the base,
is assumed to be radius; B C is the tangent
of A, and a c its secant, to that radius; or di-
viding each of these by the base, we shall
have the tangent and sccant of A, respec-
tively, to radius 1. Tracing.in like manner
the consequences of assuming в c, and a c,
each for radius, we shall readily obtain these expressions.
perp.
base
base
perp.
hyp.
3. sec angle at base.
base
tan angle at base.
tan angle at vertex.
4.
hyp.
perp.
5.
143
A
C
B
sec angle at vertex.
sin angle at base.
perp.
hyp.
base
6. =sin angle at vertex.
hyp.
}
SECTION II.-On the Heights and Distances of Objects.
The instruments employed to measure angles are quadrants,
sextants, theodolites, &c., the use of either of which may be
sooner learnt from an examination of the instruments them-
selves than from any description independently of them. For
military men and for civil engineers, a good pocket sextant,
and an accurate micrometer (such as Cavallo's) attached to a
telescope, are highly useful. For measuring small distances,
as bases, 50 feet and 100 feet chains, and a portable box of gra-
duated tape will be necessary.
We shall here present a selection of such examples as are
most likely to occur.
EXAMPLE I.
In order to find the distance between two trees A and в,
which could not be directly measured because of a pool
which occupied much of the intermediate space, I measured
the distance of each of them from a third object c, viz. Ac=588,
B c=672, and then at the point c took the angle A c в between
the two trees-55° 40'. Required their distance.
This is an example to case 2 of plane triangles, in which two
sides, and the included angle, are given. The work, therefore,
may exercise the student: the answer is 593-8.
#

144
HEIGHTS AND DISTANCES.
*
1
EXAMPLE II.
Wanting to know the distance between two inaccessible ob-
jects, which lay in a direct line from the bottom of a tower on
whose top I stood, I took the angles of depression of the two
objects, viz. of the most remote 25½º, of the nearest 57°. What
is the distance between them, the height of the tower being 120
feet?
The figure being constructed, as in the
margin, A B=120 feet, the altitude of the
tower, and AI the horizontal line drawn
through its top; there are given,

H A D=25° 30', hence в A D=B A H
HA C 57° 0', hence B A C =B A H
difference
multiplied by height
P
A
13
Thus, nat. tan 642°-2.0965436
nat. tan 33° 0.6494076
EXAMPLE III.
B
Hence the following calculation, by means of the natural tan-
gents. For, if A в be regarded as radius, B D and B C will be the
tangents of the respective angles B A D, B A C, and o p the dif-
ference of those tangents. It is, therefore, equal to the product
of the difference of the natural tangents of those angles into the
height A B.
I
H A D 64° 30'.
H A C= 33° 0'.
1.4471360
120
gives distance o D 173.6563200
•
deur selecýne di AMO SE ALLE DANS SA 1 st to ma ma se on key Sk
HI
C
D)
Standing at a measurable distance on a horizontal plane,
from the bottom of a tower, I took the angle of elevation of the
top; it is required from thence to determine the height of the
tower.
In this case there would be given A B and the angle a (see the
figure in Right-angled Triangles), to find в O=A в×tan A.
By logarithms, when the numbers are large, it will be, log.
Bo=log. A B+log. tan A.
;
145
HEIGHTS AND DISTANCES.
1
Note. If angle A=11° 19' then в C=
A=16 42
B C=
B C=
B C
BC=
B C
B C
B C:
34&1
A=21 48-
A=26 34
A=30 58
A=35 0
A=38 40
A=45°
1
·5
3
10
2k-lampa←
Hence, A в=
5
93
•8433752
5
7
10
1
5
AB very nearly.
A B
A B
A B
A B
A B
A B
A B, exactly.
To save the time of computation, therefore, the observer may
set the instrument to one of these angles, and advance or re-
cede, till it accords with the angle of elevation of the object;
its height above the horizontal level of the observer's eye will
at once be known, by taking the appropriate fraction of the
distance A B.
EXAMPLE IV.
Wanting to know the height of a church steeple, to the bot-
tom of which I could not measure on account of a high wall be-
tween me and the church, I fixed upon two stations at the dis-
tance of 93 feet from each other, on a horizontal line from the
bottom of the steeple, and at each of them took the angle of
elevation of the top of the steeple, that is, at the nearest station
55° 54', at the other 33° 20'. Required the height of the
steeple.
Recurring to the figure of Example II., we have given the
distance c D, and the angles of clevation at c and D. The
quickest operation is by means of the natural tangents, and the
C D
theorem A B
cot D cot c
Thus cot D=cot 33° 20'—1·5204261
cot c=cot 55 54
*6770509
Their difference 8433752
=110.27 feet.
EXAMPLE V.
Wishing to know the height of an obelisk standing at the top
of a regularly sloping hill, I first measured from its bottom a
distance of 36 feet, and there found the angle formed by
O
146
HEIGHTS AND DISTANCES.
the inclined plane and a line from the centre of the instrument
to the top of the obelisk 41°; but after measuring on downward
in the same sloping direction 54 feet farther, I found the angle
formed in like manner to be only 23° 45'. What was the height
of the obelisk, and what the angle made by the sloping ground
with the horizon ?
+
The figure being constructed as in the margin, there are
given in the triangle A c в, all the angles and the side A B, to
find B C.
It will be obtained by this pro-
portion, as sin c (=17° 15'-B — A) : A B
(=54) :: sin a (=23° 45') : B c=73·3392.
Then, in the triangle D B C are known в c
as above, B D — 36, C B D 41°; to find
the other angles, and the side c D. Thus,
first, as c B+ B D : C B
BD: tan
(D+c) (139°): tan (D-c)=-42° 24'.
Hence 69° 30′ + 42° 24′ 112° 54'
42° 24' 26° 5½': BC D. Then, sin B C D
51.86, height of the obelisk.
:
1
The angle of inclination D A EH DA C D B
22° 54'.

-
P
P
EXAMPLE VI.
II
Cap de 10.
1
A
B
= с
C D B, and 69° 30'
B
Remark.-If the line в D cannot be measured, then the angle
DA E of the sloping ground must be taken, as well as the angles
CAB and C B D. In that case D A E +90° will be equal to C D B:
so that, after C B is found from the triangle A. C B, C D may be
found in the triangle c B D, by means of the relation between
sides and the sines of their opposite angles.
B.
D
B D :: sin C B D C D
с
90°
E
Being on a horizontal plane, and wanting to ascertain the
height of a tower standing on the top of an inaccessible hill, I
took the angle of elevation of the top of the hill 40°, and of
the top of the tower 51°, then measuring in a direct line 180
feet farther from the hill, I took in the same vertical plane the
angle of elevation of the top of the tower 33° 45'. Required
from hence the height of the tower.
A C B
The figure being constructed, as in the mar-
gin, there are given a в=180, c A B=33° 45',
CBE CAÈ = 17° 15', C B D
11°,BDC=180°-(90° - DB E)=130°. And
CD may be found from the expression c D
rad¹ =
= AB Sin A sin C B D cosec A C B Sec D B E.
A
D
C
při tak

D
HEIGHTS AND DISTANCES.
147
Or, using the logarithms, it will be log. A B+log sin a+log
sin B+log cosec AC B+log sec D B E 40 (in the index) = log
CD; in the case proposed=log. of 83.9983 feet.
}
1
}
EXAMPLE VII.
In order to determine the distance between two inaccessible
objects E and w on a horizontal plane, we measured a convenient
basc AB of 536 yards, and at the extremities A and B took the fol-
lowing angles, viz. в A w=40° 16', w A E=57° 40′, a b e=42°
22', E B W=71° 7′. Required the distance ɛ w.
Y
First, in the triangle A B E are given all the
angles, and the side A в to find в E. So, again,
in the triangle A B w, are all given the angles, and
A B to find в w. Lastly, in the triangle B E W
are given the two sides E B, B W, and the in-
cluded angle E B W to find E w=939.52 yards.
E
A
B
Remark.-In like manner the distances taken two and two,
between any number of remote objects posited round a conve-
nient station line, may be ascertained.
b
W

EXAMPLE VIII.
Suppose that in carrying on an extensive survey, the distance
between two spires A and B has been found equal to 6594 yards,
and that c and D are two eminences conveniently
situated for extending the triangles, but not ad- A
mitting of the determination of their distance
by actual admeasurement: to ascertain it, there-
fore, we took at c and D the following angles, viz.
SACB=85° 46'
A D C=31° 48′
B C D=23° 56' ZAD B=68° 2'
Required c D from, these data.
In order to solve this problem, construct a similar quadrilate-
ral a c d b, assuming c d equal to 1, 10, or any other convenient
number: compute Ab from the given angles, according to the
method of the preceding example. Then, since the quadrilate-
rals a c d b, A C D B, are similar, it will be, as Ab:cd::AB: CD;
and c n is found = 4694 yards.

ď
B
D
148
HEIGHTS AND DISTANCES.
EXAMPLE IX.
Given the angles of elevation of any distant object, taken at
three places in a horizontal right line, which does not pass
through the point directly below the object; and the respective
distances between the stations; to find the height of the object,
and its distance from either station.
Let A EC be the horizontal plane, F E the
perpendicular height of the object above
that plane, A, B, C, the three places of obser-
vation, F A E, F BE, F C E, the angles of ele-
vation, and ▲ B, B C, the given distances.
Then, since the triangles a ¤ F, B E F, C E F,
are all right angled at E, the distances A E, B E, CE, will mani-
festly be as the cotangents of the angles of elevation at A,
B, and c.
Put A B=D, B c=d, E r = x, and then express algebraically
the theorem given in Geom. Triangles, 25, which in this case
becomes,

XC
A
A ȳ . È ¢ + С È³ . A B — B E³ . A CAC. AB. BC.
CE
B
The resulting equation is
d x² cot³ A+D x³ cot² c=(D+d) x² cot² в+(D+d) D d.
From which is readily found.
(D+d) Dd
d cot A+D cot³ c
Ja
papag
Ea
E
St
(D+d) cot² B'
Thus Er becoming known, the distances A E, B E, C E, are
found, by multiplying the cotangents of A, B, and c, respective-
ly, by E F.
Remark.-When D=d, or D+d=2 D=2 d, that is, when the
point в is midway between A and c, the algebraic expression
becomes,
2
x=d÷√(2 cot A+ ½ cot c
2
cot B),
which is tolerably well suited for logarithmic computation.
The rule may, in that case, be thus expressed.
Double the log. cotangents of the angles of elevation of the
extreme stations, find the natural numbers answering thereto,
and take half their sum; from which subtract the natural num-
ber answering to twice the log. cotangent of the middle angle.
of elevation then half the log. of this remainder subtracted
from the log. of the measure distanced between the first and
second, or the second and third stations, will be the log. of the
height of the object.
The distance from either station will be found as above.
HEIGHTS AND DISTANCES.
149
Note. The case explained in this example, is one that is
highly useful, and of frequent occurrence. An analogous one
is when the angles of elevation of a remote object are taken
from the three angles of a triangle on a horizontal plane, the
sides of that triangle being known, or measurable: but the
above admits of a simpler computation, and may usually be
employed.
Add
EXAMPLE X.
From a convenient station P, where could be seen three ob-
jects, A, B, and c, whose distances from each other were known
(viz. A B = 800, a c = 600, в C
B = 400 yards), I took the hori-
zontal angles a p c = 33° 45', B P c = 22° 30'. It is hence re-
quired to determine the respective distances of my station from
each object.
Here it will be necessary, as preparatory to the computation,
to describe the manner of
C
Construction.-Draw the given triangle A B C from any con-
venient scale. From the point a draw a line
AD to make with A B an angle equal to 22°
30', and from в a line в D to make an angle
DBA 33° 45'. Let a circle be described
to pass through their intersection D, and
through the points A and B. Through c and
D draw a right line to meet the circle again
in P: so shall be the point required. For, drawing r A,
P B, the angle A PD is evidently A B D, Since it stands on
the same are A D and for a like reason в P D = B A D.
So that is the point where the angles have the assigned.
value.
B

A
+
P
B
The result of a careful construction of this kind, upon a good
sized scale, will give the values of P A, P C, P B, true to within
the 200th part of each.
Manner of Computation.—In the triangle A B c, where the
sides are known, find the angles. In the triangle a в D, where
all the angles are known, and the sides A B, find one of the
other sides a D. Take в A D from в A C, the remainder, D A C
is the angle included between two known sides, a D, A c ; from
which the angles A D C and A c n may be found, by chap. iii.
The angles c A P 180° - (APC+AC n). Also,
A B C +
case 2.
BCP BCA A CD and P BCA BCP BA
1 BOP BA
sup. A D C. Hence, the three required distances are found by
these proportions. As sin A PO: AC:: Sin PAC: PC, and :: sin
21
0 2
150
HEIGHTS AND DISTANCES.
}
7
PCA: PA; and lastly, as sin BPC: BC:: sin B PC: B P. The
results of the computation are, P A = 709.33, p c = 1042·66,
P B 934 yards.
**The computation of problems of this kind, however,
may be a little shortened by means of an analytical investi-
gation. Those who wish to pursue this department of trigo-
nometry may consult the treatises by Bonnycastle, Gregory,
and Woodhouse.
Note. If c had been nearer to P than A B, the general prin-
ciples of construction and computation would be the same; and
the modification in the process very obvious.
II. Determination of Heights and Distances by approxi-
mate Mechanical Methods.
1. For Heights.
1. By shadows, when the sun shines.-Set up vertically a
staff of known length, and measure the length of its shadow upon
a horizontal or other plane; measure also the length of the
shadow of the object whose height is required. Then it will
be, as the length of the shadow of the staff, is to the length of
the staff itself; so is the length of the shadow of the object, to
the object's height.
R
2. By two rods or staves set up vertically.-Let two staves,
one, say, of 6 fect, the other of 4 fect long, be placed upon
horizontal circular or square feet, on which each may stand
steadily. Let A в be the object, as a
tower or steeple, whose altitude is
required, and A c the horizontal
plane passing through its basc. Let
CD and E F, the two rods, be placed
with their bases in one and the same
line o A, passing through a the foot
of the object; and let them be moved
ncarer to, or farther from, each other, until the summit в of the
object is seen, in the same line as D and r, the tops of the rods.
B
D
F
HE CASE THE NEED THE TE Tae y te jagter truma te turul algaetong si » et à 10 WE 10.ma

• man se
CE
II
Q
A
HEIGHTS AND DISTANCES.
151
1
CD,
Then by the principle of similar triangles, it will be, as D H
(=CE): FH:: DG (:
G (= C A) :
CA): BG; to which add A G
for the whole height A B.
3. By Reflection.-Place a vessel of water upon the ground,
and recede from it, until you see the top of the object reflected
from the smooth surface of the liquid. Then, since by a prin-
ciple in optics, the angles of incidence and reflection are equal,
it will be as your distance measured horizontally from the point
at which the reflection is made, is to the height of your eye
above the reflecting surface; so is the horizontal distance of the
foot of the object from the vessel to its altitude above the said
surface.*
4. By means of a portable barometer and thermometer.
Observe the altitude B of the mercurial column, in inches, tenths,
and hundredths, at the bottom of the hill, or other object whose
altitude is required; observe, also, the altitude, b, of the mercu-
rial column at the top of the object; observe the tempera-
tures on Fahrenheit's thermometer, at the times of the two
barometrical observations, and take the mean between them.
B b
Then 55000 ×
B+ b
perature of 55° on Fahrenheit. Add of this result for
every degree which the mean temperature exceeds 55°; subtract
as much for every degree below 55°.
height of the hill, in feet, for the tem-
This will be a good approximation when the height of the
hill is below 2000; and it is easily remembered, because 55°,
the assumed temperature, agree with 55, the effective figures in
the coefficient; while the effective figures in the denominator
of the correcting fraction are two fours.
***Where great accuracy is required logarithmic rules be-
come necessary, of which various are exhibited in treatises on
Pneumatics. The following, by the Rev. W. Galbraith, of
Edinburgh, is a very excellent approximation.
For Fahrenheit's thermometer.
H
{48
48400+60 (b + ¿') }
λ
+ h (0.00268 +0.00268 cos 2
in which
B b
B + b
S
440
-
{2.42 + 1 + 1'
300
} (^__^")
T
+ 0.00000005 h)
is the true height in feet, t the temperature of the
* Leonard Digges, in his curious work the Pantometria, published in 1571,
first proposed a method for the determination of altitudes by means of a geometri-
cal squaro and plummet, which has been described by various later authors, as
Ozanam, Donn, Hutton, &c. But as it does not seem proferable to the methods
above given, I have not repeated it here.
152
HEIGHTS AND DISTANCES.
}
air by detached thermometer at the lower station, t that at
the upper;
the temperature of the mercury in the barome-
ter at the lower station by the attached thermometer, ' that
at the upper; B the height of the mercury in the barometer at
the lower station, b that at the upper, h the height, and the
latitude.
5. By an extension of the principle of pa. 145.-Set the
sextant, or other instrument, to the angle of 45°, and find the
point c (pa. 144.) on the horizontal plane, where the object A в has
that elevation: then set the instrument to 26° 34', and recede
from c, in direction в c n, till the object has that elevation.
The distance c D between the two stations will be = A B
So, again, if c
40°, D
24° 31', CD will be = A B.
35°, D =
22° 23', C D
20°, D = 20° 6', C D
or,
if c =
if c
A B.
A B.
A B.
or,
or,
if o
14° 56', C D
cot c =
o rad. c D
A B.
20°, D
or, generally, if cot d
6. For deviation from level.-Let E represent the elevation
of the tangent line to the earth above the true level, in feet and
parts of a foot, D the distance in miles: then E = 232 D².
D2
This gives 8 inches for a distance of one mile; and is a
near approximation when the distance does not exceed 2 or 3
miles.
K
2. For Distances.
1. By means of a rhombus set off upon a horizontal plane.
Suppose o the object and o в the required distance. With
a line or measuring tape, whose length is equal to the side
of the intended rhombus, say 50 or 100 feet, lay down one
side BA in the direction в o towards the object,
and B c another side in any convenient direction
(for whether в be a right angle, or not, is of no
consequence); and put up ruds or arrows at a
and c.
Then fasten two ends of two such lines
at A and c, and extend them until the two other
ends just meet together at D; let them lie thus
stretched upon the ground,, and they will form
the two other sides of the rhombus A D, C D. Fix a mark or
arrow at R, directly between c and o, upon the line A D;
and measure RD, RA upon the tape. Then it will be as
R D : D C : : C B : B 0, the required distance.
B
7
/
A
R
D
HEIGHTS AND DISTANCES.
153
1
Otherwise. To find the length of the inaccessible line Q R.
At some convenient point в, lay down the rhombus
BAD C, so that two of its sides
B A, B C, are directed to the ex-
tremities of the line Q R. Mark
the intersections o and P, of
A R, CQ, with the sides of the
rhombus, (as in the former me-
thod) then the triangle o D P
will be similar to the triangle
RBQ; and the inaccessible dis-
tance R Q will be found
OP X BA2
OD X D P
Thus, if B A = B C, &c. = 100 f. o d =
11 f. 10 in. o P 13 f. 7 in. then Q R =
1219 feet.

* For P D:DA::AB: BR =
A
and o D:0 P:: BRR a =
either cotangle subtended
or h× cot angle
A B2
2
·
2. By means of a micrometer attached to a telescope.
Portable instruments for the purpose of measuring extremely
small angles, have been invented by Martin, Cavallo, Dollond,
Brewster, and others. In employing them for the determina-
tion of distances, all that is necessary in the practice is to
measure the angle subtended by an object of known dimen-
sions, placed either vertically or horizontally, at the remoter
extremity of the line whose length we wish to ascertain.
Thus, if there be a house, or other erection, built with bricks,
of the usual size; then four courses in height are equal to a
foot, and four in length equal to a yard: and distances measured
by means of these will be tolerably accurate, if care be taken
with regard to the angle subtended by the horizontal object, to
stand directly in front of it. A man, a carriage wheel, a win-
dow, a door, &c. at the remoter extremity of the distance we wish
to ascertain, may serve for an approximation. But in all cases
where it is possible, let a foot, a yard, or a six-feet measure, be
placed vertically, at one end of the line to be measured, while
the observer with his micrometer stands at the other. Then
if h be the height of the object,
D
9 f. 5 in., D P —
10000 × 137
12
P D
A B 2.0 P
B
9-5 X × 11101020
12
OD.DP
P
•
с
R
154
HEIGHTS AND DISTANCES.
will give the distance, according as the eye of the observer is
horizontally opposite to the middle, or to one extremity of the
object whose angle is taken.
When a table of natural tangents is not at hand, a very ncar
approximation for all angles less than half a degree, and a tole-
rably near one up to angles of a degree, will be furnished by the
following rules.
1. If the distant object whose angle is taken be 1 foot in length,
then
3437.73÷the angle in minutes
or 206264the angle in seconds will give the distance
in feet.
2. If the remote object be 3, 6, 9, &c. feet in length, multiply
the former result by 3, 6, 9, &c. respectively.
Ex. 1. What is the distance of a man 6 feet high, when he
subtends an angle of 30 seconds?
206264 × 6 ÷ 30 2062645
41252.8 feet 13750·9
*
yards, the distance required.
Ex. 2. In order to ascertain the length of a street, I put up
a foot measure at one end of it, and standing at the other found
that measure to subtend an angle of 2 minutes required the
length of the street.
3437.73
572.95 yards.
2 = 1718.86 feet
2
3. By means of the velocity of sound.
Let a gun be fired at the remoter extremity of the required
distance, and observe by means of a chronometer that measures
tenths of seconds, the interval that elapses between the flash and
the report then estimate the distance for one second by
the following rule, and multiply that distance by the observed
interval of time; the product will give the whole distance
required.
At the temperature of freezing, 33°, the velocity of sound is
1100 feet per second.
ply
Jam
I
For lower temperatures deduct
For higher temperatures add
From the 1100
for every degree of difference from 33° on
to the 1100
Fahr. therm.; the result will show the velocity of sound, very
ncarly, at all such temperatures.
Thus, at the temperature of 50°, the velocity of sound is,
1
1100 × 2 (50 33) = 1108 feet.
At temperature 60°, it is 1100 + ½ (60 — 33) —1113 feet.
-
}
}
half a foot,
CONIC SECTIONS.
155
CHAPTER V.
CONIC SECTIONS.
1. Conic Sections are the figures made by a plane cutting a
cone.
2. According to the different positions of the cutting plane
there arise five different figures or sections, viz. a triangle, a
circle, an ellipsis, an hyperbola, and a parabola : of which the
three last are peculiarly called Conic Sections.
3. If the cutting plane pass through the vertex of the cone,
and any part of the base, the section will be a triangle.
4. If the plane cut the cone parallel to the base, or make no
angle with it, the section will be a circle.
5. The section is an ellipse when the cone is cut obliquely
through both sides, or when the plane is inclined to the base in
a less angle than the side of the cone is.
6. The section is a parabola, when the cone is cut by a plane
parallel to the side, or when the cutting plane and the side of the
cone make cqual angles with the base.
7. The section is an hyperbola, when the cutting plane
makes a greater angle with the base than the side of the cone
makes.
8. And if all the sides of the cone be continued through
the vertex, forming an opposite equal cone, and the plane be
also continued to cut the opposite cone, this latter section will
be the opposite hyperbola to the former.
9. The vertices of any section are the points where the
cutting plane meets the opposite sides of the cone, or the
sides of the vertical triangular section; as A and B, in the figs.
below.
Hence the ellipse and the opposite hyperbolas have cach two
vertices; but the parabola only one; unless we consider the
other as at an infinite distance.
10. The major axis, or transverse diameter, of a conic section,
is the line or distance A B between the vertices.
Hence the axis of a parabola is infinite in length, a b being
only a part of it.
156
Hyperbolas.
F H
M
D
*Яi
B
L
R
K/A C
E
I G
A
Ellipse.
N
G
MH
I
F
L
CONIC SECTIONS.
E
N
Parabolas.
AK
N
I
b
M
E
11. The centre c is the middle of the axis.
Hence the centre of a parabola is infinitely distant from the
vertex. And of an ellipse, the axis and centre lie within the
curve: but of an hyperbola, without.
12. A diameter is any right line, as A B or D E, drawn
through the centre, and terminated on each side by the curve ;
and the extremities of the diameter, or its intersections with the
curve, are its vertices.
Hence all the diameters of a parabola are parallel to the
axis, and infinite in length. Hence also every diameter of
the ellipse and hyperbola has two vertices; but of the para-
bola only one; unless we consider the other as at an infinite
distance.
13. The conjugate to any diameter is the line drawn through
the centre, and parallel to the tangent of the curve at the vertex
of the diameter. So, F G, parallel to the tangent at D, is the
conjugate to D E; and н 1, parallel to the tangent at A, is the
conjugate to A b.
Hence the conjugate H1, of the axis A в, is perpendicular to
it, and is often called the minor axis.
14. An ordinate to any diameter is a line parallel to its
conjugate, or to the tangent at its vertex, and terminated by
the diameter and curve. So DK and E L are ordinates to
the axis A B; and м N and N o ordinates to the diameter
D C.-Hence the ordinates of the axes are perpendicular to
it
; but of other diameters, the ordinates are oblique to
them.
15. An absciss is a part of any diameter, contained between
its vertex and an ordinate to it; as a K or в к, and D N or e n.
Hence, in the ellipse and hyperbola, every ordinate has two ab-
scisses; but in the parabola only one; the other vertex of the
diameter being infinitely distant.
16. The parameter of any diameter is a third proportional
to that diameter and its conjugate.
17. The focus is the point in the axis where the ordinate is
equal to half the parameter as K and L, where DK or EL is
1
}
<



CONIC SECTIONS.
157
}
equal to the semiparameter.-Hence the ellipse and hyperbola
have each two foci, but the parabola only one.
The foci, or
burning points, were so called, because all rays are united or re-
flected into one of them, which proceed from the other focus,
and are reflected from the curve.
18. The directrix is a line drawn perpendicular to the axis
of a conic section, through an
assignable point in the prolon-
gation of that axis; such that
lines drawn from that directrix
parallel to the axis to meet the
curve, shall be to lines drawn
from the points of intersection
to the focus, in a constant ratio
B D2
K N3
s s'
E'
E
D C² + ED. BC.
ED. B C
E B
DC
w
T
1. If in the annexed diagram the
ellipse B KD N, be cut from the frustrum
of the right cone, the diameter of whose
ends are E D, B C. Then, if в D be the
transverse, or major axis, K N the con-
jugate, or minor axis, and s, s', the foci,
we shall have
•
A
for the same curve. Thus, if EM: MF:: E' M': M' F, then
TE'E is the directrix. The curve will be a parabola, an ellipse,
or an hyperbola, according as F M is equal to, less than, or
greater than, M E.
SECTION I.-Properties of the Ellipse.
19. An asymptote is a right line towards which a certain.
curve line approaches continually nearer and nearer, yet so as
never to meet, except both be produced indefinitely. The
hyperbola has two asymptotes.
2. If a в, a b, be the two axes, c the-
centre, F, ƒ, the foci, P any point in the
curve, P D, an ordinate: also, if a B =
2 t, a b 2 c c d = x', a d
x', A D = x, d p
V, F P
≈, angle P F D = 4, ✔ (1º c³)
ping,
M/
F
F
(1)
(2)
(3)
A
M

P C
E
K
H
p a
F D
C
S'
D
W
B

G

B
22
P
158
CONIC SECTIONS: ELLIPSE.
%
y³
. (I)
(II)
(III)
1
The first of these is the equation of the curve when the
abscissæ are reckoned from the extremity a of the transverse
axis the second is the equation when the abscissæ are
reckoned from the centre c: and the third is called the polar
equation, and is principally used in the investigations of
astronomy.
12, A D = 4. Required the
y³
Z
to
Co
C²
to
ޑރ) ބރ
Ex. Suppose A B =
numeral value of P D.
Here y²
C3
to
36
100
(2 t x
K
1-d
t d cos p
•
(2 t x
64=
Consequently y =
-
x²)
(1º x¹³)
x's)
20, a b
x²)
62.8⁹
102
6 x 8
10
4.8 P D.
Or, taking Equa. II. where c D = x' = 10 − 4 — 6. and t
and c as before we have
•
36
ya
100
3. In the same figure, we have
A c³: α c²:: A D DB: D P²..
also FP+Pf: A B (5); and Fƒ³
2
36
100
•
(100 36)
4. Let T K be a tangent to
the ellipse at any point к, and
let T be the point where that
tangent meets the prolonga-
tion of the axis: let also F H,
fh be perpendiculas from the
foci, F, f, upon the tangent, and
let GH FH: then
((20 × 4) — 16)
Start
T
6º.83
103
= A B³
A
G
H
"
as before.
(4)
à b³ . . . . (6)
F D
h
h
ƒ
K
•
B
}
LFPT
(7) CD: CA :: CA: C T.... (8)
LƒPK.... CT
H and h fall in circumf. of circle whose diam. is A B
If m be in the middle of p p, then am produced will meet
the two tangents T K, B K, in their point of intersection K (10.)
(9)
G
1

CONIC SECTIONS: ELLIPSE.
159
2
If D the foot of the ordinate pass through F the focus, then
the point, r, of intersection of the tangent and the prolonga-
tion of the axis will be the point r of the directrix (Def.
18) .... (11.)
FP
(12) F H²=c b² · Fr
fr
FH. ƒ h=c b²
(13)
5. If an ordinate be drawn to any diameter of an ellipse, then
will the rectangle of the abscissæ be to the square of the ordinate
in a given ratio.
ratio..... (14)
•
6. All the parallelograms that may be circumscribed about an
ellipse are equal to one another and every such parallelogram
is equal to the rectangle of the two axes
..(15)
7. The sum of the squares of every pair of conjugate diame-
ters is equal to the same constant quantity; viz. the sum of the
squares of the two axes.. (16)
8. Def. The radius of curvature of a conic section or
other curve is the radius of that circle which is precisely of the
same curvature as the curve itself, at any assigned point, or the
radius of the circle which fits the curve, and coincides with it,
at a small distance on each side of the point of contact. The cir-
cle itself is called the osculatory circle, or the equicurve circle;
and if the curve be of incessantly varying curvature, each point
has a distinct equicurve circle, the radius of which is perpen-
dicular to the tangent at the point of contact.
P
-
9. Let p c be the radius of curvature at any point p in an el-
lipse or hyperbola whose major axis is A B, minor axis a b, and
(P F P r f ) 23/3
•
foci, r, f, then is P c =
(17)
A B. a b
The radius of curvature is greatest at the extremities of the
minor axis, when it is
(18)
A B²
u b
The radius of curvature is least at the extremities of the
a ba
major axis, when it is =
(19)
A B
133
10. PROB. To construct an ellipse whose two axes are given.
Find the distance r f, from the value of
rf, given in cqua. 6, or from F =
√A B a b³. Then, let a fine thread,
FPF, in length=/+A B, be put round
two pins fixed at the points r,ƒ: then, if
a pencil be put within the cord, and the
whole become tightened so as to make three right lines F P,

A
F
P
C
k
P
B
160
CONIC SECTIONS: ELLIPSE.
Pf, f F, the point P may be carried on, the cord slipping round
the fixed pins F, f, so as to describe and complete the ellipse
A Pрвb A.
Otherwise.-Let there be provided
three rulers, of which the two F 1, ƒ H,
are of the same length as the transverse
axis A B, and the third H 1, equal in
length to rf, the focal distance. Then
connecting these rulers so as to move
freely about F f, and about H 1, their
intersection P will always be in the
curve of the ellipse: so that, if there
be slits running along the two rulers,
and the apparatus turned freely about the foci, a pencil put
through the slits at their point of intersection will describe the
curve.
B
S
HI
A
A
*
** There are various other methods, as by the elliptic com-
passes, the trammels, &c. But the first of the above methods is
as accurate and easy as can well be desired.
с
11. PROB. To find the two axes of any proposed ellipse.
Draw any two parallel lines
across the ellipse, as м L, F K : bi-
sect them in the points I and D,
through which draw the right line
N I DO, and bisect it in c. From o
as a centre, with any adequate radius,
describe an arch of a circle to cut the
ellipse in the points G, 1. Join G, H,
and parallel to the line G H draw
through o the minor axis a b; perpendicular to which through
c draw A B, it will be the major axis.
12. PROB. From any given point out of an ellipse to draw
a tangent to it.
curve.
Μ'
Let r be the given point, through it and the centre c draw the
diameter A B; and parallel to it any line HI terminated by the
Bisect H 1 in o :
and c o produced will be
the conjugate to a B.—
Draw any line T S≈T B,
and make TRT c.
Draw R A, and parallel
to it s p cutting a B in P.
Through P, draw Pм pa-
rallel to c D, and join т м,
it will be the tangent required.
M
N
R
H
1)
Stack
F
P
1

A F G
C
B

b L
A
K
B

ન
CONIC SECTIONS: HYPERBOLA.
161
1
1. If, in the annexed diagram, the
conjugate hyperbolas whose vertices
are D, в, are cut from the two opposite
right cones whose common summit is
v, and в C, D E, be the diameters of
the two circular bases of the two
cones; then D B, KN, being the
axes, and s, s', the foci, we shall have
SECTION II.-Properties of the Hyperbola.

====
A
a
D B²
K N2
s s'
D
C
curves.
2. Hence, putting a C = C B t, a c = c b
y, angle P F D = 4
ADX, CD
= Y,
x', D P
c2
=
B
D C²
DE. B C
E B = D C
If these three properties be compared with the corres-
ponding ones for the ellipse, they will be found to agree,
of the con-
with the simple difference of the signs
necting quantities in the first property. This at once indi-
catos a general analogy between the properties of the two
and
P
32
C, CF =
d,
9, ≈ = F P, we have
(I)
t²) . . . . . (II)
02
(III)
t + d cos
for the three most useful forms of the equation to the hyper-
bola, agreeing with those to the hyperbola, except in the
signs.
P
3. And hence it follows, taking this and the preceding
marginal figures to correspond with
those in arts. 3 and 4 Ellipse; that the
properties indicated by the parenthe-
tical figures (4), (5), (6), (7), (8), (9),
(11), (12), (13), (14), (15), and (16),
hold in the hyperbola; simply chang-
ing + to in (5), to + in (6),
circumscribed to inscribed between the four hyperbolas in (15),
and sum to difference in (16). Those properties, therefore,
need not be here repcated.
D X Y ²
F
Z
DE.BC..
11
12
212
N
D
S
· (2¹2
12
(2 t x + x²)
B
(1)
(2)
(3)
A
S
71
V
C
K
M
E


T B F D
p/
P 2
162
CONIC SECTIONS: HYPERBOLA.
4. Besides the above, however, there are
several curious properties which relate to
the asymptotes of the hyperbola; some of
the most useful being these: viz.
м c, m c, being asymptotes, PD p a double
ordinate, в H, P F, parallel to cm ; &c.
=
ba
B
7. To describe Hyperbolas.
H
K
k
•
D
C
K
A D
X
*
M

Copy
p m
B
parallelogram c H B K = parallelogram c FPG
- parallelogram cfp g
M P. Pp =mp.pr
P. p P = C B³
= C B³ (18) ; . . M P =
triangle CPT triangle c в K (20) former diagram.
5. Also, if the abscissæ c D, C E, c F, &c. of any hyperbola,
be taken on one of the asymptotes in an increasing geometri-
cal progression, the ordinates D B, E G,
FH, &c. parallel to the other asymp-
tote are in decreasing geometrical
progression, having the same ratio..
(21).
L K
B D
C
P
772
(17)
(19)
II
C
D E
P
6. And, when the distances c E, C F,
&c. are in geometrical progression,
the asymptotic spaces D E G B, D F H B, &c. will be in arithmeti-
cal progression, and will, therefore, be analogous to the loga-
rithms of the former. The nature of the system of logarithms
will depend upon the value of the angle made by the two
asymptotes. In Napier's logarithms L cr is a right angle: in
the common logarithms L C r is 25° 44' 27'1.
་

A


Let one end of a long ruler fм o be fastened at the point f
by a pin on a plane, so as to turn freely about that point as a
centre. Then take a thread r м o, shorter than the ruler, and
CONIC SECTIONS: PARABOLA.
163
A
fix one end of it in r, and the other to the end o of the ruler.
Then if the ruler ƒ M o be turned about the fixed point ƒ, at
the same time keeping the thread o M F always tight, and its
part м o close to the side of the ruler, by means of the pin м;
the curve line A x described by the motion of the pin м is
one part of an hyperbola. And if the ruler be turned, and
move on the other side of the fixed point F, the other part
A z of the same hyperbola may be described after the same
manner. But if the end of the ruler be fixed in F, and that
of the thread in f, the opposite hyperbola x az may be de-
scribed. Otherwise also by continued motion. Let c and
r be the two foci, and E and к the two vertices of the hyper-
bola. (See the last fig. above.) Take three rulers, c D, D G, G F,
so that C D G F =
G F E K, and D G C F; the rulers CD and
G F being of an indefinite length beyond c and &, and having
slits in them for a pin to move in; and the rulers having
holes in them at c and F, to fasten them to the foci c and
F by means of pins, and at the points D and & they are to
be joined by the ruler D G. Then, if a pin be put in the
slits, viz. at the common intersection of the rulers c D and
Gr, and moved along, causing the two rulers & F, c D, to
turn about the foci c and F, that pin will describe the portion E e
of an hyperbola.
SECTION III.-Properties of the Parabola.
G
1. Let the right cone A B C in the mar-
ginal figure, have a parabolic section, L D G,
whose focus is F, vertex D, basc L &; from
D let fall the perpendicular DP upon the
side A B of the cone; let Er be bisected in
s: also, let a plane be cut through s paral-
lel to в c, and continued to meet the plane
of the parabola, in R X.
D F=
4AE
I G²
4 ED
2 E P = 4 D F = parameter
CSP
D F=
A

N
Then.... R x is the directrix of the parabola... (1)
E D³
(2)
· (3)
R
· (4)
·
X
C
J
164
CONIC SECTIONS: PARABOLA.
2. Let p = parameter of a parabola, x = A P any absciss,
y=P K, the corresponding ordinate, z=F F K, d = A F,
angle K F P, F being the focus: then
Φ
y³ = p x
Z
Kata
2 d
1 = cos
the equations to the parabola : in the
latter of which, or the polar equation,
the sign obtains when p is between
A and F, and when p is below F.
•
(I)
(II).
Rad. of curvature at K =
A PAQ P L²: Q Nº, or
::
maraton Jan
where A P and A Q are any
ponding ordinates.
A F
F K A PAF, F M A Q + A F
2 F H
DH and DH
As p: QN+P L:: Q N
or, as p: MOON: 0 L
PLAQ
4. Again, let PT be the tangent to
a parabola at any point P, and let
HPD be drawn through r parallel to
the axis A K; let P K be
let P K be perpen-
dicular to T P then is G T the subtan-
gent, P K the normal, & K the subnor-
mal; and the following properties ob-
tain: viz.
E' E' E GEE

M
dria
Magda
K
parameter
- AP
(4 x + p)³
2 √p
(III)
ż p
(IV)
at the vertex, A x vanishes, and we have
rad. of curv. at vertex 10
3. In the same figure, where E" E' GE is the directrix, the
following properties obtain:
viz.
A F =AG, F D = DE, E K
= K E', F M
A P P L2
D
P
}
H
A Q Q NⓇ
abscissæ, and P L, Q N, their corres-
M E', &c. (5)
· (6)
angle FPT = angle FTP = angle T P n.
angle K P H = angle K Pr
FPT T
(12)
T
B
GAAT
A
A
F
F
G
P
K
Q
H
·
L
•
. (10)
(11)
(13)
2 A G
(14)
subtangent & T
subnormal & K=2 A F≈ param. a constant quan. (15)
(7)
(8)
(9)
"
N

D
L
CONIC SECTIONS: PARABOLA.
165
5. In the marginal figure also, where
c Q is a tangent to the parabola at the
point c, and I ¤, 0 м, Q L, &c. parallel
to the axis A d.

the external parts I E, T A, 0 N, P L,
are proportional to c 12, c r, co, c p³,
or to the squares c K³, C D³, C M³, c L³,
C
C
Then I EEK:: CK: KL. (16)
and a similar property obtains, whether
OL be perpendicular or oblique to T D.
The external parts of the parallels I E, T A, O N, P L, &c. are
always proportional to the squares of their intercepted parts of
the tangent; that is,
B
G'
D/
T
0'
E
A
K D
M' F
(17)
And as this property is common to every position of the
tangent, if the lines I E, T A, O N, &c. be appended to the points
I, T, o, &c. of the tangent, and moveable about them, and of such
lengths that their extremities E, A, N, &c. be in the curve of a
parabola in any one position of the tangent; then making the
tangent revolve about the point c, the extremities E, A, N, &c.
will always form the curve of some parabola, in every position
of the tangent.
The conte
The same properties, too, that have been shown of the axis,
and its abscisses and ordinates, &c. are true of those of any other
diameter.
A
6. PROB. To construct a Parabola.
Construct an isosceles triangle A B D, whose base A B shall
be the same as that of the proposed parabola,
and its altitude cp twice the altitude cy of
the parabola. Divide each side A D, D B,
into 10, 12, 16, or 20, equal parts [16 is a good
number, because it can be obtained by conti-
nual bisections], and suppose them numbered
1, 2, 3, &c. from A to D, and 1, 2, 3, &c. from
Then draw right lines 1, 1; 2, 2; 3, 3;
4, 4; &c. and their mutual intersection will
beautifully approximate to the curve of the parabola A V B.
Otherwise: by continued motion.-Let the ruler, or direc-
D to B.
C
trix в c, be laid upon a plane
with the square a Do, in such
manner that one of its sides p G
lies along the edge of that ruler;
and if the thread r M o cqual
in length to do, the other side
of the square have one end fixed
in the extremity of the ruler at
1
A
M
0
~
a s
M
D
G
L

5
7
C
Z
B

23
166
CONIC SECTIONS: PARABOLA.
1
o, and the other end in some point r: then slide the side of
the square D G along the ruler в c, and at the same time keep
the thread continually tight by means of the pin м, with its
part м o close to the side of the square D o; so shall the curve
A м x, which the pin describes by this motion, be one part of a
parabola.
And if the square be turned over, and moved on the other
side of the fixed point F, the other part of the same parabola
A M Z will be described.
7. PROB. Any right line being given in a parabola, to find
the corresponding diameter : also, the axis, parameter, and
focus.
Draw н I parallel to the given line D E.
and &, through which draw a o g for
the diameter.
Draw HR perp. to A G and bisect
it in в; and draw v в parallel to a G,
for the axis.
B;
Make V BH BH B: parameter
to the axis.
Then
H
Bisect D E, H 1, in o
A V
D
DE
G
с
II
F
B
I
the parameter set from v to F gives the focus.
8. PROB. To draw a Tangent to a Parabola.
If the point of contact c be given, draw the ordinate c в, and
produce the axis until a î AB: then join T c, which will be
the tangent.
Or if the point be given in the axis produced: take A B
A T,
and draw the ordinate в c, which will give c the point of
contact; to which draw the line r c as before.
1

If D be any other point, neither in the curve nor in the axis
produced, through which the tan-
gent is to pass: draw D E G perpen-
dicular to the axis, and take o н a
mean proportional between D E and
DG, and draw Ho parallel to the
axis; so shall cbe the point of contact,
through which and the given point.
D the tangent D C T is to be drawn.
When the tangent is to make a given angle with the ordi-
nate at the point of contact: take the absciss ▲ 1 equal to
half the parameter, or to double the focal distance, and draw
the ordinate 1 E: also draw A H to make with a 1 the angle
AHI equal to the given angle; then draw H c parallel to the
axis, and it will cut the curve in c the point of contact, where a
line drawn to make the given angle with c в will be the tangent
required.
E
A
Ꮮ
B
R

G
CONIC SECTIONS: JOINTS OF VOUSSOIRS.
167
}
SECTION IV.-General Application to Architecture.
PROB. 1. To find, by construction, the position of the joints
of the voussoirs, to a parabolic arch.
In the practice of arcuation, the voussoirs or arch-stones are
so cut that their joints are perpendicular to the arch or to its
tangent, at the points where they respect-
ively fall. Hence, if A v в be the proposed
parabola, P, r', r", &c. the points at which
the positions of the joints are to be deter-
mined draw the ordinates P M, P' M',
p'' m', and on the prolongation of the axis
set off v T=V M, V т'—V M', v т"=V M",
T':
&c. Join T P, T' P', T" P', &c. and per-
pendicular to them respectively the lines
r o, p'o', r" o", &c. ; they will determine
the positions of the joints required.
V

Q/
Q/
p/
011
pl
ין
0/
olf
O/
MA
PROB. 2. To find the same for an elliptical arch.
Let A B be the span of the arch, and
A P P P B the arch itself, of which F
and ƒ are the foci. Draw lines F P,
fr, from the foci to each of the points
P: bisect the respective angles r r f,
FP' f, FP" f, by the lines r o, p' o',
P"o"; they will show the positions of the joints at the points.
P, P', P".
A F
ग
O
P
m
C
P
p
m'
mu
à 2 =>
PROB. 3. To find the same for a cycloidal arch.*
Let A V в be the cycloid, c p v q its generating circle, and
P, P', r'', points in the
V
T
arch where joints will
fall. Draw the ordi-
nates rm, r'm', r'm",
each parallel to the
base AB of the cycloid,
and cutting the circle.
in the points p, p', p". Join v p, v p', v p", and perpendicular
to cach the lines po, p' o', p" o"; parallel to each of which
respectively draw ro, P'o', p" o"; they will mark the positions
of the joints at the several points proposed.
A
T
૧
M
M
M"

Pl
Oll
***KA SI
B

B
* This problem is introduced here, as belonging to the subject of arcuation
although it depends upon a property of the cycloid described hereafter, viz. that the
tangent to any point r of a cycloid is parallel to the corresponding chord vp of the
generating circle.
%
168
CURVES.: CONCHOID.
1
1
↓
Q
L
E
A knowledge of which is required by Architects and Engineers.
10
Conchoid, or Conchiles, is the name given to a curve by
its inventor, Nicomedes, about 200 years before the Christian
era.
D D DI
F
AB
The conchoid is thus constructed: A P and в D being two
lines intersecting at right angles: from P draw a number of
other lines, PFDE, &c. on which take always D E = D F
or BC; so shall the curve line drawn through all the points
E, E, E, be the first conchoid, or that of Nicomedes; and the
curve drawn through all the other points, F, F, F, is called the
second conchoid; though, in reality, they are both but parts of
the same curve, having the same pole r, and four infinite legs,
to which the line D B D is a common asymptote.
#
A
CHAPTER VI.
G
SECTION I.-The Conchoid.
CURVES,
F
E
B /D/D
ľ
}

E
HD
E

The inventor, Nicomedes, contrived an instrument for de-
scribing his conchoid by a mechanical motion: thus, in the
ruler DD is a channel or groove cut, so that a smooth nail
firmly fixed in the movcable ruler o A, in the point o, may slide
freely within it into the ruler a P is fixed another nail at r, for
the moveable ruler A P to slide upon.
If therefore the ruler A P
be so moved as that the nail d passes along the canal D the
style, or point in a, will describe the first conchoid.
CURVES: CISSOID.
169
Conchoids of all possible varieties may also be constructed
with great facility by Mr. Jopling's apparatus for curves, now
well known.
Let A B B C D E=D F=ɑ, P в=b, B G=E н=x, and & E
=BH=y: then the equation to the first conchoid will be
x² (b+x)² + x² y²=a² (b+x)², or x¹+2 bx³ + b² x² + x² y³
a² b² + 2 a² b x + a²x²; and, changing
only the sign of x, as being negative in the
other curve, the equation to the 2d con-
choid will be x³ (b — x)² + x² y³ a³
x)³, or x¹
2 b x³ + b² x² + x² y³
2

(6
= a² b² 2 a² b x + a³ x³.
Of the whole conchoid, expressed by
these two equations, or rather one equation
only, with different signs, there are three
cases or species as first, when B C is less.
than B P, the conchoid will be as in the 2d
fig. above; when B C is equal to в P,
the
conchoid will be as in the 3d fig.; and
when B c is greater than в г, the conchoid
will be as in the 4th or last fig.
B
▬▬▬▬▬▬▬

Newton approves of the use of the conchoid for trisecting
angles, or finding two mean proportionals, or for constructing
other solid problems. But the principal modern use of this
curve, and of the apparatus by which it is constructed, is to
sketch the contour of the section that shall represent the dimi-
nution of columns in architecture.
*
The fixed point p is called the pole of the conchoid; D D D D
the directrix: it is an asymptote to both the superior and the
inferior conchoid. In the last figure the inferior conchoid is also
nodated.
}
SECTION II.-The Cissoid or Cyssoid.
The cissoid is a curve invented by an ancient Greek geome-
ter and engineer named Diocles, for the purpose of finding two
continued mean proportionals between two given lines. This
curve admits of an easy mechanical construction; and is de-
scribed very beautifully by means of Mr. Jopling's apparatus.
}
At the extremity в of the diameter A B, of a given circle
A o Bo, erect the indefinite perpendicular e в E, and from the
Q
170
CYCLOID.
CURVES:
1
ļ
Į
1
"
other extremity A draw any number of right lines, A C, A D, A E,
&c. cutting the circle in the points R, O, м, &c. ; then, if c L be
taken=A R, DO—A 0, E N=A M, &c., the curve passing through
the points, A, L, O, N, &c. will be the cissoid.
A

t
ก
m
Q
P
A
M
0
d' C
B
C
I
1. Here the circle A O B o is called the generating circle; and
A B is called the axis of the curves A Lo N, &c. Al on, &c.
which meet in a cusp at A, and, passing through the middle
points o, o, of the two semicircles, tend continually towards the
directrix, e в E, which is their common asymptote.
2. If A o and A o are quadrants, the curve passes through o
and o, or it bisects each semicircle.
E
R
3. Letting fall perpendiculars L P, R Q, from any correspond-
ing points LR then is A P=B Q, and a L=C R.
2
4. APPB:: P L²; A P². So that, if the diameter A B of the
circle a, the absciss A Px, the ordinate P L =
y; then is
x: a x:: y³: x², or x³—(a—x) y³, which is the equation to
the curve.
5. The right line e B E is an asymptote to the curve.
6. Arch A м of the circle=arch в R, and arch a m=в r.
N
7. The whole infinitely long cissoidal space, contained be-
tween the asymptote e в E and the curves N O L A, &C. A LON,
&c. is equal to three times the area of the generating circle
A O BO.
SECTION III.-The Cycloid.
The cycloid, or trochoid, is an elegant mechanical curve first
noticed by Descartes, and an account of it was published by
Mersenne in 1615. It is, in fact, the curve described by a nail
in the rim of a carriage-wheel while it makes one revolution on
a flat horizontal plane.
F
D
V
OOQ
E

F
1
}
{
CURVES: CYCLOID.
1. Thus, if a circle E P F, keeping always in the same plane,
be made to roll along the right line A B, until a fixed point P, in
its circumference, which at first touched the line at A, touches it
again after a complete revolution at B; the curve A P V P B de-
scribed by the motion of the point P is called a cycloid.
2. The circle & r F is called the generating circle; and the
right line a B, on which it revolves, is called the base of the
cycloid.
Also, the right line, or diameter, c v, of the circle, which bi-
sects the base A B at right angles, is called the axis of the cycloid;
and the point v where it meets the curve, is the vertex of the
cycloid.
3. If P be a point in the fixed
diameter AF produced, and the
circle A E F be made to roll
along the line A в as before, so
that the point A, which first
touches it at one extremity,
shall touch it again at в, the
curve P v P, described by the
point P, is called the curtate
cycloid.
W
4. And, if the point p
be any where in the un-
produced diameter A P,
and the circle A E F be
made to roll along A B
from A to в; the curve
Pvr is, in that case,
called the inflected or prolate cycloid.
The following are the chief properties of the common cy-
cloid.
1. The circular arc v E = the
line & between the circle and
cycloid, parallel to a B.
2. The semicircumf. v E C= -the
semibase c B.
3. The arc v G=2, the corres-
ponding chord v ɛ.
4. The semicycloidal are v G B=2 diam. v c.
E
L
A
Р
E
A
F
A
P
A
G'
V
C
E'
с
5. The tangent T & is parallel to the chord v E.
6. The radius of curvature at y= 2 c v.
T
V
D
с
P
B
171
E
B
P
B
7. The area of the cycloid A V B C A is triple the circle o EV;
and consequently that circle and the spaces v EC BG, VE' CAG',
are equal to one another,
C
{
>



172
CURVES: QUADRATRIX.
I
8. A body falls through any arc L K of a cycloid reversed,
:
in the same time, whether that
arc be great or small; that is,
from any point L, to the lowest
point x, which is the vertex re-
versed and that time is to the
time of falling perpendicularly
through the axis м K, as the se-
micircumference of a circle is to
its diameter, or as 3.141593 to
2. And hence it follows that if
1
/
}
N
10

S
M
a pendulum be made to vibrate in the arc L K N of a cycloid, all
the vibrations will be performed in the same time.
9. The evolute of a cycloid is another equal cycloid, so that if
two equal semicycloids o r, o Q, be joined at o, so that o м be
=M K the diameter of the generating circle, and the string of a
pendulum hung up at o, having its length
OP; then, by plying the string round the curve o P, to which it
is equal, if the ball be let go, it will describe, and vibrate in the
other cycloid P K Q; where o P=Q K, and o Q=P K.
10. The cycloid is the curve of swiftest descent or a heavy
body will fall from one given point to another, by the way of
the arc of a cycloid passing through those two points, in a less
time, than by any other route. Hence this curve is at once in-
teresting to men of science and to practical mechanics.
M
K
L
= 10 X O
the curve
SECTION IV.-The Quadratrix.
The quadratrix is a species of curve by means of which the
quadrature of the circle and other curves is determined mechani-
cally. For the quadrature of the circle, curves of this class
were invented by Dinostrates and Tschirnhausen, and for that
of the hyperbola by Mr. Perks. We shall simply describe in
in this place the quadratrix of Tschirnhausen ; and that in or-
der to show its use in the division of an arc or angle.
Th
↓
CURVES: QUADRATRIX.
173
To construct this quadratrix, divide the quadrantal arc A B
A
into any number of equal parts, A N, Nn,
n n', nв; and the radius A c into the same
number of equal parts a P, P p, p p', p' c.
Draw radii c N, cn, &c. to the points of di-
vision upon the arc; and let lines P м, p m,
&c. drawn perpendicularly to A c from the
several points of division upon it, meet the
radii in`м, m, m', &c. respectively. The
curve A м m m'n that passes through the points of intersection
M, m, &c. is the quadratrix of Tschirnhausen.
The figure AC Dm'm м A thus constructed may be cut out from
a thin plate of brass, horn, or pasteboard, and employed in the
division of a circular arc.

1
24
1
}
1
Q 2
D
с
E
n
K
B
n
M
D
I
Thus, suppose the arc I L or the angle IK L is to be divided
into five equal parts. Apply the side A B of the quadratrix upon
IK, the point в corresponding with the
angle K. Draw a line along the curve a s,
cutting K L in F. Remove the instrument,
and from r let fall the perpendicular F E
upon I K. Divide E I into five equal parts
by prob. 5, Practical Geometry, and
through the points of division draw c м,
D N, &c. parallel to E F. Then from their
intersections, M, N, O, P, draw the lines
K M, K N, K O, K P, and they will divide
the angle I K L into five equal parts, as required.
Ι
Note 1.-If, instead of dividing the arc into equal parts, it
were proposed to divide it into a certain number of parts having
given ratios to cach other; it would only be necessary to di-
vide EI into parts having the given ratio, and procced in other
respects as above.
Note 2.-If the arc or angle to be divided exceed 90 degrees,
bisect it, divide that bisected arc or angle into the proposed
number of parts, and take two of them for one of the required
divisions of the whole arc.
m
N
I
M
1
1
Z
◄
f
A
E
A
P

L
1
174
CURVES CATENARY.
1
✩
SECTION V.-The Catenary, and its application.
The catenary is a mechanical curve, being that which is as-
sumed by a chain or cord of uniform substance and texture,
when it is hung upon two points or pins of suspension (whether
those points be in a horizontal plane or not), and left to adjust
itself in equilibrio in a vertical plane.
This curve is of great interest to practical men on account of
its connexion with bridges of suspension, or chain bridges. Its
consideration cannot, therefore, with propriety be omitted,
although it involves mechanical propositions which will be
announced subsequently.
B
Ang
B,
Let A, B, be the points of suspension of such a cord, a a cb в
the cord itself when hanging at rest in a vertical position. Then
the two equal and symmetrical portions A a c, cb в, both ex-
posed to the force of gravity upon every
particle, balance each other precisely at c.
And, if one half, as c b в, were taken away,
the other half, A a c, would immediately
adjust itself in the vertical position under
the point A were it not prevented. Sup-
pose it to be prevented by a force acting
horizontally at c, and equal to the weight
of a portion of the cord or chain equal in
length to cm; then is c м the measure of the tension at the ver-
tex of the curve; it is also regarded as the parameter of the
catenary. Whether the portion A a c hang from A, or a shorter
portion, as a c, hang from a, the tension at c is evidently the
same for in the latter case the resistance of the pin at a, ac-
complishes the same as the tension of the line at a when the
whole a a c hangs from a.
d,
Let the line c м which measures the tension at the vertex be
=p, let c d=x, a d=d b=y, c a = c b=z, C D. = h, A B
c a λ = c b B= 7.
Then
22
X
A
a
ก
C
1
Y
b
B
M
*This may casily be determined experimentally, by letting the cord hang very
freely over a pulley at c, and lengthening or shortening the portion there suspend-
ed, until it keeps A a c in its due position; thon is the portion so hanging beyond
the pulley equal in length to c M.
i
;

CURVES: CATENARY.
175
7
1. y=px hyp. log.
log.
p + x + z
p
d
ι
X
p× hyp. log.
p (sec s
p + x + √2 px + x²
p
(45°+s).
2. If the angle s of suspension made between the tangent
to the curve at A or B, and the horizon be 45°; then d: 1 ::
1 : 1.1346.
3. When 7 = 2 d, then h ·7966 d, and s = 77° 3'.
4. When the angle s of suspension is 56° 28', then p, x, y,
and ≈, are as 1, 0.81, 1.1995, and 1-5089 respectively. In this
case 1, the tension, at the point of suspension, is a minimum
with respect to y.
7. The radius of curvature
is rad curv.=p.
8. When s and p are given; then
z=p tan s
t
1)
COS S
P M log. tan (45° + ½ s).
and y
9. When s and z, or
- cot s...
5. Generally
tan S M L tans.
Where M = 2.30255851, Napier's logarithm of 10.
log. tan s + log. (log. cot & s
2
ι
d
Or, log.
10) + 3622157
10. This last formula serving to compute an approximative
result.
6. The distance of the
2 z, from the vertex =
centre of gravity of the whole curve
ру
(x + P_Y — p.)
½
Z
z + p
my p
1
t
p ≈
x = z cosec s versin s
X
(p+x) 2
p
p sec s.
p versin s
t2
=px hyp.
p м log. tan
p
are given: then
COsec s
: this at the vertex
Y
= M≈ cot s log, tan (45° + s).
10. When s and y are given: then
p=y M log. tan (45° + & s).
t=yм cos s log. tan (45° + ½ s)
=y tan sм log. tan (45°+ s)
x=y versin s÷M cos s log, tan (45° + ♣ s).
176
1
d
CURVES CATENARY.
11. When x and y are given; then
log. tan (45°
s)
Y
sec s versin s
Ꮇ Ꮖ
from which s may be found by an approximative process; also
x sin s
X
X
p
t
versin s
versin s
sec s
in all these cases t is determined in length of chain or cord of
which the catenary is actually constructed.
1
;
}
12. To draw the catenary mechanically.-If the dis-
tance A B between the points of suspension, and the depth p c
of the lowermost point, be given (see the preceding figure),
hang one extremity of a fine uniform chain or cord at one of
the points A, and, letting the chain or cord adjust itself as
a festoon in a vertical plane, lengthen or shorten it as it is
held near the other end, over a pin at B, until, when at rest,
it just reaches the point c; so shall the cord form the cate-
nary; and a pencil passing along the cord, from ▲ by a, c, b, to
B, will mark the curve upon a vertical board brought into con-
tact with it.
14. With reference to the practical
uses of the catenary, we may now blend
the geometrical and the mechanical
consideration of its properties. Taking
any portion cb of the catenary, from
the lowest point o; its weight may be
regarded as supported by tensions act-
ing in the tangential directions c N, b N.
The strains at c and b may be conceived
as acting at the point of intersection N;
|
one curve
13. All catenaries that make equal angles with their ordi-
nates at their points of suspension are similar, and have x to y
a constant ratio and of any two which do not make equal
angles, but have x to y in different ratios,
a portion may be cut from
similar to the other. Thus, let A c B. and
A'C' B', be the two curves, of which a'C' B'
is the flattest. Suppose them placed upon
one axis D C c', and the tangent r's', to
the lower curve at B', the point of suspen-
sion to be drawn. Then, parallel to r's
draw another line T s to touch the other
curve in b. Through b draw ba parallel
to B'A'. So shall the portion acb of the
upper catenary be similar to the lower cate-
nary A'c' B'.
A
T
t
A
A'
D
d
O
C/
D
'
D'
()
G
ル
​b
S'
N
B

T
B'
T'
B

kkkkk
R
CURVES: CATENARY.
177
above which, therefore, in the vertical direction N &, the weight
of the portion c b may be conceived to act at its centre of gra-
vity G.
Hence, strain at c: weight of c b:: sin & N b: sin b N C
G
:: Cos N R
sin b N R
:: radius
: tan 6 N R
:: radius
: tan d b N
}
Hence, the horizontal tension at c being constant, the weight,
and consequently the length of any portion c b of the uniform
chain must be proportional to the tangent of the inclination of
the catenary to the horizon at the extremity b of the said portion.
This may be regarded as the characteristic property of the
calenary.
15. In like manner,
F
hor. strain at c : oblique strain at b :: sin b N G : sin c N G
:: cos b N R : radius
rad sec N R.
Therefore, the strain exerted tangentially, at any point b, is
proportional to the secant of the inclination at that point.
Also, from (14) and (15)
tang. strain at 6: weight of в b c :: sec db N: tan d b N.
These properties evidently accord with the preceding equa-
tions.
I
16. Let, then, c o, in the axis produced downwards, be equal
to the parameter or the measure of the horizontal strain at c ;
and upon o as a centre, with radius c o describe a circle. A tan-
gent d drawn to this circle from d, will be parallel to the tan-
gent Nh of the curve at the point b to which db is the ordi-
nate. That tangent d t (to the circle) will also be equal in
length to the corresponding portion b'c of the curve while
the tension at b will be expressed by a length of the chain
equal to the secant o d. So again, if D T be a tangent, to the
circle drawn from D, it will be equal in length to в b c, and
parallel to the tangent to the catenary at B; while the secant
op will measure the oblique tension at B; evidently exceed-
ing the constant horizontal tension or strain at c, by the ab-
sciss c D.
17. When the parameter of the catenary, or the line which
measures the tension at the lowest point, is equal to the depres-
sion Dc; if cach of these be supposed equal to 1, then A B =
2-6339, the length of chain A C B = 34641; the strain at the
points of suspension A and B will each be 2, that at the lowest
point being 1; and the chain at A and B will make an angle of
60° with the horizon.
18. If the strain at c be equal to the weight of the chain,
and cach denoted by 1: then a B •96242, p c = •1180340,
I
178
CURVES: CATENARY.
the tension at A or B 1.118, the angle of suspension at those
points nearly 26° 34'; the width of the curve is 8.1536 times,
and the length 8-4719 times, the depression D c.
19. If the strain or tension at the lowest point be double the
weight of the chain: then if the parameter be 1, A B will be
•49493, C D 03078, the strain at A or B 1.03078, the angle
of suspension about 14º 2', the width or span 16.0816 times,
and the length of chain 16-2462 times the depression.
The magnitudes of the lines, angles, and strains in many other
cases may be seen in the table below. The whole theory may
be verified experimentally, by means of spring steel-yards ap-
plied to a chain of given length and weight, placed in various
positions; according to the method suggested subsequently
when treating of the parallelogram of forces, in Mechanics.
20. Taking A B = d, c Dh, length A C B = 1, strain at c or
parameter = p, then, in all cases where the depression is small
compared with the length of the chain, Professor Leslie shows,*
that
p
or p
d s
2
8 h
12.
8h
+h.... strain at A or B
h h.... strain at A or B =
FIQ
1
d2
8 h
72
l = d +
8 h
I
+ ½ h
8h2
3 d
In this case, too, the strains at c and A or B are nearly in the
inverse ratio of the depression.t
+ z h
* Elements of Natural Philosophy, p. 63.
For a very complete investigation of the proper forms of catenaries for suspen-
sion bridges, with remarks on the Menai bridge, and on the failure of the suspon-
sion bridge at Broughton, see Mr. Eaton Hodgkinson's paper in the Memoirs of
the Manchester Society, vol. v., New Series.
CURVES: CATENARY.
179
1
Table of Relations of Catenarian Curves, the Parameter
being denoted by 1.


Angle of
suspen-
sion.
123 10
1º 0'
0
0
0
5 0
4
6
17
0
0
9 0
10
0
11 0
12
0
13
0
14
0
15 0
16 0
17 0
18 0
19 0.
20 0
21
Q
22 0
23
24
25
26
28
30 0
32 4
34 16
36 52
39 11
0
0
41 44
44
0
46 1
48 11
50 8
52 9
54 13
56 28
58 3
60 0
64 6
67 28
67 32
DC
費
​1.0000
1·2894
1.6095
1.6168
D B
·00015
•01745
•01745
•00061
·03491
·03492
⚫00137
⚫05238
•05241
·00244
·06987
⚫06993
⚫00382
·08738
·08749
·00551
•10491
•10510
·00751
*12248
•12278
•00983
•14008
•14054
*01247
•15773
•15838
⚫01543
•17542
•17633
-01872
•19318
•19439
·02234
•21099
-21256
⚫02630
•22887
•23087
·03061
•24681
•24933
·03528
•26484
•26795
·04030
*28296
•28675
*04569
•30116
•30573
⚫05146
•31946
•32492
·05762
•33786
⚫34433
•06418
-35637
•36397
•07114
•37502
-38380
•07853
⚫39376
•40403
·08636
•41267
•42447
•09484
43169
•44523
•10338
•45087
•46631
11260
*17021
*48773
•13257
•50940
•53171
•15470
•54930 •57735
•18004
*5912
•62649
•21003
·6371
•68130
•24995
*6932.
•74991
-29011
-7443
•81510
·34004
•8029
•89201
⚫39016
.8566
•96569
43999
.9066
•19981
•9623
•56005 1.0142
·62973 1.0706
⚫71021 1.1304
•81021
•88972
C B
1-0361
1.1178
1.1974
1.2869
1.3874
1.1995 1.5089
1.2510
1.6034
1.3169
1·7321
1-4702 2.0594
1.6135 2.4102
1.6164
2.4182
tension at
A or B
1.0001
1.0006
1.0014
1·0024
1.0038
1.0055
1.0075
1.0098
1.0125
1.0154
1.0187
1·0223
1·0263
1.0306
1·0353
1·0403
1.0457
1.0515
1·0576
1.0642
1.0711
1.0786
1·0864
1·0946
1.1034
1.1126
1.1326
1.1547
1·1800
1.2100
1.2499
1.2901
1.3400
1.3902
1·4400
1.4998
1.5800
1.6297
1.7102
1.8102
1·8897
2.0000
2.2894
2.6095
2.6168
DB÷DC
114.586
57.279
38.171
28.613
22.874
19.046
16.309
14.254
12.654
11.372
10.820
9.444
8.701
8.062
7.508
7-021
6.591
6.208
5.863
5.553
5-271
5.014
4.778
4-562
4.361
4.176
3-843
3-551
3.284
3.034
2-773
2.567
2.362
2.196
2.060
1.925
1.811
1.699
1.592
1-481
1.416
1.317
1.140
1·002
0.9998
180
CURVES: CATENARY.
21. The preceding table is abridged from a very extensive
one given by Mr. Ware in his “Tracts on Vaults on Bridges.”
Two examples will serve to illustrate its use.
Ex. 1. Suppose that the span of a proposed suspension bridge
is 560 feet, and the depression in the middle 253 feet; what will
be the length of the chain, the angle of suspension at the extre-
mities, and the ratio of the horizontal pressure at the lowest
point, and the oblique pressures at the points of suspension, with
the entire weight of the chain?
Here D B÷D C=280÷25·875=10·82, a number which is to
be found in the table.
Opposite to that number, we find 11° for the angle of suspen-
sion, D B=19318, c в=·19438, tension at A or B=10187, the
constant tension at the vertex being 1. (fig. p. 174.)
Consequently, 19318: 19438 :: 560: 563 48 length of the
chain.
1
•
Also, horizon. pressure at c
-are as 1.0000
1.0187
and 39876
Ex. 2. Suppose that while the span remains 560, the depres-
sion is increased to 51.
oblique pressure at A or B
entire weight of chain
•
Here D B÷D C=280÷51=5·49.
This number is not to be
found exactly in the table. The nearest is 5·553 in the last
column, agreeing with 20°, the angle of suspension.
Now, 5.55- 5.49 •06, and 5.55 5.27
28, the former
difference being nearly one-fifth of the latter. Hence, adding
to each number, in the line agreeing with 20°, one-fifth of the
difference between that and the corresponding number in the
next line, we shall have
•36010,
Angle of suspension 20° 12', D C
20° 12', D C = •06556, D B =
=36797, tension at A=1∙10656.
Hence 36001: 36797 :: 560 572.24, length of chain.
Also, horizontal pressure at c
are as 1.0000
oblique pressure at a or B
entire weight of chain
}
•
1
a,
Comparing this with the former case, it will be seen that the
tensions at c and A, in reference to the weight of the chain, are
diminished nearly in the inverse ratio of the two values of no;
thus confirming the remark in art. 20.
1
1·10656
•73594
In practical cases with regard to bridges of suspension, it will
be easy, when the weight of the material and its cohesive
strength are known, to find the relative strength of any pro-
posed structure.
}
ISOMETRICAL PERSPECTIVE.
181
{
J
CHAPTER VII.
Professor Farish's Isometrical Perspective.
In the course of lectures which I deliver in the university
of Cambridge, I exhibit models of almost all the more im-
portant machines which are in use in the manufactures of
Britain.
The number of these is so large, that had each of them been
permanent and separate, on a scale requisite to make them work,
and to explain them to my audience, I should, independently
of other objections, have found it difficult to have procured a
warehouse large enough to contain them. I procured therefore
an apparatus, consisting of what may be called a system of the
first principles of machinery; that is, the separate parts, of
which machines consist. These are made chiefly of metal, so
strong, that they may be sufficient to perform even heavy work:
and so adapted to each other, that they may be put together at
pleasure, in every form, which the particular occasion re-
quires.
Those parts are various such as, loose brass wheels, the
teeth of which all fit into one another: axes, of various lengths,
on any part of which the wheel required may be fixed: bars,
clamps, and frames; and whatever else might be necessary
to build up the particular machines which are wanted for
one lecture. These models may be taken down, and the
parts built up again, in a different form, for the lecture of
the following day. As these machines, thus constructed for
a temporary purpose, have no permanent existence in them-
selves, it became necessary to make an accurate representa-
tion of them on paper, by which my assistants might know
how to put them together without the necessity of my con-
tinual superintendance. This might have been done, by
giving three orthographic plans of cach; one on the horizon-
tal plane, and two on vertical planes at right angles to each
other. But such a method, though in some degree in use
among artists, would be liable to great objections. It would
25
R
182
ISOMETRICAL PERSPECTIVE.
1
be unintelligible to an inexperienced eye; and even to an artist,
it shows but very imperfectly that which is most essential, the
connexion of the different parts of the engine with one another;
though it has the advantage of exhibiting the lines parallel to
the planes on which the orthographic projections are taken
on a perfect scale.
This will be easily understood, by supposing a cube to be
the object represented. The ground plan would be a square
representing both the upper and lower surfaces. And the two
elevations would also be squares on two vertical planes, parallel
to the other sides of the cube. The artist would have exhibited
to him three squares; and he would have to discover how to
put them together in the form of a cube, from the circumstance
of there being two clevations and a ground plan. This method,
therefore, giving so little assistance on so essential a point, I
thought unsatisfactory.
And
The taking a picture on the principles of common perspec-
tive, was the next expedient that suggested itself. And this
might be adapted to the exhibition of a model, by taking a
kind of bird's-eye view of the object, and having the plane of
the picture, not as is most common in a drawing, perpendicular
to the horizon, but to a line drawn from the eye, to some prin-
cipal part of the object. For example: in taking the picture
of a cube, the eye might be placed in a distant point on the line
which is formed by producing the diagonal of the cube. But
to this common perspective there are great objections. The
lines, which in the cube itself are all equal, in the representation
are unequal. So that it exhibits nothing like a scale.
to compute the proportions of the original from the repre-
sentation would be exceedingly difficult, and, for any useful
purpose, impracticable: there is equal difficulty too, in com-
puting the angles which represent the right angles of the cube.
Neither docs the representation appear correct, unless the
eye of the person, who looks at it, be placed exactly in the
point of sight. It is true that, as we are continually in the
habit of looking at such perspective drawings, we get the
habit of correcting, or rather overlooking the apparent errors.
which arise from the eye being out of the point of sight,
and are therefore not struck with the appearance of incorrect-
ness, which if we were unaccustomed to it, we should feel
at once.
J
The kind of perspective which is the subject of this pa-
per, though liable in a slight degree to the last mentioned in-
convenience, till the eye becomes used to it, I found much bet-
ter adapted to the exhibition of machinery; I therefore deter-
ISOMETRICAL PERSPECTIVE.
183
mined to adopt it, and set myself to investigate its principles,
and to consider how it might most easily be brought into
practice.
It is preferable to the common perspective on many accounts,
for such purposes. It is much easier and simpler in its princi-
ples. It is also, by the help of a common drawing-table, and
two rulers,* incomparably more easy, and, consequently, more
accurate in its application; insomuch, that there is no difficulty
in giving an almost perfectly correct representation of any
object adapted to this perspective, to which the artist has access,
if he has a very simple knowledge of its principles, and a little
practice.
It further represents the straight lines which lie in the
three principal directions, all on the same scale.
The right
angles contained by such lines are always represented either
by angles of 60 degrees, or the supplement of 60 degrees.
And this, though it might look like an objection, will appear
to be none on the first sight of a drawing on these principles,
by any person who has ever looked at a picture. For, he
* It is unnecessary to describe the drawing-table any further than by observing
that it ought to be so contrived, as to keep the paper steady on which the drawing
is to be made.
Here should be a ruler in the form of the letter T to slide on one side of the
drawing-table. The ruler should be kept, by small prominences on the under side,
from being in immediate contact with the paper, to prevent its blotting the fresh
drawn lines as it slides over them. And a second ruler, by means of a groove near
one end on its under side, should be made to slide on the first. The groove should
be wider than the breadth of the first ruler, and so fitted, that the second may at
pleasure be put into cither of the two positions represented in the plate, fig. 1, so as
to contain with the former ruler, in either position an angle of 60 degrees. The
groove should be of such a size, that when its shoulders a and d are in contact with,
and rest against the edges of the first ruler, the edge of the second ruler should
coincide with de, the side of an equilateral triangle described on d g, a portion of
the edge of the first rulor; and when the shoulders b and c rest against the edges
of the first ruler, the edge of the second should lie along g e, the other side of the
equilateral triangle. The second ruler should have a little foot at k for the same
purposo as the prominences on the first ruler, and both of them should have their
edges divided into inches, and tenths, or eighths of inches.
It would be convenient if the second ruler had also another groover s, so formed
that when the shoulders and s are in contact with the edges of the first ruler, the
second should be at right angles to it. For representing circles in their proper
positions, the writer made use of the inner edge of rims cut out from cards, into
isometrical ellipses as represented in the figure; of these he had a series of different
sizes, corresponding to his wheels. Such a series might be cut by help of the con-
centric ellipses in fig. 5, but he thinks that it would be an easier way to make use
of that set of concentric ellipses as they stand, by putting them in the proper place
under the picture, if the paper on which the drawing is made be thin enough for
the lines to be traced through, as by the help of them the several concentric circles
will go to the representation of one which might be drawn at once. It is difficult to
executo them separately with sufficient accuracy to make them correspond. For
this purpose a separate plate of fig. 5 should be had, and one edge of the paper on
the drawing-table should be loose to admit of the concentric ellipsos being slid under
it to the proper place, as described p. 187.
184
ISOMETRICAL PERSPECTIVE.
K
cannot for a moment have a doubt, that the angle represented is
a right angle, on inspection.
And we may observe further, that an angle of 60 degrees is
the easiest to draw, of any angle in nature. It may be instantly
found by any person who has a pair of compasses, and under-
stands the first proposition of Euclid. The representation,
also, of circles and wheels, and of the manner in which they act
on one another is very simple and intelligible. The principles
of this perspective, which, from the peculiar circumstance of its
exhibiting the lines in the three principal dimensions on the
same scale, I denominate "Isometrical," will be understood
from the following detail:
Suppose a cube to be the object to be represented. The eye
placed in the diagonal of the cube produced. The paper, on
which the drawing is to be made to be perpendicular to that
diagonal, between the eye and the object, at a due proportional
distance from each, according to the scale required. Let the
distance of the eye, and consequently that of the paper, be in-
definitely increased, so that the size of the object may be in-
considerable in respect of it.
bakal
It is manifest, that all the lines drawn from any points of
the object to the eye may be considered as perpendicular to
the picture, which becomes, therefore, a species of orthogra-
phic projection. It is manifest, the projection will have for
its outline an equiangular and equilateral hexagon, with two
vertical sides, and an angle at the top and bottom. The other
three lines will be radii drawn from the centre to the lowest
angle, and to the two alternate angles; and all these lines
and sides will be equal to cach other both in the object and
representation: and if any other lines parallel to any of
the three radii should exist in the object, and be represented
in the picture, their representations will bear to one another,
and to the rest of the sides of the cube, the same propor-
tion which the lines represented bear to one another in the
object.
If any one of them, therefore, be so taken as to bear any
required proportion to ils object, e. g. 1 to 8, as in my repre-
sentations of my models, the others also will bear the same pro-
portion to their objects; that is, the lines parallel to the three
radii will be reduced to a scale.
I omit the demonstration of this, and some other points,
partly for the sake of brevity, and partly because a geometri-
cian will find no difficulty in demonstrating them himself
from the nature of orthographic projection; and a person,
who is not a geometrician, would have no interest in reading a
demonstration.
ISOMETRICAL PERSPECTIVE.
185
For the same reason, it is unnecessary to show that the
three angles at the centre are equal to one another, and each
equal to 120 degrees, twice the angle of an equilateral tri-
angle; and the angle contained between any radius and side
is 60 degrees, the supplement of the above, and equal to the
angle of an equilateral triangle. All this follows immediately
from Euclid, B. IV. Prop. 15, on the inscription of a hexagon
in a circle.
In models and machines, most of the lines are actually in the
three directions parallel to the sides of a cube, properly placed
on the object. And the eye of the artist should be supposed to
be placed at an indefinite distance, as before explained, in a
diagonal of the cube produced.
Definitions.
The last mentioned line may be called the line of sight.
Let a certain point be assumed in the object, as for example
c, fig. 2, pl. I. and be represented in the picture, to be called
the regulating point. Through that point on the picture may
be drawn a vertical line c E, fig. 2, and two others, c B, C G,
containing with it, and with one another, angles of 120°, to
be called the isometrical lines, to be distinguished from one
another by the names of the vertical, the dexter, and the
sinister lines. And the two latter may be called by a common
name-the horizontal isometrical lines. Any other lines,
parallel to them, may be called respectively by the same
names. The plane passing through the dexter and vertical
lines may be called the dexter isometrical plane; that passing
through the vertical and sinister lines, the sinister plane; and
that through the dexter and sinister lines, the horizontal
plane.
By the use of the simple apparatus described above in the
note, the representation of these lines in the objects may be
drawn on the picture, and measured to a scale, with the
utmost facility, the point at the extremity being first found
or assumed. The position of any point in the picture may
be easily found, by measuring its three distances, namely,
first its perpendicular distance from the regulating horizontal
plane (that is, the horizontal plane passing through the regu-
lating point), secondly, the perpendicular distance of that
point where the perpendicular meets the horizontal plane,
from the regulating dexter line; and thirdly, of the point,
where that perpendicular meets the dexter line from the
regulating point; and then taking those distances reduced to
R 2
186
ISOMETRICAL PERSPECTIVE.
the scale, first, along the dexter line; secondly, along the sin-
ister line; and thirdly, along the vertical line, in the picture.
These three may be called the dexter distance of the point, its
sinister distance, and its allitude. And it is manifest they
need not be taken in this order, but in any other that may be
more convenient to the artist, there being six ways in which
this operation may be varied.
If any point in the same isometrical plane, with the point re-
quired to be found, is already represented in the picture, that
point may be assumed as a new regulating point, and the point.
required found by taking two distances; and if the new assumed
regulating point is in the same isometrical line with the point,
it is found by taking only one distance. And this last simple
operation will be found in practice all that is necessary for the
determination of most of the points required. Thus any paral-
lelopiped, or any frame work, or other object with rafters, or
lines lying in the isometrical directions, may be most easily and
accurately exhibited on any scale required. But, if it be neces-
sary to represent lines in other directions, they will not be on
the same scale, but may be exhibited, if straight lines, by find-
ing the extremities as above, and drawing the line from one to
the other; or sometimes more readily in practice by help of an
ellipse, as hereafter described.
If a curved line be required, several points may be found
sufficient to guide the artist to that degree of exactness which is
required.
The method of exhibiting the representations of any ma-
chines, or objects, the lines of which lie, as they generally do, in
the isometrical directions; that is, parallel to the three direc-
tions of the lines of the cube, is as has been already shown; and
likewise the mode of representing any other straight lines, by
finding their extremities; or curved lines, by finding a number
of points.
}
But in representing machines and models, there are not only
isometrical lines, but also many wheels working into each other,
to be represented. These, for the most part, lie in the
isometrical planes; and it is fortunate that the picture of a cir-
cle in any one of these planes is always an ellipse of the
same form, whether the plane be horizontal, dexter, or sin-
ister; yet they are easily distinguished from each other by the
position in which they are placed on their axle, which is an
isometrical line, always coinciding with the minor axis of the
ellipse.
ן
This will be obvious from considering the picture of a cube
with a circle inscribed in each of its planes, fig. 3, and
considering these circles as wheels on an axle. The two other
ISOMETRICAL PERSPECTIVE.
187
lines (or spokes of the wheel) in the ellipse, which are drawn
respectively through the opposite points of contact of the circle
with the circumscribing figure, are isometrical lines also; for
the points of contact bisect the sides of the circumscribing
parallelogram, and therefore the lines are parallel to the
other sides. They give likewise the true diameter of the
wheels, reduced to the scale required. It further appears,
from the nature of orthographic projection, that the major
axis of the ellipse is to the minor axis as the longer to the
shorter diagonal of the circumscribing parallelogram, that is
(since the shorter diagonal divides it into two equilateral
triangles) as the square root of three to one; as appears from
Euclid, Lib. I. Prop. 47; and since the sum of the squares
of the conjugate diameters in an ellipse, is always the same,
if we put 1 for the minor axis, the 3 for the major, and i
for the isometrical diameter, we shall have 2 ¿ª³ = 1 + 3,
and i = √2.
3
4,
Therefore the minor axis, the isometrical diameter, and the
major axis, may be represented respectively by √1, √2, √3,
or nearly by 1, 1·4142, 1·7321; or more simply, though not so
nearly, by 28, 40, 49.
These lines may be geometrically exhibited by the follow-
ing construction:
Let a B, fig. 4, be equal to в D, and the angle at B, a right
angle. In BA produced, take в a to A D, draw a D, and
produce both it, and a B. Then will в D, Ba, and a D, be
respectively to one another, as √1, √2, √3, by Euclid, I. 47.
Therefore if a ß be taken equal to the isometrical diameter of
the ellipse required, 3 8 drawn perpendicular to it will be the
minor axis, and a & the major axis. The ellipse itself, there-
fore, may be drawn by an elliptic compass, as that instru-
ment may be properly set, if the major and minor axis are
known. If it is to represent a wheel on an axle, care must be
taken to make the minor axis lie along that axle. In the ab-
sence of the instrument it may be drawn from the concentric
ellipsis, fig. 5, which may be placed under the paper, in the
position above described, and seen through it, if the paper
be not too thick; and in this method the smaller concentric
circles of the wheel may be described at the same time, as they
may be seen through the paper, or if they should not be ex-
actly of the right size, it would be easy to describe them by
hand between the two nearest concentric ellipses; and thus
also the height of the cogs of a wheel in the different parts
of it may be exhibited longer and narrower towards the ex-
tremities of the minor axis. Their width may be determined
from the divisions of the ellipse. In most cases this may be
+
188
ISOMETRICAL PERSPECTIVE.
1
done with sufficient accuracy from the circumference of the el-
lipse being divided into eight equal divisions of the circle, by
the two axes, and two isometrical diameters, each of which parts
may be subdivided by the skill of the artist; and not only the
face of the wheel in front may be thus exhibited, but the parts of
the back circles also, which are in sight, may be exhibited by
pushing back the system of concentric ellipses on the minor
axis or axle through a distance representing the breadth of the
wheel, and then tracing both the exterior and the interior circles
of the wheel, and of the bush on which it is fixed, as far as
they are visible. Care should be taken to represent the top
of the teeth, or cogs, by isometrical lines, parallel to the axle,
in a face wheel, or tending to a proper point in the axle in a
bevil-wheel. And nearly in the same way may the floats of a
water-wheel be correctly represented. If a series of concentric
ellipses such as are given fig. 5, be not at hand, it will still be casy
for an artist to draw the ellipses with sufficient accuracy for
most purposes, by drawing through the proper point in the
axle, the major and minor axes, and the two isometrical
diameters, thus making eight points in the circumference to
guide him.
If in any case it should become necessary to represent
a circle, which does not lie in an isometrical plane, we
may observe that the major axis will be the same in what-
ever plane it lies and it will be the picture of that diameter,
which is the intersection of the circle with the planc parallel to
the picture, passing through its centre. And the major axis
will bear to the minor axis the proportion of radius to the sine
of the inclination of the line of sight to the plane of the circle.
We may observe further, that the diameters of the ellipse, which
are to the major axis as √2 to 3, when such exist, arc isome-
trical lines.*
And the representation of every other line, parallel and
equal to any diameter of the circle, may be exhibited by draw-
ing it equal and parallel to the corresponding diameter in the
*We may remark, that if a cone be described, having its vertex at c
which lies in the line of sight fig. 2, and passing through the three radii e B, CE,
co, all the straight lines in the superfices of that cone passing through c, and all
other lines parallel to any of thein, are isometrical, as well as those parallel to the
three principal isometrical lines CB, C, CG; and no other lines but these can
be on the same scale But though these multiply the number of isometrical
lines infinitely, it is of little practical uso, because it is only those which are parallel
to the three principal linos, that can easily be distinguished at sight, to be isometrical.
We may further remark, that if a line be drawn through the point e parallel to
any given line whatever, and that line be made to revolve round the line of sight,
at the same angular distance from it, so as to describe the surface of a cone, all
other lines parallel to it, in any of its positions, will be isometrical, as they re-
spect one another.
ISOMETRICAL PERSPECTIVE.
189
}
;
ellipse. If it should be desired to divide the circumference of
an ellipse into degrees, or any number of parts representing
given divisions of the circle, it may be done by the following
method :
Let an ellipse be drawn, fig. 6, and on its major axis, A G, a
circle described, with its circumference divided into degrees or
parts in any desired proportion, at B, C, D, E, F, &c., from which
points draw perpendiculars to the major axis. They will cut
the periphery of the ellipse in corresponding points. It would
be difficult, however, in this way, to mark, with sufficient accu-
racy, the degrees, which lie near the extremities of the major
axis. But the defect may be supplied by transferring those
degrees in a similar way, from a graduated circle, described on
the minor axis. In this manner an isometrical ellipse may be
formed into an isometrical circular instrument, or an isometrical
compass, which may show bearings or measure angles on the pic-
ture, in the same manner as a real compass or circular instru-
ment would do in nature.
It may be often useful to have a scale to measure distances,
not only in the isometrical directions, but in others also. And
this may be done by a series of similar concentric ellipses, as
in fig. 7, dividing the isometrical diameters into equal portions.
The other diameters will be so divided as to serve for a scale
for all lines parallel to them respectively.
Thus, in the isometrical squares, exhibited in fig. 2, distances
measured on the longer diagonal, or its parallels, would be
measured by the divisions on the major axis, those depending
on the shorter diagonal by the divisions on the minor axis.
To describe a cylinder lying in an isometrical direction, the
circles at its extremities should be represented by the proper
isometrical cllipses, and two lines touching both should be
drawn and in a similar way, a cone, or frustrum of a cone,
may be described. A globe is represented by a circle whose
radius is the semi-major axis of the ellipse representing a great
:
circle.
It would not be difficult to devise rules for the representation
of many other forms which might occur in objects to be repre-
sented. But the, above cases are suflicient to include almost
every thing which occurs in the representation of models, of
machines, of philosophical instruments, and, indeed, of almost
any regular production of art.
Buildings may be exhibited by this perspective as correctly,
in point of measurement, as by plans and elevations, under the
advantage of having the full effect of a picture.
A bridge, or any circular or gothic arch, consisting of por-
1
26
&
心
​190
ISOMETRICAL PERSPECTIVE.
tions of circles lying in isometrical planes, may be represented
by portions of isometrical ellipses, which will easily be adapted
and drawn upon the principles already explained, by which
wheels are exhibited on their axles. The centres of those cir-
cles must be found with which the centres of the ellipses must
be made to coincide, their minor axes lying along the lines drawn
from those centres perpendicular to the planes of the circles.
The shaft of a pillar consists of a frustrum of a cone and a cylin-
der united; or perhaps of a cylinder alone, or a congeries of
cylinders; and we have already shown the method of exhibit-
ing these, as well as their bases. And on the same principles,
the position and size of the volutes and ornaments of the capi-
tal may be found, and such guiding points as will make it easy
to trace their forms. Thus the different courts and edifices
of a cathedral, a college, or a palace, may be correctly depicted;
and even the rooms and internal structure, though less in the
form of a picture, may be exhibited in such a way as to enable
an architect, or his employer, to contemplate their situation,
their ornaments, furniture, or any other circumstance belonging
to their appearance, and to mark down exactly what he would
have done, in such a way as could hardly be misunderstood by
an attentive agent, though at a distance.
But in thus exhibiting buildings as transparent, and their in-
terior laid open, there is a danger of being confused by a multi-
plicity of lines, which is a difficulty in a building containing
many rooms that would need some address to get over. It is
better adapted to exhibit the inside of a single room, of a ca-
thedral, for instance, the aisles and transepts of which would not
cause any great perplexity.
In the same manner a plan of a city might be given, which
would not only represent its streets and squares, as well (by the
help of the scale above described, fig. 7,) as a common plan, but
also a picture of its churches and public buildings, and even its
private houses, if such were the design contemplated by the ar-
tist, as they would almost all become visible when looked down
upon from the commanding height which this perspective
supposes. And such a single exhibition, if well executed, might
give a better idea of a distant capital than a volume of descrip-
tion.
In the instances which have been given, most of the lines are
isometrical. But the art is applicable to many cases, where
there are few, or none such. It may be necessary, in many
of them, to draw isometrical lines, or isometrical ellipses,
by way of a guide, to determine the position of certain lines
and points to enable the artist to describe with accuracy what
1
ISOMETRICAL PERSPECTIVE.
191
3
he has in view. And there is scarce any form so anomalous
as to preclude the artist from taking advantage of these
methods of ascertaining such lines or points in it as will
give him much assistance in representing it with precision.
If the intention be merely to make a picture, the guiding lines
may be obliterated as soon as they have served the purpose de-
signed, or they may be retained in some cases, and their
lengths or diameters noted down in figures, if it be wished,
to give ready information. And often, if the artist wishes
to provide materials to enable him at his leisure to give accu-
rate descriptions or exact drawings, the rudest exhibition of
such lines may completely serve his purpose, provided he
notes down on the spot such measurements with accuracy,
however unexact the lines may be on which they are re-
corded. In many cases it may be expedient to take liberties
with this perspective, or with the picture, which will make it
suit the purpose designed. And this will produce no confu-
sion, provided those liberties are explained: for instance, it
may often be expedicnt to make the scale in the vertical
direction larger, sometimes very considerably so, than in the
horizontal. It may in some cases be necessary to represent on
paper what is hid in nature. What has been said on the
internal structure of buildings is an instance of this, as well
as what we shall observe on the exhibition of subterraneous ob-
jects. We shall proceed to give some examples of these ob-
High
servations.
1
To give such a representation of an Etruscan vase, as
would enable an artist to model it exactly, would be ex-
ceedingly easy. Let a vertical line be drawn to represent
the axis of the vase, fig. 8, and let points be taken in that axis,
corresponding to the centres of the principal circles of the
vase; through which the horizontal isometrical lines may be
drawn representing the radii of those circles, by the help of
which the isometrical ellipses representing them are easily
drawn. These will become a complete guide to the artist. He
may assist himself by looking at the object along the line of
sight, and then, if he has any skill in drawing, he will find no
difficulty in tracing the outline from one of these to the other,
with sufficient correctness. If he is unskilled in the art, of
course he must be at the trouble of finding a larger number
of ellipses to guide him. And in a similar manner, any solid
formed by the revolution of a plane figure round one of its
sides may be represented.
{
The laying down the timbers of a ship, or making a picture
of one, shall be another example.
192
ISOMETRICAL PERSPECTIVE.
}
Let a vertical isometrical plane be conceived to pass through
its keel, and to be intersected by the perpendicular planes pass-
ing through the ribs, and by planes parallel to the decks. The
isometrical lines, which are the intersections of these, may be
measured in the ship, and represented with their proper measures
noted down in the picture, which will afford the means of re-
presenting the ribs, and laying them down in their proper
places.
If this should be designed for the purpose of constructing a
ship from a given model, it might be sufficient to represent
the ribs only on one side; those on the other side being the
exact counterparts. If the purpose should be to make use of .
these lines for a drawing, they need be marked but very faintly,
and the artist will have little difficulty when guided by them
to fill up the representation by hand.
A regular fortification, which we will suppose to have eight
bastions, will afford another example.
A person not conversant, in such a subject, is in general puz-
zled with plans and sections, and has very little idea of what is
meant to be conveyed.
But he would easily understand it if he should see every
thing exhibited in a correct picture, especially where he has the
the view of his object varied, as in a fortification, such as has
been proposed. Let an isometrical ellipse be drawn expressing
the internal circumference of the place; and another concentric
one, which marks the salient angles of the fortification on the
principles already explained. Draw other guiding lines to every
necessary point; the lines of the fortification may be easily
transferred from a common plan to the isometrical by the help
of the scale of concentric ellipses, described above, fig. 7, which
will serve also to lay down the length of the bastions and cur-
Find the elevations
tains, &c. in whatever direction they lic.
of every part on the isometrical scale; and thus the body of the
place, the ditches, counterscarp, covered way, glacis, ravellins,
and all the outworks, will be represented to the eye as they ap-
pear in reality, and in every varied position, with the advantage
of having all the admeasurements laid down with geometrical
precision.
If the artist should think the vertical lines in such an exhibi-
tion too small, to give a correct idea of all the minute elevations,
there would be no harm in his increasing the scale in that di-
mension in any desired proportions.
The face of a hilly or mountainous country, like Switzer-
land, or the district of the lakes in the north part of England,
will afford another example.
+
เ
1
}
ISOMETRICAL PERSPECTIVE.
193
}
+
Isometrical horizontal lines may be drawn representing lines
in the level from which the height of the mountains is to be
reckoned, so that vertical lines drawn from the summits of the
mountains may meet them, on which the heights may be marked
(as well as recorded in figures, if required). And the moun-
tains themselves may be drawn in their topographical situation.
Their bearings may be marked by the help of the isometrical
compass described in page 189. It would be easy to transfer
them from a common map to the isometrical plan; and thus
the face of the country might be represented just as it would
appear from the commanding height which the isometrical per-
spective supposes.
Yet, as the slopes of hills and mountains are seldom so steep
as the line of sight, it might sometimes suit the purpose to re-
present the height of elevations as twice or three times the
reality, in order that mountains might project an outline on the
plane behind; otherwise, the summit might be projected on the
mountain itself, which would, in a degree, destroy the effect of
a picture.
This art might be advantageously employed also for tracing
what is below the surface of the earth, as well as what is above
it. It may be applied to geological purposes, and give, not only
the order of the strata, but their variations and their geographi-
cal situations. And for this purpose it might be useful to in-
crease the vertical scale, in a great proportion, above the hori-
zontal. It would be easy to mark the dip, or rise of the strata,
as well as of the earth above them to represent their various
disruptions, to show the situation and extent of fissures and me-
tallic veins, to mark the boundaries where the upper strata have
been swallowed up, or cease to appear, or where the under
strata push up towards the day. It would be easy to mark the
variations in the thickness of the strata in different places, and
to record the result of experiments made at any point, by bor-
ing or sinking shafts, which might be done by drawing a verti-
cal line downward, so as to represent the thickness of the lami-
næ, which might be marked by different colours. By such a
method, the geologist might obtain a map of the country, which
might exhibit, at one view, the general results of all the experi-
ments and inquiries that had been made relative to that science,
and the owner of an estate might record, in a small compass, all
that is known respecting its minerals, and be able, from a com-
prehensive view of them all, to judge of the probability of suc-
cess in sinking a shaft, or driving a level. He might also make
good use of this perspective in tracing his shafts and drifts in
all their windings, elevations, and depressions, and comparing
them with the surface above, marking also the veins and strata
S
->
1
194
ISOMETRICAL PERSPECTIVE.
in which they run.
For if the artist knows what is beneath the
surface, he has no difficulty in representing it as transparent.
He must be careful however not to perplex himself by lines too
much multiplied, and take advantage of his being able to paint
the lines with different colours, for the purposes of distinction ;
and he must also use a considerable address in throwing out
such lines as would be of little use, and retaining such as will
produce the effect of a picture, which should be well preserved
in order to make the exhibition easily intelligible.
If he should wish to make a drawing of minerals or crystals,
this perspective would be well suited to the purpose.
The point, however, on which the writer of this paper can
speak with the greatest confidence is on the representation of
machines and philosophical instruments; having been himself
so much in the habit of practically applying to them the princi-
ples that have been detailed and this he has exemplified in
the plates.
:
The correct exhibition of objects would be much facilitated
by the use of this perspective, even in the hands of a person
who is but little acquainted with the art of drawing; and the
information given by such drawings is much more definite and
precise than that obtained by the usual methods, and better fitted
to direct a workman in execution.*
*The author has transcribed this interesting paper from the first volume of the
Transactions of the Cambridge Philosophical Society. The method is peculiarly de-
serving of the attention, and, in many cases, the adoption, of mechanics and engi-
ncers.
Some useful exemplifications of isoperimetrical perspective are given in the 18th
and 19th volumes of the Mechanic's Magazine; and while this sheet was going
through the press, I learnt that Mr. Jopling is now printing a small treatise on the
subject, for the use of mechanics and artificers.
+k
1
1
MENSURATION.
CHAPTER VIII.
MENSURATION OF SUPERFICIES AND SOLIDS.
F
SECTION I.-Mensuration of Superficies.
The following rules will serve to find the areas or superficial
contents of the figures whose names respectively precede
them.
1. Rectangle, Square, Rhombus, or Rhomboid. Multiply
the base into the height, for the area.
{
2. Triangle. Half base into the height. Or, continual
product of two sides, and half the natural sine of their inclined
angle.
Or, when three sides, as a B, a C, B C, are given, their half sum
beings; then
K
K
✓[S × (S— A B) × (S — A C) × s-
area=
B C)].
3. Trapezium. Base into sum of the perpendiculars.
4. Trapezoid. Multiply half the sum of the parallel sides
into the perpendicular distance between them.
5. Irregular Polygon. Divide it into trapeziums, or trape-
zoids, and triangles, and find their areas separately their sum
is the area of the polygon.
:
or, diameter: circumf :: 1 : 3·141593
circumf diameter :: 1: 318309
195
6. Regular Polygon. Multiply the square of the side given
into the proper "multiplier for areas," printed in the table in
Prob. 15, Practical Geometry, the product will be the area.
7. Circle. diameter: circumf :: 113: 355
►
1
:
area=diameter squared × 785398
area circumference squared × 079577
area diameter circumf.
8. Circular arc. Radius of the circle x 017453 × degrees
in the arc its length.
9. Circular sector. arc radius=area.
10. Circular segment. Multiply the square of the radius
by either half the difference of the arc (of the segment) and its
196
MENSURATION OF SUPERFICIES.
sine, or by half their sum, according as the segment is less or
greater than a semicircle: the product will be the area.
11. Parabola. of the product of base and height
12. Ellipse. Transverse axis x conjugate axis x 785398
= area.
2
3
arca.
13. The side of a square whose area shall be equal to that of
a given circle, is nearly of the diameter, or more nearly 3d,
8
可
​or 22 d, or 193 d, or 144d; cach approximating more nearly
3 9
44
2
167
than the former.
SECTION II.-Mensuration of Solids.
i
1. Prism. 1. Superficies. Multiply the perimeter of one
end by the length or height of the solid; the product will be
the surface of the sides. To this add the areas of the two ends :
the sum is the whole surface.
J
1
2. Solidity or Capacity area of the base x the height.
Note. The same rules serve for the surface and capacity of
a cylinder.
2. Pyramid, or Cone. 1. Surface
X slant height.
2. Capacity area of base × height.
3. Frustrum of Pyramid. 1. Surface
meters of the two ends × slant height.
perimeter of the base.
I
sum of the peri-
2. Capacity. Add a diameter or a side of the greater base.
to one of the less; from the square of the sum subtract the pro-
duct of the said two diameters or sides: multiply the remain-
der by a third of the height; and this last product by 785398
for circles, or by the proper multiplier for polygons; the last
product will be the capacity.
$
That is, capacity [(1+d) - Dd] } m h.
4. Sphere. 1. Surface-diameter squared × 3·141593.
2. Capacity-diameter cubed x 5236
or circumference cubed × 016887.
5. Spheric segment. 1. Surface=circumf. sphere × height
of segment."
diam.
2. Capacity •5236 h² (3 d 2h): where d
h-height.
2
= ·5236 h² (3 r²+h³): where r-rad. of the
segment's base..
5. Paraboloid. Capacity-half base x height.
This is a figure produced by the rotation of a parabola upon
its axis.
MENSURATION OF SOLIDS.
197
6. Spheroid. This is a solid generated by the revolution of
an ellipse about one of its axes. To find its capacity multiply
the square of the revolving axis by the fixed axis, and that pro-
duct by 5236.
7. Regular or platonic bodies, are comprehended by like,
equal, and regular plane figures, and whose solid angles are all
equal.
There are only five regular solids, viz.
The tetraedron, or regular triangular pyramid, having 4 tri-
angular faces;
;
The hexaedron, or cube, having 6 square faces.
The octaedron, having 8 triangular faces;
The dodacaedron, having 12 pentagonal faces;
The icosaedron, having 20 triangular faces.
PROB. 1. To find either the surface or the solid content of
any of the regular bodies.-Multiply the proper tabular area
or surface (taken from the following table) by the square of the
linear edge of the solid for the superficies. And
Multiply the tabular solidity in the last column of the table
by the cube of the linear edge for the solid content.
I
Surfaces and Solidities of regular Bodies, the side being
unity or 1.

No. of
sides..
Name.
4
Tetraedron
6 Hexaedron
8
Octaedron
12
Dodecaedron
20 Icosaedron
Solidity.
1.7320508 0.1178513
6·0000000
1.0000000
3.4641016
0.4714045
20.6457288
7.6631189
8.6602540
2.1816950
Surface.
2. The diameter of a sphere being given to find the side of
any of the platonic bodies, that may be either inscribed in the
sphere, or circumscribed about the sphere, or that is equal to
the sphere.
Multiply the given diameter of the sphere by the proper or
corresponding number, in the following table, answering to the
thing sought, and the product will be the side of the platonic
body required.
27
$
re
198
MENSURATION OF SOLIDS.
The diam. of a
sphere being 1;
the side of a
f

That may
be That may be cir-That is equal
inscribed in cumscribed about to the sphere,
the sphere, is the sphere, is
15
0.8164966 2.4494897
1.6439480
Tetraedron
Hexaedron 0.5773503 1·0000000 0.8059958
Octaedron 0.7071068 1.2247447 1.0356300
Dodecaedron 0.3568221 0.4490279 0.4088190
Icosaedron 0·5257309
0.6615845
0.6214433
3. The side of any of the five platonic bodies being given, to
find the diameter of a sphere, that may either be inscribed in
that body, or circumscribed about it, or that is equal to it. As
the respective number in the table above, under the title in-
scribed, circumscribed, or equal, is to 1, so is the side of the
given platonic body to the diameter of its inscribed, circum-
scribed, or equal sphere.
4. The side of any one of the five platonic bodies being given
to find the side of the other four bodies, that may be equal in
solidity to that of the given body. As the number under the
title equal in the last column of the table above, against the
given platonic body, is to the number under the same title,
against the body whose side is sought, so is the side of the given
platonic body to the side of the body sought.
Besides these there are thirteen demiregular bodics, called
Solids of Archimedes. They are described in the Supplement
to Lidonne's Tables de Tous les Diviseurs des Nombres, &c.
Paris, 1808; twelve of them were described by Abraham Sharp,
in his Treatise on Polyedra.
SECTION III.-Approximate Rules.
1. When the area of a field is known in square yards, to re-
duce them to acres, instead of dividing by 4840, multiply by
0002, which is much easier.
Thus, suppose the area is found to be 56870 yards.
Then.
56870
•0002
Take of this,
Or of o
30
The sum is .
viz.
11.3740
3791
11.7531 acres: the true answer
[is 11.75 acres.
}
APPROXIMATE RULES IN MENSURATION.
199
ļ
area of trigon
1
2. For regular Polygons.—Let s be the side, then
= }} s²+7% s²
To
pentagon s²+18 s²
hexagon =2 s² + 10 s³
heptagon of square of 11s
10
6
I
octagon =10 of
I
nonagon
decagon
undecagon
dodecagon
3
100
1809
=
2
6 s²+² s²
10
=7 s² + =7/0 s³
square
4
100
1
2
11 s² + s².
5
TO
S =320 ..
From the sum
Take 3 s²
2
100
Area of the nonagon
8
100
32..
9 s²+1 of 9 s²
Sa
of 7 s
—
From of ditto 1960
IT
Take
7 s
5
28
2
100
2480
Where a table of polygons is at hand, it is best to employ it.
In other cases, one or other of these approximations may occur
to a well exercised memory.
Ex. 1. The side of a pentagon is 20. Required the area.
s² 400
S
s³
7 s
תי
From their sum = 720
10 of 180 83
100
Take 18 s³
Area required = 688: the true area is 688.19.
Ex. 2. The side of an octagon is 20. Required the area.
Square of 7 s=19600-square of 140
. 8 times s³
10
Area of the octagon-1932: the true area is 1931.37
Ex. 3. The side of a nonagon is 20. Required the area.
s³=400, 6 s²
Add s²
2
2400
80
8 = offs.
i's
2472: the true area is 2472-72.
↓
200
APPROXIMATE RULES IN ARITHMETIC.
1
Ex. 4. The side of a dodecagon is 20. Required the area.
$2
400, 11 s²
4400
80
Add s³
1
Area of the dodecagon
4480 the true area is 4478'46.
3. Length of circular arc. From 8 times the chord of half
the arc, subtract the chord of the whole arc, and of the re-
mainder will be the length of the arc, nearly.
Note. The chord of half the arc is equal to the square
root of the sum of the squares of the versed sine or height, and
half the chord of the entire arc.
Or, apply a fine flexible string to the arc, then stretch it out
straight, and measure it.
4. Arc of a quadrant nearly equal 2 time the chord of the
quadrant.
This is true within about the 4000th part.
5. Periphery of an ellipse.--Multiply the square root of
half the sum of the squares of the two axes by 3.141593, and
the product will be the periphery, ncarly.
Diminish this by its 200th part, and the result will be still
more correct.
2 1
6. Area of a circle nearly to of the square of the dia-
meter. Or, multiply D2 by 11, and divide by 2 and by 7.
This is true within the 2500th part; or, add to D² of
7 D³; the sum will be the area true to within the 5000th
part.
100
38
Area of a circle nearly to of the square of the circum-
ference. Or, multiply c" by 7, and divide by 8 and 11.
True within the 2600th part.
7. Circular segment.-To the chord of the whole arc add
of the chord of half the arc; multiply the sum by the versed
sine or height of the segment; and of the product will be
the area nearly.
10
1
Or, (c+3 √ c²+v²), v = arca.
8. To find the content of irregular plane figures from
an accurate plan.
If the plan be not upon paper, or fine drawing pasteboard
of uniform texture, let it be transferred upon such. Then cut
out the figure separately close upon its boundaries and cut out
from the same paper a square of known dimensions according
to the scale employed in drawing the plan. Weigh the two
separately in an accurate balance, and the ratio of the weights
will be the same as that of the superficial contents.
APPROXIMATE RULES: MENSURATION.
201
If great accuracy be required, cut the plan into 4 portions,
called 1, 2, 3, 4. First, weigh 1 and 2 together, 3 and 4 to-
gether, and take their sum. Then weigh 1 and 3 together, 2
and 4 together, and take their sum. Lastly, weigh 1 and 4 to-
gether, 2 and 3 together, and take their sum. The mean of the
four aggregate weights thus obtained, compared with the weight
of the standard square, will give the ratio of their surfaces very
nearly.
1
*** I have employed in this operation a balance which turns
with the 100th part of a grain. The results are proportionally
accurate.
9. Prism.-L= length, B-breadth, D=depth, all in inches:
9
then 1000 L B D = content in yards, nearly.
If L, B, D, be in feet, as suppose the dimensions of a corn bin;
then '8 L B D = the content in Winchester bushels. This is
about one bushel in 200 in defect.
Ex. Suppose L= 125 inches, B
Then 125 × 25—12500—3125;
and 3125 × 24=6250 × 12=75000
14000 of 75000
9
2) 675000
7000) 337500
7
144
25, d = 24.
4.8214 yards the true answer is 4.8225.
For wrought iron square bars allow 100 inches in length of
an inch square bar to a quarter of a cut. in weight; and so in
proportion. This is easily remembered, because the word hun-
dred occurs twice.
An inch square cast iron bar would require 9 feet in length
for a quarter of a cwt.
Or, take of the product of the breadth and thickness,
each in eighths of an inch, the result is the weight of one foot
in length, in avoirdupois pounds.
Or, one foot in length of an inch square bar weighs 33
pounds.
Bricks of the usual size require 384 to a cubic yard. A rod
of brick work, brick and a half thick, requires 4356 bricks.
10. Cylinder. One tenth diameter squared, (d, d being
taken in inches,) gives the content in ale-gallons of a yard in
length.
202
APPROXIMATE RULES: MENSURATION.
This rule gives a result defective only by the 376th part.
l d² ×·00283257 = imperial gallons in a cylinder, di-
mensions in inches.
11. Timber measuring.—Let L denote the length of a tree
in feet and decimals, and & the mean girth, taken in inches: then
the following rules given by Mr. Andrews may be employed.
Τ
G2
2304
L G2
3009
cubic feet, customary, and
cubic feet, true content.
Rule 1.-No allowance for bark.
L G2
1807
Rule 2.-To allow th for bark.
L a2
cubic feet, customary, and = cubic feet, true content.
2360
Rule 3.-To allow th for bark.
L 02
2231
Rule 4.-To allow 1th for bark.
1 2
L G2
1 0
cubic feet, customary, and
cubic feet, true content.
cubic feet, customary, and
cubic feet, true content.
2150
EXAMPLE by Rule 1.-No allowance for bark.
A tree 40 feet long, and 60 inches whole girth or circumference.
40 × 602
1807
L G2
2845
L G2
2742
40 × 602
2301
62 cubic ft. customary, and
793 cubic feet, true content.
long, and 19 inches circumference.
50 X 492
503 cubic feet, truc content.
2360
Ex. by Rule 2.-A trco feet 50
50 × 492
40 cubic ft. customary, and
3009
If G as well as L be in fect, then '08 L &² content, nearly.
** When the difference between the girths at the two ends
*
is considerable, it is best to find the content of the tree as though
it were a conic frustrum, and make the usual allowances after-
wards.
21
12. Sphere of the cube of the diameter
11 =
the component factors, 3 and 7, in dividing by 21.
This rule gives a result true to its 2600th part.
Or, multiply the cube of the circumference by 0169, for the
capacity.
13. To find the capacity, or solid content of an irregular
body.-Procure a prismatic or cylindric vessel that will hold it.
Put in the body, and then pour in water to cover it, marking
Then take out the
the height to which the water reaches,
body, and observe accurately how much the water has de-
scended in consequence. The capacity of the prism or cylinder
thus left dry by the water will be evidently equal to that of the
body.
If the vessel will not hold water, sand may be employed,
though not with quite so much accuracy.
capacity. Use
APPROXIMATE RULES: MENSURATION.
203
1
In this manner, too, a portion of a body may be measured
without detaching it from the rest, by simply immersing that
portion.
Even an irregular vessel may be employed for this purpose.
In which case it should be placed in a larger vessel, and then
filled with water. Then submerge the body whose magnitude
you wish to determine, the quantity of water that has run over,
and is caught in the exterior vessel, will be the measure. It
may be weighed, and its cubic measure estimated by allowing
1000 avoirdupois ounces to the cubic foot. (See farther, Hy-
drostatics and Specific Gravity.)
KMO
14. To find the contents of surfaces and solids not re-
ducible to any known figure, by the equidistant ordinate
method. The general rule is included in this proposition:
viz.-If any right line A N be divided into any even number of
equal parts, a c, C E, E C, &c. and at the points of division be
erected perpendicular ordinates A B, C D, E F, &c. terminated
by any curve BHO: then, if a be put
for the sum of the first and last ordi-
nates, A B, N 0, B for the sum of the
even ordinates, C D, G H, L M, &c. viz.
the second, fourth, sixth, &c. and c for
the sum of all the rest, E F, I K,
I K, &c.
viz. the third, fifth, &c. or the odd
ordinates, excepting the first and last then, the common dis-
tance A c, c E, &c. of the ordinates being multiplied into the
sum arising from the addition of A, four times B, and twice
c, one third of the product will be the area A B o N, very
nearly.
A CE GILN
That is,
A+4 B+ 2 c
3
B
D F
}
H

D= area, D being =A C=C E, &c.
The same theorem will equally serve for the contents of all
solids, by using the sections perpendicular to the axis instead
of the ordinates. The proposition is quite accurate, for all
parabolic and right lined arcas, as well as for all solids gene-
rated by the revolutions of conic sections or right lines about
axes, and for pyramids and their frustrums. For other areas
and solidities it is an excellent approximation.
The greater the number of ordinates, or of sections, that
are taken, the more accurately will the area or the capa-
city be determined. But in a great majority of cases five
equidistant ordinates, or sections, will lead to a very accurate
result.
In cask-guaging, indeed, three sections will be usually
sufficient. Thus, taking the bung and head diameters, and
204
a diameter mid-way between them; the sum of the squares of
the bung and head diameters, and of the square of double the
middle diameter, multiplied into the length of the cask, and
then into 785398, will give six times the content of the cask,
very nearly. Or, if H, B, and M, represent the head, bung
and middle diameters respectively, and L the length, all in
inches; then (H³ + 4 M² + B³) × L × ·1309=content of the
cask in inches.
APPROXIMATE RULES: MENSURATION.
1
A similar method may be advantageously adopted in all
cases of ullaging either standing or lying casks, by taking the
areas at the top, bottom, and middle, of the liquor. See Hut-
ton's Mensuration, part iv.
EXAMPLE. The bung diameter of a cask is 32 inches, the
head diameter 24, the middle 30.2, the length 40. Required
the content in ale gallons of 282 cubic inches, and in wine gal-
lons of 231.
The former multiplier, divided by 282 and 231 respectively,
gives ·00047 and 00052, for the proper multipliers.
Hence (322+24+4. 30.2) × 40 x 00047-97·44 ale gal-
lons :
And (32²+24²+4. 30.29) x 40 x 00052-118.95 wine gal-
lons:
Or 1 of 97.44 (the ale gallons) gives 119, the wine gallons,
very nearly.
Also
of 118.95 (the wine gallons)=99·125 imperial gal-
lons. See the table of factors, p. 19.
!
+
MECHANICS: STATICS.
205
1
WHAT
7
MECHANICS.
1. Mechanics is the science of equilibrium and of motion.
2. Every cause which tends to move a body, or to stop it
when in motion, or to change the direction of its motion, is
called a force or power.
3. When the forces that act upon a body, destroy or annihi-
late each other's operation, so that the body remains quiescent,
there is said to be an equilibrium.
4. Statics has for its object the equilibrium of forces applied
to solid bodies.
28
5. Dynamics relates to the circumstances of the motion of
solid bodies.
6. Hydrostatics is devoted to the equilibrium of fluids.
7. Hydrodynamics relates to their motion, and connected
circumstances.
8. The properties and operation of elastic fluids are often
treated distinctly, under the head of Pneumatics.
9. A single force, which would give to a physical point, or
to a body, the same motion both in velocity and direction as
several forces acting simultaneously, is called the resultant of
those forces, while they are called the constituents or the com-
posants of the single resulting force.
10. The action of a force is the same in which ever point of
its direction it is applied; unless the manner of its action be
changed.
11. Vis inertiæ, or power of inactivity, is defined by New-
ton to be a power implanted in all matter, by which it
resists any change attempted to be made in its state, that is,
by which it requires force to alter its state, either of rest or
motion.
T
}
206
MECHANICS STATICS.
3
{
1
1
CHAPTER IX.
STATICS.
SECTION I.-Parallelogram of Forces.
1
1. The resultant of any two forces whatever, which act
upon a physical point, and which are represented by lines
taken in their respective directions from that point, has for its
magnitude and direction the diagonal of the parallelogram con-
structed upon those forces.
2. Two forces and their resultant may each be represented
by the sine of the angle formed by the directions of the two
others.
3. If F, f, represent two forces, and A the angle made by
their respective directions; also, if R be the resultant, and a
the angle which it makes with the direction of the force F:
then
r=√ (p² + ƒº ± 2 r ƒ cos a) . . . . . (1)
R
tan a =
(2)
4. These propositions are of extensive, nay, of almost univer-
sal application, in the constructions of architects, mechanics,
and civil engineers. Two, simple examples may here suffice:
others will occur as we proceed.
f sin a
F+ƒ cos a
Ex. 1. Suppose that a weight в is at-
tached by a stirrup to the foot of a king-
post a в, which is attached to two rafters
A C, AD, in the respective positions shown
in the marginal diagram. Then if AE be
set off upon a в, in numerical value equal
to the vertical strain upon A в, and the
parallelogram A FE & be completed, a r
measured upon the same scale will show
the strain upon the rafter a c, and a & the strain upon a D.
E
Α

B
G
{
}
MECHANICS STATICS.
207
}
F
•
Ex. 2. Let it be proposed to compare the strains upon the
tie-beams ▲ D, and the struts A c, when they sustain equal
weights B, in the two different posi-
tions indicated by the figures. Let
A E in one figure, be equal to the
corresponding vertical line A E in
the other, and in each represent the
numerical value of the weight в,
that hangs from A. Through E in
both figures, draw lines parallel to
DA, A C, respectively, and let them
meet a C, and D A produced in F and
G: then A FE G in each figure is the
parallelogram of forces by which
the several strains are to be measur-
ed. A G represents the tension upon
the tie-beam A D, and A F the strain
upon the strut A c. Both these lines
are evidently shorter in the lower
figure than they are in the upper,
1
A E being of the same length in both therefore the first figure
exhibits the most disadvantageous position of the beams. It is
evident also, that while c A tends upwards and D A downwards,
the greater the angle D A C, the less is its supplement c A G, and
the less the sides r A, A G, of the parallelogram.

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The truth of the general proposition relative to the parallel-
ogram of forces admits of obvious experimental confirma-
tion by means of two spring steelyards. Let any known
weight, as 10, 15, &c. pounds, represented by в in the figure
to Ex. 1, hang from a cord E B, and from a knot at E let
two other cords, E F, E &, proceed, and hang from spring
steelyards at F and & then it will be found universally that
the weights sustained by those steelyards, will be to the
weight в hanging vertically, as the respective sides E F and
E G of the parallelogram to its diagonal E A.
It will,
hence, be easy to exhibit those relations in all desirable va-
rieties.
5. Three forces not situated in the same plane, but applied
to the same point, have a resultant represented in magnitude
and direction by the diagonal of the parallelopiped con-
structed upon the parts of the directions of those forces which
represent their respective magnitudes, and drawn from their
point of application.
6. We may always decompose or resolve a force into three
others, parallel respectively to three given lines. Each com-
posant may be found by multiplying the force which we
208
MECHANICS: STATICS.
would decompose by the cosine of the angle which its direction
makes with the axis to which that composant is parallel.
7. If any number of forces be kept in equilibrio by their
mutual actions; they may all be reduced to two equal and
opposite ones. For, any two of the forces may be reduced
to one force acting in the same plane; then this last force
and another may likewise be reduced to another force acting
in their plane and so on, till at last they be all reduced to the
action of only two opposite forces; which will be equal, as
well as opposite, because the whole are in equilibrio by the sup-
position.
8. PROP. To find the resultant of several forces concurring in
one point, and acting in one planc.
1st. Graphically. Let, for example, four forces, A, B, C, D,
act upon the point r, in magnitudes and directions represented
by the lines P ^, P B, P C, P D.
;
From the point A draw a b parallel
and equal to Pв; from b draw b c paral-
lel and equal to P C from c draw c d pa-
rallel and equal to P D; and so on, till all
the forces have thus been brought into
the construction. Then join e d, which
will represent both the magnitude and the direction of the re-
quired resultant.
Υ
+
A
∞
b
ᎠᏰ
d

This is, in effect, the same thing as finding the resultant of
two of the forces A and B ; then blending that resultant with a
third force c; their resultant with a fourth force p; and so on.
A careful construction upon a well chosen scale will give a re-
sultant true to within its 250th part.
2d. By compulation. Drawing the lines A a, a b', &c. re-
spectively parallel and perpendicular to the last force P D; we
have
б
d s = s a + bb' + cc'A sin A PD + B sin BP D + c sin c P D
Pd=Pataẞ+By+yd=^ COS APD+B COS B P D+C cos CP D+D
ds
tan d r8=
PS
℗ d= ✅✓ (p 8² +d 8²)=r 8 sec d r 8.
The numerical computation is best effected by means of a table
of natural sines.
9. The resultant of two parallel forces acting in the same way,
is equal to their sum; and the distances of the direction of that
resultant from those of the composants are reciprocally propor-
tional to those composants.
10. The resultant of several parallel forces, whether confined
to one plane or not, is equal to the sum of those forces, giving
to the forces which act in one sense the sign +, and to those
which act contrarily the sign.
STATICS: CENTRE OF GRAVITY.
209
SECTION II.-Centre of Gravity.
1. Gravity is the force in virtue of which bodies left to them-
selves fall to the earth in directions perpendicular to its surface.
2. We may distinguish between the effect of gravity and
that of weight, by observing that the former is the power of
transmitting to every particle of matter a certain velocity
which is absolutely independent on the number of material
particles; while the latter is the effort which must be exer-
cised to prevent a given mass from obeying the law of gravity.
Weight, therefore, depends upon the mass; but gravity has
no dependence at all upon it.
3. The centre of gravity of any body or system of bodies is
that point about which the body or system, acted upon only
by the force of gravity, will balance itself in all positions: or
it is a point which, when supported, the body or system will
be supported, however it may be situated in other respects.
The centre of gravity of a body is not always within the
body itself: thus the centre of gravity of a ring is not in the
substance of the ring, but in the axis of its circumscribing cy-
linder; and the centre of gravity of a hollow staff, or of a bone,
is not in the matter of which it is constituted, but somewhere
in its imaginary axis; every body, however, has a centre of
gravity, and so has every system of bodies.
4. Varying the position of the body will not cause any
change in the centre of gravity; since any such mutation will
be nothing more than changing the directions of the forces,
without their ceasing to be parallel; and if the forces do not
continue the same, in consequence of the body being supposed
at different distances from the earth, still the forces upon all
the moleculæ vary proportionally, and their centre remains
unchanged.
5. When a heavy body is suspended by any other point.
than its centre of gravity, it will not rest unless that centre is
in the same vertical line with the point of suspension for in
all other positions the force which is intended to ensure the
equilibrium will not be directly opposite to the resultant of
the parallel forces of gravity upon the several particles of the
body, and of course the equilibrium will not be obtained.
(See art. 9, on Pendulums.)
6. If a heavy body be sustained by two or more forces, their
directions must meet, either at the centre of gravity of that
body, or in the vertical line which passes through it.
7. When a body stands upon a plane, if a vertical line
passing through the centre of gravity fall within the base on
which the body stands, it will not fall over; but if that verti-
C
T 2
210
STATICS CENTRE OF GRAVITY.
I
cal line passes without the base, the body will fall, unless it be
prevented by a prop or a cord. When the vertical line falls
upon the extremity of the base, the body may stand, but the
equilibrium may be disturbed by a very trifling force; and
the nearer this line passes to any edge of the base the more
easily may the body be thrown over; the nearer it falls to the
middle of the base, the more firmly the body stands.
Upon this principle it is that leaning towers have been built
at Pisa, and various other places; the vertical line of direction
from the centre of gra-
vity falling within the
base. And, from the
same principle it may be
seen that a wagon loaded
with heavy materials, as
B, may stand with perfect.
safety, on the side of a
WE O
B
1
convex road, the vertical line from the centre of gravity fall- ·
ing between the wheels; while a wagon a with a high load,
as of hay, or of wool-packs, shall fall over, because the vertical
line of direction falls without the wheels.
8. To find the centre of gravity mechanically, it is only
requisite to dispose the body successively, in two positions of
equilibrium, by the aid of two forces in vertical directions, ap-
plied in succession to two different points of the body; the point
of intersection of these two directions will show the centre.
This may be exemplified by particularizing a few methods.
If the body have plane sides, as a piece of
board, hang it up by any point, then a plumb-
line suspended from the same point will pass
through the centre of gravity; therefore
mark that line upon it: and after suspending
the body by another point, apply the plum-
met to find another such line; then will their
intersection show the centre of gravity.
Or thus hang the body by two strings from
the same point fixed to different parts of the
body; then a plummet hung from the same
tack will fall on the centre of gravity.
Another method: Lay the body on the edge
of a triangular prism, or such like, moving it to
and fro till the parts on both sides are in equi-
librio, and mark a line upon it close by the
edge of the prism: balance it again in another
position, and mark the fresh line by the edge
of the prism; the vertical line passing through
C
}




STATICS: CENTRE OF GRAVITY.
211
the intersection of these lines, will likewise pass through the
centre of gravity. The same thing may be effected by laying
the body on a table, till it is just ready to fall off, and then
marking a line upon it by the edge of the table: this done in
two positions of the body will in like manner point out the
position of the centre of gravity.
When it is proposed to find the centre of gravity of the
arch of a bridge, or any other structure, let it be laid down
accurately to scale upon pasteboard; and the figure being
carefully cut out, its centre of gravity may be ascertained by
the preceding process.
9. If on any plane passing through the centre of gravity of
a body, perpendiculars be let fall from each of its moleculæ,
the sum of all the perpendicular distances on one side of the
plane will be equal to the sum of all those on the other side.
And a similar property obtains with regard to the common
centre of gravity of a system of bodies.
10. The position, distance, and motion of the centre of gra-
vity of any body is a medium of the positions, distances, and
motions of all the particles in the body.
11. The common centre of gravity, or of position, of two
bodies divides the right line drawn between the respective
centres of the two bodies in the inverse ratio of their masses.
12. The centre of gravity of three or more bodies may,
hence, be found, by considering the first and second as a single
body cqual to their sum and placed in their common centre of
gravity, determining the centre of gravity of this imaginary
body, and a third. These three again being conceived united
at their common centre, we may proceed, in like manner, to a
fourth; and so on, ad libitum.
Or, if в, B', B", &c. denote the masses of any bodies,
D, D', D', &c. the perpendicular distances of their respective
centres of gravity from any line or plane: then, the distance,
▲, of their common centre of gravity from any line or plane,
B D+B' D'+B" D" &C.
is found by this theorem: viz. A
B+ B+ B" &c.
13. If the particles or bodies of any system be moving
uniformly and rectilineally, with any velocities and directions
whatever, the centre of gravity is cither at rest, or moves
uniformly in a right line.
Hence if a rotatory motion be given to a body and it be
then left to move freely, the axis of rotation will pass through
the centre of gravity: for, that centre, either remaining at rest
or moving uniformly forward in a right line, has no rotation.
Here too it may be remarked, that a force applied at the
centre of gravity of a body, cannot produce a rotatory mo-
tion.
I
212
STATICS: CENTRE OF GRAVITY.
14. The centre of gravity of a right line, or of a parallelo-
gram, prism, or cylinder, is in its middle point; as is also
that of a circle, or of its circumference, or of a sphere, or of
a regular polygon; the centre of gravity of a triangle is some-
where in a line drawn from an angle to the middle of the
opposite side; that of an ellipse, a parabola, a cone, a conoid,
a spheroid, &c. somewhere in its axis. And the same of all
symmetrical figures.
15. The centre of gravity of a triangle is the point of in-
tersection of lines drawn from the three angles to the middles
of the sides respectively opposite: it divides each of those
lines into two portions in the ratio of 2 to 1.
16. In a Trapezium. Divide the figure into
two triangles by the diagonal A c, and find the
centres of gravity E and F of these triangles;
join E F, and find the common centre & of
these two by this proportion, A B C : A D C :: F G :
E G, Or A B C D A D C E F E G. Or, divide
the figure into two triangles by a diagonal в D.
find their centres of gravity; the line which joins them will
intersect E F in &, the centre of gravity of the trapezium.

A
E
B
F
17. In like manner, for any other plane figure, whatever
be the number of sides, divide it into several triangles, and
find the centre of gravity of each; then connect two centres
together, and find their common centre as above; then con-
nect this and the centre of a third, and find the common
centre of these; and so on, always connecting the last found
common centre to another centre, till the whole are included
in this process; so shall the last common centre be that which
is required.
18. The centre of gravity of a circular arc is distant from
the centre a fourth proportional to the arc, the radius, and the
chord of the arc.
19. In a circular sector, the distance from the centre of
the circle is ; where a denotes the arc, c, its chord, and
r the radius.
2 cr
3 a
20. The centres of gravity of the surface of a cylinder, of
a cone, and of a conic frustrum, are respectively at the same
distances from the origin as are the centres of gravity, of the
parallelogram, triangle, and trapezoid, which are vertical sec-
tions of the respective solids.
21. The centre of gravity of the surface of a spheric seg-
ment is at the middle of its versed sine or height.
22. The centre of gravity of the convex surface of a sphe-
rical zone, is in the middle of that portion of the axis of the
sphere which is intercepted by the two bases of the zone.
STATICS: CENTRE OF GRAVITY.
213
23. In a cone, as well as any other pyramid, the distance
from the vertex is 2 of the axis.
24. In a conic frustum, the distance on the axis from the
2
3 R² + 2 Rr + p²:
R² + Rr + jo2
height, R and r the radii of the greater and less ends.
centre of the less end, ish.
where h the
2
25. The same theorem will serve for the frustum of any
regular pyramid, taking R and r for the sides of the two ends.
26. In the paraboloid, the distance from the vertex is axis.
27. In the frustum of the paraboloid, the distance on the
2 R³ +2³:
R2+23
axis from the centre of the less end, ish.
where
h the height, R and r the radii of the greater and less ends.
Many other results are given in the first volume of my
Mechanics. The preceding are selected as the most useful.
The centre of gravity of the human body is always near
the same place, viz. in the pelvis, between the hips, the ossa
pubis, and the low-
er part of the back-
bone. Elevating the
arms or the legs will
elevate the centre of
gravity a little: still,
it is always so placed
that the limbs may
move freely round it,
and this centre moves
much less than if it
were in any other
part of the body. If
a man walked upon
wooden legs,the cen-
tre of gravity of his
body would describe
portions of circles, as

A
JUDEA
B
D
F
A B. If a man with two wooden legs were to run, the centre
of gravity would describe portions of parabolas, as c D. But
the flexibility of the joints and muscles of the human legs
serves to take away the angles from these curves, and give a
softer undulation, as E F.
The centre of gravity of a human body, is not precisely in
the same place as that of the statue of a man: for, in the for-
mer, the substance is not, throughout, of the same density, in the
latter it is.
k
29
214
STATICS: MECHANICAL POWERS.
SECTION III.-Mechanical Powers.
1. The simple machines of which the more complex ma-
chines are constituted, and which, indeed, are often employed
separately, are called Mechanical Powers.
:
2. Of these we usually reckon six: viz. the lever, the wheel
and axle, the pulley, the inclined plane, the wedge, and the
screw. To these, however, is sometimes added the funicular
machine, being that which is formed by the action of several
powers, at different points of a flexible cord.
3. Weight and Power, when regarded as opposed to each
other, signify the body to be moved, or the resistance to be
overcome, and the body of force by which that is accomplished.
They are usually represented by their initial letters, w and P.
Levers.
1. A lever is an inflexible bar, whether straight or bent,
and supposed capable of turning upon a fixed, unyielding point
called a fulcrum.
R
2. When the fulcrum is between the
power and the weight, the lever is said
to be of the first kind.
When the weight is between the
power and the prop, the lever is of the
second kind.
When the power is between the weight
and the prop, or fulcrum, the lever is of
the third kind.
The hammer lever, or the operation.
of a hammer in drawing a nail, is some-
times considered as a fourth kind.
A
K
#
F
B
W
F

W
A
B
3. In all these cases when there is an equilibrium, it is
indicated by this general property, that the product of the
weight into the distance at which it acts, is equal to the pro-
duct of the power into the distance at which it acts: the dis-
tances being estimated in directions perpendicular to those
in which the weight and power act respectively. Thus, in each
of the three preceding figures,
P. ATW
BF,
or the power and weight are reciprocally as the distances at
which they act.
And if, in the first figure, for example, the arm a F were
STATICS: MECHANICAL POWERS.
215
1
4 times F B, 4 lbs. hanging at в would be balanced by 1 lb. at
A. If a r were 5 times F B, 1 lb. at a would balance 5 lbs. at
B; and so on.
4. If several weights hang upon a lever, some on one side
of the fulcrum, some on the other, then there will be an equi-
librium, when the sum of the products of the weights into
their respective distances on one side, is equal to the several
products of weights and distances on the other side.
5. When the weight of the lever is to be taken into the
account, proceed just as though it were a separate weight sus-
pended at its centre of gravity.
6. When two, three, or more levers act one upon another
in succession, then the entire mechanical advantage which
they supply, is found by taking, not the sum, but the pro-
duct of their separate advantages. Thus, if the arms of three
levers, acting thus in connexion, are as 3 to 1, 4 to 1, and 5
to 1, then the joint advantage is that of 3×4×5 to 1, or
60 to 1 so that 1 lb. would, through their intervention,
balance 60.
7. In the first kind of lever the pressure upon the fulcrum
P+w: in the other two it is P W.
8. Upon the foregoing principles depends the nature of
scales and beams for weighing all bodies. For, if the dis-
tances be equal, then will the weights be equal also; which
gives the construction of the common scales. And the Ro-
man statera, or steel-yard, is also a lever, but of unequal
arms or distances, so contrived that one weight only may serve
to weigh a great many, by sliding it backwards and forwards
to different distances upon the longer arm of the lever. In
the common balance, or scales, if the weight of an article when
ascertained in one scale is not the same as its weight in the
other, the square root of the product of those two weights
will give the true weight.
9. From numerous examples of the power and use of the
lever, one which shows its manner of application in the print-
ing-presses of the late Earl Stanhope may be advantageously
introduced.

In the adjoining figure, let
A B C D be the general frame of the
press, connected by the cross
pieces N o, D C. E is a centre con-
nected with the frame by the bars
EN, ER, E 0. To this centre are
affixed a bar к D, and a lever E F,
to which the hand is applied
when the press is used.
A
N
G
D
-
Ti.
R
KE
L
M
B
H
F
216
STATICS: MECHANICAL POWERS.
There are also several other pieces connected by joints at
N, G, I, K, L, M, O, H, which are so adjusted to each other that
when the hand is applied to the lever E F at r, by pressing it
downwards K L is brought into a horizontal line or parallel to
G H or D C, in which situation N I G, O M H, also form each a
straight line. It is evident that the nearer these different pieces,
as above mentioned, are to a straight line the greater is the
lever E F, in proportion to the perpendicular K s at the other
end of the lever E K, formed by a perpendicular from K falling
on F E produced. Consequently a small force applied at r will
be sufficient to produce a very great effect at K, when I K, K E
are nearly in a straight line, and so on, for the other pieces
above mentioned.
Hence the force applied by hand at F must be very con-
siderable in forcing down & n, which slides on iron cylindrical
bars, or in pressing any substance placed in the aperture P Q,
between the bar or plate and the frame D c.
This contrivance is now often introduced into mechanism,
under the name of the toggle, or of the knee joint.
Wheel and Axle.
1. The nature of this machine is suggested
by its name. To it may be referred all
turning or wheel-machines of different radii ;
as well-rollers and handles, cranes, capstans,
windlasses, &c.
2. The mechanical property is the same as in
the lever viz. P . A C=W. BC: and the rea-
son is evident, because the wheel and axle is
only a kind of perpetual lever.
E
W
A
с
A
P
B
C
A
CHLOE JE J
3. When a series of wheels
and axles act upon cach other,
so as to transmit and accumu-
late a mechanical advantage,
whether the communication be
by means of cords and belts, or
of teeth and pinions, the weight will be to the power, not
as the sum, but as the continual product of the radii of
the wheels to the continual product of the radii of the axles.
Thus, if the radii of the axles, a, b, c, d, e, be each 3 inches,
while the radii of the wheels, A, B, C, D, E, he 9, 6, 9, 10,
and
12 inches respectively then w: P:: 9 × 6 × 9 × 10 × 12 :
B
B
W
1

1
STATICS: MECHANICAL POWERS.
217
3×3×3×3×3 :: 240: 1. A computation, however, in which
the effect of friction is disregarded.
4. A train of wheels and pinions may also serve for the aug-
mentation of velocities. Thus, in the preceding example,
whatever motion be given to the circumference of the axle e,
the rim of the wheel A will move 240 times as fast.
And if a series of 6 wheels and axles, each having their dia-
meters in the ratio of 10 to 1, were employed to accumulate
velocity, the produced would be to the producing velocity, as
10° to 1, that is, as 1000000 to 1.
Pulley.
1. A pulley is a small wheel, commonly made of wood or
brass, which turns about an iron axis passing through the
centre, and fixed in a block, by means of a cord passed round
its circumference, which serves to draw up any weight. The
pulley is either single, or combined together, to increase the
power. It is also either fixed or moveable, according as it
is fixed to one place, or moves up and down with the weight
and power.
2. If a power sustain a weight by means of a fixed pulley :
the power and weight are equal.

3. When a power sustains a weight by a
system of moveable pulleys, each embraced
by a cord attached on one part to a fixed point,
and on the other to the centre of the pulley
next above it, as in the margin : then if the
cords are parallel to each other, each pulley
gives a mechanical advantage of two to one;
and the whole system and advantage denoted
by that power of 2 which is equal to the num-
ber of pulleys. Here p: w:: 1: 2¹ :: 1 : 16.
Id
W
A/
16.
00
A
16
4. When there are three, four, or any other number of pul-
leys in a fixed block, and an equal number in a moveable block,
capable of ascending and descending, the system is called a
muffle; and the weight is to the power as 1 to twice the num-
ber of pulleys in each block.
.
5. The friction of a system of pulleys, or even of a single
pulley, is very great, according to the common mode of con-
struction. But it may be reduced considerably by means of
U
1
1
218
Mr. Garnett's patent friction rollers, which
produce a great saving of labour and ex-
pense, as well as in the wear of the machine,
both when applied to pulleys and to the
axles of wheel carriages. His general prin-
ciple is this between the axle and nave,
or centre pin and box, a hollow space is left,
to be filled up by solid equal rollers nearly
touching each other. These are furnished
with axles inserted into a circular ring at each end, by which
their relative distances are preserved; and they are kept paral-
lel by means of wires fastened to the rings between the rollers,
and which are rivetted to them. The above contrivance is ex-
hibited in the annexed figure.
I
6. A useful combination of the wheel and axle, a fixed and
a moveable pulley, is exhibited in the marginal diagram.
The load, as of stones or
bricks to build a wall, is
raised from L to A, thus: a
rope B P L is fixed at one end
to a hook в, and passes over
a pulley at P.
That pulley,
P, is drawn along horizontal-
ly from P to F by means of a
man who turns the handle H,
and thus winds up the cord
PF H upon the roller. As the distance P в lengthens, the por-
tion P L shortens ; and the length of rope is so adjusted, that
when the pulley P is brought to be above A, the basket L has
reached that place.
STATICS MECHANICAL POWERS.

**
H
F
Inclined Plane.
P
L
B
SE

1. A body which touches a plane only in one point, can only
remain in equilibrio so long as the forces which act upon it are
reducible to a single force which shall act in a direction perpen-
dicular to the plane at the point of contact.
2. When a power sustains a heavy body in equilibrio upon
an inclined plane, then the power, the weight, and the pressure
upon the plane, will be respectively, as the sine of the plane's
inclination, the cosine of the angle which the direction of the
power makes with the plane, and the cosine of the angle which
the direction of the power makes with the horizon.
STATICS: MECHANICAL POWERS.
219
3. When the direction of the power is parallel to the plane,
the power, weight, and pressure on the plane, are respectively
as the height, length, and base of the plane; or as the sine of
inclination, radius, and cosine of inclination.
Thus, suppose a plank 12 feet long were laid aslant from the
ground to a window 4 feet high: then, since the length is three
times the height, a power of 50 lbs. would sustain three times as
much, or 150 lbs. upon the plank; and a greater power, as of
55 or 60 lbs. would cause that weight to ascend.
The truth of these propositions may be confirmed most
readily, by attaching the weight to a chord tied to a spring steel-
yard, by which the relations between the entire weight, and
that supported by a chord either parallel to the inclined plane
or in any other direction, may at once be measured.
4. If two weights w, w', sustain each other upon two in-
clined planes AC, CB, which have a common altitude c D, by means
of a cord which runs freely over a pulley and is parallel to both
planes, then will w: w' :: A c : C B.
5. When a heavy body is support-
ed by two planes, as in the marginal
figure, then, if the weight of the body
be represented by the sine of the angle
comprehended between the two planes,
the pressures upon them are recipro-
cally as the sines of the inclinations of
those plane to the horizon: viz.

The weight
the pressure on A B
the pressure on B D
are
as
Η
B
sin A B D
sin D B I
sin A B H
I
Thus, suppose the angle A в H was 30°, D B I 60°, and con-
sequently A B D 90°: since the natural sines of 90°, 60°, and
30°, are 1, 866, and respectively, or nearly as 100, 86·6, and
50; if the heavy body weigh 100lbs. the pressure upon A B
would be 86.6 lbs. and upon D d 50 lbs.
This proposition is of very extensive utility, comprehending
the pressure of arches on their piers, of buttresses against walls,
or upon the ground, &c. because the circumstance of one of the
planes becoming either horizontal, or vertical, will not affect
the general relation above exhibited.
Wedge.
1. A wedge is a triangular prism, or a solid conceived to be
220
STATICS: MECHANICAL POWERS.
generated by the motion of a plane triangle paral-
lel to itself upon a straight line which passes
through one of its angular points. The wedge is
called isosceles, rectangular, or scalene, accord-
ing as the generating triangle is isosceles, right-
angled, or scalene. It is very frequently used in
cleaving wood, as represented in the figure, and often in raising
great weights.
2. When a resisting body is sustained against the face of a
wedge, by a force acting at right angles to its direction; in the
case of equilibrium, the power is to the resistance as the sine of
the semi-angle of the wedge, to the sine of the angle which the
direction of the resistance makes with the face of the wedge;
and the sustaining force will be as the cosine of the latter
angle.
3. When the resistance is made against the face of a wedge
by a body which is not sustained, but will adhere to the place
to which it is applied without sliding, the power is to the re-
sistance, in the case of equilibrium, as the cosine of the differ-
ence between the semi-angle of the wedge and the angle which
the direction of the resistance makes with the face of the wedge,
to radius.

4. When the resisting body is neither sustained nor adheres
to the point to which it is applied, but slides freely along
the face of the wedge, the power is to the resistance as the
product of the sines of the semi-angle of the wedge and the
angle in which the resistance is inclined to its face to the square
of radius.
Screw.
1. The screw is a spiral thread or groove cut
round a cylinder, and every where making the
same angle with the length of it. So that if the
surface of the cylinder, with this spiral thread
on it, were unfolded and stretched into a plane,
the spiral thread would form a straight inclined
plane, whose length would be to its height, as
B
CCH

C
D
MAT
E
TOPOLI
C
the circumference of the cylinder is to the distance between
two threads of the screw: as is evident by considering that, in
making one round, the spiral rises along the cylinder the dis-
tance between the two threads.
2. The energy of a power applied to turn a screw round,
is to the force with which it presses upward or downward,
setting aside the friction, as the distance between two threads
STATICS MECHANICAL POWERS.
221
is to the circumference where the power is applied: viz. as
circumf. of D C to dist. B I.
3. The endless screw, or perpetual
screw, is one which works in, and turns a
dented wheel D F, without a concave or fe-
male screw; being so called because it may
be turned for ever, without coming to an
end. From the diagram it is evident that
while the screw turns once round, the
wheel only advances the distance of one tooth.

B
4. If the power applied to the lever, or handle of an endless
screw, ▲ B, be to the weight, in a ratio compounded of the pe-
riphery of the axis of the wheel E п, to the periphery described
by the power in turning the handle, and of the revolutions of
the wheel D F to the revolutions of the screw c B, the power
will balance the weight. IIence,
#
5. As the motion of the wheel is very slow, a small power
may raise a very great weight by means of an endless screw.
And therefore the chief use of such a screw is, either where a
great weight is to be raised through a little space; or where
only a slow gentle motion is wanted. For which reason it is
very serviceable in clocks and watches.
The screw is of admirable use in the mechanism of microme-
ters, and in the adjustments of astronomical and other instru-
ments of a refined construction.
TV
6. The mechanical advantage of a compound machine may be
determined by analyzing its parts, finding the mechanical ad-
vantage of each part severally, and then blending or compound-
ing all the ratios. Thus, if m to 1, n to 1, r to 1, and s to 1,
show the separate advantages; then mnrs to 1, will measure
the advantage of the system.
ď
B
7. The marginal representation of a common construction of a
crane to raise heavy loads, will
serve to illustrate this. By human
energy at the handle a, the pinion
bis turned; that gives motion to
the wheel w, round whose axle, c,
a cord is coiled; that cord passes
over the fixed pulley,d, and thence
over the fixed triple block, в, and
the moveable triple block, P', be-
low which the load, L, hangs.
Now, if the radius of the handle
Wa
H

Fla
L
P
be 6 times that of the pinion, the radius of the wheel w 10 times
that of its axle, and a power equivalent to 30 lbs. be exerted
30
U 2
222
STATICS MECHANICAL POWERS.
at a; then, since a triple moveable pulley gives a mechanical
advantage of 6 to 1, we shall have
30 × 6 × 10×6=10800 lbs.
and such would be the load, L, that might be raised by a power
of 30 lbs. applied at a, were it not for the loss occasioned by
friction.*
SECTION IV.-General application of the principles of Statics
to the equilibrium of Structures.
Every structure is exposed to the operation of a system of
forces; so that the examination of its stability involves the ap-
plication of the general conditions of equilibrium.
J
Now, no part of a structure can be dislocated, except it be
either by a progressive, or a rotatory motion. For either this
part is displaced, without changing its form, in which case it is
as a system of invariable form, incapable of receiving any in-
stantaneous motion, which is not either progressive or rotatory;
or else it happens to be displaced, changing at the same time its
form and this, considering the cohesion of tenacity, cannot
take place, without the breaking of that part in its weakest sec-
tion; which generates a progressive motion, if the force acts
perpendicularly to the section; and a rotatory motion, if it acts
obliquely.
•
We shall here consider the most useful cases; indicating by
the word stress, that force which tends to give motion to the
structure; by resistance, that which tends to hinder it.
Equilibrium of Piers.
1. Taking the marginal figure for the
vertical section of a pier, we may rea-
son upon that section instead of the pier
itself, if it be of uniform structure.
Ch
D

A
૧
R
I
G
S
Z
Let & be the place of the centre of
gravity, S R Z the direction in which
the stress acts, meeting x 1, the ver-
tical line through the centre of gra-
vity, in 1. Then, considering the stress as resolvable into
two forces, one P, vertical, the other, q, horizontal; the pier
(regarding it as one body) can only give way either by a
C
X P
B
* We shall insert a selection of useful mechanical contrivances, after wo have
given the principles of dynamics.
STATICS: EQUILIBRIUM OF PIERS.
223
progressive motion from в towards A, or by a rotatory motion
about A.
Q
2. The progressive motion is resisted by friction. If w de-
note the weight of the pier, r the stress estimated vertically,
and q its horizontal effort, then the pressure on the base
= w + p, and its friction = ƒ (w + P), which is the amount of
the resistance to progressive motion. So that to ensure sta-
bility in this respect we must have
f(w + r) > Q
(1)
While, to ensure stability in regard to rotation, we must have
W.AX + P. AE > Q. ES...
.. (2)
3. The second condition may be ascertained by a graphical
process, thus:
•
From the point A, let fall, on the direction of the stress, the
perpendicular a z. Then, s being put for the whole stress,
W. AX > S. A Z.
Or, suppose the two forces м and s to be applied at 1, and
complete the parallelogram, having sides which represent these
forces. Then must the diagonal produced meet the base on the
side of A, towards в, to ensure stability.
4. If, as is very frequently the case, the vertical section of
the pier is a rectangle, and s represent the specific gravity of
the material of which it is constituted; then the condition of
the two kinds of equilibrium will be denoted by these two
equations: viz.
•
A B. S Q
=
Q. ... (3)
(3) ...
f.c B
.. A B² S = 2 Q . . (4)
Example. Suppose a rectangular wall 39-4 feet high, and of
a material whose specific gravity is 2000, is to sustain a hori-
zontal strain of 9900 lbs. avoirdupois at its summit on the unit
of length, 1 foot: what must be the thickness that there may be
an equilibrium, taking f=3.
Here, that the wall may not be displaced horizontally, we
must have
2000
16
3300 × 32
39400
A B > Q÷ƒ. s. c B> 9900 ÷ 4
2000 × 39 4
64
39.4
> 3300 ÷
52800
39400
> 1.34 fect.
And 2dly, that it may not be overturned, we must have
2
19800
125
A B >
✓ 1584 > 12.58 feet.
S
Here, as the thickness required to prevent overturning is
much the greatest, the computation in reference to the other
kind of equilibrium may usually be avoided.
224
STATICS: PRESSURE OF EARTH AGAINST WALLS.
Pressure of Earth against Walls.
1. Let D A Er be the vertical sec-
tion of a wall behind which is po-
sited a bank or terrace of earth, of
which a prism whose section is repre-
sented by D A G would detach itself
and fall down, were it not prevented
by the wall. Then A & is denomi-
nated the line of rupture or the natural slope, or natural de-
clivity. In sandy or loose earth, the angle B A G seldom
exceeds 30°; in stronger earth it becomes 37° and in some
favourable cases more than 45°.
A E
(1
B
G
H
P
L
D F

2. Now, the prism whose vertical section is DAG, has a
tendency to descend along the inclined plane & a by reason of
the force of gravity, g; but it is retained in its place, 1st, by
the force, q, opposed to it by the wall, and 2dly, by its cohesive
attachment to the face A G, and by its friction upon the same
surface. Each of those forces may be resolved into one, which
is perpendicular to & A, and which is inoperative as to this in-
quiry, and into another whose action is parallel to G A. The
lines P I and I н, represent these composants of p, that force
being represented by the vertical liner m, drawn from the
centre of gravity r of the prism. The direction of the force q
is represented by the horizontal line a H, and its composants
by the lines Q L, II L. The force that gives the triangle its ten-
dency to descend is I II; and the force opposed to this is L H
together with the effects of cohesion and friction. Thus,
I II = L II + cohesion + friction.
It is evident, therefore, that the solution to this inquiry must
be, in great measure, experimental.
3. It has been found, however, theoretically, by M. Prony,*
and confirmed experimentally, that the angle formed with the
vertical by the prism of earth that exerts the greatest horizon-
tal stress against a wall, is half the angle which the natural
slope of the earth makes with the vertical: and this curious re-
sult greatly simplifies the whole inquiry.
The state of equilibrium is expressed by this equation: viz.
1
½ A D . A E² . § = x 1³. s. tanº ½ D a G.
A
☎
S
s and s representing the specific gravities of the wall and carth
respectively.
Example. The wall to be 39.37 feet high, of brick, specific
* Seo a demonstration at p. 369, vol. ii. tenth edition of Dr. Hutton's Course of
Mathematics.
STATICS: PRESSURE OF EARTH AGAINST WALLS. 225
gravity 2000, and the terrace of strong earth specific gravity
1428, natural slope 53° from vertex.
Then the above equation becomes
x² × 2000 × 39.37 = 1 × 39.37³ × 1428 × tanº 26½
1428
or x 39.37 tan 26°
39.37
2
1428
3 x 2000
6000
= 19.685 × 4878-9.6 feet, thickness of the wall.
4. Of the experimental results the best which we have seen
are those of M. Mayniel, from which the following are selected;
all along supposing the upper surface of the earth and of the
wall which supports it, to be both in one horizontal plane.
1st. Both theory and experiment indicate that the resultant
HQ of the thrust of a bank, behind a vertical wall, is at a dis-
tance A Q from the bottom of the wall- A D, the height.
2dly. That the friction is half the pressure, in vegetable
earths, four-tenths in sand.
3dly. The cohesion which vegetable earths acquire, when
cut in turfs, and well laid, course by course, diminishes their
thrust by full two-thirds.
4thly. The line of rupture behind a wall which supports a
bank of vegetable earth is found at a distance D G from the in-
terior face of the wall equal to 618 h, h being the height of the
wall.
5thly. When the bank is of sand, then D G= 677 h.
6thly. When the bank is of vegetable earth mixed with small
gravel, then D G='646 h.
7thly. If it be of rubbles, then D G='414 h.
8thly. If it be of vegetable earth mixed with large gravel,
then D G 618 h.
1
Thickness of Walls, both faces vertical.
1. Wall brick, weight of cubic foot = 109 lbs. avoird. bank
vegetable earth, carefully laid, course by course, d f=·16 h.
2. Wall unhewn stones, 135lbs. per cubic foot, earth as be-
fore, D F 15 h.
3. Wall brick, earth clay well rammed, D F—∙17 h.
4. Wall unhewn stones, earth as above, D F='16 h.
5. Wall of hewn free stone, 170 lbs. to the cubic foot, bank
vegetable carth, n r=13 h; if the bank be clay D r='14 h.
6. Bank of earth mixed with large gravel,
Wall of bricks .
unhewn stone
hewn free stone
D F•19 h.
D F•17 h.
D F16 h.
226
STATICS: PRESSURE OF EARTH AGAINST WALLS.
7. Bank of sand,
Wall of bricks
M
unhewn stone
hewn free stone
D F26 h.
When the earth of the bank or terrace is liable to be much
saturated with water, the proportional thickness of wall must
be at least doubled.*
8. For walls with an interior slope, or a slope towards the
bank, let the base of the slope be of the height, and let s and
s, as before, be the specific gravities of the wall and of the
earth; then
N
D F
•
S
S
where m 0424, for vegetable or clayey earth mixed with
large gravel; m0464, if the earth be mixed with small.
gravel; m = 1528, for sand; and m = 166, for semi-fluid
earths.
padded bags
h
DF 20
√3
→
1
3 n² 2
1
1200
D F33 h.
D F30 h.
+ m
h
20
Example. Suppose the height of a wall to be 20 feet, and
of the height for the base of the talus or slope; suppose,
also, the specific gravities of the wall and of the bank to be
2600, and 1400, and the earth semi-fluid: what, then, must be
the thickness of the wall at the crown?
Here the theorem will become,
n
+166.14-28
;
20
200008333+0894
1 = (20 X 3) 1
1 5 feet while the thickness of the
wall at bottom will be 6 feet.
6
Equilibrium of Polygons.
1. Let there be any number of lines, or bars, or beams, A b,
B C, C D, DE, &c. all in the same vertical plane, connected to-
gether, and freely moveable about the joints or angles, A, B, C, D,
E, &c. and kept in equilibrio by weights laid on the angles: It
is required to assign the proportion of those weights as also
the force or push in the direction of the said lines; and the ho-
rizontal thrust at every angle.
20
11
* When weights of French cubic feet are given in kilogrammes, of them will
be the corresponding weight of an English cubic foot in pounds avoirdupois.
1
STATICS PRESSURE OF EARTH AGAINST WALLS.
227

A
Through any point,
as D, draw a vertical
line a DH g, &c.; to
which, from any point,
as c, draw lines in the
direction of, or paral-
lel to, the given lines
or beams, viz. c a pa-
G
P
rallel to A в, c b parallel to в C, ce to D E, c f to E F, cg to
F G, &c.; also c H parallel to the horizon, or perpendicular to
the vertical line a d g, in which also all these parallels ter-
minate.
B
O
مين
a
b
D
H
E
F
Then will all those lines be exactly proportional to the
forces acting or exerted in the directions to which they are
parallel, and of all the three kinds, viz. vertical, horizontal,
and oblique. That is, the oblique forces or thrusts in direc-
tion of the bars...
A B, B C, C D, D E, E F, F G,
are proportional to their parallels c a, c b, c d, c e, c f, c g ;
and the vertical weights on the angles B, C, D, E, F, &c.
are as the parts of the vertical. . . . a b, b », D e, eƒ, ƒ g,
and the weight of the whole frame A B C D E F G,
is proportional to the sum of all the verticals, or to a g; also
the horizontal thrust at every angle, is everywhere the same
constant quantity, and is expressed by the constant horizontal
line C H.
Corol. 1. It is worthy of remark that the lengths of the
bars A B, B c, &c. do not affect or alter the proportions of any
of these loads or thrusts; since all the lines c a, c b, a b, &c.
remain the same, whatever be the lengths of A B, B C, &c.
The positions of the bars, and the weights on the angles depend-
ing mutually on each other, as well as the horizontal and oblique
thrusts. Thus, if there be given the position of D c, and the
weights or loads laid on the angles D, C, B ; sct these on the ver-
tical, D н, D b, b a, then c b, c a, give the directions or posi-
tions of c B, B A, as well as the quantity or proportion c H of the
constant horizontal thrust.
Corol. 2. If с н be made radius ; then it is evident that н a is
the tangent, and c a the secant of the elevation of c a or a b
above the horizon; also нb is the tangent and c b the secant
of the elevation of cb or c B; also HD and CD the tangent
and secant of the elevation of c D; also me and ce the tan-
gent and secant of the elevation of c e or D E ; also н ƒ and
cf the tangent and secant of the elevation of E F; and so on;
also the parts of the vertical a b, b D, ef, fg, denoting the
1
EQUILIBRIUM OF ROOFS, &c.
weights laid on the several angles, are the differences of the
said tangents of elevations. Hence then in general,
228
1
1st. The oblique thrusts, in the directions of the bars, are to
one another, directly in proportion as the secants of their angles
of elevation above the horizontal directions; or, which is the
same thing, reciprocally proportional to the cosines of the same.
clevations, or reciprocally proportional to the sines of the ver
tical angles, a, b, D, e, f, g, &c. made by the vertical line with
the several directions of the bars; because the secants of any
angles are always reciprocally in proportion to their cosines.
2. The weight or load laid on each angle is directly propor-
tional to the difference between the tangents of the elevations
above the horizon, of the two lines which form the angle.
3. The horizontal thrust at every angle is the same constant
quantity, and has the same proportion to the weight on the top
of the uppermost bar, as radius has to the tangent of the cleva-
tion of that bar. Or, as the whole vertical a g, is to the line
CH, so is the weight of the whole assemblage of bars, to the
horizontal thrust.
4. It may hence be deduced also, that the weight or pressure
laid on any angle, is directly proportional to the continual pro-
duct of the sine of that angle and of the secants of the elevations
of the bars or lines which form it.
Scholium. This proposition is very fruitful in its practical
consequences, and contains the whole theory of centerings, and
indeed of arches, which may be deduced from the premises by
supposing the constituting bars to become very short, like arch
stones, so as to form the curve of an arch. It appears too, that
the horizontal thrust, which is constant or uniformly the same
throughout, is a proper measuring unit, by means of which to
estimate the other thrusts and pressures, as they are all deter-
minable from it and the given positions; and the value of it, as
appears above, may be easily computed from the uppermost or
vertical part alone, or from the whole assemblage together, or
from any part of the whole, counted from the top down-
wards.
In all the useful cases, a model of the structure may be
made, and the relations of the pressures at any angle, whether
horizontal, vertical, or in the directions of the beams, may be
determined by a spring steel-yard applied successively in the
several directions, after the manner described in Art. 4. Sect. 1.
Statics.
2. If the whole figure in the preceding problem be inverted,
or turned round the horizontal line A G as an axis, till it be
· EQUILIBRIUM OF ROOFS, &c.
229
·
Į
completely reversed, or in the same vertical plane below the
first position, each angle D, d, &c. being in the same plumb
line; and if weights i, k, l, m, n, which are respectively equal
to the weights laid on the angles B, C, D, E, F, of the first
figure, be now suspended by threads from the corresponding
angles h, c, d, e, f, of the lower figure; those weights keep
this figure in exact equilibrio, the same as the former, and all
the tensions or forces in the latter case, whether vertical or ho-
rizontal or oblique, will be exactly equal to the corresponding
forces of weight or pressure or thrust in the like directions of
the first figure.
A
B
k
U
b
D
H
O
B
مة
B
h
d
ს
→
1
}
E
e
m
}

fa
A
:
This, again, is a proposition most fertile in its conse-
quences, especially to the practical mechanic, saving the
labour of tedious calculations, but making the results of ex-
periment equally accurate. It may thus be applied to the prac-
tical determination of arches for bridges, with a proposed.
road way; and to that of the position of the rafters in a curb or
mansard roof.
3. Thus, suppose it were required to make such a roof, with
a given width A E, and of four proposed rafters A B, B C, C D, D E.
Here, take four pieces that are equal or
in the same given proportions as those
proposed, and connect them closely to-
gether at the joints A, B, C, D, E, by pins
or strings, so as to be freely moveable
about them; then suspend this from two
pins, A E, fixed in a horizontal line, and
the chain of the piece will arrange itself
in such a festoon or form, a b c d E, that
all its parts will come to rest in equili-
brio. Then, by inverting the figure, it will exhibit the form and
G

R
b
M
D
d
E
31
X
230
STATICS: MANSARD ROOF.
1
frame of a curb roof A B C D E, which will also be in equilibrio,
the thrusts of the pieces now balancing each other, in the same
manner as was done by the mutual pulls or tensions of the
hanging festoon a b c d E.
A
4. If the mansard be constituted of four equal rafters; then,
if angle c A E m, angle c A B = x; it is demonstrable that
2 sin 2 x sin 2 m. So that if the span a E, and height м c,
be given, it will be easy to compute the lengths A B, B C,
Example. Suppose A E = 24 feet, м c 12.
&c.
M
Then
1 = tan 45° angle c A M
22
M C
M A
... sin 2 m =
sin 90° 1, and sin 2 x =
.. 2 x = 30°, and x = 15° CAB
Hence M A B = 45° + 15° 60°
2 × 15°) 90° 15°
(75° +60°) = 45°
:: A M
12: A B 8.7846 feet.
and M B A ½ (180°
and A M B= 180°
Lastly, sin 75°: sin 45
Note. In this example, since A M = M C, as well as A B = B C,
it is evident that M B bisects the right angle A м C; yet it
seemed preferable to trace the steps of a general solution.
Slability of Arches.
1. If the effect of the force of gravity upon the ponderating
matter of an arch and pier, be considered apart from the ope-
ration of the cements which unite the stones, &c. the investi-
gation is difficult to practical men, and it furnishes results
that require much skill and care in their application. But,
in an arch whose component parts are united with a very
powerful cement, those parts do not give way in vertical
columns, but by the separation of the entire mass, including
arches and piers, into three, or, at most, into four parts. In
this case, too, the conditions of equilibrium are easily cx-
pressed and easily applied.
J
Let f F, fr', be the
joints of rupture, or places
at which the arch would
most naturally separate,
whether it yield in two
pieces or in onc.
Let &
be the centre of gravity of
the semi-arch ƒr к k, and
G' that of the pier a в F
Let F I be drawn parallel
to the horizon, and G Í be
demitted perpendicularly
S
ADB E
=
H
K
m.
F
E B'
75°

A'
STATICS: ARCHES AND PIERS.
231
upon it; also let &' D be a perpendicular passing through a',
and F E drawn from F parallel to it.
Then
2. PROP. If the arch fr F'f' tend to fall vertically in one
piece, removing the sections f F, f' r'; if A be the weight of
the semi-arch frкk, and P that of the pier up to the joint
fr, the equilibrium will be determined by these two equa-
tions: viz.
CI
S.PA (S)
=
P.
4
A D
FE
P.
A
CI
FI
#
A E
FE
where f is the measure of the friction, or the tangent of the
angle of repose of the material, and the first equation is that of
the equilibrium of the horizontal thrusts, while the latter indi-
cates the equilibrium of rotation about the exterior angle A of
the pier.
F H
S.2=A(S)
I
3. PROP. If each of the two semi-arches r k, k, r', tend to
turn about the vertex k of the arch, removing the points F, F',
the equilibrium of horizontal translation, and of rotation, will
be respectively determined by the following equations: viz.
(3)
FH
ADA (HAB)
FE
I FE
(1)
E
(2)
(4)
4. Hence it will be easy to examine the stability of any ce-
mented arch, upon the hypothesis of these two propositions.
Assume different points, such as r in the arch, for which let the
numerical values of the equations (1) and (2), or (3) and (4), be
computed. To ensure stability, the first members of the re-
spective equations must exceed the second; those parts will be
weakest, where the excess is least.
•
If the figure be drawn on a smooth drawing pasteboard, upon
a good sized scale, the places of the centres of gravity may be
found experimentally, as well as the relative weights of the
semi-arch and piers, and the measures of the several lines from
the scale employed in the construction.
If the dimensions of the arch were given, and the thickness
of the pier required, the same equations would serve; and dif-
ferent thicknesses of the pier might be assumed until the first
members of the equations come out largest.
232
STATICS: MODELS.
The same rules are applicable to domes, simply taking the
ungulas instead of the profiles.
f
Models.
From an experiment made to ascertain the firmness of the
model of a machine, or of an edifice, certain precautions are
necessary before we can infer the firmness of the structure
itself.
The classes of forces must be distinguished; as whether they
tend to draw asunder the parts, to break them transversely, or
crush them by compression. To the first class belongs the
stretching suffered by key-stones, or bonds of vaults, &c. : to
the second, the load which tends to bend or break horizontal or
inclined beams; to the third the weight which presses verti-
cally upon walls and columns.
PROB. 1. If the side of a model be to the corresponding side
of the structure, as I to n, the stress which tends to draw asun-
der, or to break transversely the parts, increases from the
smaller to the greater scale, as 1 to n³; while the resistance of
those ruptures increases only as 1 to nº.
The structure, therefore, will have so much less firmness than
the model as n is greater.
Ka
If w be the greatest weight which one of the beams of the
model can bear, and w the weight or stress which it actually
sustains, then the limit of n will be n =
W
W
PROB. 2. The side of the model being to the corresponding
side of the structure as 1 to n, the stress which tends to crush
the parts by compression, increases from the smaller to the
greater scale, as I to n³, while the resistance increases only in
the ratio of 1 to n.
Hence, if w were the greatest load which a modular wall or
column could carry, and w the weight with which it is actually
loaded; then the greatest limit of increased dimensions would
W
be found from the expression n =
พ
If, retaining the length or height n h, and the breadth n b,
we wished to give to the solid such a thickness x1, as that it
should not break in consequence of its increased dimensions,
we should have x = n² ✓
W
W
In the case of a pilaster with a square base, or of a cylin-
•
STATICS: MODELS.
233
drical column, if the dimension of the model were d, and of
the largest pillar, which should not crash with its own weight
when n times as high, x d, we should have
K
MyD
3
X N
2/22
n² w
These theorems will often find their application in the profes-
sion of an architect or an engineer, whether civil or military.
3. Suppose, for an example, it were required to ascertain the
strength of Mr. Smart's "Patent Mathematical Chain-bridge,"
from experiments made with a model. In this ingenious con-
struction, the truss-work is carried across from pier to pier,
so that the road-way from A to в, and thence entirely across,
shall be in a horizontal plane, and all the base bars, diagonal
bars, hanging bars, and connecting bolts, shall retain their own
respective magnitudes throughout the structure.
nexed representation of half the bridge so exhibits the con-
struction as to supersede the necessity of a minute verbal de-
scription.
The an-
Base
W
Diagonal
Hanging
Bur.
Bar.
Rar.
Connecting Buit.
my Ma – B
Now, let represent the horizontal length of the model,
(say 12 feet,) from interior to exterior of the two piers, w its
weight (say 30 pounds), w the weight it will just sustain at
its middle point в before it breaks (say 350 lbs.) Let n the
/
length of a bridge actually constructed of the same material as
the model, and all its dimensions similar: then, its weight will
be n³ w, and its resisting power to that of the model, as n° to 1,
being nº (w + w.) Hence 2 (w + ½ w) - ½½ n³ w =
3
I
W • ½ n³ (n 1) w, the load which the bridge itself would
bear at the middle point.
223
Jxz
Sale
m

x 2
234
STATICS: MODELS.
1
Suppose n = 20, or the bridge 240 feet long, and entirely
similar to the model; then we shall have (400 × 350) 200
(20 — 1) 30
) 30= 14000.0 114000 26000 lbs. 11 tons 124
cwt., the load it would just sustain in the middle point of its
extent.
1
Jag tag m
dah
Note. This bridge is, in fact, a suspension bridge, and would
require brace or tie-chains at each pier. A considerable im-
provement upon its construction, by Colonel S. H. Long, of the
American Engineers, is described in the Mechanic's Magazine,
vol. xiii. or No. 368.
1
1
Ma
+
}
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!
I
1
1
1
1
曹
​1
}
DYNAMICS.
235
{
(
CHAPTER X.
DYNAMICS.
1. THE mass of a body is the quantity of matter of which it
is composed.
The knowledge of the mass of a body is given to us by that
property of matter which we call inertia ; and which being
greater or less as the mass is greater or less, we regard as an in-
dex of the mass itself.
་
2. Density is a word by which we indicate the comparative
closeness or otherwise of the particles of bodies. Those bodies
which have the greatest number of particles, or the greatest
quantity of matter, in a given magnitude, we call most dense;
those which have the least quantity of matter, least dense;
Density and weight are regarded as correlatives; so that the
heaviest bodies of a given size, are the most dense, the lightest
bodies, the least dense.
Thus lead is more dense than freestone; freestone more
dense than oak; oak more dense than cork.
3. When bodies are impelled by certain forces, they receive
certain velocities, and move over certain spaces, in certain
times. So that body, force, velocity, space, time, are the sub-
jects of investigation in Dynamics; and in mathematical theo-
rems, they are usually represented by the initial letters, b, f, v,
s, t: or, if two or more bodies, &c. are compared, two or more
corresponding letters в, b, b', v, v, v', &c. are employed in the
formulæ. Gravity, which is a separate force incessantly acting,
is represented by g; and momentum, or quantity of motion,
by m, this being the effect produced by a body in motion.
Force is distinguished into motive and accelerative, or
retardive.
4. Motive force, otherwise called momentum, or force of
percussion, is the absolute force of a body in motion, &c. ; and
is expressed by the product of the weight or mass of matter in
the body multiplied by the velocity with which it moves. But
5. Accelerative force, or retardive force, is that which
respects the velocity of the motion only, accelerating or
retarding it; and it is denoted by the quotient of the motive
236
DYNAMICS UNIFORM MOTIONS.
force divided by the mass or weight of the body. So, if a body
of 2 lbs. weight be acted upon by a motive force of 40, the ac-
celerating force is 20; but, if the same force of 40 act upon
another body of 4 lbs. weight, the accelerating force is then only
10; that is, it is only half the former, and will produce only
half the velocity.
1
SECTION I.—Uniform Motions.
1. The space described by a body moving uniformly, is re-
presented by the product of the velocity into the time: and in
comparing two, we say
?
S:S:: T v : t v.
2. In regard to momenta, m vàries, as b v, or
M: m :: BV : b v.
Example. Two bodies, one of 10, the other of 5 pounds,
are acted upon by the same momentum, or receive the same
quantity of motion 30. They move uniformly, the first for 8
seconds, the second for 6: required the spaces described by
both.
6
บ
10
5
Here 30
3 V, and 30
Then T V = 3 X 8 = 24 s; and t v 6 X 6 36 =s.
Thus the spaces are 24 and 36 respectively,
SECTION II.-Motion uniformly accelerated.
1. Motion uniformly accelerated, is that of a material point.
or body subjected to the continual action of a constant force.
2. In this motion the velocity acquired at the end of any
time whatever is equal to the product of the accelerating
force into the time; and the space described is equal to the
product of half the accelerating force into the square of the
time.
3. The spaces described in successive seconds of time are as
the odd numbers, 1, 3, 5, 7, 9, &c.
I
4. Gravity is a constant force, whose effect upon a body fall-
ing freely in a vertical line is represented by g; and the motion
of such body is uniformly accelerated.
1
DYNAMICS: MOTION UNIFORMLY ACCELERATED.
237
5. The following theorems are applicable to all cases of mo-
tion uniformly accelerated by any constant force f
22
2 g f
gft = √2gfs
s = å t v =
บ
t
f
v
gf
2 s
g 13
2 g s
Hence, in all motions of this nature, as soon as the ratio of
the force,, to the force of gravity, g, is known, the circum-
stances of space, time, velocity, &c. may be computed; or con-
versely, knowing the space described in a given time, or the
velocity acquired at the end of such time, the value of ƒ may
be obtained.
28
t
28
V
& t
g
t
ని
v=g
Ot
11
g
6. When the force of gravity acts freely, as when a body
falls in a vertical line, ƒ is omitted in the theorems, and we
have
22
3=\ 81 ===== }
2 g
to
2 g
½ 8 ƒ to
g
V 2 S
V
t
28
t
บ
شروح
S
ž g ƒ
f
v2
28
じ
​z t v
√2 g s
60
S
g
༩༧
2 s
Madda
7. Now, it has been ascertained by very accurate experi-
ments that a body in the latitude of London falls nearly 16-1
feet in the first second of time, and that at the end of that time
it has acquired a velocity double, or of 32 fect; therefore, if
ΤΣ
1
ΤΣ
g denote 16 feet, the space fallen through in one second
of time, or g the velocity generated in that time; then, if the
first scries of natural numbers be seconds of time,
namely, the times in seconds 1", 2", 3",
the velocities in feet will be 321, 643, 96,
the spaces in the whole times 1672, 643, 1443,
and the space for each second 16, 484, 80,
4", &c.
128, &c.
257}, &c.
1127, &c.
32
238
DYNAMICS: DESCENTS BY GRAVITY.
of which spaces the common difference is 321 feet, the natural
and obvious measure of g, the force of gravity.
8. If, instead of a heavy body being allowed to fall freely,
it be propelled vertically upwards or downwards with a given
velocity, v, then
s = t v = 4 g 12;
an expression in which the upper sign must be taken when
the projection is upwards, the lower sign + when the projec-
tion is downwards.
20
-1
When only an approximate result is required with reference
to bodies falling vertically, 32 may be put for g, instead of 321:
there would then result, in motions from quiescence,
v2
s = 16 t²
½ t v
t
ช
၇)
32
mulæ of art. 5; so that
8√s-
S
+ √ s
G
W
K
8
Thus, if the space descended were 64 feet, we should have
v = 8 × 8=
X 64, and t 9 2 seconds.
If the space descended were 400: then v
160,
and t
5.
9.
The force of gravity differs a little at different latitudes;
the law of the variation is not as yet precisely ascertained; but
the following theorems are known to represent it very nearly.
That is, if g denote the force of gravity at latitude 45°, g' the
force at any other place: then
64
พบ
g'=g (1—·002837 cos 2 lat.)
g'=g (1 +·002837), at the poles.
g'g (1002837), at the equator.
w + w
2 s
t
10. Motion over a fixed pulley.-In this case let the two
weights which are connected by the cord that goes over the
}
W
พ
pulley be denoted by w and w: then
f in the for-
w + w
Added
"
G D
28
ย
W
w + w +2
A
32 1.
& g t³.
พบ
C
= 8 × 20
}
Or, if the resistance of the friction and inertia of the pulley
be represented by r; then
$
1 g tº.
T
DYNAMICS: MOTION OVER PULLEYS, &C.
239
Example. Suppose the two weights to be 5 and 3 lbs. re-
spectively, what will be the space descended in 4 seconds?
5 3
W
1612
Here
16 4. 16. 16
20
J
•
14
2
w + w
162.4 64 feet.
W
g
to
J
5 + 3
Example II.-But, suppose that in an actual experiment with
two weights of 5 and 3 lbs. over a pulley, the heavier weight
descended only 50 feet in 4 seconds.
50 feet and, as w, w, g, and t,
•
พ
Then
12 g 13
are the same in both examples,
w + w + "
we have w+w+r: w+w:: 64: 50
or, dividendo r: w+w:: 14} : 50
that is, r: 5+3 :: 14:50
8+14
50
whence r =
the measure of the resistance and the inertia.
1142
50
11. If, instead of pulleys, small
wheels and axles, as in the margi-
nal figure, be employed, to raise
weights by the preponderance of
equal weights: then, if the dia-
meter of wheel and axle A be as 3
to 2; those of wheel and axle в, as
5 to 2; those of c, as 8 to 2; it
will be found that the weight 6 will
be elevated more rapidly than
either a or c: the proportion of 5
to 2, (or, correctly, of 1+ √2 to 1)
being in that respect the most fa-
vourable.
Note. Similar principles are applicable in a variety of other
cases and by varying the relations of w, w, and r, the force
may have any assigned ratio to that of gravity; which is, in-
deed, the foundation of Mr. Atwood's elegant apparatus for ex-
periments on accelerating forces. The inquisitive reader may
see an account of it in the 2d vol. of my Mechanics, or in al-
most any of the general dictionaries of arts and sciences.
2.2933 lbs.
{
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Z
b
B
U
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RICKS (BLACK
last, a
240
J
DYNAMICS: MOTION ON INCLINED PLANES.
A
Motion on inclined Planes.
1. When bodies move down inclined planes, the accelerating
force (independently of the modification occasioned by the po-
the quo-
sition of the centre of gyration) is expressed by
2.v
1 . s = å g t² sin i
tient of the height of the plane divided by its length, or by
what is equivalent, the sine of the inclination of the plane,
that is to say, sin ¿. In this case, therefore, the formulæ be-
come
23
g sin i
g t sin i = √(2 g s sin i)
A
2$
•
g sin 2
2
!
h
400 0
64.12
193
יד
½ t v
3.t
28
V
Farther, if v be the velocity with which a body is projected
up or down a plane, then
4. v = v g t sin i
V2 ༧༧
5. s = v t = è g t sin i
2 g sin i
Making v = 0, in equa. 4, and the latter member of equation.
5, the first will give the time at which the body will cease to
rise, the latter the space.
Example.-Suppose a body be projected up a smooth in-
clined plane whose height is 12 and length 193 feet, with a ve-
locity of 20 feet per second, how high will it rise up the plane
before its motion is extinguished?
Here s
V2 vo
2 g sin i
becomes s==
E
2 s
t
400
193
3 193
1 2
400.
4
100 feet, the space required.
J.
2. With regard to the velocities acquired by bodies in fall-
ing down planes of the same height, this proposition holds;
viz. that they are all equal, estimated in their respective di-
rections. Thus, if
A D, B E, C F, be
grooves of different
inclinations, and a c,
D r, horizontal lines,
the balls A, B, C, after

Ing
F
DYNAMICS: MOTION ON INCLINED PLANES.
241
descending through those planes will have equal velocities
when they arrive at D, E, F, respectively.
3. Also, all the chords, such as A D,
B D, CD, that terminate either in the
upper or the lower extremity of the
vertical diameter of a circle, will be
described in the same time by heavy
bodies A, B, C, running down them;
and that time will be equal to the time
of vertical descent through the diame-
ter.

B
A
AUTODETT STATUTABROWOLNERABLJENI
D
4. If three
weights, as A, B,
c, be drawn up
three planes of
different inclina-
tions, by three
equal weights
hanging from
cords over pulleys at P, then if the length of the middle plane
be twice its height, the body в will be drawn up that plane,
quicker than either of the other weights A or c.
Or, generally, to ensure an ascent up a plane in the least time,
the length of the plane must be to its height, as twice the
weight to the power employed.
5. If it be proposed to construct a roof over a building of a
given width, so that the rain shall rún quickest off it, then each.
side of the roof must be inclined 45° to the horizon, or the an-
gle at the ridge must be a right angle.
:
6. The force by which spheres, cylinders, &c. are caused to
revolve as they move down an inclined plane (instead of sliding)
is the adhesion of their surfaces occasioned by the pressure
against the plane this pressure is part of the body's weight;
for the weight being resolved into its components, one in the
direction of the plane, the other perpendicular to it, the latter
is the force of the pressure; and, while the same body rolls
down the plane, will be expressed by the cosine of the plane's
elevation. Ilence, since the cosine decreases while the arc or
angle increases, after the angle of elevation arrives at a certain
magnitude, the adhesion may become less than what is neces-
sary to make the circumference of the body revolve fast enough;
in this case the body descends partly by sliding and partly by
rolling. And the same may happen in smaller elevations, if
the body and plane are very smooth. But at all elevations the
Y
P

242
DYNAMICS MOTION ON INCLINED PLANES.
body may be made to roll by the uncoiling of a thread or rib
band wound about it.
If w denote the weight of a body, s the space described by
a body falling freely, or sliding freely down an inclined plane,
then the spaces described by rotation in the same time by the
following bodies, will be in these proportions.
cylindrical surface, ss tension
1. A hollow cylinder, or
of the cord in the first case
2. In a solid cylinder, s
3. In a spheric surface,
tens. 2/3 w.
Z W.
s, tens.
w.
or thin spherical shell, s =
g(sin i—ƒ)'
s, tens.
7
—
2/4 w.
4. In a solid sphere, s
If two cylinders be taken of equal size and weight, and with
equal protuberances upon which to roll, as in the marginal
figures: then, if lead be coiled uniformly over
the curve surface of B, and an equal quantity
of lead be placed uniformly from one end to
the other near the axis in the cylinder A, that
cylinder will roll down any inclined plane
quicker than the other cylinder û. The rea-
son is that each particle of matter in a roll-
ing body, resists motion in proportion to
the SQUARE of its distance from the axis of
motion: and the particles of lead which most
resist motion are placed at a greater distance from the axis in
the cylinder в than in A.
4
5
5
7. The friction between the surface of any body and a
plane, may be easily ascertained by gradually elevating the
plane until the body upon it just begins to slide. The friction
of the body is to its weight as the height of the plane to its
base, or as the tangent of the inclination of the plane to the
radius. Thus, if a piece of stone in weight 8 pounds, just
begins to slide when the height of the plane is 2 feet, and its
base 2: then the friction will be the weight, or 4 of 8 lbs.
6 ½ lbs.
2
8. After motion has commenced upon an inclined plane, the
friction is usually much diminished. It may casily be ascer-
tained experimentally, by comparing the time occupied by a
body in sliding down a plane of given height and length, or
given inclination, with that which the simple theorem for t,
would give.
For, if be the value of the friction in terms of
the pressure, the theorem fort will be
2 s
t'
instead of t
t': to: sin i sin i
2 s
sin i
f
A
!
B
cnles
S,
Hence
DYNAMICS: MOTION ON INCLINED PLANES.
243
Example.-Suppose that a body slides down a plane in
length 30 feet, height 10, in 23 seconds, what is the value of
the friction.
60
Here t
2.366 nearly.
321 × 10 3
Hence (2.6) (2·366)2::: 27603 sin i-f
Consequently, 33333
I thoala ch
P W B
tion, the weight being unity.
9. When a weight is to be moved
either up an inclined plane, or along
an horizontal plane, the angle of
traction P w B, that the weight may
be drawn with least effort, will vary
with the value of f. The magnitude
of that angle P w B for several values
of ƒ are exhibited below.
1 45° 0'
38 40
33 41
29 45
18
G/02/20/
28
g sin i
2
PW B
26°34'
23 58
21 48
19 59
•27603 = ·0573 value of the fric-

f
A
W
11°19'
10 47
27
3
T1
4 952
23
10 18
PWBPW Bf PW Bf PW B
4.
17
18°26' 1 14° 2'
16 54 13 15
15 57
1232
414 56 4 11 53
13
7
19
P
HO7-10-10
с
B

8
9°28'
8
7 8
6 20
10. If, instead of seeking the line of traction so that the
moving force should be a minimum, we required the position
such that the suspending force to keep a load from descending
should be a minimum, or a given force should oppose motion
with the greatest energy; then the angles in the preceding
table will be still applicable, only the angle in any assigned
case must be taken below, as в w p.
This will serve in the
building and fastening walls, banks of earth, fortifications, &c.
and in arranging the position of land-lies, &c.
SECTION III.-Motions about a Centre or Axis.
Pendulum, simple and compound; Centres of Oscillation,
Percussion, and Gyration.
DEF. 1. The centre of oscillation is that point in the axis of
suspension of a vibrating body in which, if all the matter of
the system were collected, any force applied there would ge-
nerate the same angular velocity in a given time as the same
4
244
DYNAMICS: LINE OF FRACTION.
1
+
force at the centre of gravity, the parts of the system revolving
in their respective places.
Or, since the force of gravity upon the whole body may be
considered as a single force (equivalent to the weight of the
body) applied at its centre of gravity, the centre of oscillation
is that point in a vibrating body into which, if the whole were
concentrated and attached to the same axis of motion, it would
then vibrate in the same time the body does in its natural state.
COR. From the first definition it follows that the centre of
oscillation is situated in a right line passing through the cen-
tre of gravity, and perpendicular to the axis of motion. It is
always farther from the point of suspension than the centre of
gravity.
DEF. 2. The centre of gyration is that point in which, if all
the matter contained in a revolving system were collected, the
same angular velocity will be generated in the same time by a
given force acting at any place as would be generated by the
same force acting similarly in the body or system itself.
When the axis of motion passes through the centre of gravity,
then is the centre called the principal centre of gyration.
The distance of the centre of gyration from the point of sus-
pension, or the axis of motion, is a mean proportional between
the distances of the centres of oscillation and gravity from the
same point or axis.
If s represent the point of suspension, & the place of the cen-
tre of gravity, o that of the centre of oscillation, and R that of
the centre of gyration. Then
SR = √ SO. S G
(1)
and so . s G = a constant quantity for the same body and the
same plane of vibration.
DEF. 3. The Centre of Percussion is that point in a body
revolving about an axis, at which, if it struck an immoveable
obstacle, all its motion would be destroyed, or it would not in-
cline cither way.
•
•
When an oscillating body vibrates with a given angular velo-
city, and strikes an obstacle, the effect of the impact will be the
greatest if it be made at the centre of percussion.
For, in this case the obstacle receives the whole revolving
motion of the body; whereas, if the blow be struck in any other
point, a part of the motion of the pendulum will be employed
in endeavouring to continue the rotation.
If a body revolving on an axis strike an immovable obstacle
at the centre of percussion, the point of suspension will not be
affected by the stroke.
We can ascertain this property of the point o when we
give a smart blow with a stick. If we give it a motion
DYNAMICS: PENDULUMS, &c.
245
3
round the joint of the wrist only, and, holding it at one extre-
mity, strike smartly with a point considerably nearer or more
remote than of its length, we feel a painful wrench in the
hand but if we strike with that point which is precisely at
If we
of the length, no such disagreeable strain will be felt.
strike the blow with one end of the stick, we must make its
centre of motion at of its length from the other end; and then
the wrench will be avoided.
3
PROP. The distance of the centre of percussion from the axis
of motion is equal to the distance of the centre of oscillation
from the same supposing that the centre of percussion is re-
quired in a plane passing through the axis of motion and cen-
tre of gravity.
DEF. 4. A Simple Pendulum, theoretically considered, is a
single weight, regarded as a point, or as a very small globe,
hanging at the lower extremity of an inflexible right line, void
of weight, and suspended from a fixed point or centre, about
which it oscillates.
DEF. 5. A Compound Pendulum is one that consists of se-
veral weights moveable about one common centre of motion,
but so connected together as to retain the same distance
both from one another and from the centre about which they
vibrate.
Or any body, as a cone, a cylinder, or of any shape, regular
or irregular, so suspended as to be capable of vibrating, may be
regarded as a compound pendulum; and the distance of its cen-
tre of oscillation from any assumed point of suspension, is con-
sidered as the length of an equivalent simple pendulum.
Any such vibrating body will have as many centres of oscil-
lation as you give it points of suspension; but when any one
of those centres of oscillation is determined, either by theory
or experiment, the rest are easily found by means of the pro-
perty that so. S G is a constant product, or of the same value
for the same body.
1
Wala
DEF. 6. When a body either revolves about an axis, or oscil-
lates, the sum of the products of each of the material elements,
or particles of that body, into the squares of their respective
distances from the axis of rotation, is called the momentum of
inertia of that body. (Sec art. 6, p. 241).
A point, or very small body, on descending along the suc-
cessive sides of a polygon in a vertical plane, loses at each angle
a part of its actual velocity equal to the product of that velocity
into the versed sine of the angle made by the side which it has
just quitted, and the prolongation of the side upon which it is
just entering.
33
Y 2
246
DYNAMICS: PENDULUMS, &c.
Therefore that loss is indefinitely small in curves.
A
D
B
7. A heavy body which descends along a curve posited in
a vertical plane, by the force of gravity, has, in any point what-
ever, the same velocity as it would have if it had fallen through
a vertical line equal to that between
the top and the bottom of the arc run
over and when it has arrived at the
bottom of any such curve, if there
be another branch either similar or
dissimilar, rising on the opposite side, the body will rise along
that branch (apart from the consideration of friction) until it
has reached the horizontal plane from which it set out. Thus,
after having descended from A to v, it will have the same velo-
city as that acquired by falling through D v, and it will ascend
up the opposite branch until it arrives at B.
V

1
8. If the body describe a curve by a pendulous motion, the
same property will be shown, free from the effects of friction.
Thus, let a ball hang
by a flexible cord s D
from a pin at s: then,
after it has descended
through the arc D E,
it will pass through
an equal and similar
arc E A, going up to A
in the same horizontal
line with D, and as-
cending from E to A in an interval of time equal to that which
it descended from D to E. But, if a pin projecting from P or p
stop the cord in its course, the ball will still rise to в or to c,
the same horizontal line with A and D; but will describe the
ascending portions of the curve in shorter intervals of time than
the descending branch.
in
9. When a pendulum is drawn from its vertical position, it
will be accelerated in the direction of the tan-

2000
gent of the curve it would describe, by a force
which is as the sine of its angular distance
from the vertical position. Thus, the acccle-
rating force at A, would be to the accelerating.
force at B, as A F to B E. (See art. 5, on the
Centre of Gravity).
This admits of an easy experimental proof.
A
n
F
E
10. If the same pendulous body descend through different
arcs, its velocity at the lowest point will be proportional to the
chords of the whole arcs described. Thus, the velocity at D,
DYNAMICS: PENDULUMS, &c.
247
after passing through A B D, will be to the velocity at D after
descending through the portion B D only, as A d to b d.
11. Farther, velocity after describing A B D, is to velocity
after describing в D, as ✅F D to √E D.
If, therefore, we would impart to a body a given velo-
city v, we have only to compute the height F D, such that
v2
V2
"
and through the point r draw the hori-
2 g
64 feet
zontal line FA; then, letting the body descend as a pendulum
through the arch A B D, when arrived at D, it will have acquired
the proposed velocity.
This is extremely useful in experiments on the collision of
bodies.
12. The oscillations of pendulums in any arcs of a cycloid
are isochronal, or performed in equal times.
F D
13. Oscillations in small portions of a circular arc are iso-
chronal.
14. The numbers of oscillations of two different pendulums,
in the same time, and at the same place, are in the inverse ratio
of the square roots of the length of these pendulums.
15. If be the length of a single pendulum, or the dis-
tance from the point of suspension to the centre of oscil-
lation in a compound pendulum, g the measure of the
force of gravity (32 feet, or 386 inches at the level of St.
Paul's in the latitude of London), t the time of one oscil-
lation in an indefinitely small circular arc, and я = 3.141593:
then
39/1/
925 inches,
16. Conformably with this we have
t = x
25
42 inches,
72
Yo
inches, length of the second
25 inches,
128
17. We have also l
half second
third of second
quarter second
•20264 × g
= 4·9348 /
g
and &
in any latitude and at any altitude.
pendulum
in lat.
of
London.
+
tabad
*At the level of the sea, in the latitude of London, g is 386-289 inches, and the
corresponding length of the second pendulum is 39-1393 inches, according to the
determination of Major Kater. Conformably with this result are the numbers in
the table following art. 30 of this section, computed at the expense of Messrs. Bra-
mahs and Donkin, and obligingly communicated by them for this work. It has
been suspected by M. Bessel, and demonstrated by Mr. Francis Bailey, that, in
the refined computations relative to the pendulum, the formula for the reduction
to a vacuum are inaccurate, and that, in consequence, we do not yet precisely
know the length of a second pendulum. Sce Phil. Transac. 1832.
248
DYNAMICS: PENDULUMS, &c.
1
In other words, whatever be the force of gravity, the length
of a second pendulum, and the space descended freely by a
falling body in 1 second, are in a constant ratio.
18. If ' be the length of a pendulum, g' the force of gravity,
and t'the time of oscillation at any other place, then
If the force of gravity be the same,
t: t' :: √l: √l'.
If the same pendulum be actuated by different gravitating
forces, we have
t: t' ::
t : t'
t' ::
t
√g': √g.
When pendulums oscillate in equal times in different places,
we have
JE V
g: g' :: 1: l'.
For the variations of gravity in different latitudes, see art. 9,
pa. 238.
π
↓
HH
g
18. If the arcs are not indefinitely short, let v denote the
versed sine of the semi-arc of vibration; then
Į
9
( 1 + s v + 2 v³+&c.)
for a semi-arc of 30°
of 15°
I
õ
In which, when the semi-are of vibration does not exceed 4
or 5 degrees, the third term of the series may be omitted.
If the time of an oscillation in an indefinitely small arc be
1 second, the augmentation of the time will be
of 10°
of 5°
of 21°
0.01675
0.00426
0.00190
0.00012
0.00003
So that for oscillations of 24° on each side of the vertical,
the augmentation would not occasion more than 2" difference
in a day.
19. If » denote the degrees in the semi-are of an oscillating
pendulum, the time lost in each second by vibrating in a cir-
cle instead of the cycloid, ist
D2
52524
; and consequently the time.
lost in a whole day of 24 hours, or 24 x 60 x 60 seconds, is
DYNAMICS, PENDULUMS, &c.
249
D nearly. In like manner, the seconds lost per day by
vibrating in the arc of A degree, is A. Therefore, if the
§
pendulum keep true time in one of these arcs, the seconds
lost or gained per day, by vibrating in the other, will be
2
(D² A³). So, for example, if a pendulum measure true
time in an arc of 3 degrees, it will lose 113 seconds a day by
vibrating 4 degrees; and 262 seconds a day by vibrating 5 de-
grees and so on.
5
3
20. If a clock keep true time very nearly, the variation in
the length of the pendulum, necessary to correct the error will
be equal to twice the product of the length of the pendulum,
and the error in time divided by the time of observation in
which that error is accumulated.
If the pendulum be one that should beat seconds, and t'
the daily variation be given in minutes, and n be the number
of threads in an inch of the screw which raises and depresses
± 2 × 39 × n t'
the bob of the pendulum, then
24 × 60
2
37
·05434 n t'=3n t' nearly, for the number of threads which
the bob must be raised or lowered, to make the pendulum vi-
brate truly,
21. For civil and military engineers, and other practical
men, it is highly useful to have a portable pendulum, made
of painted tape, with a brass bob at the end, so that the
whole, except the bob, may be rolled up within a box, which
may be enclosed in a shagreen casc. The tape is marked 200,
190, 180, 170, 160, &c. 50, 75, 70, 65, 60, at points which being
assumed respectively as points of suspension, the pendulum
will make 200, 190, &c. down to 60 vibrations in a minute.
Such a portable pendulum may be readily employed in
experiments relative to falling bodies, the velocity of sound,
&c. The pendulum and its box may go in a waistcoat
pocket.
22. If the momentum of inertia (Def. 6) of a pendulum,
whether simple or compound, be divided by the product of the
pendulum's weight or mass into the distance of its centre of
gravity from the point of suspension (or axis of motion), the
quotient will express the distance of the centre of oscillation
from the same point (or axis.)
23. Whatever the number of separate masses or bodies which
constitute a pendulum, it may be considered as a single
pendulum, whose centre of gravity is at the distance d from
the axis of suspension, or of rotation: then if к' denote
the momentum of inertia of that body divided by its mass,
the distance s o from the axis of rotation to the centre of
K
250
DYNAMICS: PENDULUMS, &c.
oscillation or the length of an equivalent simple pendulum,
will be
2
d³ + K²
d
*
s o
24. To find the distance of the centre of oscillation from the
point or axis of suspension, experimentally. Count the num-
ber, n, of oscillations of the body in a very short arc in a minute;
then
so
140850
22 B
Thus, if a body so oscillating, made 50 vibrations in a minute:
then so
56.34 inches.
140850
2500
Or, s o=39} t, in inches, t being the time of one oscillation
in a very small arc.
If the are be of finite appreciable magnitude, the time of os-
cillation must be reduced in the ratio 8+ versin of semi-arc to
8, before the rule is applied.
25. From the foregoing principles are derived the following
expressions for the distances of the centres of oscillation for the
several figures, suspended by their vertices and vibrating flat-
wise, viz.
(1.) Right line, or very thin cylinder, s o=3 of its length.
(2.) Isosceles triangle, s o=3 of its altitude.
(3.) Circle, s o= radius.
(4.) Common parabola, s o= of its altitude.
2 m + 1
Xits altitude.
5
(5.) Any parabola, s o =
3 m + 1
Bodies vibrating laterally or sideways, or in their own plane :
(6.) In a circle, s o=3 of diameter.
(7.) In a rectangle suspended by one angle, so =
&
nal.
(8.) Parabola suspended by its vertex, so =
meter.
(9.) Parabola suspended by middle of its base, s o=4 axis+ ½
parameter.
3 arc radius
4 chord
(radius of base)"
5 axis
(10.) In a sector of a circle, s 0=
(11.) In a cone, s o = axis+
of diago-
axis+para-
*For some curious and valuable theorems, by Professor Airy, for the reduction
of vibrations in the air to those in a vacuum, see Mr. F. Baily's paper referred to
in the preceding noto.
J
DYNAMICS: CENTRE OF OSCILLATION.
251
G
2 rad.2
rad. +d+
(12.) In a sphere, so
5 (d + rad.)
Where d is the length of the thread by which it is suspended.
(13.) If the weight of the thread is to be taken into the ac-
count, we have the following distance between the centre of
the ball, and that of oscillation, where в is the weight of the
ball, d the distance between the point of suspension and its
centre, r the radius of the ball, and w the weight of the thread
or wire,.
( } W + } B) 4 r²º — } w (2 dr + d³); or, if в be expressed
(½ W + B) d r w
2
ड
G 0-
B
in terms of w considered as a unit, then & O
B + ë
(14.) If two weights w, w', be fixed at the extremities of a
rod of given length w w', s being the centre of motion between
w and w'; then, if d = sw, D = s w', and m the weight of an
unit in length of the rod, we shall have
m D3 + 3 W' D² + m D³ + 3 w D³
M D* + 2 W'D m D2 2 W D
;
the radii of the balls being supposed very small in comparison
with the length of the rod.
(15.) In the bob of a clock pendulum, supposing it two equal
spheric segments joined at their bases, if the radii of those bases
=g, the height of each segment v, and d the distance
from the point of suspension to & the centre of the bob, then is
be each
-
4
2
1
§¹ + ½ s² v³ + 7% 21%
2
d.
K
SO=
2 po v z r V
3
becomes & 0 =
o
s³ + 3 v ³
centre of oscillation below the centre of the bob.
If r the radius of the sphere be known, the latter expression
1. 23
MA
1 d
6
; which shows the distance of the
pag mag
10
d (r - 3 v)
Va
(16.) Let the length of a rectangle be denoted by 7, its
breadth by 2 w, the distance (along the middle of the rectangle)
from one end to the point of suspension by D, then the distance
so, from the point of suspension to the centre of oscillation,
d³
d l + } l² + } w²
7 ½ 1°² + 3 w²
1
= { l―d+ 12
will be so=
2 7 d
። 1. it
whether the figure be a mere geometrical rectangle, or a pris-
matic metallic plate of uniform density.
G
It follows from this theorem that a plate of 1 foot long, and ½
of a foot broad, and suspended at a fourth of a foot from either
end, would vibrate as a half second pendulum.
1 6
Also, that a plate a foot long, of a foot wide, and suspended
at of a foot from the middle, would vibrate 36.469 times in 5
hours.
Ü
"
252
DYNAMICS: CENTRE OF OSCILLATION.
And hence the length of a foot may be determined ex-
perimentally by vibrations.

(17.) If a thin rod, say of a foot in length,
have a ball of an inch diameter at each end, a
and B, and a moveable point of suspension, s;
then the time of oscillation of such a pendulum
may be made as long as we please, by bringing
the point of suspension nearer and nearer to the
middle of the rod.
Or, if the point of suspension be fixed the dis-
tance s o (and consequently the time of oscilla-
tions which is as so) may be varied by placing
A nearer or farther from s. And this is the
principle of the METRONOME, by which musi-
cians sometimes regulate their time.
B
2.2
A
S
kamion
PAPERREANDIKÁTLAG
a
(18.) If the weight of the connecting rod be evanescent with
regard to the weight of the balls A and B ; then if R=radius of
the larger ball, r that of the smaller, D and d the distances of
their respective centres from s: we shall have
R³ (5 Dº +2 Rº)+7³ (5 d³ +2 µ³)
3
5 (D R³d p³)
When R and r are equal, this becomes
3
2
2
SO=
SO (D+d) +3.
5
D
ď
(19.) If the minor and major axes of an ellipse (or of an
elliptical plate of wood or metal) be as 1 to 3, or as 1000
to 1732; then if it be suspended at one extremity of the minor
axis, the centre of oscillation will be at the other extremity of
that axis, or its oscillations will be performed in the same time
as those of a simple pendulum whose length is equal to the
minor axis.
:
The same ellipse also possesses this curious and useful pro-
perly viz. That any segment or any zone of the ellipse cut
off by lines parallel to the major axis, whether it be taken
near the upper part of the minor axis, near the middle, or near
the bottom of the same, will vibrate in the same time as the
whole ellipse, the point of suspension being at an extremity of
the minor axis.
26. It is evident, from art. 17, that pendulums in different
latitudes require to be of different lengths, in order that they
may perform their vibrations in the same time; but besides
this there is another irregularity in the motion of a pendulum
in the same place, arising from the different degrees of tem-
perature. Heat expanding, and cold contracting the rod of
-
DYNAMICS: COMPENSATION PENDULUMS, &c. 253
1
1
the pendulum, a certain small variation must necessarily fol-
low in the time of its vibrations; to remedy which various
methods have been invented for constructing what are com-
monly called compensation pendulums, or such as shall always
preserve the same distance between the centre of oscillation
and the point of suspension; but of these we shall describe
two or three.
Compound or Compensation PENDULUMS have received dif-
ferent denominations, from their form and materials, as the
gridiron pendulum, mercurial pendulum, &c.
27. The Gridiron PENDULUM consists of five rods of steel,
and four of brass, placed in an alternate order, the middle rod
being of steel, by which the pendulum ball is suspended; these
rods of brass and steel are placed in an alternate order, and so
connected with each other at their ends, that while the ex-
pansion of the steel rods has a tendency to lengthen the pen-
dulum, the expansion of the brass rods acting upwards tends
to shorten it. And thus, when the lengths of the brass and
steel rods are duly proportioned, their expansions and contrac-
tions will exactly balance and correct each other, and so pre-
serve the pendulum invariably of the same length.
Sometimes 3, 7, or 9 rods, are employed in the construction
of the gridiron pendulum; and zinc, silver, and other metals,
may be used instead of brass and steel.
28. The mercurial pendulum was invented by Mr. Graham,
an eminent clockmaker, about the year 1715. Its rod was
made of brass, and branched towards its lower end, so as to
embrace a cylindric glass vessel 13 or 14 inches long, and
about 2 inches diameter; which being filled about 12 inches
deep with mercury, forms the weight or ball of the pendulum.
If, upon trial, the expansion of the rod be found too great for
that of the mercury, more mercury must be poured into the
vessel if the expansion of the mercury exceeds that of the
rod, so as to occasion the clock to go fast with heat, some
mercury must be taken out of the vessel, so as to shorten the
column. And thus may the expansion and contraction of the
quicksilver in the glass be made exactly to balance the expan-
sion and contraction of the pendulum rod, so as to preserve the
distance of the centre of oscillation from the point of suspen-
sion invariably the same.
This kind of pendulum fell entirely into disuse soon after
Graham's time; but it has lately been re-adopted with con-
siderable success by practical astronomers. A very instructive
paper on its principles, construction, and use, has been pub-
lished by Mr. F. Baily, in vol. i. part 2, Memoirs of the As-
tronomical Society of London.
T
34
Z
254
DYNAMICS: COMPENSATION PENDULUMS.
1
29. Reid's Compensation PENDULUM is a recent
invention of Mr. Adam Reid, of Woolwich, the con-
struction of which is as follows: A N is a rod of wire,
and z z a hollow tube of zinc, which slips on the wire,
being stopped from falling off by a nut N, on which
it rests; and on the upper part of this cylinder of
zinc rests the heavy ball в; now the length of the
tube z z being so adjusted to the length of the rod A N,
that the expansions of the two bodies shall be equal
with equal degrees of temperature; that is, by making
the length of the zinc tube to that of the wire, as the ex-
pansion of wire is to that of zinc, it is obvious that the
ball в will in all cases preserve the same distance from
A; for just so much as it would descend by the expan-
sion of the wire downwards, so much will it ascend by the ex-
pansion of the zinc upwards, and consequently its vibrations
will in all temperatures be equal in equal times.
N
Katik

Z
B
Τ
30. Drummond's Compensation Pendulum.
This was proposed by an artist in Lancashire more than 70
years ago. A bar of the same metal with the rod of the pen-
dulum, and of the same thickness and length, is placed against
the back part of the clock-case from the top of this a part
projects, to which the upper part of the pendulum is connected
by two fine pliable chains or silken strings, which just below
pass between two plates of brass whose lower edges will al-
ways terminate the length of the pendulum at the upper end.
These plates are supported on a foot fixed to the back of the
casc. This bar rests upon an immoveable base on the lower
part of the case, and is braced into a proper groove, which ad-
mits of no motion any way but that of expansion and contrac-
tion in length by heat and cold. In this construction, since
the two bars are of equal magnitude and like constitution, their
expansions and contractions will always be equal and in op-
posite directions; so that one will serve to correct and annihilate
the effects of the other.
An extensive and valuable table of the expansions of dif-
ferent substances is given by Mr. Baily in the paper referred
to above.
bakk
}
DYNAMICS: COMPENSATION PENDULUMS.
251
1
Table of Lengths and Vibrations of Pendulums.
[See note at foot of page 250.]
Į
Length Time of
inches. vibration.
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.1
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.-7
4.8
4.9
5.0
5.1
5.2
$
0-1598
0.1676
0.1751
0.1822
0.1891
0.1957
0·2424
0.2476
0.2527
0.2577
0.2626
0.2674
0.2721
0.2768
0.2814
0-2859
0.2903
0.2947
0.2021
0·2084
0.2144
0·2203
4.538
0.2260
4.123
0.2316
4-317
0.2370 4.217
0.2990
0.3032
0·3074
0.3115
No.Vibr. Length
inches.
per sec.
0.3236
0·3275
0.3314
0.3352
0.3390
0.3428
6.256
5.965
5-711
5.487
5.287
5.108
0.3465
0·3502
0.3538
0.3574
0.3609
0.3644
4.945
4.798
4.663
4.125
4.038
3.956
3.344
3.297
3.252
3·209
0.3157 3.167
0.3195
3.128
3.879
3.807
3.738
3.673
3.612
3.553
3.497
3-443
3.392
3.089
3.052
3.016
2.982
2.949
2-916
2.885
2.855
2.826
2-797
2.770 ·
2-743
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
17.17
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
Time of No. Vibr.
vibration,
per sec.
$
0.3679
0.3713
0.3748
0.3782
0.3816
0.3849
0·3882
0.3915
0.3947
0.3980
0.4012
0-4043
0-1075
0.4106
0.4137
0.4168
0.4198
0.1229
0.4435
0.4464
0.4492
0-4521
0.4549
0-4577
2.717
2.692
2.667
2.643
2.620
2.597
0.4259 2.347
0.4289
2.331
0.4318
2.315
0.4348
2.399
0.4377
2.284
0.4406
2.269
0-4605
0.4632
0·1660
0.4687
0.4714
0.4741
2.575
2-554
2.533
2.512
2-192
2.472

2.453
2.435
2.416
2.399
2.381
2.364
2.254
2.240
2-225
2.211
2.198
2.184
2.17]
2.158
2.145
2.133
2.121
2.108
0-4768
2.097
0.4795 2.085
0.4821
2·073
0.4848
2.062
0-1874
2.051
0.4900 2.040
256
LENGTHS AND VIBRATIONS OF PENDULUMS.

Length Time of No.Vibr. Length Time of No. Vibr.
inches.
inches. vibration.
vibration.
per sec.
per sec.
9.5
9.6
9.7
9.8
9.9
10.0
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
10.9
11.0
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
11.9
12.0
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
13.0
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
13.9
14.0
14.1
14.2
0.4926
0.4952
0.4978
0.5003
0.5029
0.5054
0.5079
0.5105
0.5130
0.5155
0.5179
0.5204
0.5228
0.5253
0.5277
0.5301
0.5325
0.5349
0.5373
0.5396
0.5420
0.5444
0.5467
0.5490
0.5514
0.5537
0.5560
0.5583
0.5605
0.5628
0.5651
0.5673
0.5696
0.5718
0.5741
0.5763
0.5785
0.5807
0.5829
0.5851
0.5873
0.5894
2.029
2.019
2.008
1.998
1.988
1.978
1.968
1.958
1.949
1.939
1.930
1.921
1.912
1.903
1.894
1.886
1.877
1.869
1.861
1.853
1.845
1.837
1.829
1.821
1.783
1.776
1.769
1.762
1.755
1.748
1.741
1.735
1.728
1.721
1.715
1.709
1.703
1.696
14.3
14.4
14.5
14.6
14.7
14.8
1.813
16.7
1.806
16.8
1.798
16.9
1.791. 17.0
17.1
17.2
0.5916
1.690
0.5938. 1.684
0.5959
1.6.78
0.5980
1.672
0.6001
1.666
0.6023
1.660
14.9
15.0
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
15.9
16.0
16.1
16.2
16.3
16.4
16.5
16.6
17.3
17.4
17.5
17.6
17:7
17.8
17.9
18.0
18.1
18.2
18.3
18.4
18.5
18.6
18.7
18.8
18.9
19.0
S
0.6044
0.6065
0.6086
0.6107
0.6128
0.6149
0.6170
0.6191
0.6211
0.6231
0.6252
0.6272
0.6293
0.6313
0.6333
0.6353
0.6373
0.6393
0.6413
0.6433
0.6453
J
0.6473
0.5493
0.6532
0.6551
0.6571
0.6590
0.6609
0.6629
0.6648
0.6648
0.6667
0.6686
1.654
1.648
1.642
1.637
1.631
1.626
1.620
1.615
1.609
1.604
1·599
1.594
0.6875
0.6893
0.6912
0.6930,
0.6949
0.6967
1.589
1.584
1.579
1.574
1.569
1.564
1.559
1.554
1.549
1.544
1.539
1.535
1.504
1.499
1.495
0.6705
1.491
0.6724
1.487
0.6743 1.482
1.531'
1.526
1.521
1.517
1.512
1.508
0.6762
1.478.
0.6781
1.474
0.6800 1.470
0.6819
1.466
0.6837
1.462
·0.6856
1.458
1.454
1.450
1.446
1.442
1.439
1.435
F
LENGTHS AND VIBRATIONS OF PENDULUMS.
Length Time of No.Vibr. Length
Length
inches.
inches.
vibration.
per sec.
19.1
19.2
19.3
19.4
19.5
19.6
19.7
19.8
19.9
20.0
20.1
20.2
20.3
20-4
20.5
20.6
20.7
20.8
20.9
21.0
21.1
21.2
21.3
21.4
$
0.6985
0.7003
0.7022
0.7040
0.7058
0.7076
0.7094
0.7112
0.7130
0.7148
0.7166
0.7184
0.7201
0.7219
0.7237
0.7254
0.7272
0.7289
0.7307
0.7324
0.7342
0.7359
0.7377
0.7394
21.5
0.7411
21.6 0.7428
21.7 0.7446
21.8
0.7463
21.9
0.7480
22.0
0.7497
22.1
0.7514
22.2
0.7531
22.3
0.7548
22.4 0.7565
22.5 10.7582
22.6
1.431
1.427
1.424
1.420
1.416
1.413
1.409
1.405
1.402
1.398
1.395
1.391
1.388
1.384
1.381
1.378
1.375
1.371
1.368
1.365
1.361
1.358
1.355
1.352
1.349
1.345
1.343
1.339
1.336
1.333
1.330
1.327
1.324
1.321
1.318
0.7598 1-315
23.3
0.7715
1.296
23.4 0.7732
1.293
23.5
0.7748
1.290
23.6
0.7765 1.287
23.7
0-7781
1.285
23.8
0.7798 1.282
23.9
24.0
24.1
24.2
24.3
24.4
24.5
24.6
24.7
24.8
24.9
25.0
25.1
25.2
25.3
25.4
25.5
25.6
25.7
25.8
25.9
26.0
26.1
26.2
26.3
26.4
26.5
26.6
26.7
26.8
26.9
27.6
22.7 0.7615 1.313
22.8 0.7632 1.310 27.7
22.9 0.7649 1.307 27.8
23.0 0.7665
1.304
27.9
1.301
28.0
23.1 0.7682
23.2 0.7699 1.298
28.1
27.0
27.1
27.2
27.3
27.4
27.5
28.2
28.3
28.4
28.5
28.6
28.7
Time of
vibration.
S
0.7814
0.7830
0.7847
0.7863
'0.7879
0.7895
0.7911,
0.7927
0.7944
0.7960
0.7976
0.7992
0-8008
0.8024
0.8040
0.8056
0.8071
0.8087
0.8103
0.8119
0.8134
0.8150
0.8166
0.8181
0.8197
0.8212
0.8228
0.8244
0.8259
0.8275
0.8290
No. Vibr.
per sec.
1.279
1.277
1.274
1.271
1.269
1.266
1.263
1.261
1.259
1·256
1.253
1.251
1.248
1.246
1.243
1.241
1.238
1.236
1.234
1.231
1.229
1.226
1.224
1.222
1.219
1.217
1.215
1.213
1.211
1.208
1.206
0.8305
1·204
0.8321
1·201
0.8336
1.199
0.8351
1.197
0.8367 1.195
0.8382
1.193
0.8397
1.191
0.8412
1.189
0.8427 1.186
0.8443
1.184
0.8458
1.182
0.8473
1.180
0.8488
1.178
0.8503
1.176
0.8518 1.173
0.8533
1-171
0.8548
1.169
0.8563 1.167
!
257



1
z 2
Z
258
LENGTHS AND VIBRATIONS OF PENDULUMS.


Į
Length Time of
inches. vibration.
1
28.8
28.9
29.0
29.1
29.2
29.3
29.4
29.5
29.6
29.7
29.8
29.9
30.0
30.1
30.2
30.3
30.4
30.5
30.6
30.7
30.8
30.9
31.0
31.1
31.2
31.3
31.4
31.5
31.6
31.7
S
0.8578
1.165
34.2
0.8593
1.163
34.3
0.8607 1.161
34.4
0.8622
1.159
34.5
1.157 34.6
1.155
34.7
0.8637
0.8652
33.6
33-7
33.8
33.9
34.0
34.1
0.8667
0.8682
0.8696
0.8711
0.8726
0.8741
0.8755
0.8769
0.8784
0.8798
0.8813
0.8827
0.8842
0.8856
0·8870
0.8885
0.8899
0.8914
0.8928
0.8912
0.8956
0.8971
0.8985
0.8999
31.8
0·9013
31.9
0.9027
32.0 0.9042
32.1
0.9056
32.2
0.9070
32.3
0.9084
No.Vibr.|| Length | Time of No. Vibr.
per scc. inches. vibration.
per sec.
33.0
0.9182
33.1
0.9196
33-2
0.9210
33.3 0.9224
33.4
0-9237
33.5
0.9251
1·154
34.8
1.152
34.9
1.150
35.0
1.148
35.1
1.146. 35.2
1.144
35.3
1.142
35.4
1.140
35.5
1.138 35.6
1.136
35.7
1.135
35.8
1.133
35.9
1.131
1.129
1.127
1.125
1.123
1.121
1.120
1.118
1.116
1.114
1.112
1111
1.109
1.107
1.105
1.104
1.102
1'100
36.0
36.1
36.2
36.3
36.4
36.5
36.6
36.7
32.4
0.9098
1.099
37.8
32.5
0·9112
1.097
37.9
32.6
0.9126
1.095 38.0
32.7
0.9140
1.094
38·1
32.8 0.9154 1.092
38.2
32.9
0.9168
1.090
38.3
0.9265
1.079
0·9279
1.077
0.9293 1.076
0·9306/ 1.074
0·9320
1:072
0-9334
1.071
36.8
36.9
37.0
37.1
37.2
37.3
37.4
37.5
37.6
37.7
1.089
38.4
1.087
38.5
1.085
38.6
1:084 38.7
1.082
38.8
1·080
38.9
39.0
39.1
39.2
I
$
0.9347
0.9361
0.9375
0.9389
0.9402
0.9415
0.9429 1.060
0.9443 1.059
0.9456
1.057
0.9470
1.055
0.9483
1·054
0.9497
1.052
0.9510
0.9523
0.9537
0.9550
0.9563
0.9577
0.9590
0.9603
0.9617
0.9630
0.9643
0.9657
0.9722
0.9736
0.9749
0.9762
0.9775
0-9788
0.9801
0.9814
1.069
1.068
1.066
1.065
1.063
1.062
0.9670'
1·034
0.9683
1.032
0.9696 1.031
0.9709
1.029
1.028
1.027
0.9827
0.9840
0.9853
0.9866
0.9879
0.9892
1·051
1·050
1.048
1·047
1.045
1.044
0.9982
0.9995
1.0001
1.04.2
1·041
1.039
1·038
1.036
1.035
1.025
1.023
1.024
1·021
1.020
1.018
1·017
1.016
1.014
1.013
1.012
1.010
0.9905
1·009
0.9918
1.008
0.9931
1.006
0.9943 1·005
0.9956 1.004
0.9969
1.003
1.001
1.000
0-9993
f
1

DYNAMICS: GYRATION AND ROTATION.
259
Centre of Gyration, Principles of Rotation.
1. The distance of R the centre of gyration, from c the centre
or axis of motion, in some of the most useful cases, is exhibited
below.
In a circular wheel of uniform thickness c R =
In the periphery of a circle revolving.
about the diam.
In the plane of a circle. . . . ditto . .
In the surface of a sphere. . ditto
In a solid sphere . .
ditto
In a plane ring formed of circles whose
radii
are R, r, revolving about
centre.
In a cone revolving about its vertex.
In a cone
In a paraboloid . .
•
}
•
•
C R
CR
its axis. . CR
CR
s = ½ fg t,. . . . . l =
.t
CR
CR
CR
CR
¿ rad.
rad. ✔?
rad.✔ 1/
=rncarly.
3
2
11
√
HO & &
In a straight lever whose arms are R and r, c R =
✓
rad. ✔✅ž.
rad. ✓.
28
ALLEN
g
√ 12
5
r✓ 3
R² + p²
2
&c.
10.
3.
3 (R+r)*
2. If the matter in any gyrating body were actually to be
placed as if in the centre of gyration, it ought either to be dis-
posed in the circumference of a circle whose radius is c R, or at
two points R, R', diametrically opposite, and at distances from
the centre each C R.
a² + 23 r³
2
3. By means of the theory of the centre of gyration, and the
values of C R = g, thence deduced, the phenomena of rotation
on a fixed axis become connected with those of accelerating
forces for then, if a weight or other moving power p act at a
radius to give rotation to a body, weight w, and dist. of centre
of gyration from axis of motion e, we shall have for the ac-
celerating force, the expression
P 13
f
P 2+3 + w §*
and consequently for the space descended by the actuating
weight or power P, in a given time 1, we shall have the usual
formula
3
R³ +2+3
introducing the above value of f
4. In the more complex cases, the distance of the centre of
gyration from the axis of motion may best be computed from
an experiment. Let motion be given to the system, turning
upon a horizontal axis, by a weight p acting by a cord over a
260
DYNAMICS GYRATION AND ROTATION.
pulley or wheel of radius r upon the same axis, and let s be the
space through which the weight e descends in the time t, the
proposed body whose weight is w turning upon the same axis
with the same angular velocity: then
g P to p³ 2 s P ra
?
CR
CR
?
2 s w
Example. A body which weighs 100 lbs. turns upon a ho-
rizontal axis, motion being communicated to it by a weight
of 10 lbs. hanging from a very light wheel of 1 foot diameter.
The weight descends 2 feet in 3 seconds. Required the dis-
tance of the centre or circle of gyration from the axis of
motion.
A deng
Here, I take g = 32, instead of 32, and obtain an approxi-
mative result. Whence
4 X 10 X 1
32 × 10 × 9 × 4
4 × 100
720 10
,
400
26.646
20 ✓ 710
1.3323 f. the answer.
20
5. When the impulse communicated to a body is in a line
passing through its centre of gravity, all the points of the body
move forward with the same velocity, and in lines parallel
to the direction of the impulse communicated. But when the
direction of that impulse does not pass through the centre of
gravity, the body acquires a rotation on an axis, and also a
progressive motion, by which its centre of gravity is carried
forward in the same straight line, and with the same velocity,
as if the direction of the impulse had passed through the centre
of gravity.
The progressive and rotatory motion are independent of
one another, each being the same as if the other had no ex-
istence.
.
6. When a body revolves on an axis, and a force is im-
pressed, tending to make it revolve on another, it will revolve
on neither, but on a line in the same plane with them, divid-
ing the angle which they contain, so that the sines of the parts
are in the inverse ratio of the angular velocities with which
the body would have revolved about the said axis sepa-
rately.
7. A body may begin to revolve on any line as an axis that
passes through its centre of gravity, but it will not continue to
revolve permanently about that axis, unless the opposite rota-
tory forces exactly balance one another.
}
K
Fut
w
This admits of a simple experimental illustration. Suspend
a thin circular plate of wood or metal by a cord tied to its
edge, from a hook to which a rapid rotation can be given.
DYNAMICS: AXES OF ROTATION.
261
The plate will at first turn upon an axis which is in the con-
tinuation of the cord of rotation. As the velocity augments,
the plane will soon quit that axis, and revolve permanently
upon a vertical axis passing through its centre of gravity, itself
having assumed a horizontal position.
The same will happen if a ring be suspended, and receive ro-
tation in like manner.
And if a flexible chain of small links be united at its two
ends, tied to a cord and receive rotation, it will soon adjust itself
so as to form a ring, and spin round in a horizontal plane.
Also, if a flattened spheroid be suspended from any point,
however remote from its minor axis, and have a rapid rotation
given it, it will ultimately turn upon its shorter axis posited
vertically.
This evidently serves to confirm the motion of the earth upon
its shorter axis.
I
8. In every body, however irregular, there are three axes of
permanent rotation, at right angles to one another. These are
called the principal axes of rotation: they have this remark-
able property, that the momentum of inertia with regard to any
of them is either a maximum or a minimum.
}
1
T
Central Forces.
{
Def. 1. Centripetal force is a force which tends constantly
to solicit or to impel a body towards a certain fixed point or
centre.
2. Centrifugal force is that by which it would recede from
such a centre, were it not prevented by the centripetal force.
1
3. These two forces arc, jointly, called central forces.
4. When a body describes a circle by means of a force direct-
ed to its centre, its actual velocity is everywhere equal to that
which it would acquire in falling by the same uniform force
through half the radius.
5. This velocity is the same as that which a second body
would acquire by falling through half the radius, whilst the
first describes a portion of the circumference equal to the whole
radius.
6. In equal circles the forces are as the squares of the times
inversely.
7. If the times are equal, the velocitics are as the radii, and
the forces are also as the radii.
8. In general, the forces are as the distances or radii of the
circles directly, and the squares of the times inversely.
9. The squares of the times are as the distances directly, and
the forces inversely.
35
262
DYNAMICS: CENTRAL FORCES.
10. Hence, if the forces are inversely as the squares of the
distances, the squarcs of the times are as the cubes of the dis-
tances. That is,
if F :ƒ :: d³ : D³, then т² : 1º :: D³ : d³.
11. The right line that joins a revolving body and its centre
of attraction, called the radius vector, always describes equal
areas in equal times, and the velocity of the body is inversely
as the perpendicular drawn from the centre of attraction to the
tangent of the curve at the place of the revolving body.
}
12. If a body revolve in an elliptic orbit by a force directed
to one of the foci, the force is inversely as the square of the
distance and the mean distances and the periodic times have
the same relation as in art. 10. This comprehends the case of
the planetary motions.
13. If the force which retains a body in a curve increase in
the simple ratio as the distance increases, the body will still
describe an ellipse; but the force will in this case be directed
to the centre of the ellipse; and the body in each revolution
will twice approach towards it, and again twice recede from
that point.
14. On the principles of central forces depend the operation
of a conical, pendulum applied as a governor or regulator to
steam engines, water mills, &c.
This contrivance will be readily comprehended from the
marginal figure, where A a is a vertical shaft capable of turn-
ing freely upon the sole a.
c D, C F, are two bars which move
freely upon the centre c, and carry
at their lower extremities two equal
weights P, Q; the bars c D, c r, are
united, by a proper articulation, to
the bars G, H, which latter are at-
tached to a ring 1, capable of sliding
up and down the vertical shaft ▲ a.
When this shaft and connected ap-
paratus are made to revolve, in vir-
tuc of the centrifugal force, the balls.
PQ fly out more and more from ▲ a,
as the rotatory velocity increases ;
if, on the contrary, the rotatory velocity slackens, the balls
descend and approach ▲ a. The ring 1 ascends in the former
case, descends in the latter and a lever connected with I may
be made to correct appropriately the energy of the moving
power. Thus, in the steam engine, the ring may be made to
act on the valve by which the steam is admitted into the cylin-
der; to augment its opening when the motion is slackening,
and reciprocally diminish it when the motion is accelerated.
P
(1
A

H
F
DYNAMICS: GOVERNOR.
263
1
f
The construction is, often, so modified that the flying out of
the balls causes the ring I to be depressed, and vice versa; but
the general principle is the same.
Here, if the vertical distance of p or ą below c, be denoted by
d, the time of one rotation of the regulator by t, and 3.141593
by я, the theory of central forces gives
↓
1.10784 d.
1 = 2 x
d
Hence, the periodic time varies as the square root of the alti-
tude of the conic pendulum, let the radius of the base be what
it may. Also, when I c Q=I C P-45°, the centrifugal force of
each ball is equal to its weight.
32/F
6
Inquiries connected with Rotation and Central Forces.
1. Suppose the diameter of a grindstone to be 44 inches, and
its weight half a ton; supposc also that it makes 326 revolu-
tions in a minute. What will be the centrifugal force, or its
tendency to burst?
π
2
2 g x²w_44×3.141593º w
Here F=
½ g 1º 16 × (3)
the measure of the required tendency.
60 2
2. If a fly wheel 12 feet diameter, and 3 tons in weight, re-
volve in 8 seconds: and another of the same weight revolves
in 6 seconds: what must be the diameter of the last, when their
centrifugal force is the same?
By art. 8, Central Forces, F:ƒ::
"
or d
D d
713 12
D
D 13
is = f,
d
T2 t2
62 feet, the answer.
T
6-1
3. If a fly of 12 feet diameter revolve in 8 seconds, and another
of the same diameter in 6 seconds: what is the ratio of their
weights when their central forces are equal?
By art. 6, Central Forces, the forces are as the squares of
the times inversely when the weights are equal: therefore when
the weights are unequal, they must be directly as the squares of
the times, that the central forces may be equal.
47.22 w = 23.6 tons,
12 × 36
Therefore, since F
Hence w: w:: 36: 64 :: 1:17
That is, the weight of the more rapidly to that of the more
slowly revolving fly, must be as 1 to 13, in the case proposed.
264
DYNAMICS: FLY-WHEELS.
4. If a fly 2 tons weight and 16 feet diameter, is sufficient
to regulate an engine when it revolves in 4 seconds; what
must be the weight of another fly of 12 feet diameter re-
volving in 2 seconds, so that it may have the same power upon
the engine?
Herc, by art. 8, Central Forces, we must have
therefore w =
2
W D 19 40 cwt. X 16 × 4 160
d r³
12 X 16
12
weight of the smaller fly.*
Note. A fly should always be made to move rapidly. If it
be intended for a mere regulator, it should be near the first
mover. If it be intended to accumulate force in the working
point, it must not be far separated from it.
5. Given the radius R of a wheel, and the radius r of its axle,
the weight of both, w, and the distance of the centre of gyration
from the axis of motion, g; also a given power p acting at
the circumference of the wheel; to find the weight w raised
by a cord folding about the axle, so that its momentum shall be
a maximum.
162
R
2
Here w=
J
4
4
2
√(R¹ p²+-2 R² Pę² w+g¹w²+P ‚w R r ç²+P² R³ r) — R³ P
Р
202
4
Cor. 1. When R, as in the case of the single fixed pul-
ley then w= √(2 p² R³+2 R P ç² w + W² + P W R g³)
R
W
2
P.
P
v2
Jaba
2
R[p w²+ p³ (q+w)] P w
W
P (q+w)
; therefore r=
v2 w
gr
}
2
Cor. 2. When the pulley is a cylinder
R², and the express, becomes w =
+ ‡ w²)] — ½ w
ያ
2
P.
/
6. Let a given power r be applied to the circumference of a
wheel, its radius R, to raise a weight w at its axle, whose radius
is r, it is required to find the ratio of R and 7, when w is raised
with the greatest momentum; the characters w and e denoting
the same as in the last proposition.
2
Here r
* Since
W
gr
r, and velocity », are given.
If r and t, the time, are given, then r
other of these r the force may be found.
1
22 20
32 r
W D
m³
13} cwt., the
w d
t?
}
1
4 π2 r w w´ 1.2273 r
812
12
;
พ.
of uniform matter
[R³ (2 p² + P w
2
32 w.
nearly; when the weight w, radius
From one or
1
DYNAMICS: INQUIRIES IN ROTATION.
265
1
Cor. When the inertia of the machine is evanescent, with
respect to that of p+w, then is r = R √(1+)-1.
W
7. If any machine whose motion accelerates, the weight will
be moved with the greatest velocity when the velocity of the
power is to that of the weight as 1+P √(1+) to 1; the in
P
W
ertia of the machine being disregarded.
8. If in any machine whose motion accelerates, the descent
of one weight causes another to ascend, and the descending
weight be given, the operation being supposed continually re-
peated, the effect will be greatest in a given time when the as-
cending weight is to the descending weight as 1' to 1618, in
the case of equal heights; and in other cases when it is to the
exact counterpoise in a ratio which is always between 1 to 1½
and 1 to 2.
9. The following general proposition with regard to rotatory
motion will be of use in the more recondite cases.
If a system of bodies be connected together and supported at
any point which is not the centre of gravity, and then left to
descend by that part of their weight which is not supported,
2 g multiplied into the sum of all the products of each body into
the space it has perpendicularly descended, will be equal to the
sum of all the products of each body into the square of its ve-
locity.
Percussion or Collision.
1. DEFS. In the ordinary theory of percussion, or collision,
bodies are regarded as either hard, soft, or elastic. A hard
body is that whose parts do not yield to any stroke or percus-
sion, but retains its figure unaltered. A soft body is that whose
parts yield to any stroke or impression, without restoring them-
selves again, the shape of the body remaining altered. An elas-
tic body is that whose parts yield to any stroke, but presently
restore themselves again, so that the body regains the same
figure as before the stroke. When bodies which have been sub-
jected to a stroke or a pressure return only in part to their
original form, the elasticity is then imperfect: but if they re-
store themselves entirely to their primitive shape, and employ
just as much time in the restoration as was occupied in the
compression, then is the elasticity perfect.
J
It has been customary to treat only of the collision of bodies
perfectly hard or perfectly elastic but as there do not exist
in nature any bodies (which we know) of cither the one or the
:
}
2 A
266
DYNAMICS: COLLISION OF BODIES.
F
other of these kinds, the usual theories are but of little service
in practical mechanics, except as they may suggest an extension
to the actual circumstances of nature and art.
2. The general principle for determining the motions of bo-
dies from percussion, and which belongs equally to both elastic
and non-elastic bodies, is this: viz. that there exists in the bo-
dies the same momentum, or quantity of motion, estimated in
any one and the same direction, both before the stroke and after
it. And this principle is the immediate result of the law of
nature or motion; that reaction is equal to action, and in a con-
trary direction; from whence it happens, that whatever motion
is communicated to one body by the action of another, exactly
the same motion does this latter lose in the same direction, or
exactly the same does the former communicate to the latter in
the contrary direction.
From this general principle too it results, that no alteration.
takes place in the common centre of gravity of bodies by their
actions upon one another; but that the said common centre of
gravity perseveres in the same state, whether of rest or of uni-
form motion, both before and after the impact.
3. If the impact of two perfectly hard bodies be direct, they
will, after impact, cither remain at rest, or move on uniformly
together with different velocities, according to the circumstances
under which they met.
Let в and b represent two perfectly hard bodies, and let the
velocity of в be represented by v, and that of b by v, which
may be taken either positive or negative, according as b moves
in the same direction as B, or contrary to that direction, and it
will be zero when b is at rest. This notation being understood,
all the circumstances of the motions of the two bodies, after
collision, will be expressed by the formula:
B v + b c
velocity
B+ b
which being accommodated to the three circumstances under
which v may enter become
velocity
B v + b v
B+ b
B V b v
velocity
{when both bodies moved in
same direction
{ when the bodies moved in
B + b
contrary
B+b when the body b was
at rest.
velocity
B V
These formula arise from the supposition of the bodies
being perfectly hard, and consequently that the two after
impact move on uniformly together as one mass.
In cases
I
DYNAMICS: COLLISION OF BODIES.
267
1
}
of perfectly elastic bodies, other formulæ have place which ex-
press the motion of each body separately; as in the following
proposition.
4. If the impact of two perfectly elastic bodies be direct,
their relative velocities will be the same both before and after
impact, or they will recede from each other with the same
velocity with which they met; that is, they will be equally
distant, in equal times, both before and after their collision,
although the absolute velocity of each may be changed. The
circumstances attending this change of motion in the two
bodies, using the above notation, are expressed in the two fol-
lowing formulæ :
2 b v + (B
B+ b
2 B V + (B — b) v
{
2 b v + (B
B + b
(B
2 B V
b) v
B+ b
which needs no modification, when the motion of b is in the
same direction with that of B.
5. In the other case of b's motion, the general formulæ be-
come
2 B V
the velocity of B
the velocity of b
b) v
b) v
the velocity of B
the velocity of b
B+ b
when b moves in a contrary direction to that of B, which arises
from taking v negative.
(B — b) V
A
B+ b
And
the velocity of B
the velocity of b
B + b
when b was at rest before impact, that is, when v — 0.
6. If a perfectly hard body в, impinge obliquely upon a per-
fectly hard and immoveable plane A D, it will after collision
move along the plane in the direction c a.
And its velocity before impact
Is to its velocity after impact
As radius
B
Is to the cosine of the angle в O D.
But if the body be elastic, it will re-
bound from the plane in the direction
c E, with the same velocity, and at the
same angle with which it met it, that is,
the angle A & E will be equal to the an-
gle B C D.
E
HUVUDSTANDORLIKDO
A
Wh

C
B
FLAIKATAULIFLANDI
U
268
DYNAMICS: COLLISION OF BODIES.
1
pla
7. In the case of direct impact, if в be the striking body, b
the body struck, v and v their respective velocities before im-
pact, u and u their velocities afterwards; then the two follow-
ing are general formulæ : viz.
V
U = V
Jak
UV
N
u = v + n
JA
B+
V
21) B
B+
In these, if n 1, they serve for non-elastic bodies; if n =
2, for bodies perfectly elastic. If the bodies be imperfectly
elastic, n has some intermediate value.
When the body struck is at rest, the preceding equations
become,
U
+ b
N V B
U (B + b)
B + b
B V
n v b
B + b
from which the value of n may be determined experimentally.
b
8/14 × 7
4.
}
8. In the usual apparatus for experiments on collision, balls
of different sizes and of various substances are hung from differ-
HR M N is an arc
ent points of suspension on a horizontal bar.
of a circle whose centre is s; and its graduations, 1, 2, 3, 4, 5, &c.
indicate the lengths of cords, as
measured from the lowest point.
Any ball, therefore, as P, may be
drawn from the vertical, and
made to strike another ball hang-
ing at the lowest point, with any
assigned velocities, the height to
which the ball struck ascends on
the side A M furnishes a measure
of its velocity; and from that the
value of n may be found from the
last equation. Balls not required in an individual experiment
may be put behind the frame, as shown at a and B.
The cup c may be attached to a cord, and carry a ball of
clay, &c. when required.
Example.-Suppose that a ball weighing 4 ounces strikes
another ball of the same substance weighing 3 ounces, with a
velocity of 10, and communicates to it a velocity of 84: what,
in that case, will be the value of n?
U (B + b)
Here n =
141375, the
B V
4 X 10
index of the degree of elasticity; perfect elasticity being indi-
cated by 2.
H
M
572
40
N
1
A
S

B
R
N
DYNAMICS: PRINCIPLES OF CHRONOMETERS.
269
Principles of Chronometers.
1. Clockwork, regulated by a simple balance, is inadequate
to the accurate mensuration of time.
2. Clockwork, regulated by a pendulum vibrating in the arch
of a circle, is of itself inadequate to the accurate mensuration of
time.
1st. Because the vibrations in greater or smaller arches are
not performed in equal times. 2dly. Because the length of
the pendulum is varied by heat and cold.
1
3. Clockwork, regulated by a pendulum vibrating in the
arch of a cycloid, is inadequate to the accurate mensuration of
time.
The isochronism of the vibrations of a cycloidal pendulum
in greater and smaller arches is true only on the hypothesis,
that the pendulum moves in a non-resisting medium, and that
the whole mass of the pendulum is concentrated in a point,
both of which positions are false. For these reasons the
application of the cycloid in practice has been entirely relin-
quished.
4. Modern time-keepers owe almost the whole of their su-
periority over those formerly made to two things; 1st, the ap-
plication of a thermometer; 2dly, the particular construction
of the escapement.
36
5. Metals expand by heat and contract by cold. This is
proved experimentally by the pyrometer. Metallic bars of
the same kind are found to expand in proportion to their length.
Metals of different kinds expand in different proportions: thus
the expansion of iron and steel are as 3, copper 44, brass 5, tin
6, lead 7. Hence pendulum rods, expanding and contracting
by the successive changes of temperature, affect the going of
the clocks to which they are applied.
Various have been the contrivances to correct the errors of
pendulums from their contraction and expansion by heat and
cold; the principal of these are described under the subject of
pendulums (p. 253).
6. The balance of a watch is analogous to the pendulum in
its properties and use.
The simple balance is a circular annulus, equally heavy in
all its parts, and concentrical with the pivots of the axis on
which it is mounted. This balance is moved by a spiral
spring called the balance-spring, the invention of the ingenious
Mr. Hook.
7. The pendulum requires a less maintaining power than the
balance.
2 A 2
1
270
DYNAMICS: PRINCIPLES OF CHRONOMETERS.
"
Hence the natural isochronism of the pendulum is less dis-
turbed by the relatively small inequalities of the maintaining
power.
8. The spring's elastic force which impels the circumference
of the balance, is directly as the tension of the spring; that is,
the weights necessary to counterpoise a spiral spring's elastic
force, when the balance is wound to different distances from
the quiescent point, are in the direct ratio of the arcs through
which it is wound.
9. The vibrations of a balance, whether through great or
small arches, are performed in the same time.
For the accelerating force is directly as the distance from
the point of quiescence; hence, therefore, the motion of the
balance is analogous to that of a pendulum, vibrating in cy-
cloidal arches.
10. The time of the vibration of a balance is the same as if
a quantity of matter, whose inertia is equal to that by which
the mass contained in the balance opposes the communica-
tion of motion to the circumference, described a cycloid
whose length is equal to the arc of vibration described by the
circumference, the accelerating force being equal to that of the.
balance.
Because in both cases the spaces described would be equal,
as also the accelerating forces in corresponding points, and
therefore the times of description.
11. If 1 denote the accelerating force of gravity, L the
length of a pendulum vibrating seconds in a cycloid, a the
semi-arc of vibration of the balance, T the time of vibration,
and the accelerating force of the balance, then will T =
a
LX F
12. Let g be the space which a body falling freely from
a state of rest describes in 1", and p 3.141593 the circum-
ference of a circle whose diameter is unity, then will
T
p³ ɑ
5 F
In this expression for the time of vibration, the letter a de-
notes the length of the semi-are of vibration; if this are
should be expressed by a number of degrees, cº, and r be the
and this quan-
radius of the balance, then a will be Prco
;
180°
tity being substituted for a, the time of a vibration will be r =
M
DYNAMICS: PRINCIPLES OF CHRONOMETERS.
271
P
pc r
2 g F
p³ co
grX 180°; let the given arc be 90°, in this case T =
FX
13. If the spring's elastic force, when wound through the
given angle or arc a 90° from the quiescent position, be =
P; the weight of the balance, and the parts which vibrate
with it w, the distance of the centre of gyration from the
axis of motion s, then will r =
|w p³ r
W
4 P g
w p ³ ę z
z p r g
These are expressions for the time of a vibration, whatever
may be the figure of the balance, the other conditions remain-
ing the same as above stated. If the balance be an annulus or
and the time of vibration T =
a cylindrical plate, ≤
2°
√2'
14. The times of vibration of different balances are in a
ratio compounded of the direct subduplicate ratios of their
weights and semidiameters, and the inverse subduplicate ratio
of the tensions of the springs or of the weights which counter-
poise them, when wound through a given angle.
15. The times of vibration of different balances are in a
ratio compounded of the direct simple ratio of the radii, and
direct subduplicate ratio of their weights, and the inverse
subduplicate ratio of the absolute forces of the springs at a
given tension.
16. Hence the absolute force of the balance spring, the
diameter and weight of the balance being the same, is inversely
as the square of the time of one vibration.
17. The absolute force or strength of the balance spring, the
time of one vibration, and the weight of the balance being the
same, is as the square of the diameter and the balance.
18. The weight of the balance, the strength of the spring
and time of vibration being the same, is inversely as the square
of the diameter.
Hence a large balance vibrating in the same time, with the
same spring, will be much lighter than a small one.
19. If the rim of the balance be always of the same breadth
and thickness, so that the weight shall be as the radius, the
4
272
DYNAMICS: PRINCIPLES OF CHRONOMETERS.
strength of the spring must be as the cube of the diameter of
the balance, that the time of vibration may continue the
same.
20. If a balance be made with two balls joined by a rod, and
the weights and distances of these balls from their common
centre of motion be unequal, but such that each separately
would vibrate in the same time; the centre of gravity of these
balls will not coincide with their centre of motion, nor will
they poise each other.
21. The momentum of the balance is increased better by
increasing its diameter than its weight.
23. A stronger balance-spring is preferable to a weaker.
Because the force of this spring upon the balance remain-
ing the same, whilst the disturbing force varies, the errors
arising from the variation will be less, as the fixed force is
greater.
23. The longer a detached balance continues its motion the
better.
Because, 1st. The friction in this case is less, and therefore
the natural isochronism of the vibration is less disturbed.
2dly. When applied to the watch, it requires a less maintain-
ing power, and therefore the variations in the intensity of the
maintaining power will be less. 3dly. The maintaining power
being less, the friction of the wheel-work will be less, and
therefore the motion more regular., 4thly. The pressure on the
escapement will be less, and therefore the oscillations of the
balance less disturbed.
24. The greater is the number of vibrations performed by
a balance in a given time, the less susceptible is it of external
agitations.
25. Slow vibrations are preferable to quick vibrations: but
there is a limit; for if the vibrations be too slow, the watch
will be liable to stop.
If we regarded only the effect of external agitations, balances
that vibrate quick should be preferred to such as vibrate slow;
but they are attended with two inconveniences, greater than
that which we would avoid. 1st. In two balances of the
same weight and diameter, the friction on the pivots increases
with the number of vibrations. 2dly. It appears by ex-
perience that the motion of the same detached balance con-
tinues longer, when its vibrations are slow, than when they are
quick.
Ad
26. A balance should describe as large arches as possible, as
suppose 240°, 260°, 300°, or an entire circle.
First, because the momentum of the balance is thus in-
DYNAMICS: PRINCIPLES OF CHRONOMETERS.
273
creased; and therefore the inequalities in the force of the
maintaining power bear a less proportion to it, and of conse-
quence will have less influence. 2dly. The balance is less sus-
ceptible of external agitations. 3dly. A given variation in the
extent of the vibrations produces a less variation in the going
of the machine. But care must be taken, that in these great
vibrations, the spring shall neither touch any obstacle, nor its
spires touch each other in contracting.
27. The times of vibration in larger arches are sometimes
shorter, sometimes longer, than in less arches.
28. A uniform spiral spring may be rendered perfectly iso-
chronal, by adjusting its length and number of spires.
:
This is the opinion of Mr. Berthoud. His reasoning seems
to be this if the spring forming a spiral of a certain species
be so disposed, that when wound through different angles, the
accelerating elastic forces of the spires, from the centre to-
wards the circumference, increase faster than they ought to do
in order to render the vibrations isochronal, it may be other-
wise so disposed, namely, by making the spires approach more
nearly to equality with each other in succession, that the law
shall vary in such a manner as absolute isochronism requires.
But as the fundamental property of springs, namely, that as
the tension is, so is the force, is determined by experiment,
so must this property likewise be ascertained in the same
manner. Accordingly Berthoud tells us, that having attached
to a balance a spiral of very large folds, making but three
turns, and whose diameter was 15 lines, the angles through
which it was wound being successively 5°, 10°, 15°, 20°, 25°,
30°, 35°, 40°, 45°, 60°, 120°, the counterpoising weights in
grains were 10, 21, 32, 42, 54, 65, 76, 88, 99, 134, 278.
The same spring forming very small spires, making five
turns in eight lines diameter, the angles through which it
was wound being the same as before, the counterpoising
weights were 11, 22, 33, 45, 56, 67, 78, 89, 100, 133, 250
grains. These experiments, he tells us, were made with great
care; and they show that the same spiral, its length continuing
unchanged, when folded in large and small spires, has a suf-
ficient difference in its progression to vary its isochronism:
when folded in large spires, according to the first experiment,
the vibrations in larger arcs are accelerated; and by the second
experiment, when folded in narrow spires, they are rendered
slower.
29. A spiral spring may be rendered isochronal by a
proper adjustment of its strength and thickness in different parts.
274
DYNAMICS: CHRONOMETERS.
30. A spiral spring which is not isochronal, may be rendered
such by the addition of two auxiliary springs, whose points of
quiescence are properly adjusted.
This was the ingenious invention of Mr. Mudge; the theory
of which construction is delivered in the Phil. Trans. for the
year 1794, by Mr. Atwood.
31. The influence of the maintaining power on the balance,
in restoring the motion which it loses by friction, or other-
wise, may be either constant or interrupted.
This depends on the escapement; when the action of the
maintaining power is constant, the escapement is called either
the recoil or the dead-beat; when it is interrupted, the escape-
ment is said to be detached.
32. By escapement is understood the means by which the
action of the wheels is applied to maintain the vibration of the
balance; and it consists of the balance-wheel and pallets.
33. Pallets are small plates or levers attached to the axis or
verge of the balance, which received the impulse of the balance-
wheel produced by the maintaining power, and thus continually
renew the motion which the balance loses by friction, or other
resistance.
In a recoil escapement, when one tooth of the balance-wheel
drops off the first pallet, the other acting tooth falls on the in-
clined plane of the other pallet, which, meeting it obliquely,
causes the balance-wheel to recoil, from which circumstance
this escapement derives its name.
Jag
In the dead-beat escapement, when one tooth of the balance-
wheel drops off the inclined plane of the first pallet, the other
acting tooth immediately falls upon the convex surface of the
other pallet, which surface being concentrical with the axis of
the balance, the wheel continues at rest until, by the motion
of the pallet or cylinder, the inclined plane of the tooth comes
to act upon the face of this latter pallet or edge of the cylin-
der, which then, by its pressure on that edge, throws the cy-
linder round, and thus gives motion to the balance; then
instantly entering the cavity of the cylinder, it falls upon the
concave surface, and for the same reason as before continues at
rest, until the balance spring drives the cylinder round in a con-
trary direction to what it did before, so as that the inclined plane
of the tooth may act on the second edge of the cylinder; which
pressure throws the cylinder round in the contrary direction,
and the tooth gets out of the cavity, and at that instant the sub-
sequent tooth falls upon the convex surface, and so on. From
the quiescence of the balance-wheel during the interval of time
DYNAMICS: ESCAPEMENTS.
275
:
that elapses between the falling of the acting tooth on the sur-
face and its pressure on the edge of the cylinder, this escape-
ment is called the dead-beat.
In the detached escapement the motion of the maintaining
power is suspended during almost the whole time of vibration;
just at the end of the return of the balance it unlocks the wheel-
work, and a tooth of the balance-wheel, immediately acting on
the pallet, restores the motion which the balance had lost; and
having given its impulse, the wheel-work is instantly locked
again, and the balance performs its vibration freely and dis-
engaged from all other parts of the machine.
34. In the escapement of recoil, the vibrations are quicker
than if the balance or pendulum vibrated freely.
For the recoil shortens the ascending part of the vibration by
contracting the extent of the arc and the reaction of the wheel
accelerates the descending part of the vibration.
35. In the dead-beat escapement, the vibrations are slower
than when they are performed in a detached state.
For the pressure of the tooth on the surface of the cylinder
retards that part of the vibration which is performed while the
cylinder, by the motion of the balance spring, revolves so far
as to bring the tooth to the edge of the cylinder and if the
maintaining power be increased, the pressure of the tooth on
the cylinder may become so great as entirely to stop the motion.
When the tooth has communicated its impulse to the edge of
the cylinder, it moves almost freely; and as the tooth does not
yet press with its entire force on the next surface, the cylinder
will indeed describe a larger arc, and therefore on that account
the time may be shortened; but when it has consumed all the
impulse of the wheel, it returns by the sole force of elasticity;
now the pressure of the tooth causes a friction which diminishes
the tendency to return to the point of rest, so that the balance
performs its vibrations slower.
36. In the escapement of recoil, if the maintaining power be
increased, the vibrations will be performed in larger arches, but
in less time.
Because the greater pressure of the crown wheel on the pal-
let will cause the balance to vibrate through larger arches; and
the time, on this account, will be less increased, than it will
be diminished by the acceleration of the balance by that pres-
sure, and the diminution of the time of recoil.
37. In the escapement of the cylinder or dead-beat, an in-
crease of the maintaining power renders the vibrations larger,
and at the same time slower.
Because the greater pressure of the tooth on the edge of
the cylinder throws it round through a greater arch; and its
276
DYNAMICS: ESCAPEMENTS.
increased pressure on both surfaces of the cylinder retards its
motion.

The upper part of the first marginal dia-
gram exhibits the anchor recoil; the lower
Graham's dead-beat escapement.
1
The second diagram represents Mr. Ar-
nold's watch escapement. The pin a, pro-
jecting from the verge or axis of the balance,
moving towards B, carries before it the
spring B, and with it the stiffer spring c, so
as to set at liberty the tooth D, which rests
on a pallet projecting from the spring. The
angle E of the principal pallet has then just
passed the tooth F, and is impelled by it
until the tooth & arrives at the detent. In
the return of the balance, the pin a passes
easily by the detent, by forcing back the spring B.
H serves to adjust the position of the detent, which presses
against it.
The screw

E
HO
D
38. The escapement can render those vibrations only iso-
chronal, whose inequality proceeds from the maintaining power,
and not such as are produced by external agitations.
39. The effect of external agitations on the balance may be
counteracted by the double escapement.
In this escapement, two equal balances are so connected, that
they vibrate through equal angles, but in contrary directions;
by which means, the one must always be accelerated as much
as the other is retarded by any external agitation. But as Mr.
Cummins observes, when balances are connected by means of
teeth, there arises a resistance which, however small, when ap-
plied in this most delicate part, will tend to diminish the mo-
mentum of the balances.
40. That escapement is best in which the duration of the
action of the balance-wheel on the pallets is least with respect
to the time of vibration.
Hence the detached escapement is the best, which appears
C
DYNAMICS: ESCAPEMENTS.
277
to have been the invention of the ingenious artist, Mr. Thomas
Mudge, who made a watch on this construction for the late
King of Spain, Ferdinand VI., in the year 1755.
41. The time of the vibration of the balance is increased by
heat, and diminished by cold.
First, because the length of the spiral spring is increased by
heat, and therefore its force diminished, and the contrary by
cold. 2dly. The diameter of the balance is increased by heat,
and therefore also the time of vibration; and the contrary by
cold.
42. That balance is the most perfect which, without the com-
pensation of a thermometer, is most subject to the influence of
heat and cold.
Because the obstructions from oil and friction act as a com-
pensation to the expansion or contraction of the spring and
balance; therefore that balance which is most affected, is freest
from the influence of oil and friction.
43. The errors in the going of a watch, arising from the
change of temperature, may be corrected by varying the length
of the balance spring.
Nevertheless, as it is extremely difficult to form an isochro-
nal spiral, any variation in its length is dangerous, because we
shall thus probably lose that point which determines its iso-
chronism.
44. The errors in the going of a watch, occasioned by the va-
riation of temperature, may be corrected by varying the diame-
ter of the balance.
This may be effected by dividing the
rim of the balance into two or more
separate parts, & D, I F, н E, each of
which is composed of two plates of me-
tal of different expansibility, riveted
together, the least expansible being near-
est the centre N, and carrying at one end
D, F, E, a weight; whilst the other is con-
nected either with the rim of the balance,
or one of its radii. Now if the tempe-
rature increase, the exterior plate expanding more than the in-
terior, the compound will become more concave towards the
centre; and consequently the end which carries the weight will
approach the centre of the balance, and on that account the
vibrations will be rendered quicker. At the root of cach ther-
mometer, there is a screw a, I, H, by which the diameter of the
balance may be increased or diminished, so as to alter the time
kept by the chronometer, without interfering with the adjust-
ment for heat and cold; and if the magnitude and position of

D
I
mory
F
B
H
K
37
2 B
278
DYNAMICS: SELECT MECHANICAL EXPEDIENTS.
the weights be properly regulated, they will correct the error
arising from the variation of the diameter of the balance caused
by the variation of temperature. (M. Young's Analysis.)
The reader who wishes to acquire practical knowledge on this
subject, may advantageously consult Hatton's Introduction to
the Mechanical part of Clock and Watch Work.
Select Mechanical Expedients.
Although a full account of the principal contrivances for
transmitting motion, and changing its rate, its direction, or its
character, would carry us much beyond the assigned limits; yet
it seems advisable to give a few of these, which are, therefore,
here presented.
1. Spiral Gear. In the ordinary cases, the teeth of wheels
are cut across their circumferences in a direction parallel to the
axis. But, in the spiral gear, now used a little in this country,
and still more in the American states, (especially in cotton-
mills,) the teeth are cut obliquely, so that if
they were continued they would pass round
the axis like the threads of a screw. By rea-
son of this disposition, the teeth come in con-
tact only in the line of the centres, and thus
operate, in great measure, without friction.
It must, however, be remarked, that the action
of these wheels is compounded of two forces,
one of which acts in the direction of the plane
of the wheel, the other in the direction of its
axis.
با
1
G
This spiral gearing is sometimes applied to clock-work, and
has this peculiarity, that it admits of a smaller pinion than any
other gearing.
2. Change of rotatory velocity. It is sometimes necessary
that a machine should be propelled with a velocity which is
not equable, but continually changes in a given ratio. Thus,
in cotton-mills, it is necessary that the speed of certain parts
of the machinery should continually decrease. from the begin-
ning to the end of an operation. To accomplish this, two
conical drums of equal size are placed with their axes parallel
1
1
ا.
1

DYNAMICS SELECT MECHANICAL EXPEDIENTS.
279
I
to each other, and with their larger diameters in opposite direc-
tions. They are connected by a belt, which
is so regulated by proper mechanism, that
it is gradually moved from one extremity of
the conic frustums or drums to the other;
and thus acting upon circles of different
diameter, causes a continual change of velo-
city in the driven cone with relation to that
which drives it.

4.
5
3
Thus, if the drum on the axis a b drives the wheel on the
axis A B, and the belt commences its operation at the ends a A;
the driven conic frustum will first revolve slower than that
which drives it, and will continue to move slower until the belt
has reached the middle of both, when the rotatory motions of
both will be equal: after that, the cone which is driven will
turn quickest, and will so continue, turning quicker and quick-
er both with respect to the other, and, in fact, until the belt
reaches the ends b, B.
A change of rotatory velocity, upon the same general prin-
ciple, is sometimes effected thus. A decreasing series of toothed
wheels is placed in the order of their size upon a common axis,
and fixed upon it. A corresponding series, but in an inverted
order, are placed upon another axis, and not fixed, but capable
of revolving about the axis like loose pulleys. The axis of
this second series is hollow, and contains a moveable rod,
which has a tooth projecting through a longitudinal slit in one
side of the axis; and this tooth serves to lock any one of the
wheels by entering a notch cut for its reception.
Thus, however, only one wheel can be locked
at a time, the others remaining loose. Hence,
the driven axis will revolve with a velocity
which is due to the relative size of the wheel
which is locked and that which drives it. Sup-
pose, for example, that the diameters of the
wheels are as 1, 2, 3, 4, and 5, upon the driving
axis, and the corresponding wheels, upon the
driven axis, are as the numbers 5, 4, 3, 2, 1. Then, when the
wheel 1 drives the wheel I, the rotatory velocity of the latter
will be one-fifth of that of the former. When 2 drives II, the
rotatory velocity communicated will be half that of the wheel
When 3 drives III, the velocities will be equal. When 4
drives IV, the rotatory velocity of the latter will be double that
of the former. And, when 5 drives V, the velocity of the lat-
ter will be five times that of the former. Different proportions
in the diameters of the wheels will, of course, give different
proportions in the velocities.
2.
1
2
b
A
UB

I
II
ΠΙ
V
IV
280
DYNAMICS: SELECT MECHANICAL EXPEDIENTS.
We owe this beautiful contrivance to the late Mr. Bramah.
It is sometimes requisite that a wheel or axis should move
with different velocities in different
parts of one and the same revolution.
This may be accomplished by an ec-
centric crown wheel acting upon and
driving a long pinion. Thus, if the
crown wheel in the margin rotates
uniformly upon a centre of motion c,

A
which is not the centre of the wheel, and the teeth of this crown
wheel play into the leaves of the long pinion p q, since the por-
tions of the crown wheel pass in contact with the pinion with
different velocities, as their distances from the centre of motion
c vary, the pinion p q will turn with an unequable or varying
velocity, depending upon the eccentricity of the centre of mo-
tion c.
B
to
3. Cams and Wipers. These are contrivances by means of
which beams placed vertically, or inclined aslant upwards,
may be made to advance over a small space in the direction
of their length, and then recede in the opposite direction;
and so on alternately. Eccentric circles, hearts, ellipses,
portions of circles, and projecting epicycloids, serve
communicate these kinds of motions. Thus, in the first of
the figures below, the circular eccentric cam, being put into
uniform rotation, the sliding or reciprocating part A B of the
machinery, will ascend and descend with a gentle, smooth
motion; being never at rest, unless at the very moment of
changing its direction. In the quadrant cam, represented in
the second figure, the reciprocating part A' B' will remain at
rest on the periphery of the cam during the first quarter of
the revolution, while, during the second, it will descend to
A/
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the axis of motion; during the third it will be at rest upon
the axis; and during the fourth it will return to its original
situation. The elliptical cam, in the third figure, turning upon
}
DYNAMICS: SELECT MECHANICAL EXPEDIENTS.
281
its centre, causes two alternate movements of A" B" for each re-
volution of the ellipse. In the fourth figure, a triple cam is ap-
plied to a tilt or trip hammer, turning upon a centre: there are
three epicycloidal cams, or wipers, as in this case they are often
called, causing three strokes of the hammer for one revolution
of the wheel, to whose circumference these wipers are attached
at equal distances.
4. Parallel motions is a term given to the contrivances by
which, especially in steam engines, circular motion, whether
continued or alternate, is converted into alternate rectilinear
motion, and vice versa. A moveable parallelogram is often,
and very successfully, employed for this purpose; as will be
described when we speak of the steam engine. From among
the numerous other contrivances for this purpose, we shall
select only one, which is very simple and elegant; and may
be used in saw mills, and other
reciprocating machines, as
well as in steam engines. This
is the invention of Dr. Cart-
wright. The reciprocating
motion of the piston rod or
other rod m n, in the same
rectilinear course is insured
by connecting it with two
equal cranks arranged in op-
position to each other, and
having their axes geared to-
gether by two equal teethed
wheels, w w, which play re-
gularly into each other.
5. Epicycloidal Wheel. This is another very beautiful
method of converting circular into alternate motion, or the
contrary. A B is a fixed wheel,
having teeth disposed uni-
formly on its inner rim. c is
a toothed wheel of half the
diameter of the fixed wheel, its

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centre c revolving about the
centre of the said fixed wheel.
While this revolution of the
wheel c is going on, any point
whatever on its circumference
will describe a straight line ;
or will pass and repass through
a diameter of the moving circle once during each revolution.
This is an elegant application of the well known mathematical
viv
CB
2 B 2
282
DYNAMICS: SELECT MECHANICAL EXPEDIENTS.
1
property, that if a circle rolls on the inside of another of
twice its diame er, the epicycloid described is a right line.
In practice, the piston rod, or other reciprocating part,
may be attached to any point on the circumference of the
wheel c.
6. Double Rack and Pinion.-This is a contrivance for an
alternating motion with a gradual change. A B is a double
rack, with circular ends, fixed to a beam that is capable of
moving in the direction of its
length. The rack is driven
by a pinion P, which is sus-
ceptible of moving up and
down in a groove m m', cut
in the cross piece. When the
pinion has moved the rack
and beam until the end в is

A
Applegat
m
m/
P
reached, the projecting a meets the spring s, and the rack is
pressed against the pinion. Then the pinion, working in the
circular end of the rack, will be forced down the groove
m m' until it works in the lower side of the rack, and moves
the beam back in the opposite direction; and thus the motion
is continued. The motion of the pinion in the groove will
be diminished, if, instead of a double rack, there be used a
single row of pins which are parallel to the axis of the pinion :
this plan is sometimes adopted in the machines called man-
gles.
7. The Universal Lever. This is a
French invention, and is often, from
the name of the inventor, called lever
de la Garousse. It consists of a bar
a b, moving upon a centre c, and
having a moveable catch, or hook, h h'
attached to each side, and acting upon
the oblique teeth of a double rack; so
that, as band a alternately rise and fall in
the reciprocal motion, the hooks h and h'
successively lay hold of the teeth of the
beam A B, and draw it up in the direction B a.
8. The Tachometer. This is a very ingenious contrivance,
which we
owe to Mr. Donkin, for the purpose of measuring
small variations in velocity. The contrivance is, in fact, hydro-
dynamical, but we mention it here, and the simplicity of its
principle will render it easy of comprehension.
If a cup with any fluid, as mercury, be placed upon a
spindle, so that the brim of the cup shall revolve horizon-
tally round its centre, then the mercury in the cup will
a
B

A
amg
Kad
DYNAMICS: SELECT MECHANICAL EXPEDIENTS.
283
1
assume a concave form, that is, the mercury will rise on the
sides of the cup, and be depressed in the middle; and the more
rapid the rotation of the cup, the more' will the surface of the
mercury be depressed in the middle and rise at the sides; the
figures being those of hollow paraboloids. Now, if the mouth of
this cup be closed, and a tube inserted into it, terminated in the
cup by a ball-shaped end, and half filled with some coloured
liquid, as coloured spirits of wine; then it is evident that the
more the surface of the mercury is depressed the more the fluid
in the tube will fall, and vice versa. Consequently, the velocity
of rotation of the cup, and of the spindle to which it is attached,
will be indicated by the height of the liquid in the tube; and,
indeed, absolutely measured by it, when the apparatus has been
subjected to the adequate preparatory experiments.
For a more minute description, see Trans. Soc. of Arts, vol.
xxvi., or the article Tachometer in the PANTOLOGIA.
Two plates, exhibiting a great variety of contrivances for
converting rotatory, reciprocating, and other motions, one into
the other, and thus facilitating the construction of machinery,
are given in my Treatise of Mechanics.
A large and valuable plate of the same kind, exhibiting 178
useful elementary mechanical combinations, has been lately pub-
lished at Manchester, and is sold in London by Ackerman.
I scarcely know a more interesting present than this would be
to a young mechanic.
}
284
HYDROSTATICS.
CHAPTER XI.
HYDROSTATICS.
1. Hydrostatics comprises the doctrine of the pressure and
the equilibrium of non-elastic fluids, as water, mercury, &c.
and that of the weight and pressure of solids immersed in
them.
2. DEF. A fluid is a body whose parts are very minute,
yield to any force impressed upon it (however small), and by
so yielding are easily moved among themselves.
:
Some attempt to give mechanical ideas of a fluid body by
comparing it to a heap of sand but the impossibility of giving
fluidity by any kind of mechanical comminution will appear by
considering two of the circumstances necessary to constitute a
fluid body: 1. That the parts, notwithstanding any compres-
sion, may be moved in relation to each other, with the smallest
conceivable force, or will give no sensible resistance to motion
within the mass in any direction. 2. That the parts shall gra-
vitate to each other, whereby there is a constant tendency to
arrange themselves about a common centre, and form a spherical
body; which, as the parts do not resist motion, is easily exe-
cuted in small bodies. Hence the appearance of drops always
takes place when a fluid is in proper circumstances. It is ob-
vious that a body of sand can by no means conform to these
circumstances.
Different fluids have different degrees of fluidity, according
to the facility with which the particles may be moved amongst
each other. Water and mercury are classed among the most
perfect fluids. Many fluids have a very sensible degree of tena-
city, and are therefore called viscous or imperfect fluids.
3. DEF. Fluids may be divided into compressible and in-
compressible, or elastic and non-elastic fluids. A compressi-
ble or elastic fluid is one whose apparent magnitude is dimi-
nished as the pressure upon it is increased, and increased by a
diminution of pressure. Such is air, and the different vapours.
An incompressible or non-elastic fluid is one whose dimensions
are not, at least as to sense, affected by any augmentation of
pressure. Water, mercury, wine, &c. are generally ranged un-
der this class.
M
It has been of late years proposed to limit the application
of the term fluids to those which are elastic, and to apply the
HYDROSTATICS: PRESSURE OF FLUIDS.
285
word liquid to such as are non-elastic. But it is an unnecessary
refinement.
4. DEF. The specific gravity of any solid or fluid body is
the absolute weight of a known volume of that substance,
namely, of that which we take for unity in measuring the capa-
cities of bodies.
Comparing this definition with that of density (DYNAMICS,
Def. 2), it will appear that the two terms express the same thing
under different aspects.
SECTION I.-Pressure of Non-elastic Fluids.
1. Fluids press equally in all directions, upwards, downwards,
aslant, or laterally.
This constitutes one essential difference between fluids and
solids, solids pressing only downwards, or in the direction of
gravity.
2. The upper surface of a gravitating fluid at rest is hori-
zontal.
3. The pressure of a fluid on every particle of the vessel con-
taining it, or of any other surface, real or imaginary, in contact
with it, is equal to the weight of a column of the fluid, whose
base is equal to that particle, and whose height is equal to its
depth below the upper surface of the fluid.
4. If, therefore, any portion of the upper part of a fluid be
replaced by a part of the vessel, the pressure against this from
below will be the same which before supported the weight of
the fluid removed, and every part remaining in equilibrium, the
pressure on the bottom will be the same as it would if the ves-
sel were a prism or a cylinder.
5. Hence, the smallest given quantity of a fluid may be made
to produce a pressure capable of sustaining any proposed weight,
either by diminishing the diameter of the column and increas-
ing its height, or by increasing the surface which supports the
weight.
6. The pressure of a fluid on any surface, whether vertical,
oblique, or horizontal, is equal to the weight of a column of the
fluid whose base is equal to the surface pressed, and height
equal to the distance of the centre of gravity of that surface be-
low the upper horizontal surface of the fluid.
7. Fluids of different specific gravities that do not mix, will
counterbalance each other in a bent tube, when their heights
above the surface of iunction are inversely as their specific
gravities.
A portion of fluid will be quiescent in a bent tube, when
38
286
HYDROSTATICS: PRESSURE OF FLUIDS.
the upper surface in both branches of the tube is in the same
horizontal plane, or is equidistant from the earth's centre. And
water poured down one branch of such a tube (whether it be of
uniform bore throughout, or not) will rise to its own level in the
other branch.
Thus water may be conveyed by pipes from a spring on
the side of a hill, to a reservoir of equal height on another
hill.
8. The ascent of a body in a fluid of greater specific gravity
than itself, arises from the pressure of the fluid upwards against
the under surface of the body.
9. DEF. The centre of pressure is that point of a surface
against which any fluid presses, to which if a force equal to the
whole pressure were applied it would keep the surface at rest,
or balance its tendency to turn or move in any direction.
-
10. If a plane surface which is pressed by a fluid be pro-
duced to the horizontal surface of it, and their common in-
tersection be regarded as the axis of suspension, the centres
of percussion and of pressure will be at the same distance
from the axis.
3
11. The centre of pressure of a parallelogram, whose upper
side is in the plane of the horizontal level of the liquid, is at 2
of the line (measuring downwards) that joins the middles of the
two horizontal sides of the parallelogram.
12. If the base of a triangular plane coincides with the
upper surface of the water, then the centre of pressure is at the
middle of the line drawn from the middle of the base to the
vertex of the triangle. But, if the vertex of the triangle be in
the upper surface of the water, while its base is horizontal, the
centre of pressure is at of the line drawn from the vertex
to bisect the base.
Illustrations and Applications.

1. If several glass tubes of different shapes.
and sizes be put into a larger glass vessel
containing water, the tubes being all open at
top; then the water will be seen to rise to the
same height in each of them, as is marked
by the upper surface a c, of the liquid in the
larger vessel.
2. If three vessels of equal bases, one cylindrical, the
TOURO:277387):
PANIMALES
i
HYDROSTATICS: PRESSURE OF FLUIDS.
287
second considerably larger at top than at bottom, the third con-
siderably less at top than at bottom, and with the sides of the
two latter either regularly or irregularly sloped, have their
bottoms moveable, but kept close by the action of a weight upon
a lever; then it will be found, that when the same weight acts
at the same distance upon the lever, water must be poured in to
the same height in each vessel before its pressure will force
open the bottom.
3. Let a glass tube, open at both ends (whether cylindrical
or not, does not signify), have a piece of bladder tied over one
end, so as to be capable of hanging below that end, or of rising
Pour into this
up within it, when pressed from the outside.
tube some water tinged red, so as to stand at the depth of seven
or eight inches; and then immerse the tube with its coloured
water vertically into a larger glass vessel nearly full of colour-
less water, the bladder being downwards serving as a flexible
bottom to the tube. Then, it will be observed that when the
depth of the water in the tube exceeds that in the larger vessel,
the bladder will be forced below the tube, by the excess of the
interior over the exterior pressure: but when the exterior water
is deeper than the interior, the bladder will be thrust up within
the tube, by the excess of exterior pressure: and when the water
in the tube and that in the larger vessel, have their upper sur-
faces in the same horizontal plane, then the bladder will adjust
itself into a flat position just at the bottom of the tube. The
success of this experiment does not depend upon the actual
depth of the water in the tube, but upon the relation between
the depths of that and the exterior water; and proves that in all
cases, the deeper water has the greater pressure at its bottom,
tending equally upward and downward.
130
4. The hydrostatical paradox, as it is usually denominated,
results from the principle that any quantity of a non-elastic
fluid, however small, may be made to balance another quan-
tity; or any weight, as large as we please. It may be illus-
trated by the hydrostatic bellows, consisting of two thick boards.
Dc, F E, each about 16 or 18 inches dia-
meter, more or less, covered or connected a
firmly with leather round the edges, to
open and shut like a common bellows, but
without valves; only a pipe A в, about 3
feet high, is fixed into the bellows above
F. Now let water be poured into the pipe
at A, and it will run into the bellows, gra-
dually separating the boards by raising
the upper one. Then if several weights,
as three hundred weights, be laid upon the upper board, by
*
d
Q
B
P000
Η
E
m

288
HYDROSTATICS: PRESSURE OF FLUIDS.
pouring the water in at the pipe till it be full, it will sustain all
the weights, though the water in the pipe should not weigh a
quarter of a pound: for the pipe or tube may be as small as we
please, provided it be but long enough, the whole effect depend-
ing upon the height, and not at all on the width of the pipe for
the proportion is always this,
As the area of the orifice of the pipe
is to the area of the bellows board,
so is the weight of water in the pipe, above D C,
to the weight it will sustain on the board.
5. In lieu of the bellows part of the apparatus, the leather of
which would be incapable of resisting any very considerable
pressure, the late Mr. Joseph Bramah used a very strong me-
tal cylinder, in which a piston moved in a perfectly air and
water tight manner, by passing through leather collars, and as
a substitute for the high column of water, he adopted a very
small forcing pump to which any power can be applied; and
thus the pressing column becomes indefinitely long, although
the whole apparatus is very compact, and takes but little room.
The marginal figure is a section of one
of these presses, in which c is the piston
of the large cylinder, formed of a solid
piece of metal turned truly cylindrical,
and carrying the lower board v of the
press upon it: u is the piston of the small
forcing pump, being also a cylinder of
solid metal moved up and down by the
handle or lever w. The whole lower
part of the press is sometimes made to
stand in a case xx, containing more
than sufficient water as at y, to fill both the cylinders; and
the suction pipe of the forcing pump u dipping into this water
will be constantly supplied. Whenever, therefore, the handle
w is moved upwards, the water will rise through the conical
metal valve ≈, opening upwards into the bottom of the pump
u; and when the handle is depressed, that water will be forced
through another similar valve a, opening in an opposite di-
rection in the pipe of communication between the pump and
the great cylinder b, which will now receive the water by which
the piston rod t will be elevated at each stroke of the pump u.
Another small conical valve c is applied by means of a screw
to an orifice in the lower part of the large cylinder, the use of
which is to release the pressure whenever it may be necessary;
for, on opening this valve, any water which was previously

11
บ
HYDROSTATICS: BRAMAH'S PRESS.
289
contained in the large cylinder b, will run off into the reservoir
y by the passage d, and the piston will descend; so that the
same water may be used over and over again. The power of
such a machine is enormously great; for, supposing the hand
to be applied at the end of the handle w, with a force of only
10 pounds, and that this handle or lever be so constructed as to
multiply that force but 5 times, then the force with which the
piston u descends will be equal to 50 pounds: let us next sup-
pose that the magnitude of the piston t is such, that the area of
its horizontal section shall contain a similar area of the smaller
piston u 50 times, then 50 multiplied by 50 gives 2500 pounds,
for the force with which the piston t and the presser v will
rise. A man can, however, exert ten times this force for a
short time, and could therefore raise 25,000 pounds; and
would do more if a greater disproportion existed between the
two pistons t and u, and the lever w were made more favour-
able to the exertion of his strength.
This machine not only acts as a press, but is capable of many
other useful applications, such as a jack for raising heavy loads,
or even buildings; to the purpose of drawing up trees by their
roots, or the piles used in bridge-building.
To find the thickness of the metal in Bramah's press,
to resist certain pressures, Mr. Barlow gives this theorem,
pr
t
where p
pressure in lbs. per square inch, r
с P
radius of the cylinder, t its thickness, and c = 18000 lbs.
the cohesive power of a square inch of cast iron.
Ex. Suppose it were required to determine the thickness of
metal in two presses, each of 6 inches radius, in one of which
the pressure may extend to 4278 pounds, in the other to 8556
pounds per square inch.
Here in the first case,
t
t
4278 × 6
18000- -4278
L
K
p
8556 × 6
18000 8556
1.87 inches, thickness.
In the second,
5.43 inches, thickness.
The usual rules, explained below (art. 10) would make the
latter thickness double the former extensive experiments are
necessary to tell which method deserves the preference.
6. If the breadth, and d the depth of a rectangular gate,
or other surface exposed to the pressure of water from top to
bottom; then the entire pressure is equal to the weight of a
prism of water whose content is b d². Or, if b and d be in
2 C
290
HYDROSTATICS PRESSURE OF SLUICE-GATES, &c.
feet, then the whole pressure
b d', in cwts.
II
7. If the gate be in form of a trapezoid, widest at top, then, if
B and b be the breadths at the top and bottom respectively, and
d the depth.
314 b d², in lbs. or nearly
6
Τι
(B
whole pressure in lbs.
whole pressure in cwts.
8. The weight of a cubic foot of rain or river water, is nearly
equal to cwt.
314 [} (в —
3 [ $ (B -
11
b) + b] d²
b) + b] de, nearly.
The pressure on a square inch, at the depth of THIR-ty feet
is very nearly THIR-teen pounds.
Pressure on a square foot, nearly a ton at the depth of thir-
ty-six feet. [The true depth is 35.84 feet.]
The weight of an ale gallon of rain water is nearly 10 lbs.
that of an imperial gallon 10 lbs.
The weight of a cubic foot of sea-water is nearly 4 of a cwt.
These are all useful approximations.
Thus, the pressure of rain water upon a square inch at the
depth of 3000 feet, is 1300 lbs.
And the pressure upon a square foot at the depth of 108 feet
is nearly three tons.
A
9. In the structure of dykes or embankments, both faces or
slopes should be planes, and the ex-
terior and interior slopes should make
an angle of not less than 90°. For if
A D' be the exterior slope, and the an-
gle D'An be acute, E D' perpendicular
to A B is the direction of the pressure
upon it; and the portion D'A E will probably be torn off. But
when DA is the exterior face, making with A в an obtuse angle,
the direction of the pressure falls within the base, and there-
fore augments its stability.
10. The strength of a circular bason confining water, requires
the consideration of other principles.
DF D'
E
B
The perpendicular pressure against the wall depends merely
on the altitude of the fluid, without being affected by the
volume. But, as professor Leslie remarks, the longitudinal
effort of the thrust, or its tendency to open the joints of the
masonry, is measured by the radius of the circle. To resist
that action in very wide basons, the range or course of stones
along the inside of the wall, must be proportionally thicker.
On the other hand, if any opposing surface present some con-
vexity to the pressure of water, the resulting longitudinal strain.
will be exerted in closing the joints and consolidating the build-
ing. Such reversed incurvation is, therefore, often adopted in
the construction of dams.
་

HYDROSTATICS: EMBANKMENTS, PIPES, &c.
291
In like manner, the thickness of pipes to convey water
h d
must vary in proportion to where h is the height of the
"
с
head of water, d the diameter of the pipe, and c the measure of
the cohesion of a bar of the same material as the pipe, and an
inch square.
A pipe of cast iron, 15 inches diameter, and of an inch
thick, will be strong enough for a head of 600 feet.
A pipe of oak of the same diameter, and 2 inches thick,
would sustain a head of 180 feet.
Where the cohesion is the same, t varies as hd or as
HDT hd: t, in the comparison of two cases.
*
Example.-What, then, must be the respective thicknesses
of pipes of cast iron and oak, each 10 inches diameter, to carry
water from a head of 360 feet?
•
HD (= 600 ×
360 × 10 X 3
600 × 15 x 4
Here, 1st. for cast iron :
15): T 4) :: h d (
10800
(
36000
1% of an inch.
10
2dly. for oak:
HD (= 180 × 15) T (= 2) :: h d (
360 × 10 × 2
98 23 inches.
180 × 15
40
15
360 × 10): t
360 × 10): t
SECTION II.-Floating Bodies.
1. If any body float on a fluid, it displaces a quantity of the
fluid equal to itself in weight.
2. Also, the centres of gravity of the body and of the fluid
displaced must, when the body is at rest, be in the same verti-
cal line.
3. If a vessel contain two fluids that will not mix (as water
and mercury), and a solid of some intermediate specific gravity
be immersed under the surface of the lighter fluid and float on
the heavier; the part of the solid immersed in the heavier fluid,
is to the whole solid as the difference between the specific gra-
* To ascertain whether or not a pipe is strong enough to sustain a proposed
pressure, it is a good custom amongst practical men to employ a safety valve, usually
of an inch in diameter, and load it with the proposed weight, and a surplus deter-
mined by practice. Then, if the proposed pressure be applied interiorly, by a forcing
pump, or in any other way, if the pipe remain sound in all its parts after the safe-
ty-valve has yielded, such pipe is regarded as sufficiently strong.
The actual pressures upon a pipe of any proposed diameter and head, may evi-
dently be determined by a similar method.
}
292
HYDROSTATICS: FLOATING BODIES.
vities of the solid and the lighter fluid, is to the difference be-
tween the specific gravities of the two fluids.
4. The buoyancy of casks, or the load which they will carry
without sinking, may be estimated by reckoning 10 lbs. avoir-
dupois to the ale gallon, or 8 lbs. to the wine gallon.
5
5. The buoyancy of pontoons may be estimated at about half
a hundred weight for each cubic foot.
Thus a pontoon which contained 96 cubic feet, would sustain
a load of 48 cwt. before it would sink.
N. B. This is an approximation, in which the difference be-
tween and, that is, of the whole weight, is allowed for
that of the pontoon itself.
ÎÎ
22
6. The principles of buoyancy are very ingeniously applied
in Mr. Farey's self-acting flood-gate. In the case of common
sluices to a mill-dam, when a sudden flood occurs, unless the
miller gets up in the night to open the gate or gates, the
neighbouring lands may become inundated; and, on the con-
trary, unless he be present to shut up when the flood subsides,
the mill-dam may be emptied and the water lost which he
would need the next day. To prevent either of these occur-
rences, Mr. John Farey, whose talent and ingenuity are well
known, has proposed a self-acting flood-gate, the following
description of which has been given in the Mechanics' Weekly
Journal.
1
A A represents a vertical section of a gate poised upon a
horizontal axis passing rather above the centre of pressure of
the gate, so as to give it a tendency to shut close.
a a is a
b

B
LIA
A
41
A
C
You
D⋅
E
HYDROSTATICS: FAREY'S FLOOD-GATE.
293
lever, fixed perpendicular to the gate, and connected by an iron
rod with a cask, b, floating upon the surface of the water, when
it rises to the line, B, D, which is assumed as a level of the wear
or mill-dam, B, C, E, F, in which the flood-gate is placed by
this arrangement it will be seen that when the water rises
above the dam, it floats the cask, opens the gate, and allows the
water to escape until its surface subsides to the proper level
at B, D; the cask now acts by its weight, when unsupported by
the water, to close the gate and prevent leakage. The gate
should be fitted into a frame of timber, H, K, which is set in the
masonry of the dam. The upper beam, H, of the frame being
just level with the crown of the dam, so that the water runs
over the top of the gate at the same time that it passes through
it to prevent the current disturbing the cask, it is connected
by a small rod, e, at each end, to the upper beam, H, of the frame,
and jointed in such a manner as to admit of motion in a vertical
direction.
Any ingenious mechanic will so understand the construction
from this brief account, as to be able to apply it to practice
when needed.
7. By means of the same principle of buoyancy it is, that
a hollow ball of copper attached to a metallic lever of about a
foot long, is made to rise with the liquid in a water-tub, and
thus to close the cock and stop the supply from the pipe, just
before the time when the water would otherwise run over the
top of the vessel.
8. This property, again, has been successfully employed in
pulling up old piles in a river where the tide ebbs and flows.
A barge of considerable dimensions is brought over a pile as
the water begins to rise a strong chain which has been pre-
viously fixed to the pile by a ring, &c. is made to gird the
barge, and is then fastened. As the tide rises the vessel rises too,
and by means of its buoyant force draws up the pile with it.
In an actual case, a barge 50 feet long, 12 feet wide, 6 deep,
and drawing two feet of water, was employed. Here,
2) × 4/
50 x 12 x 16
7
= 192 × 74
50 × 12 × (6
1344 + 279
1371 cwt. 66½ tons nearly, the measure of
the force with which the barge acted upon the pile.
न
Arts
SECTION III.-Specific Gravities.
1. If a body float on a fluid, the part immersed is to the whole
body, as the specific gravity of the body to the specific gravity
of the fluid.
39
2 c 2
294
HYDROSTATICS: SPECIFIC GRAVITIES.
•
Hence, if the body be a square or a triangular prism, and it
be laid upon the fluid, the ratio of that portion of one end which
is immersed, to the whole surface of that end, will serve to de-
termine the specific gravity of the body.
2. If the same body float upon two fluids in succession, the
parts immersed will be inversely as the specific gravities of
those fluids.
3. The weight which a body loses when wholly immersed
in a fluid is equal to the weight of an equal bulk of the
fluid.
When we say that a body loses part of its weight in a fluid,
we do not mean that its absolute weight is less than it was
before, but that it is partly supported by the reaction of the
fluid under it, so that it requires a less power to sustain or to
balance it.
4. A body immersed in a fluid ascends or descends with a
force equal to the difference between its own weight and the
weight of an equal bulk of fluid; the resistance or viscosity of
the fluid not being considered.
5. To find the specific gravity of a fluid or of a solid.
On one arm of a balance suspend a globe of lead by a fine thread,
and to the other fasten an equal weight, which may just balance
it in the open air. Immerse the globe into the fluid, and ob-
serve what weight balances it then, and consequently what
weight is lost, which is proportional to the specific gravity as
above. And thus the proportion of the specific gravity of one.
fluid to another is determined by immersing the globe succes-
sively in all the fluids, and observing the weights lost in each,
which will be the proportions of the specific gravities of the
fluids sought.
This same operation determines also the specific gravity of
the solid immerged, whether it be a globe or of any other shape
or bulk, supposing that of the fluid known. For the specific
gravity of the fluid is to that of the solid, as the weight lost is
to the whole weight.
Hence also may be found the specific gravity of a body that
is lighter than the fluid, as follows:
6. To find the specific gravity of a solid that is lighter
than the fluid, as water, in which it is pul.-Annex to the
lighter body another that is much heavier than the fluid, so as
the compound mass may sink in the fluid. Weigh the heavier
body and the compound mass separately, both in water and out
of it; then find how much cach loses in water, by subtracting
its weight in water from its weight in air; and subtract the less
of these remainders from the greater.
HYDROSTATICS: SPECIFIC GRAVITIES.
295
Then, as this last remainder,
Is to the weight of the light body in air,
So is the specific gravity of the fluid,
To the specific gravity of that body.
7. The specific gravities of bodies of equal weight, are reci-
procally proportional to the quantities of weight lost in the same
fluid. And hence is found the ratio of the specific gravities
of solids by weighing in the same fluids masses of them that
weigh equally in air, and noting the weights lost by each.
8. Instead of a hydrostatic balance, a hydrostatic steelyard
is now frequently employed. It is contrived to balance ex-
actly by making the shorter end wider, and with an enlarge-
ment at the extremity. The shorter arm is undivided, but the
longer arm is divided into short equal divisions: thus, if that
longer arm be 8 inches long, it may be divided into 400 parts,
the divisions commencing at A. Then, in using this instru-
A
C
(ULANGE VENDRELL Mone TOULOU
B
D
A** *** ** + -
*
E
Jala

ment, any convenient weight is suspended by a hook from a
notch at the end of the scale A. The body whose specific gra-
vity is to be determined, is suspended from the other arm by a
horse-hair, and slid to and fro till an equilibrium is produced.
Then, without altering its situation at p in the beam, it is im-
mersed in water, and balanced a second time by sliding the
counterpoise from A, say to c.
Here, evidently, weight in water weight in air :: BC: BA:
and loss of weight in water: weight in air :: A C : A B.
weight in air A B
Conseq.
specific gravity.
loss
A C
With such an instrument nicely balanced upon a convenient
pedestal, I find that the specific gravities of solids are ascertain-
able both with greater facility and correctness than with any
hydrostatic balance which I have seen.
*We owo this contrivance to Dr. Coates, of Philadelphia.
༧༥
296
HYDROSTATICS: SPECIFIC GRAVITIES.
Table of Specific Gravities.
Arsenic
Cast antimony
Cast zinc
Cast iron
Cast tin
Bar iron
Cast nickel
Cast cobalt
Hard steel
Soft steel
Cast brass
Cast copper
Cast bismuth
Dan
By
METALS.
Ja
Pure cast silver
Same hammered
Cast lead
Ambergris
Amber
Phosphorous
Brick
Sulphur
Opal
Te
1
1
+
រ
1
1
1
J
1
Buk
}
Id
i
STONES, EARTHS, &c.
shaw
Julietta
1
I
1
1
1
1
}
Makal
Tanah
I
I
Wat
1
thek
1
}
kated
1
Th
}
Ad
}
1
Sald
10,510
11,352
13,568
Mercury
Trinket gold
Gold coin
15,709
17,647
19,258
19,361
19,500
Pure cast gold
Same hammered
Pure platinum
Same hammered
Platinum wire
Platinum laminated,
20,336
21,041
22,069
or beat into leaves S
All metals become more dense or heavy by hammering.
1
1
bah
1
1
Spec.
Grav.
aglad
5,763
6,702
7,190
7,207
7,291
7,788
7,807
7,811
7,816
7,833
8,395
8,788
9,822
10,474
J
1,078
1,714
2,000
2.033
2,114
-
J
band
I
A
J
Ba
T
pamięć
Weight cub
inch, in
avoird. oz.
-
-А
G
J
S
Spec. Weight cub.
Grav. foot avoird. lbs.
926
3.335
3.878
4.161
4.165
4.219
4.507
4.513
4.520
4.523
4.533
4.858
5.085
5.684
6.061
6.082
6.569
7.872
9.901
10.212
11.145
11.212
11.285
11.777
12.176
12.763
125.00
127.06
}
HYDROSTATICS: SPECIFIC GRAVITIES.
297
STONES, &c. continued.
Gypsum, opaque
Stone, paving
Mill-stone
Stone, common
Flint and spar
Crystal
Granite, red Egyptian
Glass, green
white
bottle
Pebble
Slate
Pearl
-
Alabaster
Marble
I
Flint glass
Diamond
Beryl
Sapphire
Topaz
Garnet
Ruby
Porphyry
Emerald
Chrysolite, jewellers'
{
Wax
Tallow
Bees' wax
Camphor
Honey
Bone of an ox
Ivory
G
1
1
1
I
1
RESINS, GUMS, &c.
1
Chalk
Jasper
Basaltes (Giant's causey)
Hone, white razor
Limestone
J
1
I
f
1
Gay
1
I
}
1
}
}
1
1
1
1
1
1
Ca
}
1
I
1
1
T
I
I
1
}
#
1
↓
maj
Spec.
Grav.
2,168
2,416
2,484
2,520
2,594
2,653
2,654
2,642
2,892
2,733
2,664
2,672
2,684
2,730
2,742
2,765
2,775
2,782
2,784
2,816
2,864
2,876
3,179
3,329
3,521
3,549
3,994
4,011
4,189
4.283
I
f
1
H
1
B
-
M
Jeg
M
ple
-
M
Weight cub.
foot avoird. lb.
}
I
F
1
J
!
-
- 165·87
- 166.50
167.00
Gut
135.50
151.00
155.20
157.50
162.12
<span
- 171.38
172.81
- 174.00
Jada
179.00
- 179.75
198.68
Spec.
Grav.
897
945
965
989
1456
1659
1822
298
HYDROSTATICS: SPECIFIC GRAVITIES.
LIQUIDS.
Air, at the earth's surface, about
Sulphuric Ether
Alcohol absolute
Liquid Bitumen
Oil of Turpentine
Muriatic Ether
Olive Oil
Burgundy Wine
Wine of Bourdeaux
Port
Distilled Water
Sea Water
Milk
Beer
p
Poon
Willow
Cedar
Pitch Pine
Pear trée
Walnut
Mar Forest Fir
Elder tree
Beech
Orange wood
Cherry tree
Teak
Cand
Cork
Poplar
Larch
Elm, and N. E. Fir
Mahogany, Honduras
1
1
Madeira Wine
Nitric Acid
Water from the Dead Sea
Nitrous Acid
Sulphuric Acid
1
J
Maple and Riga Fir
Ash and Dan. Oak
B
WOODS.
I
1
I
t
1
Bud
I
1
1
I
}
1
Band
I
I
I
(
1
1
}
1
I
1
1
I
Kat
It
KRA
1
1
Ad
1
1
Gende
I
I
1
I
.[
I
}
1
Back
Pand
I
I
1
}
1
I
I
Pan
I
Wat
phas
1
pod
1
1
1
Spec.
Grav.
240
383
544
556
560
579
585
596
660
661
671
694
695
696
705
715
745
750
760
J
B
B
JOR
My
(d)
-
O
-
A
-
J
}
Kon
1
J
1
J
K
}
J
J
ht
J
Spec.
Grav.
Weight cub.
foot avoird. lb.
S
1/2/
716
792
848
870
874
915
991
994
997
1,000
1,028
1,030
1,034
1,038
1,218
1,240
1,550
1,841
15.00
23.94
34.00
34.75
35.00
36.18
36.56
37.25
41.25
41.31
- 41·94
43.37
43.44
- 43·50
44.06
44.68
46.56
46.87
47.50
HYDROSTATICS SPECIFIC GRAVITIES.
299
WOODS continued.
Yew, Dutch
Apple tree
Alder
Yew, Spanish-
Mahogany, Spanish
Oak, Canadian
Box, French
Logwood
Oak, English
Ditto, 60 years old
Ebony
Lignum Vitæ
-
G
-
}
1
1
1
•
-
·
}
•
Weight cub.
Grav. foot avoird. lb.
Spec.
788
793
800
807
852
872
912
913
970
1170
1331
1333
>
J
G
-
J
cast brass
cast copper
cast lead
49.25
- 49.56
50.00
50.44
53.25
54.50
57.00
57 06
60.62
73.12
Gudan
-
-
Jo
-
9. Since a cubic foot of water, at the temperature of 40° Fah-
renheit, weighs 1000 ounces avoirdupois, or 62½ lbs., the num-
bers in the preceding tables under the head Spec. Grav. exhibit
very nearly the respective weights in avoirdupois ounces of a
cubic foot of the several substances.
We have also given, in another column, the weight in ounces
of a cubic inch of each of the several metals: and, with re-
gard to different kinds of wood and stone, the weight in avoir-
dupois pounds of a cubic foot of each. These, of course, are
medium specific gravities and weights; for there are variations,
sometimes indeed considerable ones, between the specific gra-
vities of different specimens of the same kind of substance.
- 83.18
83.31
These additional columns will evidently facilitate the labour
of finding the magnitude of a body from its weight, or the
weight of a body from its magnitude. In this respect, too,
the following particulars will often be of utility.
10. (1) 430-25 cubic inches of cast iron weigh 1 cwt.
bar iron
397.60 .
368.88.
352.41
272.8
(2) 14.835 cubic feet of paving stone weigh a ton
14.222
common stone
13.505
granite
13.070
marble
12.874
chalk
11.273
limestone
64.460.
elin
64.000 .
Honduras mahogany
51.650
Mar Forest fir
51.494 .
beech
300
HYDROSTATICS: SPECIFIC GRAVITIES.
1
47.762 cubic feet of Riga fir
47.158
42.066
36.205
11. Prob. To find the internal diameter of a uniform capil-
lary or other small tube.
Let the tube be weighed when empty, and again when filled
with mercury, and let w be the difference of those weights in
troy grains, and the length of the tube in inches. Then the
diameter required, d=019252
12.
い
​Thus, if the difference of weights were 500 grains, and
the length of the tube were 20 inches: we should have d=
•019252
='019252 × 5·09626 of an inch.*
500
ash, and Dantzic oak
Spanish mahogany
English oak.
20
12. Prob. To find the weight of a leaden pipe.
If I be the length in feet, d the interior diameter, and t the
thickness both in inches and parts of an inch, w the weight in
hundred weights: then w=1382 l t (d+t).
For a cast iron pipe, the theorem is
w='0876 lt (d+t) ;
7
or nearly of the former expression.
TI
Ex. Let the internal diameter of a leaden pipe be 4 inches,
the thickness of an inch; required the weight of 12 feet in
length.
Here 1382 × 12 × 4 × 441382 × 123-1762 cwts.
13. Prob. To find the weight of the ring or rim of a cast iron
fly-wheel.
Supposing this ring or annulus to be similar to a portion of a
pipe of large diameter, the expression for the weight will be
similar to the above; but it may be advantageous to change
the notation. Let then be the interior diameter of the fly
in inches, d half the difference of the exterior and interior
diameters, r the thickness from side to side of the fly, and w its
weight in hundred weights: then w=0073 r d (D+d).
*The same thing may casily be accomplished thus:-Let a cone of box wood, or
of brass, be very accurately turned, of about 6 inches in length, and the diameter
of its base about a quarter of an inch; and let its curve surface bo very accurately
marked with a series of parallel rings, about a twentieth of an inch asunder, from
its vertex to its base. Insert this cono
carefully in the cylinder (so that their
axes shall coincide) as in the diagram:
then it will be as v A: Vu¦¦ A B¦ a
b
where, as the ratio of v a to v a is
known by means of the equidistant
rings on the surface, and A в is known,
a b becomes determined.
;

d
a
b
A
B
HYDROSTATICS: SPECIFIC GRAVITIES.
301
Ex. Let D=100 inches, d=5 inches, or the exterior diame-
110, and r the thickness = 4 inches.
ter =
Then 0073 rd (D+d) = 0073 × 4 × 5 × 105
··0073 × 4 × 5 × 105 —·0073 × 2100
=15:33 cwt., the required weight of the cylindrical rim.
Note. The reader will observe that this process is much
shorter than those usually employed, even with the aid of ta-
bles of circles already computed.
*See, farther, on kindred subjects, the approximate rules.
in mensuration, pa. 201, &c.
*
40
J
2 D
302
HYDRODYNAMICS.
CHAPTER XII.
HYDRODYNAMICS.
Hydrodynamics is that part of mechanical science which re-
lates to the motion of non-elastic fluids, and the forces with
which they act upon bodies.
This branch of mechanics is the most difficult, and the least
advanced whatever we know of it is almost entirely due to
the researches of the moderns.
Could we know with certainty the mass, the figure, and the
number of particles of a fluid in motion, the laws of its motion
might be determined by the resolution of this problem, viz. to
find the motion of a proposed system of small free bodies acting
one upon the other in obedience to some given exterior force.
We are, however, very far from being in possession of the
data requisite for the solution of this problem. We shall, there-
fore, simply present a few of the most usually received theo-
retical deductions; and then proceed to state those rules which
have flown from a judicious application of theory to experi-
ment.
SECTION I-Motion and Effluence of Liquids.
1. A jet of water, issuing from an orifice of a proper form,
and directed upwards, rises, under favourable circumstances,
nearly to the height of the head of water in the reservoir; and
since the particles of such a stream are but little influenced by
the neighbouring ones, they may be considered as independent
bodies, moving initially with the velocity which would be ac-
quired in falling from the height of the reservoir. And the ve-
locity of the jet will be the same whatever may be its direction.
2. Hence, if a jet issue horizontally from any part of the side
of a vessel standing on a horizontal plane, and a circle be de-
scribed having the whole weight of the fluid for its diameter,
the fluid will reach the plane at a distance from the vessel, equal
to that chord of the circle in which the jet initially moves.
Thus, if A s be the upper surface of the
fluid in the vessel, в the place of the ori-
fice, cr the horizontal plane on which
the fluid spouts, then cr is equal to E D,
the horizontal chord of the circle whose
diameter is a c, passing through B.
S
E
Call

A
B
It
HYDRODYNAMICS: EFFLUENCE OF FLUIDS.
303
3. When a cylindrical or prismatic vessel empties itself by
a small orifice, the velocity at the surface is uniformly retarded;
and in the time of emptying itself, twice the quantity would be
discharged if it were kept full by a new supply.
4. But the quantity discharged is by no means equal to what
would fill the whole orifice, with this velocity. If the aperture
is made simply in a thin plate, the lateral motion of the parti-
cles towards it tends to obstruct the direct motion, and to con-
tract the stream which has left the orifice, nearly in the ratio
of two to three. So that in order to find the quantity dis-
charged, the section of the orifice must be supposed to be di-
minished from 100 to 62 for a simple aperture, to 82 for a pipe
of which the length is twice the diameter, and in other ratios
according to circumstances.
5. When a syphon, or bent tube, is filled with a fluid, and its
orifices immersed in the fluids of different vessels, if both sur-
faces of the fluids are in the same level, the whole remains at
rest; but, if otherwise, the longer column of fluid in the sy-
phon preponderates, and the pressure of the atmosphere forces
up the fluid from the higher vessel, until the equilibrium is re-
stored; and the motion is the more rapid as the difference of
levels is greater provided that the greatest height of the tube.
above the upper surface be not more than a counterpoise to the
pressure of the atmosphere.
6. If the lower vessel be allowed to empty
itself, the syphon will continue running as
long as it is supplied from the upper, and
the faster, as it descends the further below
the upper vessel. In the same manner the
discharge of a pipe, descending from the
side or bottom of a given vessel, must be
increased almost without limit by lengthening
it.*

* An improvement in the construction of the syphon has
been lately proposed in the Glasgow Mechanics' Magazine,
and by M. Buntem at Paris. It might be very advantageously
used if constructed on a large scale, for lowering the water in
mill dams or canals. The improvement in the present syphon
is, that the exhausting pipe is enlarged to the some diameter as
that of the syphon, and its mouth is widened out to something
of a funnel shape, as in the figure.
In putting it into action, the short arm is immersed in the
water as in the usual manner, the bottom of the long arm is
closed, the exhausting pipe is then filled with water by the fun-
nel-shaped mouth. On the bottom of the long arm being opened,
the water flows out, exhausts the air from the syphon, when the
water, which is wished to be emptied, flows out in a continual
stream.
d
مد
b

304
HYDRODYNAMICS: EFFLUENCE OF FLUIDS.
7. If a notch or sluice in form of a rectangle be cut in the ver-
tical side of a vessel full of water, or any other fluid, the quan-
tity flowing through it will be of the quantity which would
flow through an equal orifice placed horizontally at the whole
depth, in the same time, the vessel being kept constantly full.
2
8. If a short pipe elevated in any direction from an aperture
in a conduit, throw the water in a parabolic curve to the dis-
tance or range R, on a board, or other horizontal plane passing
through the orifice, and the greatest height of the spouting fluid
above that plane, be H, then the height of the head of water
above that conduit pipe, may be found nearly: viz. by taking
R
1st, 2 cot E = ; and 2dly, the altitude of the head a R
2 H
X cosec 2 E.
Example.-Suppose that R=40 feet, and H-18 feet. Then
1·1111111=2 cot 60° 57': and A: RX cosec 2 E
R 40
2 H 36
20 × cosec 121° 54' 20 × 1.177896=23.55792 feet, height
required.
Note. This result of theory will usually be found about of
that which is furnished by experiment.
F
M
SECTION II.-Motion of Water in Conduit Pipes and Open
Canals, over Weirs, &c.
1. When the water from a reservoir is conveyed in long ho-
rizontal pipes of the same aperture, the discharges made in
equal times are nearly in the inverse ratio of the square roots
of the lengths.
It is supposed that the lengths of the pipes to which this rule
is applied are not very unequal. It is an approximation not
deduced from principle, but derived immediately from experi-
ment. [Bossut, tom. 11, § 647, 648. At § 673, he has given
a table of the actual discharges of water-pipes, as far as the
length of 2340 toises, or 14950 feet English.]
2. Water running in open canals, or in rivers, is accelerated
in consequence of its depth, and of the declivity on which it
runs, till the resistance, increasing with the velocity, becomes
equal to the acceleration, when the motion of the stream be-
comes uniform.
It is evident that the amount of the resisting forces can
hardly be determined by principles already known, and there-
fore nothing remains but to ascertain, by experiment, the velo-
HYDRODYNAMICS: PIPES AND CANALS.
305
city corresponding to different declivities, and different depths
of water, and to try, by multiplying and extending these expe-
riments, to find out the law which is common to them all.
The Chevalier Du Buat has been successful in this re-
search, and has given a formula for computing the velocity
of running water, whether in close pipes, open canals, or
rivers, which, though it may be called empirical, is extremely
useful in practice. Principes d' Hydraulique. Professor Ro-
bison has given an abridged account of this book, in his ex-
cellent article on Rivers and Water-works, in the Encyclopa-
dia Britannica.
Let v be the velocity of the stream, measured by the inches it
moves over in a second; R a constant quantity, viz. the quotient
obtained by dividing the area of the transverse section of the
stream, expressed in square inches, by the boundary or perime-
ter of that section, minus the superficial breadth of the stream
expressed in linear inches.
The mean velocity is that with which, if all the particles were
to move, the discharge would be the same with the actual dis-
charge.
The line R is called by Du Buat the radius, and by Dr. Ro-
bisou the hydraulic mean depth. As its affinity to the radius
of a circle seems greater than to the depth of a river, we shall
call it, with the former, the radius of the section.
Lastly, lets be the denominator of a fraction which expresses
the slope, the numerator being unity, that is, let it be the quo-
tient obtained by dividing the length of the stream, supposing
it extended in a straight line, by the difference of level of its
two extremities; or, which is nearly the same, let it be the co-
tangent of the inclination or slope.
3. The above denomination being understood, and the section,
as well as the velocity, being supposed uniform, v in English
feet,
}]
V
307(R)
s² log (s+18)
:(
or v R
S
When R and S are very great,
14
307
R
307
-log (s+16)
1% √(R-76);
3
10
3
10
1
10
3
-TO).
), nearly.
s²- log s
The logarithms understood here are the hyperbolic, and are
found by multiplying the common logarithms by 2·3025851.
2 D2
306
HYDRODYNAMICS: PIPES AND CANALS.
The slope remaining the same, the velocities are as ✔R
The velocities of two great rivers that have the same decli-
vity, are as the square roots of the radii of their sections.
If R is so small, that √R — 1 0, or R = 1, the velocity
will be nothing; which is agreeable to experience; for in a
cylindric tube Rthe radius; the radius, therefore, equals
two-tenths; so that the tube is nearly capillary, and the fluid
will not flow through it.
ΤΟ
101
The velocity may also become nothing by the declivity be-
coming so small, that
if
1
307
log (s+18)
1
S
500000'
mile, the water will have sensible motion.
4. In a river, the greatest velocity is at the surface, and in
the middle of the stream, from which it diminishes toward the
bottom and the sides, where it is least. It has been found by
experiment, that if from the square root of the velocity in the
middle of the stream, expressed in inches per second, unity be
subtracted, the square of the remainder is the velocity at the
bottom.
and
√ 121/1 24
S
is less than
3
10
0; but
b x
the radius after, so v =
b + 2 x
or thanth of an inch to an English
Hence, if the former velocity bev, the velocity at the bot-
tom ข 2 √ v + 1.
5. The mean velocity, or that with which, were the whole
stream to move, the discharge would be the same with the real
discharge, is equal to half the sum of the greatest and least ve-
locities, as computed in the last proposition.
The mean velocity is, therefore,
ข votě.
This is also proved by the experiments of Du Buat.
6. Suppose that a river having a rectangular bed, is increased
by the junction of another river equal to itself, the declivity
remaining the same; required the increase of depth and ve-
locity.
Let the breadth of the river b, the depth before the junc-
tion d, and after it, x; and, in like manner, v. and v' the mean
b d
velocities before and after; then
is the radius before,
b + 2 d
307 R
1
10'
&
1
2
S
supposing the
breadth of the river to be such, that we may reject the small
quantity subtracted from R, in art. 3; and, in like manner,
}
HYDRODYNAMICS: PIPES AND CANALS.
307
v'
บ
X3
307 R
v'
S
-O
307
1
S2
307
1
S2
喽
​X
Then, substituting for R and R', we have
b d
v + 2 ď
Multiplying these into the areas
of the sections b d and b x, we have the discharges, viz.
307 b d ✓ b d
307 b x ✓ b x
b d v =
X
√b + 2x
;
X
1
2
√b + 2 d
b x ✓ b x
Now the last of these is
2 b d ✓ b d
√ b + 2 ď
√ b + 2x
3 d³
b + 2 d
V X
v + 2 x
x
and
and b x v'
4 b d³
b + 2 ď
or
1832
S
double of the former; therefore,
X3
4 d³
b + 2 ď
and
b+2x
a cubic equation which can al-
ways be resolved by Cardan's rule, or by the approximating
method given at page 92.
As an example, let b 10 feet, and
1 0 and x 1·4882, which is the
river. Hence we have 1·488 × v'
v', or v: v' :: 37 to 50 nearly.
39
dad
When the water in a river receives a permanent increase, the
depth and the velocity, as in the example above, are the first
things that are augmented. The increase of the velocity in-
creases the action on the sides and bottom, in consequence of
which the width is augmented, and sometimes also, but more
rarely, the depth. The velocity is thus diminished, till the te-
nacity of the soil, or the hardness of the rock, affords a sufficient
resistance to the force of the water. The bed of the river then
changes only by insensible degrees, and, in the ordinary lan-
guage of hydraulics, is said to be permanent, though in strict-
ness this epithet is not applicable to the course of any river.
7. When the sections of a river vary, the quantity of water
remaining the same, the mean velocities are inversely as the
areas of the sections.
་་་
d
33 x3
1, then as
depth of the increased
2 v, and 1·488 : 2 :: v :
This must happen, in order to preserve the same quantity of
discharge. (Playfair's Outlines.)
8. The following table, abridged from Du Buat, serves at
once to compare the surface, bottom, and mean velocities in
rivers, according to the principles of art. 4, 5.
308
HYDRODYNAMICS: CANALS, RIVERS, &c.
VELOCITY IN INCHES.
Surface. Bottom.
4
8
1·
3.342
6.071
Velocities of Rivers.

12
16
9.
20
12.055
24
15.194
28
18.421
32
21.678
36
25.
40
28.345
44
31.742
48
35.151
52 38.564
Mean.
2.5
5.67
9.036
12.5
16.027
19.597
23.210
26.839
30.5
34.172
37.871
41.570
45.282
VELOCITY IN INCHES.
Surface.
56
60
64
68
72
76
80
84
88
92
96
100
Bottom.
42.016
45.509
49.
52.505
56.025
59.568
63.107
66.651
70.224
73.788
77.370
81.
Mean.
49.008
52.754
56.5
60.252
64.012
67.784
71.553
75.325
79.112
82.894
86.685
90.5
9. The knowledge of the velocity at the bottom is of the
greatest use for enabling us to judge of the action of the stream
on its bed.
Every kind of soil has a certain velocity consistent with the
stability of the channel. A greater velocity would enable the
waters to tear it up, and a smaller velocity would permit the
deposition of more moveable materials from above. It is not
enough, then, for the stability of a river, that the accelerating
forces are so adjusted to the size and figure of its channel that
the current may be in train: it must also be in equilibrio with
the tenacity of the channel.
We learn from the observations of Du Buat and others,
that a velocity of three inches per second at the bottom will
just begin to work upon the fine clay fit for pottery, and how-
Yet no
ever firm and compact it may be, it will tear it up.
beds are more stable than clay when the velocities do not
exceed this for the water soon takes away the impalpable
particles of the superficial clay, leaving the particles of sand
sticking by their lower half in the rest of the clay, which they
now protect, making a very permanent bottom, if the stream
does not bring down gravel or coarse sand, which will rub off
this very thin crust, and allow another layer to be worn off; a
HYDRODYNAMICS: CANALS, RIVERS, &c.
309
velocity of six inches will lift fine sand; eight inches will lift
sand as coarse as linseed; twelve inches will sweep along fine
gravel; twenty-four inches will roll along rounded pebbles an
inch in diameter; and it requires three feet per second at the
bottom to sweep along shivery angular stones of the size of an
egg. (Robison on Rivers.)
10. Mr. Eytelwein, a German mathematician, has devoted
much time to inquiries in hydrodynamics. In his investiga-
tions he has paid attention to the mutual cohesion of the liquid
moleculæ, their adherence to the sides of the vessel in which
the water moves, and to the contrac-
tion experienced by the liquid vein B
when it issues from the vessel under
certain circumstances. He obtains for-
mula of the utmost generality, and
then applies them to the motion of wa-
ter; 1st, in a cylindric tube; 2dly, in
an open canal.
11. Let d be the diameter of the cylindric tube E F, h the
total height FG of the head of water in the reservoir above the
orifice r, and 7 the length EF of the tube, all in inches: then
the velocity in inches with which the fluid will issue from the
orifice r will be

57 h d
บ 23 3 7 7 57 d
+
•
: (English measure.)
V
с
0.109 - 9582
a h
D
this multiplied into the area of the orifice will give the quantity
per second.
A
12. For open canals. Let v be the mean velocity of the cur-
rent in feet (English), a arca of the vertical section of the
stream, p perimeter of the section, or sum of the bottom and two
sides, length of the bed of the canal corresponding to the fall h,
1
all in feet: then
+0·0111.
E
G
La
p l
The experiments of M. Bidone, of Turin, on the motion of
water in canals, agree within the 80th part of the results of com-
putations from the preceding formula.
13. The following table also exhibits Eytelwein's coefficients
for orifices of different kinds; their accuracy has, in many cases,
been amply confirmed.
41
310
HYDRODYNAMICS: EYTELWEIN'S RESULTS.

No.
Jak
1
૭
3
450
6
ну
8
I
Nature of the Orifices employed.
For the whole velocity due to the
height
For wide openings whose bottom
is on a level with that of the re-
servoir
For sluices with walls in a line with
the orifice
For bridges with pointed piers
For narrow openings whose bottom
is on a level with that of the re-
servoir..
For smaller openings in a sluice
with side walls.
·
•
•
.
For abrupt projections and square
piers of bridges
For openings in sluices without
side walls
•
• ·
1
Ratio between
the theoretical
and real
discharges.
1 to 1.00
1 to 0.961
1 to 0.961
1 to 0.961
1 to 0.861
1 to 0.861
1 to 0.861
1 to 0.635
Coefficients
for finding
the velocitics
in Eng, feet.
8.04
rory
17.17
77.7
6.9
6.9
6.9
14. When a pipe is bent in one or more places, then if the
squares of the sines of the several changes of direction be added
into one sum s, the velocity v will, according to Langsdorf, be
548 d h
found by the theorem v =
being all in English inches.
l,
d + i s l + j ds; b, h, d, and v,
1
1
ठ
5.1
HYDRODYNAMICS: MOTION IN PIPES, &c.
311
Table, abridged from one by Mr. Smeaton, for showing the height of head necessary to over-
come the friction of water in horizontal pipes.
Feet in. Feet in.
0 6
1 0
0 4.5
1 4.7
0 3.0
0 11.1
0 2.2
0 8.4
0 1.8
0 6.7
0 1.5
0
5.6
0 1.3
0 4.8
0 1.1
0 4.2
0 1.0
3.7
0 0.9
0 3.3
0 0.7 0 2.8
0 0.6 0 2.4
0 0.6 0 2.1
0 0.5 0 1.9
0 0·4 0 1.7
0 0.4 0 1.4
0 0.3 0 1.2
0 0.3 0 1.0
0 0.25 0 0.9
0 0.2 0 0.8
0 0.2 0 0.8
0 0.19 0 0.7
Feet in. Feet in.
1 6
2 0
2 11.0
1 11.3
1 5.5
1 2.0
0 11·7
0 10.0
0
8.7
0 7.8
0 7.0
0
5.0
0 5.0
0 4.4
0 3.9
3.5
0
0 2.9
0 2.5
0 2.2
0 1.9
0 1.7
0 1.6
0
Velocities per second of water in the pipes.
Feet in.
2 6
Feet in. Feet in.
3 0 3 6
4 9.7
3 2.5
2 4.9
1 11.1
1 7.2
1 4.5
2
1 2.4 1
1 0.8
0 11.5
0 9.6
8.2
7.2
0 6.4
0 5.8
0 4.8
0 4.1
0 3.6
0 3.2
0
2.9
0 2.6
1.5 0 2·4
0
0
7 1.7
4
9.2
~~~ WPL
3
6.9
2 10.3
4.6
0.5
9.4
1 7.0
1 5.1
1
2.3
1
0.2
0 10.7
0 9.5
0 8.6
0 7.1
0
6.1
0 5.4
0 4.8
0 4.3
0 3.9
0 3.6
2
10
1.0 13 8.0
9 1.3
6 10.0
5 5.6
4 6.7
3 10-9
6 8.6
5 0.5
4 0·4
3 4.3
2 10.6
3 2 2 2 2 1
6.2 3 5.0
3 0.4
8.8
3.3
1 11·4
1 8.5
I
6.2
1
4.4
1 1.7
0 11.7
0 10.2
0 9.1
0 8.2
0
7.5
0
6.8
2.9
2 0.2
8.2
1
5.3
1
3.1
1 1.4
1 0.1
0 10.1
0 8.6
0
7.6
0 6.7
0 6.0
0
5.5
0
5.0
22
Feet in.
4 0
17 10.0
11 10.6
8 11.0
7 1.6
5 11.3
5
4 5.5
3 11.6
3 6.8
2 11.7
૭૭ ૭
1.1
6.6
2.7
1 11.8
1 9.4
1
5.8
1
3.3
1 1.4
0 11.9
0 10.7
0 9.7
0
8.9
Feet in.
4
6
22
6.7
15
0.5
11
3.4
9
0.3
7
6.2
6 5.4
LO LO H M M Q2 2
5
inch.
inch.
1 inch.
14 inch.
13 inch.
1 inch.
2 inch.
24 inch.
2 inch.
3 inch.
0.0 3
inch.
9.8
6.0 4 inch.
6.1 3 1.4 4
3.1
9.6 5
inch.
inch.
4.0 6 inch.
1 7.3 2 0.0 7 inch.
1 4.9
9.0 8 inch.
1 3.0
6.7
9 inch.
1 1.5
4.8
10
inch.
3.3 11
inch.
2.0
12 inch.
5
4
3
3
7.7
0.1
6.1
9.1
2.7
Feet in.
5
0
28
0.2
18 8.1
14 0·0
11 2.5
9
4.1
8
0.1
0.0
6 2.7
5 7.2
4 8.0
1 0.3
0 11.3
43 N N N
2
1 10.6 2
Bore
of the
Pipes.
Ι
I
I
1
1
গलক
{




312
HYDRODYNAMICS: TABLES, &c. FOR WEIRS.
1
I
Look for the velocity of water in the pipe in the upper row,
and in the column below it, and opposite to the given diameter
of the pipe standing in the last column, will be found the per-
pendicular height of a column or head, in feet, inches, and
tenths, requisite to overcome the friction of such pipe for 100
feet in length, and obtain the given velocity.
Table containing the quantity of water discharged over an
inch vertical section of a Weir.
Depth of the
upper edge
of the wastc-
board below
the surface
in English
inches.
1 Q3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Cubic feet of
water discharged
in a minute by
an inch of the
wasteboard, ac-
cording to Du
Buat's formula.
0.403
1.140
2.095
3.225
4.507
5.925
7.466
9.122
10.884
12.748
14.707
16.758
18.895
21.117
23.419
25.800
28.258
30.786
Cubic feet of
water discharged
in a minute by an
inch of the waste-
board, according
to experiments
made in Scotland.
0.428
1.211
2.226
3.427
4.789
6.295
7.933
9.692
11.564
13.535
15.632
17.805
20.076
22.437
24.883
27.413
30.024
32.710
Gallons
of
282 inches
correspond-
ing with
results in
col. 3.
2.621
7.417
1
13.634
20.990
29.332
38.357
48.589
59.364
70.826
83.164
95.746
109.055
122.965
137.427
152.108
167.905
183.897
200·350
Cat


+
To the above table, originally due to Du Bual, is added a
third column, containing the quantities of water discharged, as
inferred from experiments made in Scotland, and examined
by Dr. Robison, who found that they in general gave a dis-
charge greater than that which is deduced from Du Bual's
formulæ. We would recommend it therefore to the engineer
to employ the third column in his practice, or the fourth if he
wish for the result in gallons.
HYDRODYNAMICS DISCHARGES OVER WEIRS.
313
.
If they be odd quarters of an inch, look in the table for as
many inches as the depth contains quarters, and take the eighth
part of the answer. Thus, for 3 inches, take the eighth part.
of 24-883, which corresponds to 15 inches. This is 3-110.
:
15. The quantity discharged increases more rapidly than the
width to obtain a correct measure of it, if n be the width or
length of the wasteboard in inches, take (n + n) times the
quantity for one inch of wasteboard of the given depth, from the
preceding table.
In the preceding table it is supposed that the water from
which the discharge is made is perfectly stagnant; but if it
should happen to reach the opening with any velocity, we have
only to multiply the area of the section by the velocity of the
stream.
16. When the quantity of water a discharged over a weir is
known, the depth of the edge of the wasteboard, or н, may be
approximated from the following formulæ, 7 length of waste-
board
H
Q
(11-41727)³ (17) nearly.
{
=
1
20
3
2
Reciprocally, Q = 11½ 7. H nearly: or, more accurately by
adding the correction in article 15.
17. The quantities discharged for any given width are as the
3
power of the depth, or as н2.
Hence, to extend the use of the table to greater depths, we
have only for
Twice any depth, take Q × 2.828
3 times.
Q X 5.196
4 times.
Q X
8.000
5 times.
Q X 11.180
6 times.
Q X 14·697
7 times.
Q X 18.520
8 times.
Q × 22.627
9 times.
Q X 27·000
10 times.
Q X 31.623
and the results will be nearly true.
To make them still more correct, where great accuracy is
required, add to them their thousandth part.
2 E
314
HYDRODYNAMICS: DISCHARGES OF WATER.
$
Examples of the use of the Tables and Rules.
Ex. 1. Let the depth be 10 inches below the upper surface of
the water, and the width 8 inches. How many cubic feet of
water will be discharged in a minute?
By table a to depth 10, width 1
Multiply this by n =
Quantity for 1 inch width
Multiply by n =
Add of this product . .
20
Discharge in one minute
Ex. 2. Let the depth be 9 feet, and the width 1 foot. Re-
quired the cubic feet discharged in a minute.
By table Q to depth 12 inch. width 1 inch
Factor for 9 times depth is 27
3 X 9..
Add of the product . .
20
Total quantity in cubic feet
3
For 54, multiply Q at 6 by 32 or 27
For 48,
cub. feet.
13.535
8
27.525 × (6 + 2%)
106.280
5.314
3
Q at 12 by 42 or 8...
111.594
17.805
3
53.415
9
480.735
12
5768.802
286 441
Ex. 3. Let a square orifice of 6 inches each side be placed in
a sluice-gate with its top 4 feet below the upper surface of the
water how much will it discharge in a minute?
Here the quantity discharged by a slit in depth 48 inches,
must be taken from one in depth 54 inches.
•
6055.261
cub. feet.
.169.965
142-440
Difference. . . . .
173.4 cubic feet, quantity discharged.
27.525
HYDRODYNAMICS: DISCHARGES THROUGH SLUICES, &c. 315
3
Note. In an example like this, it is a good approximation,
to multiply continually together the area of the orifice, the
number 336,* and the square root of the depth in feet of the
middle of the orifice.
My
Thus, in the preceding example, it will be 1 × 1 × 336
× √4·25= 4 × 336 × 2·062=173·2 cubic feet.
The less the height of the orifice compared with its depth un-
der the water, the nearer will the result thus obtained approach
to the truth.
If the height of the orifice be such as to require considera-
tion, the principle of Art. 7 of the preceding section may be
blended with this rule.
Thus, applying this rule to Ex. 2, we shall have area ×
✔depth × 336 × 29 × 3 × 224-6048, for the cubic feet dis-
charged. This is less than the former result by about its 900th
part. It is, therefore, a good approximation, considering its
simplicity it may in many cases supersede the necessity of re-
currence to tables.
..
18. PROB. Given the vertical section of a river or other
stream, to determine the swell occasioned by the piers of a
bridge, or the sides of a cleaning sluice which contract the
passage by a given quantity, for a short length only of the
channel; the velocity of the stream being also known. Let
v be the velocity of the stream, independently of the effect
of the bridge, r the section of the river, and a the amount
of the sections between the piers; let 2 g, instead of being
taken at 64, be reckoned 58 6, to accord with the effect of ex-
perimental contractions through arches of bridges, &c., and let
s be the slope of the bed of the river, or the sine of its an-
gle with the horizon; then Du Buat (tom. i. p. 225) gives for
the swell or rise of the stream in feet, which will be occasioned
༧༠
グ
​by the obstruction (0 +s) ([2/2]² —¹):
58.6
a
a theorem, by means of which we may approximate to the said
swell in any proposed case.
The value of s will, of course, be different in different cases;
but if we assume, or '05, as a mean value, it will enable us
to compute and tabulate results, which, though they cannot be
presented as perfectly correct, may be regarded as exhibiting a
medium between those that will usually occur; and will serve
to anticipate the consequences of floods of certain velocities,
when constrained to pass through bridges which more or less
contract the stream.
*3365.6 × 60.
316 HYDRODYNAMICS: SWELLS OCCASIONED BY BRIDGES.
My
Table of the Rise of Water occasioned by Piers of Bridges
and other Contractions.*
|Velocity of current
in feet, per sec..
1
Amount of obstructions, compared with the vertical section of the River.
1-10th. 2-10ths. 3-10ths. 4-10ths. 5-10ths. 6-10ths. 7-10ths. 8-10ths. 9-10ths.
ths. 7-10ths. 8-10ths. 9-10
Proportional Rise of Water, in feet and decimals.
feet
feet
feet feet feet feet
0.11920-2012 03521
0.0157 0.0377 0.0698
20-0277 0.0365 0.1231 0.2102 0.3548 0-6208
30-0477 0.1144 0.2118 0.3618 0-6107 1.0387
40-0760 0·1822 | 0·3372 | 0.5759 | 0·97191-7008
5 0.1165 0.2793 0.5168 0.8782 | 1.4895 2.6066
60·1558 0.3736 | 0 6912
70-2078 0.4983 0.9221
8 0.2678
0.3359
feet feet feet
1.6094
0.6780
6.6389
1.1995 2-8378 11.7058
2.0580 4.8850 20-1504
3.2755 7.7750 32.0720
5.020211.9160 49.1535
1.1807 1-9925 3:4968 6.7154 15.9398 65-7518
1-5750 2.6578 4.6511 8.9578 21.2626 87.7080
0.6423 1.1884 2.0299 3-4255 5.994711-5454 27 4042 113.0422
4.2956 7-5172 14-4777 34-3646 141-7541
5.2680 9-2190 17-7551 42-1440 173-8440
9
0.8054 1.4903
2.5566
3.1218
10
0.4119 0-9877 1.8276
1

Ordinary
floods.
Violent
floods.
1
Unusually
violent
floods.
窄
​19. It will be evident, from an inspection of this table, that
even in the case of ordinary floods, old bridges with piers and
starlings, occupying 6 or 7 tenths of the section of the river,
will produce a swell of 2, 3, or more feet, often overflowing
the river's banks and occasioning much mischief. Also, that
in violent floods, an obstruction amounting to 7 tenths of the
channel will cause a rise of 7 or 8 feet, probably choking
up the arches, and occasioning the destruction of the bridge.
Greater velocities and greater contractions produce a rapid
augmentation of danger and mischief; as the table obviously
shows.
20. The same principles and tabulated results serve to esti-
mate the fall from the higher to the lower side of a bridge, on
account of an ebbing tide, &c. Thus, for old London Bridge,
where the breadth of the Thames is 926 feet, and the sum of
the water ways at low water only 236 feet; the amount of ob-
structions was 690 feet, about 7½ tenths of the entire section ;
so that a velocity of 3 feet per second would give a fall of
nearly 44 feet, agreeing with the actual result.
At Rochester Bridge, before the opening of the middle
arches, the piers and starlings presented an obstruction of
seven-tenths, and at the time of greatest fall, the velocity
* A similar table was computed by Mr. Wright, of Durham, more than 50 years
ago, and inserted in the first edition of Dr. Hutton's treatise on Bridges; but it is
not constructed upon a correct theory.
}
HYDRODYNAMICS: FALL UNDER ARCHES.
317
100 yards above bridge exceeded 6 feet per second. This,
from the table, would occasion a fall of more than 6-7 feet; and
the recorded results vary from 6 to 7 feet.
At Westminster Bridge, where the obstructions are about
one-sixth of the whole channel, when the velocity is 23 feet,
the fall but little exceeds half an inch a result which the
table would lead us to expect.
!
"
SECTION III.-Contrivances to measure the velocity of
running waters.
འ
1. For these purposes, various contrivances have been pro-
posed, of which two or three may be here described.
Suppose it be the velocity of the water of a river that is
required; or, indeed, both the velocity and the quantity
which flows down it in a given time Observe a place where
the banks of the river are steep, and nearly parallel, so as to
make a kind of trough for the water to run through, and by
taking the depth at various places in crossing make a true
section of the river. Stretch a string at right angles over it,
and at a small distance another parallel to the first. Then
take an apple, an orange, or other small ball, just so much
lighter than water as to swim in it, or a pint or quart bottle
partly filled with water, and throw it into the water above the
strings. Observe when it comes under the first string, by
means of a quarter second pendulum, a stop watch, or any
other proper instrument; and observe likewise when it arrives
at the second string. By this means the velocity of the upper
surface, which in practice may frequently be taken for that of
the whole, will be obtained. And the section of the river at
the second string must be ascertained by taking various depths,
as before.
If this section be the same as the former, it may be
taken for the mean section: if not, add both together, and
take half the sum for the mean section. Then the area of the
mean section in square feet being multiplied by the distance
between the strings in feet, will give the contents of the water
in solid feet, which passed from one string to the other during
the time of observation: and this by the rule of three may be
adapted to any other portion of time. The operation may
often be greatly abridged by taking notice of the arrival of the
floating body opposite two stations on the shore, especially
when it is not convenient to stretch a string across. An arch
42
2 E 2
}
318
HYDRODYNAMICS: STREAM MEASURERS.
of a bridge is a good station for an experiment of this kind, be-
cause it affords a very regular section and two fixed points of
observation; and in some instances the sea practice of heaving
the log may be advantageous. Where a time-piece is not at
hand, the observer may easily construct a quarter second or
other pendulum, by means of the rules and table relating to
pendulums, in the Dynamics.
2. M. Pitot invented a stream measurer of a simple con-
struction, by means of which the velocity of any part of a
stream may readily be found. This instrument is composed
of two long tubes of glass open at both ends; one of these tubes
is cylindrical throughout; the other has one of its extremities
bent into nearly a right angle, and gradually enlarges like a
funnel, or the mouth of a trumpet: these tubes are both fixed
in grooves in a triangular prism of wood; so that their lower
extremities are both on the same level, standing thus one by
the side of the other, and tolerably well preserved from ac-
cidents. The frame in which these tubes stand is graduated,
close by the side of them, into divisions of inches and lines.
1
To use this instrument, plunge it perpendicularly into the
water, in such manner that the opening of the funnel at the
bottom of one of the tubes shall be completely opposed to the
direction of the current, and the water pass freely through the
funnel up into the tube. Then observe to what height the
water rises in each tube, and note the difference of the sides;
for this difference will be the height due to the velocity of the
stream. It is manifest, that the water in the cylindrical tube
will be raised to the same height as the surface of the stream,
by the hydrostatic pressure: while the water entering from the
current by the funnel into the other tube, will be compelled to
rise above that surface by a space at which it will be sustained
by the impulse of the moving fluid: that is, the momentum of
the stream will be in equilibrio with the column of water sus-
tained in one tube above the surface of that in the other. In
estimating the velocity by means of this instrument, we must
have recourse to theory as corrected by experiments. Thus, if
h, the height of the column sustained by the stream, or the
difference of heights in the two tubes, be in feet, we shall have
v = 6.5 h, nearly, the velocity, per second, of the stream;
if h be in inches, then v = 22.47 ✓ h, nearly or farther ex-
periments made with the same instrument may a little modify
these co-efficients.
It will be easy to put the funnel into the most rapid part of
the stream, if it be moved about to different places until the dif-
ference of altitude in the two tubes becomes the greatest. In
some cases it will happen, that the immersion of the instrument
HYDRODYNAMICS: STREAM MEASURERS.
319
will produce a little eddy in the water, and thus disturb the
accuracy of the observation: but keeping the instrument im-
mersed only a few seconds will correct this. The wind would
also affect the accuracy of the experiments; it is, therefore,
advisable to make them where there is little or no wind. By
means of this instrument a great number of curious and useful
observations may easily be made the velocity of water at
various depths in a canal or river may be found with tolerable
accuracy, and a mean of the whole drawn, or they may be
applied to the correcting of the theory of waters running down
gentle slopes. The observations may likewise be applied to
ascertain whether the augmentations of the velocities are in
proportion to the increase of water passing along the same
canal, or what other relations subsist between them, &c.
Where great accuracy is not required, the tube, with the
funnel at bottom, will alone be sufficient; as the surface of the
water will be indicated, with tolerable, precision, by that part
of the prismatic frame for the tube which has been moistened
by the immersion: and the velocities may be marked against
thosé altitudes in the tube which indicate them.
:
3. Another good and simple method of measuring the ve-
locity of water in a canal, river, &c. is that described by the
Abbé Mann, in his treatise on rivers; it is this :-Take a cy-
lindrical piece of dry light wood, and of a length something
less than the depth of the water in the river; about one end
of it let there be suspended as many small weights as may
keep the cylinder in a vertical or upright position, with its
head just above water. To the centre of this end fix a
small straight rod, precisely in the direction of the cylinder's
axis in order that, when the instrument is suspended in the
water, the deviations of the rod from a perpendicularity to the
surface of it, may indicate which end of the cylinder goes
foremost, by which may be discovered the different velocities
of the water at different depths; for when the rod inclines
forward, according to the direction of the current, it is a
proof that the surface of the water has the greatest velocity;
but when it reclines backward, it shows that the swiftest
current is at the bottom; and when it remains perpendicular,
it is a sign that the velocities at the top and bottom are
equal.
This instrument, being placed in the current of a river or
canal, receives all the percussions of the water throughout the
whole depth, and will have an equal velocity with that of the
whole current from the surface to the bottom at the place
where it is put in, and by that means may be found, both
320
HYDRODYNAMICS: STREAM MEASURERS.
>
}
0
with exactness and ease, the mean velocity of that part of the
river for any determinate distance and time.
But to obtain the mean velocity of the whole scction of the
river, the instrument must be put successively both in the mid-
dle and towards the sides, because the velocitics at those places
are often very different from each other. Having by these
means found the several velocities, from the spaces run over in
certain times, the arithmetical mean proportional of all these
trials, which is found by dividing the common sum of them
all by the number of the trials, will be the mean velocity of
the river or canal. And if this medium velocity be multiplied
by the area of the transverse section of the waters of any place,
the product will be the quantity running through that place in
a second of time.
f
The cylinder may be easily guided into that part which we
want to measure, by means of two threads or small cords,
which two persons, one on each side of the canal or river, must
hold and direct; taking care at the same time neither to retard
nor accelerate the motion of the instrument.
1
4. Let A A' B B' be a hollow cylinder, open at both ends, and
let it be capable of being fixed by the side of a platform or of
a boat, so that its lower extremity в B' may be placed at any
proposed depth below HR, the upper sur-
face of the stream. Let pr' be pulleys,
fixed at opposite sides of the top and bot-
tom of the tube. To G, a globe of specific
gravity nearly the same as that of water,
let a cord & r' p s be attached, passing
freely over the pulleys r', r, and having
sufficient length towards s to allow of its
running off to any convenient distance.
Then, the bottom of the tube being immersed to any proposed
depth, let the globe & be exposed to the free operation of
the stream; and as it is carried along with it, it will in 1, 2, 5,
or 10 seconds, or any other interval of time, draw off from a
fixed point, as s, a portion of cord; from which and the
time elapsed, the velocity at the assigned depth will become
known.
S
}
II
A
P

A
R
Mr. Saumarez invented, in 1720, an instrument called the
Marine Surveyor, for the double purpose of measuring a
ship's way, and ascertaining the velocitics of streams. It is
described in the Phil. Trans. vol. 33; and in the succeeding
volume a curious example of its use is given in "tables
showing the strength and gradual increase and decrease of
the tides of flood and ebb in the river Thames, as observed
HYDRODYNAMICS: OLD LONDON BRIDGE.
321
2
in Lambeth Reach." They are too extensive to be inserted
here; but are truly interesting, and may be seen in Phil.
Trans. Abridged, vol. vii. p. 133.
SECTION IV.-Effects of the Old London Bridge on the
Tides, &c.
In the first volume of Dr. Hutton's Tracts, Svo. there are
inserted some curious papers drawn up by Mr. Robertson, of
Christ's Hospital, and others, on London Bridge, and on the
probable consequences, in reference to the tides, of erecting
a new bridge across the Thames, viz. Blackfriars. Such docu-
ments are not only interesting as matters of scientific history,
but become valuable in process of time; as the comparison of
facts with theoretic predictions is subservient to the correction
of the theory itself. With a similar object in view, I here
introduce abridged accounts of some valuable facts with re-
gard to the motion and level of the tides in the Thames at
London, collected in 1820 and 1821, when the project of a new
bridge of five arches, instead of the old bridge, originally of 20
arches, and a very contracted water-way, came first under con-
sideration.
LONDON BRIDGE.
Result of the levels of tides observed from 23d September
to 25th October, 1820, between the entrance of the London
Docks and Westminster Bridge. Also the transverse sections
of the river Thames at London Bridge, Southwark Bridge,
Blackfriars Bridge, Waterloo Bridge, and Westminster Bridge,
collected from four drawings of the above surveys, made by
Mr. Francis Giles, under the direction of Mr. James Mon-
tague, pursuant to an order of the Select Committee of Bridge
House Lands, of the 8th September, 1820.
}
London Docks to London Bridge.
The high water of spring tides at the entrance of the London
Docks, averaged a level of 1.5 inch higher and 10 minutes
earlier time, than at the lower side of London Bridge.
The low water of ditto at ditto averaged a level of 3 inches
lower, and 9 minutes carlier time, than at ditto.
The high water of neap tides at ditto averaged a level of one
inch, and 8 minutes carlier time, than at ditto.
The low water of ditto at ditto averaged a level of 2 inches
lower, and 14 minutes earlier time, than at ditto.
322 HYDRODYNAMICS: EFFECTS OF LONDON BRIDGE.
1
1
London Bridge.
High water of the highest spring tides occurs at three or four
o'clock.-High water of the lowest neap tides occurs at eight
or nine o'clock.
Spring tides flow four or five hours, and ebb seven to eight
and a half hours.-Neap tides flow five to five and a half hours,
and ebb six and a half to eight hours.
The high water of spring tides produced an average fall through
London Bridge of 8 inches, but the greatest fall at high water
was 1 foot 1 inch.-October 24th.
The low water of ditto, through ditto, of 4 feet 4 inches, but
the greatest fall at low water was 5 feet 7 inches.-Septem-
ber 27th.
And
The high water of neap tides through ditto of 5 inches.
The low water of ditto, through ditto, 2 feet 1 inch, but the
least fall at low water was 1 foot 1 inch.-October 16th.
The flood of spring tides of October 21st and 23d, produced
slack water through the bridge in about 40 minutes after low
water below bridge, from which time a head gradually in-
creased (below bridge) to 1 foot 10 inches at half flood, and
then regularly increased to about 8 inches at high water.—The
first flow of these tides, nevertheless, began above bridge about
20 minutes after low water time below bridge, although the
water was then about 2 feet 6 inches higher above than below
bridge; the time of low water below bridge averaged 10
minutes earlier than above bridge.
The ebb of these tides produced slack water at the bridge
about 30 minutes after high water, and then gradually sunk to
their greatest fall at low water.-The time of high water of Oc-
tober 21st and 23d, was the same below as above bridge; but
the average time of high water spring tides is 9 minutes earlier
below than above bridge.
The flood of neap tide, October 30th, produced slack water
through the bridge, in about two hours after low water time be-
low bridge (when there was some land flood in the river), from
which time a head gradually increased (below bridge) to 1 foot
3 inches at two-thirds flood, and then regularly decreased to 4
inches at high water. The first flow of this tide, nevertheless,
began above bridge, about 1 hour after low water time below
bridge, although the water was then 1 foot higher above than
below bridge; but the average time of low water below bridge
is 32 minutes earlier than above bridge.
The ebb of this tide produced slack water at the bridge
Ob
HYDRODYNAMICS: EFFECTS OF LONDON BRIDge.
323
1
about 15 minutes after high water above bridge, and then gra-
dually sunk to its greatest fall at low water. The time of high
water of October 30th, was 15 minutes earlier below than above
bridge, and the average time of high water neap tides is 15
minutes earlier below than above bridge.
London Bridge to Westminster Bridge.
The high water line from the upper side of London Bridge
to Westminster Bridge is generally level, unless influenced by
winds and land floods.
The time of high water is about 10 minutes earlier at Lon-
don than Westminster Bridge.
The mean low
water line has
a fall of
Ditto
Ditto
Ditto
min. (later at Westmin-
in. pts.
{
4
0 from Westminster to Waterloo Bridge, time 7
ster than at Wa-
terloo Bridge.
Ditto.
Ditto.
Ditto.
4
3
0
Total foot 10
3 from Waterloo to Blackfriars Bridge
2 from Blackfriars to Southwark Bridge
5 from Southwark to London Bridge
0
At an extraordinary high water or level of 2 feet above
the average spring tide high water mark, at the
Hermitage entrance to the London Docks, as set-
tled by the Corporation of the Trinity House, Au-
gust, 1800.
At the Trinity High Water mark, 'or
At an average
Spring Tide
High Water. S
At Ditto Spring
Tide High
Water.
At Ditto Neap
Tido High
Water.
At Ditto Neap
Tide Low
Water.
At Ditto Spring
Tide Low
Water.
Datum
Under the Datum.
ft. in.
0 6.5
Areas of the Transverse Sections in the River Thames at
London Bridge.
below Bridge, or Level of
abovo Bridge, or Level of
above Bridge, or Level of
At Ditto Spring
and Neap Tide above Bridge, or Level of
Low Water.
below Bridge, or Level of
1 2.0
V
4 3.0
14 5·0
16 5.0
below Bridge, or Lovel of 18 9.0
1
A 47
5
4
181
Fect.
8,130
7,360
7,122
6,837
5,293
1,488
1,030
540
¡Abate for
Water-
works.
Feet.
1,000
850
810
600
Feet.
7,130
6,272
6,027
4,693
1,488
1,030
540
324
HYDRODYNAMICS: EFFECTS OF LONDON BRIDGE.
At Southwark, Blackfriar's, Waterloo, and Westminster
At the above level of extraordinary
high water
At the Trinity high water mark, or
datum
•
At an average Spring tide high water,
or level of 1 foot 2 inches under
the ditto
13,170
12,975
16,447
14,015
11,135
10,590
13,116
11,380,
1
5,012 3,724
3,382
3,720 ·
London, 12th March, 1821.-(Published in a letter addressed to G. H. Sumner,
Esq. M. P. by a scientific architect.)
At ditto Neap tide high water, or level
of 4 feet 3 inches under the ditto.
At ditto Spring and Neap tide low
water.
FLOOD TIDE.
Depth of water when flood com-
menced
1st hour rise
2d hour
3d hour
4th hour
45 minutes
4 hours and 45 minutes
ABOVE BRIDGE.
Low Water at Pepper Alley," 50
minutes past nine o'clock in the
morning.
}
FLOOD TIDE.
Gradation of the Ebbing and Flowing of the tide at Lon-
don Bridge, taken above and below, on the 29th of July,
1821; being the day of the new moon.
·
·
•
Bridges.
Southwark Blackfriars] Waterloo | Westminster
Bridge. Bridge. Bridge.
Bridge.
15,26,0 15,460 19,822
16,750
13,940
14,117 17,707
15,198
•
•
- Ft. In.
、
Depth of water when flood com-
menced
1st hour rise
2d hour
3d hour
4th hour
48 minutes
•
•
BELOW
Low Water at Coxes' Quay, 30 mi-
nutes past nine o'clock in the morn-
ing.
↓
·
6 0
2 11
3 0
2 10
2 8
1 0
18 5
5
Ft. In.
110 LOQ QI
GP và di đà lo ch
3
9
5
9
2
2 5
1
4
4
4 hours and 48 minutes . 18 10
High Water at Pepper Alley, 35 mi-
nutes past two o'clock in the after-
noon.
1st hour fall
2d hour,
3d hour
4th hour
5th hour
6th hour
7th hour
55 minutes
Depth at low water
EBB TIDE.
1st hour fall
2d hour
3d hour
4th hour
5th hour
6th hour
7th hour
59 minutes
•
•
Depth left
EBB TIDE.
¿
7 hours and 55 minutes 18
+
t
•
•
·
•
BRIDGE.
High Water at Coxes' Quay, 18 mi-
nutes past two o'clock in the after-
noon.
Ft. In.
1
my
0
1 9
1 5
1 2
1 0
222---
0 11
5 6
•
Q+&&&1
3
Ft.'In.
2
1
4 4
1
2
LOO
2
5
7
3
9
1 6
7 hours and 59 minutes 18 10
*** The object of this statement was to show, that the old bridge tended to
retain the water above bridge and assist the navigation up the river,
0 11
0 4
}

$
1
1
HYDRODYNAMICS: WATERMILLS.
325
/
Difference between the Levels of High and Low Water Spring
Tides, between Rotherhithe and Battersea in the year 1820.
-
f.
-
i.
21
10
Rotherhithe, Old Horse Ferry
London Old Bridge
18
2
Blackfriars
14
9
12
6
Westminster
Vauxhall
12
2
Battersea
11
6
From Battersea Bridge to London Bridge, 5 miles; from London Bridge to Old
Horse Ferry, 14 miles. From London Bridge to the Nore, 44 miles.*
(
1
SECTION V.-Watermills.
1. The impulse of a current of water, and sometimes its
weight and impulse jointly, are applied to give motion to mills
for grinding corn and for various other purposes. Sometimes
the impulse is applied obliquely to floatboards in a manner
which may be comprehended at once by reference to a smoke-
jack. In that, the smoke ascends, strikes the vanes obliquely,
and communicates a rotatory motion. Imagine the whole
mechanism to be inverted, and water to fall upon the vanes,
rotation would evidently be produced; and that with greater
or less energy in proportion to the quantity of water and the
height from which it falls.
Water-wheels of this kind give motion to mills in Ger-
many, and some other parts of the continent of Europe. I
have, also, seen mills of the same construction in Balta, the
northernmost Shetland Isle. But wherever they are to be
found, they indicate a very imperfect acquaintance with
practical, mechanics; as they occasion a considerable loss of
power.
2. Water frequently gives motion to mills, by means of
what is technically denominated an undershot wheel. This
has a number of planes disposed round its circumference,
nearly in the direction of its radii, these floatboards (as they
are called) dipping into the stream, are carried 'round by it; as
shown in the accompanying diagram. The axle of the wheel, of
* The preceding results will always be valuable, as they supply striking evidence
of the effect of a broken dam, such as many of our old bridges present. I regret
that the contrast occasioned by the large arches of the new bridge cannot yet be
presented for though the old bridge is removed, the entire obstructions occasioned
by the starlings were not taken away when this volume went to the press.
43
2 F
326
HYDRODYNAMICS: WATERMILLS.
course, by the intervention of pro-
per wheels and pinions, turns the
machinery intended to be moved.
Where the stream is large and un-
confined, the pressure on each float-
board is that corresponding to the
head due to the relative velocities.
(or difference between the veloci-
ties of stream and floatboard): this
pressure is, therefore, a maximum when the wheel is at rest;
but the work performed is then nothing. On the other hand,
the pressure is nothing when the velocity of the wheels equals
that of the stream. Consequently, there is a certain interme-
diate velocity, which causes the work performed to be a maxi-
mum.

The weight equal to the pressure is (H— √ h)º, Q
being the quantity of water passing in a second, I the height
due to v the velocity of the water, and h that due to u the
velocity of the floatboard. Considering this as a mass attached
to the wheel, its moving force is obtained by multiplying
it into u and as H ✔h varies as v u, this mov-
ing force varies as (v U)2 u which is a max. when u — } j v.
In this case, then, the rim of the wheel moves with of the
velocity of the stream; and the effect which it produces is
K
4,
q × ( ? v ) ² × } v 24/7 Q v³ :
so that the undershot wheel, according to the usual theory,
performs work = of the moving force.
27
;
Friction, and the resistance of fluids, modify these results
but Smeaton and others have found that the maximum work
is always obtained when u is between ½ v and 1 v.
3. Where the floats are not totally immersed, the water is
heaped upon them; and in this case the pressure is that due to
M
D
2 II.
4. When the floatboards move in a circular sweep close
fitted to them, or, in general, when the stream cannot escape
without acquiring the same velocity as the wheel, the cir-
cumstances on which the investigation turns become analo-
gous to what happens in the collision of non-elastic bodies.
The stream has the velocity v before the stroke which is re-
duced to u, and the quantity of motion corresponding to the
difference, or to v u, is transferred to the wheel; this turns
with the velocity u; and therefore the effect of the wheel is as
V U U2
V
D) u, or,
; which is a maximum when y = 2 U;
V
V
being then of the moving power.
I
HYDRODYNAMICS: WATERMILLS.
327
Hence appears the utility of constraining the water to
move in a narrow channel.
5. The undershot wheel is used where a large quantity of
water can be obtained with a moderate fall. But where the
fall is considerable the overshot is almost always employed.
Its circumference is formed into
angular buckets, into which the
water is delivered either at the
top or within 60° of it: 52° is
the most advantageous distance.
In that case, if r=the full radius
of the wheel, H the whole, and h
the effective height of the fall,
h =r (1+ sin 37° 4) 1.605 r,
and r =
623 h. If the friction

be about of the moving power,
v, the velocity of the circumf. of the wheel to produce a maxi-
mum effect, will 2.67 H. Here, too, a fall of
will
give the water its due velocity of impact upon the wheel and
3
122.176 sн lbs the mechanical effect, s being the section, in
SH 2
fect, of the stream that supplies the buckets.
Mr. Smeaton's experiments led him to conclude that over-
shot wheels do most work when their circumferences move at
the rate of 3 feet in a second, and that when they move con-
siderably slower than this, they become unsteady and irregular
in their motion. This determination is, however, to be un-
derstood with some latitude. He mentions a wheel 24 feet
in diameter, that scemed to produce nearly its full effect,
though the circumference moved at the rate of 6 feet in a
second; and another of the diameter of 33 feet, of which the
circumference had only a velocity of 2 feet in a second, with-
out any considerable loss of power. The first wheel turned
round in 12". 6, the latter in 51°. 9.
6. Where the fall is too small for an overshot wheel, it is
most advisable to employ a breast-wheel (such as exhibited in
the margin), which partakes of its properties; its floatboards.
meeting at an angle, so as to be assimilated to buckets, and
the water being considerably confined
within them by means of an arched
channel fitting moderately close, but
not so as to produce unnecessary fric-
tion. But when the circumstances
do not admit of a breast-wheel, then
recourse must be had to the under-
shot. For such a wheel it is best,

261
BUELTA
T
328
HYDRODYNAMICS: WATERMILLS.
that the floatboards be so placed as to be perpendicular to the
surface of the water at the time they rise out of it; that only
one half of each should ever be below the surface, and that
from 3 to 5 should be immersed at once. The Abbe Mann pro-
posed that there should not be more than six or eight float-
boards on the whole circumference.'
7. Mills moved by the reaction of water are usually deno-
minated Barker's Mills; sometimes, however, Parent's; at
others, Segner's. But the invention is doubtless Dr. Barker's.
In the marginal diagram, where c D is a
vertical axis, moving on a pivot at », and
carrying the upper millstone m, after pass-
ing through an opening in the fixed mill-
stone c. Upon this axis is fixed a vertical
tube T T communicating with a horizontal
tube A B, at the extremities of which A, B, are
two apertures in opposite directions. When
water from the mill-course M N is intro-
duced into the tube T T, it flows out of the apertures, A, B, and
by the reaction or counter-pressure of the issuing water the
arm ▲ B, and consequently the whole machine, is put in motion.
The bridge-tree a b is elevated or depressed by turning the nut
c at the end of the lever c b.

-
M N
10
T
7'
D)
b
In order to understand how this motion is produced, let us
suppose both the apertures, shut, and the tube TT filled with
water up to T. The apertures AP, which are shut up, will be
pressed outwards by a force cqual to the weight of a column of
water whose height is rr, and whose area is the area of the
apertures. Every part of the tube A B sustains a similar pres-
sure; but as these pressures are balanced by equal and opposite
pressures, the arm A B is at rest. By opening the aperture at
A, however, the pressure at that place is removed, and conse-
quently the arm is carried round by a pressure equal to that of
a column Tr, acting upon an area equal to that of the aperture a.
The same thing happens on the arm TB; and these two pres-
sures drive the arm A B round in the same direction. This ma-
chine may evidently be applied to drive any kind of machinery,
by fixing a wheel upon the vertical axis c D.
8. Mr. Rumsey, an American, and Mr. Segner, improved
this machine, by conveying the water from the reservoir, not
by a pipe, in greater part of which the spindle turns, but by a
pipe which descends from a reservoir, as F, until it reaches
lower than the arms AB, and then turns up by a curvilinear
neck and collar, entering between the arms at the lower part, as
shown in the figure. This greatly diminishes the friction.
9. Professor Playfair has correctly remarked that the
HYDRODYNAMICS: WATERMILLS.
329
moving force becomes greater, after the machine has begun to
move; for the water in the horizontal arms acquires a centri-
fugal force, by which its pressure against the sides is increased.
When the machine works to the greatest advantage, the
centre of the perforations should move with the velocity
2
201
hg, where r is the radius of the horizontal arm, measured
from the axis of motion to the centre of the perforation, and r'
the radius of the perpendicular tube, g being put for the force
of gravity, or 32 feet.
As 2 r is the circumference described by the centre of each
is the time of a revolution in seconds.
2 x r
perforation,
√hg
p
The quantity is also the velocity of the effluent
water; therefore, when the machine is working to the greatest
advantage, the velocity with which water issues is equal to that
with which it is carried horizontally in an opposite direction;
so that, on coming out, it falls perpendicularly down.
10. The following dimensions have been successfully adopt-
ed: viz. radius of arms from the centre of pivot to the centre
of the discharging holes, 46 inches; inside diameter of the
arms, 3 inches: diameter of the supplying pipe, 2 inches;
height of the working head of water, 21 feet above the points
of discharge. When the machine was not loaded, and had but
one orifice open, it made 115 turns in a minute.
This agrees
to a velocity of 46 feet in a second, for the orifice, greater than
the full velocity due to the head of water by between 9 and
10 feet the difference is due to the effect of the centrifugal
force.
nih g
√
The theory of this machine is yet imperfect; but there can
be no doubt of its utility in cases where the stream is small
with a considerable fall.
Mr. James Whiteland, a correspondent of the Franklin
Journal, proposes to make the horizontal arms of the mill of a
curved form, such that the water will run from the centre to the
extremity of the arms in a straight line when the machine is
working. For the method of constructing the curve, see Me-
chanic's Magazine, No. 499. It is very clear, however, that
the additional efficiency of the machine will not be so great by
any means as the inventor anticipates.
2 F 2
*
330
PNEUMATICS.
CHAPTER XIII.
PNEUMATICS.
SECTION I.-Equilibrium of air and elastic fluids.
1. THE fundamental propositions that belong to hydrostatics
are common to the compressible and the incompressible fluids,
and need not, therefore, be repeated here.
2. Atmospheric air is the best known of the elastic fluids,
and has been defined an elastic fluid, having weight, and resist-
ing compression with forces that are directly as its density, or
inversely as the spaces within which the same quantity of it is
contained.
The correctness of this definition is confirmed by experi-
ment.
The weight of air is known from the Torricellian experi-
ment, or that of the barometer. The air presses on the orifice
of the inverted tube with a force just equal to the weight of the
column of mercury sustained in it.
A bottle, weighed when filled with air, is found heavier than
after the air is extracted. The pressure of the atmosphere is at
a mean about 14 lbs. on every square inch of the earth's sur-
face. Hence the total pressure on the convex surface of the
10,686,000,000 hundreds of millions of pounds.
earth
The elastic force of the air is proved, by simply inverting at
vessel full of air in water: the resistance it offers to farther
immersion, and the height to which the water ascends within.
it, in proportion as it is farther immersed, are proofs of the
elasticity of the air contained in it.*
When air is confined in a bent tube, and loaded with differ-
ent weights of mercury, the spaces it is compressed into are
*It is in virtue of this property, and ought to be known as extensively as pos-
sible, that a man's hat will serve in most cases as a temporary life-preserver, to per-
sons in hazard of drowning, by attending to the following directions:- When a
person finds himself in, or about to be in, the water, let him lay hold of his hat
between his hands, laying the crown close under his chin, and the mouth under the
water. By this means, the quantity of air contained in the cavity of the hat will be
sufficient to keep the head above water for several hours, or until assistance can be
rendered.
PNEUMATICS: AIR.
331
J
found to be inversely as those weights. But those weights are
the measures of the elasticity; therefore the elasticities are
inversely as the spaces which the air occupies.
The densities are also inversely as those spaces; therefore
the elasticity of air is directly as its density. This law was
first proved by Mariotte's experiments.
In all this, the temperature is supposed to remain un-
changed. These properties seem to be common to all elastic
fluids.
Air resists compression equally in all directions. No limit
can be assigned to the space which a given quantity of air would
occupy if all compression were removed.
:
3. In ascending from the surface of the earth, the density of
the air necessarily diminishes for each stratum of air is com-
pressed only by the weight of those above it; the upper strata
are therefore less compressed, and of course less dense than those
below them.
4. Supposing the same temperature to be diffused through
the atmosphere, if the heights from the surface be taken in-
creasing in arithmetical progression, the densities of the strata
of air will decrease in geometrical progression. Also, since the
densities are as the compressing forces, that is, as the columns
of mercury in the barometer, the heights from the surface being
taken in arithmetical progression, the columns of mercury in
the barometer at those heights will decrease in geometrical
progression.
As logarithms have, relatively to the numbers which they
represent, the same property, therefore if b be the column of
mercury in the barometer at the surface, and ẞ at any height h
above the surface, taking m for a constant coefficient, to be
determined by experiment, hm (log blog 3), or h =
m log: where m may be determined by finding trigonome-
trically the value of h in any case, where b and ẞ have been
already ascertained.
5. If b be the height of the mercury in the barometer at the
lowest station, ẞ at the highest, and t'the temperatures of the
air at those stations, f the fixed temperature at which no cor-
rection is required for the temperature of the air; and if q and
9
q' be the temperatures of the quicksilver in the two barome-
ters, and n the expansion of a column of quicksilver, of which
the length is 1, for 1° of heat; h being the perpendicular height
(in fathoms) of the one station above the other.
332
PNEUMATICS: ATMOSPHERIC ALTITUDES.
h
Ma
A
10000 ( 1 + ·00244
¿
1
10000
b
(t + "' — ƒ log.) 3 x ( 1 + n (q—q')'
2
B
n being nearly
If the centigrade thermometer is used, because the beginning
of the scale agrees with the temperature f, so that f= 0, the
formula is more simple; and if the expansion for air and mer-
cury be both adapted to the degrees of this scale,
h
b
B (1 + 00018 (q —g')
6. The temperature of the air diminishes on ascending into
the atmosphere, both on account of the greater distance from
the earth, the principal source of its heat, and the greater
power of absorbing heat that air acquires, by being less com-
pressed.
10000 (1 + ·00441 (t + t'
(t + 1') log.
2
7. Professor Leslie, in the notes on his Elements of Geome-
try, p. 495, edit. 2d, has given a formula for determining the
temperature of any stratum of air when the height of the mer-
cury in the barometer is given. The column of mercury at the
lower of two stations being b, and at the upper ß, the diminu-
b
t+t'),
2
42)
tion of heat, in degrees of the centigrade, is
This seems to agree well with observation.
8. If the atmosphere were reduced to a body of the same
density which it has at the surface of the earth, and of the same
temperature, the height to which it would extend is, in fathoms,
equal to 4343 (1 + ·00441
or, taking the expansion
according to Laplace = 4343 (1 + 1 X 1).
4
1000.
is to that of air, as b to 4343 (1 + 1660),
1000
b
vity of air
P
My
25.-
Hence ifb be the height of the mercury in the barometer,
reduced to the temperature t, the specific gravity of mercury
or the specific gra-
4 t
72 (4343) (1 + 1000
tha
The divisor 72 is introduced in consequence of b being ex-
pressed in inches.-(Playfair's Outlines.)
PNEUMATICS: PUMPS.
333
SECTION II.-Pumps.
1. Def. The term Pump is generally applied to a machine
for raising water by means of the air's pressure.
2. The common suction-pump consists of two hollow cylin-
ders, which have the same axis, and are joined
in A C. The lower is partly immersed, per-
pendicularly in a spring or reservoir, and is
called the suction-tube; the upper the body
of the pump.
At Ac is a fixed sucker con-
taining a valve which opens upwards, and is
less than 34 feet from the surface of the water.
In the body of the pump is a piston D made
air-tight, moveable by a rod and handle, and
containing a valve opening upwards. And a
spout & is placed at a distance greater or less,
as convenience may require, above the greatest
elevation of d.

A
D
B
C
3. To explain the action of this pump.
Suppose the moveable piston D at its lowest depression, the
cylinders free from water, and the air in its natural state. On
raising this piston, the pressure of the air above it keeping its
valve closed, the air in the lower cylinder A B forces open the
valve at a c, and occupies a larger space, viz., between в the
surface of the water, and D; its elastic force therefore being
diminished, and no longer able to sustain the pressure of the
external air, this latter forces up a portion of the water into the
cylinder A в to restore the equilibrium. This continues till the
piston has reached its greatest elevation, when the valve at a c
closes. In its subsequent descent, the air below D becoming
condensed, keeps the valve at A c closed, and escapes by forcing
open that at n, till the piston has reached its greatest depression.
In the following turns a similar effect is produced, till at length
the water rising in the cylinder forces open the valve at A c,
and enters the body of the pump; when, by the descent of
n, the valve in a c is kept closed, and the water rises through
that in n, which on re-ascending, carries it forward, and throws
it out at the spout &.
k
4. Cor. 1. The greatest height to which the water can be
raised in the common pump by a single sucker is when the
column is in equilibrio with the weight of the atmosphere,
that is, between 32 and 36 feet.
5. Cor. 2. The quantity of water discharged in a given time
is determined by considering that at each stroke of the piston
44
334
PNEUMATICS: SUCKING-PUMP.
a quantity is discharged equal to a cylinder whose base is a sec-
tion of the pump, and altitude the play of the piston.
6. To determine the force necessary to over-
come the resistance experienced by the piston
in ascending. Leth the height HF of the
=
surface of the water in the body of the pump
above E F the level of the reservoir; and a
the area of the section M N. Let h' the height
of the column of water equivalent to the pres-
sure of the atmosphere; and suppose the pis-
ton in ascending to arrive at any position m n
which corresponds to the height 1 r. It is evi-
dent that the piston is acted upon downwards by
the pressure of the atmosphere a³ h', and by
the pressure of the column B M == a² × HI;
therefore the whole tendency of the piston to
descenda² (h' + II I.)
But the piston is acted upon upwards by the pressure of the
air on the external surface Er of the reservoir a² h'; part
of which is destroyed by the weight of the column of water
having for its base m n, and height r 1 ;
FI);
the whole action upwards a² × (h'
whence r = a². (h'+1 1)=a³. (h' — F 1)
u³. F H = ash,
1
that is, the piston throughout its ascent is opposed by a force
equal to the weight of a column of water having the same
base as the piston, and an altitude equal to that of the sur-
face of the water in the body of the pump above that in the
reservoir. In order, therefore, to produce the upward motion
of the piston, a force must be employed equal to that determined.
above, together with the weight of the piston and rod, and the
resistance which the piston may experience in consequence of
the friction against the inner surface of the tube.*

m
1
ጎ
M
F
B
n.
N
O
H
F
When the piston begins to descend, it will descend by its
own weight; the only resistance it meets with being friction,
and a slight impact against the water.
7. Cor. 1. If the water has not reached the piston, let its
* Suppose the body of the pump to be 6 inches in diameter, and the greatest
height to which the water is raised to be 30 feet; suppose, also, the weight of the
piston and its rod to be 10lbs., and the friction one-fifth of the whole weight. Then,
by the rule at p. 201, of the square of the diameter gives the ale gallons in a
yard in length of the cylinder, and an ale gallon, p. 290, weighs 10 lbs. There-
fore (62 × 10)+ 1 62 × 10) = 360+7·4 367.4 lbs. weight of the opposing
column of water. And 367-4+ 10+¦ (377·4)=452·9 lbs., whole opposing pres-
10
BO
-
suro.
If the piston rod be moved by a lover whose arms are as 10 to 1, this pressure
will be balanced by a force of 45-29 lbs., and overcome by any greater force.
PNEUMATICS: SUCKING-PUMP.
335
level be in v z.
The under surface of the piston will be
pressed by the internal rarefied air. But this air, together with
the column of water, E v, is in equilibrio with the pressure of
the atmosphere a h'; and.. its pressure a². (h' E v). And
the pressure downwards
2
a³ h' ;
... Fa² XE V.
Hence the force requisite to keep the piston in equilibrio in-
creases as the water rises and becomes constant, and
a² h as
soon as the water reaches the constant level в н.
8. Cor. 2. If the weight of the piston be taken into the ac-
count, let this weight be equal to that of a column of water
whose base is m n and height p, =a² p
..F= a². (E v + p).
9. To determine the height to which the water will rise after
one motion of the piston; the fixed sucker being placed at the
junction of the suction-tube and body of the
pump supposing that after every elevation of the
piston there is an equilibrium between the pres-
sure of the atmosphere on the surface of the water
in the reservoir, and the elastic force of the rare-
fied air between the piston and surface of the
column of water in the tube, together with the
weight of that column.
Let a b be the surface of the water in the suc-
tion-tube, after the first stroke of the piston: if
the piston were for an instant stationary at D,
the pressure of the atmosphere would balance E b,
and the elastic force of the air in N a.
and y = h.
A F
Να
battl
h.
ܪ
πηλα
x R² b + x r². (α
h p³ a
R² b + p² a
whence h = x +
—

a,
Let A E the height of the suction-tube
DR the play of the piston = b,
h the height of a column of water equivalent to the pres-
sure of the atmosphere,
y
= the height of a column equivalent to the pressure of the
air in Na,
x = E α,
and R and r = the radii of the body and the suction-
tube.
Then x+y=h,
7.3 X
h n² a
R³ b + p³ a — p² x
r²
-
U
1)
}
A
x)
--
a
b
L
E F
I
N
C
336
PNEUMATICS SUCKING-PUMP.
糞
​2
... h R² b + h p² a
R³
or x² - (h + 2. b + a).. x
and yı
pies N a';
and x²
p x
R³
if m be
and p = h + m b + a ;
2039
.. x = }. {p ± √p² - 4 h m b},
and y = }. {2 h—p√p²-4 h m b},
;
only one of which values will be applicable, viz. that which an-
swers to the lower sign; since x and y must be less than h
and if the upper sign be used, x will be found greater than h.
10. Having given the height of the water raised, and that due.
to the pressure of air in the pump after the first ascent of the
piston; to determine them for the second, third, &c. ascents.
Let E a' represent the height of the water after the second
ascent, and let it
and let y₁
Х 11
1
the height due to the elastic force of the air ;
then x₁+y1
h;
since the air which occupied c a now occu-
ม
Y₁
1
X B
— h p² x=R² b x+r² α x = p² x² + h r² α,
R³
h. .b,
22
·
f
whence h
A b
Na"
KİŞ
· h m b,
ро
Y r³. (α
(a
R² b + p³. (a
magdada tar
x₁ +
1
petang tad
J
x)
XC 1)
y. (α-x)
mb+ax,
1
y. (a—x)
mb + a
Agat
X 1
{
;
and .*. X₁
}. {p — √ p² 4 h m b − 4 x . (p + a −
and y₁ = . {2h-p+p²-4 h mb-4 x. (h+a—x)}.
1.
J
From these are deduced values of x2, y2, X, Y3, &c.
Y
}. {p — √pº 4 h m b 4 x₁. h + a x₁)},
Y 2
1
= }. {2 h − p + √p²
'p² - 4 h m b — 4 x₁. (h+a — x₁)},
and so on. Whence if x be taken to represent the height of
the water after (n + 1) ascents,
n
2
Xn
-1
•
170 {p-
4 h m b
√p³
4 xn s⋅ (h + a
Xn-1)},
and y₁ = . {2 h − p + √ p² — 4 h m b−4 x„- 1 • (h+a-xn−1)}.
x)},
PNEUMATICS: SUCKING-PUMP.
337
11. Cor. 1. Hence may be determined the height to which
the water can rise after any given number of ascents of the pis-
ton, and the elastic force of the air in the suction-tube.
12. Cor. 2. Knowing the elevation due to each particular
stroke, the differences of those elevations, and the successive
differences in the elastic force of the remaining air, may be
known.
13. Cor. 3. If the weight of the valve c be not considered, it
is evident that after a certain number of strokes a vacuum will
be produced in the suction-tube, provided it be equal to, or not
greater than the height due to the pressure of the atmosphere,
that is, if a be not greater than ħ.
For in this case, xn= Xn-19
Xn-1
1)},
2{p
√p³.
4 h m b
and.. X-1
x
whence Xn-1
h, the greatest height of the column of water in
the tube. If therefore the length of the suction-tube do not ex-
ceed the height due to the pressure of the atmosphere, the water
will continue to ascend in it after every stroke of the piston, till
at length it will pass into the body of the pump.
But if the altitude of A r be greater than h, the water will
continue to ascend without ever reaching its maximum height.
For in this case, an actual vacuum cannot be produced; and as
xn+y=h, and y, can never become=o.; .. x, can never=h.*
But the successive values of y continually decreasing, the cor-
responding values of x will continually increase.
14. Cor. 4. If the weight of the valve c be taken into the ac-
count, a column of water must be added equal to the additional
pressure to be overcome. Let the height of this column,
/
then
x + y + 1
and .. x + y
•
4x-1. (h+u
n
h;
h - l h.
State
If therefore this value of h' be substituted for h, the preceding
equations are applicable.
15. Cor. 5. In the preceding cases, the moveable piston has
been supposed to descend to a c. If it does not, it may happen
that the water may not reach A c, though a c be less than 34
feet from the surface of the water in the reservoir.
After the first elevation of the moveable piston to its
greatest altitude, c being closed, the clastic force of the air
between DN and A c is (h—x), and its magnitude я b rº. If
* Hence it appears that it is not strictly true, that water will ascend in the suc-
tion-tube to a height equal to that of a column equivalent to the pressure of the
atmosphere. This is a limit to which it approximates, but does not reach in a
finite time.
2 G
338
PNEUMATICS: SUCKING-PUMP.
in descending, the piston describes a space b' less than b, so as
to stop at a distance b b' from a c, this magnitude becomes
K
b
Now
b- b "
in order that the pressure upwards may open the valve, this
must exceed the elastic force of the atmosphere;
2
(b — b') . π в³ ; .. the elastic force is (h-x).
—x)
B
b
(h — x) · b — b'
Katanga kataga
or (h — x). b > h . (b — b') ;
X b'
h
b'
If..
be less than the valve DN will not open;
S
b
there will therefore be the same quantity of air between a c
and the sucker: which, when the piston has reached its highest
elevation, will have the same elastic force as that between A C
and a' b'; and therefore c being equally pressed on both sides,
will remain unmoved, and the water will not ascend.
J
16. If the fixed sucker be placed at the surface of the water;
to determine the ascent of the water in the suction-tube.
and .. x
..
Xn -
... bxh b', or
Let E α, E a' be the successive heights to which the water
rises; then after the first ascent of the piston,
x+y=h,
and y=
whence x = 2. {p - √p⁹
h a
mb + a—xX
2
4 h m b }
and y. {2 h − p + √p" — 4 h m b},
which equations are the same as were determined for the first
ascent of the piston (9). Therefore, in the same manner as
before,
1
we shall have x 1. {p
√ p² — 4 h m b-4hxn-1},
4 h m b
}. { 2 h − p + √ p²
X
h'
Y n
4 h Xn-1}.
17. Cor. 1. If the water be supposed to stop after (n + 1)
ascents of the piston, then a
Xn-1 i
12
r
n--1
> h,
(
}. {p — √ p² - 4 h m b-4 h xa-1},
2.
whence . {a+mb ± √ (a + m b² — 4 h m b}.
}.
PNEUMATICS: SUCKING-PUMP.
339
2
Hence, therefore, there are two altitudes at which the water
may stop in its ascent, if (a + m b)³ is equal to or greater than
4 h m b. In the former case the two values of x„-1 are equal,
that is, there will be only one altitude §. (a + mb), at which
the water will stop. In the latter case there are two which
may be ascertained.
If 4 h m b be greater than (a + m b), the water will not
stop.
Ex. 1. If h 32 feet, a = 20, b 4, and m = 1, or the
suction-tube and body of the pump be of the same diameter,
X¸~1 =1'. {20 +4 ± √ (24)º
(24)º — 4.1. 32.4} = }. { 24±√64}
n-1
16 or 8.
25, b
32 feet, a
Ex. 2. If h
1. {25 +8 ± √ (33)³
41.8062 or 24.1938.
Xn-1
✓65}
18. Cor. 2. If m
S
makamandag
2, and m = 4,
4.32.4.2}
1. {33 ±
1, or the tubes have the same diameter,
X₁₁ = }. {a + b ± √ (a + b)³ — 4 h b},
Xn-1
which is imaginary, if (a + b) is less than 4 h b, or b greater
(a + b)³
4 h
than
In order, therefore, that this pump may produce its effect,
the play of the piston must be greater than the square of its
greatest altitude above the surface of the water in the reser-
voir divided by four times the height due to the pressure of
the atmosphere.
19. The lifting-pump consists of a hollow cylinder, the
body of which is immersed in the reservoir. It is furnished
with a movable piston, which, entering below, lifts the water
up, and is movable by means of a frame which is made to ascend
and descend by a handle. The piston is furnished with a valve
opening upwards. A little below the surface of the water
is a fixed sucker with a valve opening upwards. This is an
inconvenient construction, upon the peculiarities of which we
need not dwell.
340
PNEUMATICS: FORCING-PUMP.

20. The forcing-pump consists of a
suction-tube A EF C, partly immersed in
the reservoir, and of the body of the pump
ABG C, and of the ascending tube H M.
The body is furnished with a movable solid
sucker or plunger D, made air-tight. And
at A C and H are fixed suckers with valves
opening upwards.
21. To explain the action of this pump.
Suppose the plunger D at its greatest de-
pression; the valves closed, and the air in
its natural state. Upon the ascent of
the air in a CD occupying a greater space,
its clasticity will be diminished, and consequently the greater
elasticity of the air in a r will open the valve at A c, whilst that
at н is kept closed by the elasticity of the external air; water
therefore will rise in the suction-tube. On the descent of D
from its greatest elevation, the clasticity of the air in the body
of the pump will keep the valve A c closed, and open that at н,
whence air will escape. By subsequent ascents of the piston,
the air will be expelled, and water rise into the body. The
descending piston will then press the water through the valve
at н, which will close, and prevent its return into the body of
the pump; D therefore ascending again, the space left void will
be filled by water pressing through the valve A c; and this
upon the next ascent of D is forced into the ascending tube;
and thus by the ascents and descents of D, water may be raised
to the required height.
D9
B
dadd
M
A
E
D
N
HO
щи с
F
22. Cor. In this pump o must not ascend higher than about
32 feet from the surface of the water in the reservoir.
23. To determine the force necessary to overcome the resist-
ance experienced by the piston.
Let the height of a column of water equivalent to the
h
pressure of the atmosphere, and Eв the height to which the
water is forced. Let мM N be any position of the piston n whose
arca A, and the weight of the piston and its appendages = r.
the force necessary to push the piston upwards during
the suction, friction not being considered, and y = that em-
ployed to force it down.
=
Let x =
When the piston ascends, and H is closed
X = P + A h s. (h
M E)
= PA. M E.
Let the sucker be in the same position in its descent, and
therefore a c closed, and H open.
PNEUMATICS: FORCING-PUMP.
341
J
Y = A h + Ą . M B
P.
A. MB
pig
Hence x + y = A. E B ; or the whole force exerted in the
case of equilibrium is equal to the weight of a column of
water whose base is equal to that of the piston, and altitude the
distance between the surface of the water and the point to which
it is to be raised.
.. PA
.". P
(A h + P)
24. Cor. 1. In this pump the effort is divided into two parts,
one opposed to the suction, and the other to the forcing; whereby
an advantage is gained over the other pumps where the whole
force is exerted at once whilst the water is raised.
25. Cor. 2. In order to have the force applied uniform, let
X = Y;
"
MEA. MB
A. (M B M E).
26. Cor. 3. In the common forcing-
pump the stream is intermitting; for
there is no force impelling it during the
return of the sucker.
The piston therefore must play in such a manner that м в may
be greater than м E.
One mode of remedying this, is by
making an interruption in the ascending
tube, which is surrounded by an air-ves-
A
Ꮲ ;
T
sel r; in which when the water has risen above z, the air above
it is compressed, and by its elasticity forces the water up through
z; the orifice of which is narrower than that of the tube, and
therefore the quantity of water introduced during the descent
of the piston will supply its discharge for the whole time of the
stroke, producing a continued stream.
גן
R
27. The fire-engine consists of a large receiver A B C D,
called the air-vessel, into which
water is driven by two forcing-
pumps, E F, G н, (whose pistons
are and R), communicating
with its lower extremities at I
and K, through two valves open-
ing inwards. From the receiver
proceeds a tube M L through
which the water is thrown, and
directed to any point by means of a pipe moveable about the
extremity L. The pumps are worked by a lever, so that
N
45
2 G 2
I
IM
K
B
71
Z
0
E
I


342
PNEUMATICS: FIRE-ENGINE.
:
한
​|
whilst one piston descends the other ascends. The pumps com-
municate with a reservoir of water at N.
28. To explain the action of this engine.
The tube N being immersed in the reservoir, and the piston R
drawn up, the pump & н becomes filled; and the descent of the
piston R will, as in the forcing-pump (21), keep the valve н
close, and cause the water to pass into the air-vessel by the
valve 1, whilst by the weight of the water in the air-vessel, the
valve i will be kept shut. In the same manner, when R ascends,
a descending will force the water through к into the air-vessel.
By this means the air above the surface of the water becoming
greatly compressed, will by its elasticity force the water to as-
cend through M L, and to issue with a great velocity from the
pipe.*
29. When the air-vessel is half full of water, the air being
then compressed into half its natural space, will have an elastic
force equal to twice the pressure of the atmosphere: therefore,
when the stop-cock is turned, the air within pressing on the
subjacent water with twice the force of the external air, will
cause the water to spout from the engine to the height of
(2-1) 33, or 33 feet; except so far as it is diminished by
friction.
Or, generally, if
n
Jeg
n
1
denote the fractional height of the
1
13
water in the air barrel, then will denote the height of the
M
n
space occupied by the compressed air n times the pressure of
the atmosphere its elastic force, and (n-1) 33, the height in
feet to which the water may be projected.
Thus, if of the air barrel be the height of the water, the
elastic force of the air will be four times the pressure of the
atmosphere, and (41) 3399 feet, the height to which the
water may then be thrown by the engine.
30. The modifications in the constructions of pumps with a
view to their practical applications are very numerous.
Those
who wish to acquaint themselves with some of the most useful,
may consult the 2d vol. of my Treatise on Mechanics, and
Nos. 13, 41, 69, and 93, of the Mechanics' Magazine.
In addition to these, there may now be presented a short
* The preceding part of this section is taken from Bland's Hydrostatics; a very
elegant and valuable work, which I bog most cordially to recommend to those who
wish to obtain a comorchensive knowledge of the theory of this department of
science.
1
PNEUMATICS: QUICKSILVER PUMP.
343
account of a quicksilver pump, which has been recently in-
vented by Mr. Thomas Clark of Edinburgh, and which works
almost without friction. It has great power in drawing and
forcing water to any height, and is extremely simple in its
construction.
a a is the main pipe inserted into the
well b ; a valve is situated at c, and
another at d, both opening upwards;
a piece of iron tube is then bent into a
circular form, as at f, again turned off
at g in an angular direction, so as to
pass through a stuffing box at h, and
from thence bent outwards as at i, con-
necting itself with the ring. A quan-
tity of quicksilver is then put into the
ring filling it from q to q, and the ring
being made to vibrate upon its axis h,
a vacuum is soon effected in the main
pipe by the recession of the mercury
from g to q, thereby causing the water to rise and fill the
vacuum upon the motion being reversed, the quicksilver
slides back to g, forces up the water and expels it at the
spout e.

20
ส
b
METT LEMBELLI
CH
(100) AMEER DE
DU KAN RIDE IN
IBERI IN
CORDE KERER À LEVYAL
1 Mer cold oak
The Sam 142 DE
全​可​廿​T |
DANN KORRALES! tel
ESTUDI LOG
Thin ka sali
BEN SAN LOR
પ્રેમન
h
q
Ori
FRİDALIRI, JMLJ
134) ILIUMTHAI
(.DİSEL TESTANTE ||,
jargalkan.
allsend a 1 (0)
pekti 175lung|
|||||¬KA320FILMILÍ
filtrantCHIAJ
filistiumaidani,
น.เทป มาฝ
MIRIÇORBAIK,
THE HILERAK)
[AMJANORODATE
TUGREIDARUJEET
Hi bitnoSEAR
"Mr. Clark calculates that a pump of this description with
a ring twelve fect in diameter, will raise water the same height
as the common lifting pump, and force it one hundred and fifty
feet higher without any friction." (Mechanics' Register, and
Jamieson's Edinburgh Journal.)
31. It is usual to class with pumps, the machine known by
the name of Archimedes' screw or the water-snail. This con-
sists either of a pipe wound spirally round a cylinder, or of one
or more spiral excavations formed by means of spiral projec-
tions from an internal cylinder, covered by an external cylin-
drical case, so as to be water tight. The cylinder which carries
the spiral is placed aslant, so as to be inclined to the horizon in
an angle of from 30° to 45°, and capable of turning upon pivots
in the direction of its axis posited at each extremity. The
lower end of the spiral canal being immersed in the river or
reservoir from which water is to be raised, the water descends
at first in the said canal solely by its gravity; but the cylinder
being turned, by human or other energy, the water moves on
in the canal, and at length it issues at the upper extremity of
the tube.
Several circumstances tend to make this instrument imper-
fect and inefficacious in its operation. The adjustments
344
PNEUMATICS SPIRAL PUMP.
necessary to ensure a maximum of effectual work are often
difficult to accomplish. It seldom happens, therefore, that
the measure of the work done exceeds a third of the power
employed so that this apparatus, notwithstanding its apparent
ingenuity and simplicity, is very sparingly introduced by our
civil engineers.
32. Spiral Pump. This machine is formed by a spiral pipe
of several convolutions, arranged either in a single plane, as
in the marginal diagram, or upon a
cylindrical or conical surface, and re-
volving round an axis. The curved
pipe is connected at its inner end,
by a central water-tight joint, to an.
ascending pipe, rr, while the other
end, s, receives, during each revolu-
tion, nearly equal quantities of air
and water. This apparatus is usually'
called the Zurich machine, because it
was invented, about 1746, by Andrew
Wirtz, an inhabitant of Zurich. It has been employed with
great success at Florence, and in Russia; and the late Dr.
Thomas Young states, that he employed it advantageously for
raising water to a height of forty feet. The outer end of the
pipe is furnished with a spoon s, which contains as much water
as will half fill one of its coils. The water enters the pipe a
little before the spoon has reached its highest position, the
other half remaining full of air. This air communicates the
pressure of the column of water to the preceding portion; and
in this manner the effect of nearly all the water in the wheel
is united, and becomes capable of supporting the column of
water, or of water mixed with air, in the ascending pipe. The
air nearest the joint is compressed into a space much smaller
than that which it occupied at its entrance; so that, when the
height is considerable, it becomes advisable to admit a larger
portion of air than would naturally fill half the coil. This,
while it lessens the quantity of water raised, lessens also the
force requisite to turn the machine. The loss of power, sup-
posing the machine well constructed, arises only from the
friction of water on the pipes, and that of the wheel on its
axis and where a large quantity is to be raised to a moderate
height, both of these sources of resistance may be rendered
very inconsiderable.

F
Р
33. Schemnitz Vessels, or the Hungarian Machine. The
mediation of a portion of air is employed for raising water,
not only in the spiral pump, but also in the air vessels of
PNEUMATICS SCHEMNITZ VESSELS.
345
Schemnitz, as shown in the annexed dia-
gram. A column of water, descending
through a pipe, c, into a closed reservoir,
B, containing air, obliges the air to act, by
means of a pipe, D, leading from the upper
part of the air vessel, or reservoir, on the
water in a second reservoir, A, at any dis-
tance either above it or below it, and forces
this water to ascend through a third pipe,
E, to any height, less than that of the first
column. The air vessel is then emptied,
the second reservoir filled, and the whole
operation repeated. The air, however,
must acquire a density equivalent to the
requisite pressure before it can begin to
act so that, if the height of the columns
were 34 feet, it must be reduced to half its natural space
before any water could be raised, and thus half of the force
would be lost. But, where the height is small, the height lost
in this manner is not greater than what is usually spent in over-
coming friction, and other imperfections of the machinery em-
ployed. The force of the tide, or of a river rising and falling
with the tide, might easily be applied to the purpose of raising
water by a machine of this kind. Thus, if at low tide the ves-
sel A were filled with air; then, at high tide, the water flowing
down the tube E, would cause the water in the vessel в to as-
cend in the pipe c.
34. The Hydraulic Ram. In this hydraulic arrangement,
the momentum of a stream of water flowing through a long
pipe is employed to raise a small quantity of water to a con-
siderable height. The passage of the pipe being stopped by
a valve which is raised by the stream, as soon as its motion
becomes sufficiently rapid, the whole column of fluid must ne-
cessarily concentrate its action almost instantaneously upon the
valve. In these circumstances it may be regarded as losing
the characteristic property of hydraulic pressure, and to act
almost as though it were a single solid so that, supposing the
pipe to be perfectly elastic and inextensible, the impulse may
overcome almost any pressure that may be opposed to it. If
another valve opens into a pipe leading to an air vessel, a
certain quantity of the water will be forced in, so as to con-
dense the air, more or less rapidly, to the degree that may be
required for raising a portion of the water contained in it to a
given height.

C
D
A
E
Ma
The late Mr. Whitehurst appears to have been the first
who employed this method: it was afterwards improved by
346
PNEUMATICS: HYDRAULIC RAM.
Mr. Boulton. But, like many English inventions, it never
was adequately estimated, until it was brought into public no-
tice by a Frenchman. M. Montgolfier, its re-inventor, gave to
it the name which it now bears of the Hydraulic Ram, in al-
lusion to the battering ram.
S
The essential parts of this machine are represented in the
annexed diagram. When the water in the pipe A в (moving

D
E
In the direction of the arrows) has acquired sufficient velocity,
it raises the valve B, which immediately stops its farther pas-
sage. The momentum which the water has acquired then
forces a portion of it through the valve, c, into the air vessel, d.
The condensed air in the upper part of D causes the water to
rise into the pipe E, as long as the effect of the horizontal
column continues. When the water becomes quiescent, the
valve в will open again by its own weight, and the current
along A B will be renewed, until it acquires force enough to
shut the said valve в, open c, and repeat the operation.
The motion in the horizontal tube arises from the accelera-
tion of the velocity of a liquid mass falling down another tube,
and communicating with this.
In an experiment made upon an hydraulic ram, at Avilly,
near Senlis, by M. Turquet, bleacher, the expense of power was
found to be to the produce, as 100 to 62. In another, as 100
to 55 in two others, as 100 to 57. So that a hydraulic ram,
placed not in unfavourable circumstances, may be reckoned to
employ usefully rather more than half its force.
**For more full accounts of the three last contrivances, the
reader may consult the 2d volume of my
Mechanics.
SECTION III.-Wind and Windmills.
1. Air, when in continuous motion in one direction, be-
comes a very useful agent of machinery, of greater or less
energy, according to the velocity with which it moves.
Were
PNEUMATICS: WIND AND WINDMILLS.
347
it not for its variability in direction and force, and the conse-
quent fluctuations in its supply, scarcely any more appropriate
And even with all
first mover could generally be wished for.
its irregularity, it is still so useful as to require a separate con-
sideration.
2. The force with which air strikes against a moving sur-
face, or with which the wind strikes against a quiescent sur-
face, is nearly as the square of the velocity: or, more cor-
rectly, the exponent of the velocity determined according to
the rule given at pa. 103, varies between 2:03 and 2·05; so
that in most practical cases, the exponent 2, or that of the
If I be the
square, may be employed without fear of error.
angle of incidence, s the surface struck in feet, and v the
velocity of the wind, in feet, per second; then for the force in
avoirdupois pounds, either of the two following approximations
v³ s² sin³ I
may be used: viz. f
440
or f
·002288 v² s³ sin² 1.
Of these, the first is usually the easiest in operation, requir-
ing only two lines of short division, viz. by 40 and by 11.
If the incidence be perpendicular, sin² 1 = 1, and these be-
come,
f
3. The table in the margin exhi-
bits the force of the wind when blow-
ing perpendicularly upon a surface of
one foot square, at the several velocities
announced. The velocity of 80 miles
per hour is that by which the aeronaut
Garnerin was carried in his balloon
from Ranelagh to Colchester, in June,
1802. It was a strong and boisterous
wind; but did not assume the character
of a hurricane, although a wind with
that velocity is so characterized in
Rouse's table. In Mr. Green's aerial
voyage from Leeds, in September, 1823,
he travelled 43 miles in 18 minutes,
although his balloon rose to the height
of more than 4000 yards.
v2 52
440
•002288 v³ S³.

Tagg
Velocity of the
Wind.
Miles
lin onc
hour
-feet
in one
second
1 2 3 →
•005
⚫020
·044
4
⚫079
•123
5
10
-492
15 22.00
1.107
20
29.34
1.968
36.67
3.075
25
30
44.01.
4.429
35 51.34
6.027
58-68
7.873
40
45
66.01
9.963
50
73.35
12.300
60
88.02
17.715
80
117.36 31.490
100
146.70 49-200
Perpendi-
cular force
on one sq.
foot in a-
voirdupois
pounds.
1·47
2.93
4-40
5.87
7.33
14.67
1762, that the force
Borda found by experiment in the year
of the wind is very nearly as the square of the velocity, but
348
PNEUMATICS: WIND AND WINDMILLS.
he assigned that force to be greater than what Rouse found
(as expressed in the above table) in the ratio of 111 to 100.
Borda ascertained also, as was natural to expect, that, upon
different surfaces with the same velocity, the force increased
more rapidly than the surface. M. Valtz, applying the me-
thod of the minimum squares to Borda's results, ascertained
that the whole might be represented by the formula
0.001289 x² +0.000050541 x³
Y
and nearly as correctly by y = 0.00108 x2.2
2 representing the surface in square inches (French), and y
the force corresponding to the velocity of 10 feet per second
expressed in French pounds.
:
4. In the application of wind to mills, whatever varieties
there may be in their internal structure, there are certain
rules and maxims with regard to the position, shape, and
magnitude of the sails, which will bring them into the best
state for the action of the wind, and the production of use-
ful effect. These have been considered much at large by
Mr. Smeaton for this purpose he constructed a machine,
of which a particular description is given in the Philosophi-
cal Transactions, vol. 51. By means of a determinate weight
it carried round an axis with an horizontal arm, upon which
were four small moveable sails. Thus the sails met with a
constant and equable blast of air; and as they moved round,
a string with a weight affixed to it was wound about their
axis, and thus showed what kind of size or construction of
sails answered the purpose best. With this machine a great
number of experiments were made; the results of which were
as follow:
4
(1.) The sails set at the angle with the axis, proposed as the
best by M. Parent and others, viz. 55°, was found to be the
worst proportion of any that was tried.
(2.) When the angle of the sails with the axis was increased
from 72° to 75°, the power was augmented in the proportion of
31 to 45; and this is the angle most commonly in use when the
sails are planes.
(3.) Were nothing more requisite than to cause the sails
to acquire a certain degree of velocity by the wind, the po-
sition recommended by M. Parent would be the best. But
if the sails are intended with given dimensions to produce
the greatest offects possible in a given time, we must, if planes
are made use of, confine our angle within the limits of 72 and
75 degrees.
(4.) The variation of a degree or two, when the angle is
near the best, is but of little consequence.
PNEUMATICS: WIND AND WINDMILLS.
349
(5.) When the wind falls upon concave sails, it is an advan-
tage to the power of the whole, though each part separately
taken should not be disposed of to the best advantage.
(6.) From several experiments on a large scale, Mr. Smea-
ton has found the following angles to answer as well as any.
The radius is supposed to be divided into six parts; and th,
reckoning from the centre, is called 1, the extremity being
denoted 6.
No.
<I Q2 09
1
3
4
5
6
Angle with
that axis.
72°
71
72
74
77/1/2
83
Angle with
the plane of
motion.
18°
19
18 middle
16
12/12/
7 extremity.
(7.) Having thus obtained the best method of weathering
the sails, i. e. the most advantageous manner in which they can
be placed, our author's next care was to try what advantage
could be derived from an increase of surface upon the same
radius. The result was, that a broader sail requires a larger
angle; and when the sail is broader at the extremity than near
the centre, the figure is more advantageous than that of a
parallelogram. The figure and proportion of enlarged sails,
which our author determines to be most advantageous on a
large scale, is that where the extreme bar is one-third of the
radius or whip (as the workmen call it), and is divided by the
whip in the proportion of 3 to 5. The triangular or loading
sail is covered with board from the point downward of its
height, the rest as usual with cloth. The angles above men-
tioned are likewise the most proper for enlarged sails; it
being found in practice that the sails should rather be too
little than too much exposed to the direct action of the
wind.
Some have imagined, that the more sail the greater would
be the power of the windmill, and have therefore proposed to
fill up the whole area; and by making each sail a sector of an
ellipsis, according to M. Parent's method, to intercept the
whole cylinder of wind, in order to produce the greatest effect
possible. From our author's experiments, however, it appeared,
that when the surface of all the sails exceeded seven-eighths of
the arca, the effect was rather diminished than augmented.
Hence he concludes, that when the whole cylinder of wind is
intercepted, it cannot then produce the greatest effect for want
of proper interstices to escape.
46
2 H
350
PNEUMATICS: WIND AND WINDMILLS.
"It is certainly desirable (says Mr. Smeaton), that the sails
of windmills should be as short as possible; but it is equally
desirable, that the quantity of cloth should be the least that may
be, to avoid damage by sudden squalls of wind. The best
structure, therefore, for large mills, is that where the quantity
of cloth is the greatest in a given circle that can be on this
condition, that the effect holds out in proportion to the quantity
of cloth; for otherwise the effect can be augmented in a given
degree by a lesser increase of cloth upon a larger radius than
would be required if the cloth was increased upon the same
radius."
:
(8.) The ratios between the velocities of windmill sails un-
loaded, and when loaded to their maximum, turned out very
different in different experiments; but the most common pro-
portion was as 3 to 2. In general it happened that where the
power was greatest, whether by an enlargement of the surface
of the sails, or an increased velocity of the wind, the second
term of the ratio was diminished.
(9.) The ratios between the least load that would stop the
sails and the maximum with which they would turn, were
confined betwixt that of 10 to 8 and 10 to 9; being at a me-
dium about 10 to 8.3, and 10 to 9, or about 6 to 5; though on
the whole it appeared, that where the angle of the sails or
quantity of cloth was greatest, the second term of the ratio was
less.
On this
(10.) The velocity of windmill sails, whether unloaded or
loaded, so as to produce a maximum, is nearly as the velocity
of the wind, their shape and position being the same.
subject Mr. Ferguson remarks, that it is almost incredible to
think with what velocity the tips of the sails move when acted
upon by a moderate wind. He has several times counted the
number of revolutions made by the sails in 10 or 15 minutes;
and, from the length of the arms from tip to tip, has com-
puted, that if an hoop of the same size was to run upon plain
ground with an equal velocity, it would go upwards of 30 miles
in an hour.
(11.) The load at the maximum is nearly, but somewhat less
than, as the square of the velocity of the wind; the shape and
position of the sails being the same.
(12.) The effects of the same sails at a maximum are nearly,
but somewhat less than, as the cubes of the velocity of the
wind.
(13.) The load of the same sails at a maximum is nearly as
the squares, and the effect as the cubes of their number of turns
in a given time.
(14.) When sails are loaded so as to produce a maximum
PNEUMATICS: SMEATON'S RULES FOR WINDMILLS.
351
at a given velocity, and the velocity of the wind increases, the
load continuing the same; then the increase of effect, when
the increase of the velocity of the wind is small, will be nearly
as the squares of these velocities: but when the velocity of the
wind is double, the effects will be nearly as 10 to 274; and
when the velocities compared are more than double of that
where the given load produces a maximum, the effects increase
nearly in a simple ratio of the velocity of the wind. Hence
our author concludes that windmills, such as the different spe-
cies for draining water, &c. lose much of their effect by acting
against one invariable opposition.
(15.) In sails of a similar figure and position, the number of
turns in a given time will be reciprocally as the radius or length
of the sail.
(16.) The load at a maximum that sails of a similar figure
and position will overcome, at a given distance from the centre
of motion, will be as the cube of the radius.
(17.) The effects of sails of similar position and figure are
as the square of the radius. Hence augmenting the length of
the sail without augmenting the quantity of cloth, does not in-
crease the power; because what is gained by length of the lever
is lost by the slowness of the motion. Hence also, if the sails
are increased in length, the breadth remaining the same, the ef-
fect will be as the radius.
(18.) The velocity of the extremities of the Dutch sails, as
well as of the enlarged sails, either unloaded or even when
loaded to a maximum, is considerably greater than that of the
wind itself. This appears plainly from the observations of
Mr. Ferguson, already related, concerning the velocity of
sails.
(19.) From many observations of the comparative effects of
sails of various kinds, Mr. Smeaton concludes, that the en-
larged sails are superior to those of the Dutch construction.
M
(20.) He also makes several just remarks upon those wind-
mills which are acted upon by the direct impulse of the wind
against sails fixed to a vertical shaft: his objections have, we
believe, been justified in every instance by the inferior efficacy
of these horizontal mills.
"The disadvantage of horizontal windmills (says he) does
not consist in this, that each sail, when directly opposed to
the wind, is capable of a less power than an oblique one of
the same dimensions; but that in an horizontal windmill little
more than one sail can be acting at once: whereas in the com-
mon windmill, all the four act together; and therefore, sup-
posing each vane of an horizontal windmill to be of the same
size with that of a vertical one, it is manifest that the power of
S
Bowl
352
PNEUMATICS: SMEATON'S RULES FOR WINDMILLS.
a vertical mill will be four times as great as that of an horizon-
tal one, let the number of vanes be what they will. This dis-
advantage arises from the nature of the thing; but if we consider
the further disadvantage that arises from the difficulty of getting
the sails back again against the wind, &c. we need not wonder
if this kind of mill is in reality found to have not above one-
eighth or one-tenth of the power of the common sort; as has
appeared in some attempts of this kind."
Coulomb's Experiments,
་
5. M. Coulomb, whose experiments have tended to the elu-
cidation of many parts of practical mechanics, devoted some
time to the subject of windmills. The results of his labours
were published in the Memoirs of the Paris Academy for 1781.
The mills to which he directed his attention were in the vi-
cinity of Lille, and were, in fact, oil mills. From the outer
extremity of one sail to the corresponding extremity of the
opposite sail, was 70 feet, the breadth of each sail 6 feet, of
which the sail-cloth when extended occupies 5½ feet, being
attached on one side to a very light plank; the line of junc-
tion of the plank and of the sail-cloth, forms, on the side struck
by the wind, an angle sensibly concave at the commencement
of the. sail, but diminishes gradually all along so as to vanish at
the remoter extremity. The angle with the axis, at seven feet
from the shaft, is 60°, and it increases continually, so as to
amount to nearly 84° at the extremity. The shaft upon which
the sails turn, is inclined to the horizon, in different angles in
different mills, from 8 to 15 degrees.
Coulomb infers from his experiments,
(1.) That the ratio between the space described by the wind
in a second, and the number of turns of a sail in a minute, is
nearly constant, whatever be the velocity of the wind; the said
ratio being about 10 to 6, or 5 to 3.
(2.) That with a wind whose velocity is 21 feet (English)
per second, the quantity of action produced by the impulsion
of the wind is equivalent to a weight of 1080 pounds avoirdu-
pois raised 270 feet in a minute; the useful effect being equiva-
lent to a weight of 1080 pounds raised 232 feet in the same
time whence it results that the quantity of effect absorbed by
the stroke of the stampers, the friction, &c. is nearly a sixth
part of the quantity of action.
pl
(3.) Suppose one of these mills to work 8 hours in a day.
Coulomb regards its daily useful effect as equivalent to that
PNEUMATICS: WINDMILLS.
353
of 11 horses working at a walking-wheel, in a path of the usual
radius.
(4.) It is observable that in most windmills the velocity at
the extremity of the sails is greater than that of the wind. In
some cases, indeed, these velocities have been found in about
the ratio of 5 to 2. Now, it is evident that the impulsion of a
fluid against any surface whatever can only produce pressure,
or mechanical effect, when the velocity of the surface exposed
to the impulse is less than that of the fluid; and that the pres-
sure will be nothing when the velocity of the surface is equal
to or greater than that of the fluid. Indeed, in the latter case,
the pressure may operate against the motion of the sails, and
be injurious. It is desirable, therefore, in order to derive from
a windmill all the effect of which it is susceptible, so to adjust
the number of the turns that the velocity of the extremity of
the sails shall be less, or, at most, equal to that of the wind.
It would be highly expedient to make comparative experi-
ments on windmills, with a view to the determination of that
velocity of the extremity of the sails which corresponds with
the maximum of effect.
6. If v denote the velocity of the wind in feet per second, t
the time of one revolution of the sails, a the angle of inclination
of the sails to the axis, and n the distance from the shaft or
axle of rotation to the point which is not at all acted on by the
wind, or beyond which the sail-cloth ought to be folded
up; then theoretical considerations supply the following theo-
rem: viz.
14
•1099 t v tan. A.
Ex. Suppose v = 30 feet per second, t = 2.25 seconds, and
A 75°; then
D = 1092 × 30 × 2.25 × 3·73205
27.509 feet.
This result agrees nearly with one of Coulomb's experi-
ments, in which the velocity of the wind was 30 feet English
per second, the sails made 17 turns in a minute, and they were
obliged to fold off more than 6 feet from the extremity of
each sail, of 34 fect long, to obtain a maximum of effect.
The angle a at that distance from the tip of the sail was 75°
or 76°.
D
p
1
SECTION IV. Steam and Steam-Engines.
The whole power of the steam-engine depends on the em-
ployment of elastic vapour produced from water at high tem-
peratures.
2 H 2
354
PNEUMATICS: STEAM AND STEAM-ENGINES.
Steam, in fact, is highly rarefied water, the particles of which
are expanded by the absorption of caloric, or the matter of heat.
Water rises in vapour at all temperatures, though this is usually
supposed to take place only at the boiling point; when, how-
ever, the evaporation occurs below 212° (Fahr.) it is confined to
the surface of the fluid acted upon but, at that heat, 2120,
steam is formed at the bottom of the water, and ascends through
it, carrying off the heat in a latent form, and, therefore, pre-
venting the elevation of temperature of the water itself. At
the common pressure of the atmosphere, one cubic inch of water
produces about 1700 cubic inches (or nearly a cubic foot) of
aqueous vapour; but the boiling point varies considerably under
different pressures, and these anomalies materially affect the
the density of the vapour produced. Thus, in a vacuum water
boils at about 70°; under common atmospheric pressure at 212°;
and when pressed by a column of mercury 5 inches in height,
water does not boil until it is heated to 217°; each inch of mer-
cury producing by its pressure, a rise of about 1° in the ther-
mometer.
According to the elaborate experiments of Dr. Ure, of Glas-
gow, the elastic force of this vapour at 212° is equivalent to the
pressure of a column of mercury 30 inches high, or equal to
about 15 lbs. avoirdupois on a square inch.
At temp. 212°
226.3
238.5
248.0
257.5
273-7
285.2 :
312
30 inch. mercury 15 lbs. per sq. inch.
40
50.3
20
25.15
30
60
69.8
34.9
91.2
45.6
112.2
56.1
166.
83.*
And Mr. Woolf has ascertained that at these temperatures,
omitting the last, a cubic foot of steam will expand to about 5,
10, 20, 30, and 40 times its volume respectively; its elastic
force, when thus dilated, being in each case equal to the ordi-
nary pressure of the atmosphere.
One pound of Newcastle coals converts 7 pounds of boiling
water into steam; and the time required to convert a given
* Some recent experiments made in France, by Messrs. Dulong and Arago, do
not essentially differ in result from these of Dr. Ure. They find, at temp. 275-18
Fahr., an elasticity equal to 3 atmospheres, or 45 inches of mercury: at temps.
308-84, 320-36, 331-70, 341-96, 350-78, and 358-88, the elasticitics equivalent to
5, 6, 7, 8, 9, and 10 atmospheres, respectively. Temp. 439-34 an elasticity of 25
atmospheres, which was the limit of their experiment; but by computation they
went to a temperature of 510-60, equivalent to an elasticity of 50 atmospheres.
PNEUMATICS: STEAM AND STEAM-ENGINES.
355
quantity of boiling water into steam, is 6 times that required to
raise it from the freezing to the boiling point.
It is found, also, that if a bushel of coals per hour applied to
a well-constructed boiler, produces steam of the expansive force
of 15 lbs. per square inch, it will tend to expand itself with a
velocity of 1340 feet per second; then 2 bushels of coals, burnt
under the same boiler, are capable of giving to the vapour an
expansive force of 120 lbs. per square inch, and a velocity of
expansion of 3800 feet per second. A bushel and half of coals
would, with the same boiler, carry steam to the pressure of
50 lbs. on a square inch; which is as high as is regarded con-
sistent with safety.
From these data it will be evident that when steam is
merely employed to displace the air in a close vessel, and after-
wards produce a vacuum by condensation, no more heat is ne-
cessary than what will raise the water employed to 212°: but
if, on the contrary, steam capable of giving high pressures is
required, a considerable increase of heat, as to 260°, 280°, will
be necessary; and, of course, an augmentation of fuel, though
not one that is strictly proportional, will be required. This,
however, is a consideration upon which we cannot here en-
large.
W
We proceed to speak of the actual construction of the ma-
chine.
The principles and manner of operation of the steam-en-
gines of Savery, Newcomen and Cawley, and of Watt, may
be understood from the following brief explanations and re-
marks.
1. Let there be a sucking pipe with a valve opening upwards
at the top, communicating with a close vessel of water, not
more than thirty-three feet above the level of the reservoir,
and the steam of boiling water be thrown on the surface of the
water in the vessel, it will force it to a height as much greater
than thirty-three feet as the elastic force of the steam is greater
than that of air; and if the steam be condensed by the injection
of cold water, and a vacuum thus formed, the vessel will be
filled from the reservoir by the pressure of the atmosphere,
and the steam being admitted as before, this water will also be
forced up; and so on successively.
Such is the principle of the first steam-engine, said by the
English to be invented by the Marquis of Worcester; while
the French ascribe it to Papin though we believe the fact is
that Brancas, an Italian, applied the force of steam ejected
from a large clopile as an impelling power for a stamping-
engine so early as 1629. Brancas's was, in fact, an immense
356
PNEUMATICS: STEAM-ENGINE, NEWCOMEN'S, &c.
blow-pipe, turning a wheel. The hint so obscurely exhibited
in the Marquis of Worcester's Century of Inventions, was car-
ried into effect by Captain Savery.
2. If the steam be admitted into the bottom of a hollow cy-
linder, to which a solid piston is adapted, the piston will be
forced upwards by the difference between the elastic forces of
steam and common air; and the steam being then condensed,
the piston will descend by the pressure of the atmosphere, and
so on successively. This is the principle of the steam-engine
first contrived by Messrs. Newcomen and Cawley, of Dart-
mouth. This is sometimes called the atmospherical engine, and
is commonly a forcing pump, having its rod fixed to one end
of a lever, which is worked by the weight of the atmosphere
upon a piston at the other end, a temporary vacuum being made
below it by suddenly condensing the steam that had been ad-
mitted into the cylinder in which this piston works, by a jet
of cold water thrown into it. A partial vacuum being thus
made, the weight of the atmosphere presses down the piston,
and raises the other end of the straight lever, together with the
water, from the well. Then immediately a hole is uncovered
in the bottom of the cylinder, by which a fresh quantity
of hot steam rushes in from a boiler of water below it,
which proving a counterbalance for the atmosphere above the
piston, the weight of the pump-rods, at the other end of the
lever, carries that end down, and raises the piston of the
steam-cylinder. The steam hole is then immediately shut, and
a cock opened for injecting the cold water into the cylinder
of steam, which condenses it to water again, and thus making
a vacuum below the piston, the atmosphere again presses it
down and raises the pump-rods, as before; and so on con-
tinually.
+
3. When the cylinder is full of steam, if a valve be opened,
by which the steam is allowed to escape into another vessel,
where a jet of cold water is introduced, the condensation is
much more complete, and the heat of the cylinder being pre-
served, the steam possesses its full elasticity.
This improvement was made by Mr. Watt, and completely
changed the character of the steam-engine. In the old engines
the power was diminished to half its real value, so that the
moving force, instead of reaching 15 lbs. on each square inch
of the area of the piston, was reduced to about 8 lbs. In this
engine of Mr. Watt's the moving force is not less than 12 lbs.
upon each square inch of the piston.
4. A farther improvement has been made on this engine,
by injecting the steam into the cylinder, alternately above
1
PNEUMATICS: HIGH PRESSURE STEAM-ENGINE.
357
and below the piston, so that the whole motion is produced by
the elasticity of the steam, and has no dependance on the weight
of the atmosphere.
This improvement is also due to Mr. Watt, and could not
have been made without the previous contrivance of condens-
ing the steam in a separate vessel. It is particularly accom-
modated to the production of a rotatory motion by means of
a steam-engine. Three years before Mr. Watt introduced this
improvement, viz. in 1778, Mr. Washborough, of Bristol, took
out his patent for converting a reciprocating into a rotatory
motion; and in 1781, Mr. J. Steed effected the same thing,
for the first time, by means of what is now called a crank.
From that time Hornblower, Cartwright, Murray, Bramah,
Trevithick, Maudslay, Woolf, and others, have, in rapid suc-
cession, introduced a series of improvements which have
rendered steam-engines as efficacious and perfect as can well
be conceived.
5. Another improvement due to Mr. Watt, is that of the
Expansion Engine, invented about 1769. The principle of
this invention, as Mr. Partington correctly remarks, consists
in shutting off the farther entrance of steam from the boiler
when the piston has been pressed down in the cylinder, for a
certain proportion of its total descent, leaving the remainder
to be accomplished by the expansive force of the steam al-
ready produced. To regulate the time of closing the valve,
and as such the precise amount of steam admitted, Mr. Watt
employed a plug-frame with moveable pins, which may be so
placed that the steam-valve will shut when the piston has
descended one-half, one-third, one-fourth, &c. By the appli-
cation of this principle, the piston is made to descend uni-
formly, the pressure on it continually diminishing as the steam
becomes more and more rare, and the accelerating force is
consequently diminished.
6. The principle of the high-pressure steam-engine depends.
also on the power of steam to expand itself very considerably
beyond its original bulk, by the addition of a given quantity
of caloric, thus acquiring a considerable elastic force (equiva-
lent to from 40 to 60 lbs. on each square inch) which, in. this
case, is employed to give motion to a piston. One of the
greatest advantages attendant on employing the repellant
force of steam, as in this form of the engine, consists in an
evident saving of the water usually employed in condensation ;
and this, in locomotive engines, for propelling carriages, is
an object of considerable importance. The first description
*
* See Brewster's Robison's Mechanical Philosophy, vol. iv. p. 126.
47
358
PNEUMATICS: HIGH PRESSURE STEAM-ENGINE.
of an engine of this kind, which we have met with, is in Leu-
pold's Theatre of Machines, published in Germany, in 1724.
The apparatus consists of two cylinders placed at a moderate
distance asunder; each of them provided with a piston made
to fit air-tight, and connected with a forcing-pump. When
steam of considerable elasticity is admitted at the bottom of the
first cylinder, it is forced upwards, carrying with it the lever
of the pump; at the same time that the steam or air is expel-
led from the other. On this operation being repeated, or ra-
ther reversed, the steam is allowed to enter the second cylin-
der, which is also connected with the boiler, while the steam
in the first cylinder is allowed to escape into the air. Thus, it
may be remarked, that the process of condensation forms no
part of the principle of the "high pressure" engine; and that
even the expansion of gunpowder might be employed to pro-
duce a similar effect.
7. We have already adverted to Mr. Woolf's discovery, that
a quantity of steam having the force of 5, 6, 7, or more pounds
on every square inch of the boiler, may be allowed to expand
itself to an equal number of times its own volume when it
would still have a pressure equal to that of the atmosphere,
provided that the cylinder in which the expansion takes place
have the same temperature as the steam possessed before it be-
gan to increase.
The most economical mode of employing this principle con-
sists in the application of two cylinders and pistons of unequal
size to a high pressure boiler; the smaller of which should
have a communication both at its top and bottom with the
steam vessel. A communication being also formed between
the top of the smaller cylinder, and the bottom of the larger
cylinder; and vice versa. When the engine is set to work,
steam of a high temperature is admitted from the boiler to act
by its elastic force on one side of the smaller piston, while the
steam which had last moved it has a communication with the
larger or condensing cylinder. If both pistons be placed at the
tops of their respective cylinders, and steam of a pressure equal
to 40 lbs. on the square inch be admitted, the smaller piston
will be pressed down, while the steam below it, instead of being
allowed to escape into the atmosphere, or pass into the con-
densing vessel, as in the common engine, is made to enter the
larger cylinder above its piston, which will make its downward
stroke at the same time as that in the smaller cylinder; and
during this process, the steam which last filled the larger cy-
linder will be passing into the condenser to form a vacuum
during the downward stroke.
Jad
To perform the upward stroke it is merely necessary to
:
PNEUMATICS: HIGH PRESSURE STEAM-ENGINE.
359
:
reverse the action of the respective cylinders; and it will be
effected by the pressure of the steam in the top of the small
cylinder, acting beneath the piston in the great cylinder; thus
alternately admitting the steam to the different sides of the
smaller piston, while the steam last admitted into the smaller
cylinder passes regularly to the different sides of the larger pis-
ton, the communication between the condenser and steam boiler
being reversed at each stroke.
Mr. Partington states that a double cylinder expansion en-
gine of this kind was constructed for Wheal Vor mine in Corn-
wall, in 1815. In this, the great cylinder is 53 inches in dia-
meter, and has a nine feet stroke; the small cylinder being in
content about one-fifth of the great one. The engine works 6
pumps, which at every stroke raise a load of water of 37,982 lbs.
weight, 74 feet high. This produces a pressure of 14.1 lbs. per
square inch on the surface of the great piston, while its average
performance has been estimated at 46,000,000 lbs. raised one
foot high with each bushel of coals.
8. To render the preceding remarks and descriptions more
intelligible, we will now give fuller descriptions of a few en-
gines -Plate II. fig. 1, exhibits a vertical section of the dif-
ferent parts of a high pressure steam-engine, constructed upon
a principle in which simplicity and power are blended as far
as possible; and in which the parts are arranged in such a
manner as seemed best calculated to facilitate the comprehen-
sion of these machines to such as have not already had an op-
portunity of examining them carefully. The construction is
due to Oliver Evans; and every thing not essentially neces-
sary in a popular account of the operation, is omitted in the
drawing.
A, section of the boiler, perpendicular to its axis.
It con-
sists of two cylinders, one within the other, (the cylindric
form being most capable of resisting a great expansive force.)
The fire is made to burn in the interior cylinder which serves
for the furnace, and the water is contained between the two
cylinders. The smoke and heated air, after having traversed
the interior cylinder in the whole direction of its length,
passes out at the other extremity, and is thence conducted
under the elementary or subsidiary boiler, B, where it heats
the water which is intended to supply in the great boiler the
place of that which has been carried off in the state of vapour,
and thus given motion to the machine. c, the feeding pump,
which at every stroke of the piston, and corresponding motion
of the great beam, raises a small quantity of cold water, and
forces it into the supplementary boiler. The object of this
part of the apparatus may be explained in few words. If the
Mat
360
PNEUMATICS: OLIVER EVANS'S STEAM-ENGINE.
waste of water in the great boiler, by its conversion into steam,
were supplied at once from the cold water pump, the tem-
perature of the liquid in the great boiler would be diminished,
and the production of steam proportionally checked.
But
when water passes from the pump to the great boiler, through
the intervention of the subsidiary boiler in which it becomes
heated, it enters the great boiler without diminishing the tem-
perature of the water which it contains, and, of course, without
checking the operation of the machine. The skill and judg-
ment of the engineer are evinced in proportioning the magni-
tudes of the two boilers, and of the feeding pump, so that the
supply may be neither defective' nor superabundant, but just
sufficient.
The steam rises through the tube, and if the injection
sucker, D, is opened to permit the entrance of steam into the
machine, the valves E and F being opened, the steam will push
the piston & to the inferior extremity of the cylinder; as is
shown in the diagram. The steam which occupied the cylin-
der, being driven before the piston, escapes through the valve
F. As soon as the piston & arrives at the bottom of the cylin-
der, the valves E and r close, and the other valves I and I
open; the steam enters at I, and causes the piston to rise,
while the weaker steam, which is contained in the upper part
of the cylinder, escapes through the valve 1, presenting scarcely
any reaction to the ascent of the piston. These four pistons
are moved by two wheels к and L, carrying cams or lifters on
their surfaces, which press against four levers, to which stems
or handles are attached, and which thus cause them to open
and shut precisely at the adequate intervals. [These levers
are not represented in the figure, but are omitted to prevent
confusion.]
The motion of the piston &, of course, communicates a re-
ciprocating motion to the beam M N ; and the crank and con-
necting bar м o give a rotatory motion, the regularity of which
is maintained by means of the fly-wheel q r. After similar
operations, the valves E, F, H, I, alternately opening and shutting,
the motion of the machine is continued. The teethed wheels
s, T, of equal diameters, give motion to the wheels and lifters.
L, K ; and the beam M N communicates the requisite motion to
the forcing piston of the feeding pump c. The motion of the
whole being thus continued; the wheel v, if it have 66 teeth,
playing into the wheel u of 23 teeth, would give, say to the
mill-stone w, 100 revolutions for every 35 strokes of the pis-
ton & viz. 100 revolutions in a minute if the piston make 35
strokes in that interval of time.
G
The toothed wheel v may be applied to any other work.
PNEUMATICS OLIVER EVANS'S STEAM-ENGINE.
361
Instead of the wheel, indeed, there may be employed a crank
to move a pump, or a saw; and the machine may, by modify-
ing the power of the steam, give from 10 to 100 strokes of the
piston per minute, according to the purpose for which it is
constructed. If the steam cylinder be 8 inches in diameter it
will work a mill-stone of 5 feet diameter; or perform any
work which requires equal mechanical energy.
意
​The steam, on quitting the machine, escapes by the tube
x x, and becomes dissipated in the air above the top of the
building or, if the constructor please, it may pass into the
condenser, when a condenser is used; or it may be made to
pass along the subsidiary boiler, to heat the water which it
contains.
v, is the safety-valve, kept in its place by a lever, graduated,
like the steelyard, to weigh and thus balance the effort of the
steam. This valve rises and permits the excess of steam to
escape when its elastic force becomes more considerable than
is requisite for the work of the machine. The weight is, in
general, so adjusted upon the lever, that the valve shall open
when the pressure of the steam upon a square inch of the in-
terior of the boiler becomes about half that for which its thick-
ness and strength of material are fitted.
To prevent a recurrence of those accidents which have
sometimes occurred aboard steam-boats, and which first drew
the attention of the legislature to this important part of the
engine, it appears advisable to enclose the safety-valve in an
iron box, and thus place it beyond the control of the engine-
man.
W
The whole of the machine now described is very simple in
its construction, and easy to execute by ordinary mechanics or
mill-wrights. The orifices for the passage of the steam are
simple metallic plates, each pierced with a hole, on which the
valve intended to close it readily falls: they are very easily
cast at any common foundry.
9. Manufactories of steam-engines are now numerous, es-
pecially in those districts where machinery is much called.
for. The engines of Maudslay are much celebrated, especially
in situations where it is of consequence to erect an engine in
small space, as in steam-boats, &c. Next to Maudslay's, the
engines of Messrs. Fenton, Murray, and Wood, of Leeds, have,
perhaps, attained the highest celebrity. As the construction
of their engines serves well for the elucidation of the manner
of operation, we here present a description of one of them,
from a very useful work, Smith's Panorama of Science and
Art.
In Plate III, A represents the boiler, nearly three quarters
2 I
362
PNEUMATICS: FENTON AND CO.'s STEAM-ENGINE.
4
full of water the bottom is considerably, and the sides a little
concave, that it may receive more fully the force of the flame
circulating round it. Boilers are usually of an oblong form,
and are furnished with a part that takes off, in order that a
person may get in to clean them when needful; they have also
a valve, called the safety-valve, opening upwards, which is
loaded so that the steam escapes when it is stronger than the
engine requires, and, if retained, would hazard the bursting of
the boiler. It is not uncommon to have two boilers, one of
which is a reserve, that the engine may not be stopped, when
the other requires repair.
B, is an apparatus for regulating the fire, and giving action
to a bell, which regulates the quantity of coals and time of
firing.
c, the steam-pipe from the boiler A to the valve 1.
D, the steam-cylinder, generally called only "the cylinder ;"
it is connected at the top and bottom with the valve 1.
E, the piston, which, by its connecting rod e, gives motion
to the beam F, the other end of which, by another connecting
rod, gives motion to the heavy fly-wheel &, by means of a
crank. Thus, after the engine has begun to work, its power
is accumulated in the fly-wheel, and may be disposed of at the
pleasure of the mechanist.
н, an eccentric circle* on the axle of the fly-wheel & it gives
motion by its levers, in a manner easily understood by inspec-
tion, to the valve 1.
1, a coffer-slide valve, which requires no packing to make it
steam-tight, as there is always a vacuum under it it answers
the purpose of the four valves used in double-power engines,
and from the simplicity of its construction, when well made at
first, is not liable to get out of order.
K, the steam-admission valve and lever, connected with a
governor not shown in the figure, which regulates the speed
of the engine. See p. 262.
L, the cylinder of the discharging pump, for extracting the
water and uncondensed vapour from the condenser M.
N, a small cistern, filled with water. Into this cistern enters
a pipe from the condenser M, the top of which pipe is covered
by a valve, which is called the blow-valve, or sometimes the
snifting-valve. Through this valve, the air contained in the cy-
linder D, and passages from it, is discharged, previously to the
engine being set in motion.
* Mechanics distinguish by the appellation of "eccentric circle," a wheel, the
axle of which is purposely made to pass, not through its real centre, but on one
side of it.
I
PNEUMATICS: FENTON AND Co.'s STEAM-ENGINE.
363
•
o, the eduction-pipe, which conducts the steam from the
valve I to the condenser M.
P, the pump which supplies with water the cistern, ss, in
which the condenser and discharging-pump stand.
qq, iron columns, of which the engine has four, although only
two are shown; they stand upon one entire plate, seen edge-
way, on which the principal parts of the engine are fixed; by
this means the beam and its accompaniments are supported
without being connected with any part of the building, except
the recess below the floor on which they stand.
R R, the recess below the floor, for containing the cistern of
the discharging-pump, condenser, &c. This arrangement ena-
bles those engines to be fixed up and tried at the manufactory
before they are sent off, which renders the refixing easy and
certain. Engines are made according to this plan, from the
power of one to twelve horses.
tern N.
Before the engine is set to work, the cylinder D, the con-
denser M, and the passages between them, are filled with com-
mon air, which it is necessary to extract. To effect this, by
opening the valves, a communication is made between the
steam-pipe c, the space below the piston in the cylinder D, the
eduction-pipe o, and the condenser M. The steam will not at
first enter the cylinder D, or will only enter it a little way, be-
cause it is resisted by the air; but the air in the eduction-pipe o,
and the condenser M, it forcibly drives before it, and this part
of the air makes its exit through the valve and water in the cis-
The steam-admission valve is now closed, and the
steam already admitted is converted into water, partly by the
coldness of the condenser M, but principally by a jet of cold
water which enters it through a cock opening into it from the
well ss, in which the condenser is immersed. When this steam
is condensed, all the space it occupied would be a vacuum, did
not the air in the cylinder D expand, and fill all the space that
the original quantity of it filled; but by the repetition of the
means for extracting a part of the air, the remainder is blown
out, and the cylinder becomes filled with steam alone. Sup-
pose then the cylinder beneath the piston to be filled with
steam, and the further admission of steam to that part of it be
cut off, while the communication between it and the condenser
remains open, it is obvious that there will soon be a vacuum in
the cylinder, because as fast as the steam reaches the condenser,
it is converted into water by the coldness of that vessel and the
jet playing within it. At this moment, therefore, the steam is
admitted above the piston, which it immediately presses down,
As soon as the piston reaches to the bottom of the cylinder, the
steam is admitted to the under side of it, and as the communi-
:
1
364
cation from the upper side of the piston to the condenser is
opened, while the further admission of steam to that side
during the upper stroke is prevented, the steam which had
pressed the piston down passes into the condenser, and is con-
verted into water.
PNEUMATICS: FENTON AND CO. S STEAM-ENGINE.
The motion of the piston E, by this alternate admission and
extraction of the steam on each side of it, is thus necessarily
continued, and the distance of its upward and downward range
is called the length of its stroke. It communicates its recipro-
cating motion, by the connecting rod e, to the great beam r,
and thence, by another connecting rod and a crank, to the fly-
wheel G.
To explain the rapid accumulation of power with an increase
of the size of the engine, it must be observed, that the force of
the steam generally used, is somewhat greater than the pressure
of the atmosphere; but supposing it to be no greater, as the
atmospheric pressure is fifteen pounds on each square inch, a
piston sixteen inches in diameter, containing 201 square inches
of surface, will alternately be raised and depressed by a force
equivalent to a weight of 3015 pounds. Here no allowance is
made for friction, but after the requisite deduction on this ac-
count, which may be reckoned at one-third, the disposable part
of the engine, derived from each stroke, will still be very
great.
The condenser M, and the discharging-pump L, communicate
by means of a horizontal pipe containing a valve y opening
towards the pump; the piston, l, of this pump, also contains
two valves, and the cistern T, at the top of the pump-cylinder,
contains two other valves, which, like those of the piston 7,
open upwards. When the piston E of the cylinder is de-
pressed, the piston 7, of the discharging-pump, it will be obvious
to inspection, is depressed likewise, and its valves open, while
the valve y closes; hence the water from the condensed steam,
as well as the injection-water, and any permanently elastic va-
pour or gas, which may be present, having passed through the
valve y, passes through the piston 7; and when that piston is
drawn up, its valves close and prevent their return, as in ordi-
nary pump-work. The water and gas that have thus got above
the piston, as the latter rises, open the valves at the bottom of
the cistern T, in which the water remains till it is full, but the gas
passes into the atmosphere. As the water in the cistern T is in a
very hot state, it is sometimes, for the purpose of economizing
fuel, pumped up and returned to the boiler, the pump-rod being
attached to the great beam. The utility of the discharging-pump L
will now be appreciated, and it must be perceived how much more
1
PNEUMATICS: FENTON AND CO.'s STEAM-ENGINE.
365
materially it contributes to the perfection of the vacuum in the
cylinder D, than if the water from the condenser merely ran
off by a pipe.
The steam constantly rushing into the condenser м, has a
perpetual tendency to heat that vessel, as well as the water of
the cistern s s, in which it stands; the whole of the steam,
if this were unchecked, would not be condensed, or the con-
densation would not be sufficiently rapid, because the injec-
tion-water itself flows out of this cistern. A part of the water
is therefore allowed to flow from this cistern by a waste pipe,
and an equal quantity of cold is constantly supplied by the
pump P.
In Newcomen's, or the atmospherical engine, the cylinder
was open at the top, and therefore, during the descent of the
piston, the air exerted a great power in cooling it; but in the
modern engines, where steam is the active power both in
raising and depressing the piston, the top of the cylinder is
closed with an iron lid, and not an atom of steam can escape,
except at the proper time, into the condenser. In order that
the connecting rod e may work freely, and yet possess this de-
sirable property of being steam-tight, it passes through what
is called a stuffing or packing box. This stuffing consists of
some material which the steam will rather adapt to its office
than injure; leather, which is used for the stuffing or collars.
of machines never to be subjected to heat will not answer
here; hempen yarn is the material usually employed. The
rod of the piston 7, passes through a stuffing box of the same
kind as that of the piston ; and the pistons themselves are
surrounded with stuffing.
The cylinder D is surrounded by a case, to keep it from
being cooled by contact with the external air. The extre-
mity, or any given point removed from the centre of the great
beam, can describe only the arc of a circle; but it is neces-
sary that the piston rod e should rise and fall vertically. New-
comen effected this object, by making the end of the beam
into the arc of a circle, the radius of which was equal to the
distance from the centre of the beam: a chain went over this
are, and was fastened on the higher end of it; this simple
contrivance effectually answered his purpose, because in his
engine the effective stroke was only downwards; but here, in
a double-power engine, where the stroke is both upwards and
downwards, a chain would yield in rising, and be altogether
unsuitable. An apparatus is therefore used, called the pa-
rallel joint, which is easily understood by inspection. By
this means the rod e not only rises and falls perpendicularly,
but is perfectly rigid, and communicates all its motion to the
S
1
48
212
366
PNEUMATICS: FENTON AND Co.'s STEAM-ENGINE.
great beam in each direction of its motion. The connecting
rod g does not require the same contrivance, because it does
not rise and fall perpendicularly; its lower end, with the outer
end of the crank, describing a circle: it has therefore only a
simple joint, admitting of this deviation.
:
In order to communicate a rotatory motion to the fly-wheel,
instead of the crank may be used a contrivance giving twice
the rapidity to the fly. For this purpose, on the outside of
the axis of the fly, where the crank is shown in the plate, a
small toothed wheel is fixed, and can only be moved with the
fly at the extremity of the rod g, and on that side of it which
is next the fly-wheel, another toothed wheel is fixed, in such a
manner that it cannot turn round on its axis, but must rise and
fall. with the rod to which it is attached. These two wheels
work in each other, and that attached to the connecting rod
cannot leave its fellow, because their centres are connected by
a strap or bar of iron. When, therefore, the connecting rod
rises, the wheel upon it moves round the circumference of the
wheel upon the axis of the fly. By this means the fly makes
an entire revolution for every stroke of the piston, and some
mechanics are apt to think that they are great gainers by such
an arrangement: the contrivance is certainly elegant; but
with respect to utility, the fact is, that a crank is preferable;
for it is more simple, cheaper, and less likely to be out of order,
while, if the fly be large enough to receive, with less velocity,
all the momentum that can be communicated to it, the effect
will certainly not be inferior.
10. Locomotive steam-engines, or those which will propel
both themselves when placed upon wheels, and any suitable
carriages attached to them, were invented by Mr. Trevithick,
in Cornwall, about the year 1804. They are now in constant
use in the northern coal districts; their peculiar construction
will be very evident from the following description, by Mr.
Tredgold, of the engines employed on the Hetton rail-way.
The wheels of the coal-wagons drawn by the engines are
2 feet 11 inches in diameter, with 10 spokes, and weigh 24
cwt. Their axles are 3 inches in diameter, and revolve in
fixed bushes.
The weight of each engine is about 8 tons. It consists of
a boiler 4 feet in diameter, with an internal fire-place. The
smoke ascends from the fire by a chimney about 12 feet high;
the lower 3 feet of the chimney is formed of sheet iron of 6
Ibs. to the square foot, and the rest of iron 2½ lbs. to the
There are two cylinders, which work alternately. The
diameter of the pistons is 9 inches, and the length of the
stroke 2 feet; the pistons make about 45 double strokes per
PNEUMATICS: LOCOMOTIVE STEAM-ENGINE.
367
minute. The steam is admitted to the cylinders by slide
valves worked by eccentric wheels on the axis of the engine
carriage. The pressure of the steam in the boiler is from 40
to 50 lbs. on the square inch.
2
The wheels of the engine carriage are 3 feet 2 inches dia-
meter, with 12 spokes in each, and each weighs 3 cwt. The
axles are 3½ inches diameter, and are connected by an endless
chain working into a wheel on each axle, so that both the
axles of the carriage may be turned at the same rate. The
boiler is supported on the carriage by four floating pistons,
which answer the purpose of springs in equalizing the pres-
sure on the wheels, and softening the jerks of the carriage. A
floating piston is packed as the steam piston of a steam-engine,
and has a short piston rod of 1½ inches diameter, which rests
upon the brass bush, in which the axle of the wheel turns.
The water in the boiler presses on the upper side of the piston;
and whatever elevation or depression the wheel follows, the
pressure upon it is nearly the same. This ingenious substi-
tute for a spring, as well as the other peculiarities of this engine,
were invented by Messrs. Losh and Stephenson, of New-
castle-upon-Tyne, and made the subject of a patent in 1816.
Fig. 2. plate II. is a sketch of the steam-carriage, &c.
A is
the boiler, and в в the steam cylinders; the fireplace is within
the boiler, and r is the entrance to it; c is the chimney: DD
the floating pistons which support the carriage on the axles,
and answer as springs in making it press equally on the rails.
As the moving force is not equal at the same time on the
wheels of both axles, it is necessary to connect the axles by a
pitch chain &, working into toothed wheels on the axles. The
water for supplying the boiler, and the coals at b for the fire,
are carried by a small carriage called the tender; 1 is the water
barrel, and a is a hose pipe which conveys the water to the
force-pump н, which is worked by the engine; w w are coal
wagons, each of which carries 53 cwt. of coals.
From 13 to
17 of these wagons are drawn in a train by one steam-carriage;
they are connected by the short chains c c. The connecting rods.
which communicate the power from the pistons to the wheels
of the steam-carriage are attached to the wheels, so that one pis-
ton is at half the length of its stroke, when the other is at the
commencement of its stroke.
W
Fig. 3, is a vertical transverse section of the steam-carriage;
the comparison of which with fig. 2, will render the whole more
intelligible.
11. We shall terminate our description of steam-engines,
by exhibiting one invented by Mr. John Nuncarrow, an
American, and which is intended to give motion to water-
$
368
PNEUMATICS: NUNCARROW'S STEAM-ENGINE.
wheels, in places where there is no fall, and but a very small
stream or spring.
A, fig. 4, plate II, the receiver, which may be made either of
wood or iron.
B, B, B, B, B, wooden or cast iron pipes for conveying the
water to the receiver, and thence to the penstock.
c, the penstock or cistern.
D, the water-wheel.
E,
the boiler, which may be either iron or copper.
the hot well for supplying the boiler with water.
F,
G, G, two cisterns, under the level of the water, in which the
small bores B, B, and the condenser arc contained.
H, H, H, the surface of the water with which the steam-engine
and the water-wheel are supplied.
a, a, the steam-pipe, through which the steam is conveyed
from the boiler to the receiver.
b, the feeding-pipe, for supplying the boiler with hot water.
c, c, c, c, c, the condensing apparatus.
d, d, the pipe which conveys the hot water from the condenser
to the hot well.
e, e, e, valves for admitting and excluding the water.
ff, the injection pipe, and g, the injection cock.
h, the condenser.
It must be remarked that the receiver, penstock, and all the
pipes, must be previously filled before any water can be deli-
vered on the wheel; and when the steam in the boiler has ac-
quired a sufficient strength, the valve at i is opened, and the
steam immediately rushes from the boiler at E into the receiver
A; the water descends through the tubes A and B, and ascends
through the valve k, and the other pipe or tube B, into the pen-
stock c. This part of the operation being performed, and the
valve i shut, that at e is suddenly opened, through which the
steam rushes down the condensing pipe c, and in its pas-
sage meets with a jet of cold water from the injection
cock g, by which it is condensed. A vacuum being made
by this means in the receiver, the water is driven up to fill it a
second time through the valves e, e, by the pressure of the ex-
ternal air, when the steam-valve at i is again opened, and the
operation repeated for any length of time the machine is re-
quired to work.
There are many advantages which a steam-engine on this
construction possesses beyond any thing of the kind hitherto
invented; a few of which the inventor thus enumerates.
(1.) It is subject to little or no friction.
(2.) It may be erected at a small expense when compared
with any other sort of steam-engine.
PNEUMATICS POWER OF STEAM-ENGINES.
369
(3.) It has every advantage which may be attributed to Boul-
ton and Watt's engines, by condensing out of the receiver,
either in the penstock or at the level of the water.
(4.) Another very great advantage is, that the water in the
upper part of the pipe adjoining the receiver acquires a heat,
by its being in frequent contact with the steam, very nearly
equal to that of boiling water; hence the receiver is always
kept uniformly hot, as in the case of Boulton and Watt's en-
gines.
(5.) A very small stream of water is sufficient to supply this
engine (even where there is no fall), for all the water raised by
it is returned into the reservoir H, H, H.
From the foregoing reasons it would seem that no kind of
steam-engine is better adapted to give rotatory motion to ma-
chinery of every kind than this. Its form is simple, and the
materials of which it is composed are cheap; the power is more
than equal to any other machine of the kind, because there is
no deduction to be made for friction, except on account of turn-
ing the cocks, which is but trifling.
Its great utility is therefore evident in supplying water for
every kind of work performed by a water-wheel, such as grist-
mills, saw-mills, blast-furnaces, forges, &c.*
12. The theory of steam-power in reference to the mechani-
cal energy of engines, is, as yet, in a very imperfect state.
The best formula which we have hitherto seen, are exhibited
by the late Mr. Tredgold, in his judicious and valuable work
on Rail-Roads. As they are found to furnish results which
agree very nearly with those of experiment, we insert them
here, with the author's permission.
If ƒ be the measure of the force of steam in inches of the
mercurial column, and t the corresponding temperature mea
sured on Fahrenheit's thermometer; f' the resistance from the
friction of the steam piston, and the condensed vapour in the
cylinder, or the atmospheric pressure in high-pressure engines,
and n the bulk or capacity of the steam cylinder, when the
bulk of the steam admitted at the pressure f is unity. Then
the power of the steam generated from a cubic foot of water is
4878 (459 + 1) × (1 — "ƒ'′)+ hyp. log. n).
t)
n f'
f
·)+
When the steam does not act by expansion n 1.
* For descriptive information on the subject of steam-engines, the reader may
consult the historical treatises of Partington and Stuart; and, for still more com-
prehensive and scientific accounts, the treatises of Tredgold and J. Farey. For
some of the recent improvements by Hall and others, see the Mechanics' Maga-
zine, vol. 18.
370
PNEUMATICS: STEAM-ENGINE POWER.
When the expanding force of the steam is employed, the
above equation has a maximum, which will obtain when hyp.
nf'
log. n
is a minimum, which is evidently the case when
f
f
n = 1/ In that case, inserting, for
f n, we have
(
f
4873 (459 + 1). (hyp. log.)
cubic foot of water converted into steam.
f
f'
When ƒ = ƒ', then hyp. log.
thing.
the maximum power of a
0, and the power is no-
f'
And, when 1
ƒ
vantageous to work by expansion.
13. To calculate the quantity of fuel, let c be the quantity
which converts a cubic foot of water into steam that will bear
the pressure of the atmosphere; let s be the specific heat of
the steam, a the specific heat of the air and smoke which
escape up the chimney, and w the weight of fucl that will
heat one cubic foot of water one degree: then
is greater than hyp. log.
1
f
f'
1
5
it is disad-
-་
c+ [(t212°) × (a + s) w0]
the least quantity of fucl that will produce steam of the force
f and temperature t.
Mr. Tredgold, by assuming c = 8.4 lbs. of Newcastle coals,
·0075 lbs. S' •847, and a ·753, reduces the preceding
212°) the lbs. of coal to produce steam
20
to 8.4 +012 (t
of the temperature t.
1
14. For a high-pressure engine, taking 30 inches for the
measure of atmospheric pressure, of the pressure of the
steam for the friction of the steam piston, and for the
f
20
plus pressure in the boiler, the whole loss becomes. But,
one side of the piston of a high-pressure engine is acted
upon by the same pressure as that of the external atmos-
phere hence f' 4f+30= the resistance to the moving
force f
Consequently, when a high-pressure engine is worked ex-
pansively, we have
4873 (459 + 1) × (hyp. log.
x
f
4+ 30
the mechanical power of a cubic foot of water converted into
steam.
Hence, there is no advantage in making a high-pressure
PNEUMATICS: STEAM-ENGINE POWER.
371
1
steam-engine work expansively, when the force of the steam is
less than 60 inches of the mercurial column; because the above
4f+30
f
hyp. log. is then less than 1-
When an engine does not employ the expansive power of
steam, we have
ƒ
4873 (459 + 1) x (1− * # 30)=
×
of a cubic foot of water converted into steam.
15. Mr. Tredgold illustrates these formulæ by the follow-
ing:
30)= the mechanical power
Example. Let the force of the steam be 120 inches of mer-
cury; the corresponding temperature is 292.8°. Then
(4 × 120) + 30
4873 (459 + 292·8) ×(1
120
1,830,000 lbs. raised one foot high.
The quantity of coal is 84 +012 (292·8 — 212) 9.37 lbs.
of coal.
Now if the horse power be 16,000,000 lbs. raised one foot in
a day of 8 hours; then
1,830,000,: 9.37 lbs. :: 16,000,000: 82 lbs.
Therefore, working with steam of 442 lbs. on the square
inch on the piston, above the pressure of the atmosphere,
82 lbs. of Newcastle coal ought to do the day's work of a
horse.
But if the engine works expansively with the same force of
stcam, then
4873 (459 +2928) x hyp. log. 2) 2,540,000 lbs.
raised one foot high by 9:37 lbs. of coal; and consequently
59 lbs. of coal ought to do the day's work of a horse.
16. With regard to the maximum of useful effect in steam-
engines, it will be found, according to Mr. Tredgold, by taking
v = 120, for the working velocity of an engine in feet per
minute, being the length of the stroke in feet.
ī
If an engine has a 2 feet stroke, then v = 170 feet per mi-
nute, and the number of strokes per minute 423.
By increasing the stroke to 3-4 feet, we get a velocity of 220
feet per minute, with 32 strokes per minute.
If any variation be made from the maximum power, the de-
crease of effect is the same as in horse power; but, as Mr. T.
remarks, we have this advantage in an engine-it can be made
for any velocity, by attending to the relative proportions of its
parts; those of a horse we cannot alter.
17. A horse, when he treads a mill-path at the rate of 2
372
PNEUMATICS: STEAM-ENGINE POWER.
F
miles an hour, will, on an average, raise about 150 lbs. by a cord
hanging over a pulley; which is equivalent to 33,000 lbs. one
foot high in a minute. Boulton and Watt estimate this at
32,000. Tredgold, still lower, at 27,500. Taking the first
measure, however, as a basis of comparison; putting d for the
diameter of the piston in inches, p for the pressure of the steam
upon each square inch (diminished usually by about for fric-
tion and inertia), l for the length of the stroke of the piston in
feet, n for the number of strokes in a minute: then the power
of the engine in "Horse-powers" (H P), is
ठ
(HP) ·0000238 d² n p l, if it be a single stroke?
H P ·0000476 d³ n p l, if it be a double stroke
Example. Suppose d 20 inches, 3 feet, n=
l
50, and the engine one of double stroke. Then
•0000476 × 20⁹ × 36 × 50 × 3 102.816, or nearly 103
horse-powers, the measure of the energy of the engine.
Mr. Boullon states that 1 bushel of Newcastle coals, contain-
ing 84 pounds, will raise 30 million pounds 1 foot high; that it
will grind and dress 11 bushels of wheat; that it will slit and
draw into nails 5 cwt. of iron; that it will drive 1000 cotton
spindles, with all the preparation machinery, with the proper
velocity; and that these effects are equivalent to the work of
10 horses.
18. The rule usually given to adjust the weight of the fly-
wheel is this:
Multiply the number of Horse-powers in the machine by
2000; divide the product by the square of the velocity in feet,
per second, of the fly's circumference; the quotient will give its
weight in hundred-weights.
π
Or, 2000 (í r) ÷ (˜d n)³ =
60
engine.
36,p
t
= w. of fly.
Here
cumference per second.
20 × 2000
Whence
(2012)2
Thus, suppose the fly-wheel of a 20 horse-power engine to be
18 feet diameter, and to revolve 22 times in a minute ; what
should be its weight?
18 X 3.1416 × 22
60
20 feet nearly, velocity of cir-
cwt.
904, weight of the fly-wheel re-
quired.
**See, farther, p. 262, preceding.
STEAM-ENGINES, CANALS, RAIL-ROADS, &c.
373
Temperature of
steam.
USEFUL TABLES AND REMARKS ON STEAM-ENGINES, RAIL-ROads,
CANALS, AND TURNPIKE-Roads.
Total force in
inches of mercury.
Temperaturé of
steam.
TABLE I.—Quantity of Coals equivalent to the horse power
of 33,000 lbs. raised one foot per minute in high pressure
steam-engines, when the greatest possible effect is obtained.*
O inches.
234.5
45
251
60
275
90
292.8 120
307.7 150
320.2 180
343.6 240 104
Force in lbs. per
square inch above
the atmosphere.
Total force in
inches of mercury.
lbs.
7.4
14.8
29.7
44.5
59.3
74.2
Force in lbs. per
square inch abovel
the atmosphere.
}
CHAPTER XIV.
inches.
35
220
2.5
234.5
45
7.4
251
00
14.8
275
90
29.7
292.8 120
44.5
307·7 150
59-3
320.2 180 74.2
343.6 240 104.0
lbs.
Quantity of coal equivalent to
a horse power.
When working
at full pressure.
lbs.
480
163
98
82
74
70
65
When acting
expansively.
lbs.
Ibs.
633
62
60
143
77
59
51
48
41
Quantity of coal equivalent to
a horse power.
TABLE II.—Quantity of Coals equivalent to the horse power
of 33,000 lbs. raised one foot per minute in condensing
steam-engines, when the greatest possible effect is obtained.
When working | When acting
at full pressure. expansively.
lbs.
40
381
354
33
32
32
1
314
31
When work-
ing at full
pressure.
Pounds raised one foot high
equivalent to the immediate
power of the steam produced
by 84 lbs. of coal.
When work-
ing expan-
sively.
lbs.
When work-
ing at full
pressure.
lbs.
2,780,000
8,200,000 9,300,000
13,700,000 17,700,000
16,600,000 | 22,700,000
18,000,000 | 26,200,000
19,200,000 28,700,000
20,500,000 | 32,200,000
Pounds raised one foot high
cquivalent to the immediate
power of the steam produced
by 84 lbs. of coal.
When work-
ing expan-
sively.
lbs.
lbs.
21,000,000
33,100,000
21,400,000 35,200,000
22,400,000
37,500,000
40,000,000
41,000,000
42,400,000
42,700,000
43,500,000
*The curious tables here given, markod I. II. III., were extracted, with the au-
thor's permission, from Mr. Tredgold's work on Rail-roads.
J



49
2 K
374
STEAM-ENGINES, CANALS, RAIL-ROADS.
Remarks on Tables I, and II.-The columns showing the
pounds an engine ought to raise one foot high, by the heat of
one bushel of coals, are added chiefly for the purpose of com-
parison with actual practice. Now, it is stated, that after the
most impartial examination for several years in succession, it is
found that Woolf's engine at Wheal Abraham Mine, raised 44
millions of pounds of water, one foot high, with a bushel of
coals. And, "the burning of one bushel of good Newcastle or
Swansea coals, in Mr. Watt's reciprocating engines, working
more or less expansively, was found, by the accounts kept at
the Cornish mines, to raise from 24 to 32 millions of pounds of
water one foot high; the greater or less effect depending upon
the state of the engine, its size, and rate of working, and the
quality of the coal."
We shall further add the results of half a year's reports taken,
without selection, from Lean's Monthly Reports on the work
performed by the steam-engines in Cornwall, with each bushel
of coals. The numbers show the pounds of water raised one
foot high with each bushel, from January to June, 1818.

22 to 25 Common En-
gines average
Wheal Vor (Woolf's
Engine)
Wheal Abraham (ditto)
Ditto
January. February. March. April.
Wheal Unity (ditto)
Dalcouth Engine
Wheal Abraham Engine
United Mines Engine
Treskirby Egnino
Wheal Chance Engine
May.
June.
lbs. raised lbs. raised lbs. raised lbs. raised lbs.. raised lbs. raised
one foot. one foot. one foot. one foot. onc foot. one foot.
22,188,000 22,424,000 21,898,000 22,982,000 23,608,000 | 23,836,000
30,834,000 20,158,000 29,511,000 26,064,000 29,032,000 30,336,000
41,847,000 35,364,000 30,445,000 32,723,000 31,520,000 34,352,000
ditto) | 27,942,000 $28,000,000 | 26,978,000 | 23,626,000|29,702,000) | 34,846,000
31,900,000 |32,306,000
42,622,000 |41,354,000 | 40,499,000 41,888,000 38,233,000 38,143,000
32,239,000 36,180,000 35,715,000 33,934,000 33,714,000 34,291,000
36,390,000 31,830,000 | 31,427,000 33,564,000 | 33,967,000 | 30,105,000
38,733,000 39,375,000 41,867,000 (41,823,000 40,615,000 42,098,000
28,496,000 32,319,000 33,594,000 33,932,000
35,797,000
}
These numbers are less than the immediate power of the en-
gines, by the friction and loss of effect in working the pumps;
hence in comparing them with Mr. Tredgold's table, it may
be inferred that he made his calculations from such data as can
be realized in practice. It is known from experience, that a
cubic foot of water can be converted into steam equal in force
to the atmosphere, with 7 lbs. of Newcastle coals; but we
also know the attention necessary to produce that effect, and
therefore have assumed that 84 lbs. will be required for that
purpose.
STEAM-ENGINES, CANALS, RAIL-ROADS.
375
According to Mr. Lean's Monthly Report, for January, 1833,
the following engines raise more than 50 millions of pounds,
one foot high, by consuming a bushel of coals :
Cardrew Downs
Binner Downs
Ditto
-
Consolidated Mines
Ditto
Ditto
Ditto
Ditto
Ditto
Polgooth
Pembroke
East Crinnis
Wheal Leisure -
Wheal Vor
Ditto
Ditto
J
4
www
J
-
1
1
Diameter of Cylinder.
1
1
1
1
1
1
#
•
Poladras Downs
Great Work
Wheal Towan
Ditto
Wheal Falmouth Consols
Wheal Darlington
1
1
lbs.
66 inches, single 51,831,751
70
57,942,435
55,931,852
65
90
70
65
51,713,913
61,846,133
54,726,957
59,978,983
90
90
52,040,672
65,617,011
65
66
40
76
66
70,240,452
61,170,237
62,097,533
53,506,372
88,504,900
65,471,147
53,938,177
56,766,668
80
80
53
65,460,248
73,159,628
68,782,390
54,334,137
66,058,518
70
60
80
80
70
80
GE
A
-
1
1
I
+
1
1
I
•
t
1
་
1
1
Strokes per
1
1
min.
7.34
10.13
8.5
4.86
5.93
3.6
4.69
6.5
5.28
8.00
6.67
7.04
3.58
6.22
6.07
6.37
8.78
6.08
4.9
6.98
6.22
8.47
Of the above, the engine of greatest operation, the first at
Wheal Vor, raises the water 190 fathoms, at 7 lifts, drawing
perpendicularly 160 fathoms, and the remainder diagonally.
Main beam over the cylinder; stroke in the cylinder 10 feet;
one balance-bob at the surface, and three underground.
376
STEAM-ENGINES, CANALS, RAIL-ROADS.
TABLE III.-Showing the effects of a force of traction of
100 lbs. at different velocities, on canals, rail-roads, and
turnpike-roads." (From Tredgold.)
Velocity of Motion.
Miles Foot
per
hour.
I
23
3.66
4.40
3 +
5.13
4
5.86
5
7.33
6
8.80
my
10.26
8
11.73
9
13.20
10
14.66
13.5 19.9
૧૭.
K
per
second.
Load moved by a power of 100 lbs.
On a Canal.
Total
mass
moved.
Useful
effect.
On a level Rail-way.
Total
mass
moved.
Useful
effect.
lbs.
lbs.
lbs.
lbs.
55,500 39,400 14,400 10,800
38,542 27,361 14,400 10,800
28,316 20,100 14,400 10,800
|
|
21,680 | 15,390 14,400 10,800
13,875 9,850 14,400 10,800
9,635 6,840 14,400 10,800
7,080 5,026 14,400 10,800
5,420 3,848 14,400| 10,800
4,282 3,040 14,400 10,800
3,468 2,462 14,400 10,800
1,900 1,350 14,400 10,800
!
. On a level.
Turnpike Road.
Total
mass
moved.
lbs.
1,800
1,800
1,800
1,800
Useful
effect.
lbs.
1,350
1,350
1,350
1,350
1,800
1,350
1,800
1,350
1,800
1,350
1,800
1,350
1,800
1,350
1,800
1,350
1,800 1,350
I
This table is intended to exhibit the work that may be per-
formed by the same mechanical power, at different velocities,
on canals, rail-roads, and turnpike-roads. Ascending and de-
scending by locks or canals, may be considered equivalent to
the ascent and descent of inclinations on rail-roads and turnpike-
roads. The load carried, added to the weight of the vessel or
carriage which contains it, forms the total mass moved; and the
useful effect is the load. To find the effect on canals at different
velocitics, the effect of the given power at one velocity being
known, it will be as 32: 2.52 :: 55,500 : 38,542. The mass
moved being very nearly inversely as the square of the velo-
city; at least, within certain limits.
This table shows, that when the velocity is 5 miles per hour,
it requires less power to obtain the same effect on a rail-
way than on a canal; and the lower range of figures is added
to show the velocity at which the effect on a canal is only equal
to that on a turnpike-road. By comparing the power and ton-
nage of steam-vessels, it will be found that the rate of decrease
of power by increase of velocity, is not very distant from the
* Though the force of traction on a canal varies as the square of the velocity;
the mechanical power necessary to move the boat is usually reckoned to increase as
the cube of the velocity. On a rail-road or turnpike the force of traction is con-
stant; but the mechanical power necessary to move the carriago increases as the
velocity.
}

1
STEAM-ENGINES, CANALS, RAIL-ROADS,
377
truth; but we know that in a narrow canal the resistance in-
creases in a more rapid ratio than as the square of the velocity :*
that is, within certain limits; beyond them, there is a remark-
able change in the circumstances of resistance.
TABLE IV. Showing the maximum quantity of labour a
Horse of average strength is capable of performing, at dif-
ferent velocities, on canals, rail-ways, and turnpike-roads.
(From Tredgold.)
Velocity
in miles per
•
hour.
iniles.
22
n = K
·3
31/0
5
6
17
8
9
10
1
Duration
of the day's
work at the
preceding
velocity.
hours.
11/4/
8
or n 8.46 →
R
of camer
1 0
4/12
27%
2
H2 H8
1 1/1/0
1 //
9
ΤΟ
No
Force
of traction
in lbs.
lbs.
831
83/3/10
839/10
83/3/
83/
83/1
83:
83
BIF BIT 0),
83/1
83/31
Useful effect of one horse working one
day, in tous drawn one inile:
On a canal.
tons.
520
243
153
102
52
30
19
12.8
9.0
6.6
On a
level
rail-way.
tons.
115
92
82
72
57
48
41
36
32
28.8
On a good
level turn-
pike-road.
Lons.
14
12
10
9
7.2
6.0
5.1
4.5
4.0
3.6
Where horse power is employed for the higher velocities,
the animals ought to be allowed to acquire the speed as gra-
dually as possible at the first starting. This simple expedient
will save the proprietors of horses much more than they are
aware of; and it deserves their attention to consider the best
According to the interesting researches of Du Buat, the resistance to the motion
of boats even in canals may be regarded as proportional to the square of the velocity,
or n as v2 ncarly, provided n be made to depend upon the transverse sec-
tions of the vessel and the canal in which it moves, and a constant quantity
K determinable by experiment. If c be the vertical section of the canal, b the
vertical section of the immersed portion of the boat or barge; then
J
C
( 1 + 2). The medium of Du Buat's experiments gives K = 8.46,
+2); but those experiments were not so numerous and va-
(응
​(+2);
ried as might be wished. See Principes d'Hydraulique, tom. ii. pp. 340, 342,
&c.
<

In cases connected with these, or kindred inquiries, where velocities in miles per
hour, are to be reduced into velocities in feet per second, or the contrary; the rules
at p. 102 of this work will be found useful.
2 K 2
378
STEAM-ENGINES, CANALS, RAIL-ROADS.
1
t
I
mode of feeding and training horses for performing the work
with the least injury to their animal powers.
To compare the preceding table with practice at the higher
velocities, it will be necessary to have the total mass moved,
which is one-third more than the useful effect in this table.
Now, the actual rate at which some of the quick coaches travel
is 10 miles an hour; the stages average about 9 miles; and a
coach with its load of luggage and passengers amounts to
about 3 tons; therefore the average day's work of 4 coach
horses is 27 tons drawn one mile, or 64 tons drawn one mile
by one horse. The table gives 3.6 tons, added } of 3·6
4.8 tons drawn one mile for the extreme quantity of labour
for a horse at that speed, upon a good level road; from which
should be deducted the loss of effect in ascending hills, heavy
roads, &c., which will make the actual labour performed by a
coach-horse average about double the maximum given by the
table. The consequences are well known.
According to Mr. Bevan's observations, the horses on the
Grand Junction Canal draw 617 tons one mile, at the velocity
of 2.45 miles per hour.
According to Mr. Tredgold, if v be the maximum velocity
of a horse, and v any other velocity, the immediate power of a
horse is 250 v (1–2); and, when the weight of the vessel
V
or carriage is to the weight of the load, as n 1, we have
250 v
(1
-음​)
the effective power; and d being the hours
I + n
work will be
the horse works in one day, the day's
250 d v ( 1 —-~-~) in lbs. raised 1 mile, and 250
บ
V
(1 — 1-2) =
-음​)
(1
1 + n
the force of traction in lbs. But if the force were immediately
14.7
✓ d
when the waggons alone are moved, we have 1:
applied, the value of v would be
::
I
14.7
14.7
:
√ d' √d (1 + n)
1
250 d v
a + n
; and to find the value
v √ d (1 + n)
14.7
1
1 + n
pk Apk
v; whence the day's work is
V
; which is a maximum when
STEAM-ENGINES, CANALS, RAIL-ROADS.
379
96
v³ (1 + n)
we have
hours; and
d. Consequently, when the velocity is given,
96
v² (1 + n
equal the duration of the day's work in
8000
the effective day's work; and 250
v (1 + n) ³
834 lbs. But we may assume n to be always
72
d, and
v2
9.8
(1 - 0187)
14.7
so near, as not to affect the result; and then,
**
M
the day's work in lbs. or very nearly 2
ย
4500
ย
one mile. This being combined with the numbers of the pre-
ceding table, gives the effect of a horse on canals, rail-roads, and
turnpike-roads.
It must, however, be here added, that although the deduc-
tions from Mr. Tredgold's valuable tables, as to the effects on
canals, are tolerably accurate up to rates of 4 or 5 miles per
hour, yet, when boats are moved on canals at rates of from 9 to
12 or 14 miles per hour, the circumstances of the resistances
undergo an essential change. The resistance, in fact, becomes
so small, that passage-boats now travel at these high velocities ;
and it is hence probable that rail-roads and canals will admit of
a competition such as the supporters of rail-roads never antici-
pated.
tons raised
I shall here briefly detail some of the facts, as they have been
given in a letter widely circulated by Mr. W. Grahame of Glas-
gow, in the Nautical Magazine, and other places.
From the traffic by canal boats, which has been actually
going on during the last two years and a half, on the Paisley
canal, we learn this remarkable fact, that, while a speed of ten
miles per hour has been maintained by the canal boats, the
banks have sustained no injury whatever. The cause of injury,
in truth, has been entirely suppressed by the velocity of the
boat, which passes along the water without raising a ripple.
About two years ago, measures were adopted for increas-
ing the speed of the boats on the Paisley or Ardrossan canal.
This canal is by no means favourable to such experiments,
being both serpentine in its course, and narrow: it connects the
town of Paisley with the city of Glasgow, and the village of
Johnstone; the distance being about twelve miles. The boats
380
CANALS AND ROADS.
ל
1
employed on this canal are 70 feet in length, and 5-6 broad,
and carry, if necessary, upwards of 120 passengers. They are
formed of light iron plates, and ribs covered' with wood, and
light oiled cloth, at a whole cost of about 1257. They perform
stages of four miles in an interval of time varying from 22 to
25 minutes, including all stoppages, and the horses run three or
four of these stages alternately every day. The passengers are
under cover, or not, as they please, no difference being made in
this particular; and the fare is one penny per mile in the first,
and three farthings per mile in the second cabin.
K
The horses drawing the canal boat are guided by a boy, who
rides one of them; and, in passing under bridges at night, a
light is shown in the bow of the boat, by which he sees his
way, and which light is closed when the bridge is passed. In-
termediate passengers are also accommodated to distances even
as small as a mile; and the facility with which the boat stops
when relieved from the drawing force, is such as avoids all
danger whatever. The expense of conveying a load of cight
tons at a rate of nine or ten miles per hour, including all out-
lay, interest, and replacement of capital, is not more than 11d.
It is also ascertained that one ton weight may be
per mile.
carried on a canal at nearly the same speed as on the railway,
at about 14d. per mile, including an allowance for interest and
replacement of capital.
It is also believed, that if the breadth and curvature of the
Paisley canal admitted boats of 90 feet length, instead of 70,
they would carry more passengers by one-half without an addi-
tional expense, and a decrease of labour to the horses.
The foregoing has been deduced from calculations founded
on the observation of facts relating to the wear and tear of
boats and horses, and the absolute resistance which these
boats meet with in passing through the water. On this sub-
ject it has been observed, that, in addition to the common re-
sistance of the water to the motion of the boat, a wave, or body
of water, is also raised before it, varying in its height accord-
ing to the velocity of the boat, and constantly presenting an
obstacle to her progress, provided that she only moves through
the water at a certain slow rate. The height of this wave will
then amount to nearly two fect, often overflowing the banks
of the canal, and, from the obstruction it occasions, eventu-
ally obliging the boat to be stopped. Now, if, instead of
stopping the boat when. this wave is raised, her velocity be
increased beyond what it had then been, she advances and
passes over it, and leaves it to subside in her wake, which it
does, and the water becomes perfectly still. The same horses
}
THEN
K
CANALS AND ROADS.
381
"
drawing the boat at this increased speed, are found to perform
their work better, the resistance to their progress having become
less; and the more the velocity of the boat is thus increased,
the less resistance she meets with, merely having to cut the
still water instead of the wave. It is a curious fact, that the
wave produced by the approach of a slow canal boat is often
observed at the distance of a mile, and upwards, along the canal,
before the arrival there of the boat. But in the case of the
high wave being raised by the Paisley canal boat, it is custom-
ary to stop the boat, and after it has subsided to start again at a
greater velocity. When the boat is to be stopped for any pur-
posc, as her speed decreases the wave rises in proportion, and
washes over the banks, until the motion of the boat becomes so
small as to produce none. The discovery is doubtless a very
important one, and, if turned to account, is likely to produce a
material alteration in the rate of transport on canals. It was
not known until these experiments were made, that if a boat,
from a state of rest, was dragged along a canal, in proportion as
her speed increased to a certain limit, that the power required
was greater but that, if she were started at, and preserved a
speed excceding the same limit, the power required would be
less, and would decrease as her velocity increased. In fact,
from a certain velocity there seems to be no limit to the rate at
which a boat, as far as animal power can be applied, may thus
pass through the water; and as the rate increases, the power
required decreases. On this principle it is, that the boats on
the Paisley canal, with ninety passengers in them, are drawn by
horses at a speed of ten miles an hour; while it would kill
them to draw the same boat along the canal at six miles an
hour. A boat might indeed travel fifteen or twenty miles an
hour casier than at six miles. The former of these velocities
has already been attained by Mr. Grahame, along a distance
of two miles, and is considered by him safer both for the boat
and the canal.
As a proof of what may be done by this method of carriage,
Mr. Grahame states that he has performed a voyage of fifty-
six miles along two canals in six hours and thirty-eight minutes,
which included the descent of five, and the ascent of cleven
locks, the passage of eighteen draw-bridges where the tracking-
line was thrown off, and sixty common bridges, besides a tun-
nel half a mile long; all, of course, producing some delay. The
boat which performed this was sixty-nine feet long, and nine
broad, drawn by two horses, and carried thirty-three passen-
gers, with their luggage and attendants.
These facts furnish great encouragement to canal companies,
to improve the construction of their boats and the speed on their
50
242
382
CANALS AND ROADS
canals; and thus, probably, in some positions, supersede the
necessity of rail-roads.
Mr. Macneill, the assistant engineer upon the Holyhead
road, under Mr. Telford, in the course of his examination
before a committee of the House of Commons, on steam-
carriages, rail-roads, &c. gave the subjoined curious infor-
mation.
✡
J
Well made roads, formed of clean hard broken stone, placed
on a solid foundation, are little affected by changes of atmosphere;
but weak roads, or such as are imperfectly formed with gravel,
flint, or round pebbles, without a bottoming of stone, pavement,
or concrete, are much affected.
On the generality of roads, the proportional injury from the
weather and traffic is nearly as follows. When travelled by fast
coaches: from atmospheric changes 20; coach wheels 20; horses'
feet 60 100. When travelled by wagons: atmospheric
changes 20; wagon-wheels 35.5; horses' feet 44.5 = 100.
Has ascertained, from a number of observations, that the wear
of the iron tire of fast-going coachwheels is, compared with that
of the shoes of the horses which draw them, as 326.8 to 1000,
or as 1 to 3-4ths nearly; and infers that the comparative injury
done by them to roads is nearly in the same proportion. In
the case of slow-going carriages and horses the proportion is as
309 to 360, or as 1 to 1.16, or nearly 1 to 14. The tire of the
wheels of the fast-going coaches last from two to three months,
according to the weather, the workmanship, and quality of iron;
about 20 years ago, it did not last seven days on an average.
Coach-horse shoes remain in use about thirty days; wagon-
horse shoes about five weeks on an average. Where roads are
weak, and yield under pressure, the injury caused by heavy
wheels is far greater than on solid firm roads.
It was found, in one instance, that the wear of hard stone,
placed on a wet clay bottom, was four inches, while it was not
more than half an inch when placed on a solid dry foundation.
On the Highgate archway road, the annual wear is not more
than half an inch in depth. To the same gentleman we owe the
following useful table.
CANALS AND ROADS.
383
Rate of Inclination.
1 in 20
1 in 26
TABLE V.-The general Results of Experiments made with
a Stage Coach, weighing, exclusive of seven passengers,
18 cwt., on the same piece of road, on different inclina-
tions, and at different rates of velocity, furnish the follow-
ing statement.
1 in 30
1 in 40
1 in 600
•
•
1 in 30
1 in 40
1 in 600
•
1 in 20.
1 in
26
26.
1 in
30
1 in 40
1 in 600
1 in 20..
1 in 26.
30.
•
Rates of Travelling. Force required.
6 miles per hour
268 lbs.
6.
213
165
160
111
6
6
6.
8.
8..
8.
8.
8.
. 10. .
10
10.
10.
. 10. .
•
296
219
196
166
120
318
225
200
172
128
384
STRENGTH OF ANIMALS.
CHAPTER XV.
ACTIVE AND PASSIVE STRENGTH.
SECTION I.-Active Strength, or Animal Energy, as of Men,
Horses, &c.
1. THE force obtained through the medium of animal agency
evidently varics, not only in different species of animals, but,
also, in different individuals. And this variation depends, first,
on the particular constitution of the individual, and upon the
complication of causes, which may influence it; secondly, upon
the particular dexterity acquired by habit. It is plain, that such
a variation cannot be subjected to any law, and that there is no
expedient to which we can have recourse, but that of seeking
mean results.
Secondly, the force varies according to the nature of the
labour. Different muscles are brought into action in different
gestures and positions of an animal which labours; the weight
itself of the animal machine is an aid in some kinds of labour,
and a disadvantage in others: whence it is not surprising that
the force exerted is different in different kinds of work. Thus
the force exerted by a man is different, in carrying a weight,
in drawing or pushing it horizontally, and in drawing or push-
ing it vertically.
Thirdly, the force varies according to the duration of the
labour. The force, for example, which man can exert in an
effort of a few instants, is different from that which he can main-
tain equably in a course of action continued, or interrupted only
by short intervals, for a whole day of labour, without inducing
excessive fatigue. The former of these may be called Absolute
Force, the latter Permanent Force. It is of use to become
acquainted with them both, as it is often advantageous to avail
ourselves sometimes of the one, sometimes of the other.
Lastly, the force varies according to the different degrees of
velocity, with which the animal, in the act of labouring, moves
either its whole body, or that part of it which operates. The
force of the animal is the greatest, when it stands still; and
becomes weaker as it moves forward, in proportion to its speed ;
the animal acquiring, at last, such a degree of velocity as ren-
ders it incapable of exerting any force.
P
2. Let & be a weight equivalent to the force which a man
can exert, standing still; and let v be the velocity with which,
STRENGTH OF ANIMALS.
385
if he proceeds, he is no longer capable of exerting any force:
also, let r be a weight equivalent to the force which he exerts,
when he proceeds, equably, with a velocity v.
Then will be a function of v, such that, 1st, it decreases
whilst v increases; 2dly, when v O, then F = 9; 3dly,
when v = V, F = 0.
1. F =
3. Upon the nature of this function, we have the three fol-
lowing suppositions.
2. F
F = Ф
3. F
F
• (
Φ
B
1.
(1 — 2-)*(Ib. tom. VIII; and Act of Rowers.)
4. Coroll. 1. The effect of the permanent force being mea-
sured by the product F v, the expression for the effect will be
one of the three following, accordingly as one or other of the
suppositions is adopted.
(1 –
2. F V.
3. F V.
1. F
2. F
F v. (1 − 2), or + v (1 — 2)
-).
F
or ♣ v (1 -
(1 — 21-23).
ง √ (1-12),
-),
(1 –
1
To a lo
2). (Bouguer, Man. des Vais.)
V
3
v2
2). (Euler, Nov. Comm. Pet. tom. III.)
v2
5. Coroll. 2. To know the weight with which a man should
be loaded, or the velocity with which he ought to move, in
order to produce the greatest effect, we must make d. r v = 0.
Whence we shall have
. 3d..
; and v =
—
; and v =
; and v =
¿ V.
ΟΙ Φ v
1
V/-/S
√3
1
4
3. F
9
3
6. Coroll. 3. And the value of the greatest effect will be,
according to the 1st hypothesis.
Ꮮ è
2d.
V.
2
V
» ( 1 − })
1)
V
1
v = 0.5773 v.
2
3√3
4
27
? V.
V:
• v = 0·3836 † v :
2 L
386
ANIMAL STRENGTH: SCHULZE'S EXPERIMENTS.
But which of the three suppositions ought we to prefer?
And are we certain that any of them approximates to the true
law of nature ?
Mr. Schulze made a series of experiments with a view to the
determination of this point,* and with regard to men decided
in favour of the last of Euler's formulæ
As the experiments of this philosopher are very little known
in England, I shall here present his brief account of them.
7. To make the experiments on human strength, he took at
random 20 men of different sizes and constitutions, whom he
measured and weighed. The result is exhibited in the follow-
ing table.
Order.
Height.
1
5' 3" 4""
2
5 2
3 5 7
4
5
6 6 0 4
3
1
4
8
0 4
∞
7 5 8
8
9
10
5 5 0
5 11 2
10 10 10 10
5
viz. F p
5
5
૨૭
32
V
(1 — 2-)²?.
:)
V
Weight. Order.
122
11
134
12
165
13
131
14
177
15
158
16
180
17
117
18
140
19
126
20
4

Height.
5' 9" 7'''
5
1 4
5
3
2
5 4
1
5 10
8
5 0
3
4 11 2
5 3
9
0
5 6
5 10
1
Weight.
132
157
175
117
192
133
147
124
163
181
Here the heights are expressed in feet (marked '), inches (''),
twelfths (""), the feet being those of Rhinland, each 12.35 Eng-
lish inches. The weights are in pounds, which are to our
avoirdupois lbs. as 30 to 29.
To find the strength that each of these men might exert to
raise a weight vertically, Mr. Schulze made the following expe-
* Mem, Acad. Scienc. Berlin, for 1783.
riments:
He took various weights increasing by 10 lbs. from 150 lbs.
up to 260 lbs. ; all these weights were of lead, having circular
and equal bases. To use them with success in the proposed
experiments, he had at the same time a kind of bench made,
in the middle of which was a hole of the same size as the base
of the weights: this hole was shut by a circular cover when
pressed against the bench; at other times it was kept
at about the distance of a foot and a half above the bench
ANIMAL STRENGTH: SCHULZE'S EXPERIMENTS.
387
by means of a spring and some iron bars. To prevent the
weight with which this cover was loaded during the experiment
from forcing down the cover, lower than the level of the sur-
face of the bench, he had several grooves made in the four iron
bars, which sustained the cover, and which, at the same time,
served to hold up the cover at any height where it might arrive
by the pressure of the springs as soon as the pressure of the
weight ceased.
}
After having laid the 150 lbs. weight on the cover, and the
other weights in succession increasing by 10 lbs. up to 250 lbs.,
he made the following experiments with the men whose size.
and weight are given above, by making them lift up the
weights as vertically as possible all at once, and by observing
the height to which they were able to lift them. The annexed
table gives the heights observed for different weights marked at
its head.

150 160 170
180 | 190
200210 220 230 240 250
// /// ///// // /// // " "/ /// "/
// //// ///// ///// //
4
11 4
4 3 8 2
81
1
5
74
7 3 11 2
1 2
50
51
70 3
1 7 9 6 4
2 7 10 6 6
3 7 9 7 3 6
48 3 7 6 7
512 411 19
614 514 013
712 1111 310
811
11 910 2 9
99 5 8 3 7
10 8 1 6 5 4
55 9 4 11 4 03 03 83 11
25 105 34 74 04 73 21
78 5 7 10 7 15 106 64 10
512 811 510 18 63
5 9 3 16 95 35
81 110 2
13 21 0
38
4 8 11 8
1 6 115 10
1 2 91 3
69
64
92 5 1 70 4
15
73
7
3
1
This table proves that the size of the men employed to raise
the weights vertically has considerable influence on the height
to which they brought the same weight. We find also that
the height diminishes in a much more considerable ratio than
the weight increases and we may therefore conclude, that it
is advantageous to employ large men when it becomes neces-
sary to draw vertically from below upwards: and on the con-
trary, it is more advantageous to employ men of a considerable
weight, when it is required to lift up loads by means of a pulley
about which a cord passes, that the workmen may draw in a
vertical direction, from above downwards. To find the absolute
strength of these men in a horizontal direction, Mr. Schulze
proceeded thus:
388
ANIMAL STRength: SchULZE'S EXPERIMENTS.
Having fixed over an open pit a brass pulley extremely well
made, of 15 inches diameter, whose axis, made of well polished
stcel to diminish the friction, was inch in diameter, he
passed over this pulley a silk cord, worked with care, to give
it both the necessary strength and flexibility. One of the ends
of this cord carried a hook to hang a weight to it which hung
vertically in the pit, whilst the other end was held by one of
the 20 men, who in the first order of the following experiments
made it pass above his shoulders; instead of which, in the se-
cond, he simply held it by his hands.
Mr. S. had taken the precaution to construct this in such a
manner that the pulley might be raised or lowered at pleasure,
in order to keep the end of the cord held by the man always in
a horizontal direction, according as the man was tall or short,
and exerted his strength in any given direction.
He had made the necessary arrangements so as to be able to
load successively the basin of a balance which was attached to
the hook at the end of the cord which descended into the pit,
whilst the man who held the other end of this cord employed
all his strength without advancing or receding a single inch.
The following tables give the weights placed in the basin
when the workmen were obliged to give up, having no longer
sufficient strength to sustain the pressure occasioned by the
weight. To proceed with certainty, Mr. S. increased the weight
each time by five pounds, beginning from 60, and took the pre-
caution to make this augmentation in equal intervals of time;
having always precisely a space of 10 seconds between them.
The result of these observations repeated several days in succes-
sion, is contained in the following tables.
I. When the cord passed over the shoulders of the workmen :

Order.
1
I Q2 ♡
3
5
Ibs.
95
105
110
100
105
Order.
6
7
8
9
10
lbs. Order.
100
115
105
95
90
1
11
12
13
14
15
Ibs.
95
100
110
90
110
Order.
16
17
18
19
20
1
lbs.
95
100,
90
100
100
ANIMAL STRENGTH: SCHULZE'S EXPERIMENTS.
389
II. When the cord was simply held before the man :

Order.
1Q 3
2
4
5
1 2 3 →
4
lbs.
5
6
17
90
105
105
90
95
Order.
6
7
8
9
10
lbs.
100
110
100
90
85
Order.
40.18 4.94
8
41.12
4.85
9
39.08 5.55 10
39.40 5.04 11
34.19 5.83 12
35.11 5.68 13
38. 7 5.25 14
11
12
13
14
15
lbs.
90
90
100
85
105
Order.
16
17
18
19
20
These two tables show that men have less power in drawing
a cord before them than when they make it pass over their
shoulders; they show, also, that the largest men have not always
the greatest strength to hold, or to draw in a horizontal direc-
tion, by means of a cord. To obtain the absolute velocity of
these 20 men, Mr. S. proceeded as follows:
Having measured very exactly a distance of 12,000 Rhinland
fect in a plain nearly level, he caused these 20 men to march with
a fair pace, but without running, and so as to continue during
the period of four or five hours; the following is the time em-
ployed in describing this space, with the velocity resulting for
each of them.
Order. Time. Veloc. Ordor. Time, Veloc. Order. Time. Veloc.
Ibs.
90
90
85
100
100

40.09 4.99
36.17 5.51
15
40.20 4.96 16 41.28 4.82
40.51 4.90 17 42.25 4.71
36.17 5.51 18 40.19 4.98
38.11 5.24 19 39.57 5.01
38. 5 5.25 20 37.51 5.29
37. 1 5.40
It is necessary to mention, with regard to these experiments,
that Mr. Schulze took care to place at certain distances persons
in whom he could place confidence, in order to observe whether
these men marched uniformly, and sufficiently quick, without
running.
Having thus obtained, not only the absolute force, but the
absolute velocity also, of several men, he took the following
method to determine their relative force.
He made use of a machine composed of two large cylinders
of very hard marble, which turned round a vertical cylinder of
wood, and moved by a horse, which described in his march a
circle of 10 Rhinland feet. This machine appeared the most
51
212
390
ANIMAL STRENGTH: SCHULZE'S EXPERIMENTS.
proper to make the subsequent experiments, which serve to
determine the relative strength that the men had employed to
move this machine, and which is used hereafter to determine
which of Euler's two formulæ ought to be preferred.
To obtain this relative force, he took here the same pulley
which served in the preceding experiments, by applying a
cord to the vertical cylinder of wood, and attaching to the
other end of this cord, which entered into an open pit, a suffi-
cient weight to give successively to the machine different ve-
locities.
b
Having applied in this manner a weight of 215 lbs., the
machine acquired a motion, which after being reduced to an
uniform velocity, taking into account the acceleration of the
weight, of the friction, and of the stiffness of the cord, gave 2.41
feet velocity; and having applied in the same manner a weight
of 220 lbs., the resulting uniform motion gave a velocity of 2.47
feet. These two limits are mentioned because they serve as a
comparison with what immediately follows: Mr. S. began these
experiments with a weight of 100 lbs., and increased it by 5
every time from that number up to 400 lbs.
He made this machine move by the seven first of his work-
men, placing them in such a way that their direction remained
almost always perpendicular to the arm on which was attached
the cord which passed over their shoulders in an almost hori-
zontal direction.
Thus situated, they made 281 turns with this machine in two
hours, which gave for their relative velocity v
2.45 feet per
second. We have also the absolute force, or 4, from these 7 men,
by the above table, 730 lbs. and their absolute velocity or
v = 5.30 fect.
p,
S
Therefore, by substituting these values in the first formula,
we find the relative force r = 205 lbs., which agrees very well
with what we have just found above.
If instead of this first formula the second be taken, it gives
F = 153 lbs., which is far too little.
By this it is evident, that the last of Euler's two formulæ is
to be preferred in all respects. Mr. Schulze made a great
number of combinations, and almost always found the same
effect.
Dividing the 205 lbs. which we have just found by 7, the
number of workmen, we get 29 lbs. for the relative force
with 2.45 feet relative velocity for each man, which is rather
more than the valucs commonly adopted in the computation
of machinery. A number of other observations on different
machines have given the same result; that is to say, we
ANIMAL STRENGTH: MEN.
391
must value the mean human strength at 29 or 30 lbs. with a
velocity of 2 feet per second.
To obtain the ratio of the strength of a horse to that of a man,
Mr. Schulze proceeded in a similar manner; but his results,
in reference to that inquiry, are neither so correct nor so inter-
esting.
8. In the first volume of my Mechanics, I stated the average
force of a man at rest to be 70 lbs., and his utmost walking ve-
locity when unloaded to be about 6 feet per second; and thence
inferred that a man would produce the greatest momentum
when drawing 314 lbs. along a horizontal plane with a velocity
of 2 feet per second. But that is not the most advantageous
way of applying human strength.
9. Dr. Desaguliers asserts, that a man can raise of water or
any other weight about 550 lbs., or one hogshead (weight of the
vessel included), 10 feet high in a minute; this statement,
though he says it will hold good for 6 hours, appears from his
own facts to be too high; and is certainly such as could not be
continued one day after another. Mr. Smeaton considers this
work as the effect of haste or distress; and reports that 6 good
English labourers will be required to raise 21141 solid feet of
sea water to the height of four feet in four hours in this case
the men will raise a very little more than 6 cubic feet of fresh
water each to the height of 10 feet in a minute. Now the hogs-
head containing about 8 cubic feet, Smeaton's allowance of
work proves less than that of Desaguliers in the ratio of 6 to 8
or 3 to 4. And as his good English labourers who can work
at this rate are estimated by him to be equal to a double set of
common men picked up at random, it seems proper to state
that, with the probabilitics of voluntary interruption, and other
incidents, a man's work for several successive days ought not to
be valued at more than half a hogshead raised 10 feet high in a
minute. Smeaton likewise states that two ordinary horses will
do the work in three hours and twenty minutes, which amounts
to little more than two hogsheads and a half raised 10 feet high
in a minute. So that, if these statements be accurate, one horse
will do the work of five men.
Ka
Mr. Emerson aflirms, that a man of ordinary strength, turn-
ing a roller by the handle, can act for a whole day against a
resistance equal to 30 lbs. weight; and if he works 10 hours a
day, he will raise a weight of 30 lbs. through 32 feet in a second
of time; or, if the weight be greater, he will raise it to a pro-
portionally less height. If two men work at a windlass or
roller, they can more easily draw up 70 lbs. than one man can
30 lbs.; provided the elbow of one of the handles be at right
392
ANIMAL STRENGTH: MEN.
ļ
angles to that of the other. Men used to bear loads, such as
porters, will carry from 150 lbs. to 200 or 250 lbs. according to
their strength. A man cannot well draw more than 70 lbs. or
80 lbs. horizontally and he cannot thrust with a greater force
acting horizontally at the height of his shoulders than 27 or
30 lbs. But one of the most advantageous ways in which a
man can exert his force is to sit and pull towards him nearly
horizontally, as in the action of rowing.
M. Coulomb communicated to the French National Institute
the results of various experiments on the quantity of action.
which men can afford by their daily work, according to the dif-
ferent manners in which they employ their strength. In the
first place he examined the quantity of action which men can
produce when, during a day, they mount a set of steps or stairs,
either with or without a burthen. He found that the quantity
of action of a man who mounts without a burthen, having only
his own body to raise, is double that of a man loaded with a
weight of 68 kilogrammes, or 150 lbs. avoirdupois, both con-
tinuing at work for a day. Hence it appears how much, with
equal fatigue and time, the total or absolute effort may obtain
different values by varying the combinations of effort and
velocity.
But the word effect here denotes the total quantity of labour
employed to raise, not only the burthen, but the man himself;
and as Coulomb observes, what is of the greatest importance to
consider is the useful effect, that is to say, the total effect, de-
ducting the value which represents the transference of the
weight of the man's body. This total effect is the greatest pos-
sible when the man ascends without a burthen; but the useful
effect is then nothing: it is also nothing if the man be so much
loaded as to be scarcely capable of moving and consequently
there exists between these two limits a value of the load such
that the useful effect is a maximum. M. Coulomb supposes that
the loss of quantity of action is proportional to the load (an
hypothesis which experience confirms), whence he obtains an
equation which, treated according to the rules of maxima and
minima, gives 53 kilogrammes (117 lbs. avoird.) for the weight
with which the man ought to be loaded, in order to produce
during one day, by ascending stairs, the greatest useful effect:
the quantity of action which results from this determination has
for its value 56 kilogrammes (123½ lbs. avoird.) raised through
one kilometer, or nearly 1094 yards. But this method of work-
ing is attended with a loss of three-fourths of the total action of
men, and consequently costs four times as much as work, in
which, after having mounted a set of steps without any burthen,
1
L
1
ANIMAL STRENGTH: MEN.
393
the man should suffer himself to fall by any means, so as to
raise a weight nearly equal to that of his own body.
From an examination of the work of men walking on a hori-
zontal path, with or without a load, M. Coulomb concludes that
the greatest quantity of action takes place when the men walk
being loaded; and is to that of men walking under a load of
58 kilogrammes (128 lbs. avoird.) nearly as 7 to 4. The weight
which a man ought to carry in order to produce the greatest
useful effect, namely, that effect in which the quantity of action
relative to the carrying his own weight is deducted from the
total effect, is 50-4 kilogrammes, or 111.18 lbs. avoirdupois.
There is a particular case which always obtains with respect
to burthens carried in towns, viz. that in which the men, after
having carried their load, return unloaded for a new burthen.
The weight they should carry in this case, to produce the great-
est effect, is 61.25 kilogrammes (135lbs. avoird.) The quan-
tity of useful action in this case compared with that of a man
who walks freely and without a load, is nearly as 1 to 5, or, in
other words, he employs to pure loss of his power. By
causing a man to mount a set of steps freely and without
burthen, his quantity of action is at least double of what he
affords in any other method of employing his strength.
When men labour in cultivating the ground, the whole
quantity afforded by one during a day amounts to 100 kilo-
grammes clevated to one kilometer, that is, 220.6 lbs. raised
1094 yards. M. Coulomb, comparing this work with that of
men employed to carry burthens up an ascent of steps, or at
the pile-engine, finds a loss of about part only of the quan-
tity of action, which may be neglected in researches of this
kind.
In estimating mean results we should not determine from
experiments of short duration, nor should we make any deduc-
tions from the exertions of men of more than ordinary strength.
The mean results have likewise a relation to climate.
"I have
caused," says M. Coulomb, "extensive works to be executed
by the troops at Martinico, where the thermometer (of Reau-
mur) is seldom lower than 20° (77° of Fahrenheit). I have
executed works of the same kind by the troops in France;
and I can affirm that under the fourteenth degree of latitude,
where men are almost always covered with perspiration, they
are not capable of performing half the work they could perform
in our climate.”
ލ
10. Entirely according with these are the experiments of
Regnier, by means of a dynamometer, the results of which not
only established the superiority of civilized men over savages,
394
ANIMAL STRENGTH: MEN.
but that of the Englishman over the Frenchman. The fol-
lowing is reduced from one of Regnier's tables of mean
results.

STRENGTH.
Savages, of Van Dieman's Land.
New Holland
Timor
Civilized men: French.
English
With tho With the
hands.
reins.
lbs. OZ. lbs. OZ.
30 6
0 0
8
51 8
58
7
69
71
ર
4
14
16
22
23
∞ = 20 co
1
8
11. A porter in London is accustomed to carry a burthen of
200 lbs. at the rate of three miles an hour and a couple of
chairmen continue at the rate of four miles an hour, under a
load of 300 lbs. Yet these exertions, Professor Leslie remarks,
are greatly inferior to the subsultory labour performed by
porters in Turkey, the Levant, and generally on the shores of
the Mediterranean. At Constantinople, an Albanian porter
will carry 800 or 900 lbs. on his back, stooping forward, and
assisting his steps by a short staff. Such loads, however, are
carried for very short intervals. At Marseilles it is affirmed
that four porters carry the immense load of nearly two tons, by
means of soft hods passing over their heads, and resting on their
shoulders, with the ends of poles from which the goods are sus-
pended.
1. A man marching 7 hours on a slope of 7 degrees,
with a load of from 15 to 18 lbs.
2. Marching in a mountainous country without load.
12. With regard to the magnitude of the comparative efforts
of man in different employments, the late Mr. Robertson Bu-
chanan ascertained, that in working a pump, in turning a winch,
in ringing a bell, and rowing a boat, the dynamic results are as
the numbers 100, 167, 227, and 248.
According to the interesting experiments described in M.
Hachette's Traile des Machines, the dynamic unit being the
weight of a cubic metre of water raised to the height of
one metre [that is, 2208 lbs. avoird. or 4 hogsheads raised to
the height of 3-281 feet, or 1.3124 hogsheads to the height of
10 fect], we have the following measures, at a medium, of the
daily actions of men.
Dyn. Unit.
225
140
ANIMAL STRENGTH: MEN.
395
3. Carrier of wood up a ladder, his weight 123, his
load 117 lbs..
4. Carrier of peat up steps, his own weight com-
prised, 112 to
5. Man working at the cord of a pulley to raise the
ram of a pile engine: three examples .
6. A man drawing water from a well by means of a
cord
•
7. Man working at a capstan
8. Man working at a capstan to raise water, mean
of 24.
1. A man travelling without load on a flat road, his
weight 154 lbs. his journey 314 miles
2. A soldier, carrying from 44 to 55 lbs. travelling 124
miles, 1800 to.
3. Ditto a forced march of 25 miles
4. A French porter, weight of the man not in-
cluded, 792 to.
5. Porter with wheel-barrow, weight of the man not
included
•
110
The unit of transport being the weight of a cubic metre of
water, carried a metre (2208 lbs. 3281 feet) upon a horizontal
road, we have for the daily action.
6. Porters with a sledge
7. A man drawing a boat on a canal; 110310 lbs.
conveyed 67 miles
A drawing-knife, with a force of .
An augur, with two hands
A screw-driver, one hand
A common bench vice handle
A chisel and awl, vertical pressure
A windlass, handle revolving
109
lbs.
. 100
100
84
72
72
60
120
75
40
48
71
116
Dyn. Unit.
3500
1900
2800
800
14. Mr. B. Bevan, an able engineer, has made experiments
on the application of human energy to the use of augurs, gim-
blets, screw-drivers, &c. He has presented to the public the
following list, as a specimen; premising that many ordinary
operations are performed in a short space of time, and may,
therefore, be done by greater exertion than if a longer time
was necessary. Thus a person, for a short time, is able to use a
tool or instrument called
1015
627
550000
396
ANIMAL STRENGTH: MEN.
Pincers and pliers, compression
A hand-plane, horizontally
A hand or thumb-vice
A hand-saw
A stock-bit, revolving
•
•
Small screw-drivers, or twisting by the
thumb and fore-finger only
lbs.
60
50
45
36
16
14
15. M. Morisot informs us that the time employed by a
French stone-mason's sawyer, to make a section of a square
toise (40.89 sq. feet Eng.) in different stones, is as below :
viz.
hours.
45
Calcareous stone, equal grain, spec. grav. 2200
hard, spec. grav. 2300
Liais, ditto hard, fine grain, spec. grav. 2400.
Pyrenean alabaster, the softest of the marbles
Normandy granite.
Granite from Vosges.
Red and green porphyry
The workmen ordinarily made 50 oscillations in a minute;
each stroke about 154 inches.
16. Hassenfratz assigns 13 kilogrammes as the mcan effort of
such a man; but M. Navier, in his new edition of Belidor,
Architecture Hydraulique, regards this estimate as too high.
If this were correct, the daily quantity of action of the sawyer
would be equivalent to 376 kilogrammes elevated to a kilo-
metre (818 lbs. raised of a mile), a quantity more than triple
that of a man working at a winch. M. Navier gives, as a more
correct measure of this labour for 12 hours, 188 kilomgrammes
raised a kilometre: half the former measure. But all this is
probably very vague.
ł
17. Among quadrupeds, those which are employed to pro-
duce a mechanical effect are, the dog, the ass, the mule, the
ox, the camel, and the horse. Of these the horse is the only
one, so far as we are aware, whose animal energy has been
subjected to cautious experiments; and even with regard to
this noble animal, opinions as to actual results are very much
afloat. The dynamic effort of the horse is, however, pro-
bably, about six times that of a strong and active labourer.
Desaguliers states the proportion as 5 to 1, coinciding with
the deductions of Smeaton. The French authors usually
regard seven men as equivalent to one horse. As a fair mean
between these, I assumed in vol. I. of my Mechanics the pro-
portion of 6 to 1 and stated the strength of a horse as equiva-
62
67
56
504
700
1177
ANIMAL STRENGTH HORSES.
397
lent to 420 lbs. at a dead pull. But the proportion must not
be regarded as constant; but obviously varies much according
to the breed and training of the animal, as well as according to
the nature of the work about which he is employed. Thus the
worst way, as De la Hire observed, of applying the strength of
a horse is to make him carry a weight up a steep hill; while
the organization of a man fits him very well for that kind of la-
bour: hence three men climbing up such a hill with a weight
of 100 lbs. each, will proceed faster than a horse with a load
of 300 lbs.
18. In the memoirs of the French Academy for 1703 are in-
serted the comparative observations of M. Amontons, on the
velocity of men and of horses; in which he states the velocity
of a horse loaded with a man and walking to be rather more
than 54 feet per second, or 3½ miles per hour, and when going.
a moderate trot with the same weight to be about 8½ feet per
second, or about 6 miles per hour. These velocities, however,
are somewhat less than what might have been taken for the
mean velocities.
19. But the best way of applying the strength of horses is
to make them draw weights in carriages, &c. To this kind
of labour, therefore, the inquiries of experimentalists should
be directed. A horse put into harness and making an effort
to draw, bends himself forward, inclines his legs, and brings
his breast nearer to the earth; and this so much the more as
the effort is the more considerable. So that when a horse is
employed in drawing, his effort will depend, in some measure,
both upon his own weight and that which he carries on his back.
*
Indeed it is highly useful to load the back of a drawing horse
to a certain extent; though this, on a slight consideration,
might be thought to augment unnecessarily the fatigue of the
animal; but it must be recollected that the mass with which
the horse is charged vertically is added in part to the effort
which he makes in the direction of traction, and thus dispenses
with the necessity of his inclining so much forward as he must
otherwise do and may, therefore, under this point of view,
relieve the draught more than to compensate for the additional
fatigue occasioned by the vertical pressure.
Carmen and wa-
goners in general are well aware of this, and are commonly
very careful to dispose of the load in such a manner that the
shafts shall throw a due proportion of the weight on the back
of the shaft horse. This is most efficaciously accomplished at
Yarmouth, in Norfolk, where a number of narrow streets con-
nccting the market-place with the quay, have led to the inven-
tion and use of the low, strong, narrow carts, thence denomi-
52
2 M
398
ANIMAL STRENGTH HORSES.
J
nated Yarmouth carts, drawn by one horse; and on which
the loads are frequently shifted, especially when the vehicles
pass over the bridge, in order to give the animals better foot
hold, and consequently a greater dynamic effort.
20. The best disposition of the traces during the time a horse
is drawing is perpendicular to the position of the collar upon
his breast and shoulders: when the horse stands at ease, this
position of the traces is rather inclined upwards from the direc-
tion of the road; but when he leans forward to draw the load,
the traces should then become nearly parallel to the plane over
which the carriage is to be drawn; or, if he be employed in
drawing a sledge, or any thing without wheels, the inclination
of the traces to the road should (from the table at p. 243) be
about 1820, when the friction is one-third of the pressure.
If
the relation of the friction to the pressure be different from this,
the same table will exhibit the angle which the traces must
make with the road.
21. When a horse is made to move in a circular path, as is
often practised in mills and other machines moved by horses, it
will be necessary to give the circles which the animal has to
walk round the greatest diameter that will comport with the
local and other conditions to which the motion must be sub-
jected. It is obvious, indeed, that since a rectilinear motion is
the most casy for the horse, the less the line in which he moves
is curved, with the greater facility he will walk over it, and the
less he need recline from a vertical position: and besides this,
with equal velocity the centrifugal force will be less in the
greatest circle, which will proportionally diminish the friction
of the cylindrical part of the trunnions, and the labour of mov-
ing the machine. And further, the greater the diameter of the
horse-walk, the nearer the chord of the circle in which the
horse draws is to coincidence with the tangent, which is the
most advantageous position of the line of traction. On these
accounts it is that, although a horse may draw in a circular
walk of 18 fect diameter, yet in general it is advisable that the
diameter of such a walk should not be less than 25 or 30 feet;
and in many instances 40 feet would be preferable to either.
I
22. It has been stated by Desaguliers and some others, that a
horse employed daily in drawing nearly horizontally can move,
during eight hours in the day, about 200 lbs. at the rate of 2½
miles per hour, or 32 feet per second. If the weight be aug-
mented to about 240 or 250 lbs., the horse cannot work more
than six hours a day, and that with a less velocity. And, in
both cases, if he carry some weight, he will draw better than
if he carried none.
M. Sauveur estimates the mean effort of a
ANIMAL STRENGTH: HORSES.
399
horse at 175 French, or 189 avoird. pounds, with a velocity of
rather more than three feet per second. But all these are pro-
bably too high to be continued for eight hours, day after day.
In another place Desaguliers states the mean work of a horse
as equivalent to the raising a hogshead full of water (or 550 lbs.)
50 feet high in a minute. But Mr. Smeaton, to whose autho-
rity much is due, asserts, from a number of experiments, that
the greatest effect is the raising 550 lbs. forty feet high in a
minute. And, from some experiments made by the Society for
the Encouragement of Arts, under the direction of Mr. Samuel
Moore, it was concluded, that a horse, moving at the rate of
three miles an hour, can exert a force of 80 lbs. Unluckily,
we are not sufficiently acquainted with the nature of the experi-
ments and observations from which these deductions were made,
to institute an accurate comparison of their results. Neither
of them ought to express what a horse can draw upon a car-
riage; because in that case friction only is to be overcome (after
the load is once put into motion); so that a middling horse,
well applied to a cart, will often draw much more than 1000
lbs. The proper estimate would be that which measures the
weight that a horse would draw up out of a well; the animal
acting by a horizontal line of traction turned into the vertical
direction by a simple pulley, or roller, whose friction should be
reduced as much as possible.
23. Mr. Tredgold, in his valuable publication on Rail-roads,
has directed his attention to the subject of "horse power." The
following is his expression for the power of a horse, 250
250 d v (1— —-=)
บ
V
V
v (1 — 1 - );
འ
V
and-
for the day's work in lbs.
14.7
1 + n
raised one mile; d being the hours which the horse works in a
day, and the weight of the carriage to that of the load as n : 1.
He also gives for the greatest speed in miles per hour,
d
when the horse is unloaded. These expressions must, at pre-
sent, be regarded as tentative. The following is his tablet of
the comparison of the duration of a horse's daily labour and
maximum of velocity, unloaded.
400
ANIMAL STRENGTH HORSES.
Duration of labour.
Hours.
1
234
1 6
5
7
8
9
10
}
I
4/2/2
5
5/1/1
J
J
I
1
1
Miles per hour.
2
3
3/
4
}│
11
1
JI
I
--
1
1 1
2
1 1/
T-10 15 20
35
1
11
1
1 / 5
1
I
1
Î│
Proportional expense.
1 1/2 or 1.125
1
1
}
or 1.125
I
1 or 1·0285
or 1.333
or 1.8
1
1
Taking the hours of labour at 6 per diem, the utmost that
Mr. Tredgold would recommend, the maximum of useful effect
he assigns at 125 lbs. moving at the rate of three miles per
hour, and regarding the expense of carriage, in that case, as
unity; then,-
1
Catalan
PAN
1
Agenda
I
SA
# 1
Max. velocity unloaded
in miles per hour.
14.7
10.4
8.5
7.3
6.6
6.0
5.5
5.2
4.9
4.6
1
gal
K
f
}
J
Ad
Moving force.
166 lbs.
125
104
83
62
412
36
That is, the expense of conveying goods at 3 miles per hour
being 1; the expense at 4½ miles per hour, will be 1; and
so on, the expense being doubled when the speed is 5 miles.
per hour.
24. Thus, according to Mr. Tredgold, we have for the day
of 6 hours 2250 lbs. raised one mile. And Mr. Bevan, who has
made many experiments on the force of traction to move canal
boats on the Grand Junction Canal, found the force of traction
80 lbs., and the space travelled in a day 26 miles; hence, it is
only equivalent to 26 × 80 2080 lbs. raised one mile for the
day's work; the rate of travelling being 2-45 miles per hour;
and the result a little less than Mr. Tredgold's, the difference
probably arising from the deviation of the angle of the cate-
nary from the horizon.
25. The following experimental data from Mr. Bevan also
deserve attention.
"In the period from 1803 to 1809, I had the opportunity
of ascertaining correctly the mean force exerted by good
horses in drawing a plough; having had the superintendence
ANIMAL STRENGTH: HORSES.
401
of the experiments on that head at the various ploughing matches
both at Woburn and Ashridge, under the patronage of the Duke
of Bedford and the Earl of Bridgewater. I find among my
memoranda the result of eight ploughing matches, at which
there were seldom fewer than seven teams as competitors for
the various prizes.
The first result is from the mean force of each horse
in six teams, of two horses each team, upon
light sandy soil
The second result is from seven teams of two horses
each team, upon loamy ground, near Great Berk-
hampstead
The third result is from six teams of four horses each
team, with old Hertfordshire ploughs
The fourth result is from seven teams of four horses
each team upon strong stony land (improved
ploughs)
The fifth result is from seven teams of four horses
each team, upon strong stony land (old Hertford-
shire ploughs)
The sixth result is from seven teams of two horses
each team, upon light loam
The seventh result is from five teams of two horses
each, upon light, sandy land
The eighth result is from seven teams of two horses
each team, upon sandy land
sig
M
lbs.
156
154
127
167
193
177
170
160
The mean force exerted by each horse from fifty-two
teams, or one hundred and forty-four horses, = 163 pounds
each horse, and although the speed was not particularly
entered, it could not be less than the rate of two miles and a
half an hour.
"As these experiments were fairly made, and by horses of
the common breed used by farmers, and upon ploughs of va-
rious counties, these numbers may be considered as a pretty
accurate measure of the force actually exerted by horses at
plough, and which they are able to do without injury for many
weeks; but it should be remembered, that if these horses had
been put out of their usual walking pace, the result would have
been very different. The mean power of the draught-horse,
deduced from the above-mentioned experiments, exceeds the
calculated power from the highest formula of Mr. Leslie, which
is as follows: (15 — v)³ lbs. avoirdupois for the traction of
a strong horse, and (12) v) lbs. traction of the ordinary
horse, velocity in miles per hour."
2
v
2 M 2
402
PASSIVE STRENGTH.
SECTION II.-Passive Strength.
}
1
1. When a weight is supported by a bar resting on two ful-
crums, the pressure on each is inversely as its distance from the
weight.
2. The strain on a given point of a bar, placed horizontally,
and supported at both ends, from a weight placed on it, is pro-
portional to the rectangle of the segments into which the point
divides the bar.
1
3. Hence that strain is greatest in the middle of the bar or
beam; or, in other words, if the bar be prismatic, it is most
likely to break in the middle, or it is weakest there.
4. The strain produced by the weight of an equable bar, at
any point of its length, is equal to the strain produced by half
the weight of one segment acting at the end of a lever equal to
the other segment.
5. DEF. A substance perfectly elastic is initially extended
and compressed in equal degrees by equal forces, and proportion-
ally by proportional forces.
6. Der. The modulus of the elasticity of any substance is
a column of the same substance, capable of producing a pressure
on its base which is to the weight causing a certain degree of
compression, as the length of the substance is to the diminution
of its length.
The modulus of elasticity is the measure of the elastic force
of any substance.
ន
A practical notion of the modulus of elasticity may be rea-
dily obtained. Let be the quantity of a bar of wood, iron or
other substance, an inch square and a foot in length would be
extended or diminished by the force; and let / be any other
length of a bar of equal base and like substance; then
1 : 7 :: . : A, or 7 & A, the extension or diminution in the
length 7.
The modulus of elasticity is found by this analogy as the
diminution of the length of any substance is to its length, so is
the force that produced that diminution to the modulus of elas-
ticity. Or, denoting the weight of the modulus in lbs. for a
base of an inch square by m; it is
:f::1:m
M.
f
&
!
And if w be the weight of a bar of the substance one inch square
and 1 foot in length; then, if м be the height of the modulus
of elasticity in feet, we have
f
W E
PASSIVE STRENGTH: MODULUS OF ELASTICITY.
403
7. When a force is applied to an elastic column of a rectangu-
lar prismatic form in a direction parallel to the axis, the parts.
ncarest to the line of direction of the force exert a resistance in
an opposite direction; those particles which are at a distance
beyond the axis equal to a third proportional to the depth, and
twelve times the distance of the line of direction of the force,
remain in their natural state; and the parts beyond them act in
the direction of the force.
8. The weight of the modulus of the elasticity of a column
being m, a weight bending it in any manner f, the distance of
the line of its application from any point of the axis D, and the
depth of the column, d, the radius of curvature will be
d² m
12 D f
9. The distance of the point of greatest curvature of a pris-
matic beam, from the line of direction of the force, is twice the
versed sine of that arc of the circle of greatest curvature, of
which the extremity is parallel to that of the beam.
When the force is longitudinal, and the curvature inconsider-
able, the form coincides with the harmonic curve, the curvature
being proportional to the distance from the axis; and the dis-
tance of the point of indifference from the axis becomes the
secant of an arc proportional to the distance from the middle of
the column.
10. If a beam is naturally of the form which a prismatic beam
would acquire, if it were slightly bent by a longitudinal force,
calling its depth, d, its length, I, the circumference of a circle
of which the diameter is unity, c, the weight of the modulus of
clasticity, m, the natural deviation from the rectilinear form, A',
and a force applied at the extremity of the axis, f, the total de-
viation from the rectilinear form will be
d's c³ A m
Δ'
d² c³ m — 12 l² ƒ
SCHOLIUM. It appears from this formula, that when the
other quantities remain unaltered, A' varies in proportion to
A, and if A o, the beam cannot be retained in a state of in-
flection, while the denominator of the fraction remains a finite
quantity; but when dem = 12 l f, A' becomes infinite,
whatever may be the magnitude of A, and the force will over-
power the beam, or will at least cause it to bend so much as to
derange the operation of the forces concerned. In this case f=
d 2 M
(4-c)°.
dⓇ
.8225 m, which is the force capable of holding
19
12'
the beam in equilibrium in any inconsiderable degree of cur-
-
1
404
}
!
STRENGTH OF ELASTIC SUBSTANCES.
vature. Hence the modulus being known for any substance, we
may determine at once the weight which a given bar nearly
straight is capable of supporting. For instance, in firwood, sup-
posing its height 10,000,000 feet, a bar an inch square and ten
feet long may begin to bend with the weight of a bar of the same
thickness, equal in length to .8225 ×
1
120 X 120
feet, or 571 feet; that is, with a weight of about 120 lbs. ; neg-
lecting the effect of the weight of the bar itself. In the same
manner the strength of a bar of any other substance may be deter-
mined, either from direct experiments on its flexure, or from the
ι
m 18
sounds that it produces. Ifƒ
"
ጎ d³
d
✓ (.8225 n) .907n; whence, if we know the force re-
quired to crush a bar or column, we may calculate what must
be the proportion of its length to its depth, in order that it may
begin to bend rather than be crushed.
f
(~ (3/4). 2) — 1).
m
× 10,000,000
11. When a longitudinal force, is applied to the extremities
of a straight prismatic beam, at the distance D from the axis,
the deflection of the middle of the beam will be D. (sec. arc
.8225 n, and
12. The form of an elastic bar, fixed at one end, and bearing
a weight at the extremity, becomes ultimately a cubic parabola,
and the depression is of the versed sine of an equal arc, in the
smaller circle of curvature.
2
3
13. The weight of the modulus of the elasticity of a bar is to
a weight acting at its extremity only, as four times the cube of
the length to the product of the square of the depth and the de-
pression.
14. If an equable bar be fixed horizontally at one end, and
bent by its own weight, the depression at the extremity will be
half the versed sine of an equal arc in the circle of curvature at
the fixed point.
15. The height of the modulus of the elasticity of a bar, fixed
at one end, and depressed by its own weight, is half as much
more as the fourth power of the length divided by the product
of the square of the depth and the depression.
16. The depression of the middle of a bar supported at
both ends, produced by its own weight, is five-sixths of the
versed sine of half the equal arc in the circle of least cur-
}
vature.
17. The height of the modulus of the elasticity of a bar, sup-
ported at both ends, is of the fourth power of the length,
32
STRENGTH OF ELASTIC SUBSTANCES.
405
divided by the product of the depression and the square of the
depth.
From an experiment made by Mr. Leslie on a bar in these
circumstances, the height of the modulus of the elasticity of
deal appears to be about 9,328,000 feet. Chladni's observa-
tions on the sounds of fir wood afford very nearly the same
result.
18. The weight under which a vertical bar not fixed at the
end may begin to bend, is to any weight laid on the middle of
the same bar, when supported at the extremities in a horizontal
position, nearly in the ratio of 5 of the length to the de-
pression.
15
10000
19. DEF. The stiffness of bodies is measured by their re-
sistance at an equal linear deviation from their natural po-
sition.
20. The stiffness of a beam is directly as its breadth, and as
the cube of its depth, and inversely as the cube of its length.
21. The direct cohesive or repulsive strength of a body is in
the joint ratio of its primitive elasticity, of its toughness, and
the magnitude of its section.
Though most natural substances appear in their intimate con-
stitution to be perfectly elastic, yet it often happens that their
toughness with respect to extension and compression differs
very materially. In general, bodies are said to have less tough-
ness in resisting extension than compression.
22. The transverse strength of a beam is directly as the
breadth and as the square of the depth, and inversely as the
length.
SCHOLIUM. If one of the surfaces of a beam were incom-
pressible, and the cohesive force of all its strata collected in
the other, its strength would be six times as great as in the
d² m
natural state; for the radius of curvature would be
Dƒ'
which could not be less than twice as great as in the natural
state, because the strata would be twice as much extended, with
the same curvature, as when the neutral point is in the axis;
and/would then be six times as great.
23. DEF. The resilience of a beam may be considered as
proportional to the height from which a given body must fall
to break it.
24. The resilience of prismatic beams is simply as their
bulk.
25. The stiffest beam that can be cut out of a given cylinder
is that of which the depth is to the breadth as the square root
of 3 to 1, and the strongest as the square root of 2 to 1; but the
53
406
STRENGTH OF ELASTIC SUBSTANCES.
most resilient will be that which has its depth and breadth
equal.
26. Supposing a tube of evanescent thickness to be expanded
into a similar tube of greater diameter, but of equal length, the
quantity of matter remaining the same, the strength will be in-
creased in the ratio of the diameter, and the stiffness in the ratio
of the square of the diameter, but the resilience will remain.
unaltered.
27. The stiffness of a cylinder is to that of its circumscribing
prism as three times the bulk of the cylinder to four times that
of the prism.
28. If a column, subjected to a longitudinal force, be cut out
of a plank or slab of equable depth, in order that the extension
and compression of the surfaces may be initially everywhere
equal, its outline must be a circular arc.
29. If a column be cut out of a plank of equable breadth, and
the outline limiting its depth be composed of two triangles,
joined at their bases, the tension of the surfaces produced by a
longitudinal force will be everywhere equal, when the ra-
dius of curvature at the middle becomes equal to half the
length of the column; and in this case the curve will be a cycloid.
When the curvature at the middle differs from that of the
cycloid, the figure of the column becomes of more difficult in-
vestigation. It may, however, be delineated mechanically,
making both the depth of the column and its radius of curvature
proportional always to ✔ ɑ. If the breadth of the column vary
in the same proportion as the depth, they must both be every-
where as the cube root of a, the ordinate.-(Young's Nat. Phil.
vol. ii.)
30. The modulus of elasticity has not yet been ascertained in
reference to so many subjects as could be wished. Professor
Leslie exhibits several, however, as below. That of white
marble is 2,150,000 feet, or a weight of 2,520,000 pounds avoir-
dupois on the square inch; while that of Portland stone is only
1,570,000 feet, corresponding on the square inch to the weight
of 1,530,000 pounds.
White marble and Portland stone are found to have, for every
square inch of section, a cohesive power of 1811 lb. and 857 lb.;
wherefore, suspended columns of these stones, of the altitude of
1542 and 945 feet, or only the 1394th and 1789th part of their
respective measure of elasticity, would be torn asunder by their
own weight.
31. Of the principal kinds of timber employed in building
and carpentry, the annexed table will exhibit their respective
Modulus of Elasticity, and the portion of it which limits their
cohesion, or which lengthwise would tear them asunder.
MODULUS OF ELASTICITY.
407
Teak
Oak
Sycamore
Beech
Ash
Elm
Memel fir
Christiana deal
Larch
A
Sycamore
Beech
J
de
S
1
I
1
Ash
Elm
Memel fir
Christiana deal
Larch
1
1
1
1
f
-
1
-
1
1
6,040,000 feet
4,150,000 feet
3,860,000 feet
4,180,000 feet
4,617,000 feet
5,680,000 feet
8,292,000 feet
8,118,000 feet
5,096,000 feet
The professor gives also this tabular view of their absolute
cohesion, or the load which would rend a prism of an inch
square; and the altitude of the prism which would be severed
by the action of its own weight.
7
Teak
Oak
12,915 lb.
11,880 lb.
9,630 lb.
12,225 lb.
14,130 lb.
9,720 lb.
9,540 lb.
12,346 lb.
12,240 lb.
K
Kat
ܚ
Go
-
G
-
J
B
- 168th
144th
108th
107th
109th
146th
205th
J
146th
121th
36,049 feet.
32,900 feet.
35,800 feet.
38,940 feet.
42,080 feet.
39,050 feet.
40,500 feet.
55,500 feet.
42,160 feet.
32. The modulus of the elasticity of hempen fibres has not
been ascertained, but may probably be reckoned about 5,000,000
feet. Their cohesion is, for every square inch of transverse
section, 6,400 lb. The best mode of estimating the strength
of a rope of hemp is to multiply by 200, the square of its num-
ber of inches in girth, and the product will express in pounds
the practical strain it may be safely loaded with; for cables
multiply by 120, instead of 200. The ultimate strain is pro-
bably double this; as will appear from the account following
of Du Hamel's experiments. If yarns of 180 yards long be
worked up into a rope of only 120 yards, it will lose one-
fourth of its strength, the exterior fibres alone resisting the
greatest part of the strain. The register cordage of the late
Captain Huddart exerts nearly the whole force of the strands,
since they suffer a contraction of only the eighth part in the
process of combining.
33. For the utmost strength that a rope will bear before it
breaks, a good estimate will be found by taking one-fifth
of the square of the girth of the rope, to express the tons
it will carry.
This is about double the rule for practice which
we have just given (art. 32); and is, even for an ulterior mea-
sure, too great for tarred cordage, which is always weaker than
white.
408
STRENGTH OF CORDAGE.
The following experiments were made by Mons. Du Hamel,
at Rochfort, on cordage of 3-inch French circumference, made
of the best Riga hemp, August 8th, 1741.
ľ
White.
Broke with strain of 4500 pounds
4000
4800
4600
5000
5000
3880
4000
4200
1746 April 14th
1747 May 18th
1747 October 21st
1748 June 19th
1748 October 2d
1749 Sept. 25th
1
August 25th, 1743.
White bore.
2645
2762
2710
2575
2425
2917
1
September 23, 1746.
A parcel of white and tarred cordage was taken out of a quan-
tity which had been made Feb. 12th, 1746.
It was laid up in the Magazine, and comparisons were made
from time to time, as-
S
Be
W
J
Tarred.
3400 pounds.
3300
3258
3500
3400
3400
3000
2700
2800
Tarred.
2312
2155
2050
*1752
1837
1865
134,256 lb.
72,064 lb.
55,872 lb.
19,096 lb.
M. Du Hamel says, that it is decided by experience, that white
cordage in continued service is one-third more durable than
tarred; secondly, it retains its force much longer while kept in
store; thirdly, it resists the ordinary injuries of the weather
one-fourth longer. These observations deserve the attention of
practical men.
34. The metals differ more widely from each other in their
elastic force and cohesive strength, than the several species of
wood or vegetable fibres. Thus, the cohesion of fine steel is
about 135,000 lb. for the square inch, while that of cast lead
amounts only to about the hundred and thirtieth part, or 1800 lb.
According to the accurate experiments of Mr. George Rennie
in 1817, the cohesive power of a rod an inch square of different
metals, in pounds avoirdupois, with the corresponding length in
feet, is as follows:
Cast steel
Swedish malleable iron
ditto
English
Cast iron
Gad
J
+
Wha
Difference.
333
607
660
823
588
1052
I
39,455 feet.
19,740 feet.
16,938 feet.
6,110 feet.
MODULUS OF COHESION.
409
Cast copper
Yellow brass
Cast tin
Cast lead
N
}
1
1
f
}
*
I
19,072 lb.
17,958 lb.
4,736 lb.
1,824 lb.
-
P
B
5,003 feet.
5,180 feet.
- 1,496 feet.
348 feet.
w
It thus appears, as Professor Leslie remarks, that a vertical
rod of lead 348 feet long, would be rent asunder by its own
weight. The best steel has nearly twice the strength of English
soft iron, and this again is about three times stronger than cast
iron. Copper and brass have almost the same cohesion as cast
iron. This tenacity is sometimes considerably augmented by
hammering or wire-drawing, that of copper being thus nearly
doubled, and that of lead, according to Eytelwein, more than
quadrupled. The consolidation is produced chiefly at the sur-
face, and hence a slight notch with a file will materially weaken
a hard metallic rod. English malleable iron has 7,550,000
feet for its modulus of elasticity, or the weight of 24,920,000
b. on the square inch, while cast iron has 5,895,000 feet, and
18,421,000 lb. Of other metals, the modulus of elasticity is
probably smaller, but has not yet been well ascertained.
35. The Longitudinal Compression which any column suf-
fers, is at first equal to the dilation occasioned by an equal and
opposite strain, being in both cases proportional to the modulus
of elasticity. But while the incumbent weight is increased, the
power of resistance likewise augments, as long as the column
withstands flexure. After it begins to bend, a lateral disruption
soon takes place. A slender vertical prism is hence capable of
supporting less pressure than the tension which it can bear.
Thus, a cubic inch of English oak was crushed only by the load
of 3860 lb., but a bar of an inch square and 5 inches high gave
way under the weight of 2572 lb. It would evidently have
been still feebler if it had been longer. On the other hand, if
the breadth of a column be considerable in proportion to its
height, it will sustain a greater pressure than its cohesive power.
Thus, though the cohesion of a rod of cast iron of a quarter of
an inch square is only 300 lb., a cube of that dimension will re-
quire 1440 lb. to crush it.
In general, while the resisting mass preserves its erect form,
the several sections are compressed and extended by additional
weight, and their repellant particles are not only brought nearer,
but multiplied. This repulsion is likewise increased by the la-
teral action arising from the confined ring of detrusion. The
primary resistance becomes hence greatly augmented in the
gress of loading the pillar.
pro-
36. The most precise experiments on this subject seem to be
2 N
410
LONGITUDINAL COMPRESSION.
those of Mr. Rennie. The weights required to crush cubes of
the quarter of an inch of certain metals, are as follow:
Iron cast vertically
11,140 lb.
10,110 lb.
7,318 lb.
966 lb.
483 lb.
Iron cast horizontally
Cast copper
Cast tin
Cast lead
S
Cubes of an inch are crushed by the weights annexed:
Elm
1,284 lb.
White deal
English oak
Craigleith freestone
Cubes of an inch and a half, and consequently presenting a
section of 24 times greater than the former, might be expected
to resist compression in that ratio. They are crushed, however,
with loads considerably less.
Red brick
Yellow baked brick
Fire brick
-
1
Purbeck limestone
Cornish granite
Peterhead granite
Aberdeen blue granite.
J
A
Craigleith stone, direction of the strata
Ditto, across the strata
White statuary marble
White-veined Italian marble
1
}
1
1
I
Band
#
-
20,610 lb.
14,302 lb.
18,636 lb.
24,536 lb.
These facts show the comparative firmness of different mate-
rials, but it is to be regretted, that such results are not of much
practical value, since they are confined to a very narrow scale,
and applicable only to cubical blocks. While the breadth re-
mains the same, the resistance appears to depend on some unas-
certained ratio of the altitude of the column.
d
1,928 lb.
3,860 lb.
8,688 lb.
J
1,817 lb.
2,254 lb.
3,864 lb.
15,560 lb.
12,346 lb.
13,632 lb.
21,783 lb.
Nay, as Professor Leslie observes, the absolute height it-
self has probably a material influence on the effect. Thus,
from some experiments made in France, it appears, that prisms
of seasoned oak, two inches square, and two, four, or six feet
high, would be crushed by the vertical pressures of 17,500
lb., 10,500 lb., and 7,000 lb. ; but, if four inches square, and
of the same altitudes, they would give way under loads of
only 80,000 lb., 70,000 lb., and 50,000 lb. In the first set of
trials, the mean cohesive power amounts to 130,000 lb., and
in the second to 520,000 lb. The vertical support is there-
fore greatly inferior to these limits. When the length of
the pillar exceeds 36 times its breadth, the resistance to lon-
MODULUS OF COHESION.
411
I
gitudinal compression appears to be diminished 18 times.-(Les-
lie's Elements, and Duhamel in Mem. Paris. Acad.)
37. Mr. B. Bevan has favoured the author with a tabular
view of his results with regard to the modulus of cohesion, or
the length in feet of any prismatic substance required to break
its cohesion, or tear it asunder.
}
Bevan's Results.
Tanned cow's skin
calf skin
horse skin
cordovan
sheep skin
Untanned horse skin
Old harness of 30 years
Hempen twine
Catgut, some years old
Garden matting
Writing-paper, foolscap
Brown wrapping-paper, thin
Bent grass, (holcus)
Whalebone
Bricks, (Fenny Stratford)
(Leighton)
Ice
Platinum
Gold (pure)
Steel
Bar-iron
Lance wood
Willow
Oak
B
1 1
Kateg
}
}
Ditto
Yellow pine
Ditto
Finland deal
Mahogany
Rose wood
Oak, dry
Fir bottom, 25 years old
Petersburg deal
I
1
1
-
(PRE
ست
Sa
1
1
Leicestershire slate
The following, also, exhibits Mr. Bevan's results as to the
modulus of elasticity.
·
1
1
1
-
BOND
*
1 1
400
ܪ
1
}
I
Going
↓
11
1
Sand
Jed
p
feet.
10,250
5,050
7,000
3,720
I
5,600
8,900
5,000
- 75,000
- 23,000
- 27,000
8,000
6,700
- 79,000
- 14,000
970
144
300
7,300
/
feet.
2,390,000
1,390,000
9,300,000
9,000,000
8,450,000
9,150,000
11,840,000
6,000,000
7,500,000
3,600,000
5,100,000
7,400,000
6,000,000
5,100,000
6,200,000
4,350,000
412
MODULUS OF COHESION, &c.
1
Satin wood
Lincolnshire bog oak
Lignum vitæ
Teak wood
Yew
Whalebone
Cane
Glass tube
Ice
Limestone
t
What
Dinton, Buck
Ketton
Jetternoe
Ryegate
Yorkshire paving
Cork
Slate, Leicestershire
Ka
I
J 1
G
1
I
I
Mad
I
}
1
feet.
2,290,000
- 1,710,000
- 1,850,000
4,780,000
· 2,220,000
- 1,000,000
· 1,400,000
- 4,440,000
- 6,000,000
G
2,400,000
-
· 1,600,000
635,000
621,000
· 1,320,000
3,300
7,800,000
www
Many other results are collected in Mr. Tredgold's Essay
on the Strength of Cast Iron, a work which may be consulted
with advantage on this and kindred topics.
Indeed, we have not aimed at more in this section than a
brief summary of the leading principles involved in the con-
sideration of passive strength, and a corresponding exhibition
of the best ascertained facts. The subject is one of great and
growing interest to all concerned in the erection of extensive
structures. Such may consult, in addition to the volume just
mentioned, Tredgold's Carpentry, Barlow on the Strength
of Timber, Gregory's Mechanics, Vol. I., the Lectures of Dr.
T. Young and Professor Leslie, already quoted, and Girard,
Traité Analytique de la Résistance des Solids.
1
TABLE I.—Useful Factors, extended to several places of
Decimals, in which p represents the Circumference of a
Circle, whose Diameter is 1.
Then
p
2p
4p
p
Ap
3.
6.
• P
`i žp
1
1
36 op
1
22/24
P
1
4 p
√/2
✓ &
1
15 √
p
1
•
•
SUPPLEMENTARY TABLES.
3.1415926535897932384626433832795028841971
6939937510582097494459230781640628620899
8628034825342117067982148086513272306647
54
0938446 +
6.283185307179586476925286766559
12.566370614359172953850573533118
1.570796326794896619231321691639
0.785398163397448309615660845819
4·188790204786390984616857844372
0.523598775598298873077107230546
0.392699081698724154807830422909
0.261799387799149436538553615273
0.008726646259971647884618453842
0.3183098861837906715377675267450
0·636619772367581343075535053490
p√2...
4-4428829381583662470158809900605
p√½... 0·2214414690791831235079404950302
0.2250790790392765173887997751
1.273239544735162686151070106980
0.0795774715459476678844418816862
1·4142135623730950488016887242097 nearly
0.707106781186547524400844362104
/2... 0·4501581580785530347775995955
√p..
1.772453850905516027298167483341
2 N 2
413
414
SUPPLEMENTARY TABLES.
X √ P
2√p
✓²
p
1
2√
p
*√
p
1101100
8
2√
p
1
22
K
22-12-135-
6p³
•12-1
6 p
13
p
2
1
1
p
6р
360
p
21201·201
p
6p³
36p2
36p
36p..
t√p
•
1
0.886226925452758013649083741670
3.544907701811032054596334966682
1.253314137315500251207882642402
0.797884560802865355879892119868
0.564189583547756286948079451560
1.128379167095512573896158903120
0.282094791773878143474039725780
9.869604401089358618834490999876
0.101321183642337771443879463209
0.050660591821168885721939731604
0.016886863940389628573979910534
1.128379167095512573896158903120
0.094031597257959381158013241926
0·070523697943469535868509931445
0.053051647697298445256294587790
114.591559026164641753596309628200
2.094395102393195492308428922186
0.130899693899574718269276807636
1.909859317102744029226605160470
1.2407009819, &c.
59.217626406536151713006945999256
3.89777707, &c.
113.097335529232556584655161798062
4-83597586204, &c.
0.221556731363189503412270935418
J
415
SUPPLEMENTARY TABLES.
}
1
2 p
p
6
T
•
0.805995977007, &c.
√231 .. 15.1986841535706636
√282 16-7928556237466 + &c.
10152. 100-75713374
Given.
Diam.
282
•785398, &c.
√277·274=16.65154647474
Log. of p=0·49714987269413385435
1.00
•25
•5
*75
2.
•25
•5
•75
TABLE II.—A Table of Circles, from which, knowing the
diameters, the areas, circumferences, and sides of equal
squares are found.
3.
•25
•5
.75
4.
•25
.5
5.
•25
•5
•75
6.
75
•25
.5
.75
77.
•25
.5
:75
8.
0.159154943091895335768883763372
•25
•5
*75
?
18.948708, &c.
Area.
0.7853981
1.22718463
1.76714586
2.40528-187
3.14159265
3.97607820
4.90873852
5.93957361
7.06858347
8-29576810
9.62112750
11·04466167
12.56637061
14-18625432
15-90431280
17.72051606
19.63495408
21.64753687
23-75829444
25.96722677
28-27433388
30-67961575
33.18307240
35-78470382
38-48451001
41.28249096
44.17864669
47-17297718
50-26548245
53-45616249
56.74501730
60.13204688
Required.'
Circumference.
3.14159265
3.92699081
4-71238898
5-49778714
6.28318530
7.06858347
7-85398163
8.63937979
9.42477796
10.21017612
10-99557428
11·78097245
12.56637061
13-35176877
14.13716694
14.92256510
15.70796326
16-49336143
17-27875959
18.06415775
18.84955592
19-63495408
20-42035224
21.20575041
21-99114857
22.77654673
23-56194490
24-34734306
25.13274122
25.91813939
26.70353755
27-48893571
side of cq. sq.
0-88622692
1.10778365
1.32934038
1.55089711
1.77245384
1.99401058
2.21556731
2-43712404
2.65868077
2.88023750
3.10179423
3-32335096
3-54490769
3.76646442
3.98802116
4.20957789
4.43113462
4-65269135
4-87424808
5.09580482
5.31736155
5.53891828
5.76047501
5.98203174
6.20358847
6.42514520
6.64670193
6.86825866
7·08981539
7-31137213
7.53292886
7.75448559
416
SUPPLEMENTARY TABLES.
1
"
Given.
Diam.
9.
•25
*5
*75
10.
*25
.5
11.
*25
·5
•75
12.
*25
.5
13.
14.
•25
•5
*75
15.
.25
.5
16.
75
•25
.5
17.
18.
75
•25
+5
•75
19.
•25
+5
•75
75
20.
•25
-5
75
21.
•25
·5
•75
22.
75
•25
.5
75
*25
.5
.75
.25
+5
.75
Area.
63.61725123
67-20063035
70-88218424
74-66191290
78-53981633
82-51589454
86-59014751
90-76257525
95-03317777
99-40195505
103-86890711
108.43403393
113.09733553
117-85881189
122-71846303
127·67628893
132-73228961
137-88646506
143-13881527
148-48934026
153.93804002
159-48491455
165·12996385
170-87318792
176-71458676
182.65416028
188-69190875
194-82783190
201·06192982
207-39420252
213-82464998
220-35327221
226.98006922
233-70504099
210-52818753
247-44950885
264-46900494
266-58667578
268.80252140
276·11654180
283-52873699
291-03910692
298-64765163
306·35437111
314-15926535
322-06233437
330-06357816
338-16299672
346-36059005
354-65635814
363-05030101
371-54241865
380-13271108
388-82117826
397·60782021
406-49263694
F
Required.
Circumference. side of eq. sq.
28.27433388
29-05973204
29.84513020
30-63052837
31.41592653
32-20132469
32-98672286
33-77212102
34-55751918
35-34291735
36.12831551
36.91371367
37.69911184
38-48451000
39-26990816
40.05530633
40.84070449
41-62610265
7.97604232
8.19759905
8-41915578
8.64071251
8.86226925
9.08382598
9.30538270
9.52693944
9.74849617
9.97005290
10-19160964
10-41316637
10.63472310
10.85627983
11.07783656
11.29939329
11.52095002
42.41 150082
43.19689898
43-98229714
11.74250675
11.96406348
12.18562021
12·40717695
44-76769531
12.02873368
45-55309347
12.85029041
46-33849163
13.07184714
47-12388980
13-29340388
47-90928796
13-51496061
48.69468613
13-73651734
49-48008429
13-95807407
50.26548245
14-17963080
51.05088062
14-40118753
51-83627878
14-62274426
52.62167694
14.84430099
53.40707511
15.06585772
54-19247327
15.28741446
54.97787143
15.50897119
55-76326960
15-73052792
56-54866776
15-95208465
57-33406592 16-17364138
58-11946409
16-39519811
58-90486225
16.61675484
59-6902604 1
16.83831157
60-47565858
17-05986830
17.28142503
61-26105674
62-04645490
17-50298177
62-83185307
17:72453850
63-61725123
17·94609524
64-40264939
18·16765197
65.18804756
18-38920870
65-97344572
18.61076543
66-75884388
18-83232216
67-54424205
19-05387889
68-32964021
19.27543562
69-11503837
19-49699235
69.90043654
19-71854908
70-68583470 19-94010581
71-47123286
20-16166255
J
J
SUPPLEMENTARY TABLES.
417
Given.
Diam.
23.
Arca.
415-47562843
72-25663103 20-38321928
424-55679467
73-04202919
20.60477601
433.73613573
73-82742735
20.82633274
443-01365154
74·61282552
21.04788945
452.38934212
75.39822368
21.26944618
461.86320745
76.18362184
21-49100291
471-43524757
76.96902001
21.71255964
481-10546239
77-75441817
21·93411637
490-87385212
78-53981634
22.15567313
500-74041655
79.32521450
22-37722986
510-70515575
80.11061266
22.59878659
520-76806971
80-89601083
22-82034332
530-92915845
81-68140899
23.04190006
541-18842196
82.46680715
23-26345679
551.54586024
83.25220532
23.48501352
562-00147328
84-03760348
23-70657025
572.55526112
84.82300164
23.92812698
583·20722369
85.60839981
24·14968371
593·95736105
86-39379797
24-37124044
604-80567318
87-17919613
24-59279717
87-96459130
24-81435390
615-75216010
626-79682177
88-74999246
25.03591063
637-93965822
89.53539062
25.25746737
649-18066943
90·32078879
25.47902410
660.51985541
91·10618695
25·70058083
671-95721616
91-89158511
25.92213756
683-49275169
92-67698328
26.14369429
695.12646198
93-46238144 26-36525102
706-85834706
94-24777960
26-58680776
718-68840688
95-03317777
26.80836449
730-61664148
95-81857593
27-02992122
742-64305085
96-60397409
27-25147794
754-76763502
97-38937226
27·47303468
766-99039394
98.17477042
27-69459141
779-31132762
98.96016858
27-91614814
791-73043607
99-74556675
28.13770488
804-24771932
100-53096491
28.35926161
816-86317729 101-31636307
28-58081834
829.57681005
102-10176124
842-38861759
102-88715940
855-29859994 103.67255756
868-30675696 104-45795573
881·41308881
105-24335389
106-02875205
894-61759542
907-92027688
106-81415022
921-32113305
107:59954838
934-82016398 108.38494654
948-41736968
109-17034471
962-11275016
109-95574287
975-90630540
110-74114103
989-79803541 111·52653920
1003-78794019
36.
1017·87601975
•25 1032-06227407 113.88273369 32-12572604
1046-34670316 114.66813185 32-34728277
+5
·75 1060-72930703 115-45353001 32-56883950
*25
•5*
24.
•25
•5
*75
25'
•25
•5
26*
27.
•25
•5
*75.
28.
*25
•5
29*
75
•25
.5
.75
30.
75
•25
•5
31.
32.
75
•25
.5
33.
•25
.5
•75
75
*25
·5
*75
75
34*
35*
*25
•5
75
•25
.5
75
*25
•5
*75
Required.
Circumference. side of eq. sq.
28-80237507
29-02393180
29-24548853
29-46704526
29.68860199
29.91015872
30-13171545
30-35327219
30-57482892
30-79638565
31-01794239
31-23949912
31-46105585
31-68261258
112-31193736
113.09733552 31-90416931
418
SUPPLEMENTARY TABLES.
1
Given.
Diam.
37.
:25
•5
•75
Arca.
side of eq. sq.
1075-21008569
116.23892818
32-79039623
1089-78903909
117·02432634
33-01195296
33.23350970
117-80972450
118.59572267 33-45506643
119.38052083 33.67662316
120·16591899 33.89817989
1104-46616727
1119-24147022
1134-11494794
1149-08660043
1164-15642768
1179-32442971
1194-59060653 122.52211348
1209-95495809
120.95131716
34-11973662
121-73671532
34-34129335
34.56285008
123-30751165 34-78440681
124·09290981
1225-41748449
35.00596354
1240-07818552 124-87830797
35.22752027
1256.63706144
125.66370614
35-44907701
1272-39411208
126-44910430
35.67063374
127-23450246
35.89219048
36-11374721
128·01990063
1288-24933751
75 1304-20273770
1320-25431266
1336·40406240 129.59069695
128.80529879 36-33530394
36.55686067
36.77841740
130-37609512
1352-65198690
1368-99808617
131·16149328
36-99997413
131.94689144
37.22153086
1385-44236022
1401.98480903
132-73228961
37-44308759
1418-62543261
133-51768777
37-66464432
1435-36423096
134-30308593
37.88620105
1452.20120412
135·08348410
38·10775779
•25
1469-13635202
135-87388226
38-32931452
136.65928042
38.55087125
•5
1486-16967468
75 1503·30117212
1520-53084433
137-44467859
38-77242798
138-23007675
38.99398471
1537-85869131
139-01547491
39.21554144
139-80087308
1556-28471306
1572-80890957
140-58627124
1590-43128088
141-37166940
1608.15182692
142-15706757
1625-97054775 142-94246573
1643-88744335
143.72786390
1661-90251374
144-51326206
1680.01575889
145.29866022
146.08405839
1698-22717880
1716-53677348
146·86945655
147-65485471
1734-94454294
1753-45048716
1772-05460615
148-44025288
149.22565104
1790-75689992
150-01104920
1809-55736847
150-79644797
151-58184553
152-36724369
*25 1828-45601175
•5 1847·45282982
•75 1866-54782267
1885-74099031
153·15264186
153-93804002
-25
1905-83233270 154·72343818
•5
1924-42184986
155-50883635
156-29423451
75 1943-90954179'
1963-49540848
•25 1983-17944995
157·07963268
157·96503084
5
2002.96166619
158-65042900
-75 2022-84205720
159-43582717
38.
*25
•5
39*
•25
•5
•75
40°
*25
.5
41.
•25
.5
*75
42.
•25
.5
.75
43.
44.
•25
•5
*75
45.
75
•25
+5
•75
46.
•25
•5
•75
47.
.25
-5
•75
48.
49.
50.
Required.
Circumference.
1
39-43709817
39.65865490
39.88021164
40-10176837
40.32332510
40-54488183
40-76643856
40.98799530
41.20955203
41-43110876
41.65266549
41.87422222
42.09577891
42·31733568
42.53889241
42.76044914
42.98200587
43.20356261
43-42511934
43.64667607
43.86823280
44.08978953
44-31134627
44-53290300
44-75445973
44.97601646
!
1
SUPPLEMENTARY TABLES.
419
I
:
Given.
Diam.
51.
Area.
2042-82062300
•5
+25 2062-89736352
2083-07227884
2103.34536893
2123.71663382
2144-18607346
2164-75368786
2185-41947703
2206-18344098
2227.04557969
164 14821615
164-93361431
165-71901247
166.50441064
167-28980880
2248-00589318
168-07520696
2269-06438143
168-86060513,
2290·22104447
169.64600329
2311-47588225
170-43140145
2332-82889481
171-21679962
2354-28008215
172-00219778
2375-82944427 172-78759594
2397-47698115
173-57299411
174-35839227
175-14379043
175·92918860
2419-22269280
2441·06657922
2463-00864041
2485·04887637
2507-18728710
2520-42387260
176-71458676
177-49998492
178-28538309
2551-75863288
179-07078125
2574-19156790
179-85617941
180.64157758
181-42697574
2596-72257781
2619-35196239
2642-07942166
2664.90505579
182.21237390
182.99777207
2687-82886464
183-78317023
2710-85084834
184-56856839
2733.97100678 185-35396656
2757-18933998
186-13936472
186.92476288
2780-50584795
2803·92058070.
282743338828
187-71016105
188-49555921
.25 2851·04442049
189-28095737
•5
2874-75362754
190-06635554
-75
2898-56100937
190·85175370
2922-46656592' 191-63715186
-25
2946-47029734
•5
2970-57220350
·75 2994-77228444
3019-07054010
52.
•25
*5
•75
53*
*25
•5
*75
54.
*25
•5
•75
55.
*25
•5
56.
57.
•25
.5
*75
58.
75
•25
•5
•75
59.
•25
-5
60.
61.
75
-25
.5
*75
62'
63*
75
-25
.5
•75
64.
192-42255003
193-20794819
193-99334635
194-77874452
3043-46697053 195-56414268
3067.96157576 196-34954084
3092-55435572
197·13493901
3117.24531051
3142-03444002
3166-92174434
3191-90722341
3216·99087728
•25 3242-17270581
+5
•75
•25
•5
•75
Required.
Circumference.
160-22122533
161·00662349
161.79202166
162-57741982
163-36281798
•
197-92033717
198-70573533
199-49113350
200-27653166
201·06192982
201-84732799
3267-45270920 202.63272615
3292-83088742 203-41812431
side of eq. sq.
45.19757319
45-41912992
45.64068665
45.86224338
46-08380012
46.30535685
46.52691358
46.74847031
46.97002704
47-19158377
47-41314050
47.63469723
47:85625386
48.07781069
48.29936743
48.52092416
48.74248089
48-96403763
49.18559436
49.40715109
49-62870782
49.85026455
50-07182128
50.29337801
50-51493474
50-73649147
50.95804820
51.17960494
51.40116167
51.62871840
51.84427513
52·06583186
52.28738859
52-50894532
52.73050205
52-95205878
53.17364552
53-39517225
53-61672898
53.83828572
54-05984245
54.28139918
54-50295591
54.72451264
54-94606937
55.16762610
55-38918283
55-61073956
55-83229629
56-05385303
56-27540976
56-49696649
56.71852322
56.94007995
57.16163668
57:38319341
K

420
SUPPLEMENTARY TABLES.
Given.
Diam.
65.
66.
Area.
3318-30724035
*25 3343-88176802
.5
•75
3369-55447036
3395·32534774'
3421-19439976
3447-16162652
3473-22702806
3499.39060438
3525-65235549
3552-01228735
212-05750411
212-84290227 -
3578-47038197
3605-02665737
3631-68110754
3658 43373248
3685-28453219
3712-23350667
213-62830044
214-41369860
215·19909676
215-98449493
216-76989309
3739-28065594
3766-42597994 217-55529125
3793-66947873 218-34068942
3821·01115229 219.12608758
3848-45100064 219·91148574
3875-98902375
220·69688391
3903-62522162
3931-35959426
3959-19214168
3987·12286386
4015-15176082
4043-27883254
221·48228207
222-26768023
223 05307840
223·83847656
224·62387472
225-40927289
4071-40507905
226-19467105
4099-82750030
226·98006921
4128-24909633 227-76546738
•25
•5
67'
•25
•5
68.
69.
*25
*5
70.
•25
•5
•75
71.
.25
.5
•75
72.
75
•25
.5
•75
73.
75
74.
75
•25
·5
75 4156-76886714
4185-38681274
•25
•5
•75
75.
-25
•5
•75
76.
1717.
78.
•25
.5
-75 4506.66374893
4536-45979178
•25 4566-35400937
·5
75
-25
•5
•75
.25
•5
75
Required.
Circumference.

62-25744151
62-47899824
62.70055497
62.92211170
63.14366843
63-36522516
63.58678189
63.80833862
64·02989536
64.25145209
228-55086554
64-47300882
64.69456555
229-33626370
4214-10293309 230-12166187
64-91612228
230·90706003
65.13767901
4242·91722821
4271-82969810
4300·84034275
231.69245819
65.35923574
232-47785636
65.58079247
4329-94916219
233-26325452
65.80234920
4359·15615638
234-04865268
66-02390593
4388-46132535
234·83405085
66·24546267
2
4417·86466910
66-46701940
235-61944901
236-40484717
4447-36618760
66.68857613
4476·96588088 237-19024534
66.91043287
67·13168960
237-97564350
238.76104166
67-35324633
67.57480306
239-54643983
240-33183799 67-79635979
241.11723615 68.01791652
241·90263432
68-23947325
242-68803248
68-46102998
243-47343064 68.68258671
244-25882881
68.90414344
245.04422697 69-12570018
69-34725691
69-56881364
69-79037037
4596-34640174
4626-43696897
4656-62571078
4686-91262745
4717-29771890
4747-78098511
4778-36242610
4809 042041835
4839-81983238
4870-79579767
side of eq. sq.
57.60475015
57.82630688
58.04786361
58-26942034
58.49097707
208.13051329 58-71253380
208.91591146
58.93409054
209-70130962
59.15564727
59.37720400
210-48670778
211.27210595
59.59876073
59.82031746
60-04187419
60·26343092
60-48498765
60-70654438
60.92810111
61.14965785
61-37121458
61.59277131
61.81432804
62.03588478
,
204·20352248
204-98892064
205-77431880
206-55971697
207-34511513
245·82962513
246·61502330
247-40042146
SUPPLEMENTARY TABLES..
421
i
Given.
Diam.
79.
Area.
side of eq. sq.
4901.66993776
70·01192710
248.18581962
248-97121779
•25
4932-74225260
70-23348383
•5
4963-91274221
249.75661595
70-45504056
.75 4995-18140659 250-34201411
70-67659729
80.
5026-54824574 251-32741228
70-89815403
252-11281044
71·11971076
5058-01325966
5089-57644835
252-89820860
71.34126749
253.68360677
71.56282422
5121.23781181
5152-99735004
254-46900493
71.78438096
5184-85506304
255-25440309
72.00593769
5216.81095081
256·03980126
72·22749442
5248-86501335 256-82579942
72-44905115
5281·01725068
257-61059758
72-67060788
5313-26766276
258-39599575
72.89216461
5345.61624962 259-18139391
73-11372134
5378·06301124
259.96679207
73·33527807
73-55683480
260-75219024
5410-60794764
5443-25105880
.25
261·53758840 73-77839153
•5
5475.99234474 262-32298656
73.99994827
74·22150500
75 5508-83180544
5541·76944092
5574-80525116 264-67918105
263·10838473
263-89378269
84.
74-44306173
74-66461846
5607-93923618 265-46457922
74-88617519
5641·17139596 266-24997738
75-10773192
5674-50173054 267·03537554
75-32928866
5707-93023987
267-82077371
75-55084539
5741-45692397
269-60617187
75.77240212
5775-08178284 269-39157003
75-99395885
5808-80481648
270·17696820
76.21551558
•25
5842·62602489
270-96236636
76-43707232
5876-54540807 271-74776452
76.65862904
272-53316269
76-88018578
.5
.75 5910-56296602
5944-67869876
5978-89260623 274·10395901
87.
273-31856085
77-10174251
77-32329924
6013-20468849 274-88935718
77-54485597
6047·61494552 275-67475534
77-76641270
6082-12337734 276-46015350
77·98796943
•25
6116-72998392 277-24555167
78-20952616
.5
6151.43476526
278-03094983
278-81634799
78-43103289
78.65263962
75 6186-23772138
6221-13885226 279-60174616
6256-13815792 280-38714432
•25
78-87419635
79-09575309
79-31730982
-5
6291-23563834 281-17254248
•75 6326-43129354 281-95794065
79.53886655
90*
6361·72512352
282-74333881
79-76042329
-25 6397-11712824 283-52873697
79·98198002
.5
6432·60730774 284-31413514
80-20353673
.75
6468-19566202 285-09953330
80-42509348
80·64669021
80-86820694
81-08976367
81-31132040
6647-61005499 289-02652412 81-53287713
6683-78745789
289-81192228
81-75443366
6720:06303556
290·59732044
81·97599060
82·19754733
·75. 6756-43678800
291-38271861
•25
•5
.75
.81.
•25
•5
•75
82.
•25
•5
•75
83.
.25
.5
85.
.25
•5
•75
86.
•25
-5
-75
88.
89.
75
91-
92.
6503-88219109 285-88493146
•25
6539-66689491
286-67032963
.5
6575-54977350 287-45572779
75 6611-53082686 288-24112595
.25
.5
Required.
Circumference.
55
20
422
SUPPLEMENTARY TABLES.
Given.
Diam.
93.
•25
•5
•75
94.
*25
+5
95.
•25
*5
•75
96.
•25
•5
•75
97.
75
•25
•5
98.
99.
.25
.5
.75
75
100°
.25
.5
75
.25
.5
Required.
Circumference.
Area.
6792-90871521
6829-47881719
292.16811677
292-95351493
.6866.14709394
293-73891310
6902.91354546
294-52431126
6939-77817177 295.30970942
6976-74097284 296-09510759
296·88050579
297·66590391
7013-80194367
7050-96109928
7088-21842465
7125·57992480
298-45130208
299-23670024
7163-02759971
300·02209840
7200.57944940
300-80749657
301-59289473
7238-22947387
7275-97767308
302-37829269
7313-82404707
303·16369106
7351-76859584
303-94908922
7389-81131940
304-73448738
7427-95221771
305·51988555
7466-19129079
306·30528371
7504-52853864
307-09068187
307-87608004
7542-96396126
7581-49755865
308-66147820
7620·12933081
309.44687635
7658-85927774
310-23227453
311-01767269
7697-68739946
7736-61369591
311.80307085
312-58846902
7775-63816715
7814-76081316
313-37386718
7853-98163397 314-15926535
7893-30062952 314-94466352
7932.71779985
•75 7972-23314494
side of eq. sq.
82·41910406
82·64066079
82.86221752
83.08377425
83·30533096
83.52688771
83-74844444
83.97000117
84-19155791
84-41311464
84-63467136
84-85622811
85·07778484
85-29934157
85.52089830
85-74245503
85.96401176
86-18556849
86-40712522
86 62868195
86-85023869
87·07179542
87-29335215
87-51490888
87.73646568
87-95802234
88-17957907
88-40113580
88.62269254
88.84424927
4
315-73006168 89-06580600
316.51545984 89-28736273
The preceding tables were computed with great care by the
author's esteemed friend, the late H. Goodwin, Esq. of Black-
heath, a gentleman whose indefatigable perseverance and re-
markable accuracy in reference to numerical computations
cannot be too highly characterized. They are inserted here to
supersede the necessity of consulting some erroneous tables of
the areas, &c. of circles recently put into circulation.
Kadak
Supposing the unit to be an inch, for example, the last table
exhibits the area, circumference, and side of equal square, cor-
responding to diameters varying by a quarter of an inch, from
1 inch to 100.
But the same table may also be made to serve for many inter-
mediate diameters, as well as for diameters beyond its apparent
reach, by simply recollecting that the circumferences and sides
of equal squares are as the diameters, while the areas are as
The squares of the diameters.
USE OF TABLE OF CIRCLES.
423
Thus, divide any
circumf.
or side of
equal sq.
20 2
3
by 5
6
10 &c.
the quotient
will give
the
circumf.
or
side of
equal
square
to
AC - Aft akomko
1
10
Or multiply by 2, 3, 4, 5, &c., the product will give the cir-
cumference or side of equal square, to 2, 3, 4, 5, &c. times the
assumed diameter.
Ex. 2. Find the area of a circle whose diameter is 8.1.
Arca to diameter 81, is 5152.99735004
Divide this by 10³= 100
The quotient is the area required 51·5299735004
the original
diameter.
1
For arcas, take 4, 4, 16, 27, &c. for 2, 3, 4, 1, &c. the as-
sumed diameter, and, contrarily 4 times, 9 times, 16 times, 25
times, 100 times, &c. an area, for the one which agrees to twice,
3 times, 4 times, 5 times, 10 times, &c. the diameter to which
the assumed area corresponds.
Example. Find the area of a circle whose diameter is 8 or
8.125.
Area to diameter 16.24 is 207.39420252
Divide this by 22 4
The quotient is the area required 51-84855063
Their diff. is the area of the ring 860207338
1
Ex. 3. The exterior and the interior diameters of a circular
ring, are 8.75 and 8.1. Required its area.
Area to diameter
875, from table 60·13204688
8.1, from above 51.52997350
Areas of circular rings thus found, being multiplied into their
thickness, will give their capacity, as in the rim. of a fly-wheel,
&c.; whence, knowing the weight of a cubic inch, or foot, the
weight of the whole becomes known: but the method already
given at p. 300, will usually be found preferable, unless great
accuracy be required; in which case recourse may be had to
this table.
424
SUPPLEMENTARY TABLES.
TABLE III-Relations of the Arc, Abscissa, Ordinate, and
Subnornal, in the Catenary; useful in the construction
of Calenarian equilibrated arches for bridges or powder
magazines.


,
Arc
A P
1 QC 100
2
3
0.02
25.00
31.22
22.17
19.51
0.08
24.97
31.74
23.02 19.36
0.18
24.94
42 32.26
23.88 19.20
24.74
0.32
24'91
43
32.77
19.05
0.50
24.86
44
33.27
25.61 18.90
0.71
24.79
45
33.76
26.48
18.75
7 6.92
0.96
24.71 46
34.24
27.36
18.61
8
7.87
1.25
34.71 28.24
18.46
24.60 47
24.50
9
8.82
1.57
48
35.18
29.12
18.32
10
9.75 1.93
24.37
49 35.64
30.01
18.18
11
10.67
2.31
24.25
50 36.09
30.90
18.05
12
11.58 2.73
24.12
51 36.53
31.80
17.91
13
12.47 3.18
23.98
52 36.97
32.70
17.77
14
13.36 3.65
23.86
53 37.40
33.60
17.64
15
14.23 4.16
23.72
54
37.82
34.51 17.51
16
15.07 4.68
55 38.24
35.42
17.38
17
15.90 5.23
23.55
23.38 56
23.22
36.33
17.25
38.65
39.06 37.24 17.13
18
57
16.72 5.81
17.52 6.40
20 18.31
19
23.05
58
39.46 38.16
17.01
59
39.85
39.08
16.89
7.01 22.89
7.65 22.72
21 19.08
40.24
40'00
16.77
40.62
40.93
16.65
60
22 19.84 8.31 22.55 61
23 20.59 8.98 22.38 62
24 21.32 9.66 22.21 63
40.99
41.85
16.53
41.36
42.78
16.42
25 22.03 10.36 22.03
64
41.73
43.71 16.30
26 22.73 11.08 21.86
65
42.09
44.65
16.19
27 23.42 11.80
21.68
66
42.45
45.58
16.08
28
24.09 12.54
21.51
42.80
46.51
15.97
67
21.34 68
29
24.76 13.29
43.15
47.45
15.86
30 25.40 14.05
21.17
69
4349
48.39
15.76
21.00
70
43.83
49.33
15.65
31 26:04 14.83
32
26.66 | 15·61
20.83
71
44.17
50.28
15.55
33 27.27
27.27 16:41
20.66
72 44.50
51.22
15.45
34
27.87
17.21
20.49
73
52.17
44.82
15.35
35 28.45
18.02
20.33
53.11
15.25
20.16
54.06
15.16
36
29.03 | 18.83
37 29.59 19:66
38 30.14 20:49
55.01
15.06
20.00
19.83 777
19.67
55.96
14.96
39
78
56.91
14.87
4
5
Abscissa | Ordinate Subnormal Arc
A G
G P
G K
A P
1.00
2·00
2.99
3.99
4.97
5.95
30.6821.33
Abscissa
A G
40
41
Ordinate Subnormal
G P
G K
7445·14
75 45.46
76 45.77
46.08
46.39
SUPPLEMENTARY TABLES.
425
Arc
A P
TABLE III.-Relations of the Catenary, continued.
Abscissa Ordinate | Subnormal
Abscissa Ordinate Subnormal
G P
A G
G K
A G
G P
G K
51.38 74.20
57.86
14.77
115 55.77
12.13
79 46.69
80 46.99 58.81
14.68
116 55.98
8147.29
117 56.19
92.68
93.66 12.07
94.64 12.01
95.62 11.95
96.59 11.90
118 56.39
59.77 14.59
.82 47.58 60.72 14.50
83 47.87 61.68 14.42 119 56.60
62.64 14.33 120 56.80
63.60
64.56
84 48.16
97.57 11.84
85 48 44
14.25 121 57.00
98.55
11.78
86 48.73
14.16 122 57.21
99.53
11.73
87 49.00 65.52
100.51
11.67
14.08 123 57.41
14.00
124 57.61
101-49
11.62
88 49.28 66.48
89 49.55 67-44
90 49.82 68.41
13.92
125 57.81 102.47
11.57
13.84
126 58.00 | 103.45
11.51
91 50 09
69.37
13.76
11.46
92 50 35
11:41
93 50.61
127 58.20 104.42
128 58.39 105.42
129 58.58 | 106.40
130 58.77 107.38
131
70.34 13.68
71.30 13.60.
72.27 13.53
73.24
11.36
94
50.87
11.30
95 51 13
13.45
58.96
58.96
108.35 11.25
96
13.38
132
109.35 11.20
97 51.63
75.17 13.31
133
110.33
11.15
98
51.88
76.14
13.23
134
111.31
11.10
99
52 13
77.11
13.16
135 59 71
112:28
11.06
136
113.28
11.01
59.89
137 60.07
114.26 10.96
138 60.25
115.25
10.91
100 52.37 78.08 13.09
101
52.61 79.05 13.02
102 52.85 80.02 12.95
103 53'09 81.00 12.89 139 60 43
104 53 32 81.97 12.82 140 60 60
105
12.75 141 60.77
116.23
10.87
117.22
10.82
53.55 | 82.94
118.20
10.77
106
53 78
83.91
12.69
60.95 119.18
10.73
142
143
107
54.01
84.89
12.62
61.12 | 120.15
10.68
108 54 24
85.86
12.56 144
10.64
109 54 47
86.83
12.50 145
122.14 10.59
87.81
61 29 121.15
6146
110 54 69
12.43 146 61 63
111 54.91 88.78 12.37 147 61.79
112 55.13 89-76 12.31 148 61.96
113 55.35 90.73 12.25 149 62.12 126'09 10.42
114 55.56 91-71 12.19 150 62:29 | 127:08 10.38
123.12 10.55
124-11
10.50
125.10 10.46
Arc
A P
1
59 15
59.15
59.34
59.52
At p. 179 of this volume there is given a table of relations
of catenarian curves, so arranged as to be useful in the erection
of suspension bridges, and in determining the weight and
tension of the several parts. The table now presented in this
supplement agrees with that in principle, but presents a totally
202
426
USE OF TABLE III.
different aspect, in order that it may be subservient to the con-
struction of catenarian arches of equilibration, whether for
bridges or for powder magazines.
It is very well known with regard to arches of equilibration,
when the substance of the structure presses vertically down-
wards, by the force of gravity, that for a parabolic arch as-
sumed as the intrados, the principles of equiponderance require
a similar and equal parabolic curve, situated throughout at. the
same vertical distance from the inner curve as the extrados;
and in that case the curves may be easily constructed and the
joints of the voussoirs found, by the methods explained in p.
165 and 167. But it is equally true, that upon the same general
principles of equipollence a catenarian curve may be assumed
for the arch, and the intrados and extrados be two similar and
equidistant curves, provided that equidistance be measured upon
the radius of curvature at cach point, and be but small compared
with the span of the arch. Constructions founded upon the
knowledge of this fact have been long rejected, but have of late
been re-introduced; on which account Table III. is given, that
the time of practical men may not be wasted in needless calcu-
lation.
Suppose that the marginal figure in the lower part of p. 164
represented a catenary, in which, as indicated in the table, a r
represented any portion of the curve A G, and & P, the corres-
ponding abscissa and ordinate, and a K, the subnormal, p repre-
senting the parameter, or constant horizontal tension at the ver-
tex of the curve. Then, in addition to the equations exhibited
in p. 175, &c., it is known that ▲ P = √ (1 g)² + 2 p ▲ G.
7
1
2 AG
and
2 (AG)3 26 (A G)5
+
3 (GP)4 45 (GP)
662 (AG)"
945 (G P)8
P
(G P)*
Now, the table presents values of a g, g p, & K, correspond-
ing to several values of a P, the arch, on the supposition that P,
the parameter, or tension at the vertex, is 25.
Suppose, as an illustration of the use of the table, it were re-
quired to construct a vault whose semibase should be 8 (yards,
for example) and the height 10; we shall find its model by
searching that part of the table where the abscissa, a G, and the
ordinate, & P, are in the ratio of 10 to 8. This is, where the are
A P is 80; for there the abscissa and ordinate are 58.81 and
46.99, which are in the ratio of 10·01 to 8, sufficiently near for
practical purposes. Hence, then, the arch may be regarded as
divided into 80 equal parts, and the table will present the com-
puted proportions for 80 voussoirs, on each side of the vertex,
or rather for 79 voussoirs and half the key stone,
Or they may
be reduced to 40, or to 20, or to 10, on each side the crown of
kh
+ &c.
USE OF TABLE III.
427
8
the structure. The actual values to the dimensions 10 and 8,
will be found by multiplying each number in the table by 46.99'
or its equivalent 0.1702; or, to have the dimensions in feet, let
the multiplier be 3 × 0.1702 or 0.5106. Thus, we should have
the values of a G, & P, A P, for every voussoir from the vertex
of the curve downwards, while the corresponding values of
AG + GK would give the points K, from which the line K P
must be drawn to give in each case the direction of the joint.
In the example assumed, the value of the parameter, p, would
be 25 × 0.1702 or 4:255; and this would be the measure of the
horizontal thrust.
2
The curve thus sketched will be posited in the middle of the
arch, half way between the intrados and the extrados. Trace
above and below this, at the distance t (half the thickness at
the crown), measured upon the respective positions of the joints,
two curves parallel to the mean catenary; so shall you obtain
the proposed arch of equilibration.
In cases where the proposed ratio of the height and semi-span
of the arch, cannot be found with sufficient accuracy in the
tables, it may be approximated to by the usual methods em-
ployed in reference to proportional parts. As, suppose the
semi-span and the altitude were to be equal; this condition lies
between 60 and 61 in the values of the arch; and, by the me-
thod just alluded to, we shall find that when the arc is equal to
60.44, the height and half base are each 40:40, the horizontal
thrust being 25.
If an equilibrated circular arch were to be erected, of the same
span and height, and the same thickness at the crown, the hori-
zontal thrust would be 31.6. Whence, by the way, it appears
that an equilibrated catenarian arch in these proportions, would
produce a less horizontal thrust than an equilibrated circular
arch contrary to the opinion of Bossut, Equilibres des Voutes.
:
THE END.
1
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