Trelissuli Library.
1. Morning Room I
Compartment C
Thely 3
QA
515
P95
A
Familiar Introduction
TO THE
THEORY and PRACTICE
O F
PERSPECTIVE.
By JOSEPH PRIESTLEY, LL.D. F.R.S.
ΚΑΙ ΑΓΕΩΜΕΤΡΗΤΟΣ ΕΙΣΙΤΩ.
LONDON:
Printed for J. JOHNSON and J. PAYNE, at
N° 8. Pater-nofter-Row.
M. DCC. LXX.
English
ΤΟ
prayton
2-11233
27939
SIR JOSHUA REYNOLDS, KNT. F.R.S.
PRESIDENT
OF THE
ROYAL ACADEMY OF PAINTING, &c.
THIS TREATISE ON PERSPECTIVE
IS, WITH GREAT RESPECT,
INSCRIBED BY
HIS MOST OBEDIENT
HUMBLE SERVANT,
LEEDS,
MARCH 20. 1770.
JOSEPH PRIESTLEY.
[ 3 ]
H. US US US US US US US US
DR sk sk sk M A M sh si sk I
W US HU WWW UF
st staste DRS DR sk sk sk SR
THE
PREFACE.
++
poof of op of op of f
T
HE art of drawing in perſpective
has fo many, and fuch obvious
ufes, that there can be no occafion
to enumerate them. Thoſe who want to
communicate their ideas to others often
feel the inperfection of founds, or of charac-
ters that only reprefent founds, for this
purpoſe; and thoſe who are defirous of re-
ceiving information find, that it is, in
many cafes, conveyed with unfpeakably
more eafe and certainty, by the eye, than
by the ear.
a 2
The
iv
THE PREFA C E.
The reaſon is, that every verbal defcrip-
tion of any object that hath the dimenfions
of length, &c. must be reduced to a lineal
one in the imagination, before any diftinct
idea can be conceived of it; and this is
often a difficult and painful operation.
It is particularly fatiguing to the imagi-
nation to put together the parts of a com-
plex object, when the defcription of each
of them is made feparately; for each part
being conceived very imperfectly, and
one part being in danger of being forgotten,
while the mind is engaged in attending to
another, the conception of the whole can-
not but be very obfcure, and inadequate.
The ſymmetry and elegance of a pile or
building is abfolutely loft without a draw-
ing, and the moſt maſterly painters are in-
capable of producing any thing but the
moft hideous and unnatural groupes of fi-
gures, without the affiftance of this art.
Of all the imitative arts, this of perfpec-
tive is capable of being brought, and in-
deed has actually been brought the neareſt
to perfection; becauſe it is wholly within
the
i
GT
THE PREFACE.
V
the ſphere of mathematical ſcience, a branch
of knowledge which has been moſt dili-
gently cultivated of late years, and with
reſpect to the objects of which, we are not
uſed to acquiefce in any thing fhort of ab-
folute certainty. Accordingly, we fee,
that there is no object of fight, be it ever
fo complex, which thofe perfons who have
applied themſelves to the ſtudy of this art
are not able to repreſent, juſt as it appears
to the eye. By this means the ideas of all
the beauties of nature and art are faithfully
preferved, and thofe perfons who have no
opportunity of feeing the objects themſelves,
may ſtudy, and be delighted with them,
in the works of travellers and natural hifto-
rians; and philoſophers can defcribe their
machines, and their apparatus for making
experiments, without any danger of em-
barraffing themſelves, or of perplexing and
confounding their readers.
As in all the other branches of mathe-
matical knowledge, the progreſs of this art
has been flow, but fure; and the Engliſh
writers (particularly Dr. Brooke Taylor)
feem to have carried it to a degree of per-
fection
a 3
vi
THE PREFACE.
}
fection we can hardly conceive it poffible
to be exceeded. As much, however, is
known, as we can imagine will ever be
made ufe of.
All that is wanting feems to be a me-
thod of facilitating the attainment of this
art; and in this refpect I cannot help think-
ing there is room to improve upon all
the books that I have yet feen upon this
fubject. Indeed the actual ftate of the
practice of this art ſeems to be a proof that
the attainment of it has not been fuffici-
ently eaſy. For of the great numbers,
both of ladies and gentlemen, who learn
to draw, and even take pains to draw with
elegance, there are very few who ſo much
as attempt to learn perſpective. Though,
therefore, they be able to copy prints, and
the defigns of others, and to execute theſe
drawings in a maſterly manner, they are
utterly incapable of defigning any thing
themſelves, or of drawing from nature; fo
that a perſon who can even draw human
figures, in every variety of attitude and
paffion, ſhall not be able to take a correct
drawing of a table, a chair, or the moſt
fimple
THE PREFACE.
vii
ſimple machine, or inftrument, that re-
quires nothing but ftraight lines.
Of thofe gentlemen who travel, or go
long voyages, how few are furniſhed with
the rudiments of this art, for the exerciſe
of which they have perpetual occafion, if
they would preferve the ideas of the me-
morable ſcenes and objects they meet with,
or communicate any juſt notion of them to
their friends. How many philofophers,
and even perſons who have been no mean
proficients in other branches, both of ſpecu-
lative and practical mathematics, do we
hear complaining of their ignorance of this
art, and of the difficulties they have met
with in their attempts to learn it; ſo that
they are incapable of making a drawing of
the apparatus they make uſe of in their ex-
periments, and are always obliged to em-
ploy a profeffed artift for this purpoſe.
Thefe difficulties have difcouraged num-
bers, who have not fcrupled to give the
very great price that books of perfpective
generally fell for, and who have been wil-
ling to take a good deal of pains in the
ſtudy of them.
Though,
viii THE PREFAC E.
Though, for my own part, I got a ge-
neral idea of the theory of perſpective pretty
early, at the time that I attended to
other branches of mathematical ſcience,
I was not capable of making a draught of
any thing, till I was under a neceffity of
having original drawings of electrical ma-
chines and apparatus, and was in a ſitua-
tion where I could not find any perfon to
make them for me. At firſt I puzzled
myſelf with feveral mechanical me-
thods of drawing; but though I made
confiderable improvements in fome of
them, I was obliged, at laft, to have re-
courſe to the rules of perſpective. I found
them, however, fo immethodically di-
gefted, or fo infufficiently explained, that,
in feveral cafes, I was able to inveſtigate
the rule myſelf, from confidering the na-
ture of the thing, fooner than I could find
it in the books; and after all, the drawings
that I did make at that time were executed
when I had a very imperfect knowledge of
the art.
The embarraffment I then found myſelf
in, made me attend to the fubject after-
wards,
THE PREFACE. Ix
wards, when I was more at leifure for it.
Having ſtruggled with the difficulties my-
felf, and writing while the idea of them is
freſh in my memory, I hope that I have
been better able to obviate, or remove them,
for the benefit of others. I have been
willing, however, to make the attempt;
and I flatter myſelf that any perſon, of the
age and qualifications of thoſe who ever
think of learning to draw, may, by
the help of this treatife, without any in-
ftructor, make themſelves maſters of
every
thing that is effential to this art. Lefs
than a week, I am pretty clearly of opi-
nion, would be fufficient for a mafter of
the art to inftruct another in it, in the me-
thod here laid down; and a few hours
would be fufficient to give a perſon, who
has a previous knowledge of geometry, a
perfect idea of all the real varieties that can
poffibly occur in the practice of it.
Nor will this be any matter of wonder
when it is confidered, that after the pre-
paration of the drawing-board, the whole
of this art confifts in drawing the perfpec-
tive appearance of lines in no more than
five
X THE PREFAC E.
five different varieties of poſition, and in
fixing points at given diſtances in thoſe
lines.
I
In the method defcribed, Part IX, no-
thing is requifite, but to take the elevation
and declination of any fingle point in an
object, fo that the whole myſtery of it
might be learned in a few minutes.
therefore cannot help wiſhing that mathe-
matical inſtrument-makers would conftruct
fome cheap inftrument for this purpoſe.
The contrivance would be very eafy, and
it would greatly facilitate the practice of
this uſeful art. Whether this method be
explained in any other treatiſe I cannot
tell. It occurred to me from confidering
the nature of the thing; I practifed it be-
fore I was acquainted with any other; and
the method, both with refpect to theory
and practice, is intirely independent of the
common method, and abundantly more
fimple. The common perſpective is founded
on the confideration of planes and lines, but
in this points only, in which they terminate,
are regarded. But it is better adapted to
the purpoſe of drawing large objects, in
the
THE
xi
PREFACE.
the open air, than ſmall objects within
doors.
The reader will obferve that, befides the
rules of orthographical perſpective, laid down
in Part X, this treatife contains a defcrip-
tion of three diftinct methods of drawing
in common or scenographic perspective, each
of which may be learned independent of
the others.
That which is of the moft general ap-
plication is contained in Parts III, IV and
V; and a fummary view of it is given in
Part VI. By this method a perſon will
be able to draw the perfpective appearance
of all objects whatever, their dimenfions
and diſtances being previously known.
There is no occafion for their being in
view, or for their ichnography being taken.
The fecond method, contained in Part
VIII, teaches how to draw the appearance
of objects from their ichnography, or
ground plan. The rules of it are exceed-
ing few and eafy, but the true dimenfions
of things must be taken before their ich-
nography can be drawn. If any perfon
would chufe to confine himſelf to this me-
thod;
xii THE
PREFACE.
thod; after learning to draw the appearance
of objects on the ground plane, he muſt
confult Part V. Sect. II. Cafe I. for a me-
thod of raiſing perpendicular altitudes upon
any part of that plane; and then the me-
thod will be complete.
The third method is that mentioned
above, which fuppofes the objects to be in
view, and by finding the fituation of points,
determines the appearance of objects. This
must be done by an inftrument, but one
ſingle rule comprehends the whole practice.
Mathematicians will perhaps be ſurpriſed
to find fo little theory in this treatiſe, but
of this I have made no parade, contenting
myſelf with giving a fatisfactory reaſon for
every effential part of the practice; and if
theſe fatisfactory reafons be given, it is all
that the reader can reaſonably require. Dr.
Brooke Taylor, and other geometricians,
appear to me to have made the theory of
perſpective much more extenfive and com-
plex, than the practice requires; and to
have introduced more technical terms than
are really neceffary for this purpoſe. Con-
fidering
THE PREFA C E. xiii
*
fidering perspective as a branch of geome-
try, I am far from blaming the conduct of
thofe great writers; but I would not have
the young artist be difcouraged, by ima-
gining that he muſt neceffarily make him-
ſelf maſter of their works, in order either
to practiſe this art, or to be fully con-
vinced of the reafons on which it is founded.
The reader is indebted for part of this
work to Mr. Joſeph Priestley, of Halifax,
from whoſe knowledge of mathematics in
general, and of this branch in particular, I
once expected a much more complete and
elegant view of the theory of this art; and
I do not yet deſpair of his undertaking fo
ufeful a work. He drew up the general
view of the theory of perspective, prefixed to
the Notes, and wrote all thofe paragraphs
in which the propofitions in that piece are
referred to. He gave me the method
of meaſuring a line oblique to the ground
plane, deſcribed p. 34, with the demonftra-
tion of it, and the method of drawing cir-
cles deſcribed Cafe I and II. Part VII. He
was alſo ſo obliging as to affift me in reviſing
and correcting the whole work.
Let
xiv THE PREFAC È.
ご
​Let it be obſerved, that I term this trea-
tiſe only a familiar introduction to the the-
ory and practice of perfpective. It is there-
fore by no means intended to fuperfede other
valuable works, that contain a greater variety
of examples, and a detail of particular pro-
ceffes, which are highly uſeful to thoſe who
have much practice in this art. I flatter
myſelf that by the help of this introduction,
thoſe books will be much better underſtood,
and more uſeful than ever.
As a friend to the arts, but more efpe-
cially to fcience, which is greatly indebted
to this particular art, I fhall think myſelf
happy, if, either by what I have written
myſelf, or by recommending the writings
of others, I fhall contribute to make the
buſineſs of perſpective more generally un-
derſtood and practifed. Every boy that is
capable of being taught writing and ac-
compts, might be perfectly inftructed in
all the rules contained in Parts III, IV,
and V, of this Introduction, in a few weeks.
They take up no more than 24 pages of
the work, not concisely written. And
every
THE PREFACE.
XV
every ſchoolmaster, who is as yet unac-
quainted with the rudiments of this uſeful
art, might make himſelf maſter of them in
a few evenings.
There is no occafion to trouble every
boy with the theory of perſpective; but I
would have all young perfons, without ex-
ception, made ready in the practice. Thoſe
who apply to any branch of the mathema-
tics, may learn the theory afterwards. And
if a perſon were taught no more than the
neceſſary rules in his youth, he might learn
particular improvements in drawing at his
leifure, whenever he ſhould have occafion
for them.
Since this Work was printed off, I have ſeen a fubftance
excellently adapted to the purpose of wiping from paper the
marks of a black-lead-pencil. It muft, therefore, be of fin-
gular uſe to those who practife drawing. It is fold by
Mr. NAIRNE, Mathematical Inftrument-Maker, oppofite the
Royal-Exchange. He fells a cubical piece, of about half an
inch, for three fillings; and he Jays it will laſt ſeveral
years.
[ 1 ]
00050009 0000 0000 0005,0005 0000 0003 0003 0003 6000 0000 3000 2000 ODDO COOD COOD COOD 0800 0000 0000 0000 0000
:
A
FAMILIAR INTRODUCTION
To the THEORY and PRACTICE of
PERSPECTIVE.
PART I.
Of the Inftruments that are of Ufe in the
Practice of PERSPECTIVE, and the
Application of them.
A
Perſon who propofes to practice the
art of drawing in perſpective muſt
provide himſelf with a drawing board
and Square, and a cafe of mathematical inftru-
ments, confifting of a pair of compaffes, a
ſet of ſcales, a fector, a protractor, and a
drawing pen.
A
The
';
A TREATISE ON
The drawing board is made in the form
of a fquare, or parallellogram; and if any
one of the angles be a right angle, it will
be fufficient, provided the fquare which is
ufed in drawing lines upon it be always
applied to one of the fides that contain that
right angle; and lines may be drawn in
all poffible directions without uſing any
other fide of the board. It will be very
convenient for a perfon who practices.
drawing much, to have feveral drawing
boards, of different fizes, for the fake of
having greater variety in the compaſs of
his draughts.
The Square to be uſed along with the
drawing board is a flat ruler, as a, Fig. 1,
at one end of which are faſtened two tranf-
verſe pieces; one of them, b, fixed at right
angles to it, and the other c, moveable, ſo
as to be fixed at any angle required. Theſe
tranfverfe pieces are applied cloſe to the
fide of the drawing board, while the ruler
lies upon it; and by fliding them along the
board, lines may be drawn parallel to one
another with much leſs trouble than by
the help of a parallel ruler. By the fixed
tranf
PERSPECTIVE.
3
tranfverfe, lines may be drawn parallel to
one fide of the board and perpendicular to
the other; and by the moveable tranfverfe,
lines that have any degree of obliquity to
the fides of the board may be drawn paral-
lel to one another; and if, without moving
the tranfverfe, the ruler be removed to the
other fide of the board, lines may be drawn
perpendicular to them. But if the obli-
quity be very great, it will be impoffible to
apply the fquare, fo as to interfect the lines
at right angles in fome parts of the board.
In this cafe, recourſe muſt be had to a pa-
rallel ruler.
A great variety of fcales is uſeful, in
order to lay down lines of any given length,
in whatever proportion is moſt convenient,
with respect to the fize of the drawing, &c.
If none of the fcales happen to fuit the
purpoſe, recourſe muſt be had to the line of
lines upon the ſector: for, by the different
openings of that inftrument, a line of any
length may be divided into as many equal
parts as a perſon chufes.
A 2
The
}
}
A TREATISE ON
4
The only lines that are neceffary to be
attended to upon the fector, befides thefe,
for the purpoſe of drawing, are the two
lines of tangents; one of them marked T,
beginning at the center of the inſtrument,
and ending with 45 at the extremity of the
leg; the other marked t, beginning at the
diſtance of about one-fourth of the radius
from the center, and reaching a little be-
yond 75, near the extremity.
Left a book explaining the uſe of the
ſector ſhould not happen to be at hand, I
ſhall juſt inform the perſon who is learn-
ing to draw, that, if he want the tangent
of an angle exceeding 45 degrees, he muſt
open the ſector till the 45 on the lines
marked t be fet at the fame diſtance,
at which the two forty-fives on the lines
marked T were placed; and then take the
tangent required, juft as he would have
done, if the tangent had been less than 45,
and it had not been neceffary to make any
other opening of the inftrument,
If the fector be fo conftructed, as that the
45 t begin exactly at the distance of one-
fourth
PERSPECTIVE.
§.
1
fourth of the radius from the center, and
confequently the tangent of 45 t, be ex-
actly one-fourth of 45 T; there will
be no occafion for making any other
opening of the fector; for the tangent of
any degree exceeding 45 being taken on t,
and repeated four times, will be the length
of the tangent required.
Befides the ſmall ſector in one of the
common pocket cafes of inftruments, I
would adviſe a perfon who propoſes to learn
to draw to get another, of one foot radius,
at leaſt. Two ſectors are, in many caſes,
exceedingly uſeful, if not abſolutely ne-
ceffary; and I would not adviſe a perſon
to be fparing of expence in procuring a
very good inftrument, the ufes of which
are fo various and important.
The protractor is generally a circle, or
femi-circle, the limb of which is divided
into 360 parts, called degrees. Sometimes,
however, a parallellogram, about the fize
of a common fcale, is ufed as a protractor.
In this cafe the edges of it are divided in
the fame manner as if it had been part of a
A 3
circular
t
6
A TREATISE ON
!
circular piece, and the lines drawn from
the center to the circumference; but it is
eaſier to lay down any angle with accuracy
from a circular, or femi-circular protractor.
A drawing pen is neceffary, becauſe a
common open pen can hardly be applied
to the edge of a ruler without blotting the
paper when it is taken up; befides, by the
help of a drawing pen, lines may be made
of preciſely the fame thickneſs throughout,
and of whatever thickneſs a perfon pleaſes.
Black lead pencils are very uſeful, in
order to draw lines that are of no fervice,
but as a direction to draw other lines by
them; becauſe, when they have anſwered
this purpoſe, they may be taken out with
a few crumbs of ſoft bread.
!
PART
PERSPECTIV E. 7
PART II.
The Definition of neceſſary technical Terms,
and the Preparation of the Drawing-
Board.
PERSPECTIVE is the art of deline-
ating objects as they would appear
upon an upright plane, interpofed between
them and the eye; for inftance, as they
appear upon a pane of glaſs when they are
feen through a window. In this manner
it is evident, that the images of objects of
the fame fize will occupy more or lefs fpace,
according as they are nearer or farther off;
and the great difficulty of drawing a num-
ber of objects confifts in giving them theſe
proportions. By this art, therefore, the
pictures of objects are made to exhibit the
fame appearance upon one plane, and at
the fame diſtance from the eye, that they
have in nature upon different planes, and
at different diſtances; which is a ſpecies of
imitation that is particularly pleaſing to all
perfons.
To
8
A TREATISE ON
To make the practice of this agreeable
art the more intelligible, I ſhall ſuppoſe that
I am about to make the perfpective drawing
of a number of objects, as a, b, Fig. 2. which
require every variety in the practice; and
ſhall minutely defcribe every part of the
proceſs I make ufe of for this purpoſe. [A]
The first thing I do is to faften a ſheet
of paper upon my drawing-board, by bits
of wafer or fealing wax, at each corner,
in order to make it lie flat and ſteady.
The ſurface of this paper is defigned to re-
preſent the plane on which the objects are
to be drawn, and which is generally called
the perspective plane. In other words, it
repreſents the glass window, which I be-
fore fuppofed to intervene between my eye
and the objects, and upon which their ap-
pearance was to be drawn. This appear-
ance I am now to copy on my paper.
In order to this, I draw, or fuppoſe to be
drawn, upon the plane on which the ob-
jects ftand, called the ground plane, a line
AB, Fig. 2, on which the perſpective
plane, through which I view the objects,
is
PERSPECTIV E.
9
1
1
I then take my
is ſuppoſed to ſtand.
drawing-board, and, by the fixed tranſverſe
of the fquare, draw the line AB, Fig. 3,
to reprefent it. This line I, therefore, call
the ground line of my piece.
After this I note the point C, Fig. 2, in
that part of this ground line which is neareſt
to my eye; and upon it I raiſe, or ſuppoſe to
be raiſed, the line CE, perpendicular to the
other. Alſo, upon my paper, Fig. 3, I
draw the line CE, to repreſent it.
The next thing I do is to meafure the
height of my eye above the plane on which
the objects ſtand; and ſuppoſing it to be fix
feet, I fet off from C to D, Fig. 3, the
length of fix divifions, from any ſcale that
I think proper, and draw the line FG pa-
rallel to the ground line AB. This line,
being drawn at the diſtance of the height
of my eye, will reprefent a plane paffing
through my eye, and feen edgeways; and
being parallel to the horizon (fuppofing
that I ftand upright) it is called the hori-
zontal line. The point D in this line, to
which my eye is directly oppofite, is called
the
10
A TREATISE ON
the point of fight. It is that point in the
perfpective plane which is the neareſt to
my eye.
per
When I have thus drawn upon my pa-
the horizontal line, I meafure the dif
tance at which I ftand from the imaginary
perſpective plane; and, fuppofing it to be
nine feet, I mark a point E, Fig. 3, in the
perpendicular line, at the diftance of nine
divifions from D in the horizontal line.
I alſo ſet off the fame diftance both ways
from the point of fight D, along the hori-
zontal line to F and G.
This being done, my board and paper
are prepared for the delineation of the
objects I propoſe to draw. Or, if I pleaſe,
I may draw the lines HIKL, to bound the
picture; but this may as well be omitted,
till the drawing be compleated.
N. B. In all that follows, I fhall ſuppoſe
the drawing-board to be prepared in this
manner; at leaſt, that as many of theſe
lines are drawn, and as many of the points
fixed,
PERSPECTIV E.
I I
fixed, as the purpoſe of the picture requires.
I fhall, therefore, hereafter omit the repe-
tition of this part of the procefs. Let it be
obferved, alfo, that the letters A, B, C, D,
E, F, and G, always keep the fame places,
and have the fame ufes in all the figures
belonging to this work.
A general idea of the nature of perſpec-
tive, and of the preparation of the drawing-
board may, perhaps, be more clearly under-
ftood by means of Fig. 4.
In this figure
the plane X, which muſt be laid flat upon
the table, is the ground plane on which the
objects ſtand; the plane Y, which muſt be
raiſed perpendicular to the other, is the
perspective plane upon which they are to
be delineated, and the part Z, which muſt
alſo be raiſed perpendicular to the table,
repreſents the ſpectator, the line ab being
the height of his eye above the ground
plane. Alſo bC is the diſtance at which he
ſtands from the perſpective plane Y, on
which the fame lines are drawn as in Fig.
3, beſides others which will be explained
in their proper place. If now the plane Y
be
12
A TREATISE ON
t
be ſuppoſed to be tranſparent, ſo that ob-
jects fituated on the plane X could be ſeen
through it, their appearance on this plane Y
would be the perſpective repreſentation re-
quired.
1
10
PART
1
PERSPECTIVE.
13
PART
III.
To find the perspective fituation of right lines
upon the ground plane.
ΤΗ
HE drawing board being prepared as
has been already deſcribed, I pro-
ceed to draw upon it the perſpective appear-
ance of all the objects that are fituated be-
yond the perſpective plane.
Since the figures of all objects are con-
tained under lines, bounded by points, i. e.
under lines terminating in different places,
it is evident, that the whole art of per-
ſpective confifts of nothing more than a
method of finding the fituation of all lines
upon the perſpective plane, and of cutting
thofe lines in any given proportion, accor-
ding to the lengths required. Curves are no
exception to this obfervation, fince they
are drawn by fixing a ſufficient number of
points in right lines, and joining them by
a ſteady hand. In geometry, they are
confidered as confifting of an infinite num-
ber of right lines. I fhall, therefore, in
the
14
A TREATISE ON
the first place, explain the method of find-
ing the perſpective fituation of right lines.
running in all poffible directions, begin-
ning with thoſe which are drawn upon the
ground plane.
All right lines drawn on the ground
plane are either parallel to the ground
line, as cd, Fig 2, perpendicular to it, as di,
or oblique to it, as ml.
SECTION I.
To find the perspective fituation of right lines
parallel to the ground line.
F the lines to be drawn in perſpective be
IT
parallel to the ground line, as cd, Fig 2,
they will be parallel to it on the perſpective
plane, and confequently parallel to one
another. [B]
Being fenfible, then, that the image of
the line H I, upon the ground plane X,
Fig. 4, muſt be drawn parallel to the
ground line A B, I only want to know at
what diſtance it must be drawn from it.
To
PERSPECTIVE
15
To aſcertain this, I meaſure the perpendi-
cular diſtance CM of the given line from
the perſpective plane; and finding it, for
inftance, to be three divifions, I fet off
that diſtance from C to g, and draw the
line Gq, cutting the line CD in m; and
the line hi drawn through m; parallel to the
ground line, AB is the true perſpective
fituation of the line HI required. [C]
If I would draw another line, as OP,
farther on the ground plane, and parallel
to the ground line, after having drawn bi,
repreſenting HI; I have no occafion to
meaſure the diſtance of this line from the
ground line, but having taken the diſtance
KM, of theſe parallel lines from one ano-
ther, I ſet it off on the ground line from 9
to r; and drawing Gr, fo as to cut DC in
k, I draw op through k parallel to hi, and
that is the line required.
If I chufe to draw a number of theſe pa-
rallel lines and at the fame diftance from
one another, as in Fig. 5, I continue to
fet the fame diftance from C to a, from a
to c, from c to e, &c. along the ground
9
line
16
A TREATISE ON
:
line AB, and drawing the lines Ga, Gc,
Ge, &c. I find their interfections with the
perpendicular CD at b, d, f, h, &c. then
drawing lines through thoſe points, paral-
lel to the ground line AB, I have their
ſpective fituations. [D]
per-
If I chufe the diſtances of theſe lines to
be unequal, I ſet off unequal divifions along
the ground line.
In this manner may the diftances be
fixed, at which
lines upon
any
the ground
plane, parallel to the ground line, may be
drawn; and when a method hath been ex-
plained of cutting thefe parallel lines, in
any length required, then all fuch lines as
cd, Fig. 2, may be determined.
SECTION
}
{
PERSPECTIVE.
I koji
:
SECTION II.
To draw the perspective fituation of right
lines perpendicular to the ground line.
F I have occafion to find the perſpective
I a
to
place of a line that is perpendicular to the
ground line, as di, Fig. 2, I meaſure its
diſtance from the perpendicular CE, and
fetting it from C, upon the ground line, I
join that point, and the point of ſight D,
which gives the fituation required.
For example; in Fig. 4, if I would
draw the perſpective appearance of a line,
QD, which is perpendicular to the ground
line AB, and parallel to CD; I take the dif
tance QC, and fet it off C to q; and then.
Dq is the line required. For the fame
reaſon, Dr repreſents the line DR; alfo
Ds repreſents DS; and Dt repreſents DT,
all drawn at equal diſtances, and parallel
to one another.
Since all lines perpendicular to the
ground line, when infinitely produced, feem
R
to
18
A TREATISE ON
to meet on the perfpective plane in the
point of fight D, it is called the vanishing
point of thofe lines. [E]
SECTION III.
To draw the perspective fituation of lines
oblique to the ground line.
:
W
Hen the line, whoſe perſpective re-
preſentation I want to fix, is oblique
to the ground line, as ml, Fig. 2, and CI,
Fig. 6. I meaſure the degree of obliquity by
the angle it makes with a perpendicular (as
CE) to the ground line; and whatever that
degree is, I make the angle at E with the
perpendicular CE equal to it, and on the
ſame fide; and to the point in which the
line containing the angle falls upon the ho-
rizontal line, I draw the line required; for
that is the vanishing point of the given
line, and of all its parallels.
Thus fuppofing the angle DCI, in the
plane X, Fig. 6, to be 30 degrees, I make
the angle DEi, in the plane Y, on the fame
fide of the perpendicular, of the fame quan
tity,
PERSPECTIVE.
19
tity, and i will be the vaniſhing point of the
line CI; i. e. it will be the point in which
that line, infinitely produced, will ſeem to
meet the horizon, and alfo that in which
all its parallels will feem to meet it.
N. B. If it be more convenient (as fup-
pofing E to be without the bounds of the
paper) the tangent of the angle of declina-
tion from the perpendicular may be ſet off
along the horizontal line, from the point
of fight D, the diſtance DE being made
the radius.
In Fig. 7, a number of parallels are
drawn to the fame point a, 30 degrees to
the right hand of the perpendicular CD;
and the points in which they meet the
ground line are equidiftant from one ano-
ther. [F]
B 2
PART
20
A TREATISE ON
PART IV.
To draw the perspective fituation of lines not
fituated upon the ground plane.
A
LL lines not fituated upon the ground
plane are either perpendicular, pa-
rallel, or oblique to it; and thoſe that are
parallel may be drawn by the help of thoſe
that are perpendicular to it; fo that there
remain only two varieties in the lines that
are the ſubject of this chapter.
SECTION I.
To draw lines perpendicular to the ground
plane.
F the line I have to draw be perpendicu-
IF
lar to the ground plane, as ce, df, ib,
and lk, Fig. 2. and HK in the plane x
raiſed upright, Fig. 6. it muſt be repre-
fented by a line perpendicular to the ground
line, wherever it is fituated. If, therefore,
the
{
}
PERSPECTIVE.
21
the feat of the line on the perſpective plane
be given, as at b, in the plane Y, Fig. 6,
repreſenting H in the ground plane X, I
raiſe the perpendicular hk, and fomewhere
in that line continued will the line required
terminate. [G]
SECTION II.
To draw the perspective fituation of lines ob-
lique to the ground plane.
IF a line, whofe perſpective fatuation 1
want, has an elevation above the ground
plane, but no declination from the perpen-
dicular; making DG equal to DF radius, I
fet the tangent of it from D to a, Fig. 9, in
the perpendicular CE; or, which produces
the fame thing, I make the angle DG a equal
to the elevation above the horizon (in this
inftance 15 degrees) and will be the va-
niſhing point of the line required, and alfo
of all other lines parallel to it, as ga, fa, ea,
ba, ca, da; all which reprefent lines ftand-
ing upon the ground line, parallel to one
another, having the fame elevation above
the
22
A TREATISE ON
the horizon, and no declination from the
perpendicular.
If the line, whoſe perſpective fituation is
wanted, be both oblique to the ground
plane, and have a declination from the
perpendicular, as mk, Fig. 2, or CK, Fig.
6
; I firſt meaſure the degree of declina-
tion (e.g. 30 degrees) and making the an-
gle DEi equal to it, find i in the horizontal
line, which is that point in which the line
upon the
the ground plane, perpendicularly un-
der the given line, would meet the horizon,
and, through i, draw the line ig perpendicular
to FG. Then taking iy equal to ¿E, I make
the angle iyg equal to the elevation (in this
cafe 20 degrees) which gives g for the va-
niſhing point of the given line CK; and,
confequently (joining C and g) Cg will be
the line required. [H]
In Fig. 10, the line Ca is made to de-
cline 25 degrees from the perpendicular CE,
to the right hand; and to have a depreffion
of 15 degrees below the ground plane of the
picture; and parallels are drawn to it at the
diſtance of one divifion from each other.
i..
The
PERSPECTIV E.
23
The depreffion is made in the fame manner
as the elevation, the angle being taken on
the contrary fide of the horizontal line.
In Fig. 11, the line Ca declines from
the perpendicular 20 degrees, it has an ele-
vation of 10 degrees, and the parallels are
drawn at equal diſtances.
PART
24
A TREATISE ON
}
PART V.
To fix points in perfpective lines, or the doc-
trine of perſpective meaſures.
N the two preceding parts, I have
IN
given directions for drawing perfpective
lines, and their parallels, in all poffible
directions. All that remains to be done,
in the whole buſineſs of this art, is to divide
theſe lines, ſo as to intercept any lengths
that may be required in them. For fince,
as was obſerved before, all objects are con-
tained in, or bounded by lines; if we can
find the fituation or direction of thofe lines,
and cut them in any length or proportion
that is required, we can draw the outlines
of any given objects.
By the interſection of lines, points may
be fixed in every poffible fituation; and
curves are drawn by finding a fufficient
number of points, and joining them with
a ſteady hand. If, indeed, it ſhould hap-
·
pen,
1
PERSPECTIVE.
25
1
pen, that the perſpective curve is a circle,
it will be moft conveniently drawn by the
help of a pair of compaffes.
SECTION I.
To divide any line lying upon the ground
plane, in any proportion required.
CASE I.
To divide a perspective line lying parallel to
the ground line.
IVIDE the ground line, and lay a
DIVI
ruler from thofe divifions to the
point of diſtance in the horizon. Thus, in
Fig. 4, let bi be the perſpective line given; Fig 4.
if from any point in it, as u, I would cut
off any particular length, e. g. two divifions,
I first draw the line Gu, and continuing it
to C, fet off two meaſures from C to s;
and then drawing Gs, I cut the line ui at n
in the proportion required; for un will be
two meaſures, correfponding to Cs.
In
26
A TREATISE ON
12
4
In the fame manner is ba in Fig. 12,
made equal de, or three divifions, and be
to two divifions.
Lines parallel to the ground line may
alſo be divided by marking the divifions
upon the ground line, as before, and draw-
ing lines to them from the point of fight
D. Thus, if in Fig. 4, I draw Ds through
u, and ſet off two divifions from ſ to t, the
line Dt will go through the point n, mak-
ing un equal to two divifions, as before.
N. B. In this cafe, and in all that fol-
low, if from any point, as u, I want a
line of any given length, and the line from
which it is to be cut be not previouſly
drawn; I muſt, firſt of all, draw, with a
black lead pencil, or fomething equivalent
to it, a faint ftroke in the direction required;
and when I have found the exact length,
muſt take out the reft. [I]
CASE
PERSPECTIVE.
27
1
CASE II.
To divide a perspective line that is perpendi-
cular to the ground line.
L
ET DC, Fig. 4, be the perpendicular
propofed, and let it be required to
cut off mC equal to two divifions. To do
this, I fet off Cq equal to two divifions,
and draw the line Gq, cutting the line DC
in m; making mC equal to two divifions,
as was required.
If, from, in the line Dt, which is
another perpendicular to the ground line,
I want to cut off two divifions towards t, I
draw the line G/, and continue it till it cuts
the ground line, which is here at C; and
ſetting two divifions from C to s, draw
Gs, which cuts the line Dt in n, the point
required.
[K]
All the perpendiculars Dd, Dc, DC, Da
and Db, in Fig. 13, are divided in the very
fame manner; the divifions being fet off
from the place where thoſe lines meet the
ground
4
13.
28
A TREATISE ON
}
ground line, and other lines being drawn
from them to the point of diſtance F or G;
and this figure will fhow, that it is a mat-
ter of indifference which of the points F or
G be made ufe of for this purpoſe. It is
only more convenient to take that point of
diſtance which is on the fame fide with
the line to be divided. Thus, if I would
divide the line Db, I fet the diviſions from
b towards a, C and d; and interfect it by
drawing Ga, GC, Gc and Gd. But if I
had wanted to divide the line Dd, I fhould
have fet off the divifions the contrary way,
and have made the interfections from F.
"
CASE III.
To divide a line that is oblique to the ground
L
line.
ET Ci, in Fig. 6, be the line given, and
let it be ſuppoſed to make an angle
of thirty degrees with the perpendicular
CD. If I want to cut off feven divifions
from the point C, I fet feven divifions from
C to b; then, taking the diſtance from the
vaniſhing
PERSPECTIVE.
29
vaniſhing point i of this line, to the point
of diſtance E in the perpendicular DE, and
ſetting it off along the horizontal line to y,
I draw the line yb, which will interſect the
line Ci in b, the point required; Ch being
equal to ſeven divifions.
If from b I have occafion to cut off
any
number of divifions more, I fet them off
from b towards A; and, drawing lines from
y to thofe divifions, I find the points re-
quired.
In the fame manner may any other pa-
rallels to this line be interfected; and if
they are to be interſected in a ſimilar man-
ner, the fame line will ferve for them all,
as in Fig. 14, where the line ad, aC, and
ab, are all divided by lines drawn from the
point x to the divifions marked upon the
ground line C, b, and e. [L]
If I want nothing but a ſingle point in a
given fituation upon the ground plane, I
muſt find its place in the perſpective plane,
by fuppofing it to be the interfection of
two lines, the fituation of which is given;
as
30
A TREATISE ON
ì
to C
as by fuppofing one of them to be drawn
through it perpendicular to the ground
line, and the other to be a line drawn from
the point of diſtance, and interfecting it at
a given diſtance from the ground line.
Thus if I want to find the perfpective place
of the point U, in the plane X, Fig. 4 (as if
I have occafion to fix the foot of a column
in that place) I meaſure the diſtance UM
from the perpendicular CD; and ſetting it
off from C to s, I draw Ds, and conclude,
that this line muſt go through the perfpec-
tive place of the given point U; becauſe
every point in this line Ds is at the diſtance
of the point U from the perpendicular CE.
I then meaſure US, the diſtance of the
point U from the ground line, and ſetting
it off from C to s I draw GC, which will
cut the line Ds in the point u required.
ป
Otherwife, this point may be determi-
ned by ſuppoſing one of theſe interfecting
lines to be parallel to the ground line; for
then I meaſure the diftance US from the
ground line, and, by a preceding rule,
draw the line ki parallel to the ground line.
Then meaſuring the diftance UM from the
perpen-
PERSPECTIVE.
31
perpendicular, I fet it off from C to s, and
drawing the line Ds to interfect the line bi,
I find the point u required.
It is evident that any point upon the
ground plane might, alfo, be found, by
drawing lines making any given angle with
the perpendicular, and interfecting them
at proper diſtances from the ground line;
but the methods here defcribed are much
eafier.
SECTION II.
To divide lines not lying upon the ground
plane.
CASE I.
To divide a line perpendicular to the ground
IF
plane.
F ac, Fig. 15, be the image of a line
perpendicular to the ground plane, and
I have occafion to cut off from the point a
in it any number of divifions; for example
5, I take at random any point d in the ho-
rizontal
32
A TREATISE ON
і
rizontal line, and generally (as being more
convenient) beyond the bounds of the pic-
ture; and from thence draw a line de to
any point e, taken at pleaſure in the ground
line. I then fet off 5 divifions from e to f,
in the perpendicular ek, and draw the line
fd. After this I draw a line from the foot
a of my object, parallel to the ground line,
cutting de in the point i; and raiſing a per-
pendicular from i to the line df, find h;
and this line ib contains the 5 meaſures re-
quired; which I, therefore, fet off from a
to b in the given line ac.
Thus ab is equal
to fe.
The line de and ek, being once drawn,
will ferve to meaſure any other lines per-
pendicular to the ground plane, always
drawing a line parallel to the ground line
from the foot of the object to be meaſured,
to the line de, and then proceeding in the
method deſcribed above.
In order to meaſure objects with the
greater exactneſs, the point d fhould be
taken at a confiderable diftance from the
perpen-
PERSPECTIVE.
33
perpendicular ek; but the meaſures will be
the very fame at whatever diſtance it be
taken.
If a confiderable number of lines are to
be raiſed, all of the fame height, ftanding
in a right line, a line drawn from the va-
niſhing point of that line, to the top of any
of them, will touch the tops of them all,
which fixes their height with very little
trouble. For their tops, being all in a line
parallel to the line that goes through their
bafes, a line going through them muſt
have the fame vaniſhing point as if it had
been fituated upon the ground plane.
Thus the lines a, b, c, d, e, f, &c. Fig.
16, repreſent rods of equal heights ſtand-
ing upon a line, the vanishing point of
which is at x.
Alſo, if a number of objects be of the
fame height in different parts of the picture;
yet, if they all ſtand upon a line parallel to
the ground line, they muft all be drawn of
the fame length. All thofe marked d, in
this figure, are examples of it.
C
[M]
CASE
34
A TREATISE ON
CASE II.
To measure a line oblique to the ground
L'
plane.
ET the line be Cg, Fig. 6, and let
it be ſuppoſed that there be occafion
to cut off 8 divifions from the point C.
Draw CN parallel to the line yg, and
confidering CN as a new ground line, lay
the given number of divifions, viz. 8,
from C to a; join the points a and
Y with a line, cutting the given one in the
point k; and Ck is the part required. Or,
From the vaniſhing point g of the given
line Cg take gy, parallel to the ground
line AB, and equal to gy; on the ground
line itſelf ſet off the given length from C
to [a] and a line joining [a] and [y] will cut
off the required part Ck as before. [N]
The laft method will be found moſt
convenient in general, not only on account
of its being more correfpondent with the
rules
PERSPECTIVE.
35
rules given in the preceding fections; but
alſo, that in this cafe, we do not encumber
that part of the picture, where the images
are drawn, with unneceffary lines and points.
In this example, the given line Cg ſtands
upon the ground line AB; but the rules
above will ferve for lines ftanding on any
part of the ground plane. Should not the
application immediately appear to the learn-
er, he may take the following example :
Suppofe Cg, Fig. 8, to be an indefinite
line, ftanding on the ground plane at
C, and let it be propoſed to cut off a
part of it, from C, equal to a given length,
for inftance, 10 divifions. Find the point
[y] by the directions given above; and
from C, parallel to the ground line, draw
Cb; then confidering Cb as the image of a
line parallel to the ground line, cut off
from C, by Part 5. Cafe 1.
a part Ca
equal the given length, viz. 10 divifions,
and a line joining [y] and a will cut the
given line Cg in k, the point fought.
C 2
If
36
A TREATISE ON
C&
If there be occafion to make feveral
other divifions in the given line Cg; in or-
der to fave the trouble of reducing thoſe
divifions to the line Cb, make Cb equal
the abfolute length of 10 divifions, and
thro' k, draw bk, cutting gy produced, in d;
and d will be a meaſuring point for the line
Cg; fo that the divifions themſelves may
be ſet upon Cb at once, and lines drawn
from thence to d, will cut the given line
Ch in the defired points.
N B. The point d may be found at once,
by taking gd to gy, as the height of the
eye is to the perpendicular CL.
!
ר
逆襲
​L
粥​粥
​FR SK DR SA DR S
丸​丸​枣
​PART
PERSPECTIVE.
37
PART VI.
A fummary account of all the effential rules
for drawing in perspective.
S I have laid down the rules for
AⓇa
drawing in perfpective at full length,
and have illuftrated them by a confiderable
number of examples, it may be uſeful to
the learner to have a general view of what
is eſſential to this art, or a fummary of ail
the rules, reduced into a fmall compafs.
This, therefore, I fhall endeavour to do
for him.
All lines that are parallel to the ground
line, or perpendicular to the ground plane,
in short, all lines that are parallel to the
perſpective plane, muſt be drawn parallel
to each other. All other parallel lines
meet, or have vanishing points in fome
part
of that plane. If the lines lie in any direc-
tion upon the ground plane, they will va-
niſh ſomewhere in the horizontal line;
which is, therefore, called the vanishing line
C 3
of
38
A TREATISE ON
of that whole plane. If the lines be per-
pendicular to the ground line, they vaniſh
in the point of fight; but if they be oblique
to it, or have a declination from the per-
pendicular, the angle of this obliquity, or
declination, muſt be ſet off from the point
of diſtance on the perpendicular, and it will
find the vanishing point on the horizontal
line.
If the line to be drawn be not in the
ground plane, but have an elevation above,
or a depreffion below it, fet off the angle
of elevation, or depreffion, from the point of
diſtance, on the horizontal line, and it will
find the vanishing point on the perpendi-
cular. If the line have both an elevation
or depreffion, and likewiſe a declination
from the perpendicular, ſet off the angle of
declination as before, and through the va-
niſhing point of declination fo found, draw
a line at right angles to the horizontal line;
lay the extent between this vaniſhing point
and the point of diſtance on the perpendi-
cular, along the horizontal line; and from
the point laſt found, ſet off the given angle
of
PERSPECTIVE.
39
}
!
of elevation or depreffion, which will cut
the line, croffing the horizontal line, at
right angles, in the vanishing point re-
quired.
All the meaſures of lines upon the ground
plane are to be laid down upon the ground
line, and the meafuring point of all lines
parallel to the ground line, is either of the
points of diſtance on the horizontal line, or
the point of fight. The meaſuring point
of any line, perpendicular to the ground
line, is in the point of diſtance, on the ho-
rizontal line; and the meaſuring point of a
line oblique to the ground line, is found by
extending the compaffes from the vanish-
ing point of that line to the point of diſ-
tance on the perpendicular, and ſetting it
off on the horizontal line.
A line perpendicular to the ground plane
is meaſured by a triangle, as was particularly
explained, p. 31, Fig. 15. Laftly, a line
oblique to the ground plane is meaſured by
drawing new horizontal and ground lines
from the vaniſhing point and foot of the
given line reſpectively, and taking a mea-
Ї
faring
4.0
A TREATISE ON
i
furing point in this new horizontal line;
fuch, that its diftance from the vanifhing
point, be to the diſtance of the eye from the
fame point, as the height of the eye is to a
perpendicular let fall from the foot of the gi-
ven line, to the original horizontal line; then
lines drawn from this meaſuring point, will
transfer divifions, fet on the new ground
line, to the given image, or the contrary.
.....
PART
PERSPECTIVE.
41
PART VII.
Some more particular directions, to facilitate
the practice of drawing in particular cafes.
I
SECTION I.
Of drawing circles.
Have obferved that all curve lines may
be drawn by finding a number of points
in their circumference, and joining them
with a ſteady hand; and this might be
deemed fufficient in a treatiſe, which pro-
poſes to contain nothing but the neceffary
rules, familiarly explained; but, fince
circles are lines that frequently occur in
drawing, I fhall give fome more parti-
cular directions concerning them.
A circle is drawn with the moſt eaſe by
firft drawing a Square in which it is infcri-
bed. If the projection of the ſquare be an
oblong trapezium, the circle will have an
oblong appearance, that is, it will be an
oval,
42
A TREATISE ON
oval, or more properly, as geometricians
have demonſtrated, an ellipfis. Having
therefore found the image of a fquare cir-
cumfcribing the given circle, draw there-
in two diagonals, and from their interfec-
tion draw lines to the two vanishing points.
of the fides of the fquare, cutting the fides
of the projected ſquare in four points, thro'
which the image of the circle will paſs.
Theſe will be fufficient, in general,
to enable a perſon to draw that image with
tolerable accuracy.
Let abcd, Fig. 17, be the image of the
fquare, in which it is required to inſcribe
a circle. Having drawn the diagonals ad
and bc, and found their interfection at e, I
draw lines through e to ƒ and &, the vaniſh-
ing points of the fides (one of them making
an angle of 40, and the other an angle of 50
degrees with the ground line) and through
the points in which theſe lines interſect the
fides I draw the curve.
If two of the fides of the fquare be pa-
rallel to the ground line, as in Fig. 18, and
confequently have no vanishing points, I
inter-
PERSPECTIVE.
43
interfect the other fides, that vaniſh in the
point of fight C, by a line parallel to the
ground line, as ef
If the figure be large, and very great ac-
curacy be requifite, the images of a greater
number of points in the circumference muſt
be found; and by increafing the number
of thoſe points, the curve may be determi-
ned to the greateſt nicety.
If I want to draw the appearance of a
number of circles, fufpended perpendicu-
larly one above another, or different ſections
of an upright cylinder, I firft draw the
image a of a fquare with a circle inſcribed,
Fig. 19, lying upon the ground; then,
raiſing the perpendiculars e, f, g, and h,
I make another fquare b, at whatever
height is required. This fquare, being
raiſed fo much above the ground plane,
will appear very narrow, and confequently
the circle infcribed in it will be very ellip-
tical. If I had made another ſquare and
circle, at the height of the eye, in D, they
would both have appeared as a right line.
Advan-
i
44
A TREATISE ON
Advancing above the horizon, I draw
the ſquare c, the lines which contain it
being ſtill drawn to the point of ſight D;
but as this ſquare and circle are above the
horizontal line, it is the under-fide of them
that we fee; and as the fquare and circle d
are ſtill higher than c, they are dilated ftill
more; and the higher I carry theſe ſquares
and circles, the lefs oval, and the more
nearly circular will their appearance be.
Another example of fquares and circles,
in a different pofition from thoſe laſt de-
ſcribed, is exhibited, Fig. 20.
The edges
of theſe reft upon the ground plane, and
the whole figure is oblique to the ground
line, making an angle of 15 degrees with
it. As the drawing of a figure like this
may be thought difficult by beginners, I
fhall briefly deſcribe the whole proceſs.
ز
Having pitched upon a point on the
ground plane d, where I would chufe
the neareſt angle in the figure to ſtand
I draw the line de to a point in the horizon
out of this picture, making at E an angle
of 75 degrees (the complement of 15) with
the
PERSPECTIVE.
45
the perpendicular DE. In order to cut
this line at right angles by another line
upon the ground plane, I find the point f,
by making the angle DEƒ equal 15 degrees,
the complement of 75 to 90. In order to
cut as much of the line fd as I want for the
fide of the fquare (fuppofe 4 meaſures) I
take fb equal to fE and thereby find b the
meaſuring point of the line fd, and of all
its parallels, feveral of which will be
wanted in this figure. Setting off 4 di-
vifions upon the ground line, from x (the
place where a line drawn through b and d
would touch the ground line) I make dg
equal to 4 meaſures. Then raiſing per-
pendiculars from d, and g, I make the line
db equal 4 divifions; and, drawing the line
fb, the firft fquare is compleated. This
done, I draw lines from all the angles of
it ij, hk, gl, to the fame vaniſhing point
with de; for theſe lines will determine the
fize of all the fquares; and now I have no-
thing to do but to fet off on the ground
line, from
the place where a line drawn
through a and d would touch it, the dif-
tances at which I would place the ſquares
from one another (fuppofe 4 diviſions.)
ولا
Thefe
1
46
A TREATISE ON
Theſe I mark upon the line de, from the
meaſuring point a; and when I have drawn
the lines mq, nr, &c. to the vaniſhing
point f, and have raifed perpendiculars from
the points m, q, n, r, &c. to the lines hk,
and , the fquares are completed; after
which the circles will be infcribed in them
with eaſe.
All this may be done in lefs time than
the deſcription of it can be tranſcribed;
and with as little trouble may the fame.
number of circles, in any other direction
whatever, be drawn.
Cylinders, being figures terminated at
both extremities by circles, they may be
drawn by firſt making a parallelopiped of
the fame length, and infcribing circles in the
fquares of each end of it; after which, the
line which belongs to the parallelopiped,
being drawn with a black lead pencil only,
may be wiped out.
If the image of a diameter of any circle,
fituated any where on the ground plane,
be
PERSPECTIVE.
47
may be
be given, the image of that circle
determined in the following methods.
CASE I.
When the given image is parallel to the
ground line.
L
ET ab, Fig. 21, be the given image;
biſect it in s, and through s, and the
point of fight C, and points of diſtance
D, G, draw 1s℃,¸tsG, 25D; from
either point of diſtance, D, draw lines,
Db, Dą, cutting 1SC, in 1 and 4, and 14
is a diameter perpendicular to the ground
line. Lay down the diſtances GE, DE on
the horizontal line to F and H, by which,
through a and b, draw Fal, Fb3 and Hb2,
Ha5 cutting tsG and 25F, in t, 3, and
2,5 reſpectively; connect the points 1, 2,
b, 3, 4, 5, a, t, with a curve, which will
be the image required. [O]
}
CASE
48
A TREATISE ON
:
CASE II.
When the given line is oblique to the ground
F
{
line.
IND the image of a diameter which
is parallel to the ground line, and
then proceed as in Cafe I.
Thus, let [ab] be the given image; con-
tinue it to the horizontal line at [g]; take
the diſtance [gE,] and lay it on the hori-
zontal line to [d] the meafuring point of
[ab]; from [d] thro' [a] and [b] draw lines
cutting the ground line in [e, t]; bifect
[et] in [1] and join [ld] cutting [ab] in [h].
the image of the center of the circle; then
through [h] draw [os] parallel to the
ground line, terminating in [de] [dt] and
[os] is the image of a diameter of the fame
circle, parallel to the ground line.
By theſe rules, the image of a circle,
lying in any
any known plane, may be defcribed
from the image of a diameter being given.
An example of one, in a plane perpendi-
cular
་ས་
PERSPECTIVË.
49
cular to the ground plane, is given in the
figure; where hgm is the vanishing line of
the plane and cd the given diameter, thus
making gh, gr⇒ gC, and hm, rk = hC,
rC; then through e, the middle of cd,
draw gef, rneq and hpet; draw hc, hd
cutting gf in o and f; draw kc, kd cutting
qnr in n and q; laftly draw mc, md cutting
tph in p and t: and the points c, t, f, q,
d, p, o, n, connected, will give the required
image.
The methods delivered above become
very uſeful when the images of a great
number of parallel circles are to be drawn;
for the vanishing points C, D, G, F, H,
ferve for drawing the reprefentation of circles
in all planes parallel to the ground plane ;
and g, k, r, h, m, for all thoſe lying in
planes parallel to the plane cfdo.
D
SECTION
50
A TREATISE ON
HTT
SECTION II.
Of different ground planes.
ITHERTO all the objects to be
drawn were fuppofed to ftand upon
the fame ground plane, whofe vaniſhing
line is the horizon; but in drawing land-
ſcapes, and other things, it fometimes hap-
pens, that there are other uniform planes,
of confiderable extent, on which objects
are to be drawn. In this cafe, it will be
convenient to draw the vanishing lines of
thoſe planes, and make uſe of them as ho-
rizontal lines, for the objects that are to be
defcribed
upon them.
Thus, in Fig. 22, the plane a, a, a, a,
is the common ground plane, the vaniſh-
ing line of which is the horizontal line,
paffing through D; and to this point all
lines upon the ground plane, perpendicular
to the ground line, are to be drawn. But
the plane b, has a depreffion of 15 degrees
below the horizon. Its vanishing line,
therefore, paffes through H, DFH, being
15 degrees,
?
PERSPECTIVE.
51
15 degrees, and the plane ccc has an ele-
vation of 15 degrees above the horizon.
Its vaniſhing line, therefore, paffes through
I, the angle DFI, being 15 degrees.
If the plane have a declination with
reſpect to the perpendicular, or be ob-
liquely fituated, with reſpect to the ground
plane, its vaniſhing line may be found by
finding the vaniſhing points of two lines,
drawn on that plane in any direction, pro-
vided they are not parallel to one another
for the vanishing line required, will pafs
through both thoſe vanishing points.
ز
D2
PART
52
A TREATISE ON
PART VIII.
A method of drawing in perſpective from the
ichnography of objects.
IF
any perſon will take the pains to draw
the ichnography, or ground plan of
what he propoſes to draw, expreffing the
true proportions of all the objects he intro-
duces into it, it will be eafy, afterwards,
to reduce it into true perspective. This
method I ſhall illuftrate by an example,
which will render it very eaſy.
Let the object to be drawn be the tri-
angle abc, Fig. 23; and let its fituation, with
reſpect to the grouud line, be the fame
that it has in this figure, being only pla-
ced on the oppofite fide of it, and therefore
in an inverted pofition.
To find the vaniſhing point of any of
the fides of this triangle, as [ac] I produce
that line, till it meet the ground line in d,
and draw Ef, parallel to ad by which
means I find f the vanishing point required;
fo
PERSPECTIVE.
53
ſo that, drawing fd, I conclude that the
perſpective repreſentation of the line [ac]
muſt be a part of it. In order to deter-
mine in what part of this line either of the
extremities, as [c] is to be fixed, I lay a ru-
ler from [c] to E, which cuts the line fd
in e the point required. In like manner,
the perſpective place of [a] will be found
с
to be in a.
By the fame procefs, G will be found
to be the vanishing point of the fide [bc]
and b the perſpective place of [b.] Theſe
three points being joined, the triangle is
completed. But there is no occafion to
find any thing more than the directions of
the lines fd, and eG, together with the
points b and c, for the interfections of the
lines will find the point a.
The vanishing point of the line [bc]
might have been found by the fame method,
but it was unneceffary; becauſe it was de-
termined, by joining the points bc, which
were found as the extremities of the other
lines.
D3
If
54
A TREATISE ON
If the body to be drawn be a folid, I firſt
lay down the ground plan, and when I
have reduced that into perſpective, I raiſe
either perpendiculars, or lines oblique to
the ground plane, from any point in the
ichnography, according to the rules laid
down before.
If I want no more than to fix the per-
ſpective place of a ſingle point upon the
ground plane, as for inftance, that of [c]
I draw a line through it in whatever direc-
tion I pleaſe, to the ground line, as to d.
Then finding, as before, f the vaniſhing
point of that line, I lay a ruler from [c]
to the point of diftance E, which cuts
that line in c, the point required.
But a more ready way of determining
the place of a fingle point as [a] Fig. 24, is
to let fall a perpendicular ab, to draw
the line from 6 to D, the point of fight (or
the vanishing point of the line ab, which
is perpendicular to the ground line) and
then, fetting off, on the ground line bd
equal to ba, to draw a line from d to F,
the
PERSPECTIVE.
55
!
the point of diſtance, cutting the other
line in a, which is the perfpective point
required.
The line ba may be transferred to the
ground line to c, as well as to d; and from
thence a line may be drawn to G, the other
point of diſtance on the horizontal line.
This figure fufficiently fhows, that both
the lines dF and CG will crofs the line bD
in the very fame place a.
[P]
When objects are very complex, confiſt-
ing of a great number of fides and angles,
it may be the moſt convenient to divide the
ichnography of them into ſmall ſquares; and
having thrown the fquares into perſpective,
to note the termination of every line, by
marking its place in the correſponding
fquare. An example, in Fig. 25 (taken
from Mr. Emerſon) will make this method
perfectly intelligible.
In the lower part of this figure, the
ichnography of a piece of fortification is
divided by fquares; and in the upper part
of
56
A TREATISE ON
of it, thoſe ſquares are thrown into per-
fpective, and exhibit a view of the fame
piece of fortification; each point being
placed in its correſponding ſquare.
This is a very common method of draw-
ing in perſpective, and is called the method
of Reticulation.
F
PART
PERSPECTIVE.
57
PART IX.
An easy method of drawing in perſpective
without previous menfuration, by means of
an inftrument to take the angles under
which objects appear.
T
HE rules before laid down for draw-
ing in perfpective, ſuppoſe a previous
knowledge of the real diftances, magni-
tudes, and relative poſitions of all the ob-
jects that are introduced into the picture.
But, in many cafes, a perfon may be
fo fituated, that this knowledge cannot be
obtained; and in many cafes, likewiſe,
the labour which this method requires
would be exceedingly troubleſome and dif-
couraging. I fhall, therefore, in this part,
deſcribe a method of drawing any objects
in true perſpective, without moving from
the place in which they are viewed. It is
alſo a method of drawing that is
very fim-
ple; it requires the knowledge of very few
technical terms; and the rationale
extremely obvious.
of it is
To
58
A TREATISE ON
To fupply the place of actual menfura-
tion, I provide myſelf with an azimuth
qua-
drant, a Siffon's theodolite, or any inftrument
by which I can find the elevation of an
object, and likewiſe its angle of declination
from the perpendicular going through the
point of fight.
Having this inftrument, and placing
myſelf at what diſtance I think moſt con-
venient, from any objects that I propoſe to
draw; I lay down, upon
the paper of my
drawing-board, only two lines, croffing one
another at right angles; one of them FG,
Fig. 26, to repreſent the horizon, and the
other CE the perpendicular, paffing through
the point of fight D. I alſo chuſe what-
ever diſtance I think proper to work at,
and fet it off from D to E.
Having thus prepared every thing for
the operation, I pitch upon any point in
an object before me, as that which cor-
refponds to x, and, by the help of my in-
ſtrument, first of all, find its declination
from the perpendicular to the right or left
hand. Suppofing it to be 10 degrees to
the
PERSPECTIVE.
59
the right, I fet off the angle DEa equal to
10 degrees, and conclude, that the point I
want to fix muſt be fomewhere in the per-
pendicular ae. To find in what part of
this line the point is, I, in the next place,
take its elevation above the horizon, and,
fuppofing it to be 20 degrees, I make ab
equal to aE, and the angle abx equal to 20
degrees, which finds x the point required.
In this method may the fituation of any
other point be found, and theſe points, be-
ing joined by lines, the whole object will
be delineated.
But if the objects to be reprefented con-
tain many right lines, that are parallel to
one another, fuch as occur in buildings,
machines, &c. there will be no occafion
to take many points; becauſe the fituations
of the lines may be determined by vaniſh-
ing points, found by the help of a few of
them. Thus having found y in the fame
manner as I found x, and knowing that
the line xy is parallel to the horizon; I
produce it till it touches the horizontal
line
60
A TREATISE ON
line in c; which is, therefore, the vaniſh-
ing point for that line, and all that are pa-
rallel to it, ſeveral of which are repreſented
in this drawing. The diſtance at which
theſe parallels are drawn, may either be
gueffed by the eye, or be determined with
more accuracy by finding a point at one of
their extremities, in the manner juft now
deſcribed.
Alſo, if I know the angle that any line,
as yz, makes with another line, as xy, the
fituation of which is known, I have no
occafion to take any point, in order to
determine the direction of it. In this
Fig. xyz repreſents a right angle, which
is that which moft frequently occurs in
buildings, machines, &c. To determine,
therefore, the fituation of the line
yz, I
confider that c, the vanishing point of yx,
makes the angle DEc equal to 20 degrees,
the complement of which, from 90, is 70.
I, therefore, make the angle DEH equal
to 70 degrees, and a line EH produced till
it meets the horizontal line (in a place
without the bounds of this picture) gives
the
PERSPECTIVE.
the vanishing point of yz, and of every
other line at right angles with the line xy.
To this point, therefore, I draw the line
yz, and all the others in that figure that
are parallel to it.
The point x is to the right hand of the
perpendicular, and it has an elevation
above the horizon; but it can require no
particular inftruction, or example, to be
able, from this, to fix any point to the
left hand of the perpendicular, and one
that has a depreffion below the horizon.
[Q]
If I would introduce measures into a
drawing made in this manner, or find a
fcale for the picture; I meaſure ſome one
line in the original, as kf. In order to
this, from c, the vanishing point of the
line kf, I take cd equal to cE, which gives d
the meaſuring point of the line; and from
this point I draw two lines, one from each
extremity of the line kf to ib in the line
that bounds the picture, or any other line
drawn parallel to the horizontal line. Then
I divide
62
A TREATISE ON
1
I divide the line ib in the fame proportion
as kf; and by this means I get a fcale,
by which I can meaſure any other line in
the picture, or infert in it other objects of
any given magnitudes, according to the
rules laid down above.
25
枣​丸 ​染
​A
4:
PART
PERSPECTIVE.
63
IN
PART X.
Of orthographical perspective.
the methods of perfpective, the rules
of which have been laid down in the
preceding parts of this treatiſe, the eye of
the ſpectator was fuppofed to be placed at
a definite diſtance from the perſpective
plane; they may, therefore, be called fte-
reographical methods, in allufion to the fte-
reographic projection of the ſphere, in
which the eye has a fimilar fituation; or
rather, fince the fituation of the eye, in the
common method of perfpective, is variable,
the term Scenographic may be more proper.
But there is another method of perfpective,
in which the eye is fuppofed to be placed
at an infinite diſtance from the perſpective
plane. It may, therefore, be called ortho-
graphic, in allufion to the method of pro-
jecting the ſphere orthographically.
This method of perſpective is peculiarly
adapted to the deſcription of machines, and
other things, the magnitudes of which bear
but
64
A TREATISE ON
but a ſmall proportion to the diſtance
which they are generally viewed.
A clearer idea may, perhaps, be concei-
ved of this method of perſpective, by con-
fidering what kind of a ſhadow of an object
would be projected upon a plane, by the
rays of the fun falling perpendicularly
upon it. For the diſtance of the fun is fo
great, that all its rays may be conſidered as
coming parallel to one another.
Since there can be no more than two
pofitions of a right line with respect to the
rays of the fun, viz. perpendicular, or ob-
lique, the perſpective of lines in this ortho-
graphic method is comprehended in two
cafes, and that of angles in another. The
poſition of a line, parallel to the rays, does
not deſerve to be called a cafe, becauſe the
image of it is nothing more than a point.
If the line be perpendicular to the fun's
rays, i. e. to the vifual ray, the image of
it upon the perſpective plane will be a line,
of exactly the fame length. Thus if AB,
Fig. 27. be a fection of the perſpective
plane,
PERSPECTIVE.
65
A
plane, and I would lay down the image of
any line, as ef, I make cd of exactly the
ſame length; for it is plain that it would
be the fhadow of the line ef; the rays
coming in the direction of ed and fe.
If the line be oblique to the vifual ray, as
ce, I lay it down with the fame degree of
obliquity to the fection of the perſpective
plane. Suppofing it to be 25 degrees, I
make the angle dce equal to 25 degrees, and
the perpendicular let fall from e, the remote
extremity of this line will intercept cd, the
image required: For it is evident that cd
would be the fhadow of the line ce in
that fituation. In other words, the image
of any line is always the co-fine of the an-
gle of its elevation above the perſpective
plane, the line itſelf being confidered as
radius.
I fhall illuftrate this method of drawing
in perſpective by examples, in which both
the methods may be compared.
Fig. 28 reprefents a front view of a
cube, drawn in the common method of
perſpective. It ftands upon the ground
plane;
E
t
66
A TREATISE ON
:
{
plane, and its upper fide is terminated by
lines drawn to a vaniſhing point in the ho-
rizontal line. Only two of the fides are
vifible in this pofition. The image of a
circle, infcribed on one of its fides is a cir-
cle, that of the other an ellipfe.
In the orthographic method, the vifu-
al rays being equally perpendicular to every
part of the fide that is oppofed to it, all
that can appear is a plain fquare, with a
circle infcribed, as in Fig. 29.
Fig. 30 repreſents the fame cube in the
common method of perſpective, in a fitua-
tion oblique to the eye, one of the fides
being placed at an angle of 25 degrees,
and confequently the other at an angle of
65 degrees. In this pofition three of the
fides appear, and the images of all the cir-
cles are ellipfes.
Fig. 31 repreſents the cube, in the fame
fituation, viewed orthographically. The
fide cd was found by drawing ce, Fig. 27,
equal to the true length of the fide, ma-
king the angle dee equal 25 degrees (being
the
PERSPECTIVE.
67
the angle that this fide was fuppoſed to
make with the vifual rays) and letting fall
the perpendicular ed; fo as to intercept the
line cd, the projected length required.
The length of the line bc, Fig. 31, was
found by taking the line de, Fig. 27, the
fine of the angle of elevation of the former
line. It might have been found by placing
the line ce at an angle of 65 degrees, and
taking the co-fine of the elevation, as before.
For, reverfing the figure, the line ce will
be feen to be in that pofition, and ef equal
to de will be the line required.
Bifecting the fides of the two parallelo-
grams made upon theſe lines, Fig. 31, points
are found for the termination of the axes of
the ellipfes, into which the infcribed cir-
cles are projected.
Only two of the fides of the cube are vi-
fible in this pofition, becauſe one line in
the figure, viz. ch, is perpendicular to the
vifual
ray.
If the pofition be changed, and dx be
confidered as the bafe, it will repreſent the
E 2
cube
68
A TREATISE ON
cube refting on that fingle line; the line
ch being drawn forwards; fo that cd, or
hx will make an angle of 25 degrees with
the perpendicular; the confequence of
which will be, that the upper fide of the
cube chyb will come into view.
:
If the figure be fuppofed to reft on the
line yb, it will reprefent the cube leaning
ftill more forwards, and more of the upper
fide hcxd in view.
Inſtead of the cube being fuppofed to
reft on a ſingle line, in thefe cafes, we
may conceive it to be viewed obliquely,
the eye being raiſed above the plane on
which it ſtands; ſo that the upper fide of
fo
the cube may be feen, without its being
raiſed towards the eye.
•
If I would reprefent three fides of this
cube as feen orthographically, it is evident
that it muſt both ſtand obliquely, as in Fig.
31; and alſo that the eye muſt be elevated
above the plane on which it ftands. Sup-
poſe then, every thing elſe in this figure
remaining the fame, the eye be raiſed 10
degrees,
PERSPECTIVE.
69
degrees, as in Fig. 32, it will be evident,
that the lines ch and dx muſt be made
ſhorter than the correfponding lines in the
preceding figure, for being inclined 10 de-
grees to the viſual ray, they muſt be leſ-
fened in the proportion of the co-fine of
that angle to radius, and reduced from the
length ci, Fig. 27, to cj. Alſo every
line in the figure, parallel to them, muſt
be made of the fame length.
The lines cd and cb muſt now no longer
be one line, for the upper fide of the cube
will be ſeen, and confequently the baſe
would be ſeen too, were the figure tranſ-
parent.
In order to determine the reduced
lengths of the fides cd and cb, and at the
fame time the angle they make with each
other, in this fituation, I lay down the
fides and angles DCb, Fig. 33, in their real
fize and poſition, and draw Ce, from the
point C, in the fituation in which a plane
would pass through the eye and that point,
in the pofition given before, i. e. fo as to
make the angle eCb equal 25 degrees, and
eCD equal 65. Perpendicular to Ce, and
through
70
A TREATISE ON
through the point b I draw the line be,
and produce it till it meet the line CD,
continued, in f. Upon the center e,
with the radius eC, I defcribe the arc Ch,
and from I ſet off hg equal to the an-
gle of the elevation of the eye.
Then pa-
rallel to th, I draw gc, cutting the per-
pendicular Ce in c, the point into which
C will be projected. Joining cb, I have
the image of the fide Cb, and drawing Dd
parallel to Ce, to cut fe in d, I get de
for the other fide. Alfo the angle dcb, is
the projected angle required. [R]
Accordingly, in Fig. 32, I make deb the
fame angle as that in Fig. 33, contained
under the fame fides; and completing all
the parallelograms, the projection of the
whole figure is determined.
If the line cd be not drawn parallel to the
ground line, the angle it makes with it, ad-
ded to the angle ncb, muſt always make 10
degrees, or any other angle that is required.
Inſtead of fuppofing the eye to be raiſed
above the plane of the bafe, the image
would
PERSPECTIVE.
71
would be the fame, if the cube were fup-
poſed to rest upon the corner c, the line
ch leaning forwards, in an angle of 10 de-
grees.
It may be convenient to apply this me-
thod of orthographic perſpective in draw-
ing very complex figures, when it would re-
quire a good deal of time to find the va-
nifhing points of all the fides. If a view
in common perſpective be neceffary, it will
be eafy to reduce this orthographic projec-
tion to it by the rules laid down, in Part 8,
to draw the perſpective appearance of ob-
jects from their ichnography, which is gi-
ven by this orthographic method.
In drawing machines in general, there is
no occafion to lay down angles and fides
with this exactneſs. It will be fufficient
to lay down the ichnography of the object
to be defcribed; as for example, the paral-
lelogram, in Fig. 34, in whatever pofition
may be thought moft convenient, and
drawing the elevation in perpendiculars over
the baſe. Thus in Fig. 35, a hollow paral-
lelopiped, or trough, is reprefented ſtanding
upon
72
A TREATISE ON
I
upon the baſe of Fig. 34. A figure like
this is perfectly intelligible, and much
more eafy for a workman to copy after
than any other drawing whatever; becauſe
all the dimenfions are taken from a com-
mon fcale, and the imagination may be
fufficiently affifted to conceive of it by good
fhading. It might not, perhaps, be amifs
for perfons who are learning to draw in
perſpective, to begin with this ſimple me-
thod, and accuftom themſelves to it for
fome time, before they attempt any
other. [S]
......
1
PART
PERSPECTIVE.
73
WH
PART XI.
Of Shadows.
WHEN objects are feen in a ſtrong
light iffuing from one place, they
project ſhadows, which it is very conveni-
ent, at leaſt ornamental, to draw in the
perſpective views of them; I propoſe,
therefore, in this part, to lay down rules
for delineating theſe fhadows, beginning
with thoſe which are projected on the
ground plane on which the objects ftand,
as being the most common, and the moſt
uſeful cafe.
SECTION I.
To draw the shadows of bodies on the fame
planes on which they are fituated.
INCE rays of light always iffue in
SING
ftraight lines, it is evident that the ſha-
dow of any object will cover juſt as much
Ipace, as it would hide from an eye fituated in
}
the
74
A TREATISE ON
;
the place of the luminous body. For the fame
reaſon, the ſhadow of any point of an ob-
ject must be fomewhere in a right line
drawn on the ground plane, from a point
perpendicularly under the luminous body,
through the point that is perpendicularly
under that whofe fhadow is required; and
the precife point of this line on which the
fhadow will fall, will be determined by a
right line drawn from the luminous body
through the point of the object.
This obfervation fupplies a rule for pro-
jecting the ſhadows of all bodies whatever.
Find the point upon the ground plane that
is perpendicularly under the luminous body.
From this draw right lines through per-
pendiculars let fall from thoſe points, the
fhadows of which are required, and inter-
fect them by right lines drawn from the
luminous body through thofe points.
Thus, in Fig. 36, if the luminous body
be at a, perpendicularly over b, upon the
ground plane, and it be required to find
where the fhadow of c, the top of the line
cd (in one angle of the folid [x]) will fall;
fince
PERSPECTIV E.
75
fince ed is a line perpendicular to the ho-
rizon, I draw a right line through b and d;
and drawing another right line through a
and c, interfect it in e; which will, there-
fore, be the place of the fhadow required.
Finding, in the fame manner, ƒ the place
of the fhadow of g, and b the place of the
ſhadow of i, and joining the points d, e, ƒ
and b, I have the out-lines of the fhadow
of the whole folid [x].
If I want the ſhadow of the rod [y]
which ftands oblique to the ground plane,
I let fall a perpendicular kl from the top
of it, k, to the ground plane, and drawing
the lines ak and bl, I find that they inter-
fect each other at m; which, therefore,
marks the place of the fhadow of k; and
fince the rod is a ſtraight one, and confe-
quently all the points of the fhadow muſt
fall on the fame right line, I draw a line
from d, the foot of the rod, to m, and thus
get
the ſhadow of the whole rod.
In the fame manner may be drawn the
fhadows of folid bodies, the baſes of which
project beyond the perpendiculars let fall
from
76
A TREATISE ON
from them. Firft find the place where the
perpendicular would fall upon the ground
plane within the folid, and proceed with
it, as if nothing folid had been in the caſe.
If the fhadows of objects be made by
the light of the fun, the fituation of that
luminary, with reſpect to the picture, muſt
be determined, and this is done by the
help of the following confiderations.
By reaſon of the immenſe diſtance of the
fun, it muſt always be ſuppoſed to be in,
or over fome point in the horizontal line,
in which the ground plane, when extended
to an infinite diftance, is conceived to va-`
niſh; and the particular point in the
horizontal line must be determined from
the number of degrees that the fun is fitu-
ated to the right or left hand of the per-
pendicular that goes through the point of
fight; in the fame manner as the vaniſhing
point of a right line upon the ground
plane, that is oblique to the ground line, is
found.
!
}
If,
PERSPECTIVE.
77
If, for inftance, the fun be 30 degrees to the
left hand of it, I make the angle DEH, Fig.
37, equal to 30 degrees, and H will be
that point in the horizontal line over which
the fun may be fuppofed to be perpendi-
cular; and nothing is now wanting to fix
its precife place above the horizon, but to
know its altitude, which is laid down in
the fame manner as the vanishing point of
a line that is elevated above the ground
plane. If, for inftance, the fun be 35 de-
grees high, I make HI equal to HE; and,
raifing the perpendicular HK, make the
angle HIK equal to 35 degrees, and K will
be the fun's place.
If, now, I would draw the fhadow of
any object made by the fun in this fituation,
I fuppofe K to be the place of the candle
a, in Fig. 34, and H to be b, the point
upon the ground plane that is perpendicu-
larly under it, and I proceed exactly as in
that example. Thus, to draw the fhadow
of the folid [x] Fig. 36, the place of the
fun being at K, I draw Ha, and interfecting
it by Kb in c, I have the place of the ſha-
dow of b; and joining ac, I have the ſha-
dow
78
A TREATISE ON
dow of the perpendicular line ab. In like
manner I find d, the fhadow of f, at the
top of the line ef; and alſo g, the ſhadow
of the vertex of the line bi.
theſe points, I have the
Then, joining
fhadow of the
whole object [x]: for it is evident, from the
poſition of it, that the fhadow of any other
line in the figure muft fall within that of
theſe.
When, as in this example, the place of
the fun is fixed above the horizontal line,
it is evident, that the ſhadows will be pro-
jected as falling towards the fpectator, and
that they will always be larger than the
objects. To draw the fhadows which are
made by the fun, having; the fame degree
of elevation behind the fpectator, which
will make the fhadows fall nearer the hori-
zontal line, and lefs than the objects; the
perpendicular from H, Fig. 37, muſt be
let fall to L, and the angle HIL muſt be
the fame as HIK in the former cafe; and
then, if the point L be made ufe of, in-
ftead of K; as theſe points are ſituated on
the contrary fides of the horizontal line,
the ſhadows of objects will be repreſented
as they would be made by the fun placed
on
PERSPECTIVE.
79
on the back of the fpectator; and this po-
fition of ſhadows is generally thought more
agreeable in a picture than the other.
To find, in this manner, the fhadow
of the line ab, in the folid [x] Fig. 39, I
draw He to the foot of it, and Lb to the
top, interfecting each other in c, the place
of the shadow. Finding, in the fame
manner, where the ſhadow of the line ef,
behind the body, would fall, I join that
point and c, by a line which cuts the fide of
the figure in d, where only the ſhadow begins
to be visible. Thus joining a, c, and d,
the ſhadow of the folid [x] is completed.
In both theſe cafes, the fun is fuppofed
to be fituated either before, or behind the
picture. If it be in the plane of the pic-
ture, it is evident that its place muſt be in
the horizontal line, at an infinite diſtance
from the point of fight. Confequently all
lines proceeding from it may be fuppofed
parallel to one another. To draw the iha-
dows of objects made by the fun in this
fituation, draw lines parallel to the hori-
zontal line from the foot of the object,
and
80
A TREATISE ON
$
3
:
and interfect them by lines making angles
with the horizontal line equal to the fun's
altitude.
This practice is very eafy by means of
the drawing board and fquare. Thus if I
would draw the fhadow of a folid [x] Fig.
40; with the ruler of the fquare parallel to
the ground line AB, I draw ac; and fixing
the moveable tranfverfe of the ruler in
the angle of the fun's elevation, I draw the
line bc, which gives c for the place of the
ſhadow for b. In the fame manner alſo,
I find d the place of the ſhadow of e; and
joining the points dc, I get the ſhadow
of the whole figure; though it is fo fitu-
ated, that part of it is hid by the object.
If there be ever fo many lines, the fhadows
of which are to be found, once fixing of
the ruler ferves for them all. I have no-
thing to do but to flide it along the draw-
ing-board, till I come to the points from
which the lines are to be drawn, both for
the lines that are parallel to the horizon,
and thoſe that interfect them.
To
PERSPECTIV E.
8 1
To draw the ſhadow of the folid [y] Fig.
40, which lies oblique, both to the horizon,
and to the ground plane, I draw ac from
the foot of the line ab, parallel to the ho-
rizon; and fixing the moveable tranſverſe
of the ruler, fo as to make an angle equal
to the fun's altitude, I draw bc. In the
fame manner I alſo find g, the fhadow of f;
then drawing gc, and joining ed, the place
where the neareſt extremity of the folid
touches the ground, I am able to compleat
the outline of the whole fhadow; for the
fhadows of all the other lines muſt fall
within theſe.
Let it be obſerved, that it makes no dif-
ference in the fhadow, whatever be the
height, or ſhape of objects, provided their
tops, and every part of them, be in the
fame right lines proceeding from the lumi-
nous body. Thus the fhadows of the ſe-
veral objects a, b, c, and d, Fig. 41, per-
fectly coincide, from the plane where each
of them begins.
F
SECTION
82
A TREATISE ON
SECTION II.
Of Shadows intercepted by other objects.
HEN the ſhadow of a line falls
WH
upon any object, it muſt neceffa-
rily take the form of that object. If it
fall upon another plane, it will be a right
line, and if upon a globe, or cylinder, it
will be circular.
If the body intercepting it be a plane,
whatever be the fituation of it, the fhadow
falling upon it might be found by pro-
ducing that plane till it intercepted the
perpendicular let fall upon it from the lu-
minous body; for then a line drawn from
that point would determine the ſhadow,
juſt as if no other plane had been con-
cerned.
But the appearance of all theſe ſhadows
may be drawn with lefs trouble, by firft
drawing it through theſe interpofed objects,
as if they had not been in the way, and
then making the fhadow to afcend perpen-
dicularly
PERSPECTIVE.
83
ל
dicularly up every perpendicular plane,
and obliquely on thoſe that are fituated ob-
liquely, in the manner that I ſhall now
defcribe.
Finding that the fhadow of the upright
pole a, Fig. 42, would have reached, in a
ftraight line, to b, if it had not been inter-
cepted by the object [x]; I firſt draw the
ſhadow through that object, and finding
that it firſt touches it at c, where it muſt
afcend perpendicularly to the ground line, I
raiſe the perpendicular cd; and fince the
upper furface of this body is a plane, pa-
rallel to the ground plane, I make that part
of the fhadow de parallel to the reft of the
fhadow upon
upon the
the ground plane.
If the firft fide of the interpofing object.
that the ſhadow falls upon be hid from the
eye, as in Fig. 43, I ftill draw the line ab,
as if it were viſible, and where the fhadow
meets with it, raiſe the perpendicular bg, as
before. Or, I may begin at the termina-
tion of the ſhadow, and tracing it back to
the place c, where it cuts the interpofing
body, find the point f, where it left that
F 2
new
84
A TREATISE ON
}
new ſurface, and draw the part fg of the
ſhadow parallel to the reft of it, as before.
If the ſurface of the body interpofed be
an inclined plane, I find, by perpendiculars,
as a and b, Fig. 44, both where the ſhadow
enters the oblique furface, at e, and
where it leaves it at d; and then joining
the points c and d, I have the direction of
the ſhadow on that oblique plane.
If theſe planes change ever ſo often, I have
only to draw perpendiculars from them all,
and the direction of the fhadow will be
found with great eaſe.
If I want a fhadow of only part of an
object, or an object not ſtanding upon the
ground, as for inftance, the part [ab] Fig.
44, I fuppofe it to be continued to the
ground, and I find where the fhadow of
[b] falls, as at e, in the fame manner as if
the pillar had been no higher; and then
the part at ƒ, lying upon the ground plane,
together with the part de, upon the incli-
ned plane, will be all the ſhadow of [ab]
required.
If
!
PERSPECTIVE.
85
If the pillar reach no higher than [b]
and it had been required to find on what
part of the inclined plane its fhadow would
terminate; I fuppofe the object to have
been continued higher, till fome part of
the fhadow had fallen upon the ground
plane, as at f; then drawing the perpen-
dicular bd, I find d, the place where the
ſhadow would have left the inclined plane,
if the object had been continued; and the
point e, where a line from [b] cuts the
line dc, is the termination of the fhadow
required.
If the object upon which the fhadow is
thrown be circular, the fhadow will be
circular. It muft, therefore, be drawn
with juſt ſuch a degree of curvature, as
the body itſelf would have had, if it had
been interfected by a plane in the direction
of the ſhadow. Thus, in Fig. 45, when
the ſhadow of the pole comes to the cylin-
der, lying obliquely to the horizon, it goes
over it, in the form of an ellipfe, fuch as
would have been made, if the cylinder had
been interfected by a plane in the direction
of the fhadow. The curvature may be
F 3
drawn
$6
A TREATISE ON
drawn with ſufficient accuracy, by obſerving
where the fhadow cuts the line repreſenting
the baſe of the cylinder, and raiſing a per-
pendicular on that point, equal to the cy-
linder's apparent diameter, for this will
find the place where the fhadow goes over
it. Thus, the fhadow croffing the baſe of
the cylinder at a, I raife the perpendicular
ac, cutting the line repreſenting the higheſt
part of the cylinder in c, and the ſhadow
will pafs over that point.
WH
SECTION III.
Of faint fhadows.
HEN objects are not fuppofed to
be viewed by the light of the fun,
or of a candle, but only in the light of a
cloudy day, or in a room into which the
fun does not fhine, there is no fenfible
ſhadow of the upper part of the object, and
the lower part only makes the neighbouring
parts of the ground on which it ſtands a
Fittle darker than the reft. This imperfect,
obfcure kind of fhadow is eafily made,
being nothing more than a fhade on the
ground,
PERSPECTIVE.
87
ground, oppofite to the fide on which the
light is fuppofed to come; and it
may be
continued to a greater or leſs diſtance, ac-
cording to the fuppofed brightneſs of the
light by which it is made. It is in this
manner (in order to fave trouble, and
fometimes to prevent confufion) that the
fhadows in the greater part of drawings are
made. Examples of theſe ſhadows may be
feen in Fig. 15, 16, 26, and fome others
in this work.
SECTION IV.
To draw the reflected images of objects in
TH
water.
HE appearance of the image of any
object, as viewed in a plain mirrour,
is exactly of the fame dimenfions with the
image itſelf, but inverted with reſpect to it.
If, therefore, from the foot of the repre-
ſentation of any object, ftanding near the
water fide, a fimilar image be drawn down-
wards, in an inverted pofition, of the ſame
length and form with the repreſentation.
itſelf, the part that falls upon the water is
the
88
A TREATISE ON
the reflected image, or as much of it as can
be ſeen in that fituation.
Thus b, Fig. 46, is the reflected image
of the object a, ſtanding cloſe to the water
fide; c is as much of the image of d,
ſtanding upon the plane, at fome diſtance
from the water, as can be ſeen in that po-
fition; and e is as much as can be ſeen of
the image of f, ftanding upon an eminence
near the water.
If the object be oblique to the ground
plane, as h, give the image g an equal de-
gree of obliquity, and that will exhibit the
true image required.
Theſe objects are uniform throughout.
When objects of other forms are to be re-
preſented, let it be remembered, that,
fince theſe images must be inverted, that
part which is higheſt in the object muſt be
loweſt in the image, and vice versa. Thus
the tops of trees, and the heads of men
muſt be drawn the fartheft from the brink
of the water.
:
PART
PERSPECTIVE. 89
$
PART XII.
General advices and directions, relating to the
art of drawing in perspective.
It di
T diſcourages many perfons, who would
be glad to learn the art of perfpective,
to think that they muſt be obliged to mea-
fure every thing that they draw; and, it is
true, that all the rules of perfpective,
except thoſe laid down in Part ix. do
fuppofe the objects to have been mea-
fured, and no drawing can be perfectly
accurate without it. But where extreme
accuracy is not required (as indeed it very
feldom is) there is little occafion to mea-
fure any thing; and yet a perfon who has
a juſt idea of the nature of perſpective, will
make a drawing infinitely more juſt, and
agreeable, than another perfon can, who
ſhall even meaſure every thing.
This is more eſpecially the cafe where a
great number of parallel lines, cutting each
other at right angles, or any given angles,
are to be reprefented; fuch as always oc-
cur
go
A TREATISE ON
cur in drawing buildings, machines, fur-
niture of houſes, &c. For fince all pa-
rallel lines, not parallel to the picture, have
the fame vaniſhing points, if a perfon only
know where to fix thofe points, fo that
lines drawn from them fhall make any
given angle with one another, his drawing,
by the help of thoſe points, will be far more
agreeable to truth, and look infinitely
better, than if he had gone to work with-
out that previous knowledge to guide him.
Befides a perſpective drawing is always
made in far leſs time than any other.
A perſon who underſtands this art, has
nothing to do but to judge by his eye of
the proportion of objects, and all his
lines are fure to be perfectly true upon that
fuppofition; and, therefore, the whole will
be confiftent with itſelf. But the random
defigner may, in one draught, make a
hundred different fuppofitions; and as it
is a great chance if any two of them agree,
the whole will be very inconfiftent with
itſelf, and, therefore, muft make a very
aukward appearance.
This
PERSPECTIVE.
91
This confideration, I think, is a matter
of conſequence for the encouragement of
young defigners. I fhall, therefore, give
them an example of it, in my own practice.
The drawing, plate vii, which is a per-
fpective view of an electrical machine of
my conſtruction, was made when I had
very little knowledge of the theory or
practice of perſpective. I only knew how
to fix vanishing points in the horizontal
line, according to any pofition of lines
upon the ground plane; but as all the
planes in this machine were either per-
pendicular to the ground plane, or parallel
to it, I was fenfible that I wanted no other;
and though I knew but little of the doctrine
of perſpective meaſures, I could judge
nearly enough of the proportion of the
feveral lines by my eye, and therefore
could do without them.
All that I did, therefore, was to con-
fider the center of the globe as the point
of fight, and to draw the horizontal line
and perpendicular croffing one another at
right angles in it. On the perpendicular
I fet
92
A TREATISE ON
I fet off the diſtance at which I choſe to
work, and from the fame point ſet off
angles of 35 degrees on one fide, and 55
on the other, which gave me two vaniſh-
ing points, from which I could draw lines
cutting one another at right angles. Then
obferving the angle under which the whole
machine was viewed in the fituation I choſe
for myſelf, and having the machine before
me, I made every thing as nearly in pro-
portion to it as I could judge by my eye.
As I meaſured nothing, I drew no ground
line, but, when I had done, cloſed the
drawing where I thought proper. It is a
fault in this drawing, that I fixed the point
of diſtance too near the point of fight,
But this doth not make the drawing lefs
juſt, or leſs uſeful.
The electrical battery, plate iii, was
drawn at the fame time. Both the fides
have equal degrees of obliquity to the ho-
rizontal line; and therefore I drew all the
lines to one or other of the points of
diſtance (or the tangents of 45 degrees) on
each fide of the point of fight. Here,
alſo, I had no meaſures, or gound line.
With
PERSPECTIVE.
93
With more knowledge of the art of
perſpective I might have made more elegant
drawings of theſe figures, but theſe are
perfectly juſt, and fufficiently plain. In
theſe, and in many other cafes, I have often
been agreeably furprized to find how much
uſeful practical knowledge may be derived
from a very little theory.
I could engage to communicate to any
perfon all the knowledge that is requifite
to make theſe two drawings in two or three
minutes, whereby he might finiſh them, at
his leifure, in little more than an hour each
and without that little knowledge, a perfon
might puzzle himſelf a month about them,
and, after all, produce nothing that ſhould
not be quite prepofterous.
;
It is a maxim with defigners, founded
upon experience, that no picture fhould
take in quite fo much as is comprehended
within an angle of 90 degrees. In other
words, the figure fhould not extend to both
the points of diſtance on the horizontal
line. The reafon of this rule is, that the
eye cannot, at one time, diſtinctly take in ſo
great
94
A TREATISE ON
ジン
​great a compaſs, but the extremities of the
profpect will be confuſed and indiftinct.
In general, the leaft diftance fhould be
about one-third more than the height of
the eye.
In many drawings it ſhould be
much greater.
In drawing landſcapes, a low horizon is.
particularly agreeable. It fhould be about
one-third of the depth of the picture.
The rule laid down above for chufing
the diſtance, ſhould be obſerved even where
very high horizons are chofen, in order to
give what is commonly called a bird's view
of objects. Thus [a] Fig. 47, is a view of
a plain ſquare inclofure, with a pretty low
horizon, and [b] is a view of the fame
ſquare incloſure with a higher-horizon, and
a proportionably greater diftance. By this
means more of the infide of the incloſure is
feen.
I fhall
PERSPECTIVE.
95
I fhall cloſe this part of the work with
the following extracts from Mr. Emerfon's
Perſpective.
"If any print or perfpective view, be
looked at through a lens, whoſe focal diſ-
tance is equal to the principal ray, and the
print placed in its focus; it will be ſo mag-
nified, as to have the very fame appearance,
as at the place it was drawn for.
"Glaffes which are not of a due focal
length, will not give the exact appearance
of a place. Shorter glaffes make the dif
tances lefs, and fo contract the view. And
longer glaffes make the diſtances greater,
and extend the view.
"Since glaffes of a long focal diſtance,
give a large and extenfive proſpect of a
country, therefore they are better than
fhorter glaffes. And when the profpects.
are well drawn, and properly coloured; it
is very delightful to view them through a
good glaſs, as they fo nearly imitate nature.
And though there is but one focal length
that
}
96
A TREATISE ON
EX
that will give a true appearance, yet the
draught will always appear a regular piece
of perſpective, though it may not exactly
repreſent any place in the world, fuppofing
the eye placed fomewhere in the principal
ray. And the draught will feem longer
in proportion to the focal diſtance of the
glaſs made uſe of; or in proportion to the
apparent diſtance of the neareſt part of the
picture.
"As there is nothing more pleaſant
than viewing the draughts of countries,
towns, cities, magnificent buildings, and
other grand objects, when well drawn: to
fee them to the beſt advantage, the focal
diſtance of the glaſs ſhould be juſt ſo long,
as not to fhew the fcratches and coarfenefs
of the engraving; or not much longer; for
then the view will be narrow, and the parts
too small to be feen fo far off. And if it
be far off, it will hide the beauties as well
as deformities. And to get a proper glaſs,
obferve at what diſtance the fcratches dif-
appear to the naked eye, and that is the
focal length of the glafs. Perfpective views
ſhould be drawn, ſo that the point of view
be
PERSPECTIVE.
97
if
be farther off, than is generally practiſed,
you would have the piece to be a true
copy of nature. The principal ray ſhould
not be less than two feet, and then the
draught, being looked at through a lens of
that focal diſtance, will appear in per-
fection, and give a true reprefentation of
the place it was drawn for. The view
fhould be fo large as to fubtend an angle at
the lens of about 30 degrees. And it is
proper to put the lens in a ſhort ſquare tube;
which will confine the fight, and direct it
to the perſpective draught; which, to
complete its beauty, ought to be coloured
with the fame colours as the natural objects
appear in. For this purpoſe water co-
lours need only be uſed."
G
PART
98
A TREATISE ON
PART XIII.
The definition of all the technical terms made
ufe of in this treatije.
T
HE ground plane; the plane on which
both the fpectator, and the objects
that are to be drawn, ſtand.
The perspective plane; a plane ſtanding
perpendicularly upon the ground plane,
interpofed between the eye and the ob-
jects. On this plane (as on à glaſs window)
the images of objects are ſuppoſed to be
intercepted; fo that their perspective ap-
pearance is the appearance they have on
this plane.
The ground line; the line on which the
perſpective plane is ſuppoſed to reſt.
per-
The point of fight; that point in the
ſpective plane which is neareſt to the eye,
and at the ſame diſtance from the ground
line with the height of the eye above the
ground plane. N. B. A line drawn from
the eye to the point of fight is fometimes
called the principal ray.
The
1
PERSPECTIVE.
99
The horizontal line; a line upon the per-
ſpective plane, drawn through the point of
fight, parallel to the ground line.
The perpendicular; a line on the per-
ſpective plane, drawn through the point of
fight, perpendicular to the ground line
and the horizontal line.
Points of diſtance; points on the perfpec-
tive plane, ſet off from the point of fight,
fometimes on the horizontal line, fome-
times on the perpendicular; at the fame
diſtance from the point of fight that the
eye is ſuppoſed to be at from the perſpective
plane.
Vanishing points; points on the perfpec-
tive plane, in which parallel lines, infinitely
produced, feem to meet.
Meaſuring points; points from which any
lines in the perſpective plane are meaſured,
by laying a ruler from them to the divi-
fions laid down upon the ground line.
G 2
A GE-
A
GENERAL VIEW
}
O F
THE THEORY OF
PERSPECTIVE.
WITH
NOTE S,
RELATING TO IT.
G 3
A
[ 103 ]
tttttttttttttttet tek tettet
※
**;
A GENERAL VIEW of
THE THEORY
OF
PERSPECTIVE,
To which fome of the particular DE-
MONSTRATIONS in the following
NOTES refer.
DEFINITION I.
The vanishing line of an original plane, is
a right line in the picture, formed by the
interfection of an imaginary plane paffing
through the eye, and parallel to the ori-
ginal one, with the picture.
T
HUS, in Fig. 4, having raiſed the
planes Y and Z, if we ſuppoſe a
plane to go through the eye at a, and to be
parallel to the ground plane X, that plane
will
104
A TREATISE ON
will cut the picture Y, in the horizontal
line FG; which, therefore, is the vanish-
ing line of the plane X, or ground plane.
A little attention will make it evident,
that no point in the original plane X, can
have its image in the picture, above the
line FG, fo long as the poſition of the
picture and the eye remain the fame; and,
that if the plane X were infinitely extended
beyond the picture, yet its perſpective ap-
pearance, in the prefent example, would
be a finite ſpace, bounded by the vaniſhing
line FG.
DEFINITION II.
The vanishing point of an original line, is
that point in the picture, where an ima-
ginary line, drawn from the eye, parallel
to the given line, cuts it.
HE planes Y and Z, Fig. 4, being
THE
raiſed as before, the vanishing point
of the line CD, on the plane X, perpen-
dicular to the picture, will be the point D ;
for it is plain, that an imaginary line drawn
from
PERSPECTIVE. 105
from the eye a, to D, is parallel to the
line CD, fince CD ba, and the line
aD is alſo perpendicular to the picture.
It will not be difficult to conceive, that
were the original line CD, extended be-
yond the picture, to any diſtance whatever,
the image of no point in it could appear
higher in the picture, than the point D.
PROPOSITION I.
Suppoſing the picture a plane, which may be
extended every way at pleasure, the line in
which any original plane, not parallel to
the picture, cuts the picture (and which
may be called the interfecting line of that
plane,) is parallel to the vaniſhing line
of the fame plane.
1
FOR, fince the imaginary plane, paffing
through the eye, and producing the
vaniſhing line, is parallel to the original
plane, the lines formed by their interfec-
tions with any other plane, as the picture,
will be parallel (16 Eu. 11.) Thus the
ground line AB, Fig. 4, which is the in-
4:
terfecting
}
106
A TREATISE ON
terfecting line of the ground plane, is pa-
rallel to the horizontal, or vanishing line of
that plane.
Corollary. Planes parallel to the picture,
can have no vaniſhing line.
For a plane paffing through the eye pa-
rallel to fuch planes, is alſo parallel to the
picture, and therefore can never cut it.
PROPOSITION II.
The image of every original line, perpendi-
cular, or oblique to the picture, tends to its
vaniſhing point: and the images of fuch
original lines, as are parallel to the picture,
can have no vanishing points; they will,
therefore, be parallel to their respective
originals.
L
ET the planes, Fig 4, be prepared as
before; and let any original line, as
KM, be produced till it cut the picture in
C (which point is called the interfecting
point of the line KM) join this point and
the vaniſhing point D, of the line; then,
the appearance of every point in the ori-
ginal line, infinitely extended beyond C,
will
PERSPECTIVE.
107
will be found in this line, and confequently,
the image km of any part KM thereof,
muſt tend to the point D.
A line parallel to the picture, will have
the imaginary line drawn from the eye,
which fhould find its vanishing point, pa-
rallel to the picture alfo; fo that fuch
a line can have no vanifhing point. The
image of fuch a line will, therefore, be
parallel to this imaginary line drawn from
the eye, and, of confequence, parallel to its
original.
Corollary. The images of parallel lines,
not parallel to the picture, tend to the ſame
point; and parallel lines, parallel alfo to
the picture, will have their images parallel
one to another.
PROPOSITION III.
The vanishing point of every original line,
lying in the fame original plane, will be
found in the vanishing line of that plane.
FOR the vanishing line of a plane, is the
boundary of the appearance of that
plane,
>
108
A TREATISE ON
plane, infinitely extended before the eye;
and the vaniſhing point of a line, is alſo
the boundary of the image of that line,
extended in like manner. It is plain, there-
fore, that the latter must be found fome-
where in the former.
So the vaniſhing point of the line KM,
Fig. 4, is a point D,
FG, of the plane X,
in the vanishing line
in which it lies.
Corollary I. Hence, if we know the
vaniſhing points of two lines, not paralle
the one to the other, and which lie in
the fame plane; we know, alſo, the va-
niſhing line of that plane.
For a line joining thoſe vanishing points,
is the vanishing line required.
Corollary II. The imaginary line
drawn from the eye, which finds the va-
niſhing point of any original line, is pa-
rallel to the plane in which that line lies,
PRO-
PERSPECTIVE.
109
1
PROPOSITION IV.
The image of a line, parallel to the picture,
will be fimilarly pofited with its object;
and have its length in proportion to that
of its original, as the diſtance between
the eye and the image, is to the distance
between the eye and the object.
LEt Y, Fig. 48, be the picture, E the place
of the eye, AB an original line, parallel
to the picture; I fay, ab: AB :: Eb :
EB.
For, ab being parallel to AB, (Prop. II.)
the triangles Eab and EAB are fimilar;
whence, ab: AB :: Eb: EB :: Ea: EA.
!
}
PROPO-
I-10
A TREATISE ON
་
PROPOSITION V.
The image of any point, in a given original
line, not parallel to the picture, is a point
in the indefinite image of that line, which
divides the faid indefinite image in the pro-
portion of the diſtance between the eye
and vanishing point of the given line, to
the diſtance between the given point and
point of interſection.
Et Y, Fig. 49, be the perſpective plane,
E the place of the eye, AB an original
line, whofe vaniſhing and interfecting points
are v and A, reſpectively; I fay, vb : bA ::
Ev: Ab.
For, fince Ev and AB are parallel, (Def.
II.) the triangles Evb and BAb are fimilar;
whence ub: bA :: Ev: AB.
NOTES,
[ 11 ]
sk ste st ste si ste s S DR sta
k
FR SA SA SA IR SA MA R
N
O TE S,
Relating, chiefly, to
THE THEORY OF
PERSPECTIVE.
IF
NOTE A. Page 8.
N dwing the few objects compriſed in
Fig., there is occafion to introduce every
real variety in the practice of perſpective.
It will, therefore, be of ufe to the learner
to attend to it; and when he is mafter of
all the neceffary rules, to draw it over fre-
quently, varying the lengths and fituations
of the lines contained in it. For when
once a perſon can draw any line in this
fmall picture, independent of the reft, from
the fituation and meaſure of it, previouſly
given, he will be able to draw the repre-
fentation
112
A TREATISE ON
ſentation of any object, or any number of
objects whatever, be they ever fo complex.
B. Page 14.
Since theſe lines are parallel to the ground
line, they are parallel to the picture; hence,
(per Prop. II.) their images will be paral-
lel to the original, and, therefore, muft
be parallel to the ground line alſo.
This is clearly exemplified in Fig. 4,
where OP and HI, upon the ground plane
X, are repreſented by op and bi on the per-
fpective plane Y. For if the perſpective
plane be perforated in k, l, m and n,
and a ſtraight wire be put through a in the
plate Z, repreſenting the eye of the ob-
ferver, and alſo through the holes k or l,
it will fall upon OP; and if it be put
through the holes m and n, it will fall
upon the lines HI.
C. Page 15.
That this method of determining the
diſtances of theſe parallel lines muſt bẹ ac-
curate,
PERSPECTIVE.
113
curate, will be evident from confidering
that G, Fig. 4, being ſituated at the height
of the eye from the ground line AB, and
GD being the diſtance of the eye from the
perſpective plane, the line DC may be ima-
gined to be the perſpective plane, feen
edgeways; and then any point q, fituated
three meaſures beyond it, muſt appear at
m, in a ſtraight line drawn from the eye at
G to it: m, therefore, is the proper dif-
tance from C, at which that point in the line
required muſt be drawn. Now, fuppofing
the perſpective plane to revolve, and to
be reſtored to its former fituation, with the
eye at a; for the fame reaſon that m muſt
be placed at the diſtance of mC from the
ground line AB, every point in the fame
line muſt be placed at the fame diſtance;
and therefore a parallel to the ground line,
drawn through m, will be the line re-
quired.
To explain this more clearly, put wires
through a and m, when the planes Y and
Z are fet upright; and it will be ſeen, that
this wire is exactly in the fame fituation,
and of the fame length, with the line Gq;
H
both
114
A TREATISE ON
both of them paffing through m, and cut-
ting off the fame length, Cm, from the line
CD.
D. Page 16.
The more of thefe lines I draw upon
the ground plane, equidiftant from one an-
other, the nearer they ſeem to approach,
when reduced to the perfpective plane;
because every interval between them is
continually farther from the eye. The
lines CA, Fig. 9, being continued ad in-
finitum, and the fame proceſs repeated, it
is plain that every fucceeding parallel would
approach nearer to the preceding one, till
at laſt they coincided with the horizontal
line FG.
This is evident from Def. II. for the
original lines being perpendicular to the
ground line, are perpendicular to the pic-
ture; therefore, the imaginary line paf-
fing through the eye, which gives the va-
nifhing point of theſe lines, will be perpen-
dicular to the picture alſo; but it muſt be
parallel to the ground plane (Cor. II. Prop.
III.) confequently will cut the picture in
the point of fight.
E. Page
PERSPECTIVE.
115
E. Page 18.
That all fuch lines, perpendicular to the
ground line, will meet on the perſpective
plane in the point of fight D, is alſo evi-
dent from confidering the fituation of the
ſpace included between the parallel lines
QD and SD, each of which is equally
diftant from the perpendicular CD. The
extremity of this fpace QS, being neareſt
to the eye, muſft appear wider than the
diſtance between thefe lines at any place
more remote from the eye; and the farther
off theſe diſtances are taken, the leſs they
will appear. Confequently the fides which
bound this ſpace muft appear to approach
nearer and nearer continually, till they
meet ſomewhere; and this point, in which
they meet or vanish, muſt be fomewhere
in the line CD; becauſe each fide of the
ſpace is every where equidiftant from the
eye.
It muft, likewife, be fomewhere in the
horizontal line FG; for raifing the planes
Y and Z, the point M on the ground plane
H 2
will
116
A TREATISE ON
will
appear in m on the perſpective plane,
to an eye fituated at a; the point K will
appear in k; and fo CD on the ground
plane, being infinitely continued, and
every point in it, more remote from the
eye, appearing to rife higher and higher in
the perſpective plane, they muſt at laſt
reach as high as D. And they can never
rife higher than D;
fual ray
becaufe, were the vi-
and the line CD actually parallel,
ftill the former could only pafs through D.
For the fame reaſon that the lines QD
and SD will meet at D on the perfpective
plane, the repreſentations of RD, TD, and
all other lines that are parallel to them, at
whatever diftance they be taken, on either
fide of the perpendicular CD, muft meet
in the fame point of fight D.
F. Page 19.
It is evident, that CI, Fig. 6, being fi-
tuated upon the ground plane, muſt have
its vaniſhing point fomewhere in the hori-
zontal line DF, for the reafon given before
Prop. V. It will alſo be that point in the
hori-
:
PERSPECTIVE.
117
horizontal line where an imaginary line
drawn from the eye, parallel to CI, cuts it
(Def. II.) This imaginary line will, there-
fore, form an angle, at the eye a, with the
perpendicular aD, equal the angle ICD.
The part Di intercepted between D, and
this vaniſhing point muft, confequently, be
the tangent to that angle, the radius being
aD; or, which is the fame thing, the
radius being DE.
Let the plane Y be interpofed between
the eye at a and the line CF upon the
ground plane, and it will be evident that
the bafe of the infinite triangle (formed by
the lines CD and CI) in the true horizon,
and the line Di upon the perfpective plane,
which repreſents it, are equally tangents to
the fame angle, under which they are both
viewed; all the difference between them
being in the radius of the circle.
Wherever Ci meets the horizontal line,
all parallels to it muſt be drawn to the fame
point, for the reafon given before (Cor. I.
Prop. II.)
•
H 3
To
**
118
I I
A TREATISE ON
To illuftrate the preceding demonftration
by experiment, let a ſtraight wire pafs from
a to H, Fig. 6, and it will pass through
the perſpective plane Y in b. If a more
diſtant point in the line CI be taken, the
plane Y will be pierced in the line Ch pro-
duced, which shows that it tends to vanish
in i.
For the fame reaſon that the vaniſhing
point of Ci, and of all its parallels, is in
the horizontal line FG, the vanishing points
of all lines fituated upon the ground plane,
in whatever direction they be drawn, muſt
be fomewhere in the fame line. For this
reafon the horizontal line is called the
vanishing line of the ground plane.
And for the fame reafon that the ground
plane hath its vanishing line in the horizon,
any other plane, making any angle with the
ground plane, will have a vaniſhing line,
making the ſame angle with the horizontal
line, which the interfecting line of the new
plane makes with the ground line; and
every operation may be performed with
this, as with the firſt ground line, and the
first
:
PERSPECTIVE.
119
firft horizon.
This confideration throws
the greateſt clearneſs into the theory of per-
ſpective, and makes the practice exceeding
eafy; and this great improvement we owe
to the fagacity of Dr. Brook Taylor.
G. Page 21.
Since lines perpendicular to the ground
plane are parallel to the picture, their ima-
ges must be parallel one to another and to
their originals. (Prop. II.) Thoſe images
will, confequently, be perpendicular to the
ground line.
}
To illuftrate this, let pins, and other bo-
dies, be fixed upright on the ground plane
X, and confidered as viewed from a in Z,
while the repreſentation is taken upon the
plane Y.
H. Page 22.
Imagine the given line CK to be fituated
in a plane [x] perpendicular to the ground
plane, fuch a plane will cut the ground
plane in the line CH, making the angle
DCH
120
A TREATISE ON
DCH equal the given declination of the
line CK; the point i is, therefore, the va-
niſhing point of CH; and, fince CH lies
in the plane [x] as well as in the plane X,
the vanishing line of the plane [x] will pafs
through the point i (Prop. III.) Again,
the plane [x] being perpendicular to the
ground plane, would, if extended, inter-
fect the picture in the line CDE; hence
its vaniſhing line will be parallel to DE
(Prop. I.) therefore the vanishing point of
CK will be found in ig. We are, laftly,
to prove that g is that vanishing point.
Since the imaginary line ag is parallel to
CK (Def. II.) it muſt make the angle iag
the angle HCK, or angle of elevation
of the given line. The line ig is, there-
fore, the tangent of the angle HCK to
radius equal ai. But ig is the tangent of
the angle yg (= HCK) to radius y and
Confequently, the point
ży
iE =ia.
g is truly determined.
The imagination may be affifted in con-
ceiving of this, by fuppofing the horizon
to be raiſed as much as the angle of eleva-
tion
PERSPECTIVE.
121
tion of the given line. For, in this caſe,
the repreſentation Ck of the line produced,
muſt as certainly meet this new horizontal
line, paffing through g, as any lines upon
the ground plane, continued, muſt meet
the common horizon. Indeed, upon this
fuppofition, a plane paffing through CK,
and the foot of the obſerver, would take
place of the old ground plane.
I. Page 26.
Raife the planes Y and Z, Fig. 4.
Then, fince the given line HI is parallel to
the picture, the image un of any part UN
thereof, will be parallel to its original;
whence per Prop. IV. un: UN:: au: aU;
un : (Cs =) UN :: Gu: GC, per conftruc-
tion; but fince DG aD, Gu= au; and
GC=aU, therefore the image un is juftly
determined, by means of the lines GC,
Gs. Again, we have un: (st = UN) : :
Du: Ds; but Du: Ds: : Gu: GC; hence,
un is the image of UN, as found by this
method alſo.
By
122
A TREATISE ON
}
By raiſing the moveable plates in Fig.
4, and putting a ftraight wire through a,
and the feveral perforations in the perfpec-
tive plane Y, thefe rules will be occularly
demonftrated. For a ftraight wire extended
from a to U, will pass through u, and being
extended to M it will pass through m, ſo as
fo
to intercept um in the perfpective plane,
correfponding to UM on the ground plane.
And fince triangles ftanding upon equal
baſes and between the fame parallels are
equal, the triangle aUN will be equal to
the triangle GCs, or Dst, a being the fame
height from the ground plane with D and
G. Alfo if thefe equal triangles be cut by
a plane parallel to their bafes, as by the
plane bi, the new bafes will ſtill be equal;
fo that un, in this cafe, will be the bafe to
both the triangles. For the fame reaſon,
they would have been as well divided by
lines drawn from F, the other point of
diſtance, as from G.
Univerfally, perfpe&tive meaſures are de-
termined by drawing lines from the points
of diftance, or other meafuring points, to
the meaſures, previoufly laid down upon
the ground plane, or to lines fituated in a
fimilar
PERSPECTIVE.
123
fimilar manner with regard to the perſpec-
ive plane.
K. Page 27.
Suppoſing a line to be drawn from a to
M, it will pass through the perſpective
plane in m. But Cq being equal to CM,
and DG equal to aD, the triangles aCM
and GqC will be exactly equal; the latter
being, in fact, the very fame with the former,
when the lines aD and aM have been ſup-
poſed to revolve, till a came into the fituation
of G. Confequently m being an immove-
able point during the revolution, its place
will be the very fame in which ever of the
triangles it be taken.
In the very fame manner may it be
ſhown, that any other point, as u or n,
will be found to have the fame place upon
the perſpective plane, whether it be fixed
by a line drawn through it from the eye
to the object, or by a line drawn through
it from the point of diſtance in the horizon
to a point on the ground line, at the fame
diſtance from the perpendicular with the
object itſelf. For by Prop. V. we have Du :
us
!
124
A TREATISE ON
I
us :: (aD =) DG: SU; but from the
fimilarity of the triangles DuG and suC, it
is, as Du is to us, fo is DG to Cs; but Cs
SU per conftruction. Therefore the
point u is the image of U. The fame
reaſoning will prove, the point n to be the
image of N.
-
L. Page 29.
Since the triangles iyb and bCh are fimi-
lar, we have ib: hC:: iy: Ch. But Cb
CH and ży iEia per conſtruction.
Therefore ib: hC :: ia: CH; whence per
Prop. V. the point his the image of H, as
was required.
M. Page 33.
The proceſs in this fection hardly needs
any demonſtration, it being evident that
any row of objects, of the fame height,
ſtanding upon the ground plane, on a line
perpendicular or oblique to the ground
line, muſt appear to diminiſh till they
vaniſh in the horizontal line. Confequently
the triangle dfe, Fig. 15, will contain in
it perpendicular lines equal to the apparent
height
!
PERSPECTIVE.
125
::
height of all objects that are of the height
of fe; and this will be the caſe wherever
the point dis taken in the horizontal line.
For ib: ef: di: de; but di : de :: dm : def
that is, as Ha: HI; and fince this propor-
tion will hold wherever the point d be
taken in the horizontal line, it follows,
that the length of the perpendicular ih will
not be altered thereby; it being always to
ef in the conftant proportion of Ha to HI.
N. Page 34-
In the firft method, we have the tri-
angles kgy and kCa fimilar; whence gk :
kC::gy: Ca; but gy is equal to ga, the
diſtance of the eye from the vanishing
point g, and Ca is equal CK per conftruc-
tion. Therefore gk : kC :: ga : CK.
Hence (per Prop V.) the point is the
image of K. In the fecond method, fince
gy is drawn parallel to Ca, we have gk :
kCgy: Ca; but gy and Ca are equal
gy and Ca, and of confequence equal ga and
CK refpectively. Therefore, &c. as be-
fore,
0.
:
126
A. TREATISE ON
:
O. Page 47.
Let AB, Fig. 21, be the original of ab, on
which defcribe the circle AIB IIII, and
draw the diameters I, IIII; LIII, IIV, di-
viding the circle into 8 equal parts. Then
it is plain, the diameters I IIII, V II, L III,
will have their vanishing points at C, D,
G; and therefore 1C, 2D, and /G are their
indefinite images; and fince AI and IIII B
are parallel to V II, they will have the
fame vaniſhing point with V II, viz. D.
Therefore, lines drawn from D to a and b
will cut IC in 1 and 4, the images of I
and IIII.
Since F is the meaſuring point of G,
and H the meaſuring point of 25D, lines
drawn from F to a and b will cut off the
parts ls, s3 reſpectively, equal to as and sb.
For the fame reafon, lines drawn from H
to a and b will cut off the parts s5 s2 equal
as, sb. Confequently, 13, and 5 2 are
the images of L III and V II, diameters of
the original circle A I B IIII. Q. E. D.
P.
PERSPECTIVE.
127
P. Page 55.
The demonſtration of this rule is made
eafy by the confideration of Fig. 6. Let
HC, in the plane X, be the line to be deter-
mined. The vanishing point is fixed by
the line ai, drawn parallel to HC, or Ei,
which, making the fame angle with the
perpendicular, meets the horizontal line in
the fame place i. And the point ʼn is de-
termined by drawing a line from H to a,
the place of the eye.
In both theſe caſes
the picture is placed between the eye and
the object; and if the elevation of the eye,
in Fig. 6, occafion any difficulty to the
perſon who compares thefe figures, let him
lay the plane Y flat down, fo as to be a
continuation of the plane X, and let him
fuppofe E and [h] to be visible through it
on the backfide of the paper (as they are
marked in the figure for that purpoſe) and
he will find that a ruler laid from H to E,
will paſs through the point b. In this fi-
tuation of the plates X and Y, all the lines
have the very fame poſition to one another
as the lines in Fig. 32.
.....
128
A TREATISE ON
Q Page 61.
The rationale of this practice is very ob-
vious, depending on this one principle,
that every line in the perfpective plane
makes a tangent to the angle of vifion; and
it may be fully illuftrated by the help of
Fig. 6. Let g, in the plane Y, be the point
in the perſpective plane, reprefenting the
point of any object that is required to be
drawn. When the planes. Z and Y are
raiſed, it will be evident that the angle
¿ED is equal to iaD. Confequently i, the
foot the of perpendicular ig, is truly fixed
by this rule. Alſo the angle yg is equal
to iag, fo that the point g is likewife truly
fixed by it.
R. Page 70.
To demonſtrate this cafe,, I shall make
uſe of a larger figure, Fig. 50.
Let the triangle to be projected be Chi
in the plane Y, equal to CBI in the plane
X. Proceed as in the former example,
and both of the triangles will be com-
pleted
PERSPECTIVE.
129
pleted by producing the fides bi, and BI to
A. If the vifual rays be perpendicular to
the perſpective plane Y, and the triangle
Abc be made to turn upon its bafe AC, as
in the plane X, it is evident that, in every
pofition of the plane X, the ſhadow or
image of the point B will always fall upon
fome part of the line bD; and, at what-
ever angle the eye be elevated above the
plane X (fuppofe 30 degrees) it is alfo evi-
dent, that (raifing the plane Z) the per-
pendicular BH will fall upon the fame
point H, in the line ¿D, as the parallel
GH (FG being taken 30 degrees
DBH) for fince the triangles BDH and
GDH have BD equal to GD, DBH equal
to DGH, and the angles at H right ones,
the whole triangles will be equal; and
confequently the line DH will be the fame
in both; Ab will, therefore, be projected
into AH, C into HC, and the angle A¿C
into AHC.
=
If the fide bA terminate in i, and BA
in I; draw ik parallel to bD, and it will
cut HA in K, the place of the image of I.
I
It
$30
A TREATISE ON
It is evident from what was advanced
before, that the image of the point [I]
muft fall fomewhere in the line HA: It
is equally evident, that it muſt fall fome-
where in the perpendicular ik, in every po-
fition of the plane X; and therefore can be
no where but in the interfection, K, of theſe
two lines.
An Addition to PART XII. p. 94.
HAT the young artiſt may be a proper
judge of the importance of chufing a
proper diſtance in perfpective drawings, I
have given a duplicate of Fig. 20. in Fig. 51.
In order to take in all the lines and points.
that were principally requifite to the de-
fcription, the diſtance, DE, in the former,
was taken very fmall; in confequence of
which, the more remote fquares feem to be
out of proportion, though to an eye in a
proper
fituation for viewing them (viz. at
the diſtance DE, perpendicularly over the
point D) the angles they fubtend will be
found
PERSPECTIVE.
31f
found to be equal, and therefore the draw-
ing is juft. Fig. 51 differs from the former
in nothing but the diſtance, which is taken
near three times as great. By this means
all the fquares fall within the meaſuring
points a and b, and the diſtortion, obſerva-
ble in the former figure, is avoided.
What we call the perfpective plane is
only an imaginary thing; for, in reality,
we do not refer the objects of fight to any
plane, fome parts of which are more dif-
tant from the eye than others. Whenever,
therefore, the diſtance in drawing is fo
ſmall, that one part of the picture is con-
fiderably more remote than another, a diſ-
tortion will be perceived by an eye in the
fituation in which drawings are generally
viewed. No drawing can appear perfectly
natural except on the ſurface of a ſphere,
the eye being placed in the center. But
this inconvenience, attending the common
perfpective, will be fufficiently obviated, if
the diſtance be taken large, in proportion
to the fize of the figures to be drawn. It
is becauſe theſe things are better underſtood
by
1
J
132
A TREATISE ON, &c.
by example than precept that I have given
the two drawings of theſe figures.
Upon the principle mentioned above,
the problem concerning the perſpective
diameter of columns, ftanding at equal
diſtances from the ground line, admits of a
folution; but it is not to my purpoſe in
this treatiſe to diſcuſs fuch particulars.
THE
THE
CONTENTS.
PART I.
Of the Inftruments that are of use in the
practife of PERSPECTIVE, and the ap-
plication of them,
PART II.
page I
The definition of neceſſary technical terms,
and the preparation of the Drawing-
board,
PART III.
7
To find the perspective fituation of right lines
upon the ground plane,
13
SEC. I. To find the perfpective fituation of
right lines parallel to the ground line, 14
SEC.
THE CONTENT S.
SEC. II. To draw the perspective fituation
of right lines perpendicular to the ground
line.
SEC. III. To draw the perspective fituation
of lines oblique to the ground line,
PART IV.
page 17
18
To draw the perspective fituation of lines not
fituated upon the ground plane,
20
ib.
SEC. I. To draw lines perpendicular to the
ground plane,
SEC. II. To draw the perspective fituation of
lines oblique to the ground plane,
PART V.
21
To fix points in perspective lines, or the doc-
trine of perſpective meaſures,
24
SEC. I. To divide any line lying upon the
ground plane, in any proportion required, 25
ib.
CASE I. To divide a perspective line lying
parallel to the ground line,
CASE II. To divide a perspective line that
is perpendicular to the ground line, 27
CASE III. To divide a line that is oblique
to the ground line,
28
SEC.
THE CONTENT S.
SEC. II. To divide lines not lying upon the
ground plane,
page 31
CASE I. To divide a line perpendicular to
the ground plane,
ib.
CASE II. To measure a line oblique to the
ground plane,
PART
VI.
34
A fummary account of all the effential rules
for drawing in perspective,
PART VII.
37
Some more particular directions, to facilitate
the practice of drawing in particular cafes,
SEC. I. Of drawing circles.
41
ib.
CASE I. When the given image is parallel
to the ground line,
47
CASE II. When the given line is oblique to
the ground lire,
SEC. II. Of different ground planes,
PART VIII.
48
50
A method of drawing in perspective from
the ichnography of objects,
52
PART
1
THE CONTENTS.
PART IX.
An easy method of drawing in perspective
without previous menfuration, by means of
an inftrument to take the angles under
which objects appear,
page 57
PART X.
Of orthographical perspective.
63
PART XI.
Of shadows,
73
SEC. I. To draw the shadows of bodies on
the fame planes on which they are fituated,
ib.
SEC. II. Of Shadows intercepted by other
objects,
SEC. III. Of faint fhadows,
82
86
SEC. IV. To draw the reflected images of
objects in water,
PART XII.
87
General advices and directions, relating to the
art of drawing in perspective,
PART XIII.
89
The definition of all the technical terms made
ufe of in this treatiſe,
98
A general
THE
CONTENT S.
A general View of the Theory of Perfpec-
tive,
page 103
Notes relating chiefly to the Theory of Per-
Spective,
III
THE
LETTERS
Referring to the
NOT E S.
A. from
from page 8 to page
III
*
B.
14
112
C.
15
ib.
D.
16
114
E.
18
115
F.
19
116
G.
21
119
H.
22
119
I.
26
K.
27
L..
29
121
123
124
K
M.
1
THE CONTENTS.
M. from page 33 to page
124
N.
34
125
0.
P.
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47
55
126
127
128
R.
70
ib.
THE EN D.
ERRAT
A.
Page 15. line 5. from the top, for, C to g, read, C to q
30.
34.
36.
72.
77.
110.
114.
117.
125.
127.
129.
129.
114.
18. for, C to s, read, s to C.
14. for, gy, read, gy..
11. for, Cb, read, Cg:
13. děle [S.]
19. for, 34, read, 36.
12—16. for v, read a.
11. for, lines CA, Fig. 9, read, line CA,
Fig. 5.
12. for, CF, read, CD.
22. for, Ca, read, Ca.
25. for, Fig. 32, read, Fig. 23,
4. for, Abc, read, AbC.
21. for, C, read, Cb.
This is evident, &c. This paragraph ſhould be
at the End of the Note E, and inferted nex
the paragraph, For the fame reaſon, &c. in
page 116.
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Jofeph Priestley, LL.D. F.R.S.
And fold by
J. JOHNSON and J. PAYNE, at No. 8.
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