QA
35
M23
1772
•
ARTES
1837
SCIENTIA
VERITAS
LIBRARY
OF THE
UNIVERSITY OF MICHIGAN
ZURION
UUTMIHUUH
7.
TUEBOR
SI-QUÆRIS PENINSULAMANŒNAM
CIRCUMSPICE
ARITHMETI C,
RATIONAL and PRACTICAL.
WHERE IN
The properties of NUMBERS are clearly pointed out,
the THEORY of the ſcience deduced from firſt prin-
ciples, the methods of OPERATION demonſtra-
tively explained, and the whole reduced to PRAC-
TICE, in a great variety of uſeful RULES.
Confifting of THREE PARTS, viz.
I. VULGAR ARITHMETIC.
II. DECIMAL ARITHMETIC.
IIL PRACTICAL ARITHMETIC,
By JOHN MAIR, A. M.
PART I.
VULGAR ARITHMETIC.
The SECOND EDITION.
+
3
EDINBURG H:
Printed for A. KINCAID & W. CREECH, and J. BELL.
MDCCLXXII.
3
!
Entered in Stationers Hall, according
to act of parliament.
*
Directions to the BOOKBINDER.
Place the plate of Napier's rods to front
p. 78. of vol. 1.
Place the four plates on Menfuration in
fucceffive order, at the end of vol. 3.; and
few them fo as that every plate may, when
opened out, be wholly without the book,
то
THE RIGHT HONOURABLE
THOMAS EARL OF KINNOUL,
THE FOLLOWING
TREATISE OF ARITHMETIC,
IS,
WITH GREAT RESPECT,
INSCRIBED,
BY
HIS LORDSHIP's
MUCH OBLIGED,
MOST OBEDIENT, AND
VERY HUMBLE SERVANT,
JOHN MAIR.
PREFACE.
A
N encomium on the uſefulneſs and excellency of
arithmetic would, in this place, be vain and idle.
The world is already abundantly fenfible of the worth
and importance of this noble art. All, then, that ſeems
neceffary by way of preface, is to give fome fhort ac-
count of the following treatife, and point out the im-
provements the reader may expect to meet with in per-
uling it.
And, in general, I perfuade myſelf it will be found
fully to answer the title page; the properties of numbers
being exhibited in a clear and inftructive manner, and
the whole tho y built on firft principles; that is, on
axioms, or fuch propofitions as are felf evident. The
rules of operation, deduced immediately from the theo-
ry, are conceived in a few expreffive words, and illu-
ſtrated by a great variety of fuitable examples. Care too
has been taken to range things every where in fuch or-
der, that what is prior paves the way for what is to
follow.
By connecting things in this manner, the rugged path
is rendered ſmooth, and the learner will proceed with
pleafure from what is more fimple, to what is more
complex; and rife by eaſy ſteps from the lowest to eve-
ry ſuperior part of the fcience. And to furniſh him
with materials for exerciſe on every branch, and at the
fame time fave labour to the teacher, fets of unwrought
examples, with their anſwers, are inferted in all places
where this appeared to be any way neceffary, or of ad.
vantage. But to be more particular :
This treatiſe confifts of three parts, under the titles of
Arithmetic Vulgar, Decimal, and Practical. The
firſt of theſe parts begins with an Introduction; wherein
arithmetic is defined, its limits ſtated, and the reader
taught to diſtinguiſh betwixt a ithmetic, and the practical
purpoſes to which its operations may be applied. After
this, notation is explained at large, the various kinds of
numbers
vi
PREFACE.
numbers deſcribed, and the axioms, or first principles,
laid down, on which are founded the theory and rules
of operation. Addition, fubtraction, multiplication, and
divifion, are taught at full length, with all the modern
improvements, and the reafon of the rules every where
affigned.
The properties of proportional numbers, or numbers
in geometrical proportion, are briefly pointed out; and
thence is derived the rule of three: in which a fimple
and eafy method of ſtating the queſtions is laid down;
whereby the puzzling diftinction of direct and inverſe
is precluded, or rendered needlefs, and the multiplicity
of rules uſually adhibited on this head, is avoided.
Nor is this method confined to the fimple rule of three,
but extended to the rule of five, feven, nine, eleven,
&c. The doctrine of vulgar fractions, and the rules of
practice therewith connected, are delivered in a way
fimple and complete, without prolixity.
The ſecond part contains decimal arithmetic, or the
doctrine of decimal fractions. Here the origin of deci-
mals of every kind, viz. finite, repeating, and circula-
ting, is inquired into, their properties pointed out and
defcribed at large; the various forts and ways of reduc-
tion are minutely explained, and properly exemplified:
complete tables too are conſtructed for converting the
parts of integers to decimals by infpection; and alfo an-
other fet of tables for reducing readily decimals of e-
very fort to value. In addition, fubtraction, multipli-
cation, and divifion, the operations are conducted by a
few plain eafy rules; and thefe illuftrated by a copious.
fet of proper examples.
Some lovers of arithmetic have alledged, that authors,
in treating of interminate decimals, are always curt and
fcanty in their explications, and leave this new part of
decimal arithmetic fo much involved in obfcurity, as to
be ftill, in their opinion, far above the reach of vulgar
readers. But I flatter myfelf, that all complaints of
this kind will now be removed; the methods of operation
in
PREFACE.
vii
in decimals of this fort being here fet before the learner,
with fuch clearnefs and perfpicuity, that one, though of
a low capacity, may foon acquire ſkill enough, not only
to work examples wherein repeaters or circulates occur,
but may even underſtand the reafon of the proceſs.
This fecond part concludes with a large decimal prac
tice, fetting forth the fuperior excellency of decimal a-
rithmetic above every other method of numerical compu-
tation. Here is fhewn the utility and conveniency of
decimals in conftructing tables of various kinds; how
they may be uſed with great advantage in multiplication
and divifion of duodecimals and fexagefimals; how they
often rival vulgar fractions in point of accuracy, and
furpaſs them in point of brevity; and, finally, that the
decimal operation is frequently the ſhorteſt and eaſieſt in
the rule of three, the work being done that way in leaſt
time, and with feweft figures.
The third part contains the application of the other
two to practical purpoſes, in a great variety of uſeful
rules. Here facility is ftudied, the mercantile and
faſhionable methods of computation are introduced, and
adapted to the feveral branches of commerce and bufi-
nefs. Sometimes different ways of operation are exhi
bited, and the choice left to the practitioner. Large
and correct tables on compound intereft and annuities
are inferted, with their conſtruction and uſes. Menfu-
ration of furfaces and folids, artificers work, board and
timber, &c. is taught in an caly manner, and at more
than ufual length.
To the above account I fhall only add, that many
things of importance, feveral fmall improvements, and
fome things curious, as well as ufeful, not hitherto ta-
ken notice of, will be found fcattered here and there in
different parts of the book. To conclude, no pains.
have been ſpared in collecting materials, and working
them up, with a view to render this fyftem of arithmetic
inftructive to the tyro, entertaining to the fcholar, and
uſeful to the man of buſineſs.
CON.
EONTENT S.
Pag.
Introduction
Notation
Numeration
The fpecies of Numbers
Arithmetical Axioms
Explanation of Characters
Addition of Integers
Addition of the parts of Integers
The proof of Addition
Subtraction of Integers
Subtraction of the parts of Integers
The proof of Subtraction
Multiplication of Integers
Contractions in Multiplication
Select methods of multiplying Integers
Multiplication of the parts of Integers
The proof of Multiplication
Divifion of Integers
Contractions in Divifion
Select methods of dividing Integers
Divifion of the parts of Integers
The proof of Divifion
Reduction defcending and aſcending
Mixt Reduction
The Rule of Three
I
5
II
J 2
13
19
21
46
50
52
62
66
69
73
79
88
91
103
114
115
i19
122
137
151
The fimple Rule of Three direct
154
The fimple Rule of Three inverfe
168
The Rule of Five direct
176
The Rule of Five inverſe
183
The Rule of Seven, Nine, Eleven, &c.
189
Contractions in the Rule of Three
193
Vulgar Fractions
207
Reduction of Vulgar Fractions
209
Addition of Vulgar Fractions
226
Subtraction of Vulgar Fractions
230
Multiplication of Vulgar Fractions
Divifion of Vulgar Fractions
The Rule of Three in Vulgar Fractions
Rules of Practice
VOL. I.
b
236
239
243
246
$
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P. 39. In the Table of Long Meafure, r. make I pole,
perch, or rod.
p. 55. and 56. in the titles, inflead of grs. r. Q:
P. 74. in Ex. I Method 2. r. the product 31392
p. 86. Rule VIII. I. 4 r. folid lines.
·Rule IX. I. 4. r. folid inches.
Þ.90. queſt. 7.1 9. for 36 r. 57 fq. inches.
p. 112. l. I. r. Here
in. F. in.
p 119 /. 10. r. C 295
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p. 120. queſt. 5. 1. 5. r. 14697123087375.
p. 151. guest. 5. l. 1. r. 26 acres,
p. 167. queſt. 25. I. 2. r. coft him in all L. 201 : 12:8,
for 86 yards &c.
p. 232. for Note I. r. Note.
p. 234. for Note 2. r. Note.
p. 238. Ex. 5. r. Prod. 3%.
p. 240. 1. penult. r. of the quot,
P. 241. 1. 20. for multiplicand r. dividend
2
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1. 26. r. I + I +1+3=÷ + + + } =
풍
​11 of 14 = 11/
p. 242. 1. ult. r. Anf. 423.
p. 244 guest. 4. 1. 2. for 34 1. 3r•
p. 247./20. r. the quots.
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3
p. 260. Rule II. /. 2. r. divided into fhillings, and fome a-
liquot part or parts, divide it, &c.
p. 261. Rule III. / 4. r. pound, or fome number of fhil-
lings, and the fubfequ ni, &c.
p. 270. in Ex. 2. on the margin for yd r. yd.
ź
On the head of every left-hand page, inflead of Chap. r.
Part I.
Fellowship is now carried to Part III.
1
ARITHMETIC,
RATIONAL and PRACTICA L.
A
INTRODUCTION.
RITHMETIC is a fcience explaining the pro-
perties of numbers, and fhewing the method
or art of computing by them.
The divifion of arithmetic into theory and practice
is natural and juft. The theory, or fpeculative part,
treats of the nature and affections of numbers, and this
is properly feience. The practical part confifts in the
application of the rules gathered from the theory to the
folution of queſtions or problems; and arithmetic in this
reſpect is rightly termed an art.
Number, which is the object of arithmetic, is that
which anfwers directly to the queftion, How many?
and is either an unit, or fome part or parts of an unit,
or a multitude of units.
To a perfon having the idea of number in his mind
the following queftions naturally occur, viz. 1. How is
fuch a number to be expreffed or written? Hence we
have Notation. 2. What is the fum of two or more
numbers? Hence Addition. 3. What is the difference
of two given numbers? Hence Subtraction. 4. What
will be the refult or product of a given number repeated
or taken a certain nun ber of times? Herce Multiplica-
tion. 5. How often is one given number contained in
another? Hence Divifion.
Thele fie, viz. Notation, Addition, Subtraction,
Multiplication, and Divifion, are the chief parts, or ra-
A
ther
2
INTRODUCTION.
ther the whole of arithmetic. Every arithmetical ope
ration requires the uſe of ſome of theſe, and nothing but
a proper mixture of them is neceffary in any operation
whatever; and, by an Arabic term, theſe are called the
algorithm.
Theſe may be fitly compared to the five mechanic
powers, which, varioully combined, produce engines of
different forms, and of amazing force; and which, at first
fight, are apt to ftrike us with furprife; but upon examina-
tion, their effects are all found to flow from a proper com-
bination of the fimple powers. In like manner, we meet
with a great variety of complex arithmetical operations,
curious calculs, and aftonishing folutions, which yet are
all effected by a due application of the different parts of
the algorithm.
Numerical computations are neceffary almoft in e-
very affair; but the manner of the operations depends.
upon the nature of the ſubject to which they are applied.
Arithmetic is a handmaid to moft of her fifter arts
and ſciences; and while fhe is employed in their ſervice,
ſhe muſt be under their direction, and obſerve the laws
they prefcribe. Thus the calculation of a folar or lunar
eclipfe is performed by numbers; but the operation is
directed by the precepts of aftronomy: a reckoning at
fea is wrought by numbers; but the fteps of the work
are conducted by the rules of navigation: the contents
of a cafk is caft up by numbers; but the art of gauging
guides the operation: the perpendicular height of a hill
is computed by numbers; but it is trigonometry that di-
rects the manner of their application.
The canons directing the application of numbers in o-
ther fciences are not to be reckoned arithmetical rules:
no rule is to be efteemed fuch, but where the operation.
entirely depends on arithmetical principles, and proceeds
without any foreign aid or direction.
Hence from the number of arithmetical rules is exclu-
ded the rule of Three, with its numerous train of de-
pendents, fuch as, Fellowſhip, the rule of Falle, Intereſt,
Annuities, Alligation, Exchange, Barter, &c. The doc-
trine of proportion is delivered in the elements of geo-
metry,
INTRODUCTION.
'3
:
metry. It is Euclid, lib. 6. prop 16. who demonftrates,
that if four quantities be proportional, the rectangle, or
product of the extremes, will be equal to that of the
means it is this property of proportionals that gives
birth to the rule of Three; it is this that directs the a-
rithmetical operation. For the like reafon, Involution
and Extraction of roots are no arithmetical rules; it is
algebra that teaches the manner of thefe operations.
Reduction and the rules of Practice are likewife to be ex-
terminated; as alfo, addition, fubtraction, &c. of the parts.
of integers, ſuch as, fhillings, pence, farthings, ounces, &c.
The ſtatutes of a country, with refpect to coins, weights,
and meaſures, prefcribe the method of operation on theſe
heads. It remains, then, that the five parts of which the
algorithm confifts make the whole of arithmetic: here
the limits of this fcience are to be fixed, otherwife it
would prove endleſs and inexhauftible.
True it is, that writers on this fubject go generally
beyond theſe bounds: they frequently introduce into
their printed treatifes all the rules excluded above, and
feveral others. Nor is this any fault: authors are at
liberty, and in the right, to ſtore their writings with all
fuch numerical operations as may be uſeful to the pu-
blic. The only thing blameable in their conduct is, their
not ſtating the limits of the ſcience, and teaching their
readers to diftinguish betwixt arithmetic, and the appli
cation of its operations to practical or uſeful fubjects.
The neglect here taken notice of has been attended
with this remarkable confequence, that vulgar readers.
have generally miſtaken theſe adventitious rules for parts
of arithmetic; and as they obferved new rules publiſhed
every now and then in new books, they were apt to i-
magine that arithmetic admitted of no limits, and could
never be attained to any fort of perfection.
Arithmetic, as a fcience, ought to demonftrate her
own rules; that is, the rules of operation in addition,
fubtraction, multiplication, and divifion, both of inte.
gers and fractions; but this is all: to employ her in de-
monſtrating the rules borrowed from other arts and
fciences, would be vain and fruitlefs. An attempt of
A 2
this
INTRODUCTION.
this kind would be making her act out of fphere, and
putting her upon an office fhe is by no means fit for.
Numerical demonftrations are fuccefsfully applied to fub.
jects purely arithmetical, and they can be carried lit-
tle further. The proper demonftrations of theſe fo-
reign rules are only to be had from geometry, algebra,
or fome other branch of ſcience; and as theſe are much
beyond the reach of the young arithmetician, he cannot
at preſent expect entire fatisfaction on this head, but
muft wait with patience, till, in the courſe of his ſtudies,
he come to be acquainted with the fuperior fciences.
The following treatiſe is defigned as a rational, and
at the fame time a practical fyftem of arithmetic; wherein
the five parts of this fcience are fully but fuccinctly
handled, and all the arithmetical rules numerically de.
monftrated The adventitious rules depending on other
fciences are likewife explained, and exemplified in the
way that appears beſt for anſwering the purpoſes of the
merchant, the accomptant, and mechanic. Some few
things too are introduced for the entertainment of the
ſcholar and the curious.
1
CHAP.
5
NOTA
CHA P. I.
NOTATION.
OTATION is that part of arithmetic which explains
the method of waiting down, by characters or
fymbols, any number expreffed in words; as alſo the
way of reading or expreffing, in words, any number gi
ven in characters or fymbols. But the firit of theſe is
properly notation, and the laſt is more ufually called nu-
meration.
The ancient Romans, for this purpoſe, made choice
of a few of the letters of their alphabet; and this is called
the literal, or Roman notation. This way is ftill of confi.
derable uſe, and neceffary to be underſtood. But the me-
thod which now prevails, and is practifed over a great
part of the known world, is effected by characters or
fymbols, called figures; and is named figural notation,
or notation of numbers by figures.
The things then proper to be compriſed in this chap-
ter are, 1. The literal or Roman notation. 2. The fi.
gural notation. 3. Numeration, or the way of read-
ing numbers. 4. Defcriptions of the kinds or fpecies
of numbers. 5. Arithmetical principles, or axioms.
6. The fignification of ſuch characters as are ſometimes
uſed for brevity's fake. 7. Some practical queſtions.
I. The Roman Notation.
The letters uſed by the ancient Romans, in their no-
tation of numbers, were only thefe five, C, I, L, V, X.;
whofe values and order are as follows.
One, Five, Ten, Fifty, An hundred, Five hundred, A thouſand.
I, V, X, L, C,
IƆ
CIO
By the repetition or combination of theſe they ex-
preffed any other number, according to the following
rules,
I. The repetition of a letter repeats its value.
Thus II fignifies two, III three; and XX twenty,
XXX
*
NOTATION.
Chap. I.
XXX thirty; and CC two hundred, CCC three hun-
dred, &c.
II. The annexing a letter of a lower value to a letter
of a higher value, adds their values, or denotes their fum.
Thus, VI fignifies fix, VII ſeven, VIII eight; and
XI denotes eleven, XII twelve, XV fifteen, XVI fix-
teen; and LV fifty-five, LX fixty, LXX ſeventy, LXXX
eighty; and CV one hundred and five, CX one hundred.
and ten, CL one hundred and fifty, CLV one hundred
and fifty-five; and IOC fix hundred, IOCC feven hundred;
and I, one thouſand five hundred, CI, one
thouſand fix hundred, CIƆ, IƆCC one thouſand ſeven
hundred, &c.
III. The prefixing a letter of a lower value to a letter
of a higher value, fubtracts their values, or denotes their
difference.
Thus, IV fignifies four, IX nine, XL forty, XC nine-
ty, IC ninety-nine, &c.
IV. The annexing of D to the number ID makes its.
value ten times greater.
Thus, I
fand.
fignifies five thousand, and IƆƆƆ fifty thou
V. The prefixing of C, together with the annexing
of, to the number CIƆ, makes its value ten times
greater.
Thus, CCOO denotes ten thoufand, and Cɔɔɔ
an hundred thouſand.
The ancients, as Pliny obferves, carried this kind of
notation no higher. If at any time they had occafion to
expreſs a greater number, they did it by repetition;
thus, CCCIƆƆ,CCCɔɔɔɔ fignified two hundred thou-
fand, and CCCCCC denoted three
hundred thouſand, &c.
The Romans, in later times, inſtead of Ɔ uſed D;
and inſtead of CIƆ they introduced M.
They likewife fometimes expreffed thouſands by a
line drawn over the head of numbers. Thus, III fignifies
three
Chap. I.
7
NOTATION.
three thouſand, Vfive thoufand, X ten thoufand, LX
fixty thouſand, Can hundred thoufand, M a thoufand
thouſands, or a million, MM two millions, &c.
In ſome editions of Cæſar's commentaries we find co
ufed, inſtead of CIƆ, to fignify a thouſand.
Any thing further neceffary to be known in the Roman
notation may be learned from the following table; in
which, with a view to exerciſe the learner, as well as
for the fake of brevity, I have chofen to expreſs the va
lue of the literal numbers by figures, rather than words.
I.
II.
TABLE.
I
XC.
90
2
C.
100
III.
3
CI.
101
IV. or IIII.
4
V.
VI.
VII.
VIII. or IX.
111
1X.
688 ana
CII, &c.
102
5
CC.
200
6
CCC.
300
7
CCCC. or CD.
400
IƆ. or D.
500
9.
IOC. or DC.
600
X..
XI.
10
IDCC. or DCC.
700
II
XII.
12
XIII.
13
XIV. or XIIII
14
XV.
15
XVI.
16
1 2 3 4 5 6
IDCCC. or DCCC
800
IJCCCC. or DCCCC, or CM.
900
CID or M.
1,000
CID,C. or MC.
1,100
XVII.
17
XVIII.
18
XIX.
19
XX
20
78 9 o
CID,CC. or MCC. &c.
MM. or II.
1,200
2,000
XXI.
21
XXII.
22
MMM. or ITI
MMMM or IV.
IƆƆ. or V.
IƆƆM, or VI.
3,000
I
4,000
5,000
6,000
XXIII.
23
ƆƆMM. or VII.
XXIV. or XXIIII.
7,000
24
IƆƆMMM, or VIII.
XXV.
25
8,000
XXVI.
26
ƆƆMMMM or IX.
9,000
XXVII.
27
CCD or X.
10,000
XXVIII.
28
CCIƆOM. or XI.
11,000
XXIX.
29
XXX.
30
XL.
40
L
50
LX.
60
LXX.
70
LXXX.
80
CCIƆOMM. or XII. &c.
1ɔɔɔ. or Ì.
ƉƆƆƆM. or LI.
ɔɔɔMм. or LII. &c.
CCC or C.
CIDIƆCC,LXIII. or M,DCC,LXIII. 1763
12,000
1
50,000
51,000
52,000
100,000
II. Figural
8
Chap. I.
NOTATION.
II. Figural Notation.
An unit, or unity, is that number by which any thing
is called one of its kind. It is the firit number; and if
to it be added another unit, we fhall have another num.
ber, called two; and if to this laſt another unit be add-
ed, we fhall have another number, called three; and
thus, by the cont nual addition of an unit, there will a
riſe an infinite increaſe of numbers. On the other hand,
if from unity any part be fubtracted, and again from
that part another part be taken away, and this be done
continually, we ſhall have an infinite decreaſe of num-
bers.
But though number, with respect to increaſe and de-
creaſe, be infinite, and knows no limits; yet ten figures,
variouſly combined or repeated, are found fufficient to
express any number whatſoever. Thefe, with the me-
thod of notation by them, were originally invented by
fome of the eattern nations, probably the Indians; af-
terwards improved by the Arabians; and at laſt brought
over to Europe, particularly into Britain, betwixt the tenth
and twelfth century. From the ten fingers of the hands,
on which it had been uſual to compute numbers, the
figures were called digits. Their form, order, and va-
lue, are as follows.
I one, an unit, or unity, 2 two, 3 three, 4 four, 5
five, 6 fix, 7 feven, 8 eight, 9 nine, o cipher, nought,
null, or nothing. Of thefe, the fift nine, in contradif-
tinction to the cipher, are called fignificant figures.
The value of the figures now affigned is called their
fimple value, as being that which they have in them-
Telves, or when they ftand alone. But when two or more
figures are joined as in a line, the figures then receive
allo a local value from the place in which they ftand,
reckoning the order of places from the right hand to-
wards the lift, thus,
Fif
Chap. I.
9
NOTATIO N.
&c.
Twelfth place.
Eleventh place.
Tenth place.
Ninth place.
Eighth place.
Seventh place.
Sixth place.
Fifth place.
Fourth place.
Third place.
Second place.
Firſt place.
7 7 7 7 7 7 7 7 7 7 7 7
A figure ftanding in the first place has only its fimple
value; but a figure in the ſecond place has ten times the
value it would have in the firſt place; and a figure in the
third place has ten times the value it would have in the
ſecond place; and univerfally a figure in any ſuperior
place has ten times the value it would have in the next
inferior place.
Hence it is plain, that a figure in the first place fimply
fignifies ſo many units as the figure expreffes; but the
fame figure advanced to the fecond place will fignify ſo
many tens; in the third place, it will fignify ſo many
hundreds; in the fourth place, fo many thouſands; in
the fifth place, ſo many ten thouſands; in the fixth place,
fo many hundred thouſands; and in the ſeventh place,
fo many millions, &c. Thus, 7 in the first place, will
denote feven units; in the fecond place, ſeven tens, or
feventy; in the third place, ſeven hundred; in the fourth
place, feven thouſand, &c.
Every three places, reckoning from the right hand,
make a half-period; and the right hand figures of thefe
half-periods are termed units and thousands by turns;
the middle figure is always tens, and the left-hand fi-
gure always hundreds. But here obferve, that the right-
hand figure of every half-period, properly and ftrictly
ſpeaking, is always units; for the place called thouſands,
when expreffed at more length, is termed units of thou
fands, or the units place of thousands.
Two half-periods, or fix places, make a full period;
and the periods, reckoning from the right hand towards.
the left, are titled as follows, viz. the firft is the period.
of units; the fecond, that of millions; the third is titled
bimillions, or billions; the fourth, trimillions, or trillions;
B
the
10
Chap. I.
ΝΟΤΑΤΙΟ Ν.
the fifth, quadrillions; the fixth, quintillions; the fe
venth, fextillions; the eighth, feptillions; the ninth,
octillions; the tenth, nonillions, &c.
Half-periods are ufually diſtinguiſhed from one another
by a comma, and full periods by a point or colon; as in
the table following.
TABL E.
4th Period.
3d Period.
Trillions.
Billions.
2d Period.
ift Period.
Millions.
Units.
Hundred thouſands.
Ten thouſands.
Thouſands.
-
o Hundreds.
∞ Tens.
Units.
Hundred thouſands.
Ten thouſands.
w Thouſands.
Hundreds.
o Tens.
• Units.
Hundred thouſands.
w Ten thouſands.
Thouſands.
Hundreds.
• Tens.
+ Units.
•
678
Hundred thouſands.
Ten thouſands.
* Thouſands.
o Hundreds.
▲ Tens.
o Units.
4 6 8 5 8 1 3,7
>
2 3 7, 89 4
0 4
The table may be expreſſed in a more concife form
thus,
4.
3.
Billions.
2.
1. Per.
Units.
Millions.
Trillions.
CXM,CXU: CXM,CXU: CXM,CXU : CXM,CXU.
964,085 8 13,700 237,894: 678,040.
}
From the table it is obvious, that though a cipher fig-
nify nothing of itſelf, yet it ferves to fupply vacant pla-
ces, and raiſes the value of fignificant figures on its left
hand, by throwing them into higher places. Thus, in
the first period, by a cipher's filling the place of units,
the figure 4 is thrown into the place of tens, and figni-
fies forty. And a cipher likewife fupplying the place of
hundreds, the figure 8 is advanced to the next higher
place, and fignifies eight thouſand, &c. But a cipher
does not change the value of a fignificant figure on its
right hand. Thus, 07, or 007, is the fame as 7.
From the table it likewife appears how any number
expreffed
Chap. I.
NOTATION.
expreſſed in words, may be written down by figures;
in the doing of which obferve the following
RULE.
Beginning at the left hand, and writing towards he
right, put every figure in fuch place and period as the
veibal expreſſion points out, fupplying the omitted pla
ces with ciphers. Examples follow.
Millions.
Units.
Nine hundred and eighty-feven millions
Forty-five millions and feven hundred thou
fand
Six millions four hundred and thirty-two
thouſand
Eight hundred and five thousand and nine
hundred
Seventy-three thousand and ten
Five thousand and four
Four hundred and twenty
Ninety-five
Seven
is written thus,
CXM,CXU: CXM,CXU.
987:000 OGO
45:700,000
6:432,000
805,900
73,0 10
5,00 4
420
95
4t
7
III. Numeration.
Notation and numeration are fo nearly allied, that he
who underſtands the former cannot fail foon to acquire
the latter. The method of reading numbers when ex-
preffed by letters, is fufficiently clear from the table fub-
joined to the Roman notation; and the way of reading.
any number expreffed by figures, may be eafily learned
from the table of the figural notation; in the doing of
which obferve the following
RULE.
Beginning at the left hand, and reading toward the
right; to the fimple value of every figure join the name
of its place, and conclude each period by expreffing its
title, every where omitting the ciphers. Examples fol-
low.
B 2
Millions.
12
NOTATION.
Chap. I.
Millions.
Units
CXM,CXU: CXM,CXU.
900:000,000)
7 8:00 9,000
0,80
90 3,005
20,034
is read thus,
Nine hundred millions.
Seventy-eight millions and nine thousand.
Four millions and eight hundred.
Nine hundred and three thouſand and five.
Twenty thousand and thirty-four
8,469 Eight thouſand four hundred and fixty-
708
96
3
nine.
Seven hundred and eight.
Ninety fix.
Three, or Three units.
N. B. The name of the unit's place, in all periods,
and the title of the right-hand period, are commonly o
mitted, or very rarely expreffed.
IV. Defcriptions of the kinds or fpecies of numbers.
The fpecies of numbers are very various and mani-
fold; but in this place it will be fufficient to deſcribe ſuch
as appear moſt uſeful and neceffary; particularly thoſe
that will occur in the enfuing treatife.
1. An integer, or whole number, is an unit, or any
multitude of units; as 1, 7, 48, 100, 125.
2. A fraction, or broken number, is any part or parts
of an unit; and is expreſſed by two numbers, which are
feparated from one another by a line drawn betwixt
them; the under number being called the denominator,
and the upper one the numerator, of the fraction; as
I 3
2'4' TO'
9/9
3. A mixt number is an integer with a fraction joined
to it; as 4, 7, 485.
4. A number is faid to meaſure another number, when
it is contained in that other number a certain number of
times, or when it divides that other number without any
remainder. Thus, 3 meafures 6, 9, or 12.
5. An even number is that which is meaſured by 2,
or which 2 divides without any remainder; as 2, 4, 6, 8,
10, 12.
6. An odd number is that which 2 does not meaſure,
or which cannot be divided by 2, without a remainder;
as 1, 3, 5, 7, 9, 11, 13.
7. A
Chap. I.
13
NOTATION.
7. A prime number is that which unity, or itfelf, only
meaſures; as 3, 5, 7, 11, 13, 17, 19.
II
8. A compofite number is that which is meaſured by
ſome other number than itſelf, or unity; as 12, which
which is meaſured by 2, 3, 4, or 6.
9. Numbers are called prime to one another, when
unity only meaſures them. Thus 13 and 36 are prime to
one another; for no number, except unity, meaſures
both.
10. Numbers are called compofite to one another,
when fome number, befides unity, meaſures them. Thus
12 and 18 are compofite to one another; for 3 or 6
meaſures both of them.
11. A number which meaſures another is called an
aliquot part of that other. Thus 6 is an aliquot part of
and 3 of 12, and 5 of 20.
18,
12. The number meaſured, or which contains the a-
liquot part a certain number of times, is called a mul-
tiple of that aliquot part. Thus 18 is a multiple of 6,
and 12 of 3.
13. A number is called an aliquant part of another,
when it does not divide that other without a remainder.
Thus 7 is an aliquant part of 24.
14. Two, three, or more numbers, which, multiplied
together, produce another number, are called the com
ponent parts of the number produced. Thus 3 and 4, or
2 and 6, are the component parts of 12; and 2, 3, and
4, are the component parts of 24.
15. The product of a number multiplied into itſelf is
called the fquare, or fecond power, of that number; and
the number it felf is in this cafe called the root. And if
the ſquare be multiplied into the root, the product is
called the cube, or third power, of that number. And if
the cube be multiplied into the root, the product thence
arifing is called the biquadrate, or fourth power, &c.
V. Arithmetical Principles or Axioms.
An axiom or firft principle in any fcience is a felf-evi-
dent propofition which needs no demonftration. The
following
14
Chap. I.
NOTATION.
following ones flow neceffarily from the nature of num-
bers, the manner of their notation, or their ſpecific pro-
perties. On them is founded the theory of arithmetic,
and from them are deduced the rules to be obferved in
every operation.
1. Any given number may be increaſed or diminiſhed
at pleaſure.
For there is no number fo great, but a greater may be
given; nor is there any number fo fmall, but a ſmaller
may be affigned.
II. The figures of any number increaſe in value from
the right toward the left hand.
Hence in addition, fubtraction, and multiplication,
the operation begins at the units figure or loweſt place
on the right hand, and proceeds, with the flux of num.
bers, to the higheſt place on the left.
III. Ten in any inferior place makes one or an unit in
the next higher place, and the reverſe.
Hence when the fum of any column added, or the pro-
duct of any two figures multiplied, amounts to or exceeds
ten, we carry an unit for every ten to the next higher
place for the higheſt digit being (9), any number above
9 is compound, and requires more than one place to
exprefs it; which is done by removing or carrying away
the tens, as ſo many units, to the next fuperior place.
IV. If the right-hand figure of any number be cut
off, the remaining figure or figures are a juft number of
tens, and the right-hand figure fo cut off is the overplus.
Hence in addition and multiplication, the right-hand
figure of the fum of any column, or of the product of
any two figures, is always fet down, and the remaining
figure or figures are carried on to the next higher place.
Thus if a fum or product amount to 10, the o is fet
down, and I is carried; if it amount to 72, the 2 is
fet down, and 7 is carried; if it amount to 100, then
o is fet down, and 10 is carried; if it amount to 136,
then 6 is fet down, and 13 carried, &c.: but if the ſum
or product amount only to a fingle digit, then the digit
is fet down, and nothing is carried.
V. A number, by having one, two, three, &c. ci-
phers
$
1
Chap. I.
NOTATION.
15
phers annexed to it, becomes ten, a hundred, a thou-
fand, &c. times greater.
Becauſe the figures of the number are, by this means,
thrown into higher places, thus 7 by a cipher annexed
becomes 70, by two ciphers 700, and by three ciphers
7000, &c.
VI. Any number is naturally refolved into as many
conſtituent parts as it has fignificant figures, by annexing
to each fignificant figure as many ciphers as there are
figures on its right hand.
Thus 345 is refolved into 300, 40, and 5; and 6082
is refolved into 6000, 80, and 2; and 70090 into
70000 and 90.
VII. The feparate value of a fingle figure in any num-
ber is greater by one at leaſt, than the value of all the
figures on its right hand.
Thus, in the number 199, the ſeparate value of 1,
viz. 100, is greater by 1 than the figures on its right
hand 99.
VIII. An unit is an aliquot part of every whole num-
ber, and every whole number is a multiple of unity:
For unity meaſures every whole number.
IX. Numbers equally augmented or diminished, con-
tinue to have the fame difference.
Let any numbers, fuch as 6 and 8, be equally aug-
mented, by the addition of 4 to each of them, the fums
10 and 12 will have the fame difference as 6 and 8, viz.
2. Again, if from each of them 4 be taken away, the
remainders 2 and 4 will have the fame difference, viz.
2.
X. The difference of two unequal numbers added
to the leffer, gives a fum equal to the greater; or fub.
tracted from the greater, leaves a remainder equal to the
leffer.
Let 3, the difference of two unequal numbers, 5 and
8, be added to 5, the leffer, and the fum will be 8, e-
qual to the greater; or, if the difference be fubtracted
from the greater, 8, the remainder will be 5, equal to
the leffer.
3
XI. None
16
*
NOTATION.
Chap. I..
*
XI. None but fimilar or like things can be added or
fubtracted.
That is, units only can be added to units, tens only
to tens, hundreds only to hundreds, &c.; for if unlike
things were added together, the fum would be falfe.
Thus, if 4 units were added to 5 tens, the ſum would
be 9; but neither 9 units, nor 9 tens, nor 9 of any o-
ther name.
In like manner, farthings only can be added to far-
things, pence only to pence, thillings only to fhillings,
pounds only to pounds, ounces only to ounces, &c.
for if 3 farthings were added to 4 pence, the fum would
be 7, but neither 7 farthings, nor 7 pence.
For the fame reafon, sheep can only be added to
fheep, cows only to cows, horfes only to horses, &c.:
for if 7 theep were added to 8 cows, the fum would be
15; but neither 15 fheep, nor 15 cows.
The fame way of reaſoning may be uſed with reſpect
to fubtraction.
XII. Any whole is equal to all its parts.
COROLLARIE S.
1. If a ſignificant figure, by itſelf, or with a cipher
or ciphers annexed, be divided by 9, the remainder
will be equal to the fignificant figure taken in its fimple
value.
Thus, if 8, 80, 800, 8000, &c. be divided by 9,
the remainder will be 8: for if 8 be divided by 9, the
quotient will be o, and the remainder 8. Again, 80
is equal to 8 10's; that is, to 8 9's, and 8 1's, or 8.
And, in like manner, 800 is equal to 80 10's; that is,
to 80 9's, and 80 1's, or 80, whofe remainder is al-
ready ſhown to be 8.
The ſame way of reaſoning may
be applied to any other fuch number.
2. If any number be divided by 9, the remainder will
be equal to the ſum of the figures of the ſaid number ta-
ken in their ſimple value, or to the excefs above the 9's
contained in the faid fum.
For let the number be reſolved by axiom VI. into its
conftituent
Chap. I.
17
NOTATION.
conflituent parts, then, by the preceding corollary, the
fignificant figure of each part will be the remainder of
that part when divided by 9, and fo the remainders of
the feveral parts will be the figures of the given number;
out of which, if need be, let the 9's be taken away,
and the excefs will be the remainder.
A-
3. Hence the remainder arifing from a number divided
by 9 is found by adding the figures of the faid number.
Thus, the remainder of the number 231 divided by 9
is 6; for 2, 3, and 1, added together, make 6.
gain, the remainder of 84632 is 5; for 8, 4, 6, 3, and
2, added together, make 23, whoſe excefs above the
9's contained in it is 5, found by adding 2 to 3.
4. Numbers expreffed by the fame figures, however
different the value of the numbers be, have the fame
remainder when divided by 9. Thus, 436, 46;, 346,
364, 643, and 634, have all the fame remainder
when divided by 9, becaufe the fum of their figures is
the fame.
VI. An explanation of characters fometimes used for
the fake of brevity.
= Denoting equal to, is the fign or mark of an e-
quation; and fignifies, that the numbers betwixt which it
ſtands are equal to one another.
+Denoting plus, more, or added to, is the fign or mark
of addition; and fignifies, that the numbers it is placed
between, are to be added together; as 6+2=8;
that is, 6 plus 2, or 6 having 2 added to it, is equal to 8.
-Denoting minus, or lefs, is the fign of fubtraction;
and fhews, that the latter of the numbers betwixt
which it ftands is to be fubtracted from the former; as
734; that is, 7 minus 3, or 7 having 3 fubtracted
from it, is equal to 4.
X Denoting into, or multiplied into, is the fign of
multiplication, and fignifies, that the numbers betwixt
which it flands are to be multiplied together; as
3X5=15; that is, 3 into 5, or 3 multiplied into 5,
is equal to 15.
C
- De.
18
Chap. I.
ΝΟΤΑΤΙΟ Ν.
};
Denoting divided by, is the mark of divifion; and
fignifies, that the former of the numbers betwixt which
it ftands is to be divided by the latter; as 124
that is, 12 divided by 4, is equal to 3. Or this mark
may be confidered as fetting the dividend above and the
divifor below the line; and fo it becomes a fractional
expreffion, fignifying, that the numerator is to be divi
ded by the denominator; as 23; that is, 12 divi-
ded by 4 is equal to 3.
Divifion is alfo marked thus, 4)12(3.
:: Denoting fo is, is the fign of geometric propor•
tion; and is placed betwixt the fecond and third terms
of four proportional numbers; thus, 4:8:12:24;
that is, as 4 is to 8, fo is 12 to 24-
This mark is called the radical ſign; and has a fi
gure or index fet over it, to denote fuch a root of the
number before which it ftands.
Thus, √25=5 fig-
Thus,√25 3
nifies, that the ſquare root of 25 is 5; and 64 = 4
↓ =
denotes the cube root of 64 to be 4. When the fquare
root is meant, the radical fign is ufually left blank, or
has no figure placed over it.
VII. Practical Queſtions.
Q. I. Troy was taken and deftroyed by the Greeks,
two thoufand eight hundred and twenty years after the
creation, and one thouſand one hundred and fixty-four
years after the flood, and four hundred and thirty-fix
years before the building of Rome, and eleven hundred
and eighty-four years before the birth of Chriſt, and two
thouſand nine hundred and forty-four years before the
acceffion of King George III. to the throne of Britain :
How are theſe dates expreffed in letters?
2. The inhabitants of a certain country are twenty-
five millions four thousand and feventy men, thirty-four
millions ſeven hundred and five thouſand eight hundred
and two women, fifty millions and four hundred boys,
fifty-fix millions fix thouſand and eight girls: How are
thefe numbers expreſſed in figures ?
3. A
Chap. II.
19
ADDITION.
3. A farmer has in his granary 470900608 grains of
wheat, 80769000240 of barley, 702000678000 of rye,
and 27300845302912 of oats: How are thefe numbers
read, or expreffed in words?
A
CHA P. II.
ADDITION.
Ddition is the collecting of two or more numbers
into one fum or total.
I. Addition of Integers.
RULES.
I. Set figures of like place under other, viz. units
under units, tens under tens, &c. See axiom XI.
II. Beginning at the loweft place, fet down the right
hand figure of the fum of every column, and carry the
reft as fo many units to the next fuperior place. See
axioms II. III. IV.
EXAMPLE
I.
234
687
Becauſe fimilar or like things only can be added, I
place the numbers as directed in Rule I. viz. units un-
der units, tens under tens, &c. as in the margin.
Then, beginning at the loweſt place, viz. that of 453
units, I fay, 4 units and 3 units make 7 units,
which I fet below in the place of units; then 3
tens and 5 tens make 8 tens, which I ſet below in
the place of tens; then 2 hundreds and 4 hundreds
make 6 hundreds, which I fet below in the place of
hundreds, and find the fum or total to be 687.
From the repetition of the former example on
the margin, it is plain, that it is a matter of in- 453
difference which of the numbers be placed upper-
moſt.
1
EXAMPLE II.
234
687
Having placed the numbers, units under units, &c.
C 2
as
20
Chap. II.
ADDITION.
5974
9803
7541
862
24180
as in the margin, I fay, 2 and I make 3, and
3 make 6, and 4 make 10; which being juſt 1
ten, and nothing over, I fet the right-hand fi-
gure o in the place of units; and becaufe ten in
any lower place makes but one in the next fupe-
rior place, I carry my ten, as directed in
Rule 11. faying, 1 ten, collected out of the u·
nits, and 6 tens make 7 tens, and 4 make 11, and o
makes but ftill 11, and 7 make 18; here again I fet
down the right-hand figure 8, in the place of tens, and
carry the remaining figure 1, being i hundred, to the
next place, viz. that of hundreds; and having in like
manner added up this column, the amount is 31; fo I
fet down the right-hand figure 1 in the place of hun-
dreds, and carry the remaining figure 3 to the next
place or column; which being alfo added, amounts to
24; I fet the right-hand figure 4 below, in its proper
place, and the remaining figure 2, which belongs to the
next place, I fet on the left hand, there being no figure
in the next place to which it can be carried. So the
fum or total is 24180,
The reafon of the operation will
ftill further appear by taking the fum
of each column feparately, and then
adding them into one total, as in
the margin.
5974
9803
7541
862
Sum of the
10 units
17. tens
30.. hundreds
21... thouſands
24180 total
In adding integers, fome teach young beginners to dot
at every ten, and ſetting down the excefs, to carry for
every dot. Thus, in adding the example in the margin,
I fay, 4 and 8 make 12; where I dot,
fet down the exceſs 2, and carry 1 for the
dot to the next column; faying, 1 carried
and 6 make 7, and 8 is 15: where I again
dot, fet down the excefs 5, and carry to
7 8.8.
9.6 4
I 7 5 2
the
Chap. II.
21
ADDITION.
10;
the next column; faying, I carried and 9 is
where I dot, and fet down the 7 as the excefs. The
I carried for the dot is placed on the left hand. This
method is too low for any but a child.
MORE EXAMPLE S.
Ex. 3.
78943745
607 38937
84976858
Ex. 4.
8467472
7893968
47637
93543945
83745
48732790
4897
85476437
548763
48763728
674376
7845963
9848937
94769
53700480
II. Addition of the parts of integers, ſuch as ſhillings,
pence, farthings, ounces, &c.
RULES.
I. Place like parts under other; viz. farthings under
farthings, pence under pence, &c. See axiom XI.
II. Begin at the loweft of the parts, and carry ac-
cording to the value of an unit of the next fuperior deno.
mination; viz. for every four in the fum of farthings
carry 1 to the pence, and for every twelve in the pence
carry to the fhillings, &c. The reafon of this rule
is plain from the tables of coin, weights, and meaſures.
III. If you carry at 20, 30, 40, 60, or any juft
number of tens, as in adding fhillings, degrees, poles,
minutes, feconds, &c. proceed with the column of u-
nits as in addition of integers, and from the fum of the
column of tens carry 1 for every two, or 1 for every
three, &c. according as 20 or two tens, thirty or three
tens, &c. make an unit of the next fuperior denomina-
tion. The reafon appears plain in the following opera-
tions.
I. MO.
22
Chap. II.
ADDITION.
"}
1. MONE Y.
TABLE.
4 farthings
I penny
12 pence
make
20 fhillings
I fhilling
1 pound
1.
S.
d. f. or q.
Marked thus.
1 = 20 = 240 = 960
T
1. is put for libra, a pound; d. for denarius, a penny;
and
9. for quadrans, a fourth-part; but f is now the
more ufual mark for farthings.
There are a few other denominations of money, but
never uſed in keeping accounts, viz. a groat, in value
4 d. a tefter 6d. a crown 5s. half a crown 2 s. 6d. a
noble 6 s. 8 d. an angel 10 s. a mark 13 s. 4 d. a gui
nea 21 s. half a guinea to s. 6d. and a quarter of a
guinea 5 s. 3 d.
That the learner may proceed in addition of money
with the greater eafe, it will be proper he get the fol-
lowing table by heart.
MONEY.TABLE.
f.
d.
d.
S.
S.
1.
4
I
12 = I
20=
8
2
24 = 2
40
12
3
36
3
60
23
16
4
48
4
80
4
20
5
60
5
100
24
72
120
6
28
7
84
7
140
7
32
8
96 =
8
160 =
8
36
Q
108
9
180
9
O 120 =
:0
200 =
10
EX.
Chap. II.
23
ADDITION.
(10) (20) (12) (4)
L. S. d. f.
74
96
58
63
18
I I 3
9
10 2
17
8
I
I I
9
2
18
4
EXAMPLE.
Having, according to Rule I. pla-
ced like parts under other, viz. far-
things under farthings, pence under
pence, &c. and in each of theſe
denominations, units under units,
tens under tens, as in the margin,
I begin with the loweſt of the parts, 293
viz. the farthings; and fay, 2 far-
things and farthing make 3 farthings, and 2 make 5,
and 3 make 8; which, by the money-table, is 2 fours,
or 2 pence, and nothing over; wherefore I place o be-
low in the place of farthings, or rather leave that place
blank, and carry my 2 pence to the place of pence, as
directed in Rule II. faying, 2 pence, collected out of
the farthings, and 9 make 11, and 8 make 19, and 1
(paffing the o) make 20; to this fum of my units I add
my tens. Thus, 20 and 1 ten make 30, and 1 ten more
make 40 pence; which, by the money-table, is 3 twelves,
or 3 fhillings, and 4 pence over; thefe 4 pence 1 ſet
below in the place of pence, and carry my 3 fhillings to
the place of fhillings. Thus, 3 fhillings, collected out
of the pence, and i fhilling make 4, and 7 make 11, and
9 make 20, and 8 make 28; and becauſe in (hillings we
carry at a juſt number of tens, viz. at 20, I ſet my right-
hand figure 8 below in the place of units, as directed in
Rule III. and carry my 2 tens to the place of tens.
Thus, 2 tens collected out of the units, and i ten make
3 tens, and I make 4, and 1 make 5 tens, or 2 twenties,
and i ten over; and becauſe 2 tens, or i twenty, make
an unit in the next place, viz. that of pounds, I fet my
I ten below in the place of tens, and carry my z twenty
thillings, or 2 pounds, to the place of pounds; which, being
integers, are added as taught in addition of integers.
It is ufual to fubjoin the farthings to the
L. s. d.
pence by way of fraction, as in the mar-
gin, where the former example is tranfcri
bed in this form for the learner's inftruc.
tion; in which denotes one farthing,
two farthings, and 3 three farthings.
4
74 18 113
96 9101
58 17
63 11
81
a
293 18 4
The
24
Chap. II.
ADDITION.
}
The method of adding money explained above is the
moft approved, and generally practifed by men of buſi
nefs; but yet other methods are fometimes uſed; fuch
as,
(10) (20) (12) (4)
d. f.
L. s.
L.
1. Some, in inſtructing young children, teach them to
dot at the figure which makes an unit of the next ſuperior
denomination; and, after ſetting down the overplus, they
carry for every dot. Thus, in
adding the example in the margin,
I begin with the farthings, and fay,
1 and 3 make 4; where I dot, and,
fetting down the 2 as an overplus,
I carry for the dot, and ſay, 1
carried and 8 make 9, and 9 is
eighteen; where I dot, and fay, 192
6 of overplus and 10 make 16;
4
47
98
45
~ 3.
17.
10. 2
14.
13
9.
8 I
6
2
+
where I again dot, and, fetting down the overplus 4, I
carry 2 for the dots, and proceed, ſaying, 2 carried and
I make 15,
13
and make 19,
19, and i ten on the left make
29; where I dot, and go on, faying, 9 of overplus and
7 make 16, and I ten on the left make 26; where I a-
gain dot, and, fetting down the overplus 6, I carry 2
for the dots to the pounds, which are integers, and may
be added either by dotting at every ten, or by the ufual
method of adding integers. This method of dotting in
the addition of money is childish, and ought not to be
too far indulged.
(10) (20) (12)
L. S. d.
84
18
II
2
63
16
10
10
2. Some follow the common method in adding the
farthings and pence; but in adding
the fhillings proceed thus: 2 fhillings
carried from the pence and 4 make
6, and 6 make 12, and 8 make 20;
and to this fum of the units I add the
tens on the left hand, ſaying, 20 and
I ten is 30, and 1 is 40, and 1 is 50;
which, by the money-table, is 2 l.
10 s.; wherefore I fet down the 10 s. 322
and carry the 21. to the pounds,
which are integers, and added accordingly.
78
95 4
IÓ
7 1/1/1
3. In adding up large accounts, fome dot at 60 in
the
Chap. II.
ADDITION.
25
(10) (20) (12)
L. S.
45
48 17
83
d
19
6 플
​10
14.
I
15
9 ž
84
13
4
9
10.
I I
10
72
86
78
64
56 18.
79
80
85
10
16
I
I [
10
10
8
93/
00
the pence, and for every dot carry
5 to the fhillings; and in adding the
fhillings they dot likewiſe at 60, and
for every dot carry to the pounds.
3
Thus, in adding the example in the
margin, I begin with the farthings,
which I add in the ufual manner,
and find they amount to 23; which,
by the money-table, is 53 d.; where.
fore I fet down the 3, and carry 5
to the pence, which I proceed to
add, by ſaying, 5 carried and 8 is
13, and 10 is 23, and 11 is 34, and
10 is 44, and 11 is 55, and so is
65; where I dot, and go on, fay.
ing, the overplus 5 and 4 is 9, and
9 is 18, and 1 is 29, and 10 is 868
39, and 6 is 45; which, by the
money-table, is 3 s. 9 d.; accordingly I fet down the
9 d. and
3 s.
added to 5 s. for the dot make 8 to be
carried to the fhillings; which I proceed to add, by fay-
ing, 8 carried and 10 is 18, and 11 is 29, and 6 is 35,
and I ten on the left is 45, and 8 is 53, and I ten on
the left is 63; where I dot, and go on, faying, 3 of o-
verplus and 10 is 13, and 9 is 22, and 3 is 25, and 1 ten
on the left is 35, and 5 is 40, and i ten is
50, and 4 is
54, and I ten is 64; here again I dot, and go on, faying,
4 of overplus and 7 is 11, and 1 ten is 21, and 9 is 30,
and ten is 40, which is juſt 21 and no overplus to be
fet down; and theſe 2 1. with 6 1. for the two dots, make
8 to be carried to the pounds, which are integers, and
may be either added in the ufual method, or by dotting
at every 100.
4. In cafting up large accounts or bills, fome chuſe
to divide them into parcels, and then, by any of the
methods taught above, they caft up each parcel fepa-
rately, and afterwards add the fums of the feveral par-
cels into one total; as is done by the following ex-
ample.
D
(10.)
A
26
Chap. II.
ADDITION.
(10) (20) (12)
L. S. d.
42 14
8
36
15
4
4 /
38
17102 L.
118 7 11/
S.
d.
32
18 9
48 16 11
50
19
ΙΟ
132 15 62
56
8
6 3
54
6
57
8
H
52 14 7
220 9 10
Total 471 13
4
It remains now to be obferved, with refpect to the
empty places, whether in fhillings, pence, or farthings,
or thofe in the left-hand columns of the fhillings and
pence, that, inftead of filling them up with ciphers, as
the common practice was fome time ago, the general
faſhion now, in moft cafes, is, to leave them blank; and
this remark holds, not only in addition of money, but
in every other kind of addition; nay, extends to every
part of arithmetic, whether in addition, fubtraction,
multiplication, or divifion.
MORE
Ex. I.
(10) (20) (12)
L. S. d.
789 17 10
476 14 II
894 18 9
438 13 10
Hamit
*+
EXAMPLES.
Ex. 2
(10) (20) (12)
L. S. d.
483 15 11/1/
II
724 13 4
865 18
7
8
978 14
685 19 6
Hamk
4 /
643 12
784 15
8
538 17
435 6
10
945
9
8
947
8
6 3
782
5
IO
Ex. 3.
Chap. II.
27
ADDITION.
Ex. 3.
Ex. 4.
(10) (20) (12)
(10) (20) (12)
L. S.
d.
L.
f. d.
5748 19
I
11 2
4786 17
IO
4875 17
10 1/1/20
8490
14
9387 13
6453 14
5874 10
487
5786 13
4
8327
15
8
7509
9
7 1/1/15
9482 9 II
6742
8
6
6093
10
894
10 2
7800 17
6
672
18
508 15 10 4/
95
16
I I
11 //
75
8 19
10
2. A VOIRDUPOIS WEIGHT.
TAB L
E.
16 drams
16 ounces
I ounce.
I pound.
28 pounds
4 quarters
20 hundreds
make
Marked thus.
I quarter.
I hundred.
i tun.
T.
C.
2
lb.
02.
I
1 = 20 = 802240
]= = 4 4 = 112 =
dr.
35840 = 57 3440
1792 = 28672
By Avoirdupois weight are weighed, butter, cheeſe,
rofin, wax, pitch, tar, tallow, foap, falt, hemp, flax,
beef, brafs, iron, fteel, tin, copper, lead, allum, and all
grocery wares,
Note, 19 C. of lead make a fodder.
In adding the following example, I begin with the
ounces, and ſay, 15 and to make 25; which being a-
bove 16, 1 dot, and carry away the excefs 9, faying, 9
of excefs and 6 make 15, and 8 make 23; where I a-
gain dot, and carry away the excefs 7, faying, 7 and 2
is
9, and I ten on the left is 19; where I dot, and proceed
with the excess 3, faying, 3 and 4 is 7, and 1 ten on the
left is 17; where I dot, and carry the excefs 1, faying,
D 2
I
1
28
Chap. II.
ADDITION.
I
1 and 5 is 6, and I ten on the left is 16; where I again.
dot, and there being no exceſs, I have nothing to ſet
down.
(10) (20) (4) (28) (16)
T. C. Q lb.
QZ.
74
19
85
17
68
13
52
50
20
18
738
321
27.
15.
24.
14.
I
20
I 2.
3
19.
8.
10 2
18.
6
48
9 3
16
IO.
97 5
3 15
20
478 15 3
I now proceed to add the pounds; faying, 5 carried
from the ounces, viz. one for every dot, and 3 make
8, and 6 make 14, and 1 ten on the left is 24, and 8
make 32; which being above 28, I dot, and go on,
faying, 4 of exceſs and I ten on the left is 14, and is
23, and I ten on the left is 33; where I again dot, and go
on, faying, 5 of excefs and 20 is 25, and 4 is 29; where
I dot, and proceed, faying, I of excess and 2 tens on
the left make 21, and 7 make 28; where I dot, and the
2 tens, or 20, on the left, I fet below.
I fhould now proceed to add the quarters; faying, 4
carried from the pounds and I make 5, &c; but as you
carry here I for every four, the quarters are added ex.
actly as the farthings, in addition of money. In the hun-
dreds you carry at 20; which, therefore, are added as
fhillings. The tuns are integers; and added accord.
ingly
The excess may fometimes be known without actual
addition. Thus, in cafting up the ounces of the above
example, it is eafy to perceive, that I, ounces wants
only i to complete the pound; accordingly I imagine 1
to be taken from 10; and, after dotting, I go on with
the exceſs 9, faying, 9 and 6 make 15; and again, as
15 wants only I, I imagine this to be taken from 8,
and
Chap. II.
29
ADDITION.
and fo I dot, and proceed with the excefs 7, faying, 7
and 2 make 9, &c.
The accounts may be kept clean by making the dots.
on a blotter, or piece of wafte paper laid under your
hand.
Men of buſineſs frequently add both the ounces and
the pounds as integers; divide the fum by 16 or 28;
and, fetting down the remainder as the excefs, carry the
quotient.
Retailers of filk, worfted, thread, and fach other
fmall wares, have occafion only for pounds, ounces,
and drams, and this they call avoirdupois ſmall weight;
and in retailing fome of thefe, it is ufual to divide the
cunce into four quarters inſtead of fixteen drams.
The denomination on the left hand, whether tuns,
hundreds, or pounds, is always integers, and added as
fuch; which in the following examples is pointed out
by (ro) fuperinſcribed, being the number at which you
carry in integers. This remark may be extended to o•
ther forts of addition; as will appear in what follows.
MORE EXAMPLES.
Ex. I.
Ex. 2.
(10) (20) (4) (28)
T. C. Q: lb.
(10)(4) (28) (16)
C. Q. lb. oz.
34 14 3
15
25
1 18
12
36 18 1
32 16
14
12
3
14
14
2
12
45
2
24
15
2 2 2 2 2
0046 20
30
28
25
26
22
38
12
I
IO ·
36
I
18
10
15
3
23
64 3
22
13
19 2
20
62
I 20
II
13
3 24
10 1 18
48 3
25
9
52
1 19
8
15 3
12
54
2
8
6
Ex. 3.
30
Chap. II.
ADDITION.
Ex. 3.
(10) (16) (16)
lb. oz. dr.
10
4 14
3456 78 4
6
Ex. 4.
(10)(16)(4)
lb. oz. qrs.
4 12
3
7 10 2
14
I
5 13 3
II 2
I I
5
13
14
12
ΙΟ
10
8
15
14
8
10 3
8
12
13
2
2 6 10
8
5
7 9
15 3
8 I
356∞ 4∞ 3
3. TROY WEIGH T.
24 grains
TABL E.
20 penny-weights
12 ounces
}
I penny-weight
make I ounce
1 pound Troy
Marked thus.
lb. Oz. dw.
gr.
I = 12 = 240 = 5760
By Troy weight are weighed gold, filver, jewels, am.
ber, electuaries, and liquors.
Note. That 1 lb. Avoirdupois is equal to 14 oz. 1 1 dw.
15 gr. Troy.
+
(10) (12) (20) (24)
oz. dw. gr.
lb.
48
11 18 21.
42
10
14
18.
40
9
16
20
36
8
15
22.
38
53
6 17 13
In adding the example in the
margin, I begin with the grains,
and ſay, 13 and 4 make 17, and
I ten on the left make 27, which
being above 24, I dot, and go on
with the excefs 3, faying 3 and
2 make 5, and 2 tens, or 20, on
the left, make 25; where I again
dot, and proceed with the exceſs
1, ſaying, 1 and 20 make 21, and 261
8 make 29; where I dot, and go
10 10 14.
10 14 12
on with the excefs 5, faying, 5 and 1 ten on the left
make 15, and 1 makes 16, and 20 on the
left make
36.
Chap. II.
ADDITION.
31
36. I ſet down the excess 12, and carry away 4 for
the dots, to the penny-weights.
In
In cafting up the penny-weights you carry at 20,
which therefore are added exactly like fhillings.
the ounces you carry at 12, which are therefore added
as pence. The pounds are integers, and added as fuch.
Some men of buſineſs add the grains as integers, and,
fetting down the fum on a feparate paper, divide it by
24, and then place the remainder as the exceſs under
the grains, and carry the quotient to the penny-weights.
MORE
Ex. 1.
(10)(12) (20) (24)
EXAMPLES.
Ex. 2.
lb.
oz. dw. gr.
lb.
(10)(12) (20) (24)
02. dw. gr.
72
I I 19 23
42
ΙΟ
14
22
74 ΙΟ
13
21
44
9
16
18
70 9
16
18
40
I I
19
2I
54
8
12
22
36
8
15
17
56
6
10
20
32
6
8
12
63 10
8
16
35
ΙΟ
62 II 6
10
38
noo
6 13
I I 17 8
60 10
18
14
40
11 18 23
4. APOTHECARIES WEIGHT.
TABLE.
20 grains
3 fcruples
8 drams
1 fcruple
I dram
make
I ounce
I pound
12 ounces
Marked thus.
tb 3 3
Э
gr.
1 = 12 = 96 = 288 = 5760
By this weight apothecaries compound their medi-
cines, their pound being the fame as the pound Troy,
but differently divided; but yet they buy and fell their
drugs by Avoirdupois.
In
32
Chap. II
ADDITI O N.
(10) (12) (8) (3) (20)
3 3 Э 870
44
43
In adding the example in
the margin, I begin with the
grains; and becauſe in grains
we carry at 20, they are added
as fhillings; you add the ſcru-
ples as integers, and carry 1
for every 3 of that fum; you
add the drams in the fame man-
ner, and carry 1 for every 8;
in the ounces you carry at 12,
which are therefore added as
pence; the pounds are integers, and added as ſuch.
MORE
Ex. I.
I I 7 2 19
ΙΟ 5
I 15
48
9 3
2
14
36
8
2
I
18
32
IO 6
2
12
30
II 4
I
17
238
2 7 I
I 15
EXAMPLES.
(10)(12) (8) (3) (20)
1b 33 3 → gr.
IO 6 2
Ex. 2.
*
(10) (3)(20)
3
Э
gr.
36
10
18
1 4
122
∞ + 3 CO
8
32
18
14
23
28
45 7 3 I
II 7
I
14
10
I I
9
5
3
2
I 2
I
19
4
I
18
7 2
17
14
48 2
956
234322
I
18
28
32
4I
35
2
14
I
16.
2
19
I
12
26
2 15
25
I 17
53
6 2
18
53 2 13
7
pounds
WOOL
5. WOOL WEIGH T.
[ I clove.
2 cloves
I ſtone.
1 2
2 ftones
6 todds
2 ways
facks
I todd.
make
I wey.
I fack.
I laſt.
Marked thus.
Laft. fack. wey.
todd. ftone. clove.
lb.
1 = 12 = 24 = 156 = 312 = 624 = 4368
&
Ex. 1.
Chap. II.
ADDITION.
33
Ex. I.
(10) (12) (2) (61) (2)
Laft. fack.wey.tod. fto.
72 II 1 6 I
Ex. 2.
(10) ( ) (7)
Ston.clov. lb.
14
I 6
70 10 I 3
I
28 I
5
74 9 I
5
I
23
1
4
63
8
I
3
I
18
I
3
82
I I
I
5
I
17
I
2
86
10
4 I
15
I 5
In cafting up the
todds, I fay, 3 carried from the
ftones, and 4, is 7, and 5 is 12, and 3 is 15, and 5 is
20, and 3 is 23,
and 6 is 29; in which I find 6 four
times, making 26, and 3 of excefs: accordingly I fet
down the exceſs 3, and carry 4 to the weys. The man-
ner of adding the other denominations is obvious from
the table, or from the fuperinfcribed numbers.
Note. If the fum of the todds contain an odd num.
ber of weys, as 1, 3, 5, 7, &c. then the exceſs will be
either, 1, 2, 31, 41, or 51.
6. DRY
MEASURE.
TABL E.
2 pints
I quart.
2 quarts
I pottle.
I gallon.
2 gallons
I peck.
make
1 bushel.
I coomb
2 pottles, or 8 pints
4 pecks, or 8 gallons
4 bushels
2 coombs, or 8 bufhels
5 quarters, or 40 bushels
Marked thus.
Lo. Qrs.coom. bush. peck. gall. pot.
I quarter.
I load.
grts.
pts.
1=5=10 = 40 = 160 = 320 = 640 = 1280 = 2560
320—640—1280
By this is meaſured grain; as wheat, barley, peaſe,
&c.; alfo falt, fand, fruit, oyfters, &c.
Note. That 33 cubic inches make a corn-pint,
E
208€/
f
34
Chap. II.
ADDITION.
268 cubic inches make a corn-gallon, and 2150 cubic
inches a Winchefter bufhel, which is a cylindrical
veffel 18 inches wide, and 8 inches deep.
Ex. I.
(10) (5) (8) (4)
Ex. 2.
(10) (40) (8)
Load. bu. gall.
Load. qrs.bu. pec.
28
4 7
3
34 39 7
18
3 6
2
36 32
6
14 2 5
I
24
28
5
15 3 4
3
18 36 4
42 4 7
2
25 38 3
48
2 6
I
20 35 5
32 34 6
36 3 5 3
In cafting up the bushels in Ex. 2. becauſe we carry
at 40, a juſt number of tens, I add up the right-hand
column as integers, fet down the right-hand figure of the
fum, and carry the reſt to the left hand column; which
I alfo add up as integers, and carry 1 for every four of
the fum.
In meaſuring coals the following table takes place.
4 pecks
bufhels
9
4 quarters
Ex. I.
}
make
I buſhel.
I quarter.
I chaldron.
Ex. 2.
(10) (36) (4)
(10) (4) (9)
Chal. gr. bu.
Chal. bu. pec.
54
3
8
36 35 3
28 2 7
52
I
6
46
3
5
48
2
32
3
36 2
5
32 2 2 2
578
32
30. 2
28
34. 3
25 28
26
2.
32. 3
24 25. I
18 16 2
In cafting up the bushels in Ex. 2. where we carry at
36,
Chap. II.
35
ADDITION.
36, I ſay, 4 carried from the pecks and 16 make 20,
and 25 make 45; which being above 35, I dot, and
carry the excefs 9, faying, 9 and 32 is 41; where I a-
gain dot, and go on with the excess 5, faying, 5 and
28 make 33, and 4 make 37; where I dot, and proceed
with the exceſs 1, faying, 1 and 3 tens, or 30, on the
left, make 31, and as 31 wants only 5 to complete the
chaldron, I imagine that taken from 30; where I dot
again, and proceed with the excefs 25; and as the next
number 35 wants only 1, I fuppofe that I taken from
25, and dotting again, I fet down the excefs 24, and
carry 5 for the dots to the chaldrons, which are integers.
The man of buſineſs may either follow the above me-
thod; or he may
or he may add up the bufhels as integers, di-
vide their fum by 36, fet down the remainder as the
excefs, and carry the quotient to the chaldrons.
The dry meaſures in moſt parts of Scotland differ from
thoſe uſed in England, and are divided as in the follow-
ing table.
4 lippies or forpats
4 pecks
4 firlots
16 bolls
Ex. I.
[ 1 peck.
I firlot.
make
I boll:
(10) (16) (4) (4)
Ch. b. f.
b. f.
25 15 3 2
i chalder
Ex. 2.
(10) (4) (4) (4)
B. ƒ p. l.
231
3
p.
28
2
18 14 2
I
34
I
3
32
35
12
3 2
30
I 2
I
28
10 I
3
26
3 I
2
30
I I 3 I
28
I
3
3
42
13 I 3
45
2
I
I
40 10 2 1
33 3
2
2
The bolls in Ex. 1. are added as ounces or drams in
Avoirdupois weight; but the bolls in Ex. 2. are inte-
gers, and added as fuch.
E 2
7. WINE.
36
Chap. II.
ADDITION.
2
7. WINE-ME A SU R E.
TABL E.
2
pints
I quart.
4 quarts
I gallon.
10 gallons
18 gallons
I anchor.
I runlet.
}
31
gallons
make
1 barrel.
42
gallons
63
gallons
2 2
hog heads
pipes
Marked thus.
I tierce.
I hogshead.
I pipe or butt.
I tun.
Tun. pipe. bhd. gall. grt.
pt..
1 = 2 = 4252 = 1008 = 2016
A puncheon, ftrictly speaking, is 4 anchors, or 80
gallons; but any cafk betwixt a hogshead and a pipe is
called a puncheon.
All fpirits, mead, perry, cyder, vinegar, oil, and
honey, are meaſured as wine.
cubical inches, and the
The wine pint contains 28
wine-gallon contains 231 cubical inches,
Ex. I.
(10) (2) (2) (63) ´
Tuns. pip.hhd gall.
Ex. 2.
(10) (63) (8)
Hhd. gall. pts.
48 I I 36
45 23 7
64 I
I
48.
52
38 6
52 1
I 30.
43
46
46 5
43
I 1 23
25
18 7
32 I I 56.
73
25 6
45 I I 38
65 30 7
In cafting up the gallons, where we carry at 63, I
proceed as follows, viz. in Ex 1. I readily perceive
that 46 wants only to complete the hogfhead; accord-
ingly I imagine 7 taken from 38, and, after dotting at
56, I go on with the excefs 31, faying, 31 and 3 make
- 1
34
Chap. II.
37
ADDITION.
34, and 2 tens, or 20, on the left, make 54; and as
54 wants only 9, I imagine 9 taken from 30, fo I dot
at 30, and proceed with the excefs 21, faying, 21 and
8 make 29, and 40 on the left make 69; which being
above 63, I dot again, and go on with the exceſs 6,
faying, 6 and 36 make 42; which I fet down, and
carry 3 for the dots to the hogfheads. The manner of
operation is the fame in Ex. 2.
The man of buſineſs may either caft up the gallons as
directed above; or he may add them as integers, divide
the fum by 63, fet down the remainder as the exceſs,
and carry the quotient.
8. BEER and ALE MEASURE.
TABLE.
4
2 pints
quarts
9 gallons
firkins
kilderkins
1 barrels
227 22
hogfheads
butts
I quart.
I gallon.
1 firkin.
1 kilderkin.
make
1 barrel.
Marked thus.
I hogshead.
I butt.
I tun.
Tun. butt. hhd. barr. kild. fir k. gall.
qrt.
Pt.
1 = 2 = 4 = 6 = 12 = 24 = 216 ≈ 864 = 1728
4
The beer or ale pint contains 35 cubic inches, and
the gallon 82. ´
In fome places, 8 gallons of ale is efteemed a firkin;
and in fome places 34 gallons make a barrel.
Ex. I.
1
38
Chap. II.
ADDITION.
1
Ex. I.
(10)(3) (2) (9)
Hhd.kild.firk.gal.
32 2 I 8
Ex. 2.
(10) (3) (36)
But. bar. gall.
68
2
35
30 I I
7
42
I
32
28 I I
5
40
2 28
40 2 1
6
34
1
15
42
I I
7
22
2
12
36 2
1
6
18
I 24
9
2 18
37 I I 7
The gallons in Ex. 2. are added like the bushels in
Ex. 2. of coal-meaſure.
In Scotland the denominations of liquid meaſure are
as in the following
TABL E.
4 gills
I mutchkin.
2 mutchkins
1 chopin.
2 chopins
make
1 pint.
2 pints
I quart.
4 quarts
16 gallons
I gallon.
į I hogfhead.
The Scots mutchkin is ſomewhat leſs than the Engliſh
pint.
Ex. 1.
Ex. 2.
42
(10) (16) (4) (2)
Hhd. gall. qrt. pt.
14 3 I
48
I 2
50
52
15
I
ΙΟ
45
13
42
14
28
I 2
22 10 3
32H 3-2 33
(10) (2) (2) (4)
Pt.chop.mut.gil
14 I I 3
2
I
23
II
2
I
25
I
I
3
3
I
26
II
2
1
I
15
I I I
I
18
I
I
3
1
20
I
I 2
I
24 II
3
9. CLOTH.
Chap. II.
39
ADDITION.
9.
CLOTH-ME A SURE.
TABLE..
4 nails
I quarter.
4 quarters
I yard.
make
3 quarters
I ell Flemish.
5 quarters
I ell Engliſh.
A nail is equal to 24 inches, and confequently a quar-
ter is equal to 9 inches.
Hollands are generally meaſured by the ell English;
tapiſtry by the ell Flemish; and moft other things by the
yard.
Ex. I.
(10) (4) (4)
1
Ex. 3.
Ex. 2.
(10) (3) (4)
(10) (5) (4)
rd.. qrs. n.
42 3 3
El. Fl.
qrs. n.
El. Eng. qrs. n.
63
2
3
69
4 3
36 2
I
52
I
3
62
3
2
30 I 13
62
2
2
40
I
3
28 2 2
48
I
I
45
4
2
24 I 3
42
2
I
36
- 2
25 3 I
40
2
2
34
21
I
I 3
10. LONG MEASURE.
TABL
3 barley-corns, or 12 lines]
12 inches
3 feet
2 yards, or 6 feet,
5 yards, or 16 feet,
4 poles, or 66 feet,
10 chains, or 40 poles
8 furlongs, or 80 chains,
3 miles
E.
I inch.
i foot.
1 yard.
I fathom.
make pole, or perch.
I chain.
1 furlong.
I mile.
i league.
A hand, or hand's breadth, in horfemanſhip, is 4 inches.
A common pace is 2 feet 6 inches.
A geometrical pace is 5 feet.
The
40
Chap. II.
ADDITION.
The chain is divided into 100 equal parts, or links;
each link being 7-22 inches.
Ex. 1.
(10) (3) (8) (40)
Leag. miles. fur. poles.
Ex. 2.
(10) (52) (3) (12)
Poles. yds. feet. inches.
36 4 2
20
2 7
38
I I
18
2
5
24
28
3
[
ΙΟ
25
I
6
35
25
I
2
I [
24
2
7
32
16
2
I
- 9
18
1
4
26
24
3
I
12
2
3
20
36
2
2
8
17
2
3
33
23
4
2
2
976 n
5
In caſting up the poles in Ex. 1. where we carry at
40, a juſt number of tens, you add as in integers; and
from the fum of the left-hand column carry for every
four, fetting down the excess. The poles in Ex. 2. are
integers, and added as fuch.
The denominations of long meaſure uſed in Scotland
are as in the following
TABLE.
37 inches
6 ells, or 18 feet,
4 falls, or 74 feet,
10 chains, or 40 falls,
8 furlongs, or 80 chains,
I ell.
1 fall.
make I chain.
i furlong.
I mile.
The chain is divided into 100 links; each link being
8,88 inches.
Ex. I.
(10) (8) (10) (4)
Miles. fur. ch. falls.
24 7 9
8
32
Ex. 2.
(10) (4) (6) (37)
Ch. falls. ells. inches.
3 5 36.
3.
36
22 5
32
2
4
32.
18 4
6
3
45
I
3
25.
14 6
5
3
23
2
4
18
28 3 7
2
23
2 8 3
1 2
18
3
5
16.
24
2
2
24
In
Chap. II.
41
ADDITION.
In cafting up the chains in Ex 1. becauſe you carry
at 10, you work as in addition of integers. In Ex. 2.
the chains are integers.
In adding the inches in Ex. 2. I fay, 24 and 6 make
30, and I ten on the left make 40; which, being above
37, I dot at 16, and go on with the excefs 3, ſaying. 3
and :8 make 21, and 25 inake 46; ſo I dot at 25, and
proceed with the excefs 9, faying, 9 and 32 make a1;
accordingly I dot at 32, and go on with the excefs 4,
faying, 4 and 36 make 40; fo I dot at 36, ſet down
the excess 3, and carry 4 for the dots to the ells.
Men of bufinefs are at liberty to add the inches as a-
bove directed; or they may caft them up as integers,
divide the fum by 37, and, fetting down the remainder
as the exceſs, carry the quotient.
II.
LAND-ME A SU RE.
TABL
30 fquare yards
E.
I fquare pole.
40 fquare poles make 1 rood of land.
4 roods
I acre.
Note, Cuſtomary meaſure in ſome places differs from
the ftatute meaſure contained in the above table.
Land is ufually meaſured by a chain of 100 links,
whoſe length is 4 poles, or 66 feet;
chains is equal to an acre.
Ex. 1.
(10) (4) (40) (301)
Acres. roads. fq. p. fq. y.
and fo 10 ſquare
Ex. 2.
(10) (4) (40)
Acres. roods. fq.p.
25 3 39 15
48
2
35
23 2
25
24.
34
[
27
48
32
18
30
3
25
64 3
35
14.
46
3
38
52 I 31 22.
42
I
34
43
2 36 26
39 2
32
In adding the fquare yards in Ex. 1. I fay, 26 and
22 make 48; which being above 304, I dot at 22, and
F
go
42
Chap. II.
ADDITION.
?
go on with the excess 173, faying, 173 and 14 make
313; fo I dot again at 14, and proceed with the exceſs
I 11, faying, 1 and 18 make 19, and 24 make 4,1;
fo I dot at 24, and go on with the excefs 134, ſaying,
131 and 15 make 284; which I fet down as the excefs,
and carry 3 for the dots to the poles.
I
In cafting up the fquare poles, whether in Ex. 1. or 2.
you carry at 40, a juſt number of tens, which therefore
are added as integers; and from the ſum of the left-hand
column you carry for every four.
I
The denominations of land-meaſure uſed in Scotland
are contained in the following
TABL E.
36 fquare ells
40 fquare falls make
4 roods
i fquare fall.
rood of land.
I acre.
Note, That cuftomary meaſure in fome places differs
from this table.
The Scots chain conſiſts of 100 links, being in length
4 falls, or 74 feet; and confequently 10 fquare chains.
make an acre.
Four Scots acres are ſomewhat more than five English
acres.
Ex. I.
Ex. 2.
(10) (4) (40) (36)
Acr. roo. fq.f. fq.el.
(10) (4) (40)
Acr. roo. fq.f.
68
72 3 38 34
I 35 26
42
28
í
62
2 24 30
23
I
231
34
3 32
35
43 3 37 35
46 3
3 26
45
4I I 33 28
2 28 32
35
2 28
53
I 25
The fquare ells in Ex. 1. are added like the bushels
in coal-meaſure.
12. SU.
Chap. II.
43
ADDITIO N.
12.
SUPERFICIAL
MEASURE.
TABL E.
144 fquare inches
9 ſquare feet
}
■ fquare yard.
By this meaſure are meaſured board, glafs, pavements,
plaiſtering, wainſcotting, tiling, flooring, &c.
{
make Şi fquare foot.
Ex. 1.
Ex. 2.
Ex. 3.
(10) (9)
Yar. fee.
(10) (9)
(10) (9)
Yar. fee.
Yar. fee.
72 8
42
43
7
36
53
38
32
34
5
22
7
74
5 97
5
6
68
6
18
8
85
4
35 4
24
6
42
8
32
7
35 4
26
7
If there be inches, they muſt be added as integers,
then dividing their fum by 144, the remainder will be
the excefs, and the quotient muſt be carried to the feet.
13. SOLID
MEASURE.
1728 folid inches
27 folid feet
}
make
{
I folid foot.
I folid yard.
By this is meaſured timber, ſtone, digging, &c.
Ex. I.
Ex. 2.
Ex. 3.
(10) (27)
(10) (27)
(10) (27)
Yar. fee.
Yar. fee.
Yar. fee.
52 26.
43 25
67 23
48 22.
42 18
25 24
34
18.
34
14
32
18
25
23
26 23
27
23
68
12.
38
20
16 15
32
16
45 16
14 14
In adding the feet, I ſay, (Ex. 1.), 16 and 12 make 28;
F 2
which
44
Chap. II.
ADDITION.
which being above 27, I dot at 12, and go on with the
exceſs 1, ſaying, 1 and 23 make 24, and 8 make 32;
where I dot again, and proceed with the exceſs 5, ſaying,
5 and I ten on the left, make 15, and as 22 wants only
5, I take that 5 from the 15, and, dotting at 22, I go
on with the exceſs 10; and as 26 wants only 1, I take
that I from 10; fo I dot at 26, and, fetting down the
excefs 9, I carry 4 for the dots to the yards.
If inches be given, they muſt be added as integers,
and their fum being divided by 1728, the remainder will
be the excess, and the quotient must be carried to the
feet.
14. A CIRCLE.
TABL E.
60 ſeconds 1
Circ.
I minute.
60 minutes
make
I degree.
30 degrees
I fign.
12 figns
I circle.
Marked thus.
fig.
1 = 12 = 360 = 21600 = 1296000
Ex. I.
(10) (12) (30) (60)
Circ. fig.
II 29 59
5 24 56
7 18
Ex 2.
(10) (30) (60) (60)
Sig.
10
25 48 54
9 23 40 36
5
15
24
36
42
18
34 48
52
6 20 35
2
22
32 40
19
10 15
28
7 20
30 25
28 4 12 45
3
18
15 18
In cafting up the feconds, minutes, and degrees,
where you carry at 60 or 30, a juſt number of tens, you
add as in integers, and from the fum of the left-hand
column, you carry 1 for every 6 in the ſeconds and mi-
nutes, and 1 for every 3 in the degrees.
I
15. TIME.
Chap. II.
45
ADDITION.
15. TIM E.
TABL E.
60 ſeconds 1
I minute.
60 minutes
24 hours
7 days
make
4 weeks
13 months
1 hour.
I day.
1 week.
1 month.
1 year of 364 days.
But the year complete confifts of 365 days and 6
hours; and is divided into twelve unequal calendar
months; whoſe names, with the number of days they
contain, are as in the margin.
The 6 hours, in the space of 4
years, make a day`; and is accord-
ingly every fourth year added to Fe•
bruary, which then confifts of 29
days; and this year is called Biffex-
tile or Leap-year, and contains
366 days.
Names.
Days.
January
31
February
28
March
31
April
30
May
31
June
30
July
31
Auguſt
31
September
30
October
31
November
30
December
31
365
Ex. I.
(10) (13) (4) (7)
Year. mon. wee.da.
6
Ex. 2.
(10) (24) (60) (60)
Da. ho. min. fec.
38.23 56 48
25 II. 3
26 10. 2
4
34 20 45 54
32
12. 3
2
32
34
II. I 5
25
18 42 36
I 2 25 18
38
9
2
6
18
16
15 45
42
7. 3
3
20
21
38 42
45
4
2
4
15
14 50 25
In
46
Chap. II.
ADDITION.
In adding the months in Ex. 1. I fay, 5 carried from
the weeks, and 4 make 9, and 7 make 16; which being
above 13, I dot at 7, and go on with the exceſs 3, fay-
ing, 3 and 9 make 12,
make 12, and I make 13; where I dot,
and proceed, faying, I ten on the left, and 12 make 22;
where I again dot, and go on with the exceſs 9, faying,
9 and 10 make 19; where I dot, and proceed with the
excefs 6, faying, 6 and 11 make 17; where I dot, and
fetting down the excess 4, I carry 5 for the dots to
the years.
In Ex. 2. the hours are added like the grains in Troy
weight, and the minutes and feconds are added exactly
like thofe of a circle.
The learner, by this time, may be fuppofed to be
pretty well ſkilled in addition; and will henceforth, in
order to his cafting up any account, want only to know,
how many of any inferior denomination makes an unit
of the next fuperior; and this he will acquire partly by
reading books, and partly by converfation and practice;
and as a large detail of this kind would be tedious, and
does not properly belong to arithmetic, I fhall only add
two or three things more.
نظر
12 things of any kind
12 dozen
12 ſmall groſs
24 fheets of paper
20 quires
20 things of any fort]
5
ſcore
6 ſcore
make
}
make
{
}
make {
I dozen.
I fmall grofs.
I great grofs.
I quire.
I ream.
I ſcore.
Joo, or the ſhort hundred.
120, or the long hundred,
!
III. The Proof of Addition.
Addition may be proved ſeveral ways.
1. Merchants and men of buſineſs uſually add each
column firſt upwards and then downwards, and upon
finding
Chap. II.
47
ADDITION.
finding the fum to be the fame both ways, they conclude
the work to be right: and this is all the proof that their
time, or the hurry of buſineſs, will admit of.
2. It is a common practice in ſchools, to prove the
work by a fecond fumming without the top-line; and if
this fum added to the top-line make the firſt total, the
work is fuppofed to be right; as in the two following
examples.
Ex. I.
Ex. 2.
L.
S.
d.
8745
9378
4764
5876
28763
Top-line
748 15 103
67- 13
835 17
} I
94
90 18
8
Total
2350
3/1/2
20018 Total without the top-line 1601
ΙΟ
42
28763
Proof
2350
6
347 5
684 X
1031
5
+1=2, and
3. Addition is alfo proved by cafting out the y's: for if
the excefs above the 9's in the total be the
fame as the excefs in the items, the work
may be prefumed right. Thus, to prove
the example in the margin, I begin with the
items, and fay, 3 + 4 = 7, and 7 + 7 =
14=1++=5; with this 5 1 pafs to the
I
next item, and ſay, 5+6=1=
2 +8 = 10=1, and 1+4= 5:
<: which , being the
excess of the items, I place at the top of the cross, and
proceed to caft the g's out of the total, faying, 1+1=
4, and 4 + = 5: which 5, being the excel of the
total, I place at the foot of the crois; and becauſe it is
the fame with the figure at the top, I conclude the work
to be right.
1
If the items are of different denominations; as pounds,
fhillings, pence, &c.; you muſt begin with the higheſt
denomination;
48
Chap. II.
ADDITION.
denomination; and, after cafting out the 9's, reduce the
exceſs to the next inferior denomination; and then caſt-
ing out the 9's, reduce the exceſs to the next inferior
denomination; proceed in like manner with this, and all
the other lower denominations, placing the laſt exceſs at
the top of the croſs; then, in the fame manner, caſt the
9's out of the total, placing the excels at the foot of the
crofs; and if the figure at the foot and top be the ſame,
the work may be prefumed right.
16
d. f.
10 1
7 2
5 3
2
X
2
Thus, to prove the example in the margin, I begin
with the pounds, and fay, 4+8
=12=1+2 = 3, and 3 +5 L. s.
= 8, and 8 +5=13=1+ 48 17
34; and the excefs 4 redu- 55 18
ced to fillings is 4 X 2080
=8, and 8 +1=9=0; then 104 10
7+ 1 = 8, and 8 + 8 = 1
=1+6=7; and the exceſs 7 reduced to pence is
7 X 12 = 84 = 8+ 4 = 12 = 1 +2 2 = 3, and
3+1 = 4, and 4+ 7 = 11="+ = 2; and the
exceſs 2 reduced to farthings is 2 X 4 = 8, and 8 + !
=90, and o + 2 = 2: fo this 2, being the exceſs
of the items, I place at the top of the crofs, and pro-
ceed to caft the 9's out of the total, faying, I + 4 = i
and the exceſs 5 reduced to fhillings is 5 X 20 100
= 1, and 1 + 1 = 2, and 2 + 6 = 8; and this exceſs
8 reduced to pence is 8 X 12 = 96 = 6, and 6+5
===0+1=2; and this exceſs 2 reduced to far.
things is 2 X 4 = 8, and 8 + 3=1=1+1=2:
which excefs 2 I place at the foot of the cross; and be-
cauſe it is the fame with the figure at the top, I conclude
the work to be right.
The method of proof by cafting out the y's is found-
ed on the corollaries deduced from the axioms; and if
any operation, whether in addition, fubtraction, multi-
plication, or divifion, be right, this kind of proof will
always how it to be fo; but if an operation be wrong,
by a figure or figures being mifplaced, or by mifcounting
9, or any juft number of y's, this kind of proof will not
diſcover the miſtake.
IV. Practical
Chap. 11.
49
ADDITION.
3
IV. Practical Questions.
1. A perfon dying, left for the uſe of his widow
L. 4000; to each of his four fons L. 4256; to each of
his five daughters L. 3678; to each of fix near relations
L. 245: What was his eftate? Anf. L 40,884.
2. H of Hamburgh is debtor to L of London :
For broad cloth, L. 428:14:10; for kerfeys, L. 273,
17. 6.; for fuftians, L. 364: 19:8; for druggets,
L. 568: 18:43; for muflin, L. 392 : 12 : 81; for
grocery-wares, L. 683: 1,8; the factorage came to
L. 104, 6s.; cuftom, wharfage, and incident charges,
L. 4, 15 S.
For what fum muft L draw on H? Anf.
L. 2821: 19:10.
3. B buys fix bags of hops, of which N° 1. weighed
C. 2:3:14; N° 2. C. 2: ¡ : 24; Nº 3 C. 2: 2:20;
N° 4. C. 2 : 1 : 22; N° 5. C. 2 : 0 : 26; N° 6. C 2,
39. 12 lb.; as alfo a couple of pockets ditto, that
weighed 54lb. each. How many hundred weight did
he purchaſe? Anf. C. 16:2: 3.
4. A goldfmith fells five dozen of filver fpoons,
weighing 12 lb. 100z. 14dw.; two tankards, weighing
1 lb. 8 oz. 18 dw.; ten falts, weighing 4 lb. 11 oz. 16dw.;
forty-two plates, weighing 38 lb. 6oz. 10dw.; a pair
of juggs, weighing 2 lb. 4oz. 13 dw., two tea kettles,
weighing 13lb. 5 oz. 12 dw. What quantity did he fell?
Anf. 74lb. 3 dw.
5. A wine-merchant imports 12 tuns 2 hhds 4 gal-
lons of claret; 14 tuns 3 hhds 48 gallons of Malaga;
16 tuns 1 hhd 54 gallons of Port; 20 tuns 2 hhds 56
gallons of Canary; 18 tuns 3 hhds of Madeira; and
10 tuns 2 hhds 42 gallons of fherry. What quantity
did he import in all? Anf. 94 tuns 56 gallons."
6. A certain farm confifts of fix inclofures, whereof
the first contains 42 acres 3 roods 36 poles; the fecond,
30 a. 2 r. 24 p.; the third, 45 a. 1 r. 38 p.; the fourth,
52 a. 2 r. 28 p.; and each of the other two contains-
How many acres in all? Anf. 231 a.
26 a. 3 r. 14 p.
1 r. 34 P.
7. A gentleman had feven fons: from the birth of
G
the
N
50
Chap. III.
SUBTRACTION.
the first to the birth of the ſecond there intervened 384
days 18 hours 50 minutes; from the birth of the ſe-
cond to that of the third, 582 d. 22 h. 58 m; from the
birth of the third to that of the fourth, 623 d. 12 h.
48 m.; from the birth of the fourth to that of the
fifth, 592 d. 16 h. 36 m.; from the birth of the fifth to
that of the fixth, 745 d. 19h 45 m.; from the birth of
the fixth to that of the feventh, 864 d. 17 h. 54 m.
What was the age of the eldeſt when the youngeſt was
born? Anf. 3794 days, 12 hours, and 51 minutes.
CHA P. III.
SUBTRACTIO N.
SUBTRACTION der te difcover or
UBTRACTION is the taking a leffer number from a
greater, in order to diſcover their difference, or the
remainder.
I. Subtraction of Integers.
RULE S.
I. Set figures of like place under other, viz. units under
units, tens under tens, &c. and the greater of the given
numbers uppermoft. See axiom X1.
II. Beginning at the place of units, take the lower fi-
gures from thofe above, borrowing and paying ten, as
need requires, and write the remainders be.ow. See
axioms 11. III. IX.
EXAMPLE
I.
Becauſe fimilar or like
things only can be fub-
tracted, I place the num
bers as directed in Rule I.
viz. units under units, tens under tens, &c. and the
greater uppermoft, as in the margin. Then, beginning
at the place of units, I fay, 2 units from 7 units, and 5
units remain; which I fet below, in the place of units;
then 6 tens from 6 tens, and nothing remains; where-
fore I fet o below, in the place of tens; then 5 hundred
867 major or minuend.
562 minor or fubtrahend.
305 difference or remainder.
from
痔
​L
}
Chap. III.
51
SUBTRACTION.
from 8 hundred, and 3 hundred remain; which I fet be-
low, in the place of hundreds, and find the total differ-
ence or remainder to be 305.
EXAMPLE II.
f
7432
2785
Having placed the numbers, units under units, &c.
as in the margin, I fay, 5 units from 2 units I
cannot, but, becauſe an unit in the next fuperior
place makes ten in this place, I borrow 1, viz.
I ten, from the faid next place, as directed in
Rule II.; which ten being added to 2 makes 12; 4647
then I ſay, 5 from 12, and 7 remains; which 7
I fet below, in the place of units: then 1 proceed, and
pay the unit borrowed, either by eſteeming 3, the next
figure in the major, to be only 2, or, which is more u-
fual, and the fame in effect, by adding to the next fi
gure in the minor, thus, that I borrowed and 8 make
9, from 3 I cannot, but, borrowing as before, I fay, 9
from 13, and 4 remains; which 4 I fet below: I pro-
ceed, and fay, I that I borrowed and 7 make 8, from
4 I cannot, but from 14, and 6 remains; which 6 I fet
below: I go on, and fay, I borrowed and 2 make 3,
from 7, and 4 remains; which 4 I fet below. So the
difference or remainder is 4647. By borrowing and pay-
ing in this manner the major and minor are equally aug-
mented, or have the fame number added to each of them;
and confequently continue to have the fame difference,
by axiom IX.
EXAMPLE
III.
17452
Some, instead of adding to to the upper leffer figure,
fubtract directly from 10, and add the difference
to the upper figure for the remainder, thus,
from 2 I cannot, but 3 from 10, and 7 re-
mains; which 7 added to 2 gives 9 for a re-
mainder then I go on, faying, I borrowed and
3
:
873
16579
7 make 8, which from 5 I cannot, but from 10, and 2
remains; which 2 added to 5 gives 7 for a remainder :
fo I proceed, faying, I borrowed and 8 make 9, which
from 4 I cannot, but from 10, and I remains; which
G 2
I
52
Chap. III.
SUBTRACTION.
I added to 4 gives 5 for a remainder: fo I go on, and
fay, I borrowed from 7, and 6 remains, and o from 1,
and I remains. This method is the fame in effect with
the other; but not fo ufual, and fomewhat childiſh.
MORE
EXAMPLES.
Ex. 4.
Ex. 5.
Ex. 6.
From
84765
94307
Take 20857
5708
742680
8794
II. Subtraction of the parts of integers; ſuch as ſhillings,
pence, farthings, ounces, &c.
RULE S.
I. Place like parts under other, viz. farthings under
farthings, pence under pence, &c. and the greater of the
given numbers uppermoft. See axiom XI.
II. Begin at the loweft of the parts, and borrow ac-
cording to the value of an unit of the next fuperior de-
nomination; viz. in farthings borrow 4, in pence bor-
row 12, &c. as the tables of coin, weights, and mea.
fures direct.
III. If you borrow
borrow 0, 30, 40, 60, or any juſt num.
ber of tens, as in fubtracting fhillings, degrees, poles,
minutes, feconds, &c. proceed with the right-hand co-
lumn, as in fubtraction of integers; and then ſubtract
your tens, borrowing, if need be, the number of tens
contained in an unit of the next fuperior denomination.
The reaſon appears plain in the following operations.
I.
MON E Y.
Having, according to Rule I. (10) (20) (12) (4)
placed like parts under
O. L.
S.
d
f.
ther, viz farthings under far-
things, pence under pence, &c. 48
and in each of thefe denomi-
nations, units under units, tens
under tens, and the greater of
the
25
15 10 2 major.
I 2
62 minor.
3 4
rem,
Chap. III.
53
SUBTRACTION.
I
(10) (20) (12)
L. S. d.
14
6 major.
708
278 17
103 minor.
429 16 7 rem.
the given numbers uppermoft, as in the margin, I be-
gin with the farthings, and fay, 2 from 2, and o re.
mains; fo, having nothing to fet down, I leave the place
blank, and proceed to the pence, faying, 6 from 10 and
4 remains; which 4 I fet down, and go on to the fhil-
lings, faying, 2 from 5, and remains, and 1 from 1,
and o remains; or I may fay at once, 12 from 15, and
3 remains; which 3 being fet down, I proceed to the
pounds, which are integers, and ſubtracted as ſuch.
Here I ſay, 3 farthings from 1 farthing I cannot, but,
as directed in Rule II. I fay,
3 from 4, the number of far-
things in 1 penny borrowed,
and remains; which I add-
ed to I in the major gives 2
farthings for a remainder;
which I fet down, and pro.
ceed to the pence, ſaying, I
penny borrowed and to make 11, which from 6 I can-
not, but from 12, the number of pence in 1 fhilling,
and remains; which I added to 6 in the major gives a
remainder of 7; which I fet down, and go on to the
fhillings; and becauſe in fubtracting fhillings we borrow
a juſt number of tens, viz. 2 tens, or 20, I work as di-
rected in Rule III; and in the right-hand column ſay, 1
borrowed and 7 make 8, which from 4 I cannot, but
from 14, and 6 remains; which being fet down, I go
on to the left-hand column, and fay, I borrowed and
make 2, which from I cannot, but from 2, the num-
ber of tens in 1 pound, and o remains, which o added
to 1 in the major gives for a remainder; which I ſet
down, and proceed to the pounds, faying, I borrowed
and 8 make 9, which from 8 I cannot, but from 18,
&c.
Note 1. Some add the number borrowed to the figure
or number in the major, and then fubtract from their
fum. Thus, in the farthings they add the 4 borrowed
to in the major, and then from the fum 5 they fub-
tract the 3 in the minor; and in the pence they add the
12 borrowed to 6 in the major, and fubtract from the
fum
54
Chap. III.
SUBTRACTION.
fum 18, &c.; but the method taught above is the eaſieſt
and moſt uſual.
Note 2. A great many people, in fubtracting the fhil-
lings, inſtead of working as directed in Rule III. proceed
thus; faying, I borrowed and 17 in the minor make
18, which from 14 I cannot, but from 20, the number
borrowed, and 2 remains; which 2 added to 14 in the
major gives 16 for a remainder. The learner may chuſe
any of the two methods he likes beft.
MORE
EXAMPLES.
Ex. I.
(10) (20) (12)
L. S. d.
Ex. 2.
(10) (20) (12)
L. 5. d.
From 743
Sub. 375 15
16
4
82
13
+/-
10
54
14
73
Rem.
Ex. 3.
L. s. d.
From 95
95 8 7
68 13 61
Sub.
Rem.
Ex. 4.
L. s. d.
34 10 63
20
:
27
Ex. 5.
L.
A borrowed of B
150
L. s. d.
A paid B at one time
20 10 6
At another time
18 13 4
At another
12 16 8
Sold him goods to the value of 25 14 7
In all
Balance due to B
Ex. 6.
Chap. III. SUBTRACTION.
55
A lent B
Ex. 6.
A received at one time
At another time
At another a bill of
At another a draught of
Goods to the value of
L. s. d.
36 14 8
40 10
34 13 4
25 16 10
16 14 63
L.
200
In all
Balance due to A
Ex. 7.
L. 2000
300
250
A fteward collected of rents
Remitted to his mafter at one time
At another
Paid ſmall bills amounting to
Paid accounts amounting to
Paid of taxes and repairs
68 13 4
45 16 8/
47 15 61
In all
2.
Balance due to the maſter
AVOIR DUPOIS
¿
WEIGHT
(10) (4) (28)
C. qrs. lb.
84 1 22 major.
49
3
24 minor.
I begin with the pounds, and fay,
24 from 22 I cannot, but from 28,
the number of pounds in quarter,
and 4 remains, which added to 22
in the major, gives 20 for a remain-
der; which 1 fet below, and pro. 34
ceed to the quarters, faying, 1 quar-
ter borrowed and 3 make 4, which from 1 I cannot, but
from 4, the number of quarters in C. and o remains,
which o added to 1 in the major gives 1 for a remainder;
which I fet down, and go on to the C. which are inte-
I
126 rem.
gers,
56
Chap. III.
SUBTRACTION.
gers, ſaying, 1 C. borrowed and 9 make 10, which from
4 I cannot, but from 14, &c.
MORE EXAMPLES.
Ex. I.
Ex. 2.
(10) (20) (4) (28)
T. C. qrs. lb.
From 38 14 2 18
(10)(4) (28) (16)
G. qrs. lb. oz.
48 3
16 IO
Take 25
14 3 21
23 3
19 13
Rem.
The C. in Ex. 1. are fubtracted like fhillings; that is,
either as directed in Rule III. or by the method men.
tioned in note 2. above.
Ex. 3.
A merchant buys of ſugar
C. grs. lb.
80 I 14
15
16
2 2
56
14
14
Sells at different times
5
3 '14
116 3
Sold in all
Remains on hand
3. TROY WEIGHT.
Ex. I.
Ex. 2.
lb.
(10)(12)(20) (24)
oz. dw. gr.
lb.
03.
(10) (12) (20) (24)
drv. gr.
From 74 10 I 2
Sub. 28 11 15 19
16
03
8
1 4
ΙΟ
54
10
17
14
Rem.
Ex.
Chap. III. SUBTRACTION.
57
Ex. 3.
lb. oz. dw. gr.
Bought of filver
684
IO 12
18
(25
8 14
I 2
14
10
16
10
26
[[
18
14
Sold at feveral times
35
ΙΟ
I 2
15
42
6
5
17
38
II
14 16
Sold in all
Remains unfold
4. APOTHECARIES
Ex. I.
WEIGH T.
Ex. 2.
(10) (12) (8) (3) (20)
(10) (3) (20)
b 3 3 gr.
Э
3
→ gr.
From 42
8 4 I 15
24
1 13
Sub. 22 7
5 2 19
13
2 17
Rem.
Ex. 3.
Ex. 4.
tb
3 3 Э grо
3
ǝ gr.
37
2
4
28
2
18
From 58 5 3 I 10
Sub. 37 10 5 I 16
Rem.
5. WOOL. WEIGHT.
Ex. I.
Ex. 2.
(10) (2) (7)
Ston. clo. lb.
I
25 I 3
4 I
13 I 5
(10) (12) (2) (61) (2)
Laft. fac.wey. tod. fton.
From 84 7 I 3
Sub. 73 4 I
Rem.
H
6. DRY
58
SUBTRACTION. Chap. III.
6. DRY
Ex. I.
MEASURE.
(10) (5) (8) (4)
Load. grs.bu pec.
From 36 3 3 4 2
Sub. 24 3 6 3
Rem.
Ex. 2.
(10) (40) (8)
Load. bu. gal.
83 25 5
63 34 6
In fubtracting the bushels in Ex. 2. becauſe you bor-
row a juſt number of tens, viz. 4 tens, or 40, work as
directed in Rule III.; that is, proceed with the right-
hand column, as in fubtraction of integers, and in the
left-hand column borrow 4, when the figure in the ma-
jor is less than the figure to be fubtracted.
COAL-MEASURE.
Ex. I.
(10) (4) (9)
Ex. 2.
(10) (36) (4)
Chal. gr..bu.
Chal. bu. pec.
From 68 I 3.
78 23 I
Sub. 38 2 5
49 27 3
Rem.
SCOTS DRY MEASURE.
Ex. I.
(10) (16) (4) (4)
Chal. bo. fir. pec.
From 36 II I 2
Sub. 25 13 2 3
Rem.
Ex. 2.
(10)(4) (4) (4)
Bo. fir. pec. lip.
28
I 2 I
17 2
3 2
7. WINE.
Chap. III. SUBTRACTION,
59
1
7. WINE-MEASURE.
Ex. 1.
(10)(2) (2) (63)
Tun. pip. hhd gall
From 56 I
I
Sub. 48
II
Ex. 2.
(10) (63) (8)
Hhd.gall.pts.
35
63 45 3
47
38 53 7
Rem.
8. BEER and ALE MEASURE.
Ex. I.
(10) (3) (2) (9)
Hhd. kil. firk. gal.
Ex. 2.
(10)(3) (36)
But. bar. gal.
From 43 I I 3
Sub. 35 2 1 5
57 I 21
47
I
30
Rem.
SCOTS LIQUID MEASURE.
Ex. I.
(10) (16) (4) (2)
Hhd. gal. qrt. pt.
From 37 II 2 I
Sub. 28 12
Ex. 2.
(10)(2) (2) (4)
Pt. cho. mu.
.gil.
47 1 I 2
2 I
33
1 1
3
Rem.
9. CLOTH MEASURE.
·
Ex. 2.
Ex. 3.
Ex. I.
(10) (4) (4)
Yds. qrs. n.
(10) (3) (4)
E.FI qr. n.
From 38 1 2
Sub. 23 2 3
48 I 2
(10) (5) (4)
E.En.qr. n.
78 3 2
29 I
3
57 4 3
Rem.
H 2
10. LONG
مة
SUBTRACTION. Chap. III.
F
10. LONG
MEASURE.
Ex. 1.
Ex. 2.
(10) (3) (8) (40)
Lea. mi. fur. po.
(10)(5)(3) (12)
Po. yd. ft. inc.
From 33
Sub. 25 I
I 4 29
25 3 I IO
5
37
17
4 2
II
Rem.
SCOTS LONG MEASURE.
Ex. I.
(10) (8) (10) (4)
Mi. fur. ch. fal.
Ex. 2.
(10) (4) ( ) (37)
Ch. fal. el. inc.
From 35 3 7 I
Sub. 23 5 7 2
45 2 3 24
27 3 5 32
Rem.
II. L AND MEASURE.
Ex. I.
·
(10) (4) (40) (301)
Acr roo. fq.p. Sq⋅y.
From 38 I 27 17
Sub 28 3 33 25
Ex. 2.
(10) (4) (40)
Acr. roo fq. p.
44 2
32
27 2
36
Rem.
SCOTS LAND-MEASURE.
Ex. I.
(10) (4) (+0) (‹6)
Act.roo.fq.f fq.el.
From 72 2 24 30
Sub. 35 3 27 31
Ex. 2.
(10)(4) (40)
Acr. roo. fq.f.
48
I 33
28
2
34
Rem.
12. SU.
1
Chap. III.
SUBTRACTION.
61
}
12.
SUPERFICIAL
MEASURE.
Ex. I.
Ex. 2.
(10) (9) (144)
Yds. fee. inc.
From 723 3 128
Sub. 478 5 132
Rem.
(10) (9) (144)
rds. fee. inc.
654 5
96
96 7 118
When the number of inches in the minor is greater
than in the major, fubtract, as in integers, from 144,
and the difference added to the inches in the major
gives the remainder.
t
13. SOLID MEASURE.
Ex. I.
(10) (27)(1728)
Yds. fee. inc.
Ex. 2.
(10) (27) (1728)
Yds. fee. inc.
From 76 18 378
48 16 1000
Sub. 48 21 964
35 22 1432
Rem.
The laft direction takes place here, with this varia-
tion, that you ſubtract the number of inches in the mi-
nor from 1728.
14. A
CIRCLE.
Ex. I.
Ex. 2.
1
(10)(12)(30)(60)
Circ. fig.
From 13 5 18 46
Sub. 9 7 28 52
Rem.
(10)(30)(60) (60)
Sig.
7 14 43
"
50
4 19 50 54
In fubtracting the degrees, minutes, and feconds,
proceed with the right-hand column as in fubtraction
of integers, and in the left-hand column of degrees bor-
row
62
Chap. III.
SUBTRACTION.
row 3, and in that of minutes and feconds borrow 6, as
directed in Rule III,
15. T IM E.
Ex. I.
Ex. 2.
(10) (13) (4). (7)
Yea. mon.wee.da.
(10) (24) (60) (60)
Da. ho. min. fec.
From 24
71 4
48
14 36 28
Sub. 13 10 3 5
32
18 43 37
Rem.
III. The Proof of Subtraction.
Merchants and men of buſineſs uſe no other proof
beſides a revifal of the work, or running over it a fe-
cond time; but it is ufual in fchools to put the learner
upon proving the operation, by fome of the three me-
thods following, viz.
1. The work may be proved by addition; for if you
add the remainder to the minor, the fum, by Axiom X.
will be equal to the major, as in the two following ex-
amples.
}
Ex. 2.
Ex. I.
L.
S.
d.
5847
major
73 15
ΙΟ
2509
minor
48
12 6
3278
rem. 25 3 4
5847
proof 73 15 10
2. By ſubtraction; for if you fubtract the remainder
from the major, the difference, by axiom X. will be e-
qual to the minor, as follows,
:
5847
Chap. III. SUBTRACTI O N.
63
L. S. d.
5847 major 73 15
2569 minor 48
ΙΟ
12 6
3278 rem. 25
3 4
2569 proof
48 12 6
3. By cafting out the 9's; for the major being equal
to the fum of the minor and remainder, if you caft the
9's out of the major, and place the excefs at the top of
the cross, and then caft the 9's out of the minor and re-
mainder, as if they were items in addition, and place the
exceſs at the foot of the cross, it is plain, the figure at
the top and foot, if the work be right, will be the fame.
Only, in proving fubtraction of money, Avoirdupois
weight, &c. care must be taken to begin with the higheſt
denomination, reducing always the exceſs to the next
inferior denomination, as taught in the proof of addition.
See the following examples.
L
S. d.
6 5847 major
73 15
10
7
X 2569
minor
48
12 6
X
6
3278
7
rem. 25 3.
3 4
IV. Practical Questions.
1. America was difcovered by Columbus in the year
1492: How long is it fince, this being the year 1703?
Anfw. 271 years.
2. The fum of two numbers is 8764; the leffer is
795: What is the greater? Anf. 7969.
3. The greater of two numbers is 48 32; their differ-
ence is 967: What is the leſſer? Anf. 3865.
4. A borrowed of B L. 347: 14:4, and afterwards
paid him L. 218:16:8. What is the balance due ?
Anf. L 1 8:16 : 8.
5. What lum added to L. 6,8: 147 will make
L. 1000? Anf L. 30152
6. A grocer buys C. 17: 1:21 fugar, of which he
fells
64
MULTIPLICATION. Chap. IV.
fells G. 13: 2:24: How much remains on hand? Anf.
G. 3: 2:25.
7. The difference of the weight of two filver bowls
is 3 lb. 9 oz. 18 dw. 20 gr.; the largest bowl weighs
lb. 5 oz. 13 dw. 17 gr What is the weight of the
leffer? Anf. 9 lb. 7 oz. 14 dw. 21 gr.
13
8. A vintner had in his cellar 18 tuns 2 hhds 48 gal-
lons claret; but having fold 13 tuns 2 hhds 54 gallons,
how much remains? Anf. 4 tuns 3 hhds 57 gallons.
9. A gentleman's eſtate conſiſted of 7648 acres 1 rood
and 26 poles; but, to anſwer the demands of dunning
creditors, was obliged to fell off ſeveral large incloſures,
confifting of 3278 acres 2 roods 32 poles: What is he
now poffeffed of? Anf. 4369 acres 2 roods 34 poles.
10. The planet Venus revolves round the fun in 224
days 16 hours 49 minutes 24 feconds; and Mercury in
87 days 23 hours 15 minutes 53 feconds: What is the
difference of their periodical times? Anf. 136 days 17
hours 33 minutes 31 feconds.
11. A merchant, on balancing his books, finds, that
he has in ready money L. 348:13:4; goods to the va-
lue of L. 2635: 16:84; debts due to him L. 1784, 18 s.
63 d. : at the ſame time he owes to A, L. 275: 14: 10;
and to B, L. 384: 18 : 71: What is his neat ſtock? or
what will he be worth after all his debts are paid? Anf.
L. 4108: 15: 2.
12. The weight of two hhds tobacco when packed is
G. 9: 2:16 each; the weight of the two empty hhds is
26 lb each: What is the neat weight of the tobacco?
Anf. C. 18:37.
IN
CHA P. IV.
MULTIPLICATION,
N multiplication there are two numbers given, viz. one
to be multiplied, called the multiplicand; and ano-
ther that multiplies it, called the multiplier; theſe two
go under the common name of factors; and the num-
ber
Chap. IV. MULTIPLICATION.
65
ber arising from the multiplication of the one by the o-
ther is called the product, and fometimes the fact, or
the rectangle. If a multiplier confifts of two or more
figures, the numbers arifing from the multiplication of
theſe ſeveral figures into the multiplicand, are called par-
ticular or partial products; and their fum is called the
total product.
Multiplication then is the taking or repeating of the
multiplicand, as often as the multiplier contains unity.
Or,
Multiplication, from a multiplicand and a multiplier
given, finds a third number, called the product, which
contains the multiplicand as often as the multiplier con-
tains unity.
Hence multiplication fupplies the place of many addi-
tions; for if the multiplicand be repeated or fet down as
often as there are units in the multiplier, the ſum of
thefe, taken by addition, will be equal to the product
by multiplication. Thus, 5×315=5+5+5.
The firſt and loweft ftep in multiplication is, to mul
tiply one digit by another; and the fact or number thence
arifing is called a ſingle product. This elementary ſtep
may be learned from the following table, commonly
called Pythagoras's table of multiplication: which is con-
fulted thus; feek one of the digits or numbers on the
head, and the other on the left fide, and in the angle of
meeting you have their product. The learner, before
he proceed further, ought to get the table by heart.
To Pythagoras's table are here added, on account of
their uſefulneſs, the products of the numbers 10, 11, 12.
I
TABLE.
66
MULTIPLICATION. Chap. IV.
TABL E.
2
3
4
5
I
N.
24
3
4
5
54
46
a
Co
6718
9
II 12
68 10 12 14 16 18 20 12 24
15| 18|21|24|27|| 30 | 33 | 36
9,| 12 |
46
12 16 20 24 | 28|32|36|| 40 | 44 | 48
IO 15 20 25 30 35 40 45
612
12 18 24 30 36 42 48 54
7 1421 28 35 42 49 56 63
81624 32 40 48 56 64 72
9 | 18|27|36|45|54|63|72|81
50 560
626672
70 77 84
|
808896
90 99 108
1001101 20
10 20 30 40 50 | 60 | 70 | 80 yo
11 | 22 | 33 |44|55|66|77|| 88 | 99 110121132
12 24 36 18 6072 84|96|108 |120|132|144|
I. Multiplication of Integers.
RULES.
I Set the multiplier below the multiplicand, ſo as
like places may ftand under other, viz. units under u-
nits, tens under tens, &c.: but if either or both of the
factors have ciphers on the right hand, ſet their firſt ſig.
nificant figures under other.
The order prefcribed in this rule is not abfolutely ne
ceffary, but very convenient, as will appear in the ex-
amples.
I Beginning at the right hand, multiply each figure
of the multiplier into the whole multiplicand, carrying
as in addition, and placing the right hand figure of each
particular product directly under the multiplying figure.
See Axioms II. III. IV. VI.
III. Add the particular products, and their fum will
be the total product. See Axiom XII.
EX.
Chap. IV. MULTIPLICATION.
67
EXAMPLE
I.
Factors { 94 multiplicand.
7 multiplier.
658 product.
Having placed the mul-
tiplier under the multipli-
cand, as directed in Rule I.
I proceed to the operation,
and fay, 7 times 4 make 28; I fet the 8 below in the
place of units, and carry the 2 tens to the next place, as
directed in Rule II. faying, 7 times 9 make 63, and 2
that I carried, make 65; I fet 5 below in the place of
tens, and the 6, which belongs to the next place, I fet
on its left hand, there being no further place to which
it can be carried; fo the product is 658.
EXAMPLE
II.
742 multiplicand.
68 multiplier.
59362 particular
5936
4452 Sproducts.
50456 Total product.
Here I first multiply my right-
hand figure 8 into the whole
multiplicand, as in the former
example; then I proceed, and
multiply likewife my 6 tens in-
to the whole multiplicand, fay-
ing, 6 times 2 make 12; I fet
the 2 below under the multiply-
ing figure, viz in the place of tens, and carry my I
to the next place, as directed in Rule II. The reaſon why
I fet the 2 under the multiplying figure, or in the place
of tens, is, becauſe the multiplying figure 6 by A-
xiom VI. is really 6 tens or 60, and 60 times 2 make
120; fo that by carrying the 1 to the next place, and ſet-
ting down 20, the o would fall into the place of units,
and throw the 2 into the place of tens; but as o can
make no alteration in the addition of the partial pro-
ducts, the ſetting of it down is fafely and juftly omitted.
From the repetition of the former example on
the margin it appears, that though any of the fac
tors may be made the multiplier, the produ&
being the fame in both cafes, yet the operation
becomes eaſier and ſhorter by making that fac-
tor multiplier which confifts of the feweft figni-
ficant figures. Here likewife obferve, that in
multiplying 7 hundreds into 8 units of the mul. 50456
दर
I 2
tiplicand,
68
742
136
272
476
-68
MULTIPLICATION. Chap. IV.
tiplicand, I fet the right-hand figure of the product under
the multiplying figure, viz. in the place of hundreds, be
cauſe the multiplying figure 7 is really 700.
EXAMPLE
III.
When the multiplier has ciphers on the
right hand, as it would be evidently loft la-
bour to multiply by the ciphers, their only
ufe being to throw figures on their left hand
into higher places, I fet the firft fignificant
figures of the factors under other; and, af.
ter the operation is finiſhed, I annex the ci-
phers of the multiplier to the right hand of
the product.
EXAMPLE IV.
When the multiplicand has ciphers on the
right hand, the cafe is in effect the fame;
wherefore I proceed in the operation as be-
fore, and annex the ciphers to the product.
EXAMPLE V.
When both multiplicand and multi-
plier have ciphers on the right hand; as
the cafe is plainly a compound of the two
former, I annex to the product the ci-
phers of both factors.
853
7 2000
1706
5971
61416000
853000
72
1706
5971
61416000
853000
7200
1706
5971
6141600000
*P
EXAMPLE
VI.
When the multiplier has ciphers intermixed with fig-
nificant figures, I o nit the ciphers,
becauſe the multiplying by them would
only produce fo many lines of ciphers
and ſo be labour in vain; wherefore I
multiply by the fignificant figures only;
but I take care to place the right-hand
figure of each particular product di-
rectly under the multiplying figure.
29601847
300905
148009235
88805541
266416623
8907343771535
The
Chap. IV. MULTIPLICATION.
ION.
69
The reafon of
fetting the right-
hand figure of
each particular
product directly
under the multi-
plying figure, will
ftill further ap-
pear by refolving
the multiplier in.
to its conftituent
parts, as in the
margin.
*
29601847 multiplicand.
5]
900 partial multipliers.
300000
148009235 prod. by 5.
26641662300 prod. by 900.
8880554100000 prod. by 300000.
total product.
8907343771535
MORE
EXAMPLES.
87694 × 358
=31394452
59387 × 796
= 47274052
78464 × 4207
= 330098048
=7628400
= 59897000
978 x 7800
X
673000 × 89
X
58470 × 900700526639290co
Contractions, and fimple ways of working multiplication
of integers.
1. To multiply any number by 10, by 100, by 1000,
&c. to the given number annex one, two, three ciphers,
&c. Thus, 23 × 10230; and 384 × 100 38400;
and 745 × 1000 = 745000.
=
2. To multiply any number by 9, by 99, by 999, &c.
multiply the given number firſt by 10, by ico, by 1000,
c. that is, annex one, two, three, &c. ciphers to it;
from this ſubtract the given number, and the remainder
is the product; as in the following examples.
Ex. 3.
Ex. I.
Mult. 47
Ex. 2.
Mult. 627
Mult.
999
by 9 470
Sub.
47
by 99 62700
Sub.
by 999 999000
627
Sub.
999
Prod. 423
Prod. 62073
Prod.
998001
From
70
MULTIPLICATION. Chap. IV.
From Ex. 3. we may learn, in general, that to multi-
ply any number confifting entirely of 9's by itſelf, is to
fet in the place of units, then as many ciphers,, fave
one, as there are 9's in the given number; then 8, and
on the left hand of 8 as many g's as there are ciphers on
its right.
This method of compendizing may be further extend-
ed, thus.
If the multiplier be all 9's, except the right-hand fi
gure, or except the two or the three figures next the
right hand, annex as many ciphers to the multiplicand ast
there are figures in the multiplier; from which fubtract
the product of the multiplicand into the complement of
the right-hand figure to 10, viz. what it wants of to;
or into the complement of the two figures next the right
hand to ico; or into the complement of the three figures
next the right hand to 1000, &c.
Ex. 2.
Ex. I.
7846 × 996
54786 × 9988
7846000
547860000
313847846 × 4
65743254785 × 12
7814616 prod.
547202568 prod.
Again, if the multiplier be all 9's except the left-
hand figure, add unity to the ſaid left-hand figure, and
multiply the fum into the multiplicand; to the product
annex a cipher for each of the other figures in the mul-
tiplier; from which ſubtract the multiplicand; and the
remainder will be the product fought.
Ex. 2.
Ex. 1.
2846 × 799
4327 × 1999
2846
4327
871
2 = 1+1
2276800
8654000
2846
4327
2273954 prod.
8649673 prod.
3. To
Chap. IV. MULTIPLICATION.
71
3. To multiply any number by 5; firft multiply it by
10, that is, annex a cipher to it, and then halve it: and
to multiply any number by 15 uſe the fame method; and
add both numbers together, as in the following exam-
ples.
Multiply 7439
Multiply 9856
#
by 5-
74390
by 15-985602
49280 Sadd
Product 37195
Product 147840
4. To multiply any number by 11, 12, 13, 14, 15,
16, &c. multiply by the unit's figure, and add the back-
figure of the multiplicand to the product; and to multiply
by 21, 22, 23, 24, 25, 26, 27, &c. add the double of
the back-figure; and to multiply by 31, 32, 33, 34, &c.
add the triple of it; and to multiply by 112, 113, 114,
c. add the two back figures; and to multiply by 101,
102, 103, 104, &c. add the next back-figure fave one:
as in the following examples.
Ex. I.
876 or multiply by
876 or thus, 876
Ex 2.
694
I I
9636
II thus,
876
876
14
9636
9636
9716
Ex. 3.
435
Ex. 4.
241
27
34
11745
8194
Ex. 5.
Ex. 6.
Ex. 7.
7234
or thus,
7234
263
745
II 2
7234
119
103
7234
810208
7234
31297
76735
810208
In
72
MULTIPLICATION. Chap. IV.
48
In multiplying by 12, as in Ex. 8. it is more Ex. 8.
ufual, and equally eafy, to proceed by ſaying,
twelve times 8 make 96, and, fetting down the
6, I fay, twelve times 4 is 48, and 9 carried is
57; which I fet down, and the product is 576.
12
576
5. If the multiplier conſiſt of the fame figure repeated,
as 111, 222, 333, 777, &c. multiply by the unit's fi-
gure, and out of that product make up the total product,
thus. Begin at the right hand, and firſt take one figure,
then the fum of two, then the fum of three, &c. repeating
the operation ftill from the right hand, as often as there
are figures in the multiplier; then, neglecting the right-
hand figure, or figure in the first place, take the fum of
as many figures toward the left hand as the multiplier has
places; and if there be not fo many, take the fum of all
the figures there are; then, neglecting the figures in the
firſt and ſecond place, begin at the figure in the third place,
proceed as before; and thus go on till the laſt or left-hand
figure is taken in alone; as in the following examples.
Ex. 3.
Ex. I.
7645
33
Ex. 2.
4983
666
38
4444
22935 pr. by 3.
29898 pr. by 6.
152 pr. by 4.
252285 total.
3318678 total.
168872 total.
6. The operation may frequently be rendered ſhorter
or eaſier, either by addition, fubtraction, or a more
fimple multiplication; and the cafes of this kind are fo
numerous and various, that they admit of no limitation.
Confult the following examples and directions.
Ex. 1.
438
Ex 3.
Ex. 2.
374
746
87
56
84
3066
2244
2984
3504
1870
5968
38106
20944
62664
Ex. 4.
Chap. IV. MULTIPLICATION.
73
Ex. 4.
Ex. 5-
Ex. 6.
824
685
789
642
147
328
1648
4795
6312
3296
9590
25248
4944
100695
258792
529008
}
I work the above examples as follows.
Ex. 1. I multiply by 7, and add that product to the
multiplicand, inſtead of multiplying by 8.
Ex. 2. I multiply by 6, and out of that product I ſub,
tract the multiplicand, inftead of multiplying by 5.
Ex. 3. 1 multiply by 4, and double that product for 8.
Ex. 4. I multiply by 2; then I double, or multiply
that product by 2, for 4; and then add theſe two pro-
ducts (the right-hand figure of the one to the right-hand
figure of the other, &c.), for 6.
14.
Ex. 5. I multiply by 7, and double this product for
Ex. 6. I multiply by 8; and, becaufe 8 times 4 make
32, I multiply that product by 4, for 32.
Several other contractions might be added, but they
are rather curious than very uſeful.
Select methods of multiplying Integers.
1. Inſtead of multiplying by the multiplier, you may
multiply by its component parts, or by the component
parts of the neareſt compofite number; and to or from
the laſt product add or ſubtract the product of the mul-
tiplicand into the difference betwixt the multiplier and
the neareſt compofite number.
Thus, inſtead of multiplying by 72, you may mul-
tiply by 9, and that product by 8; and inſtead of mul-
tiplying by 56, you may multiply by 8 and 7, or by 7,
4, 2; for the choice of the component parts is arbitrary ;
K
and
74
MULTIPLICATION. Chap. IV.
ང
and inſtead of multiplying by 37, you may multiply by
4 and 9, adding the multiplicand to the laft product, for
the unit wanting in the product of the component parts;
and inſtead of multiplying by 34, you may multiply by
5 and 7, fubtracting the multiplicand from the laft pro-
duct, for the unit of excefs in the product of the com-
ponent parts; and instead of multiplying by 68, you
may multiply by 8 and 8, to the laft product adding the
product of the multiplicand into 4, the difference be
twixt the multiplier and the neareft compofite number, &c.
Here obſerve, that when three or more numbers, as
in this cafe, are given to be multiplied into one another,
the operation is called continual multiplication. And it is
of no importance which of the component parts you
make the firſt multiplier; for the laſt product will be
the fame, in whatever order the multipliers are taken:
but it is convenient that the component parts be all di-
gits, or at leaft but fmall numbers, not above 10, 11,
or 12. See the following examples,
Ex. I.
Ex. 2.
Mult. 436
436 Mult.
351
by
9
8 by
8
351
.7
72.
or thus,
56.
or thus,
3924
3488
2808
2457
8
9
7
4
31392
31392
19656
9828
2
19656
Ex. 3.
642
Multipl.
by 37.
2568
9
23112 add
9625] fub.
23754 prod.
9350 prod.
Ex. 4.
Multipl.
by 34.
27
5
1375
7
Ex.
Chap. IV. MULTIPLICATION.
75
Ex. 5.
Ex. 6.
Multipl.
by 68.
348
8
Multipl.
by 252.
324
9
22272
2784
8
1392} add
· 23664 prod.
This method may be extended to pretty high num-
bers. For,
2016
7
20412
4
81648 prod.
10 = 2
x
× 5°
50 2 × 5 ×
X
X
5•
·
100 =
× 5 ×
5•
500 4
5 5 × 5.
1000 = 8
5 X 5 X 5.
1500 =
4 ×
5 ×
5 × 5 × 3.
10000 = 8 ×
5 ×
5 × 5 × 2 × 5·
100000= 10 X 10 X 10 X 10 X IO.
And the component parts of the intermediate num.
bers may easily be difcovered. The conveniency and
proper uſe of this method of multiplying will appear in
Section 2. following.
2. Inftead of beginning with the right-hand figure of
the multiplier, you may begin with the left; only take
care to place the right-hand figure of every particular
product directly under the multiplying figure, as in the
following examples.
Multip.
by
Ex. I.
4568
Ex. 2.
Multip.
6374
234
by
7408
9136
44618
13704
prod. 1068912
25496
18272
50992
prod.
47218592
K 2
3. Mul-
76
MULTIPLICATION. Chap. IV.
3. Multiplication may be performed without any bur
den to the memory, by fetting down every fingle pro-
duct, as in the following examples.
Ex. 1.
Mult. 8742
by 8
65316
Here I fay, 8 times 2 is 16, which I
fet down; then 8 times 4 is 32, which I
likewife fet down, viz. 3 before 1 and 2
under it; then 8 times 7 is 56, I fet
5 be.
fore 3 and 6 under it; laftly, 8 times 8 is
64, which I fet down in the fame manner :
Theſe added as they ftand, give the pro- Prod. 69936
duct.
Here obſerve, that when any fingle pro-
duct is under 10, to prevent miftakes by
miſplacing, you muſt fet a cipher in the
place which a fecond product figure would
have poffeffed.
462
Ex. 2.
Mult.
Mult. 3748
by
26
14248
824
1016
648
Prod.97448
The diſadvantage attending this method of multiply.
ing is, that the addition is tedious.
I
4. Multiplication may be performed by addition in
this manner: Set the digits 1, 2, 3, 4, &c. under one
another, and oppofite to place your multiplicand;
double it for 2; and to this fum add the multiplicand
for 3; and again to this fum add the multiplicand for
4 and thus go on till you have a table of the products
of the multiplicand by all the digits, or at leaſt as many
of them as your multiplier requires; then transfer your
particular products out of this table, and their fum will
be the total product. This method is convenient in
large operations. See the following example.
TABLE.
Chap. IV. MULTIPLICATION.
77
2
TABLE.
3785946
7571892
311357838
415143784
5 18929730
622715676
7 26501622
830287568
9 34073514
Mult. 3785946
by 659847
26501622
15143784
30287568
34073514
189297 30
1
22715676
Prod. 2498145110262
5. The Noble and ingenious Lord Napier, Baron of
Merchifton in Scotland, invented a method of perform-
ing multiplication by rods; the feparate form of which,
with the figures infcribed upon them, are as in the plate,
fig. 1.
The rods, excluding the index on the left hand, and
the rod of ciphers on the right, are juſt the ſeveral co-
lumns of the multiplication-table ſeparated or cut afun-
der from head to foot; each of the little fquares in the
table being divided on the rods by diagonal lines into
two triangles. The right-hand figure of every fingle
product in the table is placed on the rods in the lower
triangle, and the other in the upper and ſuch products
as confiſt but of one digit are always fet in the lower
triangle, and the upper one left blank.
:
The rods are diſtinguiſhed from one another by their
top-figures, 1, 2, 3, 4, 5, &c. making in all ten dif-
ferent rods, befides the index: but as the fame figure
may occur feveral times in a multiplicand, it is neceflary
to have three or four rods of each kind, or to have the
four fides of each rod infcribed with a different fet of
figures.
The rods are fitted for operation thus. To the right
fide of the index apply a rod on whofe top is the left-
hand figure of the multiplicand; next to this ſet the rod
on whofe top is the following figure of the multiplicand;
and
78
MULTIPLICATION. Chap IV.
and fo on to the rihght-and figure: then right againſt a-
ny figure on the index, you have the product arifing
from the multiplication of that figure into the multipli-
cand; but to be taken out by the help of an eaſy addi-
tion, as follows.
The figure in the lower triangle on the right hand
rod is the right-hand figure of the product; the figure
in the upper triangle on this rod, added to the figure in
the lower triangle on the next rod, gives the ſecond fi-
gure of the product. Again, the figure in the upper
triangle on the fecond rod, added to the figure in the
lower triangle on the rod following, gives the third fi-
gure of the product, &c.; and the figure in the upper
triangle on the rod next the index, is the laſt figure of
the product. In this manner are the particular products
taken from the rods, the fum whereof is the total pro-
duct. See the plate, fig 2. in which the rods are fitted
or fet together for the number 9587.
Mult. 9587
by 347
Against 7 on the index I find 67109
Againſt 4 I find
Againſt 3 I find
38348
28761
Total product 3326689
6. If you make a table of the multiplicand for the di-
gits 1, 2, 3, 5, the particular products may be made out
from this finall table, almoft with the fame eafe, as from
the rods, thus.
TABLE.
I 7894
2 15788
323682
5 39470
Suppofe for a multiplicand 7894.
When the figure of the multiplier is 2,
3, or 5, you have the product by infpec.
ting the table; when the multiplying fi
gure is 4, double the number againſt 2, or
add the numbers againſt 3 and 1; when it
is 6, double the number againft 3, or add
the.
1
•
Index
To front 22.78
Napiers Rods
Fig
2 3 4 5 6 7 8 9 0
1/0 1/2 14 16 18 2
67 90
/6 8
1/2 1/6 18 21 34 34 10
74
60
4
5
1% 15 % 35 % 3% 1%
24
4
2
40
10
94 36 7%
9
0/4
Index
19587
218% 16 1/4
32 212
59491
4 362 36 2
3
10/8/2
654%
7% % %
7/27/ 15%
981 75 720
Fig
2.
•
•
Chap. IV. MULTIPLICATION.
79
the numbers againſt 5 and 1; when it is 7, add the
numbers againſt 5 and 2; when it is 8, add the numbers
againſt 5 and 3; when it is 9, triple the numbers againſt
3, or add the numbers againſt 5, 3, and 1.
II. Multiplication of the parts of Integers.
Here there are three caſes.
1. If your multiplier is a fingle digit, fet it under the
units figure of the loweſt denomination, multiply it into
all the parts of the multiplicand, beginning at the loweſt,
and carrying always as in addition, or according to the
value of the next fuperior place.
EXAM P L E.
What is the price of 7 packs of cloth, at L. 64, 8 s.
101 d. per pack?
L. 5. d.
64
Here I fay, 7 times 2 is 14, which is
3 pence and 2 farthings over; I ſet down
the 2 farthings, and carry 3 to the place
of pence, faying, 7 times 10 is 70, and
3 that I carried makes 73, which is 6 451
fhillings and penny; I fet down the
I
8 10/12/20
7
2 1/
I penny, and carry 6 to the place of fhillings, faying,
7 times 8 is 56, and 6 that I carried is 62, which makes
3 pounds and 2 fhillings; I fet down the 2 fhillings, and
carry 3 to the place of pounds, which are integers.
2. If
your multiplier confiſts of two or more figures,
multiply continually by its component parts, or by the
component parts of the compofite number that comes
neareſt to it, and then multiply the given multiplicand
by the difference of the multiplier, and the nearest com-
pofite number: the fum or difference of theſe two pro-
ducts is the anſwer.
EXAMPLE
I.
What is the price of 56 C. tobacco, at L. 2: 14: 93
per C.?
Here the component parts are 8 and 7; for 8 × 7 =
56: therefore,
Multiply
80
MULTIPLICATION. Chap. IV.
Multiply first by 8, and that product by
7; or, which will give the fame anfwer,
multiply first by 7, and then that pro-
duct by 8.
L. S.
2 14
d.
92
- 8
21 18 6
7
153 9 6
EXAMPLE II.
What is the price of 126 yards of velvet, at L. 3:8:4
per yard?
Here I multiply firft by 6, that product
by 7, and that product again by 3: but as
the component parts are various, and may
be chofen at pleaſure, I would have had
the fame anfwer, had I multiplied by 9 ×
7 × 2; or by 7 × 3 × 3 × 2.
L. s. d.
3 8 4
6
20 10
7
EXAMPLE
143 10
3
430 10
III.
What is the price of 67 tuns of iron at L. 18: 16:81
per tun?
Here the neareſt compofite num-
ber is 64, whofe component parts
are 8 x 8; by which I multiply
continually, as in the margin : then
I multiply the given multiplicand
by 3, the difference betwixt 64
and 67; and becauſe the compo-
fite number is less than the multi-
plier, I add theſe two products for
the anſwer.
L. s. d.
18 16 81
8
150 13 8
1205
8
56
10 1 1/13
} a
94, add
1261 19 5/1/20
EX-
Chap. IV. MULTIPLICATION.
81
EXAMPLE
ር.
IV.
Q: 16.
2 3 14
9
What is the weight of 77 chefts of goods, each cheft
weighing C. 23: 14 Avoirdupois ?
Here the neareſt compofite num
ber is 81; and accordingly I mul-
tiply by its component parts 9×y:
then I multiply the given multipli-
cand by 4, the difference betwixt
77 and 81; and becauſe the com-
pofite number is greater than the
multiplier, I fubtract the one pro- 232
duct from the other for the an.
fwer.
25 3 14
II 2
9
3
14}fub.
221 I 14
EXAMPLE V.
What is the price of 274 yards of linen, at 3 s. 4 d.
per yard?
Here I multiply con- L. S. d.
tinually by the com.
ponent parts 10x 0x2,
which gives the price
of 200 yards : and
then, for the price of
the reſt, I work as fol-
lows, viz. for the 70
yards, I multiply the
price of to yards by
7; and for the 4 yards,
I multiply the price of
I
3 4 price of 1 yard.
10
1 13 4 price of 10 yards.
ΙΟ
16 13 4 price of 100 yards.
33 06
11 13
13
one yard by 4; and
theſe three products
added gives the an- 45 13
fwer.
2
8 price of 200 yards.
4 price of 70 yards.
4 price of 4 yards.
4 price of 274 yards.
From the above example may be deduced a general
and eaſy rule for working all queſtions of this kind; and is
of excellent uſe when the multiplier happens to be a high
nun ber; viz.
Multiply continually ſo many times by 10 as there are
L
figures
82
MULTIPLICATION. Chap. IV.
figures in the multiplier, fave one; then multiply the gi
ven price by the right-hand figure of the multiplier; and
again, the first product of 10 by the following figure of
the multiplier; and fo on, till you have multiplied by all
the figures in the multiplier. The fum of theſe products
is the anſwer.
EXAMPLE VI.
What is the price of 8604 yards of cloth, at 19 s. 6 d.
per yard?
L. s. d. Price of
1961
L. s. d. Price of
1 yd, × 4
3 18 2
4 yds.
10
9 15 5
10 yds, x o
ΙΟ
97 14
2
100 yds, × 6 = 586 5
600 yds.
ΙΟ
977
I 8
-
1000 yds, x8 = 7816 13 4 8000 yds.
Price of 8604 yards 8406 16 6
MORE EXAMPLES.
1. What is the price of 8 yards of cloth, at 13 s. 4 d. ?
Anſ. L. 5:6 : 8.
2. What comes 12 reams of paper to, at 8 s. 61 d. ?
Anf. L. 5:2: 6.
3. What coft 96 barrels, at L. 1:14:7?
L. 166, . s.
Anf.
4. What comes 123 gallons to, at 7 s. 83 d.? Ans.
L. 47: 10: 81.
5. A gentleman, whofe yearly income is L. 250, fpends
dailys. 6 d.: How much does he ſpend in a year, or
365 days? and how much does he fave yearly? Anf
He spends L. 174: 2: 8½, and faves L. 75: 17:31.
6. What comes y760 tuns to, at 18 s. 74 d. ? Anf.
L. 9078: 16: 8.
3. If
Chap. IV. MULTIPLICATION.
83
3. If your multiplier conſiſts of integers and parts, the
operation is performed by a crofs multiplication of the
feveral parts of the multiplier into all the parts of the mul-
tiplicand.
The contents of maſon and joiners work are frequent-
ly caft up by this kind of multiplication; for underſtand-
ing of which obſerve, that
The fuperficial content of any rectangle is found by
multiplying the length into the breadth; and the content
of a right-angled triangle is found by multiplying the baſe
into half the perpendicular or height.
The dimenſions are uſually taken in lineal feet, inches,
and lines; and the operation is performed by the follow.
ing
RULE S.
I. Any lineal meaſure multiplied into the fame lineal
meaſure produces fquares of that name. Thus, lineal
feet multiplied into lineal feet produce ſquare feet; lineal
inches into lineal inches produce fquare inches, &c.
II. Lineal feet into lineal inches produce rectangles,
I foot long and 1 inch broad, which divided by 12 quote
fquare feet; and the remainder multiplied by 12, produces.
fquare inches.
I
III. Lineal feet into lincal lines produce rectangles, L
foot long and 1 line broad, which divided by 144 quote
fquare feet; and the remainders are rectangles equal to
fquare inches.
IV. Lineal inches into lineal lines produce fmall rect-
angles, 1 inch long and 1 line broad, which divided by
12 quote fquare inches; and the remainder, multiplied
by 12, produces fquare lines.
EXAMPLE
I.
In an area, pavement, or piece of plaifter-work, in
length 24 feet 7 inches, and in breadth 18 feet 5 inches,
how many fquare feet?
I 2
F.
84
MULTIPLICATION. Chap. IV.
F.in.
18×7=126
24 7
24×5120
18
5
12)246(20 by Rule II.
432 35
20 72
1+5211071
24
12×6=72)
Here I multiply 18 lineal feet into 24 lineal feet, and
the product is 432 fquare feet; then I multiply 5 lineal
inches into 7 lineal inches, and the product is 35 ſquare
inches, by Rule I.; then I multiply 18 lineal feet into 7
lineal inches, and the product is 1.6; and again I mul-
tiply 24 lineal feet into 5 lineal inches, and the product
is 120; which added to the former product gives 246
rectangles, each being 1 foot in length and 1 inch in
breadth; theſe divided by 12 quote 20 fquare feet; and
the remainder 6 multiplied by 12 produces 72 ſquare
inches, according to Rule II.; theſe I add to the former
fquare feet and inches, and find the anfwer or total pro-
duct to be 452 fquare feet, and 107 fquare inches.
EXAMPLE
II.
In an area or floor, in length 38 feet 9 inches 6 lines,
and in breadth 23 feet 8 inches 6 lines, how many
fquare feet?
F. \in. li.
38 9
23×9=207
38×8=304
28
6
12) 511(42
8747 36
48
484
By Rule II.
278
31
8 72
24
191498108
12×7=84)
38
Chap. IV. MULTIPLICATION.
85
38x628
8×6=48
23×6138
9×6=54
144)366(2)By Rule III.
12)102(8
By Rule IV.
288
96
78
12×6=72
Becauſe the fum of the inches exceeds 144, I carry
1 from them to the column of feet, and ſet down the o-
verplus, viz. 98.
The operation may be rendered eaſier and fhorter by
previously reducing the factors to two denominations,
viz inches and lines. Thus the former example may be
propofed and wrought as follows.
In an area or floor, in length 465 inches 6 lines, and
in breadth 284 inches 6 lines, how many fquare inches
and feet?
Inch. lin.
465×6=2790
465 6
284x1704
28. 6
by Rule IV.
12)4494(374 in.
132000 30
12×6=72 li.)
274 72
3434108
The answer here is 132434 fquare inches and 108
fquare lines; and if the inches be divided by 14+, you
will have 910 fquare feet and a remainder of yo ſquare
inches, as before.
Or the factors may be reduced to the loweft de-
nomination, viz. lines, and then the product will be
fquare lines, which, divided by 144, will quote fquare
inches, and the remainder will be fquare lines; and
the fquare inches divided by 144 will quote fquare
feet, and the remainder will be fquare inches.
gain, the fquare feet divided by 9 will quote fquare
yards, and the remainder will be fquare feet; and the
fquare yards divided by 36 will quote fquare roods, and
the remainder will be fquare yards,
A.
If
86
MULTIPLICATION. Chap. IV.
If this cross multiplication be extended to the menfu-
ration of folids, the content of which is found by multi-
plying the fuperficial content of the bafe into the height,
depth, length, or thickneſs, the operation muſt be con-
ducted by the following
RULE S.
Thus
V. Any fuperficial meaſure multiplied into the fame
lineal meaſure produces a folid of the ſame name.
fuperficial feet multiplied into lineal feet produce folid
feet; fuperficial inches multiplied into lineal inches pro-
duce folid inches, &c.
VI. Superficial feet into lineal inches produce pa-
rallelopipeds, whofe bafe is I fquare foot, and their
height inch; which divided by 12 quote folid feet;
and the remainder multiplied by 144 produces folid inches.
VII. Superficial feet into lineal lines produce pa-
rallelopipeds, whoſe baſe is 1 ſquare foot, and their
height line; which divided by 144 quote folid feet;
and the remainder multiplied by 12 produces folid inches.
VIII. Superficial inches into lineal lines produce pa
rallelopipeds, whofe bafe is 1 fquare inch, and their
height line; which divided by 12 quote folid inches;
and the remainder multiplied by 144 produces folid inches.
IX. Lineal feet into fuperficial inches produce pa-
rallelopipeds, whofe bafe is 1 fquare inch, and their
height 1 foot; which divided by 144 quote folid feet
and the remainder multiplied by 12 produces folid lines.
X. Lineal feet into fuperficial lines produce parallelo-
pipeds, whoſe baſe is 1 fquare line, and their height 1
foot; which divided by 12 quote folid inches; and the
remainder multiplied by 144 produces folid lines.
XI. Lineal inches into fuperficial lines produce paral-
lelopipeds, whofe bafe is 1 fquare line, and their height
I inch; which divided by 144 quote folid inches; and the
remainder multiplied by 12 produces ſolid lines.
EXAMPLE III.
In a piece of timber, whoſe length is 18 feet 6 inches,
breadth
Chap. IV. MULTIPLICATION.
87
breadth 2 feet 4 inches, and thickneſs
how many fold feet?
F. in.
18 6 2×6=12
2
36
7
+3
32
4/18×4=72
24
12)84(7 F.
24 fuperficial
3 lineal
72
861
101296
576
97 216' folid
·
by Rule II.
feet 3 inches,
43×3=129
by
12)129(10 F. Rule
144×9=1296 in.] VI.
12 × 48 = 576 in.§ IX,
2 x 2448
by R.
Here I first multiply 18 feet 6 inches into 2 feet 4i1-
ches, as formerly, and the product is 43 feet 24 inches.
fuperficial; which I next multiply into 2 feet 3 inches.
lineal, thus, 43 fuperficial feet into 2 lineal feet produce
86 folid feet, and 24 fuperficial inches into 3 lineal in-
ches produce 72 folid inches, by Rule V.; then 43 fu-
perficial feet into 3 lineal inches produce 129 parallelopi-
peds, whoſe bafe is 1 fquare foot, and their height 1
inch; which divided by 12 quotes 10 folid feet; and
the remainder 9 multiplied into 144 produces 1296 folid
inches, by Rule VI. Again, 2 lineal feet into 24 ſuper-
ficial inches produce 48; which, being less than 144, I
eſteem a remainder, and multiplying it into 12 I have a
product of 576 folid inches, by Rule IX.
Becauſe the fum of the inches exceeds 1728, I carry
I from them to the feet, and the overplus 216 I ſet
down.
EXAMPLE
IV.
How many folid feet in a polifhed ftone that is 8 feet.
9 inches 5 lines long, 7 feet 3 inches broad, and 3 feet.
5 lines thick?
F.
88
MULTIPLICATION. Chap. IV.
F. \in. l. 7×9=63
8 9
7 3
58×3=24
12)87(7 F.
56 27
7 36
35
36
I
12 × 3 = 36 in.
by Rule II.
7x535 in. by Rule III.
3635=15
12)15(1 in. by Rule IV.
63 99 30fup. 12x3=36 li.]
5lin.
18063x5=315, and 144)315(2 F. by R. VII.
12×27 = 324 in.
:
3×99=297, and 44)297 (2F.
189
2324
2108
9
19348261 2 folid
}
12 × 108 in S
X
I
by R. IX.
414324×36=108, and 12) 108 (y in. by Rule X.
5x99=495, and 12)495(41 in 1
}
144×3=432 lines. by R. VIII.
The operation may be facilitated by previously redu-
cing the three factors to two denominations, viz. inch-
es and lines, as was done in Example II. on fuperficial
meaſure.
Or the three factors may be reduced to the loweſt de.
nomination, viz. lines, which being multiplied conti-
nually, will produce folid lines; which divided by 1728
will quote folid inches, the remainder being folid lines;
and the folid inches divided by 17.8 will quote folid
feet, the remainder being folid inches; and the ſolid feet
divided by 27 will quote folid yards, the remainder being
folid feet; and the folid yards divided by 216 will quote
folid roods, the remainder being folid yards.
I ſhall only further obſerve, that as the rules for work-
ing questions by crofs multiplication are numerous, and
the operation tedious, it is eaſier to convert the parts in-
to a decimal fraction of their integer, and then work as
taught in multiplication of decimals.
III. The Proof of Multiplication.
Multiplication may be proved feveral ways, viz. by
multiplication, by divifion, and by cafting out the y's.
I.
Chap. IV. MULTIPLICATION.
89
1. By multiplication: Change the places of the fac-
tors, and make that the multiplier which before was the
multiplicand; and if the work be right, you will have
the fame product as before; but this method is tedious.
2. By divifion: When the work is right, the product
divided by the multiplier quotes the multiplicand; or,
divided by the multiplicand, quotes the multiplier. But
this fuppofes the learner acquainted with divifion.
3. The moſt uſual method therefore of proving multi-
plication is by caſting out the 9's; which is done thus :
Caft the y's out of the multiplicand and multiplier, and
place the exceffes on the right and left fides of a cross;
multiply theſe two figures into one another, cafting the
9's out of their product, if need be, and place the excels
at the top of the cross; then cafting the 9's alſo out of
the product of your multiplication, place its exceſs at the
bottom; and if the work be right, the figures at top
and bottom will agree, or be the ſame.
EXAMPLE I.
754
38
1
6032
2262
Here I caft the 9's out of the multi-
plicand, and place the excefs 7 on the
right fide of the cross; then I caft the
9's out of the multiplier, and place the
exceſs 2 on the left-fide of the cross;
next I multiply theſe exceffes 2 and 7
into one another, caft the y's out of 28652
their product, and place the exceſs 5 at
5
2
X7
5
:
the top of the crofs; laftly, I caft the 9's out of the pro-
duct, and place the excels at the foot of the cross
which being the fame with the figure at the top, I con-
clude the work to be right.
EXAMPLE II.
L. So d.
43 8 4
7
8X²
2
7
6 10
Here in cafting the 9's out of
the multiplicand, and out of the
product, I begin with the pounds,
and reduce the excefs to fhillings,
and in like manner the excess of 347
the fhillings is reduced to pence,
ΝΙ
and
90
MULTIPLICATION. Chap. IV.
and that of the pence to farthings. The multiplier being
an abſtract number, needs no reduction; but if a multi-
plier be a mixt number, or confift of integers and parts,
as feet and inches, &c. the excefs of the higher deno-
mination muſt always be reduced to the lower.
IV. Practical Questions.
1. The continual multiplication of the nine digits will
give the number of changes that may be rung on nine
bells How
:
many changes are there? Anf. 362880.
2. What is the fum, and what the difference, of fix
dozen dozen, and half a dozen dozen? Anf. Sum 936.
Diff. 792.
3. What number taken from the fquare of 86 will
leave 17 times 34?
Anf. 6818.
4. The leffer of two numbers is 284, their difference
is 132; what is the fquare of their product, and what
the cube of their fum? Anf. Square of their product is
13958004736; cube of their fum 243000000.
5. Each of nine pui fes contains L. 81:13:4; how
much money in all? Anf. 7351.
6. What is the price of 1000 hhds tobacco, at
L. 12:36 per hhd? Anf. 121751.
7. The wainſcotting of a rectangular room meaſures
156 feet 4 inches about, and is 1 feet
4
inches high;
3
the door is 7 feet by 3 feet 8 inches; eight window-
fhutters are 7 feet 2 inches by 4 feet 6 inches; the door
and window.ſhutters, being wrought on both ſides, are
reckoned as work and half; the chimney, 3 feet 9 inches
by 3 feet 3 inches, not being incloſed, is to be deduct
ed: How many yards of wainſcotting in the room?
Anf. 261 fq. yards, 8 fq. feet, and 36 fq. inches.
8. How many folid yards of digging in a cellar that is
27 feet 4 inches long, 17 feet 8 inches broad, and 8
feet 2 inches deep? Anf. 146 folid yards, 1 folid foot,
and 1024 folid inches.
CHAP.
Chap. V.
91
DIVISION.
D
CHAP. V.
DIVISION.
IVISION difcovers how often one number is con-
tained in another: or,
Divifion, from two numbers given, finds a third,
which contains unity as often as the one given number
contains the other.
The number to be divided, or which contains the o-
ther, is called the dividend; the number by which we
divide, or which is contained in the dividend, is called
the divifor; and the number found by divifion, or which
expreffes how often the dividend contains the divifor, is
called the quotient or quot.
18
6
12
As multiplication fupplies the place of many additions,
fo divifion, which is the reverſe of multiplication,
ſerves inſtead of many ſubtractions; as will thus ap-
pear Suppoſe it were required to divide 18 by 6,
that is, to find how often 6 is contained in 18, the
work by ſubtraction will ſtand as in the margin: 6
by which it appears that 6 is contained 3 times in
the number 18. But this, by diviſion, may be
found at one trial: thus,
êlaal
18
I fet the divifor on the left of the dividend,
leaving room on the right hand for the quo- 6)18 (3
tient, as in the margin; and then I fay, How
often 6 in 18? Anf. 3 times: this 3 I fet in
the quotient; then I multiply the quotient fi-
gure 3 into the divifor 6, faying, 3 times 6
make 18; which I fet down below the dividend, and
fubtract it from the dividend, and o remains.
I. Divifion of Integers.
RULE S.
(0)
I. From the left-hand part of the dividend point off
M 2
the
92
Chap. V.
DIVISION.
the firſt dividual, viz. fo many figures as will contain
the divifor.
II. Aſk how often the divifor is contained in the divi-
dual, and put the anſwer in the quotient.
III. Multiply the divifor by the figure fet in the quo
tient, and ſubtract the product from the dividual.
IV. To the right of the remainder bring down the
next figure of the dividend for a new dividual; and then
proceed as before.
The ſubſtance of theſe rules is briefly expreffed in the
following monoftich.
· Dic quot? multiplica, fubduc, transferque fequentem.
Firſt aſk how oft? in quot the anſwer make;
Then multiply, fubtract, and down a figure take.
EXAMPLE
1.
Divi- Divi- Quo-
for. dend. tient.
7)875(125
7
17
14
35
35
Here, becauſe the divifor 7 is con
tained in 8, the left-hand figure of
the dividend, I point it off, as my
firft dividual, according to Rule I.;
and then I fay, How often 7 in 8 ?
Anf. I time; which I fet in the
quotient, as dirctected in Rule II.;
then I multiply the divifor 7 by this
quotient figure 1, and fubtract the
product 7 from the dividual 8, as di-
rected in Rule III.; to the remainder
1 I bring down the following figure
of the dividend, for my fecond divi-
dual, as directed in Rule IV.; then I proceed as before,
and fay, How often 7 in 17? Anf. 2 times; where-
fore, ſetting 2 in the quotient, I multiply and fubtract,
and find the next remainder to be 3; to which I bring
down the following figure of the dividend, and have 35
for my third dividual; then I fay, How often 7 in 35?
Anf 5 times; which 5 being placed in the quotient, I
multiply and fubtract, and o remains; fo the quotient
is 1 25.
(0)
By reviewing the fteps of the preceding operation,
and reducing, by Axiom VI. the dividuals and quotient-
figures
Chap. V.
93
DIVISION.
*
figures to their feparate values, the reafon of the rules.
will be obvious; for,
The ſeparate value
of the first dividual 8
is 800; and the fe.
parate value of
1, the
first figure put in the
quot, is 100; for
as 8 contains 7, the
divifor, 1 time, fo
800 contains it 100
times, and 100 re.
mains; to which I
bring down the fol
lowing figure of the
dividend 7, whoſe ſe-
7)875(100)
ftdividual 800 20
partial quots.
700 5J
rem. 100 125 total quot.
add
70
2d dividual 170
I 40
rem.
30
add
5
parate value is 70; 3d dividual 35
and my ſecond divi-
dual is 170; and as
7 is contained 2 times
in 17, fo it is con-
35
(0)
tained 20 times in 170, and 30 remains; to which I
bring down the next or laft figure of the dividend 5;
and my third dividual is 35, in which the diviſor 7 is
contained 5 times. Now it is evident, that the fum of
the partial quots, 125, is the total quot, or a number ex-
preffing how often the dividend 875 contains the divi-
for 7.
From the above example we may learn, that there
are always juſt fo many figures in the quotient as there
-are dividuals; or the first dividual, with the number
of fubfequent figures in the dividend, is equal to the
number of places or figures in the quotient.
Hence likewife may be inferred, that no divifor is
contained in any dividual oftener than 9 times; for the
dividual, excluding the right-hand figure, is always lefs
than the divifor by 1 at leaft; and if both be multiplied.
by 10, or have a cipher annexed to each of them, the
product of the dividual will be less than the product of
the divifor by 10 at leaft; but no right-hand figure can
ſupply
94
Chap. V.
DIVISION.
i
fupply this defect of 10; therefore the divifor is not
contained 10 times in any dividual, and conſequently not
oftener than 9 times.
Here too obferve, that the right-hand figure of the firſt
dividual, and all the fubfequent figures of the dividend,
have a point or dot fet below them, as they are brought
down; which is done to prevent miftakes, by diftinguiſh-
ing them, in this manner, from the figures not yet brought
down.
EXAMPLE II.
Here, becauſe 8
is not contained in
5, I point off 56 as
my first dividual, and
fay, How often 8 in
56? Anf. 7; which
I put in the quo
tient; then multi-
ply 7 into the divifor
8, and fubtract the
product 56 from the
dividual; and as no-
thing remains, I
8) 56032897 (70041121 num.
56
·032
32
•8
8
'9
8
bring down the next
figure of the divi-
17
dend, which hap-
16
pens to be a cipher;
and as I cannot
に
​have 8 in o, I put
8 denom,
(1)
o in the quotient; and, as multiplying and ſubtracting
is in this cafe needlefs, I bring down the next figure of
the dividend 3; and as I cannot have 8 in 3, 1 put
another o in the quotient, and bring down the next fi
gure of the dividend 2: then I fay, How often 8 in 32 ?
Anf. 4; which I put in the quotient: then I multiply
and fubtract; and as nothing remains, I bring down
the next figure of the dividend 8, and fay, How often 8
in
Chap. V.
95
DIVISION.
in 8? Anf. 1; which I put in the quotient: then I mul-
tiply and ſubtract; and as nothing remains, I bring down
the next figure of the dividend 9, and fay, How often 8
in 9? Anf. 1; which I put in the quotient: then I mul
tiply and ſubtract; and to the remainder 1 I bring down
the next and laft figure of the dividend 7, and fay, How
often 8 in 17? Anf. 2; which I put in the quotient =
then I multiply and fubtract, and remains.
To complete the quotient, I draw a line on the right
hand, and ſet the remainder above the line, and the di-
vifor 8 below it, fignifying that remains to be divided
by 8; or this part of the quotient may be confidered as
a fraction, whofe numerator is 1, and its denominator
8; and the quotient thus completed fhews, that the di
vidend contains the divifor 70041 1 2 times, and one eighth
part of a time.
Here obferve, that not only the laft remainder, but
every other remainder, muſt be leſs than the divifor; for
if it be either greater or equal, the divifor might have
been oftener got, and the quotient-figure is too little.
And ſhould any one in this cafe attempt to continue the
operation, the quotient figures would all be g's, the di-
viduals would prove inexhauftible, and the remainders
would conftantly increaſe.
Hence alto learn, that if any dividual happen to be leſs
than the divifor, you must put o in the quotient, and
bring down the next figure of the dividend; and if it be
fill lefs than the divifor, you must put another o in the
quotient, and bring down the following figure of the di
vidend, &c.
EX.
96
Chap. V.
DIVISION.
་
EXAMPLE
III.
Here the divifor confifts of 36) 789426 (2192838
72
69
36
334
324
102
72
306
288
(18)
two figures; and becauſe it
is contained in the two left-
hand figures of the dividend.
78, I point them off as my
firſt dividual; and fay, How
often 3 in 7? Anſ. 2, and 1
remains, which I placed, or
conceived as placed, on the
left hand of the following fi-
gure 8, makes 18 : then I ſay,
Can I have the following fi-
gure of the divifor 6 alfo 2
times in 18? Anf Yes; con-
ſequently I get 36 the divifor
2 times in 78 the dividual;
wherefore I put 2 in the quo-
tient, and multiply that 2 in-
to the divifor 36, and the product 72 I ſubtract from
the dividual 78; and to the remainder 6 I bring down
the following figure of the dividend 9, for a new divi-
dual: then I ſay, How often 3 in 6? Anf. 2, and o re-
mains; again I fay, Can I have 6 alfo 2 times in 9?
Anf. No; therefore I can have 36 in 69 only 1 time,
which I I put in the quotient: then I multiply and fubtract
as before; and to the remainder 33 1 bring down the
next figure 4 for a new dividual: then, becauſe the di-
vidual confiſts of a figure more than the divifor, I ſay,
How often the firſt figure of the diviſor 3 in the firſt two
figures of the dividual 33? Anf. 9, and 6 remains ;
which 6 placed on the left hand of the following figure
4 makes 64: again I fay, Can I have 6 alſo 9 times in
64? Anf. Yes; confequently 36 can be had 9 times in
334; wherefore I put 9 in the quotient: then I multi-
ply and fubtract; and to the remainder 10 I bring down
the next figure 2 for a new dividual: here like wife, be.
cauſe the dividual has a figure more than the divifor, I
fay, How often 3 in 10? Anf. 3, and 1 remains; which
I
Chap. V.
97
DIVISION.
I placed on the left hand of the following figure 2 makes
12: again I ſay, Can I have 6 alfo 3 times in 12 ? Anf.
No; confequently 36 cannot be had 3 times in 102;
wherefore I try if I can have it 2 times; faying, 2 times
3 is 6 from 10, and 4 remains; which 4 placed on the
left hand of the next figure 2 makes 42: and I again
fay, Can I have 6 alfo 2 times in 42? Anf. Yes; con-
fequently 6 can be had 2 times in 102; accordingly I
put 2 in the quotient, multiply and fubtract; and to the
remainder 30 I bring down the next and laſt figure of the
dividend 6, for a new dividual: then, because the divi
dual has a figure more than the divifor, I fay, How often
3 in 20? An 9. and 3 remains; which 3 placed on
the left hand of the following figure 6 makes 36: and I
again fay, Can I have 6 alfo 9 times in 36? Anf. No;
confequently 36 cannot be had 9 times in 306; therefore
I try if it can be had 8 times, faying, 8 times 3 is 24
from 30, and 6 remains; which 6 placed on the left
hand of the following figure 6 inakes 66: I again fay,
Can I have ó alſo 8 times in 66? Anf. Yes; confe-
quently 36 can be had 3 times in 306; wherefore I put
8 in the quotient, and multiply and fubtract as before:
the laſt remainder 18 is the numerator of a fra&ion, and
the divifor its denominator, to be annexed to the inte
gral part of the quotient; as was taught in the former
example.
The preceding operation points out the manner of
procedure when the divifor confifts of more figures than
one, viz. you muſt take the firſt figure of the divifor out
of the first figure of the dividual, or out of the firſt two
figures of the dividual in cafe the dividual have a figure.
more than the divifor : then imagine the remainder to be
prefixed to the next figure of the dividual, and try if
you
can have the ſecond figure of the divifor as often out of
this number; if you can, imagine again the remainder
to be prefixed to the following figure of the dividual,
and try if you can have the third figure of the divifor as
often out of this number, &c.; but if you find you can-
not have fome fubfequent figure of the divifor fo often
as you took the first, you muſt go back, and take the
firft
N
98
Chap. V.
DIVISION.
first figure of the divifor 1 time lefs, or fome number of
times lefs, out of the firſt, or out of the firſt two figures
of the dividual: then proceed as before, repeating the
trial, till you find you can have the fecond, and all the
fubfequent figures of the divifor, as often as you took the
firft.
But here obſerve, that if, in trying how often the di-
vifor can be had in the dividual, either 9, or a number
greater than 9, any where remain, you may conclude,
without further trial, that all the fubfequent figures of
the divifor can be had as often as you took the firſt; as
may be thus demonftrated.
Suppoſe the fubfequent figures of the divifor to be the
higheſt poffible, that is, all g's, and the following fi
gures of the dividual the loweft poffible, that is, all o's;
again, imagine the remainder 9 prefixed to the follow.
ing figure of the dividual o, and it will make yo; now
it is plain, that the fubfequent figure of the divifor 9 can
be had in 90, the highest number of times poffible, viz.
9 times, and 9 will remain; which prefixed to the next
figure of the dividual o, makes 90, in which the ſubſe-
quent figure of the divifor 9 can again be had 9 times,
and 9 will remain as before; therefore all the fubfequent
figures of the divifor can be had as often as you took the
first; and if they can be had in this cafe, much more can
they be had when a number greater than 9 remains.
EXAMPLE IV.
Let it be required to divide 170948 I. among 234
men.
Here
Chap. V.
99
DIVISION.
Here the divifor confifts of
I
1638
714
702
128 rem.
20
234)2560(10 s.
234'
220 rem.
I 2
440
220
three places; and becauſe it is 234)170948(730L
not contained in the three left-
hand figures of the dividend, I
point off 1709 as my first divi-
dual, and fay, How often 2 in
17? Anf. 8, and remains;
which 1, placed on the left of
the following figure o, makes
10: then I ſay, Can I have the
following figure of the divifor
3 alfo 8 times in 10? Anf. No;,
confequently 234 cannot be had
8 times in 1709; wherefore I
try if I can have it 7 times,
faying, 7 times 2 is 14, from
17, and 3 remains; which 3,
placed on the left of the follow-
ing figure o, makes 30; I fay
again, Can I have the following
figure of the divifor 3 alfo 7
times in 30? Anf. Yes; and
becauſe 9 remains, I conclude,
without further trial, that the
divifor 234 may be had 7 times
in the dividual 1709; fo I put
7 in the quotient, multiply and
fubtract, and to the remainder
71 I bring down the following
figure of the dividend 4 for a
new dividual; and becauſe this
dividual confifts of the fame
number of places as the diviſor,
I fay, How often 2 in 7? Anf
3, and 1 remains; which 1, pla-
I
234)2640(11 d.
234°
300
234
66 rem.
4
234)264(120 f.
234
(30)
ced on the left of the following figure 1, makes 11
then I ſay again, Can I have 3 alfo 3 times in 11? Anf.
Yes, and 2 remains; which 2, placed on the left of
N 2
the
100.
Chap. V.
DIVISION.
the following figure 4, makes 24; then I fay aga'n, Can
I have the third figure of the divifor 4 alfo 3 times in
24? Anf. Yes; and confequently 234 can be had 3
times in 714; wherefore I put in the quotient, multi-
ply and fubtract, and to the remainder 12 I bring down
the next and laft figure of the dividend 8 for a new di-
vidual; but as the divifor cannot be had in this dividual,
I put o in the quotient, and the dividual 128 becomes
the last remainder.
128
234
Here, instead of annexing the fraction to the inte-
gral part of the quotient, I multiply the 128 1. which
remains to be divided among 234 men, by 20, the num-
ber of fhillings in a pound, and the product 2560 is fhil-
lings, which I divide by 234, and the quotient gives
Io s. to each man; the remainder 220 s. I multiply by
12, the number of pence in a hilling, and the product
2640 is pence, which I divide by 234, and the quotient
gives 11 d. to each man; the remainder 66 pence I
multiply by 4, the number of farthings in a penny, and
the product 264 is farthings, which I divide by 234,
and the quotient gives to each man 1 farthing, and 234
of a farthing: ſo each man's fhare is L. 730:10: II
12330 f.
1
EXAMPLE
V.
If 34168062 C. of goods be divided into 4875 equal
lots, what will be the weight of each lot?
!
Here
}
Chap. V.
JOI
DIVISION.
t
34125
43062
39000
4002 rem,
4
4875)16248( 3 Q
14625
Here the divifor confifts of 4875)34168062)7008 C.
four places; and becauſe it
is not contained in the four
left-hand figures of the divi-
dend, I point off 34168 as
my firft dividual, and fay, How
often 4 in 34? Anf-8, and
2 remains; which 2 placed on
the left of the following figure
1, makes 21: then I ſay, Can
I have the following figure of
the divifor 8 alfo 8 times in
21? Anf. No; confequently
4875 cannot be had 8 times
in 34168; wherefore I try if
I can have it 7 times, faying,
7 times 4 is 28, from 34, and
6 remains; which 6, placed
on the left of the following
figure 1, makes 61: then I
ſay again, Can I have the fol-
lowing figure of the divifor 8
alſo 7 times in 61? Anf. Yes;
and 5 remains; which 5, pla-
1623 rem.
28
I 2984
3246
4875)45444(1892lb.
43875
(1569) rem.
5
ced on the left of the following figure 6, makes 56:
then I fay, Can I have the third figure of the divifor 7 al-
fo 7 times in 56? Anf. Yes, and 7 remains; which 7,
placed on the left of the following figure 8, makes 78:
then I fay, Can I have the fourth figure of the divifor
alfo 7 times in 78? Anf. Yes; confequently 4875 can
be had 7 times in 34165; wherefore I put 7 in the quo-
tient, multiply and fubtract, and to the remainder 43 I
bring down the next figure of the dividend o for a new
dividual; and as I cannot have the divifor 4875 in this
dividual 430, I put o in the quotient, and bring down
the next figure of the dividend 6; but as the divifer
4875 is ſtill greater than the dividual 4306, I put ano-
thero in the quotient, and bring down the next and laſt
figure of the dividend 2; and as the dividual now con
fifts of five places, and the divifor but of four, I ſay,
How
102
Chap. V.
DIVISION.
}
How often 4 in 43? Anf. 9, and 7 remains; which 7,
placed on the left of the following figure o, makes 70;
I fay again, Can I have the following figure of the divifor
8 alfo 9 times in 70? Anf No; confequently 4875 can-
not be had times in 43062; wherefore I try if I can
have it 8 times, faying, 8 times 4 is 32, which, fub-
tracted from 43, leaves a remainder above 9; therefore
I conclude, without further trial, that the divifor can
be had 8 times in the dividual; accordingly I put 8 in
the quotient, multiply and ſubtract, and the laſt remain-
der is 4062.
4875
Here, as in the former example. inftead of annexing
the fraction 499 to the integral part of the quotient, I
4062
multiply the 4062 C. which remains, by 4, the number
of quarters in a C. and the product 16248 is quarters;
which I divide by 4875, and the quotient gives 3 Q. to
each lot the remainder, viz. 1023 Q1 multiply by
28, the number of pounds in a quarter, and the product
45444 is pounds; which I divide by 4875, and the quo
tient gives to each lot 9 pounds and 12 of a pound:
fo the weight of each lot is 7008 C. 3 Q 9492 lb.
EXAMPLE
If, as in the margin, a cipher,
4875
VI.
648
1569
or ciphers, poffefs the right 6480)89678|2(1382542
hand of the divifor,cut them off,
and cut off, as many figures,
viz. in this example, the figure
2 from the right hand of the
dividend; then divide the re
maining figures of the divi-
dend, viz. 89678, by the re-
maining figures of the divifor,
viz. 648, and you have the
integral part of the quotient;
but to the remainder 254 an-
2487
1944
5438
5184
(2542)
nex the figure cut off from the dividend, and you have
2542 for the numerator of your fraction, and the whole
divifor 6480 is the denominator.
The reaſon will appear obvious by working a queſtion
!
in
Chap. V.
103
DIVISION.
in this manner, and alfo at full length, without cutting
off the cipher or ciphers, and then comparing the two
operations.
EXAMPLE VII..
96
180
If, as in the margin, the fi. 4800)9780|00(20338
gures cut off from the right
hand of the dividend, happen to
be all ciphers; in this cafe, the
laft remainder, without regard.
ing the ciphers cut off, is the
numerator of your fraction, and
the fignificant figures of the di-
vifor the denominator. The rea
fon is affigned in the doctrine of fractions.
144
(36)
In like manner, if there be cut off from the dividend
any number of fignificant figures, with a cipher or ci-
phers on their right hand; in this cafe, the laft remainder,
with the fignificant figures cut off, make the numerator
of your fraction; and the fignificant figures of the di-
vifor, with as many ciphers as the number of fignificant
figures cut off from the dividend, make the deno-
minator. Thus, if, in the above example, the figures
cut off from the dividend had been 50, the numerator
of your fraction would have been 365, and the denomi
nator 480.
MORE
68)4768943(
397)8020094(
1679)9437856(
EXAMPLE S.
740)638007935(
89000)5765432000(
2530000) 3250+387+100(
Contractions, and fimple ways of working Divifion of
Integers.
1. To divide any number by 10, 100, 1000, &c.
you have only to point off for a remainder as many fi-
gures on the right hand of the dividend as the divifor
has ciphers, and the other figures on the left of the
point or feparatrix are the quotient. Thus, 7489634
divided by 10, 100, 1000, &c. ftands as follows.
Quot.
104
Chap. V.
DIVISION.
Quot. rem.
10)748963.4
100)7-896 34
1000)7489.634
10000)748.9634
2. If the figures of the divifor are all 9's, or all ex-
cept the units figure, as 9, 99, 999, 98, 997, 9996,
&c. work as follows.
Find a new divifor, by annexing to unity as many ci-
phers as there are figures in the given divifor, fubtract
the given from the new divifor, and the remainder or
difference is the complement. Divide the given dividend
by the new diviſor, viz point off fo many figures on
the right hand as there are ciphers in the faid divifor; the
figures thus pointed off are to be efteemed a remainder,
and the other figures on the left hand are to be accounted
a quotient; then multiply this quotient by the comple
ment, placing the units of the product under the units of
the former remainder; again, divide this product by the
new divifor, by pointing off from the right hand the fame
number of figures as in the former remainder, and the fi
gures to the left are to be eſteemed another quotient;
which quotient you are again to multiply by the comple
ment, and divide as before. And in this manner proceed
till the last quotient is nothing; then add as in addition.
of integers, obferving the carriage from the left-hand co-
lumn of the remainders; to the remainders add the pro-
duct of the faid carriage and complement, and the fum.
is the total remainder; and the fum of the ſeveral quo-
tients is the total quotient required.
EXAMPLE
Divide 74698 by 98.
New divifor 100
Given divifor yo
I.
ICO)746.98
14.92=746×2
.28=14×2
Complement 2
Tot. quot. 762.18+4=22 tot. rem.
Carriage 2x2 complement 4
}
EX.
Chap. V.
105
DIVISION.
EXPLICATIO N.
Firſt, to unity I annex two ciphers, becauſe the given
divifor conſiſts of two figures, and fo the new diviſor is
100; from which I fubtract the given divifor 98, and
there remains 2 for the complement.
Next, I divide the given dividend by the new divifor,
viz. I point off 98, the two figures next the right hand,
for the first remainder; and the figures on the left, name-
ly, 746, is the firſt quotient.
Then I multiply the faid firft quotient 746 by the com.
plement 2; and by the new divifor I divide the pro-
duct 1492, viz. I point off 92 for the fecond remain-
der, and 14 is the ſecond quotient.
Again, I multiply the fecond quotient 14 by the com-
plement 2, and the product 28, divided by ico, gives
28 for the third remainder, but nothing to the quo
tient.
Then I add the feveral remainders and quotients, and
find the total quotient amounts to 762, and the remain-
ders to 18.
Laftly, I multiply 2, the carriage from the left-hand
column of the remainders, by the complement 2; and
the product 4 I add to the remainders 18, and the fum
22 is the total remainder.
EXAMPLE II.
Divide 849675321 by 994.
New divifor
1000
Given divifor
994
Complément 6
1000) 849675.321
5098.050849675 × 6
30.5885098 × 6
.180 = 30 × 6
Total quot. 854804.139 +6 145 tot. rem,
Carriage 1 x 6 Complement
О
6.
The
}
Chap. V.
106
DIVISION
The fum of the remainders by itſelf, or with the pro-
duct of the complement and carriage, may fometimes
happen to be equal to or greater than the given diviſor;
in which cafe their difference is the true remainder; and
you muſt add to the fum of the quotients; as in the
following
EXAMPLE III.
Divide 4769649 by 997.
New divifor 1000
Given divifor 997
Complement
3
1000) 4769.649
14.3074769 × 3
42= 14 X 3
4783.998 fum of rem.
1.997 given diviſor,
Tot. quot. 4784- I true rem.
Carriage o× 3
Comp=0
When the given divifor is all y's, as y, 99, 999,
9999, &c. the complement in this cafe being 1, you
have only to transfer the feveral quotients toward the
right hand; as in the following
EXAMPLE IV.
Divide 4379806425 by 9999.
New divifor
10000
Given divifor 9999
Complement
10000) 437980.6425
43.7980
43
I
Quot. 438024-4448 + 1 = 4449 tot. rem.
Carriage Comp., to be added to the ſum of
the remainders.
The reaſon of the preceding operations is obvious.
For the quotient of any number divided by 100, is found
by pointing off the two figures next the right hand.
Thus,
;
Chap. V.
107
DIVISION.
:
Thus, 47856100, is 478.56, viz. 478 is the quo-
tient, and 56 the remainder. But 478 x 99 wants 478
of 478 × 100; wherefore the number 478 on the left of
the feparatrix is to be confidered not only as a partial
quotient, but as the difference betwixt the forefaid two
products which difference, being again divided by 100,
ſtands thus, 4.78; but 4 × 99 wants 4 of 4 × 100; where.
fore this 4 muft be added to the remainders, Again,
the carriage from the remainders not only denotes the
number of hundreds carried from them to the partial
quotients, but alſo the excefs above the 99's carried thi
ther; wherefore this exceſs muſt be added to the remain-
der, in order to complete it.
Hence, if the divifor be 98, every difference, as well
as the carriage, muſt be doubled; and if the divifor be
97, every difference, as well as the carriage, muſt be
tripled, &c. that is, multiplied by the complement; and
if the laſt remainder be equal to or exceed the given di-
viſor, I must be carried from it to the quotient.
For the like reafon, if the left-hand figure of the di-
vifor be any digit under 9, and the other figures all '9's,
as 79, 899, 6999, de yoù may work as follows.
Add unity to the given divifor, and the fum is the new
divifor; by which divide the dividend and the ſeveral
quotients fucceffively, till nothing remain; then add the
remainders and quotients, obferving to carry from the
remainders the quotient arifing from the fum of their
left-hand column, divided by the left-hand figure of the
new divifor. Here obſerve, that the left-hand column
of the remainders is that which has as many figures or
places on its right hand as the new divifor has ciphers.
EXAMPLE
Divide 47983756 by 599.
I.
New divifor 600, Complement 1
600) 479837.56
79972.556
133.172799726co
133 = 133 600
133600
Quotient 80106.261+1=262 tot. rem.
Carriage 1x1 Comp. I
O 3
Here
108
Chap. V.
DIVISION.
Here the fum of the left-hand column of the remain-
ders is 8; which divided by 6, the left-hand figure of
the divifor, gives of carriage, and 2 of a remainder ;
and the carriage 1 x I comp. gives i to be added to the
fam of the remainders.
EXAMPLE II.
Divide 1936820 by 3999.
4000) 1936. 820
484. 820
• 484
Quotient 484.1304 tot. rem.
Carriage ox I Complemento
3. When the divifor is a fingle digit, or when it may
be reduced to a digit by cutting off ciphers, the opera-
tion may be performed mentally; and in this cafe it is
convenient to ſet the quotient under the dividend, ſo as
its left-hand figure may ftand under the right-hand figure
of the firſt dividual; and the remainder may be ſubjoin-
ed by way of fraction to the quotient; as in the follow-
ing examples.
Ex. 1.
2) 568
284 quot.
Ex. 4.
4)672
168 quot.
Ex. 7.
Ex. 3.
Ex. 2.
3)693
8) 336
231 quot.
42 quot.
Ex. 5.
2) 1783
Ex. 6.
5) 4963
Ex. 8.
8yquot.
185 quot.
131% quot.
9923 quot.
Ex. 9.
700) 26945
38 345 quot.
2|0)37815
40)5317
This
Chap. V.
109
DIVISION.
This method may alſo be profitably uſed, when the
divifor is 11 or 12, or when it is 11 or 12 with a cipher
or ciphers annexed.
Ex. I.
11),561
Ex. 2.
I 1)878
Ex. 3.
12) 288
51 quot.
79& quot.
24 quot.
Ex. 4.
Ex. 5.
12) 342
110)748|3
Ex. 6.
1200)45|78
978
2812 quot.
68 T quot.
3120 quot.
3
ΙΙ
4. When the divifor is the product of two or more
factors, or component parts, the given dividend may
be divided by one of theſe factors, the quotient by an-
other, the fecond quotient by a third, &c. till you
have divided by every factor; the laft quotient is the
anfwer.
EXAMPLE I.
Divide 409248 by 84.
Becauſe the divifor 84
may be conſidered as the
product of 3 × 4 × 7, I firſt
divide 409248 by 3; and
the quotient 136416 I divide
by 4; and the fecond quo-
tient 34 04 I divide by 7;
and the laft quotient 4872
is the anſwer.
Or,
3)409248 | 12 x 784
and therefore
4)136416|
12)409248
7) 34104
4872
4872
7) 34104
If there be a remainder in each or any of the divifions,
the total remainder is found by fubtracting the product
of the given divifor and laft quotient from the given di-
vidend. Or rather let it be obtained thus: Multiply the
laſt remainder into all the preceding divifors continually,
adding the feveral remainders to the products of the divi-
fors to which they belong; and the laft product is the to-
tal remainder.
EX-
+
Chap. V.
110
DIVISION.
EXAMPLE II.
Divide 234786 by 168.
4)234786
Rem.
7) 58696
2
Here the factors, or compo-
nent parts of the divifor, may
be 4×7 × 6; fo I divide firft by
4, and 2 remains; then I divide
by 7, and I remains; laſtly I di- 6) 8385
vide by 6, and 3 remains.
3
I
3
1397
Total rem. 90
Complete 1397188
quotient
Now, to find the total remainder, I multiply the
laft remainder 3 into the immediately preceding divifor
7; and to the product 21 I add 1, the remainder belong-
ing to that divifor; and the fum 22 I multiply by the firſt
divifor 4; and to the product 88 I add the firſt remain-
der 2; and the fum go is the total remainder. So the
complete quotient is 1397.
The reafon of collecting the total remainder in this
manner is obvious. For the laft quotient multiplied back
continually into the feveral divifors, (that is, into the
component parts of the given divifor), produces the gi-
ven dividend, leffened by or wanting the total remain.
der. And in like manner, the last remainder multi-
plied back into all the divifors prior to it (taking in the
particular remainders that occur in the feveral diviſions)
will produce or make up the total remainder.
EXAMPLE III.
Divide 7427 by 24.
Here the component parts of
the divifor are 3 × 4 × 2; and as
nothing remains in the laſt divifion,
the total remainder is found by
multiplying 3, the laft remainder,
into 3, the preceding divifor, and
to 9, the product, adding 2, the
remainder of the firſt diviſion.
2) 618
3)7427
Rem.
4)2475 2
3
II
quotient.
} 30
309/11
Quot. 309
Total rem.
Complete
I I
EX.
Chap. V.
III
DIVISION.
EXAMPLE IV.
Divide 2061 2 by 56.
Here the component parts of the
divifor are 7 x 8; and as 4 in the
first divifion happens to be the laft
and only remainder, there is no prior
divifor into which it can be multi-
plied; fo this is the total remain-
+
der.
A
7)20612
Rem.
8) 2944
4
368
Tot. rem. 4
Complete 368-%
}
quotient.
EXAMPLE
V.
6)7842
Divide 7842 by 42.
Here the component parts of the
divifor are 6×7; and as 5, the on-
ly remainder, is in the laft divifion,
the total remainder is found by
multiplying it into the preceding
divifor 6, and ſo the fraction is 22.
3
7)1307
Rem.
186 5
Tot. rem. 30
Complete 1863%
quotient
or, 1865
Or rather, when the only remainder
happens in the laft divifion, make
it the numerator, and the laſt di-
vifor the denominator; and fo the
fraction here will be, which is equal in value to 22;
as will appear in the doctrine of vulgar fractions.
If the given divifor confift of any figure repeated, as
222, 333, 444, 5555, &c. the component parts may
be 111 × 2, 111 × 3, 111 × 4, &c.
EXAMPLE VI,
Divide 78943 by 444.
Here
:
1
1
T12
DIVISION.
Nere the compo-
nent parts of the divi-
for are 111 × 4; and
the total remainder is
4)
111)78943(7 11
777
made up as formerly,
viz. 3×111 = 333,
and 333+22=355•
124
I I I
Rem.
177 3
Total rem.
Chap. V.
355
355
Complete 177144
}
quotient
133
III
Rem. 22
5. In divifion the operation may frequently be render.
ed more fimple, if you divide both divifor and dividend
by any number, or numbers, that will divide both with-
out any remainder; and when by this method you can
bring them no lower, divide the remaining dividend by
the remaining divifor.
EXAMPLE I.
Divide 1692 by 468.
Here I divide
1692 | 846 | 423
141
47 (31
468234 117
39 13
both divifor and di-
vidend twice by 2,
and again twice by
3; and as I cannot proceed in this manner any further,
without a remainder, I divide the remaining dividend 47
by the remaining divifor 1, and the quotient is 3.
8
If, by this kind of procedure, the divifor at laſt dwin-
dle to unity, the laſt dividend is the quotient.
EXAMPLE II.
Divide 7896 by 84.
7896 | 3948 | 1974 | 658 | 94
Here I divide
twice by 2, then
by 3, and next by
84
42
21
7
I
7; and as the di-
vifor is now reduced to unity, the dividend 94 becomes
the quotient.
t
This
Chap. V.
113
DIVISION.
This method may likewife be uſed when the dividend,
or divifor, or both, are continual products, expreffed by
the fign of multiplication betwixt the factors. Only ob
ferve, that in this cafe the like number of factors above
and below muſt be divided by the common divifor; and
if
any factor occur both in the divifor and dividend, it
may be daſhed in both.
EXAMPLE III.
Divide 12× 36 by 18.
In the first step I di.
vide 18 and 12 by 6;
in the next I divide
12×36
2 × 36
2 x 12
I 2
24
18
3
I
I
and 36 by 3; then I mul-
tiply 2 into 12, and the divifor being reduced to unity,
the product 24 is the quotient.
EXAMPLE IV.
Divide 8 × 72 × 48 by 4×24x72.
Here I first
8 × 72 × 48
8 × 48
2 x 12 | 24
4
daſh the factor
72, both in the
divifor and di.
4 × 24 × 7 2
4 × 24
1 × 6 6 I
vidend; then I divide the two remaining factors, both
above and below, by 4; and proceed to multiply 1 into
6, and 2 into 12, &c.
EXAMPLE V.
Divide 5 × 49 × 3 × 17 by 17 × 3 × 5 × 35.
X
X X
5× 49 × 3 × 17 49 7
Here I dash the
factors 17× 3 × 5,
both in the divifor
and dividend; and
then divide by 7, &c.
17 × 3 × 5 × 35 | 35
5
6. In computations or calculations that require the
frequent use of the fame divilor, the operation may be
rendered more eafy and expeditious, by making a table
P
of
114
Chap. V.
DIVISION.
of the products of the divifor into all the nine digits; for
then you have the quotient-figures, and their products,
into the divifor, by inſpection.
EXAMPLE.
Divide 47896845 by 748.
748)47896845(64033
4488
TABLE.
{
J
748
3016
2 1496
2992
32244
2484
4 2002
2244
5 3749
6.4488
75236
2405
85984
2244
9' 6732
(161)
Select methods of dividing Integers.
The method of dividing integers hitherto explained is
generally uſed, as being the plaineft and beft; but yet a
great many other methods are practifed; the chief of which
I fhall here explain, by working a fingle example fix dif
ferent ways.
1. Here the products of the quo.
tient figures into the divifor are not ſet
down, but fubu acted mentally from the
dividuals.
"
34)72685(2137
46
128
265
(27)
2. Here
Chap. V.
115
DIVISION.
2. Here the quotient is placed
above the dividend, and the remain.
ders only are fet down; which
with the fubfequent figures of the
dividend make up the feveral divi-
duals.
3. Here the quotient is placed
below the dividend, and the feve.
ral remainders are fet above it.
2137 quotient.
34)72685 dividend.
4
12
26
(27)
12(2
426(7
34)7268; dividend.
2137 quotient.
4. Here the products of the quo.
tient-figures into the divifor are ſet
below the dividend, and the feveral
remainders above it.
12(2
426(7
34)72685(2137
68
34
5. Here too the products of the
quotient-figures into the divifor are pla-
ced under the dividend, and they, as
well as the dividuals, are daſhed or
cancelled as you fubtract; which helps.
to prevent confuſion or miſtakes in the
operation.
6. Here the divifor is repeated under
the dividend; you multiply and fubtract
at once, and cancel as before.
102
238
x7/(2
426(7
34)71685(2137
88478
303
x 2 (2
420(7
170 (2137
34444
II. Divifion of the Parts of Integers.
Here there are three cafes.
333
1. If the divifor be a digit, by it divide the integers
of the dividend, reduce the remainder to the parts of
the next inferior denomination, and add it, when thus
reduced, to the faid parts; then divide the fum, redu-
P 2
cing
716
Chap. V.
DIVISION.
cing and adding the remainder to the parts of the fol
lowing denomination, &c.
Note. If the integral part of the dividend be lefs than
the divifor, you must, in the first place, reduce it to the
parts of the next denomination.
EXAMPLE
I.
If L. 274 13 8 3 be equally divided among 8
men, what will each man's fhare be?
L. s. d. f.
8)274 13 8 3
34
dividend.
68 23 quotient.
Here I first divide the in-
tegers L. 274 by 8, and the
quotient is L. 34, and L. 2
remains; which reduced to
the next denomination makes 40 fhillings; and theſe
added to 13 fhillings make 53 fhillings; which divided
by 8 gives 6 fhillings to the quotient, and 5 fhillings re-
mains; which 5 hillings reduced make 60 d. and 60 d.
added to 8 d. make 68 d.; which divided by 8 gives 8 d.
to the quotient, and 4 d. remains, &c.
The operation may, if you pleaſe, be drawn out at
large; as in the following
EXAMPLE
II.
If C. 43 28 of tobacco be made up into 5 equal
hhds, what will be the neat weight of each hhd?
Here I divide the C. 4; by
5, and the quotient is C. 8,
and C. 3 remains; which re-
duced and added to the 2 Q
makes 14 Q which I di-
vide by 5, &c.
C.
Q. lb.
Q: lb. C. l. 16.
2 8 (8 2 24
5)+3 2
5)43
40
3 rem.
4
14
10
4 rem.
28
120
ΙΟ
20
20
(0)
2. If
Chap. V.
117
DIVISION.
2. If the divifor conſiſts of two or more figures, and
be a compoſite number, reſolve it into its component
parts, and divide the given dividend by one of theſe
parts, the quotient by another, &c. and the laft quo-
tient is the anſwer.
-
EXAMPLE I.
If 54 pieces of cloth coft L. 526: 12: 5, what is
that per piece?
L. s.
6)526 12
d. f. rem.
5
9) 87
15 4 3 2
Anf.9 15 0 254
8
Here the component parts
are 6x9, ſo I divide by 6, and
the quotient by 9; and the laſt
remainder 1 multiplied into the
preceding divifor 6, produces 6;
to which I add 2, the remainder belonging to that divi-
fion, and the fum 8 is the total remainder; fo the fraction
is 84.
EXAMPLE
II.
If i C. or 112 lb of nutmegs be valued at L. 76, 18 s.
1
8 d. what is that
that per lb?
Here the component parts
are 7×8×2, by which I divide
continually, and make up the
total remainder as before.
L. S. d. f. rem.
7)76 18
8
8) o 19 9 2.
6
2) I
6
Anf. 13 8 3482
7 5 2
But if the divifor be not a compofite number, work
as in the following
EXAMPLE
III.
73 perfons go equal fhares in an adventure, which
coft L. 3648: 15: 4:2: What is each perfon's fhare?
L.
118
Chap. V.
DIVISION.
L.
73)3648
S.
d.
f. L. s.
d. f.
15
4
2 (49 19 7 35
292
1420
576
276
I
728
1435
580
278
657
73'
511
219
71 rem. 705
69 rem. (59)
20
657
1420 S.
4
48 rem. 276 f.
12
96
48
576 d.
3. If the divifor confift of integers and parts, reduce
both divifor and dividend to the fame denomination, and
then proceed as in divifion of integers.
EXAMPLE
I.
A nobleman diſtributes among ſome poor people L. 20,
5 s.; each perfon got 7 s. 6 d.: What was the number of
the poor?
s. d. L.
S.
12
20
Here I reduce both the divi-
for and the dividend to the 7 6 20 5
fame denomination, viz pence,
then I divide, and the quotient
54 is the number of poor peo- 90 d.
ple, or it is a number denoting
how often go d. is contained
in 48 60 d.
1
18
405
12
1
810
405
9/0)486|0(
Anf. 54 perfons
E X-
Chap. V.
119
DIVISI O N.
(*
EXAMPLE
II.
The content of a rectangular floor or pavement is
452 fquare feet, and 107 fquare inches; the breadth is
18 feet and 5 inches: What is the length?
Here I reduce the di-
vidend to ſquare inches,
and the divifor to lineal
inches; then I divide,
and the quotient 295 is
lineal inches, agreeable
to the rules laid down in
cross multiplication.
F. in. fq.f. fq.in.
18
5 452
107
I 2
144
36
1808
18
1808
452
221
in.
Fin.
221)65195(295
24 7
442
2099
1989
1105
1105
More examples, in all the parts of divifion, may be
had, by reverfing the examples of multiplication.
III. The Proof of Divifion.
Diviſion may be proved feveral ways, viz. by multi-
plication, by divifion, and by cafting out the 9's.
1. By multiplication: Multiply the quotient by the
diviſor, or the divifor by the quotient; and the product,
with the remainder added to it, will be equal to the divi-
dend Or take the products of the quotient-figures into
the divifor, add them in the order they ſtand under the
dividuals; and their fum, with the remainder, will be
equal to the dividend.
:
2. By divifion: Divide the difference of the dividend
and remainder by the quotient, and your next quotient
will
120
Chap. V.
DIVISION.
will be equal to your firſt divifor, without any remainder.
But this method is tedious.
3. By cafting out the 9's: Caft the 9's out of the di-
vifor and quotient, place the exceffes on the right and
left fides of a crofs; then multiply theſe two figures in-
to one another, and caft the 's out of their product;
add the excess to the remainder; and, cafting out the
9's if need be, place the fum or excefs at the top of
the croſs; then caft the 9's out of the dividend, and ſet
the exceſs at the bottom. If the work be right, the fi
gures at the top and bottom of the cross will agree, or
be the fame.
Theſe methods of proof are a proper exerciſe to the
learner in ſchools; but, in buſineſs, the only proof uſed
is a careful revifal of the operation.
IV. Practical Questions.
1. What number, multiplied by 74308, will produce
28534272 exactly? Anf. 384
2. What number, divided by 87424, will quot
4783, and leave a remainder equal to a fourth part of
the divifor? Anf. 418170848.
3. The fum of two numbers is 4940, the greater is
4572: What is the leffer, what their difference, what
their product, and what the quotient of the greater di-
vided by the leffer? Anf. 368. 4204. 1682496. 12158.
368
4. The remainder of a divifion is 649, the quotient
113, the divifor is the fum of both, and 24 more: What
is the dividend? Anf 89467.
5. There are two numbers, the greater is 24840,
which divided by the leffer quotes 72: What is the leſſer,
what the difference of their iquares, what the fquare of
their fum, and what the cube of their difference?
345. 616y06575. 634284225. 146,72 1067 375.
6. If L. 768 be divided equally among 6 men, what
will each man's fhare be? Anf. L. 21: 6: 8.
Anf.
7. What is the value of 1 yard of cloth, when 84
yards of the fame coſt L. 36: 13:;? Anf 8 s. 8 d. 3 f.
8. A privateer takes a prize to the value of L. 2851, 4 s.;
of
Chap. V.
121
كم
DIVISION.
ड्ड नू
(
of which the captain gets, each of fix officers f
the remainder, and the private men, being 45, in run-
ber, get the reft equally divided among them: What is
each man's fhare?
"Captain's fhare
Anf. Each officer's fhare
Each private man's fhare
s. d.
L.
178 4 0
83 10 7
48
5 3
9. Suppofe a perfon in ten years fpends of his own
L. 2346: 18: 6, and of contracted debts to the value
of L. 132:7:8; what is his expence per year, per
month, per week, per day?
Per year
Per month
Anf. Per week
¡Per day
L.
s. d. f.
247 18 7 I
56
20 13 2 212
96
4 15 4 520
13 7383
84
10. The planet Mercury revolves round the fun in 88
days: How many revolutions will he perform in 17 years
and 219 days? Anf. 73 revolutions.
11. How many bricks, 9 inches long and 4 inches
broad, will floor a room that is 18 feet wide and 24 feet
long? Anf. 1728 bricks.
12. A rectangular floor contains 919 fquare feet, 98
fquare inches, 108 fquare lines; and the length of it is
38 feet, 9 inches, 6 lines: What is the breadth? Anſ.
23 feet 8 inches 6 lines.
13. The content of a parallelopiped is 146 folid yards
1 folid foot and 1024 folid inches; the length of it is
27 feet 4 inches; and breadth 17 feet 8 inches: What
is the thickneſs or depth? Anf. 8 feet 2 inches.
Q
CHAP,
العمري
}
122
Chap. VI.
REDUCTION.
R
CHAP.
VI.
REDUCTION.
EDUCTION teacheth how to bring a number of
one name or denomination to another of the fame
value; and is either defcending, afcending, or mixt.
I. Reduction defcending brings a number of a higher
denomination to a lower, when the lower is fome aliquot
part of the higher; as pounds to fhillings, pence, or far-
things; and is performed by multiplication.
II. Reduction afcending brings a number of a lower
denomination to a higher, when the lower is fome aliquot
part of the higher; as fhillings, pence, or farthings, to
pounds; and is performed by divifion.
III. Mixt reduction brings a number of one denomi
nation to another, when the one is no aliquot part of the
other; as pounds to guineas; and requires the uſe of
both multiplication and divifion.
In treating of reduction I ſhall ſhow its application to
money, and to the moſt uſual ſorts of weights and mea
fures; in doing of which I fhall conjoin reduction de-
fcending and afcending, the one ferving as a proof of the
other; and fhall afterwards treat of mixt reduction by
itſelf.
In working reduction, of whatever kind, the follow-
ing rule is to be obſerved, viz.
Multiply or divide as the tables of coin, weights, and
meaſures, direct.
Reduction defcending and afcending.
I.
MONE Y.
Queſt. 1. In L. 472 how many fhillings, pence, and
farthings?
This
Chap. VI.
123
REDUCTIO N.
}
472 pounds.
20
9440 fhillings.
I 2
This reduction is deſcending, fo
I multiply the pounds by 20, be-
caufe 20 fhillings make 1 pound,
and the product is fhillings: then
I multiply the fhillings by 12, be-
cauſe 1 2 pence make i fhilling, and
the product is pence: laftly, I mul-
tiply the pence by 4, becauſe 4 113280 pence.
farthings make penny, and the
1888
944
product is farthings.
4
453120 farthings.
Proof by Reduction afcending.
In 4531 20 farthings how many pence, fhillings, and
pounds?
Here I divide the farthings by
4, becauſe 4 farthings make 1
penny, and the quotient is
pence: then I divide the pence
by 12, becauſe 12 pence make
1 fhilling, and the quotient is
fhillings laftly, I divide the
fhillings by 20, becauſe 20
:
4)453120 farthings.
12)113280 pence.
20)944 fhillings.
472 pounds.
fhillings make 1 pound, and the quotient is pounds.
Note 1. To reduce pounds to pence at one operation,
multiply by 240, the number of pence in I pound.
Note 2. To reduce pounds to farthings at one opera-
tion, multiply by 960, the number of farthings in I
pound.
Note 3. To reduce fhillings to farthings at one ope-
ration, multiply by 48, the number of farthings in 1
fhilling.
I thall here reduce to farthings the L. 472 in Queſt. 1.
by theſe notes.
e 2
By
124
Chap. VI.
REDUCTION.
}
By note 1.
472
pounds.
240
By note 2.
472 pounds.
960
1888
2832
944
4248
113280 pence.
4
453120 farthings.
453120 farthings.
By note 3.
472 pounds.
20
9440 fhillings.
48
7552
3776-
453120 farthings.
Note 4. To reduce perce to pounds at one operation,
divide by 240, the pence in 1 pound.
Note 5. To reduce farthings to pounds at one ope-
ration, divide by 960, the farthings in 1 pound.
Note 6 To reduce farthings to fhillings at one ope-
ration, divide by 48, the farthings in 1 fhilling.
Here follows the farthings of Queft. 1. reduced back
to pounds by theſe notes.
*
By
Chap. VI.
125
REDUCTION.
By note 4.
4)453120 farthings.
By note 5.
960)45312|0(472 L.
384.
24|0)1 1328|0 d. (472 L.
691
96
672
172
192
168
48
48
192
∞ ∞
(0)
(0)
By note 6.
2/0)
48)453120(944|0 fhillings.
432 472 pounds.
2[1
192
192
192
(0)
Queſt. 2. In 4386 pounds how many groats?
4386 pounds.
20
87720 fhillings.
3 groats in 1 fhilling.
Anf. 263160 groats.
Or pounds may be reduced to
groats at one operation, if you
multiply by 60, the number of
groats in 1 pound.
4386 pounds.
60
Anf. 263160 groats.
PROOF.
126
Chap. VI.
REDUCTION.
PROOF.
In 263160 groats how many pounds?
3)263160
20)87720
Anf. L. 4386
Or thus.
6|0) 26316|0
Anf. L. 4386
Queſt. 3. In 7643 pounds how many three-pences?
7643 pounds.
20
152860 fhillings.
4
three-pences in 1 fhilling.
Anf. 611440 three-pences.
I
Or pounds many be re-
duced to three-pences at
one operation, if you mul-
7643 pounds.
80
tiply by 80, the number of Anf. 611440 three-pences.
three-pences in 1 pound.
PROOF.
In 611440 three-pences how many pounds?
4)611440
2|0)15286|0
Anf. L. 76+3
Or thus.
80)611440
Anf. L. 7643
Queſt. 4. In 48 guineas how many fhillings, pence,
and farthings?
48
Chap. VI.
127
REDUCTION.
48 guineas.
21 the fhillings in 1 guinea.
[*9
1008 fhillings.
12096 pence.
4
48384 farthings.
PROO F.
In 48384 farthings how many pence, fhillings, and
guineas?
4)+8384 farthings.
12)12096 pence.
21)10c8 ſh. (48 guineas.
84
168
168
(0)
Queſt. 5. In 853 pounds how many crowns, fhillings,
groats, and pence?
Pounds
853
4 crowns in pound.
Crowns
3412
5 fhillings in I crown.
Shillings 17060
3 groats in 1 fhilling.
Groats
51180
4 pence in I groat.
Pence
2047 20
PROOF.
128
Chap. VI.
REDUCTION.
PROOF.
In 204720 pence how many groats, fhillings, crowns,
and pounds?
4)204720 pence.
3)51180 groats.
5)17060 fhillings.
4)3412 crowns.
853 pounds.
Queſt. 6. In L. 458:16:72 how many fhillings, pence,
and farthings?
L. s. d.
458 16 72
20 I take in the 16 s.
Shillings 9176
12 I take in the 7 d.
Pence
110119
4 I take in the 3 f.
Farthings 440479
RROOF.
In 440479 farthings how many pence, fhillings, and
pounds?
The remainders in reduction a
ſcending are of the fame name or
denomination with the dividends.
Rem.
4)440479 3 f
12)110119
7 d.
20)9176 16s.
L. 458 16 72
MORE
*
Chap. VI.
REDUCTION.
129
MORE
EXAMPLES.
Queſt. 7. In L. 78342 how many groats and farthings?
Quest. 8. In 69453 guineas how many fhillings, three-
pences, pence, and farthings?
Queſt. 9. In L.4568 2 how many crowns, half-crowns,
fixpences, and farthings?
Queſt. 10. In L. 57843: 19: 10ž how many shillings,
pence, and halfpence?
2. AVOIRDUPOIS
WEIGHT.
Queſt. 1. In C. 47 : 1 : 20 how many ounces?
Method 1.
Method 2.
C. Q: lb.
C. Q. lb.
47 I 20
47
1 20
47
47
189 Q
47
2. 16.
28
48 lb. in 1 20
1512
5312 lb.
380
16
5312 lb.
31872
16
5312
31872
84992 oz.
5312
84992 oz.
Method 3.
G.
Q: lb.
47 I 20
564
2. 16.
48 lb. in 1 20
5312 lb.
16
31872
5312
84992 oz.
R
In
130
Chap. VI.
REDUCTION.
In method 1. in multiplying the C. by 4, I take in
the Q.; and in multiplying the Q. by 28, I take in the
20 lb.
In method 2. the operation is the fame as multiplying
the C. by 112. For 47 placed below 47, with units un-
der units, is the fame as multiplying 47 by 2; and what
follows is the multiplication of 47 by II.
In method 3. I multiply the C. by 12; and the pro
duct 564, being placed two figures to the right hand of
47, ftands fo, that being added, their fum will be equal
to the product of 47 multiplied by 112. Or, the reaſon
of the operation will ſtill appear plainer by confidering,
that 47 C. is 4700 lb. and 47 times 12 lb. and there-
fore to 4700 lb. I add 12 times 47.
PROOF.
In 84992 ounces how many lb. Q. and C.
28) 4) G. Q. lb.
16)84992 (5312 (189 (47 I 20
80... 28.. 16
49
251
29
48
224
28
19
272 (1) Q
16
252
32 (20) lb.
32
(0)
MORE
EXAMPLES.
Quest. 2. In C. 485: 3: 24: 12 oz. how many Q. Ib.
oz. and dr.?
Quest. 3. In 764 tuns 15 C. 2 Q. 16 lb. how many
C. Q. and lb. ?
III. TROY
Chap. VI.
131
REDUCTIO N.
1
3. TROY WEIGH T.
Quest. 1. In 482 lb. 7 oz. 13 dw. 21 gr. how many
ounces, penny-weights, and grains?
lb. oz. dw. Gr.
482 7
7 13 21
12
5791
Ounces.
20
115833 penny-weights.
24
463333
231668
2780013 grains.
PROOF.
In 2780013 grains, how many penny-weights, oun
ces, and pounds ?
2/0)
24)2780013 (115833 13 penny-weights.
24.
12)5791 7 ounces.
38
24
482 pounds.
140
120
200
192
81
lb.
oz. dw. gr.
72
In all 482
7 13 21
93
72
Rem. (21) grains.
R 2
MORE"
132
Chap. VI.
REDUCTION.
1
MORE
EXAMPLE S.
Queſt. 2. In 7694 lb. 8 oz. how many grains?
Queſt. 3. In 8546 lb. 10 oz. o dw. 18 gr. how many
grains?
4. DRY
MEASURE.
Queſt. 1. In 48 loads 12 bufhels, how many bufhels,
pecks, and gallons?
Loads.
Buſh.
12
48
40
1932 bufhels.
4
7728 pecks.
2
15456 gallons.
PROOF.
In 15456 gallons, how many pecks, bufhels, and
loads ?
MORE
EXAMPLES.
Quest. 2. In 74 bushels 2 pecks, how many pecks,
gallons, and pottles?
Queſt. 3. In 58 chalders 6 bolls 2 firlots, Scots mea-
fure, how many bolls, firlots, and pecks?
5.
WINE. MEASURE.
Queſt. 1. In 72 hogfheads of wine, how many gallons,
and pints?
72
Chap. VI.
133
REDUCTION.
72 hogsheads.
63
216
432
4536 gallons.
8
36288 pints.
PROOF.
In 36288 pints of wine, how many gallons and hogf.
heads?
MORE
EXAMPLE S.
Quest. 2. In 45 tuns of wine, how many gallons?
Queſt. 3. In 354 tuns 1 hhd 42 gallons of wine, how
many hogsheads, gallons, quarts, and pints?
6.
BEER-MEASURE.
Queſt. 1. In 54 butts of beer, how many hogsheads
and gallons?
54 butts.
2
108 hhds.
54
432
540
5832 gallons.
PROO F.
In 5832 gallons of wine, how many hogfheads and
butts?
MORE
134
Chap. VI.
REDUCTION.
MORE
EXAMPLES.
Quest. 2. In 96 tuns of beer, how many hogfheads,
gallons, and pints?
Quefi. 3. In 25 hogfheads of beer, Scots meaſure,
how many gallons, pints, and gills?
7. CLOTH.ME A SURE.
Queſt. 1. In 34 yards 3 quarters, how many quarters
and nails?
rds. qrs.
34
3
4
139 quarters.
A
556 nails.
PROOF.
In 556 nails, how many quarters and yards ?
MORE EXAMPLES.
Queſt. 2. In 38 ells Flemish and 2 quarters, how
many quarters and nails?
Queſt. 3. In 45 ells English and I quarter, how ma-
ny quarters and nails?
8. LONG
MEASURE,
Quest. From London to York is 151 miles, how ma-
ny furlongs, poles, half-yards, inches, and barley corns,
will reach from the one city to the other?
151
Chap. VI.
735
REDUCTION
I
151 miles.
8
1208 furlongs.
40
48320 poles.
I I
4832
4832
531520 half-yards.
18
425216
53152
9567360 inches.
3
28702080 barley.corns.
PROOF.
In 28702080 barley-corns, how many inches, half-
yards, poles, furlongs, and miles?
MORE
EXAMPLES.
Quest., 2. In 876 miles, how many chains, poles,
half-yards, and inches.
Quest 3. In 950 Scots miles, how many furlongs,
chains, feet, and inches?
9 LAND-ME A SURE
Queft. i. In 75 acres, how many roods and poles?
75 acres,
4
*
300 roods.
40
1 2000 poles.
Or thus. 75 acres.
I 60
450
75
1 2000 poles.
PROOF.
1-36
Chap. VI.
REDUCTION.
PROOF.
In 12000 ſquare poles, how many roods and acres?
MORE EXAMPLE S.
Queſt. 2. In 84 acres 3 roods and 25 poles, how
many roods and poles?
Quest. 3. In 96 Scots acres, how many roods, falls,
and ells?
10.
TIME.
Quest. 1. In 28 years 24 weeks 4 days 16 hours 30
minutes, how many minutes?
Years. weeks. days. hours. minutes.
28
24
4
16
30
52
60
142
1480 weeks.
7
10364 days.
24
41462
20729
248752 hours.
60
14925150 minutes.
PROOF.
In 14925150 minutes, how many hours, days, weeks,
and years?
The above reduction allows only 364 days to the
year; but if you incline to be accurate, find the hours
in
Chap. VI.
137
REDUCTION.
+
in 365 days 6 hours, a complete year, and then reduce
as follows.
Days. hours.
Weeks. days. hours. min.
365
6
24
4
16 30
24
7
1466
172 days.
730
24
8766 hours in one year. 694
28
70128
17532
345
4144 hours.
245448
245448 h. in 28 years. 249592 hours in all.
60
14975550 minutes.
PROO F.
In 14975550 minutes, how many hours and years?
MORE
EXAMPLES.
Quest. 2. How many hours and minutes from the
creation to the birth of Chriſt, it being accounted 4004
years ?
Quest. 3. How many hours and minutes from the
birth of Chrift to the end of the year 1764?
Mixt Reduction.
In working mixt reduction obferve the following
RULE.
By reduction defcending bring the given name to fome
fuch third name as is an aliquot part both of the name
given and of the name fought, and then by reduction a-
fcending bring the third name to the name fought.
S
Mixt
1
138
Chap. VI.
REDUCTION.
Mixt reduction, as well as reduction defcending and
afcending, extends to money, and to things weighed, or
meaſured, as follows.
I.
MONE Y.
Quest. 1. In 764 1. how many guineas?
Here the name given is
pounds, the name fought is
guineas, and the third name,
to which the pounds are re-
duced, is fhillings; for a
fhilling is an aliquot part
both of a pound and of a
guinea.
21)
764 pounds.
20
15280 (727 guineas.
1 47 * *
58
42
160
I 47
(13) fhillings.
PROOF.
In 727 guineas 13 fhillings, how many pounds?
Guineas. Thill.
727
21
་༣
730
1455
2|0) 1528|0
764 pounds.
Queſt. 2. In 832 moidores, how many pounds Ster-
ling?
Here
Chap. VI.
139
REDUCTION.
FRA
Here I reduce the moi-
dores to fhillings, a fhil-
ling being an aliquot part
both of a moidóre and of
a pound Sterling.
l.
832 moidores.
27 fhill. in 1 moidore..
5824
1664
I
2|0)2246|4 fhillings.
1123 L. Ster.
PROOF.
In 1123 1. 4 s. how many moidores?
L.
S.
1123 4
20
27)22464(832 moidores.
216..
86
81
54
54
(0)
ន ឌ
Quelt.
140
Chap. VI.
REDUCTION.
Queſt. 3. In 968 1. how many merks?
Method 1.
968 1.
Method 2.
3
2) 2904 half-merks.
1452 merks.
1936
16|0)23232|0 (1452 merks.
16..
72
968 1.
240
3872
64
83
80
32
32
(0)
In method 1. I reduce the pounds to half-merks, a
half-merk being an aliquot part both of a pound and of
a merk.
In method 2. I reduce the pounds to pence, and then
divide by 160, the number of pence in 1 merk; but
method 1. is the eaſier and ſhorter way.
PROOF.
Chap. VI. REDUCTION.
147
PROOF.
In 1452 merks, how many pounds?
1452 merks.
2
3) 2904 half-merks.
Or thus. 1452 merks.
160
8712
968 1.
1452
24|0)23232|0(968 L
216..
163
144
192
192
(0)
Queſt. 4. In 540 dollars, at 4 s. 4 d. per dollar, how
many pounds Sterling?
s. d.
4 4
Method 1.
540 dollars
s. d.
Method 2.
4.4
540 dollars.
I 2
52
3
13
52 d. in 1 dol. 108
Groats 13in1 dol. 162
270
54
24|C)2808|0(117k
610)7020
24**
40
24
168
168
(0)
117 L
142
Chap. VI.
REDUCTION.
In method 1. I reduce the dollars to pence, and then
divide by 240, the number of pence in I pound.
In method 2. I reduce the dollars to groats, and then
divide by 60, the number of groats in pound.
PROOF.
In 117 1. Sterling, how many dollars, at 4 s. 4 d. per
.*
dollar?
117 1.
240
468
Or thus. 1 17 1.
60
13) 7020(540 dollars.
65
52
52
(0)
234
52)28080(540 dollars.
200·
208
208
;
(0)
Queſt. 5. In 47851. 13 s. how many pieces of 13 d.
per piece?
13 d.
2
L. 4785 13 S.
20
27 half-pence. 95713 fhillings.
24 half-pence in I fhilling.
382852
191426
27)2 297112(85078 pieces of 131 d.
216·
, 137
135
21L
189
222
216
1
Rem. (6) half-pence, or 3 d.
PROOF.
Chap. VI.
143
REDUCTION.
PRO O F.
1
In 85078 pieces of 13 d. each, and 3 d. how many
pounds?
Pieces. d.
85078 3
27
170156
595552
20
24) 2297112 (9571|3
216.
137
I 20
s.
L. 4785 13 S.
171
168
31
24
72
72
(0)
Queſt.
144
Chap. VI.
REDUCTION.
Queſt. 6. In L. 872, how many pieces of 6 d. of 5 d.
and of 4 d. of each an equal number?
d.
{
6
L.
872
5
4
Sum 15 pence.
240
3488
1744
15) 209280(13952 pieces of 6 d. of
15 * * * *
5 d. and 4 d.
59
45
142
135
78
75
30
30
(0)
PROOF.
In 13952 pieces of 6 d. of 5 d. of 4 d. of each an e,
qual number, how many pounds?
240)20928|0(872 L.
13952 pieces.
15
192.
69760
172
13952
168.
Total 209280 pence.
48
21/
48
(0)
This
Chap. VI.
145
REDUCTION.
This fort of reduction could easily be extended much
further, and might be uſed to reduce Sterling money to
any fort of foreign money, or any fort of foreign mo-
ney to Sterling; but as this is a wide field, it is more
ufual, and the better way, to affign a place for this by
itſelf, under the title of Exchange I fhall therefore con-
clude mixt reduction of money by fubjoining a few
MORE EXAMPLE S.
Queft. 7. A gentleman was robbed of 57 guineas 33
half-guineas and 48 moidores, how many pounds Sterling
did he lofe? Anf. L. 141 : 19: 6
Queſt. 8. In 685 merks 234 nobles and 142 quarter-
guineas, how many pounds Sterling? Arf. L. 571,
18 s. 10 d.
Queſt. 9. In L. 148, how many pieces of 131 d. of
12 d. of 9 d. of 6 d. of 4 d. and of each an equal num.
ber? Anf. 798 pieces of each fort, and 9 d. over.
2. AVOIRDUPOIS
WEIGHT.
Queſt. 1. In 8 bags cotton, each containing C. 12: 1:20,
how many tuns?
C. Q. lb.
12 I 20
8
Tuns. C. & lb.
I
20 Anf.
2|0)9|9 1 20 ( 419
2100g0
1
8
Rem. (19) C.
PROO F.
I
If 4 tuns 19 C. 1 Q 20 lb. of cotton be made up in-
to 8 equal bags, how much will each bag contain?
T
Tuns.
146
Chap. VI.
REDUCTION.
Tuns. C. Q. 16. C. Q. lb.
8) 4 19 1 20
20
12 1 20 Anf.
99
8.
t
(5)
28
19
160
16
16
rem. 3
(0)
4
13
| |
8
(5)
MORE
EXAMPLES.
Queſt. 2. In 24 equal cheeſes, weighing 3 Q. 15 lb.
each, how many C. Anf. C. 21 : 0 : 24.
Queft. 3. One buys of a grocer C. 4 : 1 : 14 of pep-
per, and orders it to be made up into parcels of 14 lb. of
12 lb. of 8 lb. of 6 lb. of 2 lb. and of each an equal num-
ber: Required the number of parcels of each fort?' Anf.
11 parcels of each fort, and 28 pounds over.
3. TROY WEIGHT.
Queſt. 1. In 96 pair of filver fpurs, weighing 3 oz. 12
dw. per pair, how many pounds?
#
Oz.
Chap. VI.
147
REDUCTION.
02. dw.
3
20
12
72 dw.
96
432
648
2|0)691|2 12 dw,
12) 345 9 oz.
lb. oz. dw.
28 lb. Anf. 28 9 12
PROOF.
How many pair of fpurs, each pair weighing 3 oz. 12
dw. may be made out of 28 lb. 9 oz. 1 2 dw. of filver?
oz. dw.
3
2210
I 2
72 dw.
lb. oz. dw.
28 9 12
I 2
345
20
72) 6912 (96 pair of spurs Anf
648
432
432
(0)
MORE
EXAMPLES.
←
Quest. 2. In 7 dozen filver candleſticks, each weigh
ing 10 oz. 14 dw. how many pounds? Anf. 74 lb. 10 oz.
16 dw.
Quest. 3. A merchant fent to a goldſmith 16 ingots of
T 2
filver,
148
Chap. VI.
REDUCTION.
filver, each weighing 2 lb. 4 oz. and ordered it to be
made into bowls of 2 lb. 8 oz. per bowl, and tankards
of 1 lb. 6 oz. per piece, and falts of 10 oz. 10 dw. per
falt, and ſpoons of 1 oz. 18 dw. per ſpoon, and of each
an equal number, how many of each fort will there be?
Anf. 7 veffels of each fort, and 224 dw. over.
I
4.
WINE-MEASURE.
Quest. 1. In 34 runlets of wine, each runlet contain-
ing 18 gallons, how many hogfheads?
In
୨
34 runlets.
18
272
34
63) 612(9 hhds.
567
Rem. (45) gallons.
Anf. 9 hhds 45 gallons.
PROOF.
hhds 45 gallons of wine how many runlets?
Hhds. gall.
9 45
63
18)612 (34 runlets. Anf
54
72
72
(0)
MORE
Chap. VI.
REDUCTIO N.
149
MORE
EXAMPLE S.
Queſt. 2. In 752 runlets, how many tierce of wine?
Anf. 322 tierce, and 12 gallons.
Queft. 3. In 840 hogfheads, how many puncheons of
80 gallons per puncheon? Ans. 661 puncheons, and 40
gallons.
5. CLOTH MEASURE.
Quest. 1. In 96 yards, how many ells Flemish ?
96 yards.
4
3) 384
Anf. 128 ells Flemiſh.
PROOF.
In 128 ells Flemish, how many yards?
MORE
EXAMPLES.
Queſt. 2. In 450 yards, how many ells Engliſh? Anf.
360.
Queſt. 3. In 785 ells Flemish, how many ells Engliſh?
Anf. 471.
6. LONG MEASURE.
Queſt. 1. In 72 miles, how many furlongs, poles, half
yards, yards, and feet?
1
72
150
Chap. VI.
REDUCTION.
72 miles.
8
576 furlongs.
40
23040 poles.
II
2304
2304
2) 253440 half-yards.
126720 yards.
3
380160 feet.
PRO O F.
In 380160 feet, how many yards, half-yards, poles,
furlongs, and miles?
MORE
EXAMPLE S.
Quest. 2. Suppofing the circumference of the earth to
be divided into 360 degrees, and each degree to contain
60 miles, I demand how many miles, furlongs, poles,
half-yards, yards, feet. inches, and barley-corns, will
reach round the globe of the earth? Ans. The barley-
corns are 4105728000.
A
Quest. 3. In 150 Engliſh miles, how many Scots miles?
Anf. 133 miles, 6 furlongs, 10 falls, and 15 feet.
Mixt reduction of weights and meaſures, as well as that
of money, might be carried a great way farther than what
is done here; but the above ſpecimens, it is hoped, will
be found fufficient inſtruction to the learner.
Practical Questions.
1. In a purſe containing 84 guineas, 56 half-guineas,
and
Chap. VII.
151
The Rule of THREE.
and 48 moidores, how many pounds Sterling? Anf.
L. 182, 8 s.
2. A gentleman ſetting out on a journey carried along
with him 8 Joannes's, being L. 3, 12 s. each, 17 gui-
neas, 13 half guineas, 14 crowns, 10 half-crowns, 1 5 fhil-
lings, and 9 fixpences; and finds, on his return, by his
pocket-book, that he had ſpent L. 49, 4 s.: How many
pounds Sterling did he carry out, and how many brought
he home? Anf. He carried out L. 59, 4 s. and brought
home L. 10.
3. A merchant bought 30 hogheads fugar, viz. 12
hhds containing C 16: 1:14 each, 12 hhds weighing
C. 14 3 7 each, and 6 hhds containing C. 15: 2:21
each: How many tuns of fugar did he purchaſe? Anf.
23 tuns 8 C. 1 Q. 14 lb.
4. In 96 firkins of beer, 9 gallons each, how many hhds
of 54 gallons per hhd? Anf. 16 hhds.
5. A piece of ground, confifting of 27 acres, is to be
laid out in ſmall divifions, of 3 roods and 10 poles each;
how many ſmall diviſions will there be?
Anf. 32.
6. Queen Elifabeth came to the throne of England the
17th of November 1558, and died the 24th of March
1603, in the 70th year of her age: What year was the
born? and how many months, of 28 days each, did the
reign, reckoning 365 days 6 hours to a year? Anf. She
was born in the year 1533, and reigned 578 months 2
weeks and 1 day.
CHA P.
VII.
THE RULE OF THREE.
HE Rule of Three, called alfo, on account of its
TH excellence, the Golden Rule, from certain num.
bers given, finds another; and is divided into fimple and
compound, or into fingle and double.
SEC-
*
152
1
Chap. VII.
The Rule of THREE.
SECTION I
The Simple or ſingle Rule of Three.
THE fimple rule of three, from three numbers given,
finds a fourth, to which the third bears the fame propor-
tion as the first does to the ſecond.
The nature and properties of proportional numbers
may be underſtood fufficiently for our purpofe from the
following obfervations.
In comparing any two numbers, with respect to the
proportion which the one bears to the other, the firſt
number, or that which bears proportion, is called the
antecedent; and the other, to which it bears proportion,
is called the confequent; and the quantity of the propor
tion or ratio is eſtimated by the quot arifing from divi-
ding the antecedent by the confequent. Thus the ratio
or proportion betwixt 6 and 3 is the quot arifing from
dividing the antecedent 6 by the confequent 3; namely,
2; and the ratio or proportion betwixt 1 and 2 is the
quot arifing from the diviſion of the antecedent 1 by
the conſequent 2; namely, or one half.
Four numbers are faid to be proportional when the
ratio of the firft to the fecond is the fame as that of the
third to the fourth; and the proportional numbers are
uſually diſtinguiſhed from one another as in the follow-
ing examples.
4: 2 :: 16 : 8.
6 : 9 :: 12 : 18.
Proportional numbers, or numbers in proportion, are
uſually denominated terms; of which the firſt and laſt
are called extremes, and the intermediate ones get the
name of means, or middle terms.
If four numbers are proportional, they will alſo be
inverfely proportional; that is, the firft confequent will
be to its own antecedent as the fecond confequent is
to its antecedent; or the fourth term will be to the
third as the ſecond is to the firft. Thus, if 6: 3 :: 10:5,
then by inverſion, 3 : 6 :: 5 : 10, or 5 : 10 :: 3 : 6.
Euclid
Chap. VII.
153
The Rule of THREE.
Euclid v. 4. cor. By either of theſe kinds of inverſion
may any queſtion in the rule of three be proved.
If four numbers are proportional, they will alſo be al-
ternately proportional; that is, the firft antecedent will
be to the fecond antecedent as the first confequent is to
the ſecond confequent; or the firft term will be to the
third term as the fecond term is to the fourth. Thus,
if 84: 24: 12, then, by alternation, 8: 24 :: 4 : 12.
Euclid v. 16.
But the celebrated property of four proportional num-
bers is, that the product of the extremes is equal to the
product of the means. Thus, if 2 : 3 :: 6 : 9,
6:9, then
2 × 9 = 3 × 6 = 18, Euclid vi. .6.
Hence we have an eafy method of finding a fourth
proportional to three numbers given, viz.
Multiply the middle number by the laft, and divide the
product by the firft, the quot gives the fourth propor
tional.
EXAMPLE.
Given 6, 5, and 36, to find a fourth proportional ;
put x equal to the fourth proportional, then 6::: 36:x,
and 5 × 36
36 = 180 = 6 x x; wherefore, dividing the
product 180 by the factor 6, the quot give the other
factor x, namely 30, the fourth proportional fought.
Every queſtion in the rule of three may be divided
into two parts, viz. a fuppofition and a demand; and
of the three given numbers, two are always found in the
fuppofition, and only one in the demand.
EXAMPLE.
If 4 yards coft 12 fhillings, what will 6 yards coſt at
that rate?
In this queſtion the fuppofition is, if 4 yards ceft 12
fhillings; and the two terms contained in it are 4 yards
and 12 fhillings; the demand lies in thele words, What
will 6 yards coft? and the only term found in it is 6
yards,
The fuppofition and demand being thus diftinguiſhed,
U
proceed
$54
The Rule of THREE. Chap. VII.
proceed to ſtate the queftion, or to put the terms in
due order for operation, as the following rules direct.
RULE I.
Place that term of the fuppofition, which is of the
fame kind with the number fought, in the middle. The
two remaining terms are extremes, and always of the
fame kind.
RULE II.
Confider, from the nature of the queftion, whether the
anſwer muſt be greater or less than the middle term;
and if the arfwer muft be greater, the leaft extreme is
the divifor; but if the answer muſt be leſs than the
middle term, the greatest extreme is the divifor.
RULE III.
Place the divifor on the left hand, and the other ex-
treme on the right; then multiply the fecond and third
terms, and divide their product by the first; and the quot
gives the anſwer; which is always of the fame name with
the middle term.
When the divifor happens to be the extreme found
in the fuppofition, the proportion is called direct; but
when the divifor happens to be the extreme in the de-
mand, the proportion is inverfe.
The three rules delivered above are indeed fo framed,
as to preclude the diſtinction of direct and inverſe, or
render it needlefs, the left-hand term being always the
divifor; but yet the direct queſtions being plainer in their
own nature, and more easily comprehended by a learn-
er, I fhall, in the first place, exemplify the rules by a
fet of queflions of the direct kind, and fhall afterwards
adduce a collection of fuch as are inverſe.
1. The fimple Rule of Three direct.
Queſt. 1. If 4 yards coft 12 fhillings, what will 6
yards coft at that rate?
The fuppofition and demand of this queſtion have
already been diftinguished, and the two terms in the
former
Chap. VII.
155
The Rule of THREE.
former are 4 yards 12 fhillings, and the only term in
the latter is 6 yards.
The number fought is the price of 6 yards, and the
term in the fuppofition of the fame kind is the price of
4 yards, viz. 12 fhillings, which I place in the middle,
as directed in Rule I. and the two remaining terms are
extremes, and of the fame kind, viz. both lengths.
It is eafy to perceive
that the anſwer muſt be
greater than the middle
term; for 6 yards will coſt
more than 4 yards; there-
fore the leaſt extreme,
viz. 4 yards, is the divifor,
according to Rule II.
yds. S. yds.
If 4: 12 :: 6
6
4) 72(18 fhillings Anf.
4
32
32
(0)
Wherefore I place the divifor 4 yards on the left
hand, and the other extreme 6 yards on the right; and
multiplying the fecond and third terms, I divide their
product by the firft term, and the quot 18 is the anſwer,
and of the fame name with the middle term, viz. fhil-
lings, according to Rule III.
And becauſe the divifor is the extreme found in the
fuppofition, the proportion is direct.
Quest. 2. If 7 C. of pepper coft 21 1. how much will
5 C. coft at that rate?
The fuppofition in this question is, that 7 C. of pep-
per cofts 16 1. and the two terms in it are 7 C. and 161.;
the demand is, How much will 5 C. coft? and the term
in it is 5 C.
The number fought is the price of 5 C. and the term
in the fuppofition of the fame kind is the price of 7 C.
viz. 21 1. which I place in the middle. The two re-
maining terms are extremes, and of the fame kind, viz.
quantities of pepper.
U 2
It
156
Chap. VII.
The Rule of THREE.
It is obvious, that the anſwer
muſt be leſs than the middle
term; for 5 C will coſt leſs than
7 C.; and therefore the greateſt
extreme, viz. 7 C. is the divifor.
If 7
C. L.
C.
:
21:5
5
7)105(15l. Anf.
7
35
35
(0)
Accordingly I place the divifor 7 C. on the left hand,
and the other extreme 5 C. on the right; and having
multiplied the fecond and third terms, I divide their pro-
duct by the firſt term, and the quot 15 is the anſwer, of
the fame name with the middle term, viz L. Sterling.
And becauſe the divifor happens to be the extreme in
the fuppofition, the proportion is direct.
Quest. 3. If 13 yards of velvet coft L. 21, what will
27 yards coft at that rate?
T.
1
Chap. VII.
157
The Rule of THREE.
r. L. r.
If 13:21:27
27
147
42
# Rem. 4 s.
12
13)48 (3 d.
39
When there happens
to be a remainder, it
may be reduced to the
next inferior denomi.
nation, and the opera.
tion continued, as in
the margin; and in this
caſe the quot will con-
13) 567 (43 L.
Rem. 9 d.
52
4
47
13) 36 (2 f.
39
26
1
fift of two or more
Rem. 8 L.
Rem. 10 f.
parts.
20
13) 160 (12 S.
13
L.
s. d. f.
30 Anf. 43 12 3 20
26
* Rem.
4 S.
Such remainders are always of the fame name with
the preceding part of the quot. Thus, the firſt remain-
der 8, and the first part of the quot 43, are both pounds
and the ſecond remainder 4, and the ſecond part of the
quot 12, are both fhillings; and the third remainder
and the third part of the quot 3, are both pence; and
the fourth remainder 10, and the fourth part of the quot
2, are both farthings.
9,
As we have no money under farthings, the laſt re-
mainder cannot be reduced any lower; fo there remains
10 farthings to be divided by 13; that is, there is want-
ing to complete the quot the thirteenth part of 10 far-
things, or the thirteenth part of every remaining farthing;
that is, ten thirteen parts of one farthing; ſo I ſet the
remainder 10 above and the divifor 13 below a line
drawn between them, in the form of a fraction, of which
1
the
}
158
Chap. VII.
The Rule of THREE.
f
the remainder is the numerator and the divifor the de-
nominator.
Queſt. 4. If 14 lb. of tobacco coſt 27 fhillings, what
will 478 lb. coſt at that rate?
lb. s. lb.
If14:17::478
27
3346
956
Here the middle term being
fhillings, the first part of the
quot is alſo fhillings; which I
divide by 20, and ſo reduce it
to pounds.
There remains at laſt 2 far-
things, to be divided by 14; fo
I fubjoin theſe by way of frac
tion to the preceding part of the
quot; as in the former exam-
ple.
2/0)
14) 12906 ( 92|1
1 26**
30
Rem.
28
26
14
L. 46 I
I 2 S.
12
14) 144 (10 d.
14
L. s. d. f.
Anf. 46 1 10 12
Rem.
4 d.
4
14)16(1 f.
14
Rem. 2 f.
ue ft.
1
Chap. VII. The Rule of THREE.
159
Quest. 5. If 15 ounces of filver be worth L. 3, 15 s.
What are 86 ounces worth at that rate?
If the middle term be
02. L S.
Oz.
complex, or mixt, and fo If 15: 3—15::86
confift of two or more
parts, reduce it to the low-
eft, and the anſwer will
come out of the fame name
with that of the loweſt part..
Thus,
The middle term here
is complex, confifting of
two parts, viz. pounds and
fhillings; fo I reduce it
to fhillings, and the quot
comes out in fhillings;
which I reduce to pounds.
20
75 fhill.
86
450
600
210)
15) 6450 ( 43|0 fhillings.
60.
- L. 21 10
45
45
(0)
L.
S.
Anf. 21
IO
The parts of a complex term may be diftinguished from
one another by a ftroke; as in this example.
Quest.
160
Chap. VII.
The Rule of THREE.
Queſt. 6. If 18 C. fugar coft L. 54, what will 7 C. 3 Q.
14 lb. coft at that rate?
·C..
L. C. Q: lb.
If 18: 54: 7 — 3 — 14
18
7
18
7
18
798
2016 lb.
882 lb.
54
3528
When any or both of the
extremes happen to be com-
plex, or mixt, and ſo confift
of two or more parts, reduce
both extremes to the loweſt
of the parts; for in the ope-
ration both extremes muſt be
equally low, or of the fame.
name.
Thus, in this example the
extreme on the right hand is
complex, confifting of three
parts, viz. C. Q. and lb.; and
fo I reduce both this extreme,
and alfo the other extreme on
the left hand, to lb. the low-
eft of the parts.
4410
2016)47628 (23 L.
4032°
7308
6048
Rem. 1260 L.
20
2016) 25200 (12 s.
2016
5040
40,2
Rem. 1008 s.
I 2
2016) 12096(6 d.
12096
L.
S.
d.
Anf. 23
I 2
6
(0)
Quest.
Chap. VII.
161
The Rule of THREE.
3
: 1 : 14 of raiſins coft L. 10: 2 : 6,
Queſt. 7.
If C.
what will 6 C. 3 Q coft at that rate?
C. Q. lb. L.
If 3—1—14:10—2—6 :: 6—3
s. d. C. Q.
3
3
342
20
6
6
202
684
12
Here all the terms 378 lb.
756 lb.
are complex, and
the two extremes are
both reduced to lb.
2430 d.
756
that being the low-
eft of the parts.
The middle term,
for the like reafon,
is reduced to pence,
and accordingly the
1458
1215
1701
12) 210)
378)1837080(4860 (405
1512 48..
L. 20 5
quot or anſwer
3250
comes out in pence.
3024
1188
2268
2268
(0)
(0)
L.
S.
Anf. 20 5
Quest. 8. A goldſmith bought a wedge of gold, which
weighed 14 lb. 3 oz. 8 dw. for the fum of L. 514, 4 S. :
I demand what it ftood him per ounce?
X
lb.
162
The Rule of THREE.
Chap. VII.
OZ.
lb. oz. dw. lb. S.
If 14—3—8 : 514—4 :: I
12
20
20
171 OZ.
10284 fhill. 20 dw.
20
20
dw. 3428 3428) 205680 ( 60 fhill.
2/c)
20568
L. 3
(0)
Anf. L. 3 per ounce.
Here the left-hand extreme being complex, both it
and the other extreme are reduced to penny weights, the
loweſt of the parts.
The middle term, being likewiſe complex, is reduced
to fhillings; and fo the answer comes out in fhillings;
which I reduce to pounds.
Queſt. 9. A grocer bought 4 hogfheads of fugar, each
weigning neat C. 6: 2: 14, which coft him L. 2 : 8 : 6
per C.: Required the value of the 4 hogfheads at that rate?
f
હ
G.
Chap. VII.
163
The Rule of THREE.
C. Q. lb.
6 2 14
26 2
26
26
2656
4
2968 lb. in 4 hhds,
lb.
L. s. d.
lb.
If 112: 2—8—6 :: 2968
20
48
12
582 d.
When part of a term only
is given, and the term to be
completed by multiplica-
tion, it is generally con-
venient to complete the
term, or even to reduce it,
before you ftate the que
ftion.
582
5936
23744
14840
12) Rem.
112). 1727376 (15423d 3
Í 12 · · ·
607
2|0)1 285
560 L. 64 5 3 Anf.
473
448
257
224
336
336
(0)
Queſt. 10. If 4 s. 6 d. is the price of 1 yard, how
many yards will L. 16, 4 s. buy?
X 2
,
If
164
Chap. VII.
The Rule of THREE.
1
s. d. yd.
L.
S.
If
4—6
·6: 1 :: 16–4.
2
20
9
324
2
When the middle term hap-
pens to be 1, or unity, in this
cafe, as I neither multiplies nor
divides, you have only to di-
vide the third term by the firſt,
and the quot is the anſwer.
9) 648 (72 y.
63
18
18
(0)
Anf. 72 yards.
In this example both extremes are complex, and re-
duced to fixpences.
Quest. 11. If the yearly rent of a houſe be L. 73, how
much is that per day?
When the third term happens to
be 1, you have only to divide the
middle term by the firft, and the
quot is the anſwer.
Anf. 4 s. per day.
Days.
L. day.
365
73: I
20
365) 1460 (4 s.
1460
(0)
But if in this cafe the middle term be less than the
first, it must be reduced to fome lower denomination,
viz. till the middle term thus reduced may be divided
by the firſt.
Quest. 12. A merchant bought 14 pieces of broad.
cloth, each piece containing 28 yards, at 13 s. 61 d. per
yard: How much did the 14 pieces coſt?
14
Chap. VII.
165
The Rule of THREE.
14 pieces.
28
1
112
yd
s. d. yds.
If 1 : 13—61 :: 392
28
392 yds in 14 p.
I 2
162
325
1960
2
1
325
784
1176
24)127400)53018 s.
I 20*
2|0}
L. 265 8
74
72
200
192
Rem. (8) half-pence.
L.
s. d.
Anf. 265 8 4
When the first term happens to be 1, the product of
the ſecond and third is the anſwer
The middle term here being complex, is reduced to
halfpence; and accordingly the product, or anſwer,
comes out in halfpence, which I reduce to pounds.
Queſt. 13. A draper buys 50 pieces of kerſey, each
piece containing 34 ells Flemish, to pay at the rate of
8 s. 4 d. per ell Engliſh: How much did the 50 pieces
coft?
grs.
qrs. s. d.
If 5: 8-4: 5100
12
100
34
50
1700 ells Flemiſh
100
5)510000
12) 102000
2|0)8500
Anf. 425 1.
3
5100 quarters
Queſt.
166
Chap. VII.
The Rule of THREE.
Quest. 14. What is the yearly intereft of L. 78, 16 s.
6 d. at 5 per cent.?
pr..
pr. int. L. S. d.
If 100: 5 :: 78—16—6
5
Int.
L. s. d. f.
100)394—2—6(3 18 9 31% Anf.
20
1882
I 2
9190
4
3160
MORE
EXAMPLE s.
Quest. 15. If 17 yards of cloth coft L
19: 2:6,
what will 35 yards coft at that rate? Anf. L 9:7:6.
Quest. 16. If 24 lb. of raifins coft 6 s. 6d. what will
18 frails coft, each frajl weighing 3 Q 18 lb? Anf.
L. 24: 17: 3.
I
Quest. 17. If 40 pieces of cloth coft 750 l. 15 s. what
is the price of 1 piece? Anf. L. 18: 15:41.
Queſt. 18. If i ell of cloth coſt 8 s. 4 d. how many
ells will L. 10: 16: 8 buy? Anf 26 ells.
Quest. 19. A gentleman expends daily L. 5, 15 s.
How much is that in a year? Anf. L. 2098, 15 S.
Quest. 20. A merchant buys cloth to the value of
L. 537, 12 s. at 5 s. 4 d. per yard: How many yards
did he buy? Anf. 2016 yards.
Quest 21. If I fell 14 yards for L. 10, 10 s. how
many ells Flemiſh ſhall I feil for L. 283 : 17 : 6, at that
rate? Anf. 504 ells Flemish.
Quest. 22. What is the price of 14 ingots of filver,
each ingot weighing 7 lb. 5 oz. 10 dw. at 5 s. per
ounce? Anf. L. 313, 5 S.
Quest. 23. A grocer mingles 17 C. 3 Q. 14 lb. of ſu-
gar,
1
Chap. VII.
167
The Rule of THREE.
gar, at L. 4, 4 s. per C. with 8 C. 3 Q. 21 lb. other ſu-
gar, at 6 d. per lb.: What is 1 C. of this mixture worth?
Anf. L.
3 : 14: 8.
Quest. 24. A draper buys 8 packs of cloth, each pack
containing 4 parcels, and each parcel 1 pieces, and
each piece 26 yards, for which he paid at the rate of
L. 4, 16 s. for 6 yards: How much did the whole coſt?
Anf. L. 6656.
1
Queft 25. A merchant bought 242 yards of broad
cloth, which coft him in all L. 209, I s.; for 96 yards
of which he paid at the rate of 13 s. 4 d. per yard: How
much did he give per yard for the reft? Anf. 18 s. 6 d.
per yard.
Quest. 26. An oilman bought 3 tuns of oil at the rate
of L. 50: I
4 per tun; and fo it happened that it leak-
ed out 85 gallons; but he is minded to fell it again, fo
as to be no lofer: How muft he fell it per gallon? Anf.
At 4 s. 6174 d. per gallon..
Queft. 27. A fhopkeeper bought 4 bales of cloth,
each bale containing 6 pieces, and each piece 27 yards,
at L. 16, 4 s. per piece: What was the price of the
whole? and what the rate per yard? Anf. The whole
coft L. 388, 16 s. and the rate per yard was 1 2 S.
:
Quest 28. A bankrupt, who owed L. 1490 5: 10,
compounds with his creditors for 12 s. 6d. per L.: How
much did he pay in all? Anf. L. 931873.
Queſt. 29. If the rent of a ſmall eftate be L. 250,
what will this afford to fpend every calendar month, in
order to fave, or lay up, L. 80 at the year's end? Anf.
L. 14: 3:4•
Quest. 30. The rents of a whole parish amount to
L. 3500, and a rate is granted of L. 131, 5 s.: What is
that per pound? Anf. 9 d.
Quest. 31. Received 42 C. 2 Q. cotton, at L. 3, 15s.
per C. and 12 lb. cloves, at 9 s. 1 d. per lb.; for which
I have given in part 1oco yards linen, at 2 s. 9 d. per
yard: What balance do I owe? Anf. L. 27:6 : 6.
Quest 32. A chapman bought a parcel of ferge and
fhalloon, which together coft him L. 61:9: 2; the
quantity of ferge he bought was 236 yards, at 3 s. 4 d.
per
168
Chap. VII.
The Rule of THREE.
3
per yard; and for every 2 yards of ferge he had yards
of fhalloon: How much ſhalloon had he? and what did
it coft him per yard? Anf. 354 yards of fhalloon, at
15 d. per yard.
II. The fimple Rule of Three inverſe.
Queſt. 1. If 8 men can do a piece of work in 12
days, in how many days will 16 men do the fame ?
In this queftion the fuppofition is, If 8 men do a
piece of work in 12 days, and the two terms contained
in it are 8 men and 12 days: The demand lies in theſe
words, In how many days will 16 men do the fame?
and the only term contained in it is 16 men.
•
The number fought here is the days in which 16 men
will do the work, and the term in the fuppofition of the
fame kind is 12 days; wherefore I place 12 days as the
middle term, according to Rule I the two remaining
terms are extremes, and of the fame kind, viz. both
of them men.
It is obvious that the an-
fwer muſt be less than the
middle term; for 16 men will
do the work in fewer days than
8 men; and therefore, by Rule
II. the greateſt extreme, viz.
16 is the divifor; which I place
on the left hand, and the other
extreme on the right, as di-
men. days. men.
16: 12 :: 8
8
16)96(6 days. Anf.
96
(0)
rected in Rule III. Then multiplying the fecond and
third, and dividing their product by the first, the quot
comes out in days; that is, of the fame name with the
middle term.
And becauſe the extreme found in the demand hap、
pens to be the diviſor, the proportion is inverſe.
Queſt. 2. If 30 yards of cloth that is 5 quarters
broad, be required to hang a bed, how many yards of
3 quarters broad will ferve the fame purpoſe ?
The fuppofition here is, that 30 yards of cloth 5
quarters broad are fufficient to hang a bed, and the two
terms
Chap. VII.
169
The Rule of THREE.
terms in it are 30 yards of length, and 5 quarters of
breadth: The demand is, How many yards of cloth
that is 3 quarters broad will hang the fame bed? and
the only term in it is 3 quarters of breadth.
The thing fought is the length or number of yards of
cloth 3 quarters broad requifite to hang the bed, and
the term in the ſuppoſition of the fame kind is 30 yards,
which therefore I place in the middle; the two remain-
ing terms are extremes, and of the faine kind, viz.
both breadths.
It is eafy to perceive
that the anſwer muſt be
greater than the middle
term; for more yards of
cloth 3 quarters broad will
be neceffary to hang a bed
than of cloth 5 quarters
broad, and the narrower
the cloth is the more length
breadth. length. breadth.
qrs. yds.
3: 30 ::
5
qrs.
5
3)150(50 yards. Anf.
15.
will be required; where-
fore the leaft extreme is
(0)
the divifor, which I place on the left hand, and fet the
greater extreme on the right.
And becauſe the extreme found in the demand is the
diviſor, the proportion is inverſe.
Quest. 3. If the penny-loaf weigh 8 ounces, when
flour is fold at 2 s. or 24 d. per peck, what ought to be
the weight of the penny-loaf when flour is fold at 18 d.
per peck?
It
170
The Rule of THREE.
Chap. VII.
d.
d. oz.
18: 8 :: 24
8
It is obvious that this an-
fwet muſt be greater than the
middle term; becauſe the
cheaper the flour is, the hea-
vier ought the penny-loaf to
be and therefore the leaſt
extreme is the divifor; which
I accordingly place on the
left hand.
18)19(10 oz.
18.
Rem. 12 oz.
20
18) 240(13 dw.
18
60
3
54
Rem. 6 dw.
24
18) 144(8 gr.
144
(0)
Anf. 10 oz. 13 dw. 8 gr.
Quest. 4. If 1200 lb. weight be carried 36 miles for
a fum of money, how many miles ought 1800 lb. weight
to be carried for the fame fum ?
Here the anſwer muft
be less than the middle
term; becauſe the heavier
the load is, the fewer miles
ought it to be carried for
the fame money; and fo
the greater extreme is the
divifor, and placed on the
left hand.
lb.
lb.
miles.
1800 : 36 :: 1200
1 200
1800) 43200 (24 miles. Anf
36.
72
72
(0)
2
Quest.
Chap. VII.
171
The Rule of THREE.
Queft. 5. If L. 100 in 12 months gain a fum of in-
tereft, what principal will gain the fame fum of intereſt
in 8 months?
Here the anſwer muſt be great-
er than the middle term; becauſe
the fewer the months are, the
more principal will be required
to gain the ſame fum of inter-
eft; and therefore the leaſt of
the extremes is the divifor.
m. L.
8 : 100 ::
Quest. 6. If 40 poles in length, and
make an acre, what muſt be the length to
when the breadth is 15 poles?
of
Here the anſwer muſt
be less than the middle
term; becauſe the
broader a piece
ground is, the lefs
length is required to
make the acre; and fo
the greateſt of the ex-
tremes is the divifor.
m.
1 2
100
8) 1200
Anf 150 L.
4 in breadth,
make an acre
breadth. length. breadth.
15 : 40 :: 4
4
15) 160 (10 poles.
15
Rem. 10 poles.
33 half-feet in 1 pole.
2)
15) 330(22 half feet.
30
II feet.
30
Quest. 7. How much
line a cloak that hath in it
30
(0)
poles. feet.
Anf. 10 I I
pluſh of 3 quarters wide will
4 yards of 7 quarters wide?
Y 2
Q:
172
Chap. VII,
The Rule of THREE.
Here the anſwer muſt be
greater than the middle term;
for the pluſh being narrower
Q. yd. Q.
3:47
7
thin the cloth of which the 3) 28 (9 yards. Ans.
cloak is made, will require
more length.
27
(1)
feet
Quest 8. If 36 yards be a rood of mafon-work, at 3
feet high, how many yards will make a rood at 9
high?
Feet. yards. feet.
9.: 36 :: 3
3
9) 108
MORE
Anf. 12 yards.
EXAMPLES.
Queft. 9. If a meffenger makes a journey in 24 days
when the day is 12 hours long, in how many days may
he make out the fame journey when the day is 16 hours
long? Anf. in 18 days.
Quest. 10 If 360 men be in garrifon, and have pro-
vifions only for 6 months, how many men muſt be turn-
ed out, that the fame ftock of provifions may laſt 9
months? Anf. 240 men are to be retained, and the reſt,
viz. 120, muſt be turned out.
Quest. 11. If I have 1200 lb. weight carried 36 miles
for a fum of money, how many pound-weight fhall I
-have carried 24 miles for the fame fum? Anf. 1800 lb.
Quest. 12. If I borrow of a friend L. 64 for 8 months,
what fum ought I to lend him for 12 months in order
to requite the favour? Anf. L. 42: 13: 4.
Quest. 13. A fabric is reared in 8 months by 120
workmen; and another fabric, on the fame plan, and of
the fame dimenſions, is to be reared in 2 months: How
many workmen muſt be employed? Anf. 480.
Quest.
Chap. VII.
173
The Rule of THREE.
Quest. 14. A regiment of foldiers, confifting of 1000
men, are to have new coats, each coat containing 3 yards
2 quarters of cloth that is 6 quarters wide, and they are
to be lined with fhalloon that is yard wide: How many
yards of cloth will the coats take? and how many yards
of fhalloon will line them? Anf. 3500 yards of cloth,
and 5250 yards of fhalloon.
Queft 15. How many yards of paper that is 3 quarters
wide will be fufficient to line a room that is 24 yards
round and 4 yards high? Anf. 128 yards.
Queſt. 16. If I lend my friend L. 650 for 22 months,
how long ought he to lend me L. 953: 6: 8 to requite
my kindneſs? Anf 15 months.
Queft. 17. If 50 horſes are maintained a year in corn
for a certain fum, when oats are fold at 9 s. per boll, how
many horfes may be maintained a year in corn for the
fame fum, when oats are at 10 s. per boll? Anf. 45
horfes.
Queft. 18. If the penny-loaf weigh 10 oz. when wheat
is fold at 12 s. per bufhel, what ought to be the price of
wheat when the penny-loaf weighs 8 oz.? Anf. 15 s. per
bufhel.
Queſt. 19. If 12 inches in length and 12 inches in
breadth make a ſquare foot, what length of a plank that
is 9 inches broad will be equal to a fquare foot? Anf.
16 inches.
Queſt. 20. If 40 poles in length and 4 poles in breadth
make an acre, what muſt be the breadth to make an
acre when the length is only 32 poles? Anf. 5 poles.
SECT. II.
The Compound Rule of Three.
THE Compound Rule of Three, from five given
numbers finds a fixth, or from feven given numbers
finds an eighth, or from nine given numbers finds a
tenth, or from eleven finds a twelfth, &c.
This rule eaſily and naturally admits of fubdivifions,
which, from the number of the terms given, may be
denominated
74
Chap. VII.
The Rule of THREE.
denominated the rule of Five, the rule of Seven, the rule
of Nine, the rule of Eleven, &c.
Queſtions in the compound rule of three are alſo re-
folved into two parts, viz. a ſuppoſition and a demand.
If five terms be given, three of theſe are always found
in the fuppofition, and two in the demand; if feven
terms be given, four of theſe are in the ſuppoſition, and
three in the demand; if nine terms are given, five of
theſe are in the fuppofition, and four in the demand;
if eleven terms be given, fix of theſe are in the fuppo
fition, and five in the demand, &c.
The fuppofition and demand being diftinguifhed, pro-
ceed to ſtate the queftion; that is, to put the terms in
due order for operation, as the following rules direct.
RULE I.
Place that term of the fuppofition which is of the
fame kind with the number fought, in the middle. The
remaining terms are extremes, which must be claffed
into fimilar pairs, by making each pair conſiſt of one
term taken from the fuppofition, and another of the
fame kind taken from the demand.
RULE II.
Out of each fimilar pair, joined with the middle term,
form a umple queſtion; and in each ſimple queſtion, fo
formed, fi d the divifor; viz. confider from the nature
of the queſtion, whether the anſwer muſt be greater or
lefs than the middle term; and if the anſwer muſt be
greater, the leaft extreme is the divifor; but if the an
fwer muſt be lefs than the middle term, the greatelt ex-
treme is the divifor.
RULE III.
Place all the divifors on the left hand, and the other
extremes on the right; then multiply the divifors, or ex-
tremes on the left, continually, for a divifor, and mul-
tiply the extremes on the right hand and the middle.
term, continually, for a dividend; and lastly, divide
the
Chap. VII.
175
The Rule of THREE.
the dividend by the divifor; and the quot is the anſwer,
of the fame name with the middle term.
The anſwer to queftions in the compound rule of
three may alſo be had by working the fimple queſtions
feparately, or by themſelves, in the following manner,
viz.
The middle term, with any one pair of fimilar ex-
tremes, make the firft fimple queftion, and the anſwer
to this queſtion muſt be made the middle term to the
next fimilar pair of extremes; and the anſwer to this
fecond queſtion muft, in like manner, be made the
middle term to the following fimilar pair of extremes,
&c.; and the anſwer to the laft fimple queftion is the
number fought.
But the joint operation preſcribed in Rule III. is the
ſhorter as well as the eaſier method; for in working ſome
of the fimple queftions, there may happen to be a re-
mainder, and confequently the middle term of the next
Simple queſtion will have fome fractional part; which in-
conveniency is avoided by working jointly.
In every fimple qucftion, when the divifor is an ex.
treme found in the fuppofition, the proportion is direct;
but when the divifor is an extreme found in the demand,
the proportion is inverfe.
The three rules delivered above are indeed fo calcula-
ted, as to make no difference betwixt direct and in-
verſe, or ſo as to render that diſtinction needlefs, the
left-hand extremes being all divifors; but yet, as que.
ftions confifting entirely of direct proportions are the
plaineſt and eaſieft, it will be proper, in the firſt place,
to exemplify the rules by queftions of the direct kind,
and afterwards introduce fuch as are inverfe.
And as queftions in the rule of five are by far more
numerous, and occur much oftener, than queſtions
in the rule of feven, nine, or eleven; I fhall, first
of all, adduce a fet of queftions in the rule of five,
wherein both proportions are direct; then propoſe ano.
ther fet, wherein one or both proportions are inverſe;
and, laftly, give a few examples of the rules of feven,
nine, and eleven.
I. The
176
Chap. VII.
The Rule of THREE.
1
I. The Rule of Five direct.
Quest. 1. If 14 horfes eat 56 bushels of corn in 16
days, how many bufhels will 20 horſes eat in 24 days?
The fuppofition in this queftion is, If 14 horfes eat
56 bushels in 16 days; and the three terms contained in
it are, 14 horfes, 56 bufhels, and 16 days: The de-
mand is, How many bushels will 20 horfes eat in 24
days? and the two terms contained in it are 20 horſes,
and 24 days.
The number fought is bufhels, and the term in the
fuppofition of the fame kind is 56 buſhels; wherefore,
according to Rule I. I place 6 bufhels in the middle.
The remaining four terms are extremes, which I claſs
into fimilar pairs, by making each pair confift of one
term taken from the fuppofition, and another of the
fame kind taken from the demand. Thus, 14 horſes
and 20 horſes make one pair; again, 16 days and 24
days make another pair.
Out of the feveral fimilar pairs, joined with the mid-
dle term, I form fo many fimple queſtions, according to
Rule II. viz. I fay,
1. If 14 horfes eat 56 bufhels in a certain number of
days, how many bushels will 20 horfes eat in the fame
tinie?
2. If 16 days eat up, or confume, 56, or any other
number of bushels, how many bushels will 24 days con-
fume ?
In the firft fimple queftion it is obvious, that the an-
fwer will be greater than the middle term; for 20 horſes
will eat more bushels than 14 horfes will do in the ſame
time; and fo the leaſt extreme, viz. 14, is the divifor;
and becauſe 14 is an extreme found in the ſuppoſition,
the proportion is direct.
In the fecond fimple queftion it is alfo plain, that the
anſwer will be greater than the middle term; for 24 days
will confume more bushels than 16 days; and confe-
quently the leaft extreme, viz. 16, is the divifor; and
becauſe
Chap.VII.
177
The Rule of THREE.
becauſe 16 is an extreme found in the ſuppoſition, the
proportion is direct.
According to Rule III. I
place the divifors on the left
hand, and the other ex-
tremes on the right, and
both of them under one an-
other; ſo that the two up-
per ones make a pair, or be
of one kind, and the two
lower ones make another
pair, or be of one kind;
and no matter which of the
pairs be uppermoft: then
I multiply the divifors, or
the extremes on the left
hand, for a divifor; and a-
gain I multiply the extremes
on the right, and the mid-
dle term, continually, for
Foint operation.
Horfes bufhels. horſes.
: 56 :: 20
If 14
da. 16
24 da.
8+
480
56
288
240
14
224
224)26880(120
224
448
448
a dividend; and dividing the Anf. 120 bushels.
dividend by the divifor, the
(0)
quot, or anſwer, comes out of the fame name with the
middle term, viz. 120 buſhels.
The two fimple queftions into which the compound
queſtion is refolved, are ſtated, and wrought ſeparately,
as follows.
H. B. H.
If 14: 56:: 20
Days. B. Days.
If 16: 80:: 24
20
80
14) 1120 (80 B.
16) 1920 (120 B.
112
16.
(0)
32
32
Anf. 120 bushels, as before.
(0)
Z
Quest.
178
Chap. VII.
The Rule of THREE.
Queſt. 2. If 40 acres of grafs be cut down by 8 men
in 7 days, how many acres fhall be cut down by 24 men
in 28 days?
Foint operation.
Men. acres.
Men.
If 8 : 40:
::
24
da. 7
28 da.
56
19 2
48
672
40
56) 26880 ( 480 Acres. Anf
224
448
448
(0)
The fame refolved into two fimple queftions, and
wrought ſeparately.
M. a. M.
If 8: 40: 24
Days. a. Days.
If 7: 120 :: 28
40
I 20
8) 460
120 acres.
7)3360
Anf. 480 acres.
Queſt. 3. If L. 100 in 12 months gain L. 5 intereſt,
what will L. 75 gain in 9 months?
Joint
Chap. VII.
179
The Rule of THREE,
Foint operation.
Pr. int. Pr.
L. L.
L.
'If 100 : 5 :: 75
months 12
9 months.
1200
675
5
L. S.
d.
12|00) 33|75 (2 16 3 Ans.
24
975 L. rem.
20
19500
12.
75
72
3 s. rem.
12
36
36
(0)
The fame queſtion refolved into two fimple ones, and
wrought ſeparately.
麈
​Z 2
Pr.
180
Chap. VII.
The Rule of THREE.
Pr. int. Pr.
L. L. L.
If 100
:
575
Mo. L. s. Mo.
If 12:3-15:9
5
20
L.
S.
1500
1|00) 3|75 ( 3—15
20
12) 675 ( 3|6 ( 2
60
4
75 16 s.
75
9
20 L.
S.
d.
16 3 Anf.
72
Rem.
3 S.
12
36(3 d.
36
| ©
(0)
Quest. 4. If a carrier receive 42 fhillings for the car-
riage of 3 C. weight 150 miles, how much ought he to
receive for the carriage of 7 C. 3 Q 14 lb. 50 miles ?
Before the queſtion is ſtated, it will be convenient to
reduce one of the pairs as follows.
Umm
3
3
G. & lb.
7—3—14
7
| ww
7
798
336lb. 882 lb.
}
+
Foint
Chap. VII.
181
The Rule of THREE-
Joint operation.
lb.
S.
lb.
If 336:42 :: 882
miles 150
1680
336
50 miles.
44100
42
50400
882
1764
S. d.
504|00)18522|00 (36-9 Anf.
1512.
3402
3024
Rem. 378 fhill.
12
4536(9 d.
4536
(0)
The ſame queſtion refolved into two fimple ones, and
wrought ſeparately.
lb.
182
Chap. VII.
The Rule of THREE.
lb. S.
N.
If 336:42::882
42
If 150: 110-3 :: 50
12
m.
S. d.
m.
1764
1323
3528
50
s. d.
12)
336) 37044(110--3 15|0)6615|0(441
336••
60..
344
336
1 5 8 | 9 9 1 0
36-9
Rem. 84 hill.
12
1008 (3 d.
1008
(0)
MORE
(0)
S. d.
Anf. 36-9
EXAMPLE S.
Quest. 5. If a regiment of 936 foldiers eat up 351
quarters of wheat in 168 days, how many quarters of
wheat will 11,232 ſoldiers eat in 56 days at that rate? Anf.
1404 quarters.
Queſt. 6. If 48 bushels of corn yield 576 bufhels in 1
year, how much will 240 bushels yield in 6 years at that
rate? Anf. 17,280 bushels.
Queſt. 7. If 40 fhillings be the wages of 8 men for 5
days, what will be the wages of 32 men for 24 days?
Anf. 768 fhillings, or L. 38, 8 s.
Queft. 8. If 8 cannons in 1 day fpend 48 barrels of
powder, how many barrels will 2 cannons ſpend in 12
days at that rate? Anf. 1728 barrels.
Quest. y. An ufurer receives L. 37: 6 as the inter-
eſt of L. 75 for 9 months: How much is that per cent.
per annum? Anf. 6 per cent.
Quest. 10. If 18 roods of ditching be done by 3 men
in
Chap. VII.
183
The Rule of THREE.
in 6 days, how many roods will be done by 8 men in 4
days at the fame rate of working? Anf 32 roods.
Queft. 11. If 4 ftudents ſpend L 19 in 3 months, how
much will 8 ſtudents ſpend in 9 months? Anf. L. 114.
Queſt. 12. If 600 feamen in 1 week eat 1500 lb. of
beef, how much will ferve 120 feamen 12 weeks? Anf.
3600 lb.
11. The Rule of Five inverse.
The queſtions that fall under this rule have commonly
one of the proportions inverfe, and the other direct, and
fometimes the upper, and fometimes the lower, is the
inverſe proportion; and in fome few queſtions both pro-
portions are inverfe. Now, though the three rules de-
livered above make no difference betwixt direct and in-
verſe; yet, to bring the learner to fome meaſure of ac-
quaintance with this ufeful diftinction, I fhall, in ftating
the following queſtions, expoſe the fame to view, by af-
fixing an aſteriſk to the extremes of every inverſe pro-
portion.
Queſt. 1. If 1 14
horſes eat 56 bushels of corn in 16
days, in how many days will 20 horfes eat 120 buſhels
at that rate?
In this queſtion the fuppofition is, that 14 horfes eat
56 bushels in 16 days; and the demand is, In how ma-
ny days 20 horfes will eat 120 buſhels?
The number fought is days, and the term in the fup-
pofition of the fame kind is 16 days; and accordingly I
place 16 days in the middle. The remaining four terms
are extremes; which 1 clafs into fimilar pairs, by ma-
king each pair confift of one term taken from the fuppo-
fition, and another of the fame kind taken from the de-
mand Thus, 14 horfes and 20 horfes make one pair:
again, 56 buſhels and 120 bushels make another pair.
Out of the fimilar pairs, joined with the middle term,
I form fo many fimple queſtions; namely,
1. If 14 horfes eat a certain number of bushels in 16
days, in how many days will 20 horfes eat the fame
quantity?
2. If
184
Chap. VII.
The Rule of THREE.
2. If 56 bushels are eat up in 16 days, in how
days will 1 20 bushels be eat up by the fame eaters?
many
In the firft fimple queftion it is plain, that the anſwer
muſt be less than the middle term; for 20 horfes will
eat the fame number of bushels in fewer days than 14
horfes; and fo the greateſt extreme, viz. 20, is the di-
vifor; and becauſe 20 is an extreme found in the de
mand, the proportion is inverſe.
In the fecond fimple queftion it is alfo obvious, that
the anſwer muſt be greater than the middle term; for 120
bufhels will require more days to be eat up in, than 56
buſhels; and therefore the leaft extreme, viz. 56, is
the divifor; and becauſe 56 is an extreme found in the
fuppofition, the proportion is direct.
I now proceed to
flate the queſtion, by-
placing the div fors
Foint operation.
Horf day Horf.
*
20: 16 :: 14
on the left hand, and bufh. 56
the other extremes
on the right; then I
multiply and divide,
as directed in R. III.
and the anſwer comes
out of the fame name
with the middle term,
viz. 24 days.
*
120 buſh.
II 20
1680
16
1008
168
days.
11 |0)2688,0(-4 Anf.
224
448
448
(0)
The two fimple queftions into which the compound
question is refolved, are ftated and wrought feparately,
as follows.
Horf.
Chap. VII.
185
The Rule of THREE.
Horf day. Horf.
* 20: 16 :: 14
14
64
16
2|0)22|4 (11 days.
24
9/6(4 hours.
1000
48
22
1
Buſh. D. h. m.
56: 1≥—4—48 :: 120
24
Buſh.
268
60
16128
16
I 20
60
60)
56(1935360(3456|0
960(48 min.
168...
24)576(24 days.
255
48
224
96
313
96
280
(0)
336
336
(0)
Queſt. 2. If 40 acres of grafs be cut down by 8 men
in 7 days, in how many days will 480 acres be cut
down by 24 men?
Аа
Joint
186
Chap. VII.
The Rule of THREE.
Foint operation.
Acres. days. Acres.
40: 7: :: 480
Men * 24
8 * Men.
960
3840
7
960)2688|0(28 days. Anf.
192
768
768
(0)
The ſame queſtion reſolved into two ſimple queſtions,
and wrought feparately.
Acr. da.
Acr.
Men. da. Men.
40: 7 :: 480
24
848 *
7
8
40)3360(84 days. 24)67 2(28 days. Anf.
32
16
16
(0)
48
192
192
(0)
Quest. 3. If 100 l. principal, in 12 months, gain
51. intereft, what principal will gain L. 2:16:3 in-
tereſt in 9 months?
Joint
Ghap. VII.
187
The Rule of THREE.
L.
L.
S.
d.
Foint operation.
5
2—163
240
20
Pr.
Mon.
L.
Mon.
1200d. 56
*
9: 100 :: 12
*
12
Int. d. 1200
675 d. Int.
675 d.
10800
8100
100
10800)810000(75 1. Ans.
756.
540
540
(0)
The ſame queſtion refolved into two fimple queſtions,
and wrought ſeparately.
Mo. L.
Mo.
D. L. s. d.
D.
9: 100 :: 12 *
12
9) 1200(133).
3
20
9)60(6 s.
54
6
I 2
9)72(8 d.
72
(0)
1200: 133—6—8 :: 675
20
2666
12
32000
675
1350
2025
12|00)21 6000|00(18000
12)
12
20) 1500
96
96
751. Anf.
(000)
A a 2
Quest.
188
The Rule of THREE.
Chap. VII.
Quest. 4. If 12 inches of length, 12 of breadth, and
12 of thickneſs, make a folid foot, what length of a
plank that is 6 inches broad and 4 inches thick will
make a folid foot?
Joint operation.
Br. leng. Br.
* 6: 12 :: 12 *
Th. *
4
12 *Th.
24
144
12
24)1728(72 inches. Anf
168.
48
48
(0)
The fame queſtion refolved into two fimple queſtions,
both inverſe, and wrought feparately.
Br. leng.
Br.
*
* 6: 12 :: 12
12
*
Th. leng. Th
4: 24: 12 *
1 2
4)288
6)1 44
24 inches.
Anf. 72 inch. in leng.
3
MORE EXAMPLES.
Quest 5. If a carrier receive 42 fhillings for the car.
riage of C. weight 150 miles, how many miles ought
he to carry C. 7:3:14 for 36 s. 9 d. Anf 50 miles.
y
Queſt. 6. If a regiment of 936 foldiers eat up 351 quar-
ters of wheat in 168 days, how many foldiers will eat
up 1404 quarters in 56 days at that rate? Ans. 11,232
foldiers.
Queſt. 7. If 8 men in 5 days earn 40 fhillings of wa-
ges, in how many days will 32 men earn 381. 8 s.?
Anf. In 24 days,
Queſt.
Chap. VII.
139
The Rule of THREE.
Quest. 8. If 8 cannons in 1 day ſpend 48 barrels of
powder, in how many days will 24 cannons ſpend 1728
barrels at that rate? Anf. In 12 days.
Quest. 9. If 75 1. principal, in 9 months, gain
L. 3:7:6. intereft, in how many months will 100 l.
principal gain 61. intereſt? Ans. In 12 months.
Quest. 10. If the penny-loaf weigh 12 ounces when
wheat is fold at 3 s. 4 d. per bufhel, what ought to be
the weight of a loaf worth 9 d. when wheat is fold at
10 s. per bufhel? Anf. 36 ounces.
Quest. 11. If one travels 300 miles in 10 days when
the day is 12 hours long, in how many days may he
travel 600 miles when the day is 16 hours long?
Anf. In 15 days.
Quest. 12 If 48 pioneers, in 12 days, caft a trench
24 yards long, how many pioneers will caft a trench
168 yards long in 16 days? Anf. 252 pioneers.
III. The Rule of Seven, Nine, Eleven, &c.
Quest. 1. If 15 men eat 156 d. worth of bread in 6
days, when wheat is fold at 12 s. per buſhel, in how
many days will 30 men eat 520 d. worth of bread when
wheat is at 10s. per buſhel?
This question belongs to the rule of feven, the num-
ber fought is days, and the term of the fame kind in
the fuppofition is 6 days, which I place in the middle.
The remaining fix terms are extremes, which I clafs into
fimilar pairs, by taking one term of each pair out of
the fuppofition, and another of the fame kind out of the
demand.
Out of the fimilar pairs, joined with the middle term,
I form fo many fimple queftions, in each of which I find
the divifor by Rule II.; then I place the divifors on the
left hand, and the other extremes on the right, as di-
rected in Rule III. and multiply and divide, as follows.
Foint
190
Chap. VII.
The Rule of THREE.
S.
Foint operation.
Men. days. Men.
*
*
30: 6:15
* 10: d. 156
S.
520 d. 12 *
:
4680
30
IO
75
46800
7800
I 2
93600
6
3
468|00)561600(12 days. Anf.
468⚫
936
936
(0)
This compound queſtion is refolved into three fimple
ones, as follows.
30: 6:: 15: 3
156: 3 :: 520: 10
10 10 :: 12 12 days. Anf.
Quest. 2. If 18 men build a wall 40 feet long 3 feet
thick and 16 feet high in 12 days, how many men will
build a wall 360 feet long 8 feet thick and 10 feet high
in 60 days?
This queftion belongs to the rule of nine, and is fta-
ted and wrought in the ſame manner with the preceding
queftion in the rule of feven, as follows.
Joint
Chap. VII.
191
The Rule of THREE.
d.
Foint operation.
Length. men. Length.
h. 40: 18: 360
*60: 16: Th. 3
I 20
16
72
12
h. d.
8 Th.: 10:12
2880
10
28800
12
1920
345600
60
18
115200
27648
3456
1152|00)62208|00(54 men. Anf.
5760
4608
4608
(0)
This compound queſtion is refolved into four fimple
ones, as follows.
40: 18:: 360 : 162
3: 162 ::
16: 432 ::
8:432
10:270
60 270 :: 12: 54 men. Anf.
Queſt. 3. If 12 men caſt a ditch 30 feet long 6 deep
and 3 broad in 15 days, when the day is 12 hours long;
in how many days will 60 men caft a ditch 300 feet
long 8 deep and 6 broad, when the day is but 8 hours
long?
This queſtion belongs to the rule of eleven, and is
ftated and wrought in the fame manner with the prece-
ding queſtions in the rules of feven and nine, as follows.
Joint
192
Chap. VII.
The Rule of THREE.
Joint operation.
Men. days. Men.
h. b. d. * 60: 15 :: 12 *
d. b. h.
* 8 : 3 : 6 : 1. 30
300l.:8:6: 12
***
And 60 × 30×6×3×8=259200 the diviſor.
And 12×300×8×6×12×15=31104000 the dividend.
259200) 31104000 (120 days. Anf.
2592..
5184
5184
(0)
The work may be contracted by omitting the equal
extremes, 6 and 8, found both on the left hand and the
right, thus.
60 × 30×35400 the divifor.
And 12×300×12×15=648000 the dividend.
54|00) 6480|00 (120 days. Anſ.
54**
108
108
(0)
This compound queſtion is refolved into five fimple
ones, as under.
60: 15 :: € 12:
3
30: 3 :: 300: 30
6:30 ::
8: 40
3:40 :: 6: 80
8:80:: 12: 120 days. Anf.
MORE EXAMPLE S.
Queſt. 4. If 18 roods of ditching be wrought by 3
men in 16 days, when the day is 15 hours long; how
many
Chap. VII.
193
The Rule of THREE.
many roods will be wrought by 8 men in 4 days when
the day is 9 hours long? Anf. 7 roods.
Quest. 5. If 12 men build a wall 30 feet long 6 feet
. high and 3 feet thick in 15 days, in how many days will
60 men build a wall. 300 feet long 8 feet high and 6 feet
thick? Anf. In 80 days.
Quest. 6. If 10 men, by working 6 hours a day, can,
in 4 days, dig a cellar that is 24 feet long 16 wide and
12 deep; how many men, by working 8 hours a-day, will,
in 2 days, dig a cellar that is 32 feet long 24 wide and
8 deep? Anf. 20 men.
Contractions in the Rule of Three.
1. If the firft term is an aliquot part of the fecond or
third, divide the ſaid ſecond or third term by the firſt,
and the product of the quot and the other term is the an-
fwer. Thus, if 9: 27 :: 42, then divide 27 by 9, and
the quot 3 × 42≈ 126, the anſwer. Again, if 8:5: : 64,
divide 64 by 8, and the quot 8 x 540, the anfwer.
X
EXAMPLE.
A, B, C, and D, have L. 100 to be divided among
them, in fuch manner, that for every L. 3 A receives,
B muſt have 5, C 7, and D 10; that is, their ſhares are
to be as the numbers 3, 5, 7, 10: Required their ſhares?
Add the proportional numbers, by faying, 3+5+7
+10=25: then fay,
If 25: 100:
:
X
3 × 4
12 A's fhare.
5×420 B's fhare.
| 7×428 C's fhare.
(10x440 D's ſhare.
100 proof.
Becauſe 25 meaſures 100, I multiply the quot 4 into
all the extremes on the right hand, and the products are
the anſwers or ſhares.
2. If the firſt term be a multiple of the fecond or third,
B b
divide
194
Chap. VII.
The Rule of THREE
divide the firſt term by the faid fecond or third; and the
remaining term, divided by the quot, gives the anſwer.
EXAMPLE
I.
If 60 yards coft L. 20, what will 45 yards coft?
Here the first term fo is a multiple of the ſecond term
20, and 3, and 45 15 L. Anf.
EXAMPLE
11.
If-18 yards coft L. 12, what will 6 yards coſt?
Here the first term 18 is a multiple of the third term
6, and 183, and 124 L. Anf.
ર
3. In compound queſtions the operation may frequent-
ly be rendered more fimple, by placing the component
parts of the divifor and dividend in a fractional form, re-
jecting fuch parts or factors of the numerator and deno-
minator as happen to be equal, or cutting off an equal
number of ciphers, or dividing by fome common di-
viſor.
EXAMPLE
I.
If 35 ells of Vienna make 24 at Lyons, and 3 ells at
Lyons make 5 at Antwerp, and too ells at Antwerp
make 125 at Frankfort, how many ells at Frankfort make
42 at Vienna ?
This question belongs to the rule of feven; and be
cauſe the number fought is ells at Frankfort, therefore
the given ells at Frankfort, viz. 125, is the middle term:
the remaining terms are extremes; which I claſs into fi-
milar pairs, and ſtate as follows.
Ant.
Chap. VII. The Rule of THREE.
195
!
Vienna
35 Lyons 3
Ant. Frank. Ant.
100
: 125: 5
Vienna
24 Lyons: 42
300
I 2Q
35
42
10500
5040
125
2520
Ioc8
504
105|00)6300|00(60 ells. Anſ.
630
The fimple queftions are,
100: 125
(0)
5: 61
3: 61 :: 24: 50
50 :: 42: 60 ells at Frankfort. Anf.
35: 50
But the queſtion becomes more fimple, by being ſta
ted and wrought in the fractional form, as follows.
42×24 X 5× 125 42 × 24×5X125
35×3×100
3x35x100
42X8X1X25 8400 840
60 ells at Frank.
1X7X20 140 14
fort. Anf.
EXAMPLE
II.
If 100 lb. of Venice weigh 70 lb. of Lyons, and 120
lb. of Lyons weigh 100 lb. of Roan, and 80 lb. of Roan
weigh 100 lb of Toloufe, and 100 pound of Tolouſe
weigh 74 lb. of Geneva, how many pounds of Geneva
will 100 lb. of Venice weigh?
B b 2
This
}
196
Chap. VII.
The Rule of THREE.
This queſtion belongs to the rule of nine; and becauſe
pounds of Geneva is the number fought, the given pounds
of Geneva, viz. 74, muſt be the middle term: the re-
maining terms are extremes; which may be claſſed into
fimilar pairs, and ftated as follows,
Ly. Ven.
Tol. Gen. Tol.
Ven. Ly.
100:74::100
100: 120: Roan 80
100 Roan : 70: 109
8000
10000
I 20
960000
70
700000
I CO
100
96000000 70000000
74
96000000)5180000000(532 lb. of Ger
480.
neva. Ans.
380
288
(92)
But the queftion becomes more fimple, and is wrought
with greater eaſe and advantage, by being ftated in the
fractional form, as follows.
100 X 70 X 100 X 100 X 74
100 X 120 x 80 x 100
7 × 10 × 74
X
5180
12 × 8
96
70 X 100 X 74
120 X 80
=532 lb. of Geneva. Anf.
I shall now conclude, by obferving, that every com-
pound queftion, whether in the rule of five, feven, nine,
or eleven, &c. properly and frictly speaking, conſiſts
but of three given terms. For the firft term, or divifor,
is
Chap. VIII.
197
FELLOWSHI P.
is to be confidered as one compound term made up, or
produced, by the continual multiplication of the ex-
tremes on the left hand, as fo many component parts.
In like manner, the third term is to be conſidered as one
compound term, made up by the continual multiplica-
tion of the extremes on the right, as component parts.
Suppoſe the queſtion to be,
If L. 100 in 12 months gain L. 5 intereft, what will
L. 75 gain in 9 months?
Here it is obvious, that it is neither the L. 100 prin-
cipal, nor the 12 months of time, taken feparately, that
gains the L. 5 intereft, but both contribute their ſhare;
that is, they confpire, as joint caufes, to produce one
effect; and therefore their product, viz. the firſt term,
is to be confidered as the caufe producing the effect;
that is, the firft term, viz. 100 x 12, caufeth, produ-
ceth, or gains, L. 5 of intereſt. And in like manner,
the product of the extremes on the right hand, or the
third term, viz. 75 × 9, is to be eſteemed the cauſe that
produceth a fimilar effect; that is, gains a like fum of
intereft, namely, the fourth term, or anfwer. In re-
ference to this way of confidering the first and third
terms, the queſtion might be ſtated as under.
F
If 100 × 12:5:75×9
CHAP VIII.
FELLOWSHIP.
ELLOWSHIP, called alſo Company, or Partnership,
is, when two or more perfons join their ſtocks, and
trade together, dividing the gain or lofs proportionally
among the partners.
Fellowſhip is either without or with time, called alſo
fingle or double.
I.
198
FELLOWSHIP. Chap. VIII.
I. Fellowship without time.
Queſtions in fellowſhip without time are wrought by
the following proportion.
As the total ſtock
To the total gain or loſs,
So each man's particular ſtock
To his ſhare of the gain or loſs.
Quest 1. A and B make a joint ftock: A puts in 121.
and B 81.; they gain 51.: What is each man's ſhare?
L.
Stock. gain. Stock.
A's ftock
12 |
A. If 20 : 5 :: 12
B's ftock
8
5
Total ſtock 20
Stock. gain. Stock.
B. If 20: 5 :: 8
8
210)610
A's gain 31.
L.
A's gain 3
5
210)410
B's gain 2
B's gain 21.
Total gain 5 proof.
Quest. 2. A, B, and C, make a joint ftock: A puts
in 781. B 1171. and C 2341.; they gain 2651.: What
is each man's ſhare?
L.
Chap. VIII. FELLOWSHIP.
199
L.
(A 781
Stock of B 117
[C 234
Stock. gain. Stock.
A. If 429: 265 :: 78
78
Total ftock 429
2120
1855
429)20670(481.
1716
3510
3432
78
20
429) 1560(38.
1287
273
12
429)3276(7 d.
3003
273
4
429) 1092(2 f.
858
(234)
Stock.
200
Chap. VIII.
FELLOWSHIP.
Stock. gain. Stock.
B. If 429 : 265 :: 117
117
· 1855
265
265
429)31005(72 l.
3003.
975
858
Stock. gain. Stock.
C. If 429: 265 :: 234
234
1060
795
530
429)62010(1441
429..
1911
1716
117
1950
20
1716
429)2340(5 s.
2145
234
20
195
12
429)2340(5 d.
2145
195
4
429)780(1 f.
429
(351)
429)4680(10 s.
429
390
- 12
429)4680(tod.
429
390
4
429)1560(3 f.
1287
Anf.
(273)
Chap. VIII. FELLOWSHIP.
201
。L. S. d. f. Rem.
A 48- 3- 7—2—
Anf. Gain of
B
{ A = = 7 =
72—5—5—1-
C 144
-234
·351
144—10—10—3—273
Proof 265-oo-o
-858
Note 1. When in any queſtion there happen to be:
remainders, they muſt be reduced equally low, ſo as to
be all of one name; and then their fum will be either
equal to the divifor, or exactly double, triple, &c. of
it and accordingly 1, 2, 3, &c. carried from the ſum
of the remainders, and added to the particular gains,
will make up the total gain; or the divifor will always
divide the fum of the remainders exactly, and the quot
added to the particular gains will give the total gain.
Note 2. When the partners have equal ſhares of ſtock
or capital, their flares of gain, lofs, or neat proceeds,
is found readily by dividing the total gain, lofs, &c. by
the number of partners, thus.
Suppoſe A, B, and C, were equally concerned in a
voyage to Virginia, and that the neat proceeds turned
out to 5751. each partner's fhare will be L. 191:13:4,
caft up as under.
L. S. . d.
3)575(191 13 4
Note 3. When the fractions denoting the partners
ſhares of ſtock or capital have all the fame denominator,
their ſhares of gain, lofs, &c. is quickly found, by di-
viding the total gain, &c. by the denominators, and
multiplying each quot by its refpective nnmerator, thus.
Suppofe A, B, C, and D, buy a fhip for a fum of
money, whereof A, B, and C, pay each, and D 3 or
; and that her freight on a voyage amounts to 260l.
the partners ſhares are found as follows.
C C
ठ
6)260
202
FELLOWSHIP. Chap. VIII.
6)260
L. 43:6:8 to A.
4368 to B.
4368 to C.
130:00 to D.
Proof 260
Note 4. The operation may ſometimes be ſhortened
or facilitated by firſt finding the gain, lofs, or neat pro-
ceeds, per cent. or per pound, and then working by the
rules of practice, thus.
Suppoſe A, B, and C, join in an adventure to Bar-
badoes, which, with all charges, amounted to 2000 l.;
whereof A paid 4001. B 600 l. and D 1000l.; they
have returns in goods, which being difpofed of, the
neat proceeds amounted to L. 4333: 6: 8: What ſhould
each partner get?
If 2000: 43331
20
10
6-8: 100
I
· 4333—6—8 ::
: 216|6—13—4 :: Į
20
I
'n
133
I 2
40
216 13 4 per cent.
4
L.
s. d.
400 866 13 4
A gets 866 13 4
200=
433
68
600 1300
1000 — 2166 13 4
B gets 1300
C gets 2166 13 4
Proof 4333 68
Again,
1
Chap. VIII.
203
FELLOWSHI P.-
Again, Suppoſe A, B, and C, join in an adventure to
Jamaica, and (hip off, on their joint credit, goods to
the value of 3000 l. ; and, to complete the cargo, the
partners, from their own warehouſes, fhip off fuch
goods as they had proper for the voyage, viz. A goods
to the value of 100 l. B 200 l. C 300 l.; the neat
proceeds of their returns in fugar and rum amounted to
6930 1. What fhare of this belongs to each partner?
3600: 6930: D
360: 093 :: I
120: 231 :: I
40: 77
:: 177 per 1.
I : 40
Shares.
L.
S.
A's ſtock 1100x77
B's ſtock 1200 x 77
C's ſtock 1300 × 77
=
84700,
402117
10
92400, 402310
=
100100, ÷ 40—2502
IO
Proof 6930
MORE
EXAMPLES.
Quest. 3. A, B, and C, make a joint ftock: A puts
in 460 l. B 510l. and C 4801.; they gain 3401.: What
is each partner's fhare?
A 107-17-2—3
Anf. Gain of B 119—11—8—2—
I..
S. . d. f.
Rem.
85
I 10
—0—I
95
•290
C 112-II —
Proof. 340-00
340-00—0—0
Quest. 4. Three perfons trade together: A puts in
2301. B 5291. and C 3441. 10 s.; they gain 5201.:
What is that to each ?
L.
S. . d.
Rem:
A 108-7-73-
Anf. Gain of B 249-5-7
C 162-6-9
271
844
~1092
Cc 2
Quest.
204
Chap. VIII.
FELLOWSHIP.
Quest. 5. A, B, and C, make a joint ſtock of 32561.:
whereof A puts in 10261. B 9851. and C the reft; by
misfortune they lofe 2000l.: What part of that muſt
each bear?
L. s. d.
A 630 45
Anf. B 605-0-83
C 764-14-10
Rem.
928
1 240
1088
Quest. 6. Four partners, A, B, C, and D, built a
ſhip, which coſt 17301.; and the freight for her first
voyage amounted to 370l.; of which A's fhare was
741. B's 1111. C's 148 1. and D's 371.: What was each
partner's ſtock?
L.
A's ftock was
346
B's
519
Anf.
C's
692
D's
173
Total flock 1730 Proof.
Quest. 7. A, B, and C, in company, gain 361. A
put in 201. B 30 1. C a fum unknown; C took up 161.
of the gain: What did A and B gain? and what did C
put in?
L.
A's gain
8
Anf. B's gain
C's ſtock
12
40
certain fum; of which A,
C, and D, go; A, C, and
What was the ſtock and
Queſt. 8. A, B, C, and D, in partnerſhip, had a joint
ftock of 600l. and gained a
B, and C, took up col.; B,
D, 80; A, B, and D, 70:
gain of each partner ?
Stock.
Chap. VIIL FELLOWSHIP.
205
Stock. Gain.
L.
L.
A's
60-
10
B's
120-
20
Anf.
C's
180—30
į D's
240- 40
II. Fellowship with time.
In fellowſhip with time, the gain or lofs is divided a-
mong the partners, both in proportion to the ſtocks
themſelves, and alfo in proportion to the times of their
continuance in company: for the fame ftock continued
a double time, procures a double thare of gain; and
continued a triple time, procures a triple ſhare of gain;
that is, the ſhares of gain or lofs are as the products of
the ſeveral ſtocks multiplied into their respective times:
and accordingly queftions belonging to this rule are
wrought by the following proportion.
As the fum of the products of the ſeveral ſtocks into
their reſpective times
To the total gain or loſs,
So the product of each man's ftock into his time
To his fhare of the gain or loſs.
Quest. 1. A put into company 401. for 3 months,
B 751. for 4 months; they gain 70l.: What ſhare muſt
each man have?
A 40×3=120, third term for A's thare.
B 75×4300, third term for B's ſhare.
420, first term,
L.
L.
A. If 420 : 70 :: 120 B. If 420 70 :: 300
120
420)8400(201.
84
(0)
300
42|0) 2100|0(50 1.
210
(0)
1
A's
206
FELLOWSHIP. Chap. VIII.
A's gain
B's gain
L.
20
50
Total gain 70 proof.
Quest. 2. A put into company 560 l.
B 2791. for 10 months, and C 7351.
they gained 1000l.: What ſhare of the
have?
for 8 months,
for 6 months;
gain muſt each
A 560 × 8=4480, third term for A's fhare.
B 279×10=2790, third term for B's ſhare.
C 735× 6=4410, third term for C's ſhare.
L.
11680, firft term.
L. 5. d. f. Rem.
208
80
A If 11680: 1000 :: 4480: 383-11-2—3—
B If 11680: 1000 :: 2790 : 238-17 —4—3—·
C If 11680: 1000 :: 4410 : 377-11-4—1—— 880
Proof
MORE
1000—00—0—0—· 1168
EXAMPLES.
Queſt. 3. A, B, and C, hire a paſture for 241.: A
puts in 40 cows for 4 months, B 30 cows for 2 months,
and C 36 cows for 5 months: What ſhare of the rent
muſt each pay?
L. S.
A's fhare of rent
9-12
3-12
C's fhare
10—16)
Anf. B's fhare
Proof 24
Quest. 4. A, B, and C, agreeing to trade together, A
puts in 529 1. for 4 months, B 3291. for 7 months, and
C gool. for 2 months; they gain 540 1.: What is each
man's fhare?
Anf.
•
Chap. IX. VULGAR FRACTIONS.
207
L. s. d.
Rem,
A
183-14-8
2304
Anf. B
199—19— 5
1222
-2583
C 156— 5-102
Quest 5. A and B enter into partnerſhip for a year:
accordingly A put in the first of January 501.; but B
could not put any money in till the firft of May: What
fum muſt В then put in to be intitled to an equal ſhare
of the gain at the year's end? Anf. 751.
Queſt. 6. A, B, and C agree to trade in company for
12 months: A put in at firſt 3641. and at the end of 4
months he put in 40 l, more; B put in at firſt 408 1. and
at the end of 7 months he took out 861.; C put in at
firſt 1481. and at the end of 3 months he put in 861.
more, and at the end of other 5 months he put in 100l.
more; at the year's end their gain was 14361.: Whạt
is each man's fhare?
A's gain
Anfį B's gain
C's gain
L·
s. d. Rem.
556- 3—6—6192
529-16-9-
349-19—8-
5496
416
Note. Queſtions in double fellowſhip are proper exer.
cife for the learner, but ſeldom occur in real buſineſs;
differences in point of time being uſually adjuſted by an
intereft-accompt.
A
CHA P. IX.
VULGAR FRACTIONS.
FRACTION is a part or parts of an unit, or of
any integer or whole; and is expreffed by two
numbers, one above and the other below a line drawn
between them; as, I
4.
The number under the line fhews into how many
parts the unit or integer is divided; and is called the
denominator, becauſe it gives name to the fraction: the
number above the line fhews or tells how many of theſe
parts the fraction contains; and is therefore called the
numerator.
In the fraction 1. a pound Sterling is the unit, in-
teger,
208
REDUCTION of Chap. IX.
ร
teger, or whole; and the denominator 4 fhews that the
pound is broken or divided into four equal parts, viz.
4 crowns; and the numerator 3 fhews that the fraction
contains three of theſe parts, that is, three crowns;
and fo the value of this fraction is fifteen fhillings.
COROLLARY
I
Hence it follows, 1. When the numerator of a frac
tion is less than the denominator, the value of ſuch a
fraction is less than unity, or the integer. 2. When
the numerator is equal to the denominator, the value of
the fraction is exactly an unit or integer. 3. When the
numerator is greater than the denominator, the value of
the fraction is more than an unit; and fo often as the
denominator is contained in the numerator, ſo many u-
nits or wholes are contained in the fraction. If, there.
fore, the numerator of a fraction be divided by the de-
nominator, the quot will be a number of units or in-
tegers, and the remainder fo many parts.
The numerator of a fraction is to be confidered as a
dividend, and the denominator as a divifor; and the
fraction itſelf may be taken to denote the quotient.
COROLLAR Y II.
From this view of a fraction, it is evident, that if the
numerator and denominator of a fraction be either both
multiplied or both divided by the fame number, the
products or quotients will retain the fame proportion to
one another; and confequently the new fraction thence
arifing will be of the fame value with the given one.
Thus the numerator and denominator of the fraction 2
multiplied by 2 produces, and divided by 2 quots
both which fractions are of the fame value with
Fractions having 10, 100, 1000, or
I with any
number of ciphers annexed to it, for a denominator,
are called decimal fractions; and fractions having any
other denominator are called vulgar fractions.
But the doctrine of vulgar fractions, or the rules there-
in laid down for reducing, adding, fubtracting, multi-
tiplying, and dividing fractions, are equally applicable
to
Chap. IX. VULGAR FRACTIONS.
200
to fractions of every kind; to the decimal as well as the
vulgar; and therefore, in what follows, the decimal and
vulgar fractions are promifcuously uſed in the exempli-
fication of the rules.
Decimal fractions, indeed, may be added, ſubtracted,
&c. and have their operations conducted, not only by
the rules affigned in the doctrine of vulgar fractions,
which are of a general or univerfal nature, and extend
alike to fractions of every fort; but alſo by a more eafy
and ſimple method, peculiar to themſelves, which vulgars
fractions do not admit of; and this may be called the
doctrine of decimals; the explication whereof is referved.
to ſome ſubſequent part of this treatife. In the mean
time, we proceed to defcribe the other claffes or kinds
into which fractions in general are divided.
1. A proper fraction is that whofe numerator is lefs
than its denominator, and confequently is in value lefs
than unity; as
4
2. An improper fraction is that whofe numerator is
equal to or greater than its denominator; and confequent-
ly is in value equal to or greater than an unit; as ‡, 7.
3. A fimple fraction is that which has but one nume-
rator, and one denominator; and may be either proper
or improper; as or 2.
4. A compound fraction is made up of two or more
fimple fractions, coupled together with the particle of,
and is a fraction of a fraction; as of 2, or of 3 of 3.
1
5. A mixt number confifts of an integer, and a frac-
tion joined with it; as 73.
Becauſe in moſt caſes fractions can neither be added
nor fubtracted, till they be reduced, we begin with re-
duction,
Reduction of Vulgar Fractions.
PROB.
I.
To reduce an improper fraction to an integer, or mixt
number.
1
D d
RULE.
210
Chap. IX.
REDUCTION of
{
RULE.
Divide the numerator by the denominator, the quo-
gives integers; and the remainder, if there be any, plat
ced over the divifor or denominator, gives the fraction
to be annexed.
-
EXAMPLE S.
1. 52585 integers, there being no remainder.
2. 437,48, the remainder being 5.
5
ΙΟ
3. 1382=9814, the remainder being 10.
48
4. 23370=173, the remainder being 48.
The reafon of the rule is evident from Corollary I.
For the denominator of every fraction is one whole;
and if the numerator be divided by the denominator, the
quot will be units, integers, or wholes, and the remain-
der fo many parts.
PROB.
II.
To reduce a mixt number to an improper fraction.
RULE.
Multiply the integer by the denominator; to the product
add the numerator: the fum is the numerator of the im.'
proper fraction; and the denominator is the ſame as be-
fore.
EXAMPLE S.
1. 54437; for 54 x 8432
+5
Numerator 437
2. 9810=1382; for 98 x 14=1372
}
+10
Numerator 1382
3.
Chap. IX. VULGAR FRACTIONS.
2 L
3. 173,48=23376, for 173×136=23528
+48
Numerator 23576
The reafon of this rule needs not be affigned, both
the rule and the examples being the reverſe of the fore-
going.
PRO B. III.
To reduce a whole number to a fraction of a given de-
nominator.
RULE.
Multiply the whole number by the given denominator;
and place the product by way of numerator over the gi-
ven denominator.
EXAMPLES.
1. Reduce 9 to a fraction whoſe denominator is 5.
9×5=45; ſo the fraction is 45
2. Reduce 30 to a fraction whofe denominator is 4.
36×4=144; ſo the fraction is 144.
3. Reduce 8 to a fraction whoſe denominator is 1.
8×18; ſo the fraction is .
8
It is eaſy to perceive, that the fractions of this pro-
blem will all be improper; and from Ex. j. we may learn,
that any whole number may be reduced to the form of
a fraction, by making unity the denominator.
The reafon of the rule appears by reverfing the ope-
ration; for if the numerator be divided by the denomi-
nator, it will quot the integer, or whole number.
PROB.
IV.
To reduce a compound fraction to a fimple one.
RULE.
Multiply the numerators continually for the numera
Dd 2
tor
-
242
Chap. IX.
REDUCTION of
tor of the fimple fraction; and multiply the denomina-
tors continually for its denominator.
Ex. I.
EXAMPLES.
8
I5
송
​3 of 4 = 1.
Ex. 2.
6
1 of 3 of 2 = 24.
An operation may fometimes be fhortened by con-
necting the numerators and denominators with the fign
of multiplication, and then rejecting fuch of the factors
above and below as happen to be the fame. Thus,
of 2 of =
4
2×3×4
3×4×5;
and rejecting 3 and 4, both a-
bove and below, the fraction is quickly reduced to 2.
The reafon of the rule will appear by reviewing Ex. 1.;
in which, by corollary II. is equal to
4× 3
5×3
12; and
I5
=
one third of is; confequently two thirds is
I 2
Ι
2 × 4
3 x 5.
COROLLAR Y.
From this problem may be deduced a method of re-
ducing a fraction of a leffer denomination to a fraction of
a greater denomination: namely,
Form a compound fraction, by comparing the given
fraction with the fuperior denominations; and then re-
duce the compound fraction to a fimple one.
EXAMPLE S.
1. What fraction of a pound Sterling is of a penny?
I
I 20
3 d. is of of L. & L.
2 2 25 す ​6
2. What fraction of a C. is of a pound?
8
7 lb. is of of 4 C. =337 C.
/ 28
896
3. What fraction of a pound Troy is of a penny-
weight?
I
I
4
4 dw. is of of 12 16. = 125 lb. Troy.
4 2
PROB.
1
Chap. IX. VULGAR FRACTIONS.
213
3
PROB.
V.
To reduce a fraction of a greater denomination to a
fraction of a leffer denomination.
RULE.
Multiply the numerator of the given fraction, as in
reduction of integers deſcending; and the product is the
numerator, to be placed over the denominator of the
given fraction.
EXAMPLE S.
1. What fraction of a fhilling is of a pound?
Here, as in reduction defcending, I multiply the nu
merator 3 by 20, becauſe 20 fhillings make a pound; as
under.
L.
3×20
= Go fhilling.
4
2. What fraction of a penny is L.?
L.
4 X 20 X 12
5
= 200 d.
3. What fraction of a farthing is L.?
L.
2 X 20 X 12 X 4
3
= 1920 f.
4. What fraction of a pound is C.?
C.
7 × 4 × 28
784 lb.
8
5. What fraction of a grain is lb. Troy?
lb.
5 × 1 2 × 20 × 24
ΤΣ
I 2
A
= 28800 gr.
12
}
6.
214
Chap. IX.
REDUCTION of
}
6. What fraction of an inch is yard?
Yard.
1 × 3 × 12
Yard.
1 × 36
=36 inch. Or
36 inch.
2
2
The reaſon of this rule will appear by obferving, that
every fraction may be confidered in two views. Thus,
may either be confidered as expreffing three fourths of
one unit, or as denoting the fourth part of three units.
Now, if the unit be a pound Sterling, the fraction, in
the latter view, will denote the fourth part of three
pounds; and by reducing the numerator L., to fhil.
lings, we have s.; and again reducing 60 fhillings to
pence, we have 20 d. equal to 40 s. or to L.
60
4
PRO B. VI.
To find the value of a fraction.
RULE.
Reduce the numerator to the next inferior denomina-
tion; divide by the denominator; and the quot, if no-
thing remain, is the value complete.
If there be any remainder, it is the numerator of a
fraction whoſe denominator is the divifor. This frac-
tion may either be annexed to the quotient, or reduced
to value, if there be any lower denomination.
EX.
Chap. IX. VULGAR FRACTIONSONS.
.
215.
EXAMPLES.
1. What is the value of 2 L.?
3
Here I confider L. as expreffing
the fourth part of three pounds Ster-
ling; f I reduce L. 3, the numera-
tor, to fhillings, and divide by the
denominator 4; and as nothing re-
mains, the quot, viz. 15 fhillings, is
the value complete.
L.
20
4)60 (15 s.
4
20
20
1 1
(0)
S.
2=15
2. What is the value of L.?
9
20
Here there is a remainder of 4, which
makes of a fhilling; this I reduce to
value, viz I multiply 4 by 12, the
pence in a fhilling, and divide the pro-
duct by the denominator 16; and the
quot gives 3 d.
16) 180 (11 s.
16.
20
16
4 rem.
12
16) 48 (3 d.
48
(0)
L. s. d.
1/8=11-3
3. What
216
Chap. IX.
REDUCTION of
3. What is the value of £C.?
Here the laſt remainder
7 makes
16
4
21) 64 (3 Q:
63
I
of a pound; which I annex to the
quotient.
28
21) 28 (1,7 lb.
21
(7)
21
C. Q. lb.
1124=3-121
7
EXAMPLE S.
MORE
What is the value of 27 L. Sterling?
Anf. 18 s. 7 d. 123 f.
5. What is the value of 53 of a tun?
Anf. 11 C. I Q. 222 lb.
6. What is the value of
Anf. 4 oz. 10 dw.
To
lb. Troy?
7. What is the value of 41 of a year?
Anf. 299 days 7 hours 12 minutes.
The reafon of this rule is the fame with that in the
preceding problem. It is by the practice of this pro-
blem that remainders in the rule of three are reduced to
value.
PROB. VII.
To reduce a fraction to its lowest terms.
RUL E.
Divide both numerator and denominator by their
greateſt
Chap. IX. VULGAR FRACTIONS.
217
greateft common divifor; the two quots make the new
fraction.
The greateſt common divifor of the numerator and
denominator of a fraction is found by the following
rule.
Divide the greater of theſe two numbers by the leffer ;
and again divide the diviſor by the remainder; and ſo on,
continually, till o remains. The laft divifor is their great-
eft common divifor.
EXAMPLE S.
1. Reduce 78 to its lowest terms.
95
Firſt I find the greateſt common divifor of the numera-
tor and denominator, as follows.
784)952(I
784
168)784(4
672
112) 168(1
II 2
Greateſt common divifor 56(112(2
112
(0)
I now proceed to reduce the given fraction to its
lowest terms, by dividing both numerator and denomi-
nator by 56, the greateſt common diviſor.
56)784(14 new num.
56
224
224
56)952(17 new denom.
56
1
392
392
(0)
So 794 = 14
Ее
(0)
2. Reduce
.
Chap. IX,
218
REDUCTION of
2. Reduce 846 to its lowest terms.
4 ठ
Firſt find the greateſt common divifor, as under.
468)846(I
468
378)468(I
378
90)378(4
360
Greateſt common divifor 18)90(5
༡༠
(0)
Now reduce the given fraction to its lowest terms.
18)846(47 new num.
72
}
18)468(26 new denom.
36
108
126
126
(0)
108
(0)
So 846=47=12/
#
3. Reduce
Chap. IX. VULGAR FRACTIONS. 219
3. Reduce
to its lowest terms.
147)323(2
294
29)147(5
145
2)29(14
2
8
Greateſt common divifor 1)2(2
2
(0)
The greateſt common divifor being unity, the given
fraction is already in its loweſt terms.
MORE EXAMPLE S.
208
4. What is 30 in its lowest terms?
684
5. What is 2542 in its loweſt terms?
2912
6. What is 2728 in its lowest terms?
2976
Ans.
52
1.
Anf. z.
7
8
Ans. 1
ΙΣ
This way of abbreviating or reducing a fraction to its
lowest terms, by finding the greateſt common divifor,
being tedious, you may ufe the following method.
Divide the numerator and denominator of the given
fraction by any number that will divide both without any
remainder, and you have your fraction in lower terms. In
like manner, reduce this new fraction to lower terms ſtill,
and ſo on continually, till no common divifor, except
unity, is found, and you have your fraction in its loweſt
terms.
In order to diſcover what number will divide both nu-
merator and denominator without a remainder, obſerve
the following practical rules or directions.
1. If the units, or right hand figure, of both nume‣
E c 2
rator
220
REDUCTION of Chap. IX.
rator and denominator are even numbers, you may al
ways halve them, or divide by 2.
2)
2)
7)
418 1 + 1 +
196
98
Ex. 1. 784 | 323 | 128 | 28 | 14
95
In the above example I proceed by halving, till the
new fraction is, where 9, the unit of the denomi
nator, is an odd number, fo 2 cannot meaſure both;
confequently neither can 4, 6, or 8, the multiples of
2. By trial I find 3 cannot meaſure both, and ſo neither
can 6 or 9, the multiples of 3. Then I try 7, and find it
meaſures both So the given fraction is reduced to I 7)
and is in its lowest terms, unity being the only number
that meaſures both numerator and denominator.
14
The divifors in the above example may be confidered
as component parts of the greateſt common diviſor, and
accordingly the product arifing from their continual mul-
tiplication, viz. 2×2×2×7 = 56, is the greateſt com-
mon divifor.
2) 2) 2) 2) 2. 2) 3)
Ex. 2. 123 1 203 1 14 | 34 | 33 | 38 | 3 |
92
576
48
72
I 2
2. If the right-hand figure of both numerator and de-
nominator be 5, or if the right-hand figure of the one
5 and of the other o, you may in either caſe divide
be
by 5.
5)
5) 5) 3)
75
5
Ex. 1. 375 125 | 1 | 2 | 3
1125
45
3)
Ex 2,921911 2
I 40
7)
2 17/
2
Ex. 3. 148 128 14
245
3. If there are ciphers on the right hand of both nu
merator and denominator, cut off an equal number of
ciphers from both; for to cut off one cipher is the
fame
Chap. IX. VULGAR FRACTIONS.
221
fame thing as to divide by to, and to cut off two is the
fame as to divide by 100, &c.
10)
2) 3)
Ex. 1. 2018 | 22/21 2
24
100) 5) 7) 3)
210
Ex. 2. 31983 | 38 | 31913
315
Ex. 3. 31000=}
The reaſon of this rule has been already affigned in
Corollary II. viz. When the numerator and denomina-
tor of a fraction are both divided by the fame number,
the new fraction formed by the quots is of the ſame va-
lue with the given one.
The rule for finding the greateſt common divifor may
be thus demonftrated: If any number meaſure both the
remainder of a divifion, and alfo the divifor, it will like-
wife meaſure the dividend; for the dividend conſiſts of
two parts, whereof one is a multiple of the divifor, pro-
duced by multiplying the quot into the divifor; and the
other part is the remainder itfelf: now, fince the fuppo
fed number meaſures the remainder, and alfo meatures
the divifor, and all its mutiples, it will meaſure both
parts of the dividend; that is, it will meaſure the divi
dend.
Thus, in Ex. 1. fince every number meaſures itſelf,
the remainder 56, in the third divifion, will meaſure it-
felf; but it alfo meaſures the divifor 112; and therefore
will alfo meaſure the dividend 168; that is, it meaſures
the remainder and the divifor of the fecond divifion; and
conſequently meaſures the dividend 784; that is, it mea-
fures both the remainder and the divifor of the firft divi
fion; and therefore will alſo meaſure the dividend 952,
and at the fame time is the higheſt or greateſt number
that will do fo.
PROB. VIII.
To reduce fractions of different denominators to a
common denominator.
RULE
222
of Chap. IX.
REDUCTION of
RULE
Multiply the denominators continually for the com.
mon denominator; and multiply each numerator into all
the denominators, except its own, for the feveral nu-
merators.
EXAMPLE S.
1. Reduce and to a common denominator.
4×520, the common denominator.
3×5=15, the firſt numerator.
4× 416, the ſecond numerator.
So the new fractions are 15 and 10.
2. Reduce 3, §, and 7, to a common denominator.
3 × × 8 = 144, the common denominator.
2×6×8 = 96, the first numerator.
5 × × 8=120, the ſecond numerator.
2
7 × 3 × 6 = 126, the third numerator.
The new fractions are,
3. Reduce
6
3 G
I 20
2, and 120
144'
39,7%, to a common denominator.
2: 5
2 × 5 × 7 × IC=700, the common denominator.
1 × 5 × 7 × o 0=350, the firſt numerator.
3 x =
×
X
6 × × 1
X
× 10 = 410, the ſecond numerator.
×10=600 the third numerator.
9 × × 5 × 7=630, the fourth numerator.
New fractions, 450, 420, 900, 930.
MORE
4. 3, 4, 5, 7,
휴​, 송​, 등​,
동​,
5 2
5. 3., 1, 3,
I 5
6.1, 4.1%, 1,
품​. ㅇ​,
7.
EXAMPLES.
720 768, 800.840.
ठ
300,000.
420
315, 128.675, 378.
945 945 945' 945°
140, 400, 504, 280.
When the denominator of one fraction happens to be
an aliquot part of the denominator of another fraction,
the former may be reduced to the fame denominator
with the latter, by multiplying both its numerator and
denominator by the number which denotes how often
the leffer denominator is contained in the greater.
5
5
Thus, += ½ + ½·
ΙΣ
1 2'
Here
Chap.IX. VULGAR FRACTIONS.
223
Here 3 is contained in 12 four times; fo I multiply
both 2 and 3 by 4, and I have 23.
Again,╋十 ​ㄨ​ˊ = ㄔ​ㄨ​ˊ + ㄓ​ㄢ​ˊ +
Sometimes, too, the fraction that has the greater de.
nominator may, in like manner, be reduced to the fame
denominator with that which has the leffer, by diviſion.
Thus, 3+}+}·
7
And 19+3/+½=½+&+√2•
The reaſon of the above rule for reducing fractions
to a common denominator is evident from Corollary II.;
for both numerator and denominator of every fraction
are multiplied by the fame number, or by the fame num-
bers.
After fractions are reduced to a common denominator,
they may frequently be reduced to lower terms, by di-
viding all the numerators, and alfo the common deno-
minator, by any divifor that leaves no remainder, or by
cutting off an equal number of ciphers from both.
The lower fractions are reduced, the more fimple and
eafy will any operation be; if, therefore, it be required
to reduce fractions to the loweſt or leaſt common deno-
minator poffible, it may be done as follows.
The leaſt multiple of all the given denominators is the
leaft common denominator.
Divide this common denominator feverally by the de-
nominators of the given fractions; and multiply each
numerator reſpectively by the quot arifing from its own
denominator; and the products will be the numerators.
The leaft multiple of two or more numbers is found
thus.
Place the given numbers in a line; then divide two
of them, or as many of them as you can, by 2, or 3, or
any ſmall divifor that leaves no remainder; place the
quots, and the numbers not divided, in a line below; a-
gain, divide the numbers in this line, either by the for-
mer, or by fome other divifor, placing the quots, and
the numbers not divided, in a line below; proceed by
dividing, in the fame manner, till the laft quot of every
number
224
REDUCTION of Chap. IX.
{
number be unity; then multiply the divifors into one an-
other continually, and their product is the leaſt multiple
required
EXAMPLE
I.
Required the leaft multiple of 24 and 36?
Divifors 624. 36.
1
21 4.
6.
2 2.
3.
3 1.
3.
I.
´6 × 2 × 2 × 3 = 72, the leaſt multiple.
EXAMPLE II.
Required the leaſt multiple of 24, 36, 54?
Divifors 6 24. 36. 54.
34.
6.
9.
2 4.
ત
2.
3.
21 2.
3.
3 I.
3.
I.
6×3×2×2×3=216, the leaſt multiple.
EXAMPLE III.
Required the leaſt multiple of 2, 4, 6, 10, 12, 15 ?
Divifors 2 2. 4. 6. 10. 12. 15•*
21
3.
ふ
​6. 15.
3
1.
5.
3. 15.
I.
?
i.
5.
f.
I.
2×2×3×560, the leaft multiple.
The
}
Chap. IX. VULGAR FRACTIONS.
225
The reaſon of the operation is obvious. For the con-
tinual product of the divifors uſed in dividing any of the
given numbers is equal to the faid given number; and if
this product be multiplied by the remaining divifor or
divifors, the product will be a multiple of the given num-
ber. Thus, in Ex. 1. the divifors ufed in dividing 24
are 6, 2, 2, and 6×2×2 24, the given number; and
if we multiply this by 3, the remaining divifor, we ſhall
have 6×2×2×372, a multiple of 24. In like man-
ner, the divifors ufed in dividing 36 are 6, 2, 3, and
6×2×336, the given number; and if we multiply
this by 2, the remaining divifor, we fhall have, as be
fore, 6×2×2×372, a multiple of 36, which, at
the fame time, is the leaft common multiple of 24 and
36.
If any of the ſmaller given numbers be an aliquot part
of fome of the higher given numbers, you may, if you
pleaſe, neglect the aliquot parts. Thus, in Ex. 3. you
may neglect 2, 4, and 6, as being aliquot parts of 12;
and by dividing 10, 12, 15, you will have 60 for the
leaſt common multiple, as before.
I now proceed to the exemplification of the above
rule.
I
Ex. 1. Reduce and to their leaſt common deno-
minator.
Divifors 2. 18. 6x 2 x 336, the leaft
2
ત્ય
2.
3.
common denominator.
I
I.
12) 36(3 × 5 = 15 the first numerator.
18) 30( 2 × 7 = 14 the ſecond numerator.
The new fractions are 15 and 1.
I
4
14
Ex. 2. Reduce, 72, 1, 3, 5, 4, to their leaft com-
mon denominator.
Ff
Div.
226
Chap. IX.
ADDITION of
Div. 28. 12. 9. 3. 6. 4.
24.
6. • 9. 3. 3. 2.
22.
3. 9. 3.
3. 1
34.
3. 9.
3. 3.
3
I. 3.
I. I.
2 × 2 × 2 × 3 × 3 = 72,
the common denomi
nator.
I.
8)72(9 × 5=45 first numerator.
12)72(6 × 7=42 fecond numerator.
9)72(8 × 4
32 third numerator.
3)72(24 × 2 = 48
6)72(12 × 5 = 60
fourth numerator.
fifth numerator.
4)72(18 × 1 = 18 fixth numerator.
32 48
72
18
New fractions 4돌​, 속글​, 무릎​, 숲​을​, 속​을​, 녹음​.
72
The method of finding the new numerators may be
thus accounted for. The value of a fraction is not al-
tered by multiplying both its numerator and denomina-
tor by the fame number; but when the common deno-
minator is divided by the denominator of the given
fraction, the quot gives the number into which the de-
nominator of the given fraction is multiplied; and con-
fequently its numerator must be multiplied by the fame.
Thus, in the laſt example, the common denominator,
72, divided by 8, the denominator of the given fraction
§, quotes 9; and fo the new fraction is formed thus,
5
5×
45
; in like manner, the new fraction
7
is
I 2
8xy
72
7×6 42
formed thus,
&c.
12×6 72,
Addition of Vulgar Fractions.
Rule I. If the given fractions have all the fame deno-
minator, add the numerators, and place the fum over
the denominator.
Ex. 1. What is the fum of +3? Anf. §.
Ex.
Chap. IX. VULGAR FRACTIONS.
227
15
Ex. 2. What is the fum of +13?
6
of+
by Prob. VII.
Ans. 1/3=};
2=3,
Ex. 3. What is the fum of +&+7? Ans.
15-17, by Prob. I.
Rule II. If the given fractions have different denomi-
nators, reduce them to a common denominator, by
Prob VIII. then add the numerators, and place the
fum over the commmon denominator.
Ex. 1. What is the fum of 2 + 3 ?
16
I
2+1=10+15, by Prob. VIII.
and 10+15=25.
Ex. 2. What is the fum of + 3 + § ?
3+3+3=3/4+54 +92, by Prob. VIII.
and 24 +54 +12=-3222
28
72
을​을
​66
72
72
721
ΙΙ
and 132199, by Prob. I = 1, by Prob. VII.
72
The operation becomes more fimple, if the given
fractions be reduced to the leaſt common denominator.
The former example wrought in this manner follows.
What is the fum of 5+ 3 + §?
4
3+2+3=12+&+12=}}=1, by Prob. I.
· Rule III. If mixt numbers be given, or if mixt num-
bers and fractions be given, reduce the mixt numbers
to improper fractions, by Prob II; then reduce the
fractions to a common denominator, by Prob. VIII.
and add the numerators.
Ex. 1. What is the fum of 7 + 5 } ?
3
72 +53 = 24 +7, by Prob. II.
8
and 31+2+, by Prob VIII.
and 23 +98=16113, by Prob. I.
I2
Ff2
Ex:
228
Chap. IX.
ADDITION of
Ex. 2. What is the fum of 6+÷?
63=20, by Prob. II
and 3+10+12, by Prob. VIII.
and 400+13=127, by Prob. I.
I 5
I5
5
When mixt numbers, or mixt numbers and fractions,
are given, you may, with greater expedition, work by
the following rule, viz. reduce only the fractions to a
commmon denominator, and add the fum of the frac-
tions to the integers. The above two examples wrought
in this manner follow.
Ex. 1. What is the fum of 73 +53 ?
3
8
5
2 + 3 = 1/2 + 1/2 = 12 = 1/2/2·
3
I 12་
and 7+5+12=13 1/2.
Ex. 2. What is the fum of 63 + /?
10
I
22
3+4=18+12=33 = 17/3
I 5
15
and 6+17=715
I 5
I
Rule IV. If any, or all of the given fractions, bet
compound, firft reduce the compound fractions to fimple
ones, by Prob. IV.; then reduce the fimple fractions to
a common denominator, by Prob. VIII. and add the
numerators.
Ex. 1. What is the fum of 3 of 4 +3
3/4
of, by Prob. IV.
8
and +232 + 45, by Prob. VIII.
I5
and 3+45=72=17, by Prob. I.
Ex. 2 What is the fum of of 3+3 of 1?
I
3
1 of 3 + 3 of 12+2%, by Prob. IV.
and
and
40
18
3+4+, by Prob. VIII.
I 8
120
58
ΤΣΟ
40+15=22, by Prob. VII.
Rule V. If the given fractions be of different deno-
minations, firſt reduce them to the fame denomination,
by Cor. of Prob. IV. or by Prob. V.; then reduce the
fractions,
Chap. IX. VULGAR FRACTIONS.
229
fractions, now of one denomination, to a common de.
nominator, by Prob. VIII. and add the numerators; or
reduce each of the given fractions feparately to value,
by Prob. VI. and then add their values.
EXAMPLE.
What is the fum of 2s and 1. ?
METHOD
I.
2 s. of 1.1. by Cor. Prob. IV.
== 3/ I
7
20
and+3=24560, by Prob. VIII.
8
=
and 24+ 500=5841, 18 s. 3 d. by Prob, VI.
840
METHOD II.
11/11. ====
7 × 20
8
s.140 s. by Prob. V.
and 3+140=8+140, by Prob. VIII.
and +140146s.
18 s. 3 d. by Prob. I. and VI.
METHOD III.
al
3×12
3 s. =
d. = 36 d. =
أو
4
by Prob. VI.
S.
7 × 20
21.
11
=
5. — 1405. = 17-
17-61
8
18—
3
MORE
EXAMPLES.
1. What is the ſum of +3+§? Ans. 17.
2. What is the ſum of 7+11+17? Anf. 253.
3. What is the ſum of 133 +248? Anf. 3814,
or 38 72.
I
4. What is the fum of 3 of 3+2 of 9+3? Anf.
237
280
5. What
230
Chap. IX.
SUBTRACTION of
<. What is the fum of
3231. in value 15 s. 9 d. 3 f.
$20
Anf.
1.+s. + 3d.?
S.
The reaſon of the above rules appears from Axiom XI.
becauſe none but fimilar or like things can be added.
Thus, third parts cannot be added to fourth parts, for
the fum would neither be third parts nor fourth parts;
third parts can only be added to third parts, and fourth
parts only to fourth parts; and confequently fractions
to be added muſt have all the fame denominator. In
like manner, the fraction of a pound cannot be added
to the fraction of a fhilling, or of a penny, being unlike
things; and therefore fractions to be added muſt be all
of the fame denomination.
Subtraction of Vulgar Fractions.
Rule I. If the given fractions have the fame denomi
nator, fubtract the leffer numerator from the greater,
and place the remainder over the denominator.
Ex. 1. From
fubtract 3.
Ex. 2. From 13 fubtract 4.
ΙΟ
4
I 3
6
I 3
Rule I. If the given fractions have different denomi
nators, reduce them to a common denominator, by
Prob VIII.; then fubtract the leffer numerator from
the greater, and place the remainder over the common
denominator.
Ex. 1. From & fubtract 3.
8
2,3 = 12, 12, by Prob. VIII.
4
8
and 1/2 1/2 = 1/2
I
Ex. 2. From fubtract Z.
1%, 7=12, 78, by Prob. VIII.
2,7
I
and 72-70=32, by Prob. VII.
80
8
80
Rule
Chap. IX. VULGAR FRACTIONS.
231
Rule III. If it be required to fubtract one mixt num-
ber from another, or to fubtract a fraction from a mixt
number, reduce the mixt numbers to improper fractions,
by Prob. II.; then reduce the fractions to a common
denominator, by Prob. VIII. and fubtract the one
numerator from the other.
Ex. 1. From 73 ſubtract 5.
I
72, 51=3, 4, by Prob. II.
2
and 31, 11=62, 44, by Prob. VIII.
62
8
and 92-44-18 = 2 = 24, by Prob. I. and VII,
Ex. 2. From 8
26
ſubtract 4.
832, by Prob. II.
and 26, 4 = 130, 12, by Prob. VIII.
20
5
and
I
5
5
I 5
130-12=118713, by Prob. I.
15
In fubtracting one mixt number from another, or in
fubtracting a fraction from a mixt number, you may,
with greater eafe and expedition, proceed thus, viz. Re-
duce only the fractions to a common denominator, and
then work by one or other of the two rules following.
1. If the numerator of the fraction to be fubtracted
be less than the numerator of the fraction from which
you are to ſubtract, then ſubtract the leffer numerator
from the greater, place the remainder over the com-
mon denominator, and to this fraction prefix the differ-
ence of the integers of the two mixt numbers; or the in-
tegral part of the mixt number when a fraction is ſub.
tracted.
Ex. 1. From 73 fubtract 51.
$, 10, 4 by Prob. VIII.
and === by Prob. VII.
9/
and 7
4
5=2.
So the difference of the two mixt numbers is 24.
Ex:
232
Chap. IX.
SUBTRACTION of
Ex. 2. From y ſubtract 3.
8
2, 3, by Prob. VIII.
ΤΣ
and; and to this fraction I prefix the
integral part of the mixt number, and the difference, or
remainder, is y I
Note 1. In fubtracting a fraction from an unit, you
have only to fubtract the numerator from the denomina.
tor; and the remainder, placed over the denominator,
is the difference, or anfwer. Thus, -=: for
5/
1 = 3, and — 3 =}.
I
ड
2
2. If the numerator of the fraction to be fubtracted
be greater than the numerator of the fraction from
which you are to fubtract; in this cafe fubtract the great-
er fraction from an unit borrowed; that is, fubtract the
greater numerator from the common denominator, add
the remainder to the numerator of the leffer fraction,
and place the fum over the common denominator, for
the fractional part of the answer; or, which is the fame
in effect, fubtract the greater numerator from the ſum of
the numerator and denominator of the leffer fraction,
and place the remainder over the common denominator
for the fractional part of the anſwer: then, for the unit
thus borrowed, add to the integral part of the leſſer mixt
number; fubtract this fum from the integral part of the
greater mixt number; and prefix the difference to the
fractional part of the anſwer. But in fubtracting a frac-
tion from a mixt number, for the unit borrowed, take
1 from the integral part of the mixt number, and pre-
fix the remainder to the fractional part of the anſwer.
Ex. 1. From 8 ſubtract 53.
T
중
​1
3,4, by Prob. VIII.
Here having reduced the fractions to a common de-
nominator, I fay, from 3 I cannot; wherefore I fub-
tract from an unit borrowed, viz. I fubtract the nu-
merator 4 from the common denominator 6; and add
the remainder 2 to the numerator of the leffer fraction
3 ;
Chap. IX. VULGAR FRACTIONS.
233
5
3; and the fum 5, placed over the common denomina-
tor, gives for the fractional part of the anſwer. Or,
which is the fame in effect, I fubtract 4 from 3+6=9,
and there remains for the fractional part of the an-
fwer, as before. Then, for the unit borrowed, I ſay,
I borrowed and 5 make 6; which, fubtracted from 8,
leaves 2 of a remainder; and this prefixed to the frac-
tional part, gives 23 for the difference, remainder, or
anfwer.
Ex. 2. From 9
3,
=
I 2
fubtract 4.
ΤΟ
18, 13, by Prob. VIII.
15
ΙΟ
I 5
Here having reduced the fractions to a common de-
nominator, I lay, 1 from 1g I cannot; but, borrowing
an unit, I fubtract the numerator 12 from the common
denominator 15, and 3 remains; which 3, added to the
leſſer numerator 10, gives 13 for the fractional part of
the anſwer. Or, I fubtract 12 from 10+15=25,
and 13 remains; which 13, placed over the com-
mon denominator, gives for the fractional part,
as before. Then, for the unit borrowed, I take from
the integral part of the mixt number; and the remain-
der, prefixed to the fractional part, gives 813 for the
difference, or anſwer.
I 3
15
Rule IV. If it be required to fubtract a mixt number,
or a fraction, from an integer, firſt fubtract the fraction
from an unit borrowed; that is, fubtract the numerator
from the denominator, and place the remainder, as a
numerator, over the denominator, for the fractional part
of the anſwer: then, for the unit borrowed, add i to
the integral part of the mixt number; fubtract the fum
from the given integer; and prefix the remainder to the
fractional part of the answer. But when a fraction is
fubtracted from an integer, for the unit borrowed, take
1 from the given integer, and prefix the remainder to
the fractional part of the anſwer.
Ex. 1. From 14 fubtract 73.
Here I fay, 5-32; fo is the fractional part of
G g
the
234
Chap. IX.
SUBTRACTION of
1
the anſwer: then I fay, I borrowed and 7 make 8, and
8 fubtracted from 14 leaves 6; which I prefix to the
fractional part: fo the difference or anſwer is 63.
Ex. 2. From 12 fubtract 3.
Here I fay, 7-3=4; fo is the fractional part :
then I fay, I borrowed from 12, and 11 remains: fo
11 is the difference, or anfwer.
Note 2. When an integer is given to be fubtracted
from a mixt number, you have only to ſubtract the given
integer from the integral part of the mixt number; and
to the remainder annex the fractional part. Thus,
93—5—43/3•
Rule V. If one or both of the given fractions be com-
pound, firſt reduce the compound fractions to fimple
ones, by Prob. IV.; then reduce the fimple fractions to
a common denominator, by Prob. VIII.; and ſubtract
the one numerator from the other.
Ex. 1. From
4
ſubtract 3 of 2.
6
23
3 of 3, by Prob. IV.
and 4,48, 30, by Prob. VIII.
and 48-30-1830, by Prob. VII.
Ex. 2. From 3 of ſubtract 3 of 4.
3 of 1, 3 of 1 = 2, 3, by Prob. IV.
49
and 2, 240, 18, by Prob. VIII.
중​,
I I
and 40-182261, by Prob. VII.
Rule VI. When the given fractions are of different de-
nominations, firft reduce them to the fame denomina-
tion, by Cor. of Prob. IV. or by Prob. V; then reduce
the fractions, now of, one denomination, to a common
denominator, by Prob. VIII.; and fubtract the one nu-
merator from the other. Or, reduce each of the given
fractions, feparately, to value, by Prob. VI.; and fub
tract the one value from the other.
EX.
Chap. IX. VULGAR FRACTIONS.
235
EXAMPLE.
From 1. fubtract 3 s.
METHOD
I.
// =
s. of 1.1. by Cor. Prob. IV.
3
3 20
I
and 2, 2=180, 8, by Prob. VIII.
I
180-28-1721.14 s. 4 d. by Prob. VI,
and
METHOD
II.
3×20
$ 1.
s.s. by Prob. V.
4
and 60, 3-180,, by Prob. VIII.
으우​,
12'
=
and 180-8=172s. 14 s. 4 d. by Prob. I. and VI,
METHOD III.
S. d.
3×20
3.1. =
S.
s.=00s. = 15-0
4
by Prob. VI.
cylca.
S. =
2 X 12
3
d. = 24 d. = 8
MORE
8
144
EXAMPLES.
1. From fubtract.
ΙΣ
2. From & ſubtract 3.
3. From 7 fubtract 5.
4. From 1 fubtract 4.
5. From 73 fubtract 55.
6. From 127 fubtract .
7. Froni 133 fubtract.
8. From 50 fubtract 97.
9. From 7 fubtract 3.
10. From 15 fubtract 7.
11. From & fubtract of
I
Rem. ₁ = 1.
3
I 2
Rem. .
Rem. 22.
Rem. .
3.
Rem. 1.
Rem. 1223.
Rem. 1252.
Rem. 407.
Rem. 6.
Rem. 83.
.
Rem. .
G g 2
12. From
236
MULTIPLICATION of Chap..IX
12. From 3 of fubtract of. Rem. .
ΣΤ'
13. From 1. fubtract Zd.
Rem. 21.6s. 7 d.
1982 71
The reafon of the rules in fubtraction is the fame as
in addition. For like things only can be fubtracted from
one another; and therefore in fubtraction the fractions
muſt have all the fame denominator, and be of the fame
denomination.
Multiplication of Vulgar Fractions.
In multiplication of fractions there is no occafion to re
duce the given fractions to a common denominator,
as in addition and fubtraction: only if a mixt number be
given, reduce it to an improper fraction; if an integer
be given, reduce it to an improper fraction, by putting
an unit for its denominator; if a compound fraction bẹ
given, you may either reduce it to a fimple one; or, in
ftead of the particle of, infert the fign of multiplica.
tion: then work by the following
RUL L E.
Multiply the numerators for the numerator of the pro-
duct, and multiply the denominators for its denomi-
nator.
EXAMPLE S.
1. 3 × 3 = 1/8/8.
5
2.2 × 53 = 2 × ==4=44.
17 I 2
3. 4×8=4×4 = 5.
88
4. 3/4 of 2 × = 1/2 × 3 = 3; = {//·
5
Or, × 2 × =38 = 18.
3
NOTE S.
1. If any number be multiplied by a proper fraction,
the product will be lefs than the multiplicand; for mul
tiplication is the taking of the multiplicand as often as
the multiplier contains unity; and confequently, if the
multiplier be greater than unity, the product will be
greater
Chap. IX. VULGAR FRACTIONS.
237
greater than the multiplicand; if the multiplier be unity,
the product will be equal to the multiplicand; and if
the multiplier be lefs than unity, the product will, in
the fame proportion, be lefs than the multiplicand.
Thus, fuppofing the multiplier to be or, the pro-
duct, in this cafe, will be equal to one half or to one
third of the multiplicand.
54
242
216
108
2. Mixt numbers may be multiplied without
reducing them to improper fractions, by work-
ing as in the margin; where I firſt multiply the
integral parts, viz. 54 by 24; then I multiply
the integral parts cof-ways into their al-
tern fractions, viz. 54 by 1, and the product
27 I fet down; in like manner I multiply 24
by I. and the product 6 I likewife fet down;
then I add; and to the fum I annex, the 1329
product of the two fractions.
4
27
6
3. In multiplying a fraction by an integer, you have
only to multiply the numerator by the integer, the put-
ting for the denominator being only matter of form.
And to multiply a fraction by its denominator is to take
away the denominator, the product being an integer,
the fame with, or equal to the numerator.
1/2
×8 = 7. For 7x=7.
8
I
4
Thus,
4. If the numerators and denominators of two equal
fractions be multiplied crofs-ways, the products will be
equal. Thus, if 3, then will 3×129×4;
for multiplying both by 9, we have 3=94; and
=
4
I 2
=
multiplying thefe by 12, we have 3 × 129× 4.
Hence, if four numbers be proportional, the product of
the extremes will be equal to the product of the means:
for if 3 9 4: : 12, then 3; and it has been
proved that 3 × 1 2 — 9 × 4• Therefore if, of four-
proportional numbers, any three be given, the fourth
may easily be found, viz. when one of the extremes
is fought, divide the product of the means by the
given extreme; and when one of the means is fought,
divide the product of the extremes by the given mean.
5. In multiplying fractions, equal factors above and
below
238
MULTIPLICATION of Chap. IX.
below may be dafhed or dropt. Thus of x of =
1 × × ×; and dropping the factors 2, 3, 4, both
above and below, the product is . In like man-
ner, to facilitate an operation, a factor above and ano-
ther below may be divided by the fame number: Thus,
9 × 532=4×2 =
X
12
X
5
5
14
7 × 2
•
numerator for another: Thus,
5
14'
×
8
I
5
Or we may exchange one
X
x
5
I 2
= 1/2 × = 1 × 3
5
Ι
6. To take any part of a given number, is to mul
tiply the faid number by the fraction. Thus, & of 320
is found thus, 320×320x40=200=200.
In like manner, 3 of 45, is × 458 × 303 = 3
3
× 363 = 1 × 121 = 121 = 30.
121 30. Hence, to reduce a
compound fraction to a fimple one, is to multiply the
parts of it into one another.
X
I
7. If a multiplicand of two or more denominations be
given to be multiplied by a fraction, reduce the higher
part or parts of the multiplicand to the loweſt ſpecies,
and then multiply. Thus, to multiply 81. 102 s. by 3,
I fay, 81.8 x 20 s. 160 s. and 160 + 10317035.
-683, and 3×683—136611312s. L. 5:13:10.
Or, without reducing, you may multiply the given mul
tiplicand by the numerator of the fraction, and divide
the product by the denominator.
4
MORE
by
1. Multiply
8
2. Multiply 7 by 4.
EXAMPLES.
Prod. 45.
Prod. 614.
3. Multiply 8 by 92.
Prod. 84.
4. Multiply 64 by 8.
Prod. 542.
5. Multiply 9 by 1 of 2.
Prod. 48.
6. Multiply 12 by 2 of 4.
Prod. 7.
7. Multiply of 3 by 2 of 4 of 5. Prod.
8. Multiply 13 by 18.
9. Multiply L. 3:12:6 by 2. Prod. L. 2: 14:47.
Prod. 13.
The
Chap. IX. VULGAR FRACTIONS.
239
The reaſon of the rule may be fhewn thus: ×
for, and of is; and confequently
12
12 is 185.
8
8
of
I 5 The truth of the rule may alſo be proved thus: Af-
fume two fractions equal to two-integers, fuch as, and
, equal to 2 and 3, and the product of the fractions
will be equal to the product of the integers; for
=486, and 2 × 3 = 6.
х
Divifion of Vulgar Fractions.
In divifion of fractions, if a mixt number be given,
reduce it to an improper fraction; if an integer be given,
put an unit for its denominator; if a compound fraction
be given, reduce it to a fimple one, and then work by
the following
RULE.
Multiply cross-ways, viz. the numerator of the divi
for into the denominator of the dividend, for the deno-
minator of the quot; and the denominator of the divifor
into the numerator of the dividend, for the numerator
of the quot.
EXAMPLE S.
I
1. 3) 4 (12 = 1/2=1}•
8
2. 3) 44 (= 3) 4 (12=512 = 53.
3. 43) 71 (= 1334) 7 (21/12 = 33.
I
4. 34) 6 § (=-43) 53 (212 =214=226.
5. 4) 8 (= 4) (40 10.
÷ =
6.7) 2 (= 7) 2 (23.
=
7. 21) 15 (= ½) 15 (30 = 6.
I
(등
​8.5) 10 (=) 31 (25 =
5
2 15.
9. 7) 3 of 3.(= 7) 14 (13 8 = 4•
ΤΟ
10. 1 of 2) 41 (= 1 ) 2 ( 2 ²
(2) (22² = 12.
II. of +) 8 (옳​)을 ​(그룹​으
​3 4) = 15.
12. 2 of 3 of 1) # of & (=24) 30 (48023.
6 좋음
​NOTES.
240.
Chap. IX.
DIVISION of
NOTE S.
1. Instead of working divifion of fractions as taught
above, you may invert the divifor, and then multiply
it into the dividend. Thus, in Example 1. inſtead of
3)4(12, you may fay, 2x-12=12=13
I
ΙΟ
:
I
2. If any number be divided by a proper fraction, the
quor will be greater than the dividend for in divifion
the quot fhews how often the divifor is contained in the
dividend; and confequently if the divifor be greater than
unity, the quot will be less than the dividend; if the di-
vifor be unity, the quot will be equal to the dividend;
and if the divifor be lefs than unity, the quot will, in
the fame proportion, be greater than the dividend. Thus,
fuppofing the divifor to be, or, the quot in this caſe
will be double or triple of the dividend.
I I
3
3. To divide a fraction by an integer, is only to mul
tiply the integer into the denominator of the fraction, the
numerator being continued. Thus, 7). See Ex. 6.
4. A mixt number may fometimes be divided by an
integer, with more cafe, in the following manner. Di-
vide the integral part of the mixt number by the given
integer and if there be no remainder, divide likewife
the faction of the mixt number by the given integer, and
annex the quot to the integral quot formerly found.
Thus, in working Ex. 8. by this method, I fay, 5)10(2,
and 5); and fo the complete quot is 2, as be.
fore. But if, in dividing the integral part, there happen
to be a remainder, prefix this remainder to the fraction
for a new mixt number; which reduce to an improper
fraction then divide the improper fraction by the given
integer; and annex the quot to the integral quot for-
merly found. Thus, if it be required to divide 15 by
8, I fay, 8)15(1, and 7 remains; which 7, prefixed to
the fraction, gives 73 for a new mixt number; and this,
reduced to an improper fraction, is 31, and 8)21(31:
fo the complete quot is 3
:
Ι Ο
I
32
I
5. If the factors of the numerator and denominator
of the quots, inſtead of being actually multiplied, be
only connected with the fign of multiplication, it will be
eafy
Chap. IX. VULGAR FRACTIONS.
241
eaſy to drop fuch factors, above and below, as happen
to be the fame, thus: 3) of 3 (1 x 35 x 3
4 × 4 × 3
3 ×
4×4
8 5×8
16
40
Or a factor above and below may be divided by
6×7 7
the fame number, thus: 8)72(5x12
X
5×2
7. Or
the factors of the numerator of the quot may be ex-
3×5
changed, thus: 3) §
2xY
5×3_5
2×9 2×3
6520
6. To divide an integer by a fraction, is to divide the
product of the denominator and integer by the numera-
tor, thus: 4) 8 (
#) (=
5×8
-= 5 × 2 = 10.
=5×
See Ex, 5.
4
7. If the divifor and dividend have the fame denomi-
nator, you have only to divide the numerator of the di-
vidend by the numerator of the divifor, thus: ) = 3;
for
8 × 3
5×8
8. If a dividend of two or more denominations be
given to be divided by a fraction, reduce the higher part
or parts of the dividend to the loweft fpecies, and then
divide. Thus, to divide 61. 93 s. by 3, I fay, 61.=
6 x 20 s. 120; and 120+9=1292.12; and
3) 512 (1557 — 19485 91. 145. 7 d.
8
Or, Divide the given multiplicand by the numerator
of the fraction, and multiply the quot by the denominator.
EXAMPLE.
Divide L. 276: 16: 8 among four men, A, B, C, D,
fo that A, B, C, may have equal fhares, and D only two
thirds of one of their ſhares.
I
1+1+1+3=3+3+3+3=11
L. 5. d. L. S. d.
L. S. d.
11) 276 16 8 (25 3 4×3 = 75
JO
A.
× 3 = 75
75
10
B.
× 3 = 75
10
C.
Hh
×2=50
50 6 8 D.
Proof 276 16 8
MORE
242
Chap. IX.
DIVISION of
MORE EXAMPLES.
1. By
divide §.
2. By 3 divide 2 2.
3. By 9 divide 19.
4. By 5 divide 18.
5. By 7 divide of 5.
6. By divide 121. 6s.
2
8 d.
I
Quot 25= 1 24
Quot 1.
25'
Quot 2 z.
Quot 37.
Quot 45.
Quot 11840 d. 161. 8 s.
10 3 d.
The reaſon of the rule will appear by confidering,
that the method here ufed is nothing elſe but the redu
cing the divifor and dividend to a common denominator,
and then dividing the one numerator by the other.
Thus, 3) (3, for reducing the divifor and dividend to
a common denominator, we have 2).
8
The truth of the rule may alſo be proved by affuming
two fractions equal to two integers, fuch as, and 10,
equal to 2 and 4, and the quot of the fractions will be
equal to the quot of the integers. Thus, )( =
2, and ¿)4(2.
6
Practical Questions.
Quest. 1. The leffer of two numbers is 368; their
difference is 113: What is the greatest?
25
Anf. 48772.
Queſt. 2. What number added to 4, will give 12
Anf. 737.
ΙΟ
15
Quest. 3. A has 7 of a fhip, and B has of the
fame veffel: Which of them has the greateſt ſhare? and
what the difference? Anf. A has the greateſt ſhare;
and the difference of their ſhares is.
Quest 4 Three purfes contain 1301.; in one
purfe is 55g1 in another 42 41.: How much is in the
third purfe? Anf. 3231.
15
Queft. 5. What is & parts of 130?
Quest 6. What number multiplied by
353? Ans. 42 13.
Anf. 81.
will produce
The
Chap. IX. VULGAR FRACTION S.
243
The Simple Rule of Three in Vulgar Fractions.
The queſtion is ftated as formerly taught in the rule
of three. The extremes muſt be of one denomination.
Reduce mixt numbers and integers to improper fractions,
compound fractions to fimple ones, and then work by
the following rule, viz.
Multiply the fecond and third terms, and divide the
product by the firſt term: that is, multiply the numera-
tor of the firſt term into the denominators of the ſecond
and third, for the denominator of the answer; and mul
tiply the denominator of the first term into the numera-
tors of the ſecond and third, for the numerator of the
anfwer.
I. Direct Questions.
Queſt. 1. If a yard coſt §1. what willyard coft?
4 × 5 × 9
Anf.
3 × 8 × 10
21.=
3 X 20
4
6
rd. L. rd.
If 2 : 3 :: 1.
5×9
3 × 2 × 10
5. === 155.
9
3
3 × 2 × 2 2 × 2
ΙΣ
12
Queſt. 2. If § yard coft 1. what will yard coſt?
rd. L. rd.
If ½ : 3/
: // : : 11/11/1
12
6×2×11
XII
I I
Anf.
==
풍 ​1.
5 × 3 × 12
5 × 3 × 2
5×3
I I X 20
s. 220
14 s. 8 d.
15
Quest. 3. If 3 yards coft 2 what will 14 yards
coft?
Hh a
rds.
244
The Rule of THREE in Chap. IX.
:
14
Yds. L.
If 3 2
rds.
2.
3 : 14 ::
50.
L.
s. d.
I X 14 × 59
7 × 59
Anf.
11
=4131=13-15-4.
X X
3 × 5 × 4
3 × 5 × 2
MORE EXAMPLES.
Queſt. 4. If 2 & lb. tobacco coft 3s. what will
2424 lb cost? Anf. L.15:8:32.
Queſt. 5. If 3 of C. fugar coft 1. what will
82 C. cost? Anf. 262 1. 8 s.
pieces of filk, cach
ells, at 6 s. 2 d. per ell: Required
pieces at that rate? Anf. 261.
Quest. 6. A mercer bought 3
piece containing 24
the value of the 3
3 s. 4 3 d.
2
Quest. 7. If 2 oz. filver coft 2 s. what will be the
price of 11 lb. at that rate? Anf. 3.1
I
Queſt 8. ` If 1 ½ lb of gold is worth 611. Sterling,
what is a grain worth at that rate?
Anf 1 d.
Quest. y. If 3 yard of filk coft of 1. what is the
동
​price of 15 ells Flemish? Anf L. 9:15:10.
2
Queſt. 10. If 3 of 3 lb. of cloves coft 6 s. 2 d. what
coft the C. at that rate? Anf. L. 6y: 6:8.
II. Inverse Questions.
Quest. 1. If 3 yard of cloth that is 2 yards wide,
will make a garment, how much of any other cloth
that is yard wide will make the fame garment?
Bread. len.
Bread.
953
3
:: 24.
5 ×? × 2
5 × 2
X
Anf.
Na
2 yards.
X
3 × 4 × I
4
Quest. 2. If I lend my friend 481. for 2 of a year,
how much ought he to lend me for of a year?
ΤΣ
Yea.
Chap. IX. VULGAR FRACTIONS.
245
Yea. L.
rea.
ΤΣ
152:48
48 :: 3.
Anf
12 × 48 × 3 =
12 X 12 X 3
432861, 8 s.
5 × 1 × 4
5
MORE EXAMPLES.
Quest. 3. If yard of cloth that is 2 yards wide,
will make a coat, what is the breadth of the cloth
whereof yard will make the fame coat? Anf. 50
yard, or 3 qrs 2 nails.
56
63
Quest. 4. How may inches in length, of a plank that
is inches broad, will make a foot fquare? Anf. 16
inches in length.
9
Quest. 5. If the penny-loaf weigh 10 ounces when
the bushel of wheat cofts 43 s. what ought the penny-
loaf to weigh when the bufhel of wheat cofts 82 s.?
Anf. 518 ounces.
2
Queſt. 6. If 12 men do a piece of work in 10 days,
in how many days will 6 men do the fame? Anf. In
21 days.
The Compound Rule of Three in Vulgar Fractions.
་
Quest. 1. If 2 acre of grafs be cut down by 2 men in
day, how many acres fhall be cut down by 6 men in
3 // days?
Men. acr.
2244
::
Men.
읏​.
day 2/20
3
I
days.
45
: 4 ::
50
3 x 3 x 60
3 × 60
3 × 15
Anf.
=45=11acr.
X
4 × 4 × 3
4 X 4
4
Or
thus:
3 × 1 × 3 × 6 × 10
3×6×10
Anf.
11
45
II
2 × 2 × 4 × 1 × 3
acres.
2 × 2 × 4
3 × 3 × 5
2 × 2
Queſt.
11
246
Chap. X.
Rules of PRACTICE.
Quest. 2. If 4 men caft a ditch 8 feet long, 3 deep,
and 24 broad, in y days, in how many days will 8 men
feet long, 63 deep, and 4
caſt a ditch 25
I
Men. days.
b. d. *8
231. 8.
: 3 3 : 1.8 1/1/1.
ΙΟ
Men.
: 9/1/1 :: 4
*
broad?
d.
b.
25 ½ 1. : 6 3 : 4 1.
8 : 28 :: 4
옷​: 물 ​: 꽃
​9 × 10 X 17 × 8
4 × 3 × 2 ×
X X
I
521 : 20 : 2.
3 × 5 × 17 x 2
510 firft term.
X
1
I
=2060 third term.
4 × 51 × 20 × 9
I X 2 X
༨
× L
210: 28:: 2060
2 X 5 1 × 10 × 3
1
1 × 28 × 3060
14 X 1020
1428056 days.
510 × 3 × I
X
255
Or thus:
4× 3 × 2 × 1 × 28 × 4 ×
9 × 10 × 17 × 8 × 3 × 1 ×
4 × 3 × 2 × 28 × 4 × 51 X 20
1 × 20 × Q
2 ×
ܐ x
2 X 28 X
X 51 X 2
28 × 4 × 51
10 X 7 × 8 × 3 × 2 × 3
14 × 4 × 51
× 2
17 X 2 X
× 3 × 2
14 X
I
× 4
17 × 3 × 2
17 × 3
17
I
56 56 days.
CHAP. X.
Rules of Practice.
HEN the firft term of a queſtion in the rule of
WHE
three happens to be unity, the anſwer may
frequently be found more ſpeedily and eaſily than by a
formal ftating or working of the rule of three; and the
directions to be obferved in fuch operations are called
Rules of Practice.
The rules of practice naturally follow the doctrine of
vulgar
Chap. X.
247
Rules of PRACTICE.
vulgar fractions, the operation being nothing elſe but a
multiplying the number whofe price is required, by ſuch
a fraction of a pound, of a ſhilling, or of a penny, as
denotes the rate or price of one.
Thus, if the price of 24 yards, at 6s 8 d. per yard,
be demanded, the anſwer is found by multiplying 24
by, the fraction of a pound equivalent to 6s. 8 d. viz.
24x!=31=81
Hence it is obvious, that to, multiply a number by a
fradion whofe numerator is unity, is to divide the ſaid
number by the denominator of the fraction. But if the
numerator of the fraction be not unity, you muſt firſt
multiply the given number by the numerator, and then
divide the product by the denominator. Thus, if the
rate be 13 s. 4 d. 1. the price of 24 yards is found
by faying, 24x4161.; or take of the given
number twice.
48
When the fraction denoting the rate happens to be
compound, the product or anfwer is found by dividing
the given number by one of the denominators of the
compound fraction, the quot by another, and the next
quot by the third, &c. Thus, if the rate be 2 far-
things of of the price of 1440 yards is
을
​1
found by faying, 1440 7.0, and 220 60, and
2=31.
I 2
1
20
2
When the rate is expreffed by two or more ſimple
fractions, connected with the fign+, the product or
anſwer is found by dividing the given number fuccef-
fively by the feveral denominators, and then adding
the quot. Thus, if the rate be 3 s.+2. the
price of 80 yards is found by faying, 208, and
204, and 8+ 4 = 12).
I
ΤΟ
I
The fractions equivalent to any number of farthings
under 4, to any number of pence under 12, and to any
number of fhillings under 20, are exhibited in the fol-
lowing tables.
TABLE
248
Chap. X.
Rules of PRACTICE.
TABLE I.
of a fhilling.
Farthings. of a penny. | of a ſhilling.
I
2
3
TABLE II.
HIGHOMH
of a pound.
옷
​of I
1 of
1
of
12
20
1
of I
ΤΣ
3 of 12
of
ΙΣ
I 2
20
I
ZO
I2
ofof
I of
TABLE III.
Pen. of a fhill. s. d of a pound.
I
3
4
ΤΣ
тло
Si 2 MESO NO
8
+++++
||
I 8
20
1
2
2
3 4 ठ
mt i van
ง
•
9
10
I
8
7, or 1
I O'
s. d. of a pound.
9
ΙΟ
I
12
1 2 3 m +
13
13 4
14
2+2/10 or $15
I
3
ΙΟ
.6
17
I
4 + 25/0
5
To'
5
or
Or 플
​3 + 22/0
ΙΟ
6
or
10/80 + 2/1/00
I O
Ι
7
1 + 2/6, or 30
ΤΟ
20
8 or 4
TO'
ΤΟ
20
18+ 2/5
9/2/
19 1/+25
+
I
18
ur /
The fractions in Table II. become compound frac
tions of a pound, by annexing (of) to each of them.
Thus, 1 d. is of
&c.
I
12
2
I
I
1.; and 5 d. is
of ofl.
+ ½
20
The variety that occurs in the rules of practice arifes
chiefly from the different rates, or prices, of one thing,
as a yard, a pound, an ounce, &c. and may be reduced
to the eight cafes following, viz.
The rate may be, 1. Farthings under four. 2. Pence
under twelve. 3. Pence and farthings. 4. Shillings
under twenty. 5. Shillings, pence, and farthings. 6.
Pounds. 7. Pounds, fhillings, pence, and farthings.
8. The given number may conſiſt of integers and parts.
CASE
Chap. X.
249
Rules of PRACTICE.
1
CASE I.
When the rate is farthings, under four.
RULE.
Divide the given number by the denominator of the
fraction denoting the rate, as contained in Table I. viz.
if the rate be or 2 farthings, divide by 4 or 2, the
quot will be pence; and the remainder, in dividing by
4, will be farthings, and in dividing by 2, it will be
I halfpenny then divide the pence by 12, the quot will
be (hillings, and the remainder pence: laftly, divide the
fhillings by 20, the quot will be pounds, and the re-
mainder fhillings. But if the rate is 3 farthings, first
multiply the given number by the numerator 3, and
then divide as above directed.
I
Ex. 1.
EXAMPLE S.
4859, at 1 f.
I
3/21214-3 f.
ΙΣ
I 10/1-2 d.
20
(L.5 1 23.
Ex. 3.
1753, at 3 f.
3
25259
21314-3f.
I 109-6 d.
20
L.5 9 63.
H24+
Ex. 2.
8347, at 2 f.
I
4173-1d.
234/7-9d.
L. 17 7 9/1
Or, becauſe 3 f. = ½ d.+4d. or
= d.+ 1 of d. you may work
Ex. 3. either of the two ways fol-
lowing.
1753, at 3 f.
I
1753, at 3 f.
876-2 f.
438-1 f.
2
14
876-2 f.
438-1 f.
ΙΣ
1314-3
I
༡
109-6
L.5
9
9 63.
I i
L. 5 9 63.
More
121; 14—3 f.
I
2010/9-6d.
250
Chap. X.
Rules of PRACTICE.
More ways of working the above examples might be
affigned, but the moſt eaſy and fimple methods are the
beft.
CASE II.
When the rate is pence, under twelve.
RULE.
Divide the given number by the denominator of the
fraction denoting the rate, as contained in Table II.
and you have the anſwer in fhillings; which reduce into
pounds, by dividing by 20.
EXAMPLE S.
Ex. I.
I
I2
8:8, at 1 d.
I
20
2
6/8 - 2 d.
-10
1
L. 3 8 2
Ex. 2.
5316, at 2 d.
2 886
L.44 6
Ex. 3.
879, at 3 d.
21/9-9
L. 10 19 9
I
20
20
d.
2 d.
3 2
2
2
HN HO
I
Ex. 4.
3097, at 4 d.
1032-4
L.51 12 4
Ex. 5.
439, at 5 d
109- 9
73- 2
182-11
L. 9
2 II
Ex. 6.
78642, at 6 d.
3932/1
L. 1966 I
Es
Chap. X.
251
Rules of PRACTICE.
Ex. 7.
587, at 7 d.
4 d.
I
3
1958
3 d.
146-9
20
342-5
6 d.
4 d.
12H3
Ex. 10.
386, at 10 d.
193
128-8
I
20
32/1-8
4 d
4 d.
L. 17 2 5
Ex. 8.
836, at 8 d.
278-8
278-8
L. 16 I 8
Ex. II.
534, at 11 d.
I I
443
ddd
d.
178
d.
178
4
133-6
I
20
5517-4
I
20
48|9—6
L. 27 17 4
L. 24 9 6
ㅎㅎ
​6 d.
3 d.
20
Ex. 9.
417, at gd.
208-6
104-3
312-9
L.15 12 9
I
Note 1. The remainders at the first diviſion in all
the above examples is the fame with the rate. Thus, in
Ex. 1. every remainder is 1 d. ; in Ex. 3. every remain-
der is 3d.; and in Ex. 5. every remainder, in dividing
by 4, is 3 d.; and, in dividing by 6, every remainder
is 2 d. &c.
Note 2. In all the examples wherein the rate is no a-
Liquot part of a fhilling, the queſtion may be folved as
I i 2
many
252
Chap. X.
Rules of PRACTICE.
1
등
​many different ways as you can find different fractions
equivalent to the rate. Thus,d.ss.or
= s. 4 of 5/5s. Or 5 d. of half a crown, and 1
+
half crow = 1; that is, 5 d. — // of.; and ac-
cordingly Ex. 5. may alſo be folved the three ways fol
lowing.
439, at 5d|439, at 5 d.
439, at 5 d.
I
12
146 — 4
36— 7
+/+
146— 4
1100
73 h. cr. 5d.
h.cr.
36-7
L. 9
2 II
182—1 F
I
20
I
2.0
18|2—1 1
L9211
L.9 2 11
I
1
2
In like manner, 9 d. = ½s. + ½ of ½ s. or 9 d. =
11. + 1 of 70; and accordingly Ex. 9. may alſo be
folved other two ways, as under.
40
I
40
I
414
2 417, at 9 d.
124
208-6
104-3
200
31/2-9
L. 15 12
9
I
40
417, at 9 d.
HIN
10—8—6
5-4-3
L.15 12 9
if
Note 3. When the rate is 11 d. you may work as
it were 12 d.; that is, from the given number you may
fubtract of itſelf, and the remainder will be the price
I2
in fhillings. Thus,
534 s.
ΙΣ
44 s. 6d. 489 s. 6 d.
= —
44 s. 6d. and 534 s.
L. 24:9:6. See Ex. 11.
CASE
III.
When the rate is pence and farthings.
RULE.
Chap. X.
253
Rules of PRACTICE.
RUL E.
The pence muſt be fome aliquot part of a fhilling,
and at the fame time the farthings fome aliquot part of
the pence; and if they be not fo given, divide the pence
into two or more fuch parts, ſo as the farthings may be
ſome aliquot part of the loweſt divifion of the pence.
Then, beginning with the higheſt divifion of the pence,
divide by the denominators of the fractions denoting the
aliquot parts.
EXAMPLE S.
Ex. I.
Ex. 3.
d. I
ΙΣ
532, at 14 d.
1
1753, at 13 d.
1 d. 1
id.
1 d.
1½d.
rld.
20
I 2
플
​Ha
I
ΣΟ
-100
20
44-4
II- I
55-5
L.2 15 5
Ex. 2.
1753, at 1½d.
146-1
73-0
219-1/3/20
I
L. 10 19 1/2/2
Or rather thus.
1753, at 1 d.
21/01/12/20
L. 10 19 11/
I
20
219-1/12/2
36-611
255-73/
L.12 15 7.
3
Ex. 4•
I d. 859, at 1 d.
I
I 2
d.
I
20
71-7
8—11 - 1/1/ f.
86-13/2
L. 4 0 6 1f
Ex. 5.
2 d. 485, at 24 d.
d.
2
80-10
10- 1/4/
9/2-114
L. 4 10 11
Ex
254
Chap. X.
Rules of PRACTICE.
Ex. 6.
3 d. 1/20
♡ Hk
ૐd.
20
3471, at 31 d.
4 d.
867-9
144-7
101|2—4/2/2
L. 50 12 41
Ex. 7.
4d. | 475, at 4¾-d.
cafe parley and fando
4 d.
1}d.
3d.
3
d.
miky Hlomker
181
1 58—4
19-9
I
9-103
20 1818-01
1348
I
༡༠
L.9 8 04
Ex. 8.
976, at 5 d.
325-4
122
447-4
L. 22 7 4
Ex. 9.
3506, at 6 d.
3 d.
3 d.
334+
6 d.!
2 d.
21
d.
Boller mi kimekt
20
1/2 H/3H/00
20
6 d. 11/12
3 d.
3d.
-244
dd
ㅎ
​6 d.
2
876-6
876-6
3 d.
4
d.
12
73-c
Id.
I
20
18216-01
L. 91 6 0
10/2/2
HAHIN HO
20
Ex. 10.
520, at 73 d.
173-4
130
32-6
33/5-10
L. 16 15 10
Ex. II.
783, at 8 d.
391-6
130-6
16-33/2
538-33
L.26 18 32
Ex. 12.
7012, at 93 d.
3506
1753
438-3
5f97-3
L.284 17 3
Ex. 13.
1
137, at 10 d.
68—6
34-3
17—1/1/20
119-10/1/20
L. 5 19 10/2/2
}
Ex.
Chap. X.
255
Rules of PRACTICE.
Ex. 14.
2753, at 11 d.
6 d.
I d.
I d.
d.
1376-6
917-8
229-5
57-44
2580-111
20
L. 129 0 114
EXPLICATION.
In Ex. 1. I work firft for 1d.; which being
2
s. I di.
vide the given number by the denominator 12, and the
quot is fhillings, and the remainder pence: then, be-
cauſe farthing is 4 d. I divide the former quot by 4, and
the fum of the quots is the price in (hillings; which I di-
vide by 20.
1
In Ex. 2. the rate 1 d. being an aliquot part of a
fhilling, the fecond method is fhorter and better than the
first.
In Ex. 3. I work firſt for 1½ d. and then for 4 d. by
taking of the former quot.
4 d. then I work for d
of the first quot: laftly, I
of 1 d. I take of the fe-
In Ex. 7. I work firſt for
which being of 4d. I take
work for d.; which being
cond quot; and, adding the three quots, I have the price
in fhillings.
2
In Ex. 8. I divide the rate into two parts, viz. 4 d.
and 1; each of which being an aliquot part of a fhil.
ling, I take firſt, and then, of the given number, and
add the quots.
I
12
In Ex. 9. I divide the 6 d. into two parts, viz. two
3 d. and then the 4 d. is of 3 d. which makes an eafy
operation; but had I taken of the given number for
6 d. then 4 d. would have been of 6 d. and 24 would
have been a troubleſome divifor.
4
24
Note 1. The above examples, and all others of the
{
like
25.6
Chap. X.
Rules of PRACTICE.
1
t
like kind, may alſo be folved by affuming fractions of a
I
Thus, in Ex. 6. the rate
d. and 3 d. l. and
d. 1 of 1.
I
81+ ½ 80
80
pound equivalent to the rate.
3d. is divided into 3 d. and
½ 3
d = of 3 d.; wherefore 3
Again, in Ex. 10. the rate 73 d. is divided into 4 d. 3 d.
and d. and 4 d. 1. and 3 d. 1. and 2 d=
2
d. 1+1+1 of 1.
73 =
And theſe two examples wrought in this manner will ſtand
4
of 3 d.; and fo 7
as under.
I
I
80
I
I
80
3 d.
22 d.
I
8
80
Ex. 6.
Ex. 10.
I
3471, at 31 d.
d.li
I
4 d. 52, at 72 d.
43-7-9
3 d. 8%
8-13-4
7-4-7
d.
6-10
1-12-6
L.50 12 4
L.16 15 10
Note 2. Some, in working the above, or like exam
ples, proceed in the following manner.
Ex. 6.
L. S. d.
86 15 6
3471, at 12 d. is 3471, or 173 11
at 6 d. is
at 3 d. is
atd. is
3471, at 3 d. is
43 7 9
7 4 72
L. 50 12
4/2/2
Ex. 10.
S.
L. S. d.
520, at 12
d. is 520, or 26
at 6
d. is
13
at I d. is.
2
3
4
at
2
I d. is
II
8
at
4 d. is
10 10
520, at 73 d. is
L. 16 15 10
CASE
Chap. X.
257
Rules of PRACTICE.
CASE · IV.
When the rate is fhillings under twenty.
R U L E.
Multiply the given number by the numerators of the
fractions contained in Tab. III. and divide the product
by the denominators. Or, inftead of this general rule,
take the two particular ones following.
1. If the rate be an even number of fhillings, multiply
the given number by half the number of fhillings in the
rate, always doubling the right-hand figure of the pro-
duct for fhillings, and the reft are pounds.
2. If the rate be an odd number of fhillings, work
for the next leffer even number of fhillings, as above;
and for the odd fhilling take of the given number.
EXAMPLES.
1. When the rate is an even number of fhillings.
Ex. 1.
436, at 2 s.
I
Ex. 4.
48, at 8 s.
4
Ex. 7.
326, at 14 s.
7
L. 43, 12 S.
L. 19, 4 S.
L. 228, 4 S.
Ex. 2.
127, at 4 s.
2
Ex. 5.
87, at Io s.
5
Ex. 8.
48, at 16s.
8
L. 25, 8 s.
Ex. 3.
56, at 6s.
3
L. 43, 10 s.
Ex. 6.
L. 38, 8 s.
Ex. 9.
420, at 12 s.
6.
52, at 18 s
9
L. 16, 16 s.
L.252
L. 46 16 s.
K k
2. When
258
Rules of PRACTICE.
Chap. X.
:
2. When the rate is an odd number of fhillings.
Ex. 10.
Ex. 14.
Ex. 17.
635, at Is.
124, at 9 s,
248, at 15 S.
L. 31, 15s.
Ex. 11.
422, at 3 s.
42 4
21 2
L.63, 6s.
Ex. 12.
' 206, at 5 s.
121 ΙΟ
12 3
49 12
6 4
173 12
12 8
L.55, 16s.
L. 186
Ex. 15.
Ex. 18.
243, at II s.
324, at 17 s.
259 4
16 4
41 4
IO 6
L.51, 10s.
L. 133, 13 S.
L. 275, 8 s.
Ex. 19.
Ex. 13.
516, at 7 s.
Ex. 16.
431, at 13 S.
425, at 19 Sa
154 16
25 16
L. 180, 12s.
258 12
21 II
L. 280, 3 S.
382 10
21 5
L. 403, 15S.
Note 1. The reaſon of multiplying by half the num-
ber of fhillings in the rate will appear by confidering,
that theſe are the numerators of the fractions denoting
the rate.
Thus, 2 s. is. and 4 s. is 21. and 6s.
is 1. and each unit in the product is two fhillings.
The divifion by the denominator 10 is performed by
cutting off the right-hand figure of the product, and the
figure fo cut off is the remainder; and as each unit in
the remainder is two fhillings, the double of them is the
remainder in fhillings.
Note 2. From Ex. 1. we may learn, that when the
rate
{
Chap. X.
259
Rules of PRACTICE,
rate is 2 s. the price is found by doubling the right hand
figure of the given number for fhillings, and the other
figure or figures are pounds.
Note 3. In Ex. 2. the price may alſo be had by ta-
kingof the given number; and in this way every re-
mainder will be 4 s.
Note 4. When the rate is 10 s. as in Ex. 5. the price
may be obtained more easily by taking
of the given
number; and the remainder, if any, will be 10 s.
Note 5. When the rate is 5 s. as in Ex. 12. the price
is more readily had by taking of the given number;
and in this cafe every remainder is 5 s.
In like manner,
when the rate is 15 s. as in Ex. 17. the price may be
found by taking of the given number, and then
that quot.
of
Note 6. By reverfing the operation, from the price.
and any even rate given, we may readily find the quan-
tity of goods, viz. Multiply the price by 10, that is,
to the price annex a cipher, and divide the product by
half the rate.
Ex. 1. How many yards, at 14 s. may be bought
for 491. 7)490(70 yards. Anf.
Ex. 2. How many gallons, at 8 s. may be bought
for 500l. 4)5000(1250 gallons. Anf.
CASE V.
When the rate is fhillings and pence, or fhillings,
pence, and farthings.
RULE I.
If the rate be fhillings and pence which make an ali-
quot part of a pound, divide the given number by the
denominator of the fraction denoting the rate; the quot
is pounds, and each unit of the remainder is equal to the
rate.
Kk 2
EX.
260
Chap. X.
Rules of PRACTICE.
f
EXA M P LE S.
I
Ex. 4.
Ex. 1.
I
ΙΣ
354, at 1 s. 8 d.
L. 29, 10 s.
Ex. 2.
443, at 2 s. 6 d.
L. 55 7 6
Ex. 3.
| 346, at 3 s. 4 d.
L. 57 13 4
439, at 6 s. 8 d.
L. 146 6 8
Ex. 5.
766, at 13 s. 4 d.
255 6 8
255 6 8
L. 510 13 4
Note. The fraction denoting the rate in Ex. 5. be-
ing, you may alſo work thus: 766×2 =1532, and
3)1532(510 13 4.
RULE II.
If the rate be no aliquot part of a pound, but may
be divided into fuch parts, divide it accordingly, work
for the parts ſeparately, and then add.
EXAMPLES.
Ex. I.
427, at 8s.
6 d.
6 s.
2s. 6d.
128
2
53 7 6
L. 181 9 6
Ex. 2.
540, at 5 s.
4 d.
Ex. 3.
386, at 8 s.
8 d.
6 s. 8 d.
I
3
2 S.
ΤΟ
128 13 4
38 12
L.167 5 4
Ex. 4.
386, at 14 s.
8 d.
3 s. 4 d.
8 s.
90
154 8
S.
25. To
6s. 8 d.
54
128 13 4
L. 144
L.283 I 4
Ex.
Chap. X.
261
Rules of PRACTICE.
Ex. 5.
Ex. 6.
796, at 3 s.
iod.
394, at 17 s.
4 d.
6s.8 d.
3s. 4 d.
131 6 8
132 13
6 d.
46s. 8 d.
19 18
131 6 8
4S.
4 s. 20
78 16
L. 152 11
L. 341 9 4
III.
RULE
If the rate be no aliquot part of a pound, and cannot
readily be divided into fuch parts, divide it into parts
whereof one at leaſt may be an aliquot part of a
pound, and the fubfequent part, or parts, each an ali-
quot part of ſome prior part. Or,
Multiply the given number by the fhillings, and then,
for the pence and farthings, work as in Cafe III.
EXAMPLE S.
Method 1.
Ex. I.
Method 2.
Ex. I.
3506, at 1 s. 3 d. I S.
is j
3506, at 1 s. 3 d.
3 d. I
876 6
I S.
175 6
43 16 6
I
20
4382 6
3d.
L.219 2 6
Ex. 2.
9/5, at
I S.
3 d.
I
1½d.
I S.
6 d.
163HID
Z
10½d.
4 15
2 7 6
I S.
6 d.
sddd
HO 3Ha
3 d.
d.
HaHaHa
L.219 2 6
Ex. 2.
95, at I s. 10 d.
47 6
23 9
11 102
I
I
3 9
17/8 12/2
20
1플
​11 10/21/20
L.8 18 1
1플
​L.8 18 11/
Ex.
262
Chap. X.
Rules of PRACTICE.
Method 1.
Ex. 3.
485, at 2s. 24d
48 10
2 S.
IO
2 d.
I
I2
4 00
10
d.
10
I
11
2244
1
Method 2.
Ex. 3.
485,at 2 s. 24 d.
2 S.
2 d.
I
970
80 10
d
2244
10 12/
4
L. 53 0 11
1144
4 S.
6 d.
2 d.
d
4
21-H
8 s. 4
Ex. 4.
480, at 4 s. 83 d.
96
12
4
I
O IO
L. 113 10
Ex. 5.
136, at 9s. 21 d.
I
20
1060 11
L: 53
I I
11/4/
Ex. 4.
480, at 4 s.
83 d.
4 S.
1920
6 d.
240
21214
2 d.
80
20
ΙΟ
22710
9 s.
a 2 Hlld
I
20
L.113 10
Ex. 5.
136, at 9 s. 21 d.
1224
22
8
∞ ∞
5
8
| 2
4
ΙΟ
54 8
I S.
2
6 16
2 d.
2 d.
I
2 8
플​d.
मात
½ d.
5
8
2
L.62 12 4
I 24
L.62 12 4
Ex.
Chap. X.
263
Rules of PRACTICE.
Method 1.
Method 2.
I 2 S.
2 S.
6
ΙΟ
ΙΟ
2 d. Ι
is vij j ť
I2
d.
d.
2
Ex. 6.
370, at 14s. 23 d.
222
37
3 1 8
15 5
7 8 플
​1
L.263 4 912
Ex. 6.
370, at 14s.
22 d.
148
37
14 S.
5180
6
8
15
5
7
플
​OHN HO
20
5:04 02/2
L.263 49/1/
Ex. 7.
18s.
Ex. 7.
1504, at 19 S.
9 d.
1353 12
1 504, at 19 S.
9 d.
13536
1404
I S.
75 4
6 d.
19 s.
37
12
od.
3 d.
L. 1485 4
18 16
3 d.
20
mamia ma
28576
752
376
I
297-4
L.1485 4
Note 1. When the rate is more than I s. and lefs
than 2 s. as in Ex. 1. and 2. there is no occaſion, in
working by Method 2. to draw a line under the given
number: we eſteem it ſo many fhillings, and the
parts for the pence or farthings are added up with it.
Note 2. In working the above or like examples, fome
people of inferior ſkill proceed in the following manner.
Ex.
264
Chap. X.
Rules of PRACTICE.
*
Ex. 2. 95, at 1 s. 10 d.
L. s. d.
95 fhillings make
4 15
95 fix-pences make
2 7 6
95 three-pences make
I
3 9
95 pence make
95 half-pence make
7 II
3 11/1/20
Anf. 8 18 1
CASE VI
When the rate is pounds.
RULE.
Multiply the given number by the rate, and the pro-
duct is the price in pounds.
EXAMPLES.
Ex. I.
Ex. 2.
42, at 21.
13, at 81.
L.84
L. 104
Ex, 3.
Ex. 4.
30, at 3 l.
48, at 121.
L.go
L. 576
VII.
CASE
When the rate is pounds and fhillings, or pounds,
fhillings, pence, and farthings,
RULE
I.
If the rate be pounds and fhillings, multiply the given
number by the pounds, and work for the fhillings as in
Cafe IV.
EX.
Chap. X.
265
Rules of PRACTICE.
EXAMPLES.
Ex. I.
Ex. 3.
58, at 31. 7 s.
1 l.
46, at 1 1. 4 s.
4 S.
9 4
3 1.
174
6s.
17 8
L.55 4
IS.
Ex. 2.
2 18
L.194 6
Ex. 4.
82, at 4 1. 10 s.
26, at 31. 15s.
41.
328
3 l.
78
ios.
41
14 S.
18
4
I S.
I 6
L. 369
L.97 10
I
Note. When the rate is more than 11. and leſs than
21. as in Ex. I. we have no occafion to draw a line
under the given number, it being efteemed fo many
pounds, and the parts for the fillings or pence are
added up with it.
RULE II.
If the rate be pounds, with fhillings and pence that
make ſome aliquot part of a pound, or are diviſible into
aliquot parts, or into fhillings and ſome aliquot part or
parts; then multiply the given number by the pounds
and work for the fhillings and pence as in Cale V.
Rule I. or II.
EXAMPLE S.
Ex. 1.
Ex. 2.
54, at L. 3:2:6.
43, at L. 5:3:4.
31.
2s. 6d.
162
51.
215
6 15
3 s. 4 d.
7 3 4
L. 168 15
L., 222 3.4
LI
Ex.
266
Chap. X:
Rules of PRACTICE.
Ex. 3•
Ex. 6.
92, at L. 3:5:4
I
1.
73, at L. 1:6:8.
6 s. 8 d.
24 6 8
31.
276
L.97 6 8
S.
3 s. 4 d.
2 S.
15 6 8
9 4
ΙΟ 8
Ex. 4.
76, at 71. 13 s.
4 d.
L. 300 10
Ex. 7.
37, at L.4:8:8.
4 1.
148
71.
532
6 s. 8 d.
12
6 8
6 s. 8 d.
25
6 8
2 S.
6 s. 8 d.
3 14
25
68
L.582 13 4
Ex. 5.
83, at L. 2:8:6.
L. 164 0 8
Ex. 8.
26, at L. 2:17:4.
2 1.
21.
166
52
6 s.
6 s. 8 d
24 18
8 13 4
2s. 6d.
6 s. 8 d.
10 7 6
8 13 4
4 S.
5 4
L. 201 5 6
L.74 10 8
RULE III.
If the rate be pounds, with fhillings, pence, and far-
things, that cannot readily be refolved into aliquot parts
of a pound; multiply the given number by the pounds;
and then work for the fhillings, pence, and farthings,
as in Cafe V. Rule III. Method 1. But if you propoſe
to work for the fhillings, pence, and farthings, by Me-
thod 2. it will be convenient to do that in the firſt place,
and then work for the pounds.
EX.
Chap. X.
267
Rules of PRACTICE.
EXAMPLE S.
Method 1.
Method 2.
Ex. 1.
Ex. I.
11.
213, at 11. 13 s.
4½ d.
213, at 11. 13 s.
4d.
IOS.
106 10
639
2 S.
21 6
213
I S.
3 d.
1½d.
ůj j
10 13
2 13
3
13 s.
2709
1 6 7
3 d.
53
3
1 d.
26
7플
​7/2/20
L. 355 8 101/2
1
20
2.8418
Il.
213
Ex. 2.
37, at 31. 8 s.
142 8 10
L.355 8 10/1/1
Ex. 2.
37, at 31. 8 s.
1.01 d.
I
101 d.
6 s.
31.
242
II [
2. S.
6 s.
74
II
2
2s. 6d.
3 d.
6 d.
18
6
4 12 6
9 3
I d.
d.
L. 127 7 7
3
I
91
MAAK
H
3 d
9 3
I d.
3
I
d.
9
20
32/7
7 1/
I
L12
10 7 7
III
L. 127 7 7
Method
268
Chap. X.
Rules of PRACTICE.
Method 1.
Ex. 3.
416, at 21. 9s,
Method 2.
Ex. 3.
416, at 21. 9s.
32 d.
32 d.
21.
832
8 s.
I S.
ů ůi j j
3 d.
Doug pr. mit
166 8
20 16.
5 4
d.
I 6
}
a 33H
9 s.
3744
3 d.
104
d.
26
3874
193
14
21.
832
L. 1025 14
L. 1025 14
Note. A troubleſome fraction in the rate may fome
times be avoided, by multiplying the rate, and dividing
the quantity by the denominator of the fraction, and
then computing the price of the quot from the new rate.
8
Ex. I.
600, at 77 d.
5 s.
3 d.
I
75, at 5s. 3 d.
375
18 9
3913 9
L 19 13 9
d.
S. d.
7 1/3 × 8 = 63 d. = 5 3
63d=53
Ex
Chap. X.
Rules of PRACTICE,
269
I
72
540, at 31 d.
Ex. 2..
45, at 3 s. IId.
s. d.
31× 12 = 47 d. = 3 1 1
4 d.
4 d.
3 d.
ůj j j
3 S. 135
15
15
II 3
201716 3
L.8 16 3
Ex. 3.
924, at 4 5
s. d.
s. d.
L. s. d.
4 5×11=294
84, at L. 2:9:4
9 s.
756
4d.
28
784
39 4
21.
168
L. 207 4
CASE
VIII.
When the given number confifts of integers and parts.
RULE.
Work for the price of the integers as already taught;
and for the part or parts, take a proportional part, or
parts, of the rate,
EX.
2
270
Chap. X.
Rules of PRACTICE.
6s. 8 d.
yd.
EXAMPLE S.
Ex. 1.
Yards.
Ex. 4.
7204, at 6 s. 8 d.
per yd.
Yards.
2371, at 18 s. per
yd.
240
18 s.
20 14
0 I 8
L. 240
I 8
+02010
# yd.
9
y.d.
4
6
2 3
2s. 6d.
Ex. 2.
Yards.
116, at 4 s. 6d.
per yd,
14. 10
yd.
L.21 9 9
Ex. 5.
C. Q
365 2, at 51.
5 s. p. C.
2 S.
II 12
yd.
2 3
5 l.
5 s.
1825
L.26 4 3
2 Q
91 5
212 6
Ex. 3.
L. 1918 17 6
Yards.
2283, at 12 s.
11 d. p.yd.
12 S.
2736
1 l.
4 d.
76
4d
76
10 S.
3 d.
57
2 Q.
O 15
yd.
6 5/1/
yd.
3 22
IQ.
14 lb.
I
295/4 8
4
Ex. 6,
G. Q. lb.
28 3 14 at 1l.
10 s. per
14
7
6
3 9
L.43 6 3
C.
L. 147 14 81
Ex.
Chap. X.
278
Rules of PRACTICE.
Ex. 7.
C. Q. lb.
144 2 21, at 31.
17s. 6d. p.C.
Ex. 9.
lb. Tr. oz. dw."
36 8 16, at
23 l. 16 s.
3 1.
432
4 d. per lb.
15 s.
108
2s. 6d.
18
108
2 Q
I 18 9
72
14 lb.
9
81
7 lb.
23lb
828
105.
18
L. 560 13 33
5 s
9
Ex. 8.
I S.
I
16
T. C. Q.
73 17 2, at 91.
4 d.
I 2
d.
I 6
12s. Iad.
4 Oz
7 18
per tun.
4 OZ.
7 18
16 dw.
91.
IOS
2 S.
7 6
657
1 11 91%
36 10
L.874 18 10
6 d.
I 1 16
6
2 d.
12
2
2 d.
10 C.
5C.
2 C. 2 Q.
12 2
4.16 5
2 8 2
I 4 I
L.712 5 63
An operation in the rules of practice may be proved
by running over the feveral ſteps a fecond time, by
working the fame queſtion a different way, or by the
rule of three.
Practical Questions.
1. 126, at 143 d.
2. 478, at 4 s. 10 d.
Anf.
L. s. d.
7 14 101
Anf. 116 10 3
3.50,
272
Chap. X
Rules of PRACTICE.
3. 50, at L. 3:15:6
4. 419 yards,at 4s. 10ž d. p. yd.
5 75 C. 2Q 14 lb. at 15 s.
41 d. per C.
6. 8 T. 15 C. 3 Q 27 lb. at
141. per tun.
L. s.
Anf. 188 17 I
d.
Ans. 101 16
3/7
f.
Anf. 58 2 8 3
Anf. 123 3 10
7. A bankrupt's effects amount to 8111. 10 s.; and
he owes
L. S. d.
To A
220 16 6
To B
312
To C
117 12 6
To D
106 12 6
To E
200 6
To F
124 12 6
In all 1082
How much can he afford to pay per pound, and what
muft each creditor have?
Anf. 15s. per pound; and the creditors get as fol-
lows:
L. S. d.
A
165 12 4
B
234
C 88 4 4
D. 79 19 42
E
150 4 6
F
93
94
Total 811 10
The End of the Firſt Volume.
D. Jingles wy g
L
to
urratian tgue
UNIVERSITY OF MICHIGAN
3 9015 03684 9589
A 548552