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JZ.JIj. dfMii.etfecit ji.i].F(?rt· v j% .JJzippa, pe/feoü.
Tidttifiid(hr du^icl: dii-ec&r Jan'jë-i8o3, én/Ze^h-^duUy SaSon.Tork Strau ö>vait (rarde/t.
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A
PRACTICAL TREATISE
O F
PERSPECTIVE,
ON THE PRINCIPLES OF
Dr. BROOK TAYLOR.
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BY
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EDWARD EDWARDS,
ASSOCIATE AND TEACHER OF PERSPECTIVE
IN THE
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Pamphilus----primus in Picturd omnibus uteris eruditus,
prascipue Arithmetice et Geometrice, sine quibus negabat
Artem perfici posse. Pliny, Nat. Hist.
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LONDON:
Printed by Luke Hansard,
FOR, LEIGH, SOTIIEBY AND SON, YORK-STREET, COVENT-OARDJ;S, 1803,
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Luke Hansard, l'rinter,
Creat Turnstile, LincoinVlnn Fields |
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TO THE
Ê IN G.
SIR,
AS it is principally by the patronage of"
Your Majesty that the polite Arts have been raised to their present state of prosperity in the British Empire, I am induced to hope that any the humblest attempt towards their further cultivation or improvement will not be deemed unworthy of é our Majesty's favour. It is from this persua- sion, that I am encouraged to present to Your a. 2 Majesty
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Majesty the following Treatise, on the science
and practice of Perspective. That Your Majesty may long be enabled to
continue to the Arts that patronage and protec- tion under which they have hitherto flourished., and may thence derive the satisfaction which must result from private gratitude and public esteem; is the humble, but ardent wish of Your Majesty's
Most dutiful and devoted
I ,';'-■
Subject and Servant,
'Edward Edwards,
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PREFACE.
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Al THOUGH various treatifes on the fcience of Perfpeétive
have been written, fome of which have great excellence, yet it may with truth be aiferted, that not one of them is calculated to be ufeful, or even intelligible, to fuch artifts as do not underftand Euclid, yet wiih to gain a knowledge of the fcience, fufficient to qualify them to conduce their works upon true principles. The defects which deftroy the utility of thofe treatifes will Be clearly demonftrated, by arranging them in two claffes, and then confidering each under its fpecific character. In the firft clafs are thofe which, by their ex- amples, appear clear and inftruotlve at the firft view, hut when ex- amined are found deficient in fcience and theory, and are even want- ing in the explanation of the few principles which they contain. Such are the defeds of the works of Maralois,. the Jesuit, and of Pozzo. In the fecond clafs are thofe treatifes which are the beft, and con-
tain the trueft principles of the fcience; but are fo mathematical in their ftructure, and confequently fo abftrufe to thofe who are not verfed in the Elements of Euclid, that they contain no examples of forms or figures in Perfpe&ive, and confequently have nothing that pan invite the eye of a practical artift to examine their principles. Such
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vi PREFACE.
Such are the difadvantages attending the elegant work publiihed
by Dr. Brook Taylor, of Cambridge, firft in the year 1715, and again, with improvements, in 1719. The fame inconveniencies attend the treatife "by Mr. Hamilton, which, added to its magnitude, deters the artift from its perufal, rather than invites him to ftudy the fcience it contains. There is alfo another work that has infinite merit, written by
T. Malton, fenior, and publiihed in 1775; which contains fome excel- lent and mafterly examples ; but he has deftroyed their utility ,by entangling the vaniiliing points, and croifmg the diagrams in fo confufed a manner, that it is almoft impoffible for a young practi- tioner to trace and distinguish the different figures. Thefe faults, which are too frequent in books of inftruéHon, have
arifen from two caufes; the firft is, that the authors of them, though perfect mailers of the fcience on which they wrote, had not acquired the art of explaining it to thofe who are unacquainted with it; they feem alfo to have forgotten, that thofe who would inftruet, muft .de- fcend to that language for explanation, and apply thofe figures for il- luftration, which are fuitable to the powers and comprehenfion of their pupil, rather than to the difplay of their own fcience and abilities. The fecond caufe of the defects before mentioned arifes from the
following circumftance; which is, that excepting Pozzo and High- more, there is no author who has written on the fubjecl; of Perfpec- tive, that can be .confidered as a painter; confequently they were deficient in the knowledge of the forms of objects, and thereby unable to apply their fcience to the ufes required by the artift. Having experienced and confidered the difadvantages before men-
tioned, the author prefumed to think that a work might be produced, better calculated than any one that has hitherto appeared, for the fervice of thofe artifts who have neither time nor refolution fufircient •to inveftigate the fcience of Perfpeétive, under its prefer) t obfcurities and
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PREFACE.
and difficulties. Whether the following treatife, which he has at-
tempted in conformity to his idea, will anfwer the end propofed, mail be left to the reader to determine. The arrangement of the work is as follows :
As a preliminary apparatus, a feleclion of definitions and problems
in geometry is given, all of which are abfolutely neceflary to be underftood by thofe who mean to practife Perfpeétlve; they are in- ferted not to increafe the fize of the volume, but that the ftudent may not be compelled to feek for other books before he can make ufé of this. After the Geometry follows-the Perfpeclive, which is divided into
fix feelions; The fir ft is introductory, and contains all the terms that are em-
ployed in the practice, together with their definitions, illuftrated by proper examples; the diiFerence between the center of the piclure and point of fight is defined; and the various pofitions in which objects may be difpofed to the piclure: it alfo contains the rudiments of practice for lines, parallel and perpendicular, to the picture. The fecond feétion contains inftruftions, with examples for draw-
ing objects, the fronts and fides of which are parallel and perpendi- cular to the piaure. Γ he third fection treats of objects, the fronts of which are inclined
to the piaure. η the fourth feaion are examples, with inftruéiions for delineat-
, 8 o -jects, when the planes or faces of which they are compofed are inclined both to the piBure and to tile horizon. It muft be obferved, that the aforenamed feaions contain all the
praaical principles neceffary for the delineation of objeas in Perfpec- tive, however their different planes may be difpofed to the eye of the fpeaatOT. The
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Μ------L'
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Ρ R E F Α C E.
The fifth feétion treats of ihadows, in which the author has at-
tempted to explain the leading principles of that part of the fcience in the clearer! manner he was able; but whatever his fuccefs may have been, it muft not be expected that this part can be clear and eafy to thofe who do not well underftand the preceding feclions of the work; therefore the ftudent muft make himfelf mafter of thofe, before he attempts ihadows. The fixth and laft feótion contains methods for facilitating opera-
tions in difficult cafes, as alfo fome theoretic inftru<5lions, together with obfervations by way of praxis; all of which will be found ex- tremely ufeful to the ftudent. In the technical language of the fcience, the terms adopted by
Dr. Brook Taylor are united with thofe employed by the old writers on Perfpeólive, by which means it is expected that the ftudy of the fcience will be facilitated to thofe who chufe to refer to the works of that great mafter and his principal fucceflors. In the plates are f elected the moft ufeful and familiar examples,
fuch as are moft generally wanted in the common courfe of practice, yet fuch as will include all the pofitions in which objects may be placed to the picture or fpeclator; omitting the inclined picture, for which the ftudent is referred to the fenior Malton, Hamilton, &c. Moft of the examples are drawn to a fcale, the ufe of which is
explained in the firft feótion, and applied in moft of the following. This circumftance has never before been attended to by writers on the fubject; and therefore it may be hoped, that this will operate as an improvement, and greatly facilitate the ftudy of the fcience in its practical part: but the reader muft obferve, that the author does not mean to offer any new method of procefs, founded on any fuperior theory of the fcience ; he only wiilies to teach the readieil mode of practice, directed by the principles of Dr. Brook Taylor, whofe 1 writings |
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PREFACE.
writings on Perfpeólive are certainly the ne plus ultra of the fcience,
and do infinite honour to his country. At the end of the fixth feaion is added a Difcourfe on the conduce
and compofition of apifture; which, if attended to, will not only help to explain the principles of Perfpeftive, but alfo prevent much error m the future works of thofe artifts who have not opportunity to enter deeply into the fcience. The author cannot conclude this preface without obferving, that
m the courfe of the work he certainly would have endeavoured to; give more copious inftructions concerning the theory of the fcience, were he not of opinion, that this cannot be done with fufficient effect without perfonal explanation, and that with an apparatus adapted to thepurpofe, which is abfolutely neceffary to thofe who are not ac- quainted with Geometry; but to fuch as have received a mathema- tical education, and comprehend the eleventh book of Euclid, Dr. Brook Taylor's treatifes aFe fufficient for the theory, and fuch perfons will require but little afiiftance in the praaice.. If, after ftudying him as the great theorift of the fcience, they find any help in the praaical part from this- work, the author will rejoice in having faci- 1 ed the ftudy 0f a fcience which is ufeful to the fcholar, orna-
a to t5le gentleman, and indifpenfably neceffary to the artift, |
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Qyi&fcst»^
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[ xi ] .... ··,.
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CONTENTS.
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GEOMETRY:
Definitions and Propositions --------- Page é
PERSPECTIVE:
SECTION I.
icrnis, Definitions, and Rudiments of Practice ----- i$
SECTION II.
Terms and Definitions ------------ ai'
SECTION III.
samples of Objects, the Sides of which are inclined to the
Picture -...............94 SECTION IV.
Examples, the Lines and Planes of which are inclined both to
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the Picture and Hor
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"izon........... lei
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SECTION
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CONTENTS.
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SECTION V.
Of Shadows.............Page 211
SECTION VI.
Examples, with Instructions, for facilitating Operations in-
some Cases of Difficulty; together with Observations, as a Praxis for illustrating the Principles of the Science - - 243 A DISCOURSE
On the Application of the Science of Perspective, in the Com-
position of a Picture, and other JVorfcs of Art - -- - - 279 List of Plates - - - - - - -.....- - - 309
Index........--------- 311
List of Publications referred to in this Work - - - - - 315
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GEOMETRY.
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DEFINITIONS and PROPOSITIONS.
Ë Straight Line, or right line, is that which lies evenly between
its extreme points. It is evidently the ihorteit diilance be- tween them. A plane, or plain furface, is that which lies evenly between its
extreme boundaries, which may be one or more curved lines, or any number of right lines, not lefs than three. It has evidently this property, that any two points being taken in
it, the ftraight line between them lies wholly in that furface. If two or more ftraight lines on the fame plane, are fo fituated
that they would if continued never meet, they are parallel to each other. If they are fo placed that they would cut each other, if continued,
they are then faid to incline to each other, and they form, at the point of interfeclion, what is called the angle of their inclination. The point of interfeétion may be fuppofed in either of the lines, for it is common to both. Â . If
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GEOMETRY.
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If two lines, when cutting each other, form equal angles on every
fide of them, fuch angles are called right angles, and the lines are laid to be at right angles, or perpendicular to each other. Examples of Lines.
Fig. 7, N* 2. The lines Áé, Â % C3, are parallel to each other,
for if they were continued or produced they would never meet. Fig. 8 and 9. The lines A B, and A C, are inclined to each
other : the point A is their interfectlon. Fig. li. The lines A a, B b, crofs or interfecl; each other at right
angles, therefore  b is perpendicular to A a, or A a is perpendicular toBb. The point of interfection C may be fuppofed in the line A a, of
in  b, for it is common to both. Of Angles, that which is lefs than a right angle is called an-
acute angle; that which is greater than a right angle is called an obtufe angle. A plane figure bounded by three ilraight lines is called a rectilineal
triangle. An equilateral triangle, is that which has all its fides equal: it
has alfo all its angles equal. Fig. 6. An ifofceles triangle has only two of its fides equal to each other :
the angles oppofite to thefe equal fides are alfo equal. Fig. 12. A fcalene triangle has all its fides unequal. Fig. 7, N° 1.
The angles of a fcalene triangle are all unequal; of any two angle^
that one being the greater, which is oppofite to the greater fide. The converfe of the three laft propofitions is alfo true; viz.
If a triangle has its three angles equal, its fides are alfo equal.
é If
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GEOMETRY.
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If a triangle has two angles equal, the fides oppofite to them are PLATE
alfo equal. If a triangle has three unequal angles, its fides are alfo unequal;
of any two fides, that one being the greater, which is oppofite to the. greater angle. Examples of Angles.
The angle A B D, Fig. 4. is a right angle, as are alio the angles
of Fig. 11. Fig. 8, is an acute angle, and the angles of the triangles, Fig. 7
and 12, are all acute. Fig. IS, is an obtufe angle.
Having defcribed the properties of right lines and angles, it will
be neceffary to fpeak of the circle. A Circle is a plain figure contained by one line, which is called
the circumference, and is fuch, that all ftraight lines drawn from a certain point within the figure to the circumference, are equal to one another. The point from which fuch lines are drawn, is called the center of
the circle. A diameter of a circle, is a ftraight line drawn through the center,
and terminated both ways by the circumference. A radius is any right line, drawn from the center of the circle to
the circumference, and is equal to half the diameter. A femicircle is the figure contained by a diameter, and the part of
the circumference cut off by the diameter. A fegment of a circle is the figure contained by a ftraight hue,
and any portion of the circumference it cuts off. Â 2 A chord
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GEOMETRY.
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Λ chord of an arc is a right line, joining the extremes of that ara
The longeft chord that can be drawn in a circle is the diameter, which always paffes through the center: and any portion of a circle cut off by a right line, may be confidered as a fegment, whether greater or lefs than a femicircle. Examples..
Fig. 14, is a circle; the point, a, is its center, and the line b e is
the diameter, the half of which is the radius, as from a to b, or from a to e. The femicircle is that portion of the circumference which lies
between the points e and b, either above or below the line b e. Fig. 15, is the fegment of a circle, the portion of the circum-
ference lying between a and b, marked e, being the arc, and the line a b, the chord of that fegment or arc; In Fig. 15, that part of the circle expreffed by the dotted line, i*
the greater fegment; the part marked a, b, e, the lefs fegment. Of Planes, in the ftudy of perfpe&ive, it is required to have a
clear and accurate knowledge,- In the theory of the fcience, imaginary planes are of the utmoft
confequence; and in practice, the consideration of real planes is abfo* lutely neceffary, becaufe the forms of moil objects produced by art are compofed of planes. Planes may be fituated in the fame relation to each other as lines.
If they are fo placed that if continued they would never meet, they
are then parallel to each other: But if they are fo difpofed that they would crofs each other, they
are then faid to incline to each other; and the line in which they meet or crofs each other, is called their interfeótion. The
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GEOMETRY. s
The interferon of two or more planes is a right line, and that PLATE
line may be fuppofed in either of thofe planes. If two planes crofs or cut each other, fo that they have equal
fpaces between them, they are then at right angles, or perpendicular. to each other. Example of Planes.
Fig.. 2. The faces, or fides of the block B, are planes; as alfo the PLATE,
faces or fides X l, X 2, of the building R a; and thofe planes are at right angles, or perpendicular to each other. PROBLEMS.
ïig. l. To bifeét a right line. See Euclid, Problem 10, Book l. ■ PJLATI
Upon either of the points A or B^ in the right line A B, fix the compaffes, and with any diftention, draw an arc, as the arc l D 2; then with the compafles on the oppofite point, as on B, draw another arc, as the arc é Ca; through the interfeétions é, % of the two arcs, draw the right line é E 2, and the line A B will be equally divided, or bifected in E. Fig. 2. To erect a perpendicular from a given point in a given
right line. Euc. P. II, Â. é. Let A B be the right line given, and C the point at which the per-
pendicular is to be erecled.. With one foot of the compaifes on C, fet off two points at equal
diftances from C, as D and E, upon the line A B. Then with the compafles diftended, on the points D and E, draw
the arcs F and G, and from their interfeclion draw a right line to C, which will be perpendicular to the right line A B. Fig. 3.
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GEOMETRY.
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Fig. 3. To let fall a perpendicular from a given point upon a given
right line. Euc. P. 12, Â. é. The given right line is  C, and A is the point from which it i s
required to draw a right line perpendicular to  C. With the compaffes fufficiently diftended, and placed upon the
point A, draw the arc  C, cutting the given line in the points  and C. Then with the companies on the points  and C, defcribe the
arcs A D Ç and A E H, interfering each other in the points A and H. Through the points A H draw a right line, and it will be the per-
pendicular required. Fig. 4. At the extremity of a given right line, to draw a line which
ihall be perpendicular to the given line. Let A B be the given right line, to which it is required to draw a
line perpendicular at the extremity B. Take any point, as C, out of the line A B, and with the radius
C B, defcribe the arc A B D. From the point A draw a right line through the center C, that
may interfecT; the arc, as at D. From the point D, draw a line to the point B, which will be the
perpendicular required. Fig. 6. To defcribe an equilateral triangle upon a given right
line. Euclid, P. ïft, B. lft. Let the line  D, be the given line.
Upon the point  fix the compaifes, and with a radius equal to
 D, draw the arc D C. Then with the compaifes upon the point D, with the fame radius,
draw the arc  C. From
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GEOMETRY.
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From the interfeaion C, draw the lines C Â and C D, and the PLATE
triangle will be completed. ** Fig. 7. To conftrua a triangle, the fides of which ihall be equal to
three given right lines. Eu. P. 22, B. 1. Let the given lines be A 1, Â a, C 3, Fig. 7, N° 2, and.let the line
A 1, be the required bafe of the triangle. Draw the right line A, N° 1, equal to the right line A 1, Ne 2.
Take the length of the line  2 in the compaifes, and fix one foot
of them on that extremity of the line upon which the fide equal to  2 is to be placed, as on the point a, and defcribe the arc D. Then take the length of the line C 3 in the compaifes, and with one
foot upon b, the other extremity of the line A, N° 1, defcribe the
arc E. Then from the interfeaion of thofe arcs, draw the lines  and C,
and the triangle will be completed. * Fig. 8. To bifeft an angle.
Let  A C be the given angle.
With the compafles fixed on the point A, defcribe the arc  C
then on the points  and C, defcribe the arcs D d and E e of equal radii interfeaing each other. Through that interfeaion draw a right line to the vertex A, and
the angle  A C will be equally divided or bifeéted. Fig. 10. *To defcribe an arc or circle which will touch or pafs
through three given points. Let the given points be A, B, C, through which it is required to
draw an arc or circle. Upon
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GEOMETRY.
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Upon each of the points as a center, defcribe circles of equal mag-
nitude and fufficient radii to interfeól each other, as the circles a a, b b, e e. Then through the interfecüons l i, 1 2 of thofe circles, draw the
right lines D E and D F, which will interfeét each other in the point D. Then will D be the center of a circle or arc that will touch or pafs
through the given points Α Β CL—Therefore with the compafles upon the point D extend them to either of the points A, B, or C, and defcribe the circle required. As no one muft hope to become mailer of the practice of per-
fpeclive who does not underftand the conftruótion of angles, it will be proper to explain the procefs. Every circle is, by univerfal confent, divided into 360 equal parts,
called degrees *,. each of which is again divided into 60 equal parts called minutes, and thofe again into 60 equal parts called feconds, by which degrees, minutes, and feconds, every angle is meafured. When two right lines crofs each other, they conftitute an angle, to 'meafure which a circle muft be drawn from the point where the lines interfect each other, as from a center cutting the right lines, and the portion of the circle which lies between thofe right lines, determines the meafure of the angle which the lines make with each other, Example.
Fig. 9. The two lines A C and A B cut each other and form an
angle; the meafure of which is that portion of the circle marked Β C lying between the lines A C and A B. * The meafures of angles are exprefled by the following characters:
Degrees are marked by the letter °. Minutes, by a fingle ftroke '. Seconds by double ftrokes ". Thus the following characters 360 10' 6" muft be read thirty-fix degrees 10 minutes and
fix feconds. The
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GEOMETRY. 9
The fame procefs determines the inclination of planes. PLATE
It is neceifary to obferve, that the ftudent muft not conceive a dif-
ference from the fize of circles, for whether they are great or fmall, the proportionate feétion continues the fame. When lines are to be drawn inclined to each other in a given angle,
the following methods are the readieft and moil ufeful. In every cafe of inftruments, neceifary for thofe who ftudy Per-
ipeclive, there is an ivory or box ruler, upon which is marked a fcale, which is formed of double right lines parallel to each other, and diftinguiilied by the letters C Ç O, or C, for chords; it is alfo figured with a number of degrees to the amount of 90. The ufe of this fcale is as follows:
Suppofe it be required to draw an angle of 35 degrees. Fig. 9,
Plate I. Fix one foot of the compaffes on the extremity of thofe lines upon
the fcale, which are diftinguiilied by the letters C Ç O, or C, and extend the other to the point marked 60, which is the radius of the circle. Then having drawn one line for the bafe of the angle, as the line
A B, fix the compaifes on the point A, and draw the arc  C.
Then apply the compaifes again to the fame fcale, one foot on the
point C, as before, and extend the other to the figures 36, the number of degrees required. The compaifes thus extended, apply one foot to the point B, in the
line A B, and mark off the meafure at the point C in the arc  C. Then from the point A draw a right line through C, as the line
A C, and the required angle will be completed, A C being inclined to A B, or A B being inclined to A C, in an angle of 36 degrees. As it fometimes happens that there are more than one line of
chords upon the ruler, it is neceifary to take particular care that the meafures arc all taken from one line. C There
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GEO Ì E Ô R Õ.
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10
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PLATE There is alio another inftrument called a protractor, which is tome-
times made of ivoryj fometimes of brafs : if of the former, it is a pa- rallelogram ; if of the latter, a femicircle. In the femicircular inftru- ment, the bafe or ftraight line has a point in the center, and the circular part is marked with a number of degrees upon the edge. It is the fame with the parallelogram, one fide being the bafe, and the other three fides being marked with the degrees. The ufe of the foremen- tioned inftrurnents may be defcribed as follows : Let it be fuppofed that a right line is already drawn, as the line
Ë Â, Fig. 9, and it is required to draw another right line at the point A, that ihall make a given angle with the line A B, fuppofe of 65 degrees. ■
Apply the bafe or lower edge of the inftrument to the given line
A B, fo that the point or center, may coincide exactly with the point A in the given line, at which the angle is to be conftrufted ; then at the point which is numbered 65 on the edge of the inftrument, mark a point as C ; then from the point A draw a line through C, and the required angle will be completed. To one who is unacquainted with the principles and practice of
Geometry, the following remarks will not be ufelefs : First. When geometrical writers refer to triangles which are
marked with three initial letters, the middle letter always denotes the particular angle of the triangle to which the reference is made. Thus in referring to the triangle Fig. 12 Plate I, if it be faid that the angle b a e is the given angle, the point or apex a of the triangle is meant; but if the letters were arranged a e b, then the point or apex e is the angle indicated. é If
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GEOMETRY, n
If any other letters are employed, the fame rule muft be obferved PLATE
In the application. Second. When a line is faid to be perpendicular, it muft be un-
derftood to be fo difpofed in relation to fome other line, without re- gard to the pofition of the plane Upon which it is drawn, or its feem- ing fituation to the natural horizon; for a line may be perpendicular to another line, yet neither of them be vertical, or as vulgarly called upright. Thus in Fig. 11, Plate I, the line A a is perpendicular to the line
 b, or the line  b is perpendicular to the line A a; yet neither of thofe lines reprefent vertical lines, or lines perpendicular to the hori- zon, becaufe they are difpofed obliquely to the fides of the plate upon which they are drawn. The preceding remarks are offered as a caution to thofe who wiih to
ftudy the writings of Brook Taylor, Hamilton, and other authors, who often mention perpendiculars, which, by their fituation on the print, appear to the unfcientific eye as inclined ; thereby confound» ing thofe who are unacquainted with this neceifary diftinclion. Planes have already been defined; but left they ihould not be per-
fectly underftood, fome further attempts ihall be made towards their illuilration. Suppofe any fquare room with a floor and flat ceiling, with four flat
fides; all thefe together conftitute fix planes, perpendicular or pa- rallel to each other. Where any one of the fides meets the floor, that meeting is the in-·
terfeclion of thofe two planes; and the line produced by fuch inter- feaion, may be confidered as in either of thofe planes, in the floor, or in the fide, V e If
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G E O Ì E Ô R Ô.
If a right line be drawn upon the floor, exactly even to one of the
fides, that line will be parallel to two of the fides, but it will be· perpendicular to the other two fides; it will alfo be parallel to the ceiliixo·, becaufe it is in the plane of the floor which is parallel to the ceiling.
If upon any fide of the room a right line be drawn either perpendi-
cular or inclined to the floor, and another right line be drawn on the floor from the point where the firft line touches the floor · both thefe lines may be confidered as being in one plane, which paffes through* the wall and the floor,- but this may be confidered as an imaginary plane.
The author has been the more particular on this fubjeét, becaufe it
is neceffary that thofe who wiih to practife Perfpecüve ihould'per- fectly underftand the nature of planes, as they relate to the forms of objects.; and thofe who defire to ftudy the theory of the fcience muft not only confider the real, but alfo the imaginary planes;—therefore- the foregoing remarks are given, which will be found ufefuL efpe- cially to thofe who have not before confidered the fubjeét. This o-eometrical fection ihall be clofed with the beft method of
conftruéting an oval or ellipfis to any given fize by the compaffes; an operation which is but little known, but which is certainly very ufeful.
It is required to form an oval whofe tranfverfe diameter is equal
to the line A i, A 2, and whofe conjugate is equal to the line  é, B.2,
Fig. 5, Plate I. *
Draw two lines at right angles or perpendicular to each other, as,
the lines A l, A 2, and  é,  2, and their interfection C is the center.
\
* By the tranfverfe is meant the longefl: diameter, and by the conjugate the fhorteft
diameter; as the line A i, A a, is the tranfverfe, and the line  é, â a, the conjugate dia-, meter, Plate I. Fig. J. From
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GEOMETRY. is
From C the center, fet off half the given or required meafures of PLATE
ï
the tranverfe and conjugate diameters, as C A l, C A 2, and C Â é, CB2.
Take the fpace from the center C to  é, or from C to  % which.
is half the breadth of the oval, and fet it on upon the tranfverfe diame- ter, from A 1 to F, then divide the fpace between C and F into three equal parts, and fet one of thefe parts from 3 to 4. Take the fpace from C to 4, amd fet it on upon the line C A a to 5,
making C 5 equal to C 4. Then conftruft an equilateral triangle upon the fpace between 4
and 5 for its bafe, as the triangle 4 D 5 ; conitruct a funilar triangle on the contrary fide, as the triangle 4 E 5. Draw right lines through the points D, 5, and alfo through the
points D, 4, Repeat the fame through the points E, 4, and alfo E, 5.. Then will the points 4 and 5 be the centers for what may be called
the ends of the oval, and D and E the centers for what may be con- fidered as the fides of the oval; therefore upon thofe points with the compaifes defcribe the oval required, obferving that the lines D 5} J) 4, and E 4, and E 5, determine the different fegments required. It muil be obferved, that the above procefs does not produce a
true oval, but forms an excellent fubftitute, efpecially as no means have yet been difcovered by which a perfeét ellipfis can be produced, except by the tramel, or by the pins and firing, both of which are in- convenient on canvas or paper; while the foregoing method is fuffi- ciently correét for any general purpofe, and therefore will be found very ufeful to every artift. |
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The
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é4 - GEOMETRY.
PLATE The geometrical problems given in this fe&ion are fo effential to
the artift, that every ftudent in painting, fculpture, and architecture, ought to imprefs them on his memory, fo as to execute them with facility upon every occaiion, in which they may be required. |
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PERSPECTIVE.
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The Firfi or Introductory Section, containing Terms, Definitions, and
Rudiments of Practice.
r\R. Brook Taylor commences his treatife on perfpeélive in
the following; words: " Perfpeélive is the art of drawing on a plane, the appearances* of
" any figures, by the rules of geometry. " In order to underftand the principles of this art, we muil con-
" fider that a picture painted in its utmoft perfection ought ίο to " affecl the eye of the beholder, that he ihould not be able to judge " whether what he fees be only a few colours laid artificially, or " the very objeéts there reprefented." This may be confidered as a general definition of perfpeélive, and
probably as clear as words will admit. Yet to thole who have not confidered the fubjeél, fome farther explanation and illuftration may be neceifary. For the word appearances the doclor has fuffered from the rod of the critics, who infill,
that he ihould have employed the word reprefentationt; but the diftinction is perhaps too trivial to deferve cenfure; yet it is certain that the words have very different meanings, the former implying the effeft or image any object produces on the eye, but the latter, the defcnption of that effeit or image drawn upon fome plain tablet, or other furface: but fince the appearances are to be defcribed, it may be confidered as a metonymy, by which the effeit is fubftituted for the caufe. The
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IS
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SECT. The fpectator who views an object receives the impreffion of its
foTm upon the retina of the eye, by means of rays of light reflected from the object and coming from all points of it, in right lines to the center of the eye, forming what is called the cone of vifual rays. If this cone of vifual rays be interfered by a tranfparent or
opaque plane, that interfection, provided it can be delineated, will be the perfpective reprefentation of the appearance of the object viewed by the fpectator. This will be better underftood, if it be fuppofed that any
perfon ftands looking' through the glafs of a window at fome diftant object or building; let him fix himfelf at a particular fpot, and with his eye perfectly ileady, trace out upon the glafs the form of the object or building as it appears to him, and he will find then the Ifadr figure fo produced will be a true perfpective reprefentation of that
object or building, which, if ikilfully ihadowed and coloured, would not be diftinguiihed from the original object viewed at the fame point -or ftation. This experiment furniihes the great outline of the theory of Per-
fpective, and the practice is founded upon it; for if the nieafures of the original object, which was viewed through the glafs, can be ob- tained, together with the diftance between the fpectator and the window, thofe nieafures being applied by a proportionate fcale to certain lines drawn agreeably to the rules of the fcience, upon a plane or flat tablet, a reprefentation will be produced fimilar to the tracing upon the glafs. And fuch reprefentation will be fo'perfect, that all the lines will exactly coincide with the original object, pro- vided it be placed at the proportionate height and diftance; it will alfo correfpond with the tracing upon the glafs. From the confideration of the practical part of the foregoing ex-
periment, it will be evident to every ftudent in the fcience, that the perfpective
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PERSPECTIVE. *J
perfpeftive veprefentation of objects can never be truly delineated SECT,
without the knowledge of their forms, their proportions, or the meafurements of their parts. He muft alfo confider how the objects are difpofedto his view, whether their fides or faces be parallel or in- clined to the piclure, or whether they be inclined both to the horizon and piclure ; from all which circumftances it muft follow, that no perfon will ever make great progrefs in Perfpective, who is unac- quainted with Geometry and Architecture; for Geometry may be eonfidered as the foundation of the fcience, while architectural de- figns furniih the beft and grandeft fubjecls for the application of its principles, and the difplay of its deceptive powers. It may perhaps be ©bjeóted, that the art of painting requires f@
much time and application, that the ftudent can have but little teifure for the ftudy of geometry or other fciences; but this objection can only be made by the weak and indolent, and can have no weight With him who forms a right judgment of the art; an art which is jnftly efteemed not only for the .elegance and fublimity of its pro- ductions, but alfo for the extenfive knowledge which it requires the artift to potTefs, before he can excel in the fuperior exertions of the pencil. As it is the purpofe of this iiitroduótion to prepare the reader for
the following work, it will be proper here to beg his attention to the following circumftances. Firft, The author has thought it neceflary to diftinguiih the hori-
zontal line, above any other vaniihing line, contrary to the observa- tion of Dr Taylor, who, in his preface· to the fecond edition of his Principles, obferves, that he makes " no difference between the plane ' of the horizon and any other plane whatfoever ; *" a circumftance * Vide fimilar remarks in " The Elements of Linear Perfpeilive, by Edward Noble," v
frmted for T. Davies, .1771, page 46. D which
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which very much obfcured his principles, particularly to thofe who,
with no ikill in geometry, have endeavoured to improve their practice by his inftructions. For, although the reafons he gives for his conduct are perfectly juft,
namely, that " all planes, as planes, are alike in geometry ;" yet the, painter or defigner in Perfpective is obliged to have recourfe to this line, before he can poffibly determine any other vanifliing line in the picture. But of this more will be faid under its proper head. Secondly, It is neceffary to obferve, that the line which in this
work is called the parallel of the picture, is the fame with what is, called by Hamilton and Malton the parallel of the eye, the author thought it convenient to change this term, becaufe there are many other lines in the procefs which may be confidered as parallels of the eye, while there is but one that can claim this particular dif- tinction. Thirdly, All the examples and figures in the following work are
drawn by the ihorteft procefs, without having their geometrical plans drawn at the bottom of the picture, or below the bale line, at their full proportions, as was the practice with the old writers upon the fubjeél, and which has been abfurdly imitated by too many of the moderns, who have pretended to illuftrate the doctor's theory, the fu- periority of whofe principles renders that tedious procefs unneceifary j confequently they are better adapted to the painter's ufe; for as he muil ever find the lower edge of his canvas the extreme limits for operation, he can have no room below that boundary, to draw plans in their geometrical proportions and Ituations.. In the preface, it has been obferved, that deep theory is not at-
tempted in the following work; it is the practical part only upon ,which the following inftructions are employed : yet, as fome theore- tical remarks mull of neceffity be made, they are chiefly difpofed in ι notes,.. |
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PERSPECTIVE.
notes, and the ftudent will do well to confider them maturely, and s
by their affiftance to form his obfervations and reflections upon the appearances of objects. Above all things, he muft make himfelf matter of the diftinction between the center of the picture and the point of fight; alfo of the difiance of the piQure, and the difiance of a vanijhing point; for if thofe principles are not well underftood, no great progrefs can ever be made in the fcience. Laftly, The ftudent is advifed to copy the examples either by
a larger or fmaller fcale, but never to the fame fize with the given Examples ; or, he may felect different figures of his own choice; but above all things let him work conftantly to a fcale, keeping the height of the eye in due proportion to nature, and to the conftruc- tion of the object he means to delineate. This introduction ihall be concluded with obferving, that the fol-
lowing Treatife is founded upon the theory of Dr. Brook Taylor, who firii taught the real principles of Perfpeélive, or, in other words, was the firft perfon who demonilrated the true methods of finding the vaniihing points, and vaniihing lines. For the delineating of lines and planes perfpectively, in whatever poiitions the originals may be difpofed to the picture before him, the vanilhing points for oblique Tines were called accidentalpoints, and they deferved no better ap- pellation, for they were produced by chance, without rule: hence it followed, that examples of objects, the fides of which incline to the picture, were fcarcely ever given by the old writers on Perfpec- tive; and the few that may be found in their works are conftructed by a procefs fo inverted, that the vanilhing points are determined by the reprefentaticn of the objeft*, inftead of the reprefentation being produced by the vanifliing points, • See the Examples in the Jefuit's Perfpcaive, pages 102-111, both of which are very
imperfeft, particularly the example in the upper part of page 11 u which is totally falfe. D 2 It
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PERSPECTI V E,
It muit allo be obferved, that when the lines or planes to be repte»
fented were inclined both to the horizon and picture, the old writers appear to be wholly ignorant of the necefiity, or even ufe, of vaniih- ing points; and, therefore, conftructed their examples by means of the center and diftance of the picture only, firft finding the repre-? fentations of certain points in the object, then joining thofe points in the beft manner they could to produce the required figure; which method ferved them for all reprefentations of objects inclined to the picture or horizon. But this procefs, which is extremely de- fective and fallacious, is at the fame time fo laborious and tedious^ that the limpleil forms require more time and trouble to compleat their reprefentations, than would ferve to produce a very complex figure, when conduéied by the Doctor's principles, which are fo ex- tenfive, that they apply to the inclined picture equally with the ver- tical, or what may be called natural picture; which confederation leads the author to obferve, that in the following work, there are no inftructions given concerning the inclined picture, becaufe he does not think that fuch can ever he of real ufe to the artifi; and therefore recommends the works of the elder Malton and Hamilton to thofe who wiih to purfue the ftudy of Perfpective beyond the inftru&iom which are given in the following pages... |
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PERSPECTIVE.
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TERMS and DEFINITIONS.
j\ S no Science can be explained without the ufe of technical Ian- S
guage, the Student in Perfpeótive muft clearly underiland the
following Terms and Definitions: TERMS.
1. The pi&ure.
2. The vaniihing points,
3. The vaniihing lines.
The vaniiliing points are as follow :
i. The center of the picture, commonly called the point of
fight. 2. The difiance of the pifture.
3. The oblique vaniihing points.
The vaniiliing lines are the following:.
1. The horizontal line.
2. The prime vertical line.
3. The oblique vaniihing. lines.
To
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PERSPECTIVE.
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To thefe points and lines may be added, the ilation point, and
the bafe or ground line, by Dr. B. Taylor called the interfectiou of the picture. DEFINITIONS.
* In the practice of Perfpective, the term Picture, means the pa-
per, canvas, or tablet, on which the reprefentation of any object is to be drawn. The wnijhing Points.
A vaniihing point, is any point on the picture, in which the re^
prefentations of any number of right lines, the originals of which are parallel to each other, appear to meet or concentrate. 1. The center of the picture is the firft and principal vaniihing
point, and when the picture is vertical, or, as commonly called, up- right, the horizontal line conftantly pafles through it. The old writers on perfpective call it the point of fight. 2. The diftance of the picture, or point of diftance, is a point
which is generally fet off upon the horizontal line, either way from the center of the picture or point of fight, in the fame proportion- ate meafure that the painter or fpe&ator is fuppofed to fraud diftant from the picture, or from the view, or object he means to reprefent. The old writers call it the point of difiance ; but Dr. Brook Taylor, the difiance of the pi&ure. * In theory, the pi&ure is a feélion of the optic cone of vifual rays, made by a plane
paffing through that cone, perpendicular to its axis, and at any diftance from its apex. In nature, it is all that quantity of objefts which is comprehended in the cone of vifual rays, at any one point of view. Some authors call the Piclure the Perfpective Table, others the Perfpective Plane. See Ditton, Gravefande, Pritftley, &c. |
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3. The
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23
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3. The oblique vaniihing points, are thofe points to which all SECT,
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II.
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lines tend, which are to reprefent the appearances of original lines,
fituated obliquely, or inclined to the piaure; and they may be fituated on any part of the piaure. Thefe points, by the old writers on the fubjea, were called acci- ental points; but they knew no rules, and confequently gave none, for finding them. Example of vaniihing Points s
S"· !· C is a vaniihing point, and the lines 1, 2, 3, of ment F, which are the reprefentations of original lines peipendicular to the piaure, concentrate or vaniih in the point C i it is alfo the center of the piaure. In Pig. 2j of the fame plate the points V l and Vs, are oblique
vaniihing points, and all thé^ineTin the fronts X l, X 2, of the budding R 2, vaniih or meet in thofe points. The vanijhing Lines*
A vaniihing line is, any line on the plane of the piaure, in which
e rePrefentations of original planes, parallel to each other, appear to meet or concentrate.
e firft and principal vaniihing line, is the horizontal line,
which, as before obferved, when the piaure is vertical, always pafles
through the center of the piaure.
2. The next vaniihing line, is the prime vertical line, which, like
the former, always paffes through the center of the picture, and is perpendicular to the horizontal line. 3. The oblique vaniihing lineSj like the oblique vaniihifig pointSj
depend on the center and diftance of the piaure, and may be fitu- ated m any direaion upon the plane of the piaure. Of thefe lines, |
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as.
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«4 PERSPECTIVE.
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SECT, as of the prime vertical line, the old writers had no ideas; they
never ufed them, nor ever fpeak of them in their works. The bafe or ground line, which is called by Dr. Brook Taylor
the interfeCtion of the piélure, is that line which limits the bottom of the piélure, and generally indicates the firft appearance of the ground or plane, upon which the original objeéls are placed. Example of vaniihing Lines:
Plate III. Fig, ι. Η is a vaniihing line, and the floor F vaniihes
into it; as does the ceiling X, of the building A B. It is alfo the horizontal line. In Fig. 2, Plate III. the line R is the vaniihing line of the plane
■or face X 2, of the building. This is an oblique vaniihing line, the fide X 2, being oblique, or inclined to the picture ; it is alfo the va- niihing line for the fide Β of the block; for the fides or faces of the block, and of the building, are parallel to each other, consequently have the * fame vaniihing line. C D 3, in the fame figure, is the prime vertical line, and panes
-through the center of the piélure. Of the Center and Oifiance of the JPiBwre.
As it is abfolutely neceflary to poffefs a clear and perfeól idea of
the center of the picture, as alfo of the diftance of the piélure, fome farther explanation is here attempted, which the ftudent muft con- fider with attention. Dr. Brook Taylor was the firft perfon who diftinguilhed the center
of the picture from the point of fight; the latter term being con- stantly employed by the old writers upon the fubjeél, to exprefs what * - The reprefentations of all right lines vaniih in points. The reprefentations of all
planes vaniih in lines. he
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PERSPECTIVE, 2$
hejuflly calls the center of the piaure, which is a point dHlm& from PLAT5
the point of fight. L In the two treatifes upon Perfpeaive, publiihed by him, he gives
the two following Definitions: ^ 1· " The center of the piaure, is that point, where a line from
^ the fpedator's eye cuts it, (or its plane continued beyond the trame,ifneed be,) at right angles.
" Definition ift, in firft Edit. 1715."
2. ££ Tf f é.
" t' th · r°m POint °f fight there be drawn a line perpendicular
u 6 P Ure' the Point where that line cuts the picture is called (e 16 Center of the pi&ure; and the diftance bekeen that center
and the point of fight, is cailed the diftance of the piaure.
' Definition 7th, in fecond Edit. 1718."
In practice, the center of the piaure is that point which is marked upon the paper, canvas, or tablet, to indicate the point which is neareiUhe eye of the fpeaator who looks at it; hence follows the odor's fecond Definition, that, if from the point of fight, or (which is the fame thing) the eye of the fpeórator, a right line be fup- . rawn perpendicular to the piaure, the point where that right : cuts the pifture, will be its center. But the ftudent muft re-
'')er' l> Dy the center of the piaure, is not meant the center canvas, or tablet, as it can fcarcely ever happen that they will , m general, the canvas, or tablet, will not comprehend an equal but a partial portion, of the bafe of the cone of vifual rays; therefore the eentpv nfa .„ «iiei ot the pidure will be nearer to one fide than the
other of the tablet 0rm„ ÷ Ë ' 01 "lore towards the bottom than the top, iuft as
the nature of the 'view ,.,. ë-÷- , .
~ ■ , w> 01 "dpofition of the obieas to be repre-
iented, may require.
The diftance of the piaure reprefents the diftance at which the '
ipeaator u fuppofed to ftand, from the objea orlandfcape he means E to
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s6
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PLATE to reprefent; and the choice of a proper diftance is of great confe-
quence to the beauty of the reprefentation. It is almoft needlefs to obferve, that no object can be feen to ad-
vantage, unlefs the eye of the fpeciator.be withdrawn to fuch diftance as fhall admit the angle of vifual rays to comprehend the whole of the object: therefore, let it be fuppofed, the angle of virion is equal to 45°, it will be neceflary to retire fo far from the object, fuppofe in a houfe, that the fpace between it and the eye, may be at leaft equal to the height or width of that houfe. So alfo, in viewing a landfcape or ftreet, or the interior of a build-
ing, the ftudent may obferve, that he will not fee any object, nor any of the ground that is clofe to him; but on the contrary, his rays of viiion will meet the ground or floor, at a point or line coniiderably diftant from the point on which he ftands. It is this fpace, lying between the ftation or point where he ftands, and the part of the plane or floor firft feen, that conftitutes the diftance of the picture. By confidering the foregoing obfervations, it will be eafily under-
ftood, that the center of the picture, and point of fight, are two diftinci points; the former being a point on the picture, the latter a point out of the picture. Both points being the extremes of a line, pairing from the eye of the fpectator to the picture, and perpendicular, to it, the length of which line is the diftance of the picture. Example, Plate II.
PLATE -^et ^%· *· ^e confidered as reprefenting the interior of a room, one
u' of whofe fides, as A A A, is open to the country; and let the figures reprefenting the woman and boy be fuppofed two perfons, viewing the landfcape through that aperture. The
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PERSPECTIVE. 2?
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The aperture muft then be confidered as the picture, both to the PLATE
II,
woman and boy. Now the center of this opening is at C 3; but this is not the center of the picture, to either of thofe perfons who are
fuppofed to be looking at the view. For, to the woman, it is at * C, and that point muft be confidered as the reprefentation of a point in a line drawn from her eye perpendicular to the plane of the aperture, and interfering that plane in that point; but to the boy, who is looking through the fame aperture, the center of the picture is at C 2, which is very much to the right, and confiderably lower down in the picture; for the center of every picture or view is direéÏly oppofiteto the eye of the fpeélator, upon that plane which is fuppofed to intercept the vifual rays of the fpectator. Hence it will follow, that to perfons placed in the fituations reprefented by the figures, that the one placed in the fituation of the woman would have a very different view from the other, placed in the fituation of the boy. To the firil figure the horizon is elevated, to the iecorid depreifed,
becaufe it is ihorter than the other. The firft figure has the moft equal and advantageous view through the aperture; the fecond lefs on the right than on the left hand; and part of the view, which can be feen on the right by the woman, is concealed to the boy by the fide of the aperture. In Fig. 2. Plate II. the illuftration is farther attempted; and, to
the per/on who views the print, the fcheme is in profile, and the center of the pióhirea^at D % coinciding with the point of fight of that figure, but in a tranfverfe diredion, * It is alfo the center of the pifture to tbéperfort who views the print, for tire whole of
Fig. 2, is conftruaed to that center. It muft be obferved, that the initials are not exadly upon the points intended; as, C 3 fhould be at the top of the bonnet of the figure, and C in the middle of the head. C 2, alfo, fliould be in the middle of the head of the boy : but they are removed a little to the fide, to prevent confufion. Η Η is the horizontal line to the woman. E S Let
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S8 Ñ E Ê S Ñ E C Ô É V E,
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PLATE Let it be fuppofed that the figures D i, D % I) 3, reprefent three
perfons looking through the aperture, A l, 2, 3, 4, which to them muft be fuppofed the picture. The eye of each of thofe figures is the point of fight to each of them. C 2 is the center of the picture to the perfons reprefented by D 2 and D 3; but to the figure fitting, down, whofe point of fight is D l, the center of the picture is at C 1, as much below C 2, as the height of the eye of the figure D l is below that of the eye of the figure D 2. The diftance of the picture is alfo explained by this· exam pie,—far
the length of the line which paifes perpendicularly from the eye of each figure to the picture, conftitutes the different diftances to each, different figure.—Thus the fpace from C é ta D l, is the diftance of the picture to the figure D 1 j and from C a. to D 2, reprefents the diftance of the picture to the figure Da. Here it will not be improper to obferve, that at figure D 2 will be the beft diftance, as it may be pre- fumed, that the angle of vifion from that figure, will coincide with the aperture better than from either of the others; the firft being toe* near, the third too diftant. The ftation, point, and bafe line, or interfection of the picture;, are
alfo explained by thefe examples ; as the line A 3, A 4, is the inter- feron of the pi&ure, for it is the bottom of the aperture, and may be confidered as the firft place at which the figures difcover the floor., The ftation points are the points upon the floor, exactly perpendicular to the eyes of the figures, as Si, S % S3; or, they may be confidered as the points upon which thofe fpectatois ftand. Before the ftudent attempts to- delineate objects in Perfpective, it
wilt be proper for him to confider the forms of objects, and alfo their pofitions in relation to the picture. |
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PERSPECTIVE.
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The general forms of plans and elevations, upon which moil of the PLATE
objecls of art are conftruóted, are thofe of a fquare and cube,* the former being compofed of lines only, the latter of planes at right angles, or perpendicular to each other. Upon thefe principles are founded the outlines of all buildings*
whether plain or decorated; they not only apply to the conitruélion of the edifice, but alfo to its minuteft furniture. Thus, drawing the reprefentations of a fimple fquare and cube, in true Per- fpeaive, as viewed in all directions, is to apply all thofe principles of fcience with refpeft to praaice, which muft be ufed in the 1'eprefentation of all objeas, conftruaed upon the fame geometrical Principles. ■out it is not enough to underftand the form and conftruaion of the
°bjea; its poiition in relation to the piaure muft alfo be confidered, before a true perfpeaive reprefentation can be attempted.—For, let it be required to draw a cube in perfpeaive, as placed upon a plane, it muft be known whether any of its fides are parallel to the piaure, or whether any are inclined ; for thefe different pofitions require very different proceffes for the obtaining their reprefentations. Therefore, it is to be obferved, that there are four different pofitions· in which the fides or faces of a cube may be difpofed to the piaure. Thefe varieties are occafioned, not only by the form of the objeét, but alfo by the poiition of the plane upon which it is Iterated; for, in the firft and fecond pofition, the cube is fuppofed ftanding upon a horizontal plane, in the other two, as upon a plane inclined to the horizon. * A fquare is a fuperfkies, formed by four right lines of equal dimenfions, cutting each
other at right angles. A cube is a folid, formed by fix fquares of equal dimenfions, at right angles to each
ether. |
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PLATE The various pofitions are as follows: "f In the firft pofition, two of
the fides of the cube Will be parallel to the picture, and the other four will be perpendicular: this pofition is commonly called parallel to the piel ure. In the fecond pofition, two of the faces or fides will be perpen-
dicular to the picture, and four will be inclined. The third pofition will difpofe two fides perpendicular to the pic-
ture, and four inclined, both to the picture and horizon. The fourth pofition will difpofe all the faces or fides inclined, both
to the picture and horizon. To explain thefe important diftinctions, let Fig. 2, Plate II. be
confidered as an example; and let the figures D I, D 2, D 8, be fup- pofed fpectators, looking through the apertures». A l) 2, 3, 4, as before. Then fuppofe the blocks,  1 and  2, to be the objects which
thofe fpectators are viewing, the firft parallel, the fecond inclined to the picture. The block, Â l, is parallel to the picture, for the face b is parallel
to the line A 4, A 3, which line reprefents the interfection, or bafe line of the picture. j But the cube, Â 2, hath the fides inclined to the picture, for if the planes of thofe fides were continued, they would all cut or crofs the interfection A 4 of the picture. PLATE *%&* 'l& farther illuftrated by the example in Plate III., in which ni' Fig. l. reprefents objects whofe fides or faces are parallel and perpen- dicular to the picture. ■j- In theory, the parallelifm, or inclination of any line or plane to the piiture, implies
the difpofition of that line or plane to the axis of the cone of vifual rays of the fpeitator, rather than to the plane of the pifture; fo that, to the common vertical pifture, the original line that is parallel to the piclure, is perpendicular to the axis of the cone of vifual rays; and, contrarywife, that which is parallel to the axis of the cone of rays, is perpen- dicular to the piiture. é - \ But
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PERSPECTIVE.
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But in Fig. 2. in the fame Plate, all the objects have their fides or
faces inclined, or oblique to the picture. In Fig. l, Plate III. the front of the houfe E, and of the building
B, are both parallel to the piélure. They are alfo parallel to each other, while the fides A and G are perpendicular to the picture, as is likewife the ceiling X of the turret. In Fig. 2. of the fame Plate, the fides of the houfe Χ ι, Χ % and
alfo the fides of the block B, are inclined to the picture, while the roof R 2 of the houfe, is inclined both to the piélure and to the horizon. The two figures contained in Plate III. are to be confidered as ex-
amples of objects difpofed to the piélure, in the firil and fecond pofitions of the cube, as before defcribed: but the roof R 2, Fig. 2, is inclined to the horizon, and, therefore, partakes of the third po- fition of the cube, which will be explained in the fourth Section. It may be obferved, that the examples of Plate III. exhibit the
reprefentations of objects, difpofed in all the pofitions that are gene- rally required by the artift, upon the natural picture; that is, when the plane of reprefentation is perfectly vertical, and perpendicular to the axis of the cone of vifual rays; in which poiition, the reprefen- tation of objects that are drawn upon it will appear tolerably juft in many different points of view; whereas, in the inclined picture» the reprefentation of objects, when drawn upon it, will appear diftorted, ofFenfive, and fometimes unintelligible in all points of view, except that by which it was conftruéled. • It is from the foregoing circumftances that the author has before
©bferved, in the Introduélion, that reprefentations upon inclined- pictures are of no ufe to the painter; and, therefore, no inftruélions are given in this work, concerning that unneceifary part of the fcience, Rudiments
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Rudiments of Praiïice.
When objects are to be drawn in perfpeaive, the following proceft muft be employed; , |
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PLATE
III. |
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Pale 33.
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Example
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ι Firft, draw the bafe line, or interfeaion of the pidure G G.
«. * Then, at the proper height, draw the horizontal line D D 2, parallel to the bafe line.
3. Upon the horizontal line mark the center of the piaure C,
commonly called the point of fight, t < *
4. + Through the center C draw the line V perpendicular to the
bafe line.
. As the horizontal line always paffes through the fpeftator's eye, it muft always be fet
Astlienoi proportionate meafure that the eye of the fpeftator lS tZ*«tï^^ÏZA^ or plane upon which he ftands: the natural
height,' as commonly allowed, is five feet fix inches. t It is.alfo markedι center; ^ ^ objeas are
„l*^^^^ than as a guide for all the oth.hu. whic:^
required to be drawn perpendicular to the horizon ;-but, when the objefts to be repre- fented are inclined to the pidure, it is indifpenfcble. ^ ^^ |
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PERSPECTIVE.
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5. From the center of the picture C, fet off on either fide, or on
both fides,5 upon the horizontal line, the points D, D 2 for the point of diftance, or diftance of the picture. Thefe points and lines are abfolutely neceffary in all cafes, and are
fufficient, when the objects to be reprefented have their fides parallel and perpendicular to the picture; but, if their fides are inclined to the picture, additional lines are required, which will be demonftrated in the next fection. Having drawn the foregoing lines, let it be fuppofed, that a fquare
is required to be drawn in perfpective, as on the ground, and alfo a cube, of the fame dimenfions with the fquare, ftanding on the fame ground or plane, the fquare A at forne diftance beyond the picture, but the cube  clofe to it. S l, is the feat of one of the angles of the fquare upon the picture,
therefore, from S I, draw a line to the center of the picture C; then take tire dimenfions of the fquare with the compafies, (any fuppofed fpace will be fufficient in the prefent cafe, without a fcale), and fet that fpace upon the bafe line, from S é to 2, and draw another line from 2 to C, then take the known diftance that the fquare is beyond the picture, fuppofe it to be half the meaiure of one fide of the fquare, and fet it on the bafe line from 2 to 3, and from 3 draw a right line to D, then take the fpace from S l to 2, and fet it on the bafe line from 3 to 4, and from 4 draw another line to D, thefe laft lines will cut the line 2 C * at the points a and b ,\ then draw two lines from the points a and b, parallel to the ground line, till they cut the line S 1, C in the points e f, and the reprefentation of the fquare will be completed. . For the block B, proceed as follows: Suppoie the point 0, the feat * The lines S é, C, and 2 C, are the indefinite reprefentatioos of lines perpendicular to
Ae pi&ure, parts of which lines form two fides of the fquare. F. of
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PERSPECTIVE.
of one of the angles of the block ; take the meafure of the fquare A*.
that is, from S é to % and fet it oif from 6 on the bafe line to 5, and; from 5 and 6 draw right lines to C · then fet on the fame meafure, from 6 to 7, and from 7 draw a right line to D 2 ; and where it cuts the line C ä, as at e, will reprefent the depth of the bafe of the- cube; then from the points 6, 5, and e, draw lines parallel to the vertical line V, or (which will produce the fame effeót) perpendicular to the bafe line, as a, b, e 1; then upon the line a or b, fet up from the bafe line the fame meafure as from 6 to 5, and at the point d draw aline parallel to the bafe line, cutting the other perpendicular line at g, then will the front face  of the block be formed. To complete the cube, draw lines from d and g to the center of the picture C, and where thofe lines cut the perpendicular line e h, will give another fide of the block. At the interfe&ion h of the vertical line e h, by the line d C, draw
another line, parallel to the bafe line, joining the line g C at k, and the cube will be completed, Thefe figures contain the practical elements of all objects that can
be drawn in Perfpe&ive, when their faces or fides are parallel and perpendicular to the picture; therefore, the ftudent will do well to confider them attentively, and to make himfelf mafter of their conftruction. The procefs, as directed in the five different articles at the com-
mencement of this problem, is indifpenfably neceffary, and muft be the fame in all cafes where the rules of Perfpeétive are employed in the reprefentations of objects, whofe faces, or fides, are parallel and perpendicular to the picture ; but when the objects to be reprefented,. have their faces or fides inclined to the picture, then the fifth article, which relates to the diftance of the picture, muft be different, as will 'be ihewn in the next fection,. The
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PERSPECTIVE.
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S3
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JV. Â. The feat of a point upon the picture is produced by a right SECT.
line drawn from any given point in the original object, perpendicular «_^~J-^ to the picture, the interferon of which line with the picture, is Flate Ir" its feat. |
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Example I. Page 32.
S 1 is the feat upon the picture of the angle e of the fquare; it is
alfo the feat of f, becaufe both thofe points are in the line C S l, which represents a line perpendicular to the picture. Alfo in Plate III. Fig. 2, the feat of the angle ñ of the building PLATE
upon the picture is at S, which is further explained by the geome- trical plan; the angle of the building upon the ground being at O, and the line Ï S being perpendicular to the picture, therefore S is the feat of the point Ï upon the picture. The feat of a point or line upon the picture, or upon the repre-
fentation of any plane, is further considered in the fourth fection. The Scale.
As no true reprefentation of any original object; can be obtained,
unlefs that reprefentation be drawn to a fcale, fo that all the parts may be in proportion to each other, and alfo to the original object; and as many ftudents in art are quite uninformed of the neceifary principle, it will not be improper to give fome hints for the con- struction and application,of a fcale. # Upon any part of the picture, or upon a feparate paper, draw a
* It is needlefs to obferve, that the fcale muil always be proportioned to the fizè of the
canvas, tablet, or paper, upon which the objeifl is drawn. In the example, the fcale is marked by the figures, 4, 8, tz, Sec. for the fpace between
every figure is intended to exprefs four feet. F « · right
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PERSPECTIVE.
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right line, or two lines, as in Plate III.; then, with the compaffeiv
extended to any meafure, fuppofe one quarter, or one half of an inch,. as may be moil convenient, fet on upon the lines as many of thofê equal fpaces or parts as may be neceffary, fuppofe fix; then divide one of thofe fpaces into 12 equal parts, and let the greater fpaces be confidered as feet, and the leffer as inches, by fuch fcale, not only the object mull be proportioned, but alfo the height of the horizontal line> and the place of the eye, or diflance of the picture, muil be difpofed. Inattention to thefe neceffary confiderations, has not only obfcured the inflructions of many writers on the fciénce, but * alfo produced diftortion in their examples and diagrams.. In the courfe of this work, the terms given or fuppofed meafure,
are fometimes employed, it may therefore be proper to explain what is meant by the author in thofe expreffions. When any objeét or building already conftructed, is to be accu-
rately reprefented in Perfpe&ive, the meafures of all the parts of fuch original object ihould be obtained—thefe are the given meafures. The fame may be faid of an original iketch accurately figured, in both which cafes the meafures muil be applied by a fcale. But when a picture or drawing is to be made of an object, i»
which general proportions only are required, fufficient to produce a good effect, then the meafures may be fuppofed, in fuch proportion, as may bell anfwer. the intention of the artift, without regard to a fcale.. * See figure 24 in Dr. Brook Taylor's firft Treatife, which, though an elegant problem.,
is yet very ill drawn, and therefore not eafily understood. See alfo feveral of the fchemes in Noble's. Perfpeftive.. |
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PERSPECTIVE.
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3?
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SECTION II.
Containing Examples of Objects in which the Lines and Planes are
parallel and perpendicular to the Picture. Plate IV. Fig. i.
ÃÔÐÇIS figure reprefents a pavement of fquares, upon which are PLATE
-*- placed two blocks; the fides of the fquares are each £ feet in dimeniions.* The block, Â 1, is 1 feet by 2 feet 3 inches, at the bafe, or end
upon which it ftands, and 4 feet high. Its feat or fituation upon the ground, is nearly at the middle of the fecond fquare, beyond the picture; and upon part of the fifth and fixth from the ftation; which ftation may be confidered as at the point S. The prifm or pillar, Â 2, is 2 feet fquare at the bafe, and 18 feet
6 inches high;: and it ftands upon the third fquare of the pavement, beyond the picture, in the firft row next the wall. The breadth of the picture, or tablet, contains twelve fquares,
with a margin of one foot breadth on each fide, which margin isalfo expreifed at the farther end, but not in front. Ç is the horizontal line, which is drawn 6 feet f above the bafe
line of the piéture, which bafe line is A m. C is the center of the *The figures in this Plate are both drawn to the fame icale which is given tinder
Fig. i... piéture.
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PERSPECTIVE.
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38
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SECT, picture. D l, D 2, are the points of diftance which are fet oiF each
way from the center C, about 17 feet f. |
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Plate IV.
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Procefsfor tlie Pavement.
For the fquares proceed as follows : * Take é foot by the coinpaffes
from the fcale, and fet it from either end of the bafe line, or bottom of the picture, as from y to x, and from o to m, which fpace is equal to the breadth of the margin. Then take two feet from the fcale, and fet that meafure on the
fame line from ÷ to a, to b, to e, and continue thofe meafures to m, and from every one of thofe divifions, draw lines to the center of the picture C, and thofe lines will produce the indefinite appearances of thofe fides of the fquares which are perpendicular to. the picture. From the point m in the bafe line, draw a line to the point of diftance D 1, and where that line cuts or interfects the lines which are drawn to C, will give the apparent depths of the fquares, as they recede from the picture. Laftly, through all the points in which the line D l m, interfects
or cuts the lines that are drawn converging to C, (as the points % 3,) draw lines parallel to the ground line, or bafe of the picture, and the general figure of the pavement will be obtained, which requires nothing more than to be diftinguiihed by the alternate colours of the fquares. To reprefent the block  l, proceed as follows :
Let the point w, be the feat of the angle V of the block upon the
bafe line. * This example is bounded by the lines A, A i, A 2, A 3, which lines, in fubjefts
fimilar to this, fhould generally be firft drawn, that the whole view may be determined with certainty and elegance, particularly in a drawing. But if a pifture is intended, the canvas ihould be made in proportion to the width and height required, and the bafe line drawn a little above t& lower edge. From
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PERSPECTIVE.
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From w, fet on upon the ground line to p, two feet by the fcale,
and draw lines to C, the center of the picture. Then from the point p, fet on to ç the meafure equal the diftance
which the block is beyond the picture, which, in this example, is about 2 feet four inches by the fcale; and from n, fet on to æ in the bafe line, the meafure of the depth of the block, nearly 2 feet; and from the points æ and n, draw right lines to the point of diftance D 2, and their interfeótions, with the line which is drawrn from ñ to the center C, gives the apparent depth cf the block, as at the points 6 and 7; at which points draw lines parallel to the horizontal line, that may interfect the line drawn from w to C, and the plan of the block will be obtained. Complete the figure as follows: At the four angles of the Për-
fpective plan, draw lines perpendicular to the horizontal line, and· determine their heights thus; at the point w draw a line perpen- dicular to the ground line, and upon it fet up the given height of the block, four feet, as from w to F, and from F draw a line to the center of the picture C, which will interfeét the angle of the block at L, and consequently give the height required. The prifm, or block  2, Hands upon one of the fquares in the
third row beyond the picture, and the fixth, reckoning from the ftation S towards the left: the bafe is of the fame dimenfions with the fquares. Therefore, at the angles of the fquare, draw lines per- pendicular to the horizon, and determine their height as follows : At the point ÷ in the ground line, draw a line perpendicular to
the horizontal line, as the line ÷ 5, and upon it fet up the given height of the prifm, as from ÷ to 5 ; from 5, draw a line to C, the center of the picture, which will cut the angle of the prifm at Í, and determine the required height.. At the point Í draw a line parallel to the horizon, which will
: cut
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40 PERSPECTIVE.
SECT, cut the other perpendicular line at q, from which draw a line to the
»—-ã-—» center of the picture C, which will give the point z, and determine the height and form of the prifm without the cap. * The reprefentation of the Avail Y, % is obtained by fetting up
the known height from the bafe line y to Y, and from Y, drawing a line to the center of the picture C ; and as the wall ex- tends to the end of the pavement, nothing more is required than to draw a line at the farther angle of the pavement perpendicular to the horizon, which will give the point T, and determine the limits of the wall. Plate IV. Fig. 2, is an example of a building with two arches,
which may be fuppofed as intended to contain a garden feat. Its dimenfions are as follows: The height is - -' 18| feet Width, - - 15f
Depth, - - 8 f
Height of the piers from the ground to the fpring of the arch l of·
The building is three feet fix inches beyond the picture.
Ç is the horizontal line, which is fix feet above the bafe line of the
picture. C is the center of the picture, and D the diftance, which is about 171 feet. Let the point e, on the bafe line, be the feat on the picture of the
angle Ñ of the building. From e draw a right line to the center of the picture C.
Then fet on upon the bafe line from e to f, the diftance which the
objeft is beyond the picture, which is S feet 6 inches by the fcale, and from f draw a line to D, and its interferon with the line drawn from * The covering or cap of the block  ô, is omitted in thefe inftruüions, but may be feen
in another part of the work. It muft be obferved, that the whole of the procefs by which the reprefentations of thefe blocks are obtained, is but a repetition of that which was firft taught in the rudiments» page 3.2. |
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perspective:
e to C will be p, which reprefents the feat upon the ground of that
angle of the building, which is neareft to the fpectator. At the point o continue the bafe line to any convenient length, as
to x, and from the point e, fet on to ÷ the whole breadth of the building, 15 feet 6 inches, and from x, draw a line to the center of the piéture C. Then at the point p, draw a line parallel to the horizon, or to the
bafe line of the picture, as the line ñ t, which will mark the apparent width upon the ground of the front of the building. From the point f, fet on to 3 in the bafe line, the depth of the
building, 8 feet 6 inches, and draw a line from 3 to D, which will interfeel the line e C in S, confequently ñ S is the apparent depth of the building. Next determine the height as follows : At the points t, p, s, draw
lines perpendicular to the horizon, as the lines 11, ñ u, and s v. Then at the point ÷ in the bafe line, draw the line ÷ k, perpendicular to the horizon, and from x, fet up the whole height of the building, by the fcale 18 feet 3 inches, to k, and from k, draw a line to the center of the picture C, which will interfect the line 11 in 1, there* fore, 11 is the apparent height of the building. At 1 draw the line 1 u parallel to the horizon, and from the inter~
feótion u draw a line to the center of the picture, which will give the interfeetion í, confequently, the lines ñ t, 11, 1 u, u í, í s, and ñ s, determine the general form or outline of the building. Then find the proportions of the piers and arches as follows:
The piers are all of the fame dimenfions, and the arches are fimilar
to each other; their meafures are as follows ; The piers are each l foot 8 inches fquare.
The opening of the arches 5 foot 3 inches. From the point e, which is the feat of the angle ñ of the building
G , upon
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42 PERSPECTIVE.
SECT, upon the bafe line, fet on to d, the breadth of the pier, l foot 8
II.
i^-^^ji inches, and from d to e, the width of the arch, 5 feet 3 inches ; from e to d the breadth of the fecond pier, and then the fecond arch,
and laftly, the third pier, obferving that thefe different proportions
Draft all coincide with the whole meafure of the front, as indicated
from e to x. From every one' of thofe points draw lines to the
center of the picture C, and their interfeüions with the line t ñ will
mark the apparent widths of the piers and arches, as at the points
w, z, y, at which points draw lines perpendicular to the horizon,
which will exprefs the faces of the piers with the apertures of the
arches.
Their heights muft next be found as follows :
Upon the vertical line ÷ k, fet up from ÷ to h, the height of the
piers, 10 feet 8 inches by the fcale, and from h draw a line to C, and its interfection with the angle of the building at n, will be the apparent height of the piers. Then find their depths as follows : At the point f in the bafe line,
fet off to g, the meafure of the depth of the piers, which, in this example, is equal to the fronts; and from g draw a line to the diilance D, and the interfeétion q with the line e C, will determine the apparent depth of that pier. To determine the others, draw aline at q parallel to the bafe line,
which, by interfering the lines e C, and f C, will determine the apparent widths of the other piers, as expreifed by the point w. The depth of the back pier is alfo found by the fame procefs,
that is, by marking the meafure from 3 to £ upon the bafe line, which meafure being transferred by a line dra\m to the diftance of the picture D, will give the point r, the apparent width of the back pier. The places and dimeniions of the fafcia? upon the upper part of
the building, are determined by the fame procefs which was em- ployed |
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PERSPECTIVE.
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43
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ployed to obtain the height of the building, that is, by fetting the SECT
proper meafures upon the line ÷ k, and transferring them to the angle |
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Plate IV.
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of the building f 1, by lines drawn to the center of the picture C,
which is * fufficiently demonitrated by the points m and n. It would be vain to attempt an explanation of the procefs neceifary
for the conftruólion of every fmall part of the building reprefented in this example; it is fufficient to obferve, that the thicknefs of the wall is marked upon the bafe line from ÷ to a, which is transferred by the center of the picture C, and gives the lower line ■$& of the interior of the building. The ftudent will clearly comprehend the conftruction of the other
parts, by applying a ruler to the center of the piélure C in the example, which will demonftrate the methods for meafuring all heights, and to the diftance of the picture D for all depths. Thus the heights upon the building of the parts 1, m, n, are deter-
mined by lines drawn from k, m, h, to the center of the picture C. And the depths of the piers and building p, q, s, are found by lines
drawn to the diftance of the piélure D, from the meafures f, g, 3, fet on the bafe line. f Example of a Houfe whofe Front E is parallel, and wkofe Side G is,
perpendicular to the PiBture. |
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Fig. i. Plate III.
The bafe line of the picture is marked by its title. Ç Ç is the . AT
horizontal line; the points of diftance are marked D, and dift- I11· ance; either of which may be employed, but D is the moil con- venient, * The manner of finding the centers of-the arches, and defcribing the archivau'ts,
,mul**> and foflits, is given in Example F, Plate VII. Fig. .3. · . 'I G a The
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PERSPECTIVE.
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The building is 2 feet beyond the picture.
The front Eis 13 f feet wide. The iide G is 16 feet deep. The height 24 feet. Let a be fuppofed the feat upon the picture, of the angle of the
building neareft the fpectator. From a, draw a right line to the center of the pi&ure C, as the
line a É. Then from a, fet on upon the bafe line to e, 2 feet by the fcale,
equal the diftance which the houfe is beyond the picture ; and from e draw a line to the diftance D, which will cut the line a C at S, con- fequently, the point § will reprefent the angle of the building-neareft to the fpectator. From a, upon the bafe line, fet on to b, the width of the front of
the building, 13 feet 6 inches j and from b draw a right line to C, the center of the picture. At the point S draw a line parallel to the bafe line, as S g, which Ayill be the bafe line of the front of the building. Find the depth of the building' as follows :
Set on from e in the bafe line to h, 16 feet by the fcale, and
from h draw a right line to the diftance of the picture D, and its interfection K, with the line drawn from a to C, will be the apparent depth of the building. At the points S, g, K, draw lines perpendicular to the horizon, as the
lines g ÷ % S v, and E m. Then find the height of the building as follows :
At b in the bafe line, which is the feat on the picture of the angle g
of the building, draw the vertical line b m, and upon that line, fet up the height 24 feet to o, from which point, draw a line to C, the center of the piéture, and its interfection ÷ with the line ÷ 2, g will be the height of the building. At
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PERSPECTIVE.
At ÷ 2, draw a line parallel to the horizon, as ÷ 2, V, and from V,
draw another line to the center of the picture C, which will produce the interfeótion m : then will the lines g, ÷ 2, d v, and Ê m, reprefent the outlines of the building. It remains to find the reprefentations of the door and window, but it is not neceifary that every part mould be minutely defcribed, which would be but a tedious repetition of the fame procefs applied to different parts; therefore, no more will be given than to find the appearance of the door, and the height of the top of the middle window, which are obtained by the following procefs: From a in the bafe line, fet on to d the width of the pier, and from
d on the fame line, fet on to e the Ayidth of the opening of the door; and from thefe two points, d and e, draw lines to the center of the picture C, and the interfeétions n i of thofe lines, with the bafe line gs of the building, will give the dimenfions for the width of the aperture of the door; as the fpace ç i is the breadth required. Then find the height of the door as follows :
Upon the vertical line b m, fet up from the bafe line the height of
the door to y, 6 feet by the fcale, and draw a line to the center of the picture C, which will interfeci the angle g, ÷ 2, of the building at 4, then from the points ç i, draw two vertical lines to exprefs the fides of the door, and from 4, draw a right line parallel to the horizon, which will exprefs the top of the door. The middle window is of the fame width with the door, con-
fequently, the fides of the door continued upward, will produce the fides of the window, while its height is found by the fame procefs which is employed for the height of the door; the meafure being fet up upon the vertical line b m, from y to w, and transferred by the center of the picture C, to the plane E, as at the point 5, from which, a line drawn parallel to the horizon, will give the top of the window. u
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PERSPECTIVE.
* It will be proper to confider the examples in Plate III. in relation
to the planes of which the objects are compofed, and alfo to mark other circumftances, with which the ftudent in Perfpeótive ought to ■be well informed. All the buildings in Fig. l and 2, Plate III. are compofed of
planes, and the doors and windows are to be confidered as apertures in thofe planes; the dimenfions of which muft be found by marking their meafures on the interfections of the picture, and then transferring ■I
thofe meafures by the center of the picture, to the edges of the planes,
in which thofe apertures are feated. |
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46
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Plate UI.
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A Building with Wings parallel to the Picture.
Plate V. Fig. I and 2.
Both examples in the plate muft be confidered as drawn from
the fame object; Fig. 2 being an enlarged part of 1^·. i, the better to explain the procefs. The building confifts of a center, with a fmall arcade, uniting two
wings, with an area in the middle. The fcale is adapted to the fmall example, which is half the fize of
the large one; therefore, what is marked io feet in the fcale, muft be confidered as no more than 5 feet, when applied to the large example, Fig. 2. The dimenfions are as follows :
The center part of the building is 66 feet wide.
The plans of thé íéï pes are 28 feet f fquare, and have the fame
height with the center building, which is 37 feet from a to p, the top of the blocking courfe. * The femicircukr window, and the other parts of the building, are produced by rules,
which are given in the following examples. The
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PLATE
V. |
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PERSPECTIVE.
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The width of the area, between the wings, is equal to the width of
the center building, which is 66 feet, confequently, the whole extent of the wings and area is 123 feet in front, from a to b 2. Fig. l. The area beyond the wings, is 12 feet on each fide wider than in
front, fo that its width between the arcades is 90 feet. The depth of the area from the front of the wings to the front of
the center building, is 87 k feet. * The height of the eye is 8 feet, and the diftance of the picture
about 90 feet. In drawing objects of this kind, proceed as follows :
Draw the ground line, or interfeétion of the picture, a b 2, Fig. 1,
or G G 2, Fig. 2, and upon that line determine the center 1 of the whole breadth of the edifice ; and from the center 1, mark half the width of the area between the wings, 32 feet each way, as at b and a 2, Fig. 1; and from b a, each way the width of the wings b a, 28 \ feet. Then draw the faces or fronts of the wings geometrically, as X 2,
Fig. 2, and difpofe the windows and doors by their true meafures, beginning with the inward pier R, and marking off the piers and windows alternately, to the proper dimenfions, as from b to «, irt both figures. Having proceeded thus far, mark the height of the eye, and draw
the horizontal line Η Η. Then mark the center of the picture at pleafure, as at C, after which,
draw the fides of the wings. * The height of the eye is raifed two feet above the common height, for the fake of
producing a more pleafing effect in the appearance of the area, than could have been ob- tained by the common height. In this example it muil be obferved, that the center of the piéture C, is not in the center
of the building, but on one fide, as recommended in the fubfequent inftructions ; by this means, the building is better explained, and appears more piiturefque than it would do» if the center of the picture coincided with the middle of the door. It
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PERSPECTIVE.
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48
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SECT. ^* mu^ ^e obferved, that in the example, Fig. 2, the plate not
. I/V. being fuflficiently large to contain the whole diftance of the picture,
Plate V. the half only is employed; but, as the fcale is fitted to the fmall
example, the numbers with which it is marked, correfpond with the
given dimenfions, although no more than equal to half the proportion
of the larger example.
Thus the diftance of the picture is 90 feet, the half of which is 45
feet; therefore, from the center of the piclure C, fet on to D, 90 feet by the green fcale, which would be no more than 45 feet, if the fcale were proportioned to the large example, Fig. 2, Having thus fixed the point of diftance to half its length, con-
tinue to employ the fame proportion and fcale, for the meafures of the fides of the wings, as follows ; The plan of the wing X % Fig. 2, is a fquare, each fide 28 feet f,
therefore, take that meafure from the given fcale, and fet it off from the angle b, to the point 4, on the bafe line; then having drawn a line from the angle b, to the center of the picture C, draw another line from 4 to D, and their mutual interfeélion at N, will mark the depth of tlie fide of the wing, at which point draw the vertical line Í Ì,
It muil be obferved, that the application of the half diftance and
meafure, is only to proportion thofe lines which reprefent lines per- pendicular to the pióture, not thofe which are parallel to it; as the line b N, Fig. 2, is perpendicular to the piclure, reprefenting the depth of the wing; but the fronts which are parallel to the piclure, muft be determined by the whole meafure. Then from the point S, in the upper angle of the front, draw a
line to the center of the pi&ure C, which will inteifeft the line Í Ì, and produce the reprefentation of the outlines of the fide of the wing. Complete
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PERSPECTIVE. 49
Complete the fide as follows: S EC'1
II
The windows are 8 feet wide, and of the fame.height with thofe of v*-v-— the front, and the piers are 10 feet 5 inches wide each. Fiute v
Therefore, take 10 feet 5 inches from the fcaie, and fet it on from
the angle b to the point e, and from e to 5, the width of the window, by the fcaie 8 feet, and from 5 to 4, the width of the pier as before, and from the points e, 5, 4, draw lines to the point of diftance, D, and the interfection of thofe lines, with the line that is drawn from b to the center of the picture, as the points ç ñ, will give the perspective pro- portionate width of. the window, at which points, draw lines per- pendicular to the horizon, which will determine the width of all the windows on that fide. To determine their heights, continue the bottom and top lines of
the windows in the front X 2, till they interfeót the angle b S of the wing, as at the points f, f; and, from thofe points, draw lines to the center of the picture C, which will determine the height of the windows in the fide w, as marked by the example. Having thus far determined the forms and general proportions
of the wings, proceed to defcribe the center part of the building O, which, as before obferved, is equal to the fpace between the wings, 66 feet. The depth of the area, from the front of the wings, to the front of
the center building, is 87! feet, which is 57 feet more than .the depth of the wing; therefore, take 57 feet from the fcaie, and fet it upon the bafe line from the point 4 to the point G % and from G 2, draw a right line to the half diftance D, and its interferon y, with the line that is drawn from the neareft angle b of the wing, to-the center of the picture C, will give the depth of the area. At the point y, draw a line parallel to the horizontal line as í R;
which will be the bafe line of the center building; then determine |
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50 PJBRSPECTIV Ë.
SEC T. its breadth by continuing the fide of the other wing, to the center
u—v-w of the picture C, as from a % in Fig. I, which will give the inter- lection k,„ and confequently determine the width'' of the center building, as y Κ in the final! figure. To filid the apparent height, draw the line g, g; g, Fig. % perpen-
dicular to the horizon, at the interfeétion. y, and continue the line from the blocking courfe S, on the fide of the wing, to the center of the piciure C, and the interfeétion g x, Fig. l, or Fig. 2, will be the apparent height of the center building C Complete the general form of the front O, obferving the break
where the * pediment fprings, and determine the door and windows as follows : In thebafe or ground line,, the point ι is the middle of the extent of
the whole building, confequently, it may be confidered as a point in a right line, drawn from the middle of the door ; therefore,, draw a, right line from l to the center of the picture C, and its interfee~tion, with the lower line V of the center building,, will give the reprefentatkm of the middle of the door as at m. Find the width of the door by fetting its real width by the fcale,
5 feet, half of which muft be marked on the ground line, each way from the point i, as-the points 2, 3. Fig. fL Transfer that meafure to the front of the building, by lines drawn,
from 3 and 2 to the center of the piéture C, and the interfections of thofe lines* with the bafe line V of the centen building, will give the * To thofe who are not acquainted, with the principles of architecture,, it is neceiTary, to-
obfefve, that all pediments have certain proportions for their height or pitch, the beft medium, of which is, two ninths of its width. Thus in example, Plate V. Fig. 3. a b isthe width of the pediment, thererefore the dotted line a b, is divided into nine equal, parts,, two of which kre fet up from 3 to C, confequently, C is the height of the pediment, to which points. draw 'lines from a and b,. and the. general outlines of the pediment will be deteïmined. - - |
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apparent
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PERSPECTIVE.
apparent breadth of the diftant door, as is exprefled in the esarrTt-e,
Eig. .2. For the proportions of the piers and windows, purfue the fame
procefs as for the door. Thus, upon the ground line GG 2, fet off from 3 to 8, the width of the pier, and from 8 to 9, the width of the window; and transfer thofe meafures to the bafe line V m of the diftant front, by lines drawn to the center of the picture C, and the inter- fections of thofe lines, with the line V A, will give the apparent breadth of the windows; from which interfections, -draw lines perpendicular to the horizon, which will determine the breadth «f the windows. Then find the height of the windows in the diftant building, by
continuing the lines of the tops and bottoms of thofe which are on the fides of the wings, as from f f, &c. to the center of the picture -C, and the interfections of thofe lines with the angles g, g, g, will mark the dimenfions required, which dimenfions muil be transferred to the fpaces for the windows, by lines-drawn parallel to the horizon, as the lines g h, gh. *, Fig. 2. For the width of the area beyond the wings, fet the addi-
tional meafure, 12 feet, from b to X on the bafe line, and draw the vertical line X X 2. From the point X, draw a line to the center of thepiélure C, and extend the bafe line V A, of the center ø&% of the building, till it interfeéls the line X C in the point R; and draw the vertical line R T, which will be the remote angle of the arcade, the height of which is marked from X to k, in the line X X 2, and is transferred to Ô by a line drawn from k to the center of the picture. In drawing peffpeftive views of buildings, fimilar to this example,
it cannot be expecled that all the fmall parts ihould be made out by * To find the height and pitch of the roof on the wings, fee Pkte IX, Fig.-J, The
pediment is alfo illsftrated and explained in theythird feSion, Rate j@, Ç s the
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PERSPECTIVE.
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the ftriet rules of the fcience, yet the general form muft be decided?
with accuracy ; after which the fmaller parts, fuch as the members of the cornice, the dreffing» of the doors,-windows, and other inferior ornaments, may be finiilied by the eye, particularly infmall drawings ;· but in larger works, the ornamental parts muft be added agreeably to the rules which are given in the fucceeding parte of tliis-worfi. It is alfo prudent neve» to defcribe the exterior view of any fihgle·
building as diftant from the picture, but to draw it as commencing at the interferon of the picture; and afterwards to introduce a fore ground, as indicated in Fig. é, where the Mne a B % is the inter* feet ion or ground line, although drawn above the lower limits of the picture; by this method, great trouble and much time will be favedj efpecially when the object is inclined to the picture; The heights, and conftr^etions of the roofs of the wings, are deter-
mined by the procefs given for the example, Fig. 3. in Plate IX. the general form of that figure being4 exactly the fame with the wingst in Plate V. The pediment of the center part of the building having all its parts
parallel to.the picture, may be drawn almoft geometrically; efpecially, when the reprefentation isfmall and diftant from the picture, as in this, example; but when large, thofe parts muft,be determined, by the rules, given for the explanation of the figures in Plates IX. and X. which, exhibit examples of mouldings parallel and perpendicular to.the. picture. Interior o f a Room, whof& Sides are parallel ana' perpendicular to-
the Picture.
In Plate VI. is an example reprefenting the infide of a chamber,
which may be fuppofed a drawing-room, finiilied with a cove and a flat ceiling, divided, into compartments, The
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PERSPECTIVE.
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f he proportions are as follows:
The whole length of the room is 26 feet 3 inches,.
The width 20 feet..
The height, including the cove, 15 feet».
The height of the windows 12 feet.
Their width 4 feet 3 inches.
* The width of the piers s feet 9 inches.
The height of the door 7 feet 8 inches..
The width 3 feet 6 inches..
The chimney 3 feet 6 inches high, by 4 feet 3 inches wide.
Before the ftudent proceeds to operation, it will be proper for him to obferve, that in this example, the fpeétator is fuppofed to be Handing in the room, therefore, the whole of it is not fèen by him, becaufe a part lies behind the angle of his vifion; hence it follows, that all the firft pier, and more than half of the neareil window, cannot be intro- ducedlnto the piëlure, becaufe it cannot be feen by the fpeclator; this muft ever be the cafe in all interior views. The fpeelator being prevented by the limits of the room, or
building, from retiring to fiich diftance as would allow him to com- prehend the whole within the angle of vifion; therefore, a part muft be omitted, to produce a natural reprefentation, as in this example; yet, by the rules of the fciènce, the room might be reprefented to its full length, but fuch reprefentation would give a falfe idea of the dimenfions, by making it appear longer than the original,, an effeéi which ought ever to be avoided f. The operation is as follows:. •The iecond pier is rather lefsi
f· A portion of the room, equal to 6 feet 3 inches, is omitted, as not falling into the-
angle of vifion. Draw»
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PERSPECTIVE.
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Draw the boundary lines, or limits of the picture, equal to the
given meafures of the width and height of the room, as the lines  é,  2,  3,  4. Then draw the horizontal line at a proper height above the bafe
line, or bottom of the picture, as the line Ç Ç is the horizontal line, which is 5 fee| 6 inches above the line a, B 4, b. Then determine the center of the picture C, and alfo the point of
■diftance *, which, in this example, is 16 feet by the fcale, from C. Having thus determined the center and diftance, draw right lines
irom the four angles, a, b, K, e, to the center of the picture C. Then, on the bafe line, fet off from the paint a,-the meafure of the
depth of the room, as from a to b, which, in this example, is no more than 20 feet, that meafure being as much as can be feen by a fpe&ato? Handing in the room. From b, draw a right line to the point of diftance, which will
interfeót the line that is drawn from a to C in the point e, con- fequently, the fpace from a to e reprefents the vifible depth of the xooxa. At the point e, draw a Tight line parallel to the bafe line of the
picture, as the line e f, interfecting the line which is drawn from the point b to C in the point f. Then is the line e f the bafe of the farther end, and the points e f
the farther angles or corners of the room. At the points e and f draw lines perpendicular to the horizontal line,
•as the lines e g, and f h, which may interfeet thofe lines which are drawn from the upper angles e and d, in the points I and m, which lines, with thofe before drawn, exprefs the general form of the room, » The diftance of the piéture is marked by the word diflence.
Then
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Ρ E R SPECTIVE.
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Then, find the reprefentations of the piers and windows as
follows: At the point b in the bafe line, which is the point by which the
apparent length of the room was determined, fet on from b to n, the depth of the fartheft pier, equal 3 feet 6 inches by the fcale; and from n, draw a right lme to: the diilance of the pióhire, and its mteifeétion, with the line a e, as at t, gives the apparent width of the fartheft pier. Then from the point η upon the bafe line, continue the alternate
meafures of the windows and piers, as at the points Py 3, % ι; and from thofe points, as before directed, draw lines to the diftance of the picture, which will produce the interfééllons t, s, v, w, z, the apparent breadths of the windows and piers. Then determine their heights as follows::
Upon the line Β ι, fet up the meafure 12 feet by the fcale, from a
to a, and draw a right line from a to the center of the picture, and from the points z, w, % s, t, draw lines parallel to Β ι, and the inter- feclion of thofe lines, with that which is drawn from a-'to G, as at b d, will give the general forms of the fpaces for the windows. After having determined the piers and windows, defcribe the
T?eceffe's as follows : The point a, in the bafê line, expreffes the extreme width of the
Móm on the left of the fpe£tator; but the receifes extend beyond that point, nearly equal to the thicknefs of the wall of the building. Therefore, continue the bafe line from a, and fet on from a to R, the depth of the recefs equal 2 feet; and from R, draw a right line to C, the center of the picture; and from the points t, r, z, draw lines parallel to the bafe line, as the line r y in the fecond window, which &ne marks the apparent depth of the recefs upon the üoot. At d, draw the line d e parallel to the floor,, and at y,, draw the line-
È e Per-
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56 Ñ E 'R S Ñ E C Ô É V E.
SiC T, -y .£ perpendicular to the horizontal line, or, which is the fame in effect,
<—-í—j. parallel to the line  é. Plate VI.
Through the point e, draw a line from the center of the picture,
and the recefs of the.fecond window will be determined. *
Then proceed to determine the door as follows : This aperture being in the middle of the chamber, divide the bafe line into two equal parts, as at M, and from that point fet off on each fide to X X, half its width, 1 foot 9 inches, and from thofe points, draw lines to the center of the picture, and the interfections of thofe lines, with the line e f, will mark the apparent width of the door atYY. To determine its height, take the meafure from the fcale 7 feet
9 inches, and fet it up from b to Ô in the line  2, and from T, draw a line to the center of the picture C, which will cut the line f h in the point L. -At the two points Õ Õ, draw lines perpendicular to the horizon,
for the fides of the door; and from the point L, draw a right line parallel to the horizon, and the mutual interfeétions of thofe lines will determine the appearance of the width and height of the door. The architrave, with the frize and cornice, muft be determined by
the fame procefs, which produces the width and height of the door: thus the meafures of the width of the architrave muft be marked on each fide of X X, in the bafe line, and transferred by the center of the picture C, to the line e f, and the height of the frize and cornice muft be fet above the point Ô in the line  2, and transferred by the point C (which is the center of the pidure) to the line f h; and from thofe * The other windows cannot be feen, the firft being too near, .and iht third toe
adiftant. |
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points,
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PERSPECTIVE. 57
points, draw lines parallel to the horizon, which will determine; the SECT,
members both of the frize and cornice. ò—v—J Plate Õ L
The chimney in this example has no ornament, but a plain fafcia,
which furrounds the aperture. To determine the reprefentation of the chimney aperture, proceed
as follows: Through the further angle f of the room, draw a right line from
the diftance of the picture, that may interfecl; the bafe line, as the line f n, which paffes through f, and interfeóts the bafe line in the point a. Then from the point a, fet on upon the bafe line, 13 feet by the fcale, to the point 4, that being half the length of the room. On each fide of the point 4, fet off half the width of the aperture of the chimney, £ feet 1 inch to the points 5 and 6, from which points, draw lines to the points of diftance *, interfering the fide of the room in the points k i, which give the apparent width of the opening of the chimney. For the height of the aperture of the chimney, fet up the meafure
3 feet 8 inches from the point b in the bafe line, to the point 9 in the line  2, and draw a line to the center of the picture C Then from the point k and i, draw lines perpendicular to the hori-
zontal line, and their interfeclions, with the line which is drawn from 9 to C, will determine the aperture of the chimney. The height of the fur-bafe is about 3 feet 2 inches; therefore, from
the lower angle b of the room, fet up that height by the fcale, from the point b to the point 7 in the line  2, and draw a line to the center of the picture C, which will reprefent the upper line of the fur- bafe; under which defcribe the profile of the mouldings, and draw lines to C, and at the inteifeclion of thofe lines, with thé line f h, * The point fhould lie on the right fide of the example, but is omitted, for want of fpa'ce,
on the plate; but the lines drawn from the points 5 and 6 indicate its place, I draw
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PERSPECTIVE.
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58
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SECT, draw the returns of the mouldings at the farther end of the room,
í—*■—» parallel to the horizontal line.
1'iate VI. . "
The cornice muft be delineated by the following procefs :
On either fide of the room, as at G 1, or G 2, which points mark
the extreme height of the cornice from the ground, draw the -geometrical profiles of the mouldings, as the profile G i, on the line  é. And from every angle of the projection, draw lines to the center
of the picture C, producing thofe lines forward before the profile, until they terminate in the line which exprefles the limits of the fides of the room, as at G 2 on the right fide; for, as the cornice con- tinues behind the fpeétator, no break or interruption can be feen. To find the return of the cornice at the farther end of the room,
determine the mitres, as at the points g and h, by drawing lines through the angles of the mouldings from the points of diftance, as the line ] 5, which gives the mitre for the upper moulding, and will be a fufficient guide for the delineation of the parts, when the repre- sentation of the room is fmall; but in large works, where great accuracy is recpaired, the reprefentation muft be obtained by the rules given in the laft example of this Seclion, Plate X. in which is a full explanation of the procefs for reprefenting mouldings that are parallel and perpendicular to the .picture. The cove and ceiling, with its compartments and foffits, are the
next and laft confideration ; but, before the procefs is defcribed, by which their reprefentations are obtained, it will be proper to make fome obfervations upon their forms. The flat part of the ceiling is divided into nine compartments, by
eight foffits or fafcise, thus: F, K, P, O, are the foffits which pafs
lengthwife the room, and V 1, V 2, crofs the ceiling tranfverfely at the
farther end; but the -foffits at the neareft. end of the room, parallel to
4 V é and
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PERSPECTIVE.
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V 1 and V 2, are not feen, becaufe they lie behind the view of the
fbeelator. The lines Ê 1, and d m, exprefs the angles where the ceiling would
ïnterfect the fides of the room, provided there was no cove. Therefore, from K, the neareft point in that interfeélion, fet off to the point 10, the meafure equal to the projection of the cove, é foot 4 inches, and repeat the fame projection at the point d, to 14, on the contrary fide. Then divide the fpace between the points, ßï and 14, into the
proper divifions, marking off upon the line Ì 2, the places and breadths of the foffits F K, and Ñ Ï, which, in the example, are about 9 inches wide; and, from thofe divifions, draw lines to the center of the picture, as from ßï, u, 12, which lines will give the longitudinal foffits of the ceiling. Then determine the reprefentations of the tranfverfe foffits, V I
and V 2. But, before the procefs is defcribed, it will be proper to obferve, that
the foffit V l, projeéls forward as far before the wall, at the farther end of the room, as thofe marked Ê and P, projects beyond the fides; confequently, · the line æ æ, of the foffit V l, is the fame diftance from that wall, as the point 13 is from the point d, which, in the example, is 5 feet 9 inches. From the point of diftance, which lies on the right hand, draw
the line Í through the point f in the floor, that may interfecl the bale line, as at the interfeftion a. From a, fet on upon the bafe line to the point A, the diftance of the-
inner or nearer edge of the foffit V l, from the wall, equal 5 feet 9 inches ; and from A, return a line to the point of diftance, which will interfecl the fide of the room f b, in the point V, confequently^ the portion of line f V upon the floor, is equal to the apparent pro- 12 jeéüon
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6Ó PERSPECTIVE.
SECT, jection of the cove, together with the compartment S, and the two
Ir
t___,IL^j foffits, V é and V 2, upon the ceiling·.
e v At the point V on the floor, draw a line perpendicular to the horizon,
as the line í w, and its interfeetion with the line m d, which is the
interfeetion of the ceiling, with the fide  2 of the room, gives the point W. At the point W, draw a right line parallel to the horizon, which
will be the outer line of the foffit V l, projecting forward before the farther end of the room. The foffit V 2, is found by the fame procefs with the former, there-
fore the inftructians need not be repeated. The pannels of the window ihutters are determined by firft drawing
thofe of the neareft recefs, and then transferring thofe proportions to the remote pannels, by lines drawn from the neareft, to the center of the picture, as from X to C, which fufficiently explains itfelf to the (lighter! infpeetion. The pannels of the door may be determined by fetting their
widths upon the bafe line, between the points ÷ ÷, and their heights upon the line  2, and then transferring thofe meafures to the proper place, by lines drawn to the center of the picture, in the fame manner as was done for the door. It muft be obferved, that thofe perfons who underftand the con-
ftruction of pannels, need not apply real meafures, but may .pro- portion the parts by the eye, especially when the drawing or picture is fmall.
The procefs which is employed to obtain the true reprefentation of
the parts in this example, is the fame that muft determine the various features in the interior reprefentations of all buildings, whofe fides are parallel and perpendicular to the picture ; for though tbey may he very different in appearance, yet the general principles are the fame
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PERSPECTIVE.
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fame in every fpecies of architecture, feeing that all edifices are
compofed of right lines, difpofcd perpendicular, or parallel to the horizon. For, fuppofe it be required to reprefent the interior view of St.
Paul's, or Weftminfter Abbey, or any fimilar edifice, the firft cir- cumftance to be determined by the artift is, the height and width of the building; to which dimenfions, the drawing or picture iliould correfpond; then the breadths or thickneifes of the piers, or columns, and alfo the fpaces between them, iliould be propor- tioned to each other by the fame procefs, which is given in this example, for determining the piers and windows. The heights of all the fmaller parts muft alfo be found by the fame procefs, as there can be no difference in the methods employed for obtaining their reprefentations, whether the edifice be of few parts and plain, or of fuch as are more numerous and magnificent. JV". B. In the foregoing example, the meafures for the piers and
windows are arranged by the point which marks the difiant angle of the room upon the bafe line of the picture, as at b, from which point the meafures are continued to a. This procefs may be confidered as inverfe, becaufe the nearer angle of the chamber cannot be feen, and therefore the meafures are applied from that angle which can be feen. In Plate VII. Fig. l, is an example fimilar to the foregoing, but
the parts are proportioned by the half diftance which lies within the breadth of the room, and confequently within the limits of the piclure, as the point d f. It alfo demonftrates how the meafures may be transferred from
one fide to the other, fo that much trouble may be prevented, and a confufion of lines avoided. The general meafures of the room are as follows;
The
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PERSPECTIVE.
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The width' é7 feet from X to Y.
The length or depth 24 feet. The height 17 feet. The firft pier a, as alio the fartheft pier, 2 feet 6 inches wide.
Windows 3 feet wide. The other piers 5 feet wide. Ç II is the horizontal line, and the height of the eye is about 4 feet 9 inches. The diftance of the picture is at D, which is by the fcale, 12J- feet
from C, the center of the pióture. From C to d f, is exactly half the fpace from C to D, confequently,
d I is half the diftance of the picture. Therefore, to proportion the fpaces on the fides of the room, take
half the given or known meafures of the parts required, and apply them as follows: The firft pier is only q feet k wide, therefore, take half that
meafure, 1 foot 3 inches by the fcale, and fet on the bafe line from X to l, and from l to 2, fet on 1 foot 6 inches, which is half of 3 feet, the width of the window; and from thofe points, draw lines to the half diftance, d f, and the interfeclions of thofe lines, with the lineX e, as at a b, will mark the apparent widths of the piers and windows. Continue the alternate meafures of the piers and windows by their
half proportions upon the bafe line, as marked by the points 3, 4, ,5, and on to the point 11 ; and from every one of thofe points, draw lines to the point d f as before, and the interfeclions of thofe lines, with the line which is drawn from the point X to C, the center of the picture, will give the apparent fpaces of the windows, piers, and pilafters, with as much accuracy as if the whole meafures were employed. As
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PERSPECTIVE. 63
As it would occafion much trouble, and produce great confufion SECT.
of lines, to determine all the parts required on the fide  of the room t—«—^
Plate VI."
oppofite the windows, by the fame procefs which was employed to determine the parts or divifions on the fide A ; thofe parts may be
accurately marked by the following method :
From every one of the divifions, at the lower angles of the pilafters,
draw right lines parallel to the bafe line, as the line æ k, which lines will give the divifions for the parts required, as the point k is the neareft angle of the fecond pilafter on the fide B, exactly oppofite to the pilafter Æ on the fide A. The procefs which is here given, is applicable to the interior repre-
fentation of every building, whofe oppofite fides are fimilar to each other; for, having found by the center and diftance* of the picture, the neceffary divifions of the parts on either fide, as may be moft con- venient ; the parts on the oppofite are defined, by drawing lines parallel to the bafe line of the picture, from the points in the fide upon which the meafures have been before determined, to the lower line of the contrary fide, as the line Ê Æ, drawn parallel to the bafe line X 10, determines the point E exactly oppofite the point Z, therefore, the point E is the neareft angle of the fecond pilafter, on the fide B, exactly oppofite to the neareft angle Æ of the fecend pilafter, on the fide A of the room. In this example, the full length of the. room is defcribed, as from
Õ to W, but the effect of the whole, in confequence of fo much bemg ihewn, is lefs pleating, and not fo natural, as in the foregoing ex- ample ; for the fpace B, between the neareft pilafters on the left fide, appears too wide in proportion to the next fpace D, and the aperture of the chimney does not feem to be in the middle of the fides, but more remote in the room; yet, it is perfectly juft as to the rales of * Or by the half diftance, or any other proportion,
the
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PERSPECTIVE.
the fcience ; but it may be faid, that thofe rules are violently forced
into action, and therefore, the confequent effect is not pleating, nor ftnclly juft. Of Steps parallel to the Picture.
In Plate XVI. Fig'. 2, is an example of the method for deter-
mining the reprefentations of fteps, when fituated parallel to the picture. The horizontal line is II H, and G G is the bafe or ground line.
The center of the picture is marked center.
The diftance is at D P.
The neareft ftep touches, or is in the plane of the picture; its width
is marked X 2, g, and its height a o; through which points, the face is drawn geometrically. At the angle X 2, mark off, on the ground line, the meafure of the
depth or tread of the fteps, as from X 2 to d, and Trom d to e, and alfo to f. From X 2, draw a right line to the center of the picture, as the line
X 2, N, and from the points d, e, and f, draw lines to the diftance of the picture D P. At the angle X 2 of the ftep, draw a line perpendicular to the bafe
line, as the line X 2, e, and upon that line, let up the heights of the fteps, as many as may be required, as from a to b, to e, from all which points, draw lines to the center of the picture. Then from the points L, Ì, Í, draw lines perpendicular to the
bafe line, which will interfect the lines that are drawn from a, b, e, to the center of the picture, in the points k, 1, m, n, and by their mutual interferons, form the profile of the fteps. Then determine the front and tops of the fteps as follows :
At
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Ñ E R S Ñ E C Ô É V E.
At each of the points k, 1, m, n, draw lines parallel to the hori-
zontal line, which lines will exprefs the horizontal fteps. From the point o, the upper and remote angle of the loweft ftep.,
draw a line to the center of the picture, as the line o, p, which by its interfeélion with the line k p, forms the tread or furface of the loweft ftep; from p, draw another line perpendicular to the horizon, as the line ñ h, which line, by its interfeclion with the line q h, will determine the front of the fecond ftep. It is needlefs to continue the inftraótions beyond what are already
given, as the reprefentations of the other fteps are found by a repe- tition of the fame procefs, which produced the firft; and a ftricl. examination of the figure, with the application of a ruler from the center, and diftance of the picture, to the different points in the ob^ j eel;, will clearly demonftrate what is required to complete the figure. Obferving, that all the meafures for the heights or rife of the fteps muft be fet upwards upon the vertical line X 2, e, and their apparent heights determined by lines drawn from thofe points to the center of the piclxire, but the breadths or depths of thofe fteps are applied to the bafe line, as at the points d, e, f, from which lines are drawn to D P, the diftance of the picture, which produce the interfeétions L, Ì, Í, and mark the breadth of the fteps. Another method may be employed for the reprefentation of fteps>
as ihewn in Plate XVI. which is more elegant and fimple than the ■foregoing, but being more fcientific. is given in the fourth Seclion^ fee Index. |
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Ê -Qf
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PERSPECTIV E,
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Of the Repnfentations of Arches parallel to the Picture.
Examples of two different kinds are given in Plate VIE
Fig,. I,, and Fig. 2, are examples of arches which are the fegments
of a circle. Fig. 3, is the femicircular arch.
When the arch is a femicircle, the center will be within the limits
of the arch or aperture, as in Fig. .3, where the center of the arch X, a, k, is at e l, in the line X k, which line is the chord of the arch. But if the arch is the fegment or] arc of a circle, the center is not
confined to the chord of the arch, but may lie out of the tablet or canvas, as in Fig. 1, where the centers for the foffits of the ceiling are low in the plate, as at a 2, b 2. 2V. B. In Fig. 2 and 3, the center and diftance of the picture,
are the fame to both figures. The center is indicated by the word, and the diftance is at D, upon the horizontal line Ç Ç. Procefsfor the Semicircular Arch, Fig. 3.
Let X k, be the chord of the arch bifected at the point e l, which
will be the center of the arch X, a, k. Then, from the center e ],. of the arch, draw a right line to the
center of the picture, upon which line the other centers muft be found, by the following procefs : Complete the fides of the archway, as Ì, Í, as well that which
is not feen, as that which is, and then divide either fide of the arch- way
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PERSPECTIVE. 67
way perfpe&ively*, into as many divifions as there are foffits requir- s E c ô·
ed, as at o, y, and at thofe divifions draw lines parallel to the <—-v—» ' ' J' . r PlateVII.
horizon, which will interfeét the line that is drawn from e 1, to the
center of the picture, in the points e % and e 3, which, together with
the firft point e i, will be the centers for the arch and its foffits.
Therefore, with compafies on thofe centers, defcribe the femi circles
required, as o, b, m; y, d, y 2.
Of Arches which are the Segments or Arcs of Circles,
Plate VII. Fig. b. Let S and T, reprefent the fides of the aperture, which may be plate
confidered as the piers fupporting the arch, and let g, h, be the width of the arch, and k, its height. By Problem the 9th in the geometrical Section, Plate I, Fig. 10,
find the center e l, of an arc, or of a circle, paffing through the three given points g, k, h, and with, compaffes, upon e l, as a center, defcribe the arc g, k, h, which will be the reprefentation required. To find the other center, praceed as follows:
Draw aline from e l, to the center of the picture.
* By dividing a line perfpecüvely, is meant to divide the reprefentation of an original
line, that is either perpendicular or inclined .to the piiture, in a given number of divifions, reprefenting equal parts. As the line sg, is divided into two equal parts perfpeftively, the part e g, reprefenting a part equal s e. The method of dividing a line perfpeitively is as follows:
The line s g, Fig. 3, Plate VII. is the reprefentation of a right line perpendicular to the
pifture. S f is the bafe line, Ç is the horizontal line, and D is the point of diftance·. therefore, upon the bafe line fet on from the point S to ç and f, two parts equal to each other, and to the given meafures of the required divifions; then, from the points ç and f. draw lines to the point of diftance D, and the line S g, will be divided perf'pectively inte portions reprefenting equal parts. The fpace e g, being the reprefentation of a portion of the line S g, equal to S e. I · ■ |
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Ê 2
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Then
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Ñ E R S Ñ E C Ô Ã Ô Àß
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Then.at the point c J, draw the line P, parallel to the horizons,
producing the point q, in the fide M, or pier of the aperture, front; which point draw a line to the center of the picture, and divide that line perfpeótively into the fame number of divifions, with the re- quired foffits, as at u, v, and then transfer thofe points by lines drawn parallel to the horizon, producing the points e 2, e 3, by their in-. terfections with the line that is drawn from the point e l, to the center of the picture. In Fig. l, of the fame Plate, the ceiling is the portion or arc of a
circle;, but being of very faint elevation, the centers for the conftruc- tions of the foffits lie very low on the plate. To find thofe centers, proceed as in the foregoing figure. That is,
by Problem, Fig, io, Page 7, iri the geometrical Section, find the center a % of an arc paflïng through the points Si, P, S 2, The points S 1, S % arc the extreme width of the room, and the
'fummits of the neareft pilailers from, which the foffits fpring ; there- fore, at thofe points draw the line a l, parallel to the horizon, which- will be the chord of the arch, bifect this line at a l, through which, point draw the line Ñ, Ñ % perpendicular to the horizontal line, and continue it downward as to a a,. Then from a é and a a, draw right lines to the center of the pic-
ture, for the other centers muil be found in the line which is drawn, from a 2 to the center of the picture. To obtain the other centers, firft find the diyifions perfpectively
m. the fides of the room, from which the foffits fpring, as. the points S3, S4, S.5, S 6, and at thofe divirions draw lines parallel to the horizon, or to the floor of the room;. and thofe lines will cut the line that is drawn from the point a i, to the center of the piéiure, and produce the interfections b l, e, d, from all which points draw light lines perpendicular to the horizon, and where thofe lines .inter-. feft,
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PERSPECTIVE.
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69
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iëót tile line that paffes from a 2* to the center of the picture, will SECT.
II.
fee the centers required for the different foffits. <—v~J
^ Plate VII.
In this figure, no more than two centers are given, which are a 2
and b 2 ; a 2 is the center for the fegment S 1, P, S 2, as already
fhewn, and b 2, the center for the fegment S3, S 4. The reft of the
centers are omitted, becaufe the lines neceffary to produce them
would incumber the example, but if thofe which are given are well
confidered, the ftudent will have no difficulty in finding the reft,
only let him remember that every line in the foffit muft have its own
center.
In Plate IV. Fig;. 2, is the reprefentation of a building with femi- PLATE
IV.
circular arches, which are drawn by compaffes, the centers are found by the procefs taught in the preceding inftruétions, given
foï, Fig. 3, Plate VII. page 66.
The points 4 and 6 being the centers for the front of the foffits,
and the points 5 and 7, the centers for the back line of the foffits of
the arches..
Plate VIII. Fig. 1.
Which is drawn to a Scale of One Inch to a Foot,
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Reprefents a ftool fituated parallel to the picture.
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The dimenfions PLATE
VIII. |
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©f which are as follows f :
The height is 1 foot 9 inches.
The width is 1 foot 6 inches*, *· The line which is drawn from a 2 to the center of the piclure, is the indefinitive
reprefentation of the line in which the centers for all the foffits are found, for it may be confidered as the axis of the concave cylinder, of which the ceiling is a part. f The fcale is not marked on the plate, therefore the ftudent may exert himfelf, by-
taking the proper meafures from any common rule* |
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The
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PERSPECTIVE.
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The depth is l foot l f inch.
The upper rail is 2 f inches broad.
The legs are 11 inch thick.
The lower rail is 4 f inches from the ground.
And ------------if inch broad.
Its diftance beyond the picture is l foot 6 inches.
The height of the eye is 5 feet,
Ç is the horizontal line.
C is the center of the picture
And D is the diftance, which is 5 feet 3 inches.
-* Having determined the feat of the neareft leg on the bafe line G, as at lj fet on the whole width of the front of the ftool, that is 1 foot 6 inches by the fcale, from 1 to the point 4. Then determine the diftance at which the ftool is placed beyond the picture, which is l foot 6 inches; thus from the point l, fet off that diftance by the fcale to the point 5. Then, from 5, fet off the depth of the ftool, é foot two inches, to 8. Haying thus determined the meafures of what may be called the plan of the ftool, draw right lines from the points l and 4 to C, the center of the picture, and from 6 and 8, draw lines to D, the diftance of the picture, and thofe points in which the lines that are drawn from 5 and 8 to D, cut or interfect the line which is drawn from the point l to the center C, will mark the fpace that lies between the picture and the object, and alfo the depth of the object. Thus the fpace between the points é and e, reprefents the diftance
* The flool is difpofed as if (landing in a room, part of which is not feen. The lines
R R, are the interferons of the floor with the fides of the room, and the line R a, is the interfeaioa of two of the fides which are feen. Pig, j. No. 2, is the reprefentation of a fquare on the fame floor, on which the ftool
'Hands, therefore the center and diftance of the piiture are the fame with thofe by· which the ftool is drawn.—See the explanation in the Addenda. between
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PERSPECTIVE. «
between the picture and the fiool, and the fpace from e to f, is the SECT,
depth of the ftooL u—v—J Plite
Complete the plan of the object, by drawing lines parallel to the vill.
horizon, or to the bafe line, from e to e, and from f to d, then will the line e e, exprefs the apparent width of the front, and e f, the depth of the fide. Then rind the height of the ftool as . follows·-:-
At the point 4 in the bafe line, draw the line 4, 10, perpendicular
to the ground line; and upon that line, fet up the height of the ftool,-from 4 to lO, which is si inches by the fcale; and from the point 10, draw a line to the center of the picture C. Then from tha» point e, in the bafe or plan of the ftool, draw the
perpendicular line e, X 2, till it cut the line which is drawn from 10 to C, and the line e, X 2, will be the apparent height of the object, for it is the outer angle of one of the neareft legs. Complete the general form of the ftool by the following method r
At the points f, d, draw lines,.perpendicular to the horizontal
line, and at the points X 2 and × é, draw lines parallel to the hori- zontal line, as the lines X 2. V, and X 1 k. Then from the point V, draw a line to the center of the piélure C,
and the mutual interfeclions of thofe lines will produce the general outline of the ftooL Thus far it is evident, that the procefs is the fame with that which
was employed for the fimple form of a block, whofe fides are at right angles with each other, and front parallel to the picture. There remains to find the thicknefs of the legs, and the propor-
tions, and places, of the rails. For the thicknefs of the legs 'mfront, take the given dimenfions,
l i inch, from the fcale by the compaffes, and fet that meafur-e wpon the bafe line from l to 2, and from 4 to 3; and from thofe points-
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72 Ñ E It S Ñ E C Ô É V E.
SECT, points draw right lines to the center of the picture C, and thofe
e—v-»j lines will mark the apparent thicknefs, by their interfection with Plate ,, ,.
Viil. the line e, e.
Then for the thicknefs of the legs on the fides of the ftool, apply
the fame meafures in the fame manner upon the bafe line, from 5 to 6, and from 8 to 7, and from thofe points draw lines to the difiance of the picture D, and the interfections of thofe lines with the line that is drawn from l to C, will give the apparent width required. Having found the widths of two legs e and f, the others are ob-
tained of courfe, by drawing lines parallel to the bafe line from one fide to the other. To find the depth of the upper rail æ upon the line 4, 10, mark the
dimenfions of its depth, 2 f inches, which muft be fet downwards from "io to h, from which point draw a right line to C, and the interfection of that line, with the angle of the ftool, will give the apparent depth of the top rail, as at o. The fame method muft be employed to determine the place and
dimenfions of the lower fide rails, the upper lines of which are mark- ed at m, upon the line 4, io, and then transferred to the ftool by lines drawn from that point to the center of the picture C. The place of the ftretcher S, which paffes from rail to rail, is in the
middle of the fides of the ftool. Therefore to find its place divide the fpace between 5 and 8 (in the bafe line) in two equal parts, as at P, and from Ñ draw a line to the diftance of the picture D, and the interfection of that line with the line which is drawn from l to the center of the picture, as at M, will give its feat on the floor, or bafe of the ftool; therefore, transfer that point to the fide rail by a line drawn perpendicular to the floor, which will give the interfec- tion T, the place of the ftretcher. é By
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PERSPECTIVE. 7S
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By duly confidering this figure, together with the explanation, it SECT*
■will be evident that the principles are extremely fimple, and the u—v—-» Plate
procefs uniform, for obtaining the true reprefentations of the differ- VllL
ent parts of any object, in proportion to their juft dixrenfions : con- sequently, if the method employed for finding the reprefentation of the upper rail ï, × 2, is well underftood, the undent will readily perceive that a repetition of the fame procefs will determine all the other finall parts. More lines might have been employed, fuch as thofe for obtaining
the thicknefs of the fide rails; but they are omitted, that the figure might not be too much confufed *. In Plate VIII. Fig. % is alfo given an example for determining the
heights and ftations of figures, when fuch are required in a landfcape, or view of a building. What has already been obferved in the courfe of this work muft
be here repeated, namely, that the horizontal line is always drawn upon the picture, at the fame height above the bafe line, that the eye of the fpecfator is known or fuppofed to be above the ground or plane of the view; therefore, if the fpe&ator ftand upon the fame floor or plane with the perfons that are to be reprefented, then the horizontal line will pafs through the eyes of all thofe figures, provided they are all of equal heights with, each other, and of the fame fta- ture with the fpecfator; but as this can rarely happen in nature, fo in the example they are of different heights, yet all of them have their heads nearly equal to the horizontal line. Let it be fuppofed, that G is the ground line, or bafe line of
the picture, and Ç Ç the horizontal line; and that the * figure * In this Example, the line L, a, relates to the ihadows, and is explained under that
ieétion. The ufe of the vertical line 8, 9, is alfo explained in the Addenda.
L which
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7A Ñ E R S Ñ E C Ô É Õ! E.
S E C Ô. which frauds on the point 1, is of the fame ftature with the fpeclator;
then the horizontal line will pafs through the eye of that figure, as Plate
VIII. in the example.
To find the proportionate heights of the Other figures which are
on the fame plane, from any point in the horizontal line, draw a right line to the bafe line, as from S to l ; and any fpace taken be- tween the line S i, and the horizontal line H, will give the apparent height of a figure, in proportion to its diftance, as at 2, 0, 5 ; ob- ferving, that the figures not being fuppofed fo tall as that which is ftanding at l, their heads are rather below the horizon. If it be required to determine the height of any figure at a given
diftance from the picture, proceed as follows : At the point r, which is the feat of the principal figure, fet off on
the bafe line the known diftance of fuch figure, as from é to 6; and having marked the diftance of the picture from S to D, on the hori- zontal line, draw aline from 6 to D, and it willinterfecl. the line which is drawn from l to S, at the point ï ; therefore, the fpace between the point ï and the horizontal line, will be the general height of every figure equally diftant beyond the picture. If it be neceflary to transfer that height to fome other part,
equally d'iftant from the picture, draw a right line, as from o, parallel· to the ground line, as o, 3 to 9; and that line will determine the feat of all the figures which mall range parallel to the picture, at the fame diftance from it with the point o. Should the figure be required' to appear as ftanding upon a fiep,
proceed as follows : Upon the ground line fet up, from the feat of the figure, tll-e height of the ftep, and above that add the height of the * The back figure I, is placed upon the ground line, which is done to iliew where the
meafure muft be placed on the piélure ; but in finiihed: works it would be improper to /place any figureon that ftation ;. the neareft iliould ftand ibmewhat beyond the picture. figure,,]
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PERSPECTIVE. Þ
figure, as at a, A, b ; and from the ftep a, and the upper point b, «f E C T.
draw lines to S; and thofe lines will determine the heights of all u—J^j figures, that may be required as ftanding upon fteps of equal height yill. with the firft; the lower lines will determine the height of the ftep, and the upper line the height of the figure. When the fpectator is fuppofed to ftand upon an elevation, above
the plane on which the figures are ftanding, the procefs will be the fame, though the appearance will differ, for the heads of the figures will then be below the horizontal line, in the fame proportion as the horizon is above the general height of their eye. Thus, fuppofmg the height of the horizon Ç PI, twelve feet, then will the neareft figure appear about half the height of the horizon, allowing fuch figure to be fix feet high. To find the heights of figures, when the horizon is elevated, pro-
ceed as follows : , Suppofe the horizon to be 10 f feet high, correfponding with the
eye of the fpectator, and the figure to be 6 feet high. Let the line 7, S, Fig. 2, reprefent the height of the figure, Ç the
horizontal line, and S the center of the picture. Draw lines from the points 7 and 8 to the point S, and any fpace taken between thofe lines, will give the height required; as from 9 to x, will be the height for any figure placed at that diftance beyond the picture -#£■ the point 9. Any two points taken at pleafure in the horizontal line, if con-
fidered as center and diftance, will give both the proportion and place of the figure, provided thofe points are diftant from each other in the fame ratio with the diftance of the picture ; for nothing more is to be obtained than a fingle vertical line, that * ihall mark the height and ftation of the figure, after which the reftmuft be drawn by the hand. * This example fully explains the procefs, by which the hiilorical painter thou'd proportion
his figures to each other ; but of this more will be faid in the inftxuftions which are fub- joihed. L2 Tt
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Ôâ f PERSPECTIVE.
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S E C Ô. It muft be obferved, that in all the examples of this work, in which
<—v—J figures are introduced, the horizontal line paffes either through, or Plate
Vlli. juft above the heads, of thofe figures; becaufe they are intended to
reprefent adults nearly of the fame flat ure with the fpeeiator, who views them, {landing upon thé fame plane with the figures. PLATE In Plate IV. Fig. 1, there are two figures, one taller than the other;
the head of the taller rifes a little above the horizontal line, the ihorter is confiderably below: therefore the taller may be confidered as above fix feet high, and the loweft five ; or it may be fuppofed the reprefentation of a younger perfon. |
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Of Mouldings parallel and perpendicular to the Picture.
PLATE
IX. * As many of the leflons in this work are compofed of architec- tural forms, it is neceflary to obferve, that the ftudent who is unac-
quainted with the principles of architecture, will labour to no purpofe if he attempt to delineate fuch objeels, ixnlefs he firft. acquire fome knowledge of that fcience. But the limits of this work will not admit of introducing a feries of inftruétions for the ufe of fuch : yet fome hints are abfolutely neceffary, particularly in the 'article of mould- ings, becaufe their true reprefentation in perfpecüve cannot be obtained without a correct knowledge of their conftru<5tion. The ftudent muft therefore endeavour to underftand perfectly the follow- ing explanation: When mouldings, placed* on any flat furface or plane, are united or
joined to mouldings, which are difpofed upon another flat furface or plane, the joint of thofe mouldings is called the mitre; and fuch joint, or mitre, always bifecls or equally divides the angle, which the planes make at their in terfecliou with each other; hence it follows, * See fimilar obfervaiion in the elder Malton's Treatife, Vol. I. page 172, firft
edition. thai í
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PERSPECTIVE.
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that every mit^e line of a fquare, or right angular block, makes an
angle of 45 degrees with either of the fides of the block. When the object is a fquare, or regular polygon, the mitre line will
not only bifeél the angle, but it will alfo pafs through the center of the fquare or polygon; but if the figure be a parallelogram, or irregular polygon, the mitre will only bifeét the angle, but will not pafs through the center of the figure. It is neceifary to attend particularly to the foregoing circumftances,
becaufe the mitre line may be found without its vaniihing point, in the reprefentation of a fquare, or of a polygon when regular; but when irregular, then the vaniihing point of the diagonal muft be found and employed. The better to explain the foregoing infiruélions, let the under
figure be confidered as reprèfenting the geometrical elevation of the hafe of a pedeftal, together with its plan ; the mouldings of which arc the fame with the bafe B, Plate IX. Fig. l, but of fmaller dimenfions.. |
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Example Pago ηη
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Below
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PERSPECTIVE.
Below the geometrical bafe is half its plan, in which the lines
a, a, a, exprefs the dado. The lines b, e, d, are the extreme projec- tions of the mouldings, which are carried round the dado, and form the bafe. The mitre lines of two of the angles are marked Μ, Μ, upon which lines, the mouldings are united or joined together; therefore, in the perfpeótive reprefentation, it is necefiary that thofe mitre lines ihould be drawn perfpeelively, before the reprefentations of the members or mouldings can be determined. When mouldings are to be drawn in perfpeótive, their geometrical
profiles muft be^firft drawn by the fcale to the fize required; which may be done in a detached part of the drawing, as in Plate IX. Fig. 2, or upon a feparate paper. When they are difpofed parallel and perpendicular to the piclure,
the fimple profiles, drawn upon their place, is a fuincient guide by which to form their reprefentations ; but if they are inclined to the piclure, it will then be neceflary to draw the profile with the mitre line of one angle, as will be iliewn in the next feétion. As a preparatory leifon to the delineation of mouldings, it will be
neceflary, firft to confider the manner of reprefenting a plain project- ing fafcia, or plat band: which being mitred at the angles, exhi- bits the rudiments of mouldings. In Plate IX. Fig. 3, is an example for delineating* a projecting
fafcia, or plat-band, round any pedeftal or building whofe fides are at right angles with each other, and whöfe front is parallel to the piclure. Let it be fuppofed, that the fquare block or cube A is already
drawn, and that it is required to defcribe the projecting fafcia 13, Grounding the upper part. Ti€ horizontal line is marked Η Η.
C is tbc center, and D the diftance, of the piclure.
The
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PERSPECTIVE.
The upper angles of the cube are e, g, o : therefore, from the points
e g, fet downwards the dimenfions or thicknefs of the fafcia, as from e to i, and from g to k. Then draw the geometrical form or profile of the fafcia, at either
of the front angles, as at e, l, s, V, or g, k, s, V £, on the contrary- fide. Then determine the perfpective appearances of the fquares e, g, h,
o, and l, k, 1, m, * one above, anfwering to the upper furface of the block; the other below, correfponding with the thicknefs of the fafcia, as if the object were tranfparent; by which fuppofition the points e, g, h, o, i, k, 1, m, will exprefs the angles upon the block, of the thick- nefs of the plat-band, or fafcia. Through the points e, g, h, o, and i, k, 1, m, draw the diagonals,
continuing them beyond the angles of the cube, as the lines e, h, i, 1, and k, m. Then, through the points s and v, which are the geometrical pro-
jections of the profile of the fafcia, draw lines from the center of the picture C, which may interfect or cut the diagonal lines e h, and i 1,, in the points X„ × é;. which points will be the mitre angles of the^er- fpect'we projection required; from thofe points draw the lines w i, w 2, parallel to the horizon; and thofe lines will produce the repre- fentation of the front of the fafcia. The length of the fide of the fafcia is alfo determined at P, by the
interfeclion of the diagonal, m k, with the line drawn from the cen- ter of the picture C to X i. It muft be obferved, that in this example the fafcia is to be con-
fidered as projefting forwards before the picture, feeing that the front of the cube or building, as a, b, g, e, is in the plane of the picture. * It is needlefs to obferve, that the reprefentation of thefe fquares, muft be obtained by
the rules already taught in the firft leffon of this feilion. ■ Yet
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PERSPECTIVE..
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80
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Yet the operation would be the fame, if the building were placed
beyond the picture; nothing more being required, than to proportion the dimenfions of the fafcia to the fize of the building, and then to proceed as before. Ail objects that are fquare, when their fides are parallel and per-
pendicular to the picture, have their diagonals vaniihing in the points of diftance : confequently, thofe diagonals can be obtained by draw- ing lines through the angles from the points of diftaiice, as the line through e and i, from the point of diftance D, or through g and k, from the other point of diftance, provided it were in the paper or tablet: but, as it will often happen that one of thefe points cannot be introduced, as is the cafe in this example ; it will then be the beft method to employ the procefs here given, which is, firft to form the fquare of the dado, as if transparent, and then to draw the diagonals through the angles of that fquare, without regard to the points of diftance. Plate IX. Fig. l, is an example of a pedeftal with a bafe and con-
nice, whofe fides are parallel, and perpendicular to the picture. G is the ground line or bafe line.
Ç is the horizontal line.
C is the center of the picture, and D indicates the diftance, which
in this example is out of the plate. Upon any feparate paper; or upon fome remote part of the drawing,
clefcribe the geometrical profile of the mouldings, at their given fize, as at Fig. % together with the line X, X 2, which is the angle of the dado. Upon the ground line G, Fig. i, determine the perfpeétive appear-
ance of the fquare of the whole bafe, as the lines o 7, ñ é, ñ 2. Then from the points o, 7, pi, draw lines perpendicular to the
horizon, and determine the general form, which may be fuppofed to
I contain
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Plate IX,
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Ñ E R S Ñ E C Ô × V E.
contain the whole of the pedeftal, with all its mouldings, as the lines
0,0 2, 7, 8, Ñ 1, Ñ 2, 11.
From the point o, draw a line to Ñ 2, and from 7 draw a line to
Ñ é, which lines will be the diagonals of the bafe: they are marked d 1, d 2. Upon the line ï, ï 2, fet up the height of the plinth ï to 14,
equal the height X, to 4, in Fig. 2, and complete the perfpe&ive re- presentation of the plinth mark'd 14, y z. Then from the point o, fet upon the ground line the fpace 0 3,
equal to X O, in the bafe of the geometrical profile Fig. 2; repeat the fame on the other fide from 7 to 4 : then will the fpace between 4 and 3 be equal to the given or fuppofed breadth of the dado ; and the fpaces from 0 to 3, and from 4 to 7, will be equal to the projec- tions of all the members of the bafe beyond the dado. Between Ï and 3, in the bafe line of Fig. ], fet off the divilions
1, 2, 3, equal to the projection of each of the members of the bafe, as given in the geometrical pofile Fig. 2, as from 4to 1, and from 5 to 2, which is 3 to 1, and 3 to 2, in Fig. 1; repeat the fame at, the other angle, as from 4 to 6, and from 4 to 5. From the points 1, 2, 3, and 4, 5, 6, Fig. 1, draw right lines to the
center of the picture C, and they will interfeét the diagonal lines d 1, d 2, in the points a, b, e, and d, e, f, and alfo at g, h, 1; which points are the feats upon the ground of the projections of the mould ings of the bafe at their angles. Then upon the vertical line o,_o 2 Fig. 1, fet up the heights of the
mouldings, equal the heights of the members in the geometrical bafe Fig. 2, as from o to 14, Fig. 1, equal to X 4 in Fig. 2, and from 14 to 6, to 5, to 3, equal the fame figures in the geometrical profile Fig. 2. From thofe points draw right lines to the diftance of the picture D,
Ì and
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PERSPECTIVE.
and from the points a, b, e, in the diagonal d 2, draw lines perpen-
dicular to the horizon; and the interfeétions of thofe lines with the former will determine the perfpeftive projections of the mouldings, as at S, m, k, n, whofe forms or profiles muft be drawn by hand where they are qltrved, where ftraight, by the ruler. The fame pro- eefs muft be repeated at the other angles, obferving, that to com- plete the forms of the mouldings, right lines muft be drawn from thofe points parallel to the horizon, and alfo from the fame points to the center of the picture C: the former will produce the reprefenta- tions of the front mouldings, as the lines m, n, k, s ; the latter thofe of the fides m P, and S V. Cornice of Pedeflal.
For the upper mouldings, or cornice of the pedeftal, proceed as
follows; Firft, let it be obferved, that the plinth projects beyond the cornice
in Fig. % as much as from o 2, to n. Therefore, from the angle o 2, Fig. l, fet off thefpace to W, eqvial ï 2, Õ, in the geometrical profile Fig. 2; then is W the feat of the upper angle ÷ of the dado, or iliaft of the pedeftal, upon the picture. Then from W, Fig. l, fet on towards o: 2, the projections of all th@
mouldings of the cornice, equal to the projections of the fame mem- bers in the profile, Fig. 2, as ç, í ; repeat the fame at the angle 8; and from thofe points draw lines to the center of the picture C, which will interfeót the diagonals of the top in the points a, b, e. At the angle ï 2, Fig. i, fet downwards the heights of the mould-
ings, equal the meafures from Õ to 8,- 9, 10, in the profile of the cornice No. 2, as at 8, 9, 10, in Fig. 1, and as was done in the bafe; draw lines from thofe points to the difiance of the picture. 4, Then
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PERSPECTÏVi,
Then from the points a, b, e, in the diagonal lines, draw lines per-
pendicular to the horizon, that may interfect the lines which are drawn from the points 8, 9, 10, to thediftance of the ρiélure D ; *and their mutual interfections will give the angular terminations of the mouldings, from which lines muft be drawn horizontally to exprefs the front mouldings, and from the fame points to the center of the picture for thofe which exprefs the mouldings of the fide; and the form or perfpective appearance of the angles of the members muft be drawn by the hand, as in the mouldings of the bafe. In the example, Plate IX. Fig. i, the mouldings do not project
before the picture ; on the contrary, the plinth touches it; confe* quently, the other members are beyond the picture. It muft be obferved, that the mouldings in the example A, Fig. 1,
appear exaggerated, which.effect follows from the profile having too great projection in the geometrical conftruction ; but this was done that the example might be more explicit. When it is required to reprefent mouldings round any building
whofe fides are parallel, and perpendicular to the picture, it will be fufficient, in moft cafes, to iketch the geometrical profile upon the part where fuch decorations are to be placed, marking the mem- bers by a general proportion to the height of the pedeftal, or building, then by the mitre lines adjufting the perfpective appearance of the profile, as directed in the foregoing examples. Let it be particularly noticed, that in all pofitions of mouldings*
whether parallel to the picture or inclined, it is the mitre lines that muft determine the true reprefentations of their projections. * The reader will obferve, that, to avoid confufion, all the points are not diftmguiihed
by marks or initials; yet the whole will be eafily underftood by tracing the example* which, if attentively examined, will explain as much as words» |
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Of
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Μ 2
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PERSPECTIVE.
Plate X. Of Mouldings, with their returns at the Angles of Planes or Walts,
which are difpofed parallel, and perpendicular to the Picture. Example, Plate X. Fig. 1.
"The planes A and E are parallel, while thofe marked  and F are
perpendicular to the picture. They reprefent four Avails, at right angles with each other, furmounted by a cornice, having the' returns of the mouldings agreeably to the plan, Fig. 4, which is half the fize of Fig. l. The line D C is the horizontal line, C is the center of the picture,
and D is the diftance. Firft determine the perfpeclive appearance of the form of the build-
ing, by the rules already given in the preceding part-of this Section ; and having found the different angles marked by the letters f, g, L, m, and p, proceed as follows : The plane or wall which isneareftto the fpectator is A, confequently
the neareft angle is g i Ê ; therefore, upon that angle, draw th@ geometrical profile of the mouldings or cornice a, b, e, d ; and let the points g, i, K, be confidered as the mitre points of the mouldings upon the angle of the walls A and B, for thofe points are the feats of the lines which form the different members upon that angle. Through the points g, i, K, draw right lines to the center of the
picture C, which lines will interfect the line Õ L, the inner angle of the planes  and E, in the points L, o, p, which will be the points for the mitres of the mouldings, at the interior angle L Õ of the planes or Avails B, E ; for the lines which are drawn from the neareft angle g, k, of the building, to C, transfer the geometrical heiglits of the mouldings from the line g, i} S, to the line LY, Then
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The front Æ of the cornice muit then be obtained by the following
method: |
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The vertical line V f is the outer angle of the plane A, and the
points f, h, x, V, are the mitre points for the mouldings upon the angle V f. Thofe points are found by continuing the lines of the geo- metrical profile of the cornice to that angle; therefore, through the points f, h, x, draw lines from the contrary point of diftance to that which was before employed, as the lines f, Ê 2, and h, t, and alfo through the points ÷ and V, which lines will determine the pro- jection of the members of the cornice upon the angle V. Then from the points l, Q, 3, and the other points of the perfpec-
tive profile, draw lines parallel to the horizontal line, as the lines l lis, 2 t, 3, &·. and alio from the lower points. Then find the reprcfentations of the mouldings upon the plane or
wall E, as follows : From the points L, o, p, which are the feats of the mouldings upon
the line Õ L, draw lines parallel to the horizontal line, which linea rnuft
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8â PERSPECTIVE.
Å C ô· muil interfeót the line Ì m, as the lines L m, h o, and ñ q; and the
^-v—> points of interfection in, n, q, will be the feats of the mouldings on the exterior angle Ì m of the plane or wall E. Through the points m, n, q, draw right lines from the diftance of
the picture D, as the lines m s, n, 9, and fo on of the reft; and thofe lines will be the mitre lines for the mouldings upon the outer angle Ì of the plane or Wall E. Then from the points 4, 5, and 6, which points are the extreme
projeaions of the perfpeaive profile of the mouldings in the inner angle, draw lines parallel to the horizontal line ; and the interfeaion of thofe lines with'the mitre lines, as m s, and the other lines, will determine the perfpedve projeaion of the mouldings upon the angle M. To complete the farther angle Q R, find the feats of the mitre
points upon the line R Q ; which is done by drawing lines from the points m, n, and g, to the center of the piaure C, which lines will interfea the line QR in the points R, T, andS; through which points draw lines from the contrary point of diftance* to that which was before employed, and thofe lines will be the mitre lines for the termi- nation of the remote angle Q T. Then from the points 8, 9, and alfo from the other points in the
perfpeaive profile of the mouldings on the angle Ì m, draw lines to the center of the piaure C, which lines will exprefs the upper and lower fillets of the Cima ReQa, and their interfeaions with the mitre lines at w and z, will mark the projeaion of the mouldings upon the outer angle, the curved profiles of which muft be drawn by hand, as muft all the profiles of the other angles. f Obferve, that this point of diftance is not in the plate, for want of room.
The
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PERSPECTIVE.
The whole of the forgoing procefs confifts in firft finding the
geometrical feats of the mouldings upon the neareft angle of the building, as the points g, i, k, upon the line S, are the feats of the mouldings upon that angle ; Secondly, transferring thofe points or feats to the other angles of
the building, as to the angle Õ L; Thirdly, through thofe feats drawing the diagonal lines for the
mitres or joints of the mouldings, as the lines D g 1, and D i 2, in the neareft exterior angle S, and alfo the mitre lines D L 4, and D ï 5, in the interior angle Õ; Fourthly, determining the perfpeclive reprefentation of the hori-
zontal divifions of the mouldings, by drawing right lines from the center of the picture C, through the angles a, b, e, of the geometrical! profile, till they interfeót the diagonal or mitre lines, as the line 4 a; which is drawn from the point C through a, and cuts the diagonal, or mitre line D g, in the point é; which line forms the upper line of the Regala, marked 1 4, above the C'vma ReBa, and, by its interfeelion with-the diagonal D L, determines, at the point 4, the return for the fimilar member marked 4 8, which .fur-mounts the cornice upon the plane or face E. The foregoing will be ftill better underftood, by coiifidering the
plan, Fig. 4, in. the fame plate, which is half the dimenfions of Fig. i. The fides A, B, E, F, of the plan, correfponding with the fame marks in the fides of the elevation, Fig. é. The diagonals g, 1, m, in the plan, correfpond alfo with thofe in
the elevation marked with the fame characters, as do thofe marked w, f. Obferve, that the three diagonals, or mitre lines, g, L, m, are parallel
to each other'; therefore their reprefentations require the fame vaniih- ing point, which in Fig. l. is D, the point of diftance. The
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PERSPECTIVE.
The other diagonals w, f, crofs the former at right angles, and,
therefore, their reprefentations require the contrary point of diftance for their vaniihing point; but this point is not in the example, for want of room. Could it be introduced, its place would be upon the horizontal line, equally diftant from C, upon the left fide, that the point D is on the right. It is to be obferved, that the angle V, Z, of the cornice appears
diftorted in the reprefentation, although perfectly true in conftruc- tion ; this unpleafing effect is produced by the difpofition of the ob- ject, which is fo placed, that the plane A is in the plane,of the picture; confequently it follows, that all the mouldings upon that plane project forward before the picture, and produce this difagreeable eflfea In Plate X. Fig. 2, is an example for drawing diagonals or mitre
lines, for the conftruction of mouldings, when * more than one point of diftance cannot be applied upon the plane of the picture. Let A reprefent a folid pier, to which it is required to find the
mitre lines, or diagonals, for the conftruction of mouldings at the fartheft angle E ; the canvas or paper being too ihort to admit of the diftance jj) 2, which, if obtained, would be the vaniihing point for the mitre lines of the angle E. Having drawn the pier A, E, whofe plan or bafe is a parallelogram,
* To prevent mifapprehenfions it ihould be obferved, that as there can never be more
than one point of fight, fo there can never be more than one diftance of the picture. Yet, for the facility of operation, this point may have many reprefentations ; it commonly has two, which are placed upon tiie horizontal line, one on each fide of the center of the picture, and at equal diftances from it, as D é and D 2, are both the reprefentations of one point of diftance, being equally diftant from the center of the picture C, Fig. 2, and Fig. 3. This is the general practice, when the Objects to be drawn have their fiu'es pa- rallel, and perpendicular to the picture ; but in the oblique pofitions, the diitance of the picture is applied in as many different places as may be required, which will be fliewn in the next feftion. let
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PERSPECTIVE. «9
let the fartheft part of the perfpective plan be refolved into a fquare, SECT.
by the following procefs. i_ ,- » Plate X.
The thicknefs of the pier is a. b, and the fpace a 2, on the bafe
line, is the length ; therefore, from 2 fet off the to 1, the meafure of the thicknefs equal a b, and by the point of diftanee D l, transfer that meafure to the farther part of the reprefentation of the pier, as at 3, 4 ; and again, transfer that fquare at the bafe to the upper part of the Avail or pier, as at 7, 5, 8, 6, by the methods already taught. Then - through the points 7, 6, draw a right line, which line is a diagonal of the fquare dimenfions of the pier at the fartheft angle; confe- quently, it is the mitre line for any moulding or projecting fafcia that may be placed upon the pier ; as in the example it is the mitre line upon which the projection of the plat-band Μ F muft be conftructed. If more mitre lines be required, it will be neceffary to draw as
many fquares as there are mouldings; which, it muft be allowed, occa- iions much trouble, that can be avoided by no other method than by laying the picture or drawing upon a long table, and then finding the proper point of diftanee, and from that point drawing the diagonals or mitre lines required. Thus the block B, Fig. 3, in the fame Plate, being fquare, the mitre lines are found by drawing lines through the angles a, e, from the points of xliftance D l, and D 2. It muft again be repeated, that no one who is .unacquainted with
architecture, or who knows not the conftruótion of mouldings, can expeél to fucceed in the reprefentat ions of fuch objeéls ; nor is it pof- fible to give an explanation by words or drawings fufficiently clear for their inftruótion. Therefore, they muft apply to the forms, and by - examination make themfelves mailers of their conftruclions. In Plate IX. Fig. S, is an example for finding the reprefentation of
what is called a biped roof. The procefs is as follows : complete the plan of the building, as a,
Ν b, e,
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/
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ge>. PERSPECTIVE.
SEC Τ. h, e, d, and draw the diagonals a d, b e, and at their interferon %
^^-^, draw the line f h 2, perpendicular to the horizon. c ' Divide the front of the building a h, on the bafe line, as at y, and·
draw the line y g 2, perpendicular to the horizon, and upon that line fet up the whole height of the roof, as from y to g 2. Prom g 2, draw a right line to C, the center of the pifture, that
will inter-fecl; the line f h 2, in the point h 2, which willmark the height of the roof upon the center line f h 2. From the points e, g, o, draw lines to h %, and thofe lines will ex-
prefs the hips or angles of the roof. The point h 2 might alfo be obtained by the following method :
Continue the lines of either of the angles of the building, as a e,
er b g, upwards, and upon either line fet up the given height of the roof» as at g 3, from which point draw a line to the point D, the diftance of the picture, which will interfeel; the line f h2, athSj: which is the height required. This variety of operation is founded in the consideration of the
different pofitions in which a plane may be fuppofed to pafs through the center of the building; for a plane may be fuppofed to pals through the center f h 2, parallel to the fides a, e, b, d, and vaniih in the center of the pióture, which is the firft operation ; but in the fe- cond, a plane is fuppofed to pafs through the center of the building diagonally, as the line a £ d, and to vaniilx in the point D, the dif- tance of the picture. The iludent mull obferve, that in the whole courfe of this Section no
other vaniihing points are employed than the center and diftance of the
picture, which laft point, is always placed upon the horizontal line,
when the objects to be reprefented have their fides at right angles
with each other, and thofe fides difpofed parallel, and perpendicular
to, the picture.
But
1 t
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PERSPECTIVE. fi
But there is alfo another pofition of fquare objects, in which thofe SECT,
points difpofed in the fame fituation are fufficient; as when any í«^~ã—* right angular objeél. has its fides inclined to the piclure in an angle 'of 45"; for in fuch cafe, the diftance of the piAure, placed upon the horizontal line, is the vaniihing point for the fides of fuch object; and if it be a perfect fquare, the center of the picture is the vaniihing point for one of the diagonals. In Plate II. Fig. 3, is an example of fquares inclined to the pic-
ture in an angle of 45 degrees, together with a cube and block, in the fame pofition to the picture, {landing upon two of thofe fquares* Procefs for the Squares*
Wlien the center of the picture is determined, as at C, on each fide
of it fet ofFthediftance of the picture D D; then take the meafure of the * diagonal of the given fquares which is the line a b, Fig. 4, and fet on the ground line or bafe line of the picture as many of fuch meafures as may be required, as l, % 3, 4, 5; and from each of thofe points draw lines to the points of diftance D, D; and the mutual inter- ferons of thofe lines will produce the reprefentation of the fquares required. The cube E and prifm  are each fituated upon a fquare of the
pavement; therefore, at the angles of the fquare which forms the * It muft be remembered, that in all cafes fimilar to this example, where the fides of the
fquare are inclined to the picture in an angle of 45 °, that the meafure of the diagonal of fuch fquare muit be employed, not the meafure of the fide of the fquare. To obtain the meafure of the diagonal, it may be neceflary to draw the fquare geome-
trically, with its diagonal, to the proportionate fize upon a feparate paper, or upon a a detached part of the drawing, as at Fig 4, in which the line a b is the diagonal. As fquares difpofed like the example frequently form the pavement for the exterior
and interior of a building, it is proper to obferve, that in views of particular objeóts, the meafure of the diagonals of the originals muft be taken; but in compolitions, any proportion may be employed as (hall be moft agreeable to the purpofe of the artift. Í 2 bafe
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PERSPECTIVE.
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92
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SECT, bafe of each, draw lines perpendicular to the horizon, as the line P'i»·
>■—-v~^> the prifm B. Then confider where the faces of the cube or prifm would interfect,
or cut the picture, provided thofe faces were continued forward. Thus the lower line in the face  of the block interfects the picture in the point i, and the lower line of the face E of the cube interfects it at 5 : therefore, at the points l and 5 draw lines perpendicular to the. horizon, and upon thofe-lines fet up the heights required, as from l to 0, for the block B, and from 5 to n, for the cube E;. and from thofe points draw lines to the points of diftance D, D, and they will cut the perpendicular lines that form the angles of the cube and block, and determine the heights required.. Thus the line from Ï to D cuts- the angle of the prifm  at Ñ ; therefore from Ñ draw another line to the oppofite point D, and the form of the prifm will be completed. In this example the cube E and prifm  are placed exactly upon
two of the fquares of the pavement, and' therefore their bafes are thofe fquares whofe dimenfions are found by the figure of the pave- ment : but if they were not placed upon the fquares, yet their bafes muft be found' by the fame procefs, obfenohg, that their diagonal's are parallel, and perpendicular to the piclure; confequently the repre- fentations of the parallel diagonals have no vaniiliing point, while the reprefentations of thofe which are perpendicular vaniih in the center of the piclure C. The Student having gone through this feclion, it may be prefurned
that he is poiTeifed of the leading principles of the fcience, and, con- fequently, that he will be able to comprehend the following Theo- rems, which may be confidered as Axioms. FH'ft, The center of the picture muft never be out of the canvas or
tablet» |
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Secondly,
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PERSPECTIVE,
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Secondly, All lines whofe originals are parallel among themfelves,
have one and the fame vaniihing point for their reprefentations. Thirdly, All lines whofe originals are parallel to the picture, have
their reprefentations alfo parallel to each other in the picture, confe- quently have no vaniihing point- Fourthly, The center of the picture is the vaniihing point of all lines perpendicular to the picture. Fifthly, The diftance of the picture when placed upon the hori-
zontal line, is the vaniihing point of all horizontal lines that are in- clined to the picture in an angle of 45°. Let it alfo be remembered, that the point which, is vulgarly called
the point of fight, is properly the center of the picture, and therefore can only be confidered as the reprefentation of the point of fight; and this point is unique, there can be but one in the picture, although there may be innumerable vaniihing points, which are often vulgarly, but falfely, called points of fight. The author has dwelt on the above theorems with fome prolixity,
in the hope of impreifmg upon the mind of the ftudent thofe prin- ciples which are the foundation of the fcience, and which muil be clearly underftood before any great progrefs can be made by thofë who wiih to comprehend the following Sections. |
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PERSPECTIVE,
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 E C Ô É Ï Í ÉÐ,
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Containing Examples of Objects, the Sides vf which are inclined to
ike Picture.
ÔÃ Í this fecHon will be ihewn the true methods of drawing in
**- perfpeótive, right lined objects or figures, the fides or faces of which are inclined to the picture. As this feélion is founded upon a * problem of Dr. Taylor, which
may be confidered as the te'ft of the fcience, the author will at- tempt, in as familiar a manner as the fubjecl; will admit, to explain and demonftrate the true methods of finding the vaniihing points of lines which are inclined to the picture only; and the Student will do well to make himfelf throughly mailer of this feétion, for when he underftands this clearly, he may confider himfelf as pof- feifed of the mailer-key to all the difficulties of the fcience. When various original lines are parallel, and perpendicular to the
picture, the center of the piélure and the point of diilance are the only vaniihing points that are necelïary to their delineation; as has been ihewn in the foregoing feétion. But when the original lines are * See Def. 5, Brook Téylor's firft edition, 1715; and Def. 17, fecond edition,
1719. difpofed
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PERSPECTIVE.
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95
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difpofed obliquely, or inclined, to the picture, in any other angle SECT,
than 45 degrees, then the vaniihing points of thofe lines muft be found after the center and ÜÀâáçïå of the picture are fixed. But for this purpofe it is abfolutely neceffary to know, or to fuppofe, the angle which *the original line makes by its inclination to the picture. Let it be fuppofed that the reprefentation of a line is to be drawn
in perfpeólive, the original of which is inclined to the picture in, an angle of 3.5 degrees · proceed as follows.,:. |
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Draw the ground line, or interiectión of the picture, G G; then
* By original line, or original objeft, is meaat fome real line or object, of which that
in the picture is only the reprefentation. |
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the
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PERSPECTIVE.
the horizontal line Ç Ç, at its proper height above the ground
iine. "Determine the center of the picture C, and through it draw the
prime vertical line C E, perpendicular to the horizontal line. Having drawn thefe lines, upon the vertical line mark the
point E, *at the fame proportionate diftance from C, as the fpeclator is fuppofed to iland from the picture. Through the point E draw the line Ñ Ñ parallel to the horizontal
line, which line is the parallel of the picture, for it is parallel to the ground line G. At the point E, conitruct fan angle with the line Ñ Ñ, equal to 35
degrees, as the angle ç E m; and through ç draw the line E, and continue it till it cuts the horizontal line; and the interfection of that line at V will be the vaniihing point for the reprefentation of a line inclined to the picture in an angle of 35°, or any number of lines parallel to it. The foregoing are the preparatory lines, neceffary to produce the
reprefentation of the line required; which is found as follows : Let a be fuppofed the interjection of the original line with the
picture; from a draw aright line to the vaniihing point V ; then will the line a V be the indefinite reprefentation of a line inclined to the picture in an angle of 35 degrees. Thus is the indefinite reprefentation of the line a V found, the ori-
ginal of which inclines to the picture in an angle of 35 degrees ; and all lines that are inclined to the picture, have their reprefentations found by a fimilar procefs ; the only difference being in the angle * The point E is the fame point which, in the foregoing feiiion, is placed upon the
"horizontal line, and is the diftance of the piihire, and generally marked D. It is the different difpofition of this point which conftitutes the great difference in operation between lines which are inclined to the pifture, and thofe which are perpendicular to it. f See the inftruitions for conftruding angles, in problem 9th of Geometry. 1 m F ç
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PERSPECTIVE
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m E η, which muft always be made equal to the angle that the origi-
nal line makes with the picture *'. When it is required to cut off from the indefinite reprefentation of
a line, a portion equal to a given meafure, the following is the procefs: Let the line a b, which is below the bafe line GG, be equal
to the meafure which is required to be cut off from the line a V. Fix one point of the compaffes upon the vaniihing point V, and take the length of the line V" E, which bring down upon the horizontal line to the point d v; then is dv the diftance of the vaniihing point V f. Then take the length of the line a b with Compaffes, and with
one foot fixed upon the point a, in the bafe line G, mark off the point b, making a b equal the given line a b. From the point b, draw a right line to the point in the horizontal
line marked d v, which will interfeél the line a V in the point e : then will the fpace a e be the reprefentation of a line equal in length to the original line a b, and inclined to the picture in the given angle 35°. A Square inclined to the Picture.
Having well confidered the foregoing example, the Student may
proceed to Fig. 1, Plate XL which reprefents a fquare of 6 feet dimen- * Here it may be obferved, that what is fo much infifted upon in this treatife, namely,
the difference between the center of the pifture and the point of fight, may be demon- ftrated by this diagram; for C is abfolutely the center of the pi&ure, whereas the point of fight is only reprefented by the point E, The above procefs is, in the opinion of the author, the moil elegant and fimple that
can be employed : but there is alfo another method, fomewhat different in praiHce, though the fame in theory, which fhall be Ihewn hereafter.—See Addenda, Piate XXXVII. Fig. z. f This point is by Dr. Prieilley called the meafuring point. See his Treatife, page 39.
O iions,
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PERSPECTIVE.
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fions, the fides of which, D and G, are inclined to the picture irs an
angle of so degrees. The height of the eye is 6 feet, and the diftance 11 f feet.
Determine the center of the picture as marked, through which draw
the horizontal line H, and the prime vertical line * X D ; and upon that line fet up the diftance 11 f feet, as from the center to D3 which is the diftance. Through the diftance D draw the line Ñ Ñ, parallel to the hori-
zontal line H, which line is the parallel of the picture. At the point D conftruct the angle Ñ D M, equal to the given
inclination of the object to the picture, as thus : from the point D, (which is the reprefentation of the eye) draw a right line which ihall make the given angle 30 degrees with the line Ñ Ñ ; and the interfec- tion of that line with the horizontal line, as at V 2, will be the vaniihing point required. Through the eye D draw another right line as F, perpendicular
to the line D E, and its interfection with the horizontal line, will give the other vaniihing point, as V 1. Having thus found the two vaniihing points, find the diftances of
thofe vaniihing points as follows : Take the lengths of the radials, or lines D V l, and D V 2, by
the compaffes, and mark thofe lengths on the horizontal line, as from V l to b, the length of the line F, and from V 2 to a, the length of the line E; then will b be the diftance of the vanifliing point V I, and a the diftance of the vaniihing point V 2; and they are fo writ- ten in the example. Having determined the vaniihing points and their diftances, draw
the neareft fides of the fquare ; that is, from the point 1 in the bafe line, which is the feat of the angle of the fquare upon the picture, draw lines |
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• The prime vertical line is always perpendicular to the horizontal line.
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to
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PERSPECTIVE.
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99
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to the vaniihing points V l and V % as the lines  and D, which 8 EC T.
i [I.
are the indefinite reprefentations of the fides D and  of the <_—v—-»
Plate XI;
fquare.
By the points a and b proportion the reprefentations of the iides
of the fquare, thus: take the given meafure of the fquare in the compaifes, 6 feet hy the fcale, and fet each way on the bafe line from 1 to 2 for the fide D, and from é to 3 for thé fide B. Obferve that the angle 1 of the fquare touches the picture, and therefore is its feat upon the picture. From 2 draw a right line to a, and where it cuts the line drawn
from 1 to V 2, as at 5, will give the proportion of the fide D of the fquare. Repeat the fame procefs from 3 to b, and the interfeetion 4 of the
line which is drawn from l to V 1, will give the proportion of the fide  of the fquare. From 5 draw a right line to the vaniihing point V l, and from the
point 4 draw a right line to the vaniihing point V 2, and the repre- fentation of the fquare A B CD will be completed*. When the object is reprefented as removed beyond the picture, the
procefs is fomewhat different from that employed in the foregoing example. The difference of appearance is lhewn in Fig. 2 of Plate XL where the parallelogram A B C D is removed beyond the picture. The procefs is as follows: Let it be fuppofed that the angle of the parallelogram neareft to
the picture is at the diftance of 2 feet beyond it. Draw the horizontal line H, and determine the center of the pic-
* Obferve that the dimenfions of the fquare are formed by drawing lines from the
meafures placed on the bafe line, to the diflance of the vaniihing point; as from a to a; but the' figure is afterwards formed by drawing from the point 5 to the vaniihing point itfelf. as to V 1. This muft be ftriclly attended to. Ï 2 ture;
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PERSPECTIVE.
ture; draw the vertical line E, and upon it fet up the diftance of the
picture, as at D, which is alfo marked diftance. From the center bring down the diftance of the picture to the hori-
zontal line, as at D 2, which is the length of the line E, from the word center to D. Let the point 2 upon the bafe line, or interfection of the picture,
be confidered as the feat of that angle of the parallelogram which is neareft to the picture. From the point 2 draw a right line to the center of the picture. Then from the point 2 fet on upon the bafe line or interfection, 2
feet by the fcale, as from 2 to x, and from ÷ draw a right line to the diftance of the picture D 2, and it will interfect the line that is drawn from 2 to the center, in the point S; then is S the place upon the ground, or horizontal plane, of the neareft angle of the parallelogram. It is not neceffary to repeat the procefs for finding the vaniihing
points V l and V 2, that being fully explained in the foregoing figure. It needs only to be obferved, that the fides of the object have the fame inclination to the picture with thofe of the former, only reverfed ; therefore, from the point S draw right lines to the vaniihing points V é and V 2, and cut off the dimenfions of the parallelogram, as follows : Bring down the diftances of the vaniihing points upon the hori-
zontal line, as in the foregoing example, as d í é, and d í 2. Then draw a right line from the point d í l through the point S,
till it cuts the bafe line, as at the point s. From s fet off the mea- fure of the fide  upon the bafe line to 7, and from 7 return a line to d í l which will cut the.line that is drawn.from S to the vaniihing point V I, in the point 9 ; then will the fpace between S and 9 be the reprefentation of the fide  of the figure. Then find the reprefentation of the longeft fide, thus, from the point
d í 2 draw a right line through S, till it cuts the interferon of the |
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PERSPECTIVE,
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101
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Then from 4 fet on upon the interfeclion line, the meafure of the S EC T.
lone-eft fide D of the figure, twelve feet to 6, and return a line to %■-*--*
° D Plate ×Ã. d v 2, which mil iirterfecl the line that is drawn from S to the
vaniihing point V 2, in the point 8 : confequently, the fpace between
the points S and 8 reprefents the longeft fide of the parallelogram.
From 8 draw a line to the vaniihing point V f, and from 9 draw
a line to the poi ntVs; and the perfpeaive appearance of the paral- lelogram Á Â C Â will be completed according to its meafure, 12 feet by 6 feet, and 2 feet beyond the piclure, to \?hich its fliorteft fides are inclined in an angle ot SO degrees.* In the example, page 96 of this Seclion, may alio be feen the method
of cutting off a portion of a line, as follows : d í is the diftance of the vaniming point V, for it is the length
of the line V E, fet off from V, upon the horizontal line H, to d v. The required length or portion of the line to be reprefented is
then fet on upon the ground line G, from a to b, and a line drawn * As itis of the utmoft confequence that the ftudent fliouki clearly comprehend thefe
figtires, before he proceed to the following, it may be neceifary to make iome further obfervations. In the firft place, both the examples are bounded by two lines, which are to be con-
lidered as the limits of the pifture ; the inner line at the bottom is the bafe line, or inter- feftion of the picture, as expreffed by the terms, Fig. é being marked bafe line, Fig, 2 interfeftion ; which various appellations., have been employed.by .different authors to ex- prefs the fame line. The loweit of the upper lines marked Ñ Ñ is the parallel of the pifture, as written in Fig. z.
This line may fall within or beyond the upper limits of the pifture, as in many of the fol- lowing examples; for it is not limited by the d'menfions of the tablet or canvas, but depends upon the diflance of the pifture. It is always drawn through the point D, which is the diftance of the pifture, fet upon the prime vertical line, from the center of the pifture. The vaniihing points are alfo one within, and the other beyond, the limits of what may
be confidered in the example as the canvas or tablet. As this will generally be the cafe, it will be neceifary to have the paper longer than the intended dimensions of the drawing; or it may be laid upon a drawing board of fufticient length to contain the vanifhing points. If a canvas, it may be laid flat upon the floor, or placed againft the wall, and the
vaniihing. points found and employed as above direfted. See Addenda, Plate 37. |
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from
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PERSPECTIV E.
from b to d í, &c. cuts the indefinite reprefentation at e : confe-
quently, the fpace a e reprefents the length required. In the three foregoing leffons are contained the moft eflential prin-
cipes of this feclion, for they demonftrate the methods of finding the vanifbing points for all horizontal lines that are inclined to the pic- ture ; therefore the iludent is again advifed to confider thofe ex- amples with attention, obferving that the firft is of a line only; the next of a figure or plan formed of four lines horizontally fituated ; and the third is fimilar to the fecond, with the additional circum- ftance of being placed beyond the picture. Rudiments of the Perfpectwe Delineation of a Houfe inclined to the
Picture.
Fig. ], Plate XII.
Let  3, D 3, be fuppofed the ihell of a fmall houfe, with one
window, the fide â 3 being inclined to the picture in an angle of 35 degrees, and the dimensions as follows : The length of the fide  3, - - 4 feet é ï inches.
The depth of the fide D 3, - - 3 feet 9 inches. The height -------5 feet 9 inches. The height of the eye - - - 3 feet.
The diftance of the picture - - 6 feet. Let the lines A, A, be fuppofed the boundaries of the tablet or canvas. Determine the center of the picture C, and the diftance E, which
is 6 feet, by the fcale. Through the eye draw the parallel of the picture, which is the
right line fo marked, parallel to the horizontal line, and marked Ñ P. At
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PERSPECTIVE.
At the eye E, conftruet. an angle, with the parallel of the picture, of 35
degrees, equal to the inclination of the fide  3 of the object to the picture. In other words, draw the right line E  4 through E, making the given angle S5 degrees with the line Ñ Ñ, and cutting the hori- zontal line at V 2, which will be the vauiihing point for the fide  3 of the building. Through the point E draw the line D 4 perpendicular to the line
 4, and its interfeótion V é with the horizontal line will be the vaniihing point of the fide D 3 of the building. Having found the vaiiiibing points V i, and V 2, bring down
their difiances to the horizontal line, that is, take the lengths of thofe lines from the points V l to E, and alfo from V 2 to E ; and mark them on the horizontal line, as at d é and d 2. Thus are the vaniihing points and their difiances found.
Having determined the point e in the bafe line of the piclure, as
the point where the angle of the building is placed; from that point draw right lines to the vaniihing points V l and V 2, which will give the indefinite reprefentations of the bafe lines of the fides of the building, as e f V 1 and e g V 2. Then cut off the dimenfions of thofe fides as follows : with the
compafles take the given length of the fide  3 from the fcale, 4 feet 10 inches, and fet it on the bafe line of the piéiure from e to e and alfo by the meafure from the fcale 3 feet 9 inches, and fet it from e to d; from e draw a right line to d 2, which will interfeót the line that is drawn from e to V 2 at g; and a right line drawn from d to d 1 will interfeel the line that is drawn from e to V 1, in the point f: then will e g reprefent the lower line of the fide  3 of the building, and e f the lower line of the fide D 3. Then raife the fides by the following procefs :
At the point e, which is the neareft angle of the building; draw
the
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PERSPECTIVE.
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the right line e k perpendicular to the horizontal line Ç Ç, and at
the points f and g draw f h and g m parallel to e k. Upon the line e k fet up the given height of the building, 5 feet
9 inches; and from the point k draw right lines to the vanifliing points V 1 and V 2, and their interfeélions with the vertical lines drawn from f and g will form the fides of the building, as the interfeélions at e, k, f, h, produce the fide D 3, and the interfeélions e, k, g, m, produce the fide  3. Thus will the general form of the objeél be completed, in which
figure is comprehended the rudiments of almoft. all objeéls that fall within the notice of the painter or architeót; becaufe the general principles of fueh objects require, that their fides fhould be at right angles with each other, or, as vulgarly called, fquare. To find the reprefentation of the window or aperture, proceed as
follows: Upon the angle of the building e k, fet up the meafure of the
height from the ground to the fill or bottom of the window, as from e to i; and above that point the height of the window, from i to 2; and from thofe points draw lines to the vanifliing point V 2. Then fet off the width of the pier on the bafe line from e to n,
1 foot 10 I inches, and from ç to h the width of the window, é foot. Then from the points é and h draw right lines to d 2 the difiance
of the vanifliing point V 2, and the interfeélions of thofe lines with the line drawn from C to V 2, as at E w, will give the apparent width of the window. At the points Ê w draw lines parallel to e k, which will interfeél;
the lines which are drawn from the points é and 2, and produce the apparent dimenfions of the window, as æ ç h ï. If the window or aperture is placed in the middle of a fide, as in
7 this |
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PERSPECTIVE.
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lOS
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this example ; its width and place may be found by another procefs : g*E C T.
Thus, divide the fpace between e and e on the bafe line, into two u-v-^/ equal parts, which will give the middle X ; then from that point on the bafe line fet off, on each fide, half the meafure of the breadth of the window, as from X to n, and on the other fide from X to h, and draw lines from thofe points to the diftance of the vaniihing point d 2, and finiih the aperture as before. In this and the three preceding examples, the iludent muil pay
particular attention to the procefs by which the true dimenfions of the reprefentations of objects are obtained. For this purpofe let him note, that after any vaniihing point is determined, then the diftance of that vaniihing point muft be employed to meafure or proportion the parts of fuch line as is drawn to that vaniihing point. Thus, when the vaniihing point V 2 is found, then the length of
the line or radial Β 4 (which is drawn from E to V 2) muft be laid down on the horizontal line, as from V 2 to d 2. then will d 2 be the point for dividing or meafuring all the proportions which may be required in the line or lines which vanim into V 2. The plan which is below the interfeétion or bafe line, ihews the geo- metrical fituation of the objeói to the piélure. It is further explained in the Addenda, N. B. The dimenfions which are given to this example may
probably be thought frivolous; but let what is called one foot be confidered as four, and the proportion will be found equal to the pur- pofe for which it is intended ; and as the fcale is by this means lefs confufed than if more divided, the ftudent will have lefs trouble in the operation. The m miner of obtaining the appearance of the thicknefs of the wall, which is feen
through the aperture, is not explained in this example, to avoid a confuiion of lines ; it will be ihewn hereafter. * |
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Example
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PERSPECTIVE,
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Example for -drawing a Houfe, of which the Front F and Side Ê are
inclined to the Piëiure.
Plate XIII. Fig. 1.
The front F is inclined 53 degrees to the picture.
The dimenfions of the houfe are as follows :
The height ----------- 29 feet
The width of the front -------- 33 feet.
Depth ------......-22 feet.
From the ground to the top of the loweft window 9 feet.
Bottom of the upper window -----12 feet.
Height of the middle window ----,- 17 feet.
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From the top of the middle window
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4 feet.
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to the bottom of the upper
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Height of the upper window ------ 4 feet.
Width of the windows -------- 3 feet.
Door of the fame height with the windows, width 5 feet.
The height of the eye is 6 feet, and the * diftance of the picture S3 feet. That is, from the bafe line G> to the horizontal line H, is 6 feet by the fcale; and from C, tile center of the picture, to E which is the diftance, is 24 feet. Let G G be the bafe line, and C the center of the picture, and
E the place of the eye, or diftance of the picture: Ç is the horizon- tal line paffing through C, the center of the pi&ure, as before obferved. The variiihing points are V 1 and V 2; they are determined by
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*
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The diftance of the pi&ure, in this example, is rather too ihort, not being equal to the
height of the building; but it was chofen thus to ihew the confequence of fuch errors, as from this caufe the lines A M> A N, and all their parallels^ rake or flope too much, the
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PERSPECTIVE.
the fame procefs which was employed in the preceding example,
R l being the radial producing the vaniihing point of the front F of the houfe, and the radial R 2 producing the vaniihiag point of the fide Ê of the houfe. The diftances of thofe vaniiliing points are d V 1 and d V 2.
Let X be the interfeclion of the angle of the building with the picture; or, in more familiar language, fuppofe it the corner of the building, at which the picture begins* At the point X draw the the line X A perpendicular to the hori-
zon, and by the fcale fet up the whole height 29 feet from X to A. From the points X and A draw right lines to V 1, the vaniiliing
point for the front of the houfe. Then from X fet off upon the bafe line the whole length of the
front of the building 33 feet, to the point lo, and draw a right line to cl í 1, that will interfecl, the line drawn from X to V 1, in the point o, which will give the apparent depth of the front F. Draw a line from the point o perpendicular to the bafe line, or to the horizontal line, till it interfe&s the line drawn from A to V 1, as at M, and the face of the front F will be determined. Repeat the fame for the end or fide Ê ; then from X draw a right
line to V 2 ; then fet on the bafe line, from the point X to 12, the depth of the building 22 feet by the fcale, and from 12 draw aright line to d í 2, and its interfeclion with the line drawn from X to V 2 will give the apparent depth at D. At that point draw a rio-ht line perpendicular to the horizon, and where it interfeóis, at N, the line that is drawn from A to V 2, will give the perfpective appearance of the fide Ê of the building. To find the dirnenfions and places of the door and windows, pro-
ceed as follows : Ñ â For
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PERSPECTIVE.
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For their heights, upon the line X A, fet up the meafures of the
fafcis and windows, as at a, b, e, f, g, m, and ç; and from thofe points draw lines to V l, which is the vaniihing point for the front F. Then find their breadths, as follows : upon the bafe line G, from
the point X fet off the width of the outer pier, as from X to ], from 1 to 2 for the window; and continue the fame alternate meafures of piers and windows with the door, till the fpace between X and 10, in the bafe line G, is filled with the meafures of thofe different parts. From every one of thofe points é, 2, s, 4, &c. draw right lines to
the point d í l, which is the diftance of the vaniiMng· point V J, and where thofe lines interfeét the line drawn from the angle X to V l, will give the apparent widths of the piers, windows, and door upon the front F; from which interfections draw lines perpendicular to the horizon, as is ihewn in the example. To find the breadth of the window in the fide E, the fame procefs
muft be employed as was pracfifed in the front F; fetting the mea- fures of the pier and window from the point X towards 12, as at 9, 11, on the bafe line, and from thofe points drawing lines to d í % which will give the interfecf ions on the line X V 2; which interfec- tions are the places for the windows of the fide. * Many lines are omitted in this figure, to prevent confufion, for it is
prefumed that thofe which are given at thefirft diviiion of the windows, in the front F, are fufficient to demonftrate the whole, feeing that the other parts are obtained by a repetition of theprocefs already defcribed; nor was if thought neceffary to defcribe the procefs for obtaining the vaniihing points V é and V 2, becaufe it has been repeatedly given in the four preceding examples of thisfeétion. It is fufficient to obferve, • The method? for obtaining the projections of the fafciae are not given in this ex-
ample: they are explained hi a fucceeding Part of the Work. See Index. that
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PERSPECTIVE, 109
that the line or redial R l inclines to the line Ñ Ñ, which is the parallel s ^ T-
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of the piaure, in an angle of 5 3°, which is equal to the inclination of u-j^"
the front F of the building to the pi&ure : confequently the line R 1 Xïifc produces tbe vaniihing point V 1 ; and, as the fide Ê is at right angles or perpendicular to the front F, the radial R 2 is drawn at right angles or perpendicular to the line R iK thereby producing the vaniiliing point V 2. |
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Of the Diagonal when Oblique or inclined to the picture.
In drawing the forms of regular fquares, it is often neceifary to find
the reprefentations of the * diagonals of thofe fquares, particularly when mouldings are to be delineated round a fquare or right angular block or ihaft, in which cafe the vaniiliing point of the diagonal will be required. Therefore, in Fig. 2. PlaLe XII. an example is given for finding Plate
the vaniiliing point.of the diagonal of a fquare. f Let it be fuppofed that the reprefentation of the fquare 1x5 4
is already drawn; that C is the center of the pifture ; that E is the eye, and that Õ ß and V 2 are the vaniiliing points of the fides of the fquare : it is required to find the vaniihing point of the diagonal. Bifea the angle A E B, of the radials A, B; and the line E Y5
which biieas the angle, will interfeét the horizontal line at e, which ■ interferon is the vaniihing point required. Left the foregoing inftruclions ihould not be clearly underftood,
they ihall be repeated in a more explicit manner. * Diagonal is a right line drawn acrofs a parallelogram, or other quadrilateral figure, from
the vertex of one angle to that of its oppofite. Some authors call it the diameter, and others the diametral of a figure. The line X 4, fig. a, is the perfpeftive reprefentation of the diagonal. f As the procefs for finding the reprefentation of the fquare was given in the foregoing
figures, it is unneeeffary to repeat the fame for this diagram. After
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PERSPECTIVE.
After having found the radials A andB, with the compaffes, on the
point E (which is the eye) defcribe the arc ÷, æ ; and then with the compaifes on the points x, z, defcribe the arcs y Y. Then from E draw a right line through the interfections of Y, y, till it cuts the ho- rizontal line at e, which will be the vaniihing point for one diagonal of every fquare, whofe fides vaniih into the points V 1 and V 2 : there- fore from X, the neareft angle of the fquare, draw a right line to d, and the representation of a diagonal will be produced, as the line X 4 is a diagonal of the fquare X 4, 3, 5. N. B. The fludent muft obferve that the line which is called the
parallel of the picture is not drawn in this diagram; but this makes no difference, for the angle of inclination, which the fides of the object make with the picture, may be found by means of the vertical line, as follows: Every angle that is lefs than a right angle hath its complement,
which added to the given angle, produces the contents of a right angle, which is 90 degrees. Thus to an angle of 30 degrees the com- plement will be 60, which added together produces 90, the meafure of a right angle. The complement of an angle of 40 degrees will be 50, and the fame ratio of the reft. The parallel of the picture and the prime vertical line are always
drawn at right angles with each other; and as the radial producing the yaniihing point makes a given angle with the parallel of the pic- ture, it muft follow, that the angle produced by fuch radial and prime vertica line will be the complement of the primary angle produced by the radial and parallel of the picture. Therefore, if an original line is inclined to the picture in an angle
of 35 degrees, its parallel radial Mall be inclined to the prime vertical plane in an angle of 55 degrees, thefe two quantities added together producing 90", equal a right angle. 4 This
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PERSPECTIVE.
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This is demonftrated by the example, page 95, where E L is the
vertical line, and Ñ Ñ the parallel of the picture. The angle m E n, is 35 degrees, formed by the parallel of the pic-
ture P, and the radial K, the complement of which is the angle ç E L, formed by the radial Ê and the prime vertical line L, which is as. angle of 55°. Therefore, it is evident that if an angle be conftruóted with the
prime vertical line L, at the eye or diftance E, equal the comple- ment of the inclination of the original line with the picture, which la this example is 35 degrees, the radial Ê will be produced, and confe- quently the interfeélion V, which is the vaniihing point, will be the fame as if obtained by the angle m E n, made with the parallel of the piélure Ñ Ñ * Of Squares inclined to the PiBure.
In fig. % Plate XIII. is an example of a pavement of fquares in-
clined to the picture. There is alfo a reprefentation of a fquare block F, whofe fides in-
cline to the picture, but with different inclinations to that of the fquares; confequently there are four vaniihing points in the example, two for the fquares of the pavement, and two for the fides of the block. Thofe for the fquares or pavement are marked V 1, V 2, and
thofe for the block, marked F, are V S and V 4. All the vaniihing points are found by the procefs already defcribed,
every radial being drawn through the eye, with the fame inclination * Dr. Prieftley, in hisTreatife on Perfpe&ive, employs the above procefs for determining
the vaniihing points of lines inclined to the picture. Vide Sec. iii. page 18 of his work. |
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to
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PERSPECTIVE.
to the parallel of the picture P, as the fides of the original objects in-
cline to the picture. The fide A of the pavement inclines to the picture in an angle of
29°; confequently the other fide  inclines in an angle of § i.° The fide F of the block inclines to the picture in an angle óf 50",
the fide G 40°. This figure is given for two purpofes; the firft, to demonfirate (what
muft ever be particularly remembered by the ftudent in perfpecüve) that when the reprefentations of many objects are to be delineated in one picture, they are all governed by one and the fame center and diftance of picture, though there may be an infinite number of vaniihing points for the reprefentations of thofe objects. The fecond purpofe of this figure is to iliew the method of repre-
fenting a pavement of fquares, when theirfides are inclined to the pic- ture ; in which cafe their true reprefentations are beft obtained by the following procefs; Suppofe the fides of the fquares to be inclined to the picture in an
angle of 29°. "Through the eye draw the radial RXi, making the given angle
with the line Ñ Ñ, which will produce the vaniihing point V I. At the eye draw the radial R X 2 perpendicular to R X 1, pro-
ducing the vaniihing point V £. ' Then find the vaniihing point of the diagonal of the fquare, as
follows: Bifectthe right angle formed by the two radials R ÷ é andR ÷ a,
by the method employed in Fig. 2 of plate XII. {fee page l io)
which will give the line T, interfering the horizontal line in the
point a, which will be the vaniihing point for one diagonal of all the
fquares of the pavement.
Upon'
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PERSPECTIVE.
Upon a feparate paper, or remote part of the picture, draw the re-
presentation of the original fquare geometrically, at its proper dimen- fjons, by the fcale, as at Fig. 3, No. 1, together with its diagonal, as the line l, 2. Determine the point in the bafe line, where it is intended that the
angle of one of the fquares ihall touch the pifture, as at the point 3 ; from which point draw aright line to a, the vaniihing point of the diagonal, as the line 1 a. Then take the length of the diagon l 2 in the geometrical fquare
Fig. 3, and fet it upon the bafe line from the point l to 2 and 3, and continue thofe meafures for as many fquares as may be required ; and from every point, as from l, 2, 3, draw lines to the point d d, which h the diftance of the vaniihing point of the diagonal é a; and the interfections of thofe lines with the diagonal, at the points 4, 5, will determine the perspective dimenfions of the diagonals of the fquares, as they recede from the picture. Through the points 4 and 5 draw lines from the vaniihing
points of the fides of the fquare, as the points V é and V 2, and the reprefentations of the fquares will be completed. Another method may alfo be employed, as follows : determine the
point l for the angle of the fquare which touches the picture ; from which point draw right lines to the points V 1 and V 2, the va- niihing points of the fides of the fquares. Then from the point é fet off both ways the meafures of the fides
of the fquares, and from thofe meafures draw lines to the diftances of the vaniihing points; and the lines fo drawn will interfect the lines A and  in the proportionate meafures required ; through which di- ritlons draw lines to the vaniihing points V é and V 2, and the reprefentations of the fquares will be completed. In this problem it will be neceflary to obfeive, that fquares can be
i'Jttsated in no other pofitions to the picture only, than the following: Q Firftj
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PERSPECTIVE.
Firfr, The fides parallel and perpendicular to thepiclure. Example.
Plate IV. Fig. 1. Secondly, The fides inclined to the piclure in an angle of 45".
Plate II. Fig. 3. Thirdly, Inclined to the picture in any other angle than 45 de-
grees. Plate XIII. Fig. 2. In the firft pofition, the diagonals of the fquare are inclined to the
picture in an angle of 45°, and confequentlv have the diftance of the piclure, when placed upon the horizontal line, for their vaniihing points. In the fecond pofition, one diagonal is parallel to the piclure, and
the other perpendicular; therefore the latter vaniilies in the center of the piclure. In the third pofition, both the fides, and alfo the diagonals, vaniili
in oblique vaniihing points, as has juft been demonftrated. Table inclined to the Picture.
Plate XIV. Fig. é.
Example of a table, of which the front ï Ñ is * inclined to the
piclure in an angle of 49 degrees. The dimenfions of the table are as follows :
The length.......--5 feet 9 Inches.
The breadth -.-,----3 feet 9 inches.
The height.........3 feet 3 inches.
The thicknefs of the legs about - - - 0 feet 3 inches.
• hi this example the objeils are fuppofed to be ftanding in a room. The limits of the
picture are the lines marked A. The center of the pifture is marked by the word center, and the lines marked R are the interferons of the fides of the room with the floor and ceiling, and the line marked R 2 is the interfeftion of the two fides which are feen; the others are not feen, not being within the angle of vifion. The
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PEÏiSP'ECTIVE. Π*
The heia-ht of the eve is fix feet and one inch, and the diftance of SECT.
the picture about nine feet five inches. w-v—j Determine the center and diftance of the pióture, and draw the χ^
parallel of pióture as before directed. Find the vaniihing points ν l and ν % by the methods already ex-
plained in the foregoing examples. In this example the table is fuppofed at a diftance beyond' the
pióture; therefore the firft part of the procefs is to find the perfpec- tive feat upon the floor, of the angle Ρ of the neareft leg, which pro- cefs is explained in Plate XL where the angle S of the parallelo- gram A B C D, Fig. 2. is removed beyond the piclure : but that the ftudent may not be obliged to refer back to that example, the procefs iliall be here repeated. Let a in the ground line, be the feat of the angle of the leg upon
the picture. From a, draw a line to the center of the pióture : then from a fet off on the ground line the meafure of the diftance at which the leg is from the pióture, as at b, which is one foot. The diftance of the piclure, D i, being firft brought down upon the hori- zontal line to D 2, draw a right line from b to D 2; and its inter- feótion with the line drawn from a to the center of the pióture, will give the perfpeclive feat P, which will be the angle of the neareft leg of the table. Having thus found the point P, draw lines from that point to the
vaniihing points V ι and V 2; which lines will be the indefinite re- prefentations of the outer fides of the legs of the table. Then cut off the length and breadth ofthe table, by the following-
method : Through the point Ρ draw right lines, from the points marked
diftance of ν 1 and diftance of ν 2; and continue thofe lines until they cut or interfeót the bafe line., as in the points l and 3, Q « Then
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PERSPECTIVE.
Then from the two points é and 3, fet off upon the bafe line
the meafures of the front and end of the table i thus from é to 8, for the extent of the legs in front, and from 3 to 6 for the extent of the legs on the fide ; that is, from é to 8 five feet nine inches, and from S to 6, three .feet nine inches. From the point 8 in the bafe line, draw a right line to the point
marked diftance of í 2; and where that line interfeóts the line which is drawn from the angle Ñ of the leg of the table to í 2, as at o, will give the apparent limits of the front legs of the table. Then from the point 6 in the bafe line, draw a right line to the
point marked diftance of í 1 ; and its interfeclion with the line drawn from Ñ to í 1, as at Ñ 2, will mark the extent of the legs for the width of the table. Haying thus obtained the general extent of the leg of the table»
as at the points Ñ, Ñ 2, ï, draw lines from thofe points perpendicular to the horizon, which lines will be the angles of the legs of the table. Then find the height of the table as follows :
Continue the line which is drawn from í l through P, until it cut
the ground line at e ; and at that point draw the line Ì perpendi- cular to the horizontal line. Upon that line fet up the height of the table 3 feet, as at the point
J i, and from that point draw a line to the vaniihing point í 1 ; and; the iiiterfeclion of that line with the vertical line, which is drawn from P, as at the point w, will be the required height of the table. For the thicknefs of the legs, take the meafure three inches, and fet it
on the bafe line from é to % for the front of the neareft leg, and from 8 to 7, for the diftant leg. From 2 draw a right line to the diftance of í 2, which will give the apparent breadth at y ; from 7 draw aline to the diftance of í é, which will give the-apparent breadth of the farther leg at s..
The
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PERSPECTIVE
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The fame procefs muft determine the breadth of the legs at the end
of the table, the meafures being fet on the bafe line from 8 to 4, and from 6 to 5, and lines drawn from thofe points to the diftence of í l, which by their interfeólion with the line drawn from b in the bafe line to the vaniihing point í 1, will give the apparent width of the legs at the end of the table. Then find the place and breadth of the rail at the end, with the-
ftretcher in-the middle, as follows: Upon the vertical line M, fet up from the bafe line the height of
the bottom of the rail above the ground, and above that the meafure of the breadth of the rail, and transfer thofe points by lines drawn to the vaniihing point í 2 ; and the * interfeclions of thofe lines with the angles of the legs will mark the apparent dimenfions and place of the neareft rail. The ftretcher being exaétly in the middle between the legs, divide
the fpace between the points 3 and 6 in the ground line (which fpace is the real breadth of the legs at the end) into two equal parts at 7; and from the middle draw a right line to the diftance of í l ; and its interfeftion 8 with the line drawn from Ñ to í é will giye the middle of the end of the table. Transfer that interfeftion by a line drawn perpendicular to the horizon, and it will cut the rail at the point 9 ; from which point defcribe the firetcherf, by drawing- lines to the vaniihing point í 2, which will give the form of the ftretcher uniting with the farther rail.. * The pointe is the interferon of the end'of the table with the bafe line of the picture,
that is, if the end Ñ Ñ æ, be confidered as a plane continued towards the picture, it will inter* feci the picture in the line Ì; which line aifo cuts the interfeflion or bafe line of the picture in the point e. f It mull: be obferved in this example, that although the general form of the objects is
fimple in its principles ; yet it requires fo many lines to define all the parts with accuracy in perfpective, that if every line were given, the diagram would be rendered fo confufed, as rather to puzzle than inftruit the ftudent: therefore let him reflect,, that the fame procefs which gives the height of the rail muft be repeated to obtain its breadth; nothhw more being necefTary than to add the meafur.e above, and transfer it by the vaniihing point V u |
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The
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il» Ô E R S Ñ E C Ô É V E.
E- C f· The projection of the top of the table is obtained by the fame pro-
-—v^-»j cefs, by which mouldings are delineated in the example given in Plate ■·'·'■■■■ ï é t.
XIV, Plate XVIII. therefore the ftudent is referred to that for farther
inftruction. In this plate is alfo an example of a block B, the fides of which
have the fame inclination to the picture with thofe of the table: confequently the fides C and D are parallel to the front and end of the table, and therefore have the fame vaniihing points. This example is given to demonftrate the different methods of de-
termining the height of any object, the plan of which is fuppofed to be already drawn, in perfpeótive. The firft method is the fame with what is generally employed
through this work, and is as follows : Through the angle S of the plan of the block, draw a right line
from the vaniiliing point V 1, that may interfeól the bafe line of the •picture, as at X ; at which point draw the line X E, perpendicular to the bafe line, and upon that line fetup from X to E the height of the block, and transfer that height to the angle of the object, by drawing a right line from E to the vaniiliing point V 1, which will mark the height at t. The apparent height may alfo be obtained, by taking at pleafure
any point in the horizontal line, from which draw a right line through any angle of the plan, as b or S, till it interfecl the bafe line; at which interfection draw a line perpendicular to the bafe line, and upon that perpendicular line fet up the height, and transfer it back by a line drawn to the fame vaniiliing point, whence the firft line was drawn ; and the interfeclion of fuch line with a line drawn at the angle S, or b perpendicular to the horizon, will give the apparent height re- quired, 4 Another
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PERSPECTIVE.
Another method may alfo be employed, as at Fig. 3, as follows :
At any point in the bafe line taken at pleafure, as at L, draw a line perpendicular to the bafe line, as the line L N, upon which line fet up from the bafe line the required height of the object, as from L to G. Then from the points L and G, draw lines to any point in the hori-
zontal line. In the example they are drawn to the center of the pic- ture, as the lines marked m and n, which lines are to be confidered as the indefinite reprefentations of the upper and lower edges of a plane, of which the height G L is equal to that of the objecl;, the re- prefentation of which is required. Then transfer the required height, as follows :
At the neareft angle S of the plan of the block, draw a line parallel to the horizontal line, that may interfecl the line L n, as at the point g, at which point draw the perpendicular gK; then is the line g Ê equal to the apparent height required. From the point k draw the line k t, parallel to the horizontal line,
which will cut the vertical angle of the block at t, and consequently determine the height. Or the height of the line, g Ê may be taken with compafles, and
applied from S to t, which will give the height required. This laft given procefs for determining the heights of objects may
be employed to avoid a number of lines, as the operation may be per- formed on a different paper, or detached part of the fame tablet; taking- care that the height of the eye and diftance of the pi&ure be the fame as employed in the general arrangement of the other objects in the picture. |
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The front A - - - - 2 feet 8 inches.
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The fide  ----- 2 feet 2 inches.
The height - - - - - é foot.
The thicknefs ------ 3 £ inches.
It is about 9 inches beyond the picture.
The height of the eye is 2 feet, fo that the box may be fuppofed
ftanding on a table, the furface of which is 2 feet below the eye of the fpectator. C is the center of the picture, and the diftance which is marked
by the word eye, is 4 feet 10 inches. Procefs.
Draw the radial R é from the eye, making the given angle with
the parallel of the picture 41 degrees, which is the inclination of the fide  to the piélure. Draw the radial R 2, perpendicular toRl, and the interfections of
thofe lines with the horizontal line Ç Ç, "will give the points V ], V 2, which are the vaniihing points for the fides of the box. Then determine the diftance of the box beyond the picture ; for
which purpofe, fuppofe S to be the feat of the angle of the box upon the picture. From the point S draw a line to the center of the picture C, and from S on the ground line fet off to x, by the fcale, the dif- tance of the angle of the box from the picture, which is 9 inches. Then
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PERSPECTIVE. êé
Then bring down the diftance of the picture to the horizontal line, SET9TT'
as at D, by the fame procefs that hath been ihewn by Fig. % Plate <—v~«»
' J l . J t> > Plate XV. XL at the beginning of this feclion.
From ÷ draw a ris:ht line to D, and where that line interfe&s thé line
which is drawn from S to C, as at N, that point will be the place of the angle required, or the reprefentation of its feat upon the ground. From the point N, draw lines to the vaniihing points VlJ»^
and cut off on thofe lines the dimenfions of the box, by the following- method : Upon the horizontal line bring down the diftances of the vaniihing
points, as at d l, d 2, by the methods already directed. Then from d 1, in the horizontal line, draw a right line through
the angle Í of the box, till it cuts the interfeétion or ground line of the picture at i. Repeat the fame from d 2 through N, till it cuts the ground line
at 5. Then take the meafure of the fide A, 2 feet 8 inches by the fcale,
and fet off on the ground line from 5 to 8 ; and alfo the meafure of the fide B, 2 feet 2 inches, and fet off from the point é to 4; and from thofe points draw lines to the points d i, d 2, and where thofe lines interfeét the lines that are drawn from the angle of the box N, to the vaniihing points V l, V 2, thofe interfeclions will give the apparent dimenfions of the box, as at a and b. Thus the lines drawn from 5 and 8 to d 2, produce the dimenfions
of the fide A at a N ; and the lines drawn from é and 4, to d i, pro- duce the fide  at Í b. Then find the thicknefs of the box as follows :
For the fide A fet off the meafure 3 |- inches, * from 5 to 6, and alfo from 8 to 7 in the interfeclion or ground line, * Three inches and half is too great a thicknefs for fuch conftructions, but is given for.
the better denionftrationof the figure-^One inch would be fuflicient, R For
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PERSPECTIVE.
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For the fide  fet off the fame meafure from l to 2, and from 4
to 3 ; and from thofe points draw lines to d é and d a, and their interfeétion with the bafe lines of the box will give the thicknefs, as at e d on the face A gives the thicknefs for the fide  and its op- pofite, as the fimilar interfections on the fide A indicate the thick- nefs on the fide  and its oppofite. Then find the height of the box as follows :
Continue the lower line of the fide A, by a line drawn from the
vaniihing point V 1, till it cuts the interfeétion or bafe line at x, at which point draw a line perpendicular to the interfeétion or ground line, as the line m x, upon which fet up the height of the box l foot by the fcale to h, and from h draw a line to V l, which line will inter- feét the lines that are drawn perpendicular to the ground line at the angles Í a, and give the upper angle æ of the box ; from which point draw lines to the vaniihing points V 1 and V 2, which will form the two fides A and B. Transfer the thicknefs which is found at d and e in the lower
line of the box, to the upper edge, as at t, and from that point and thofe which are fimilar, draw lines to the vaniilring points V é and V 2, and the thicknefs of the top of the box will be expreffed. It muft be obferved, that the procefs employed for finding the
thicknefs of the fides of the box, is equally applicable and proper to» determine the thicknefs of the walls of ahoufe or other building *„ Of Circles in Perfective.
In Plate XV. Fig. 3, is an example for delineating circles in
yerfpeétive. * In the next Seftion this leflbn is continued, the box being completed with the addi-
tion of a Ud—See Plate XXIX» Fkft
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PERSPECTIVE,
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12JL
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Firft draw the circle geometrically at its full fize on any other SECT.
III.
paper and then circumfcribe that circle with a fquare, as in Fig. 2. w—^-_j Divide the fides of the circunrfciibing fquare into any number of ate '
equal parts, fuppofe fix; and at thofe equal divifions draw lines, which may divide the large fquare into a number of fmaller fquares, producing a reticulation, as in the example Fig. 2. In this example the fquare which circumfcribes the circle is in-
clined to the picture; confequently oblique vaniihing points, with their diftances, are employed in the conftruftion of the figure. Having determined the center of the picture and its diftance, and
alfo the oblique vaniihing points V ι and V 2, by the rules already demonstrated; draw the perfpeclive fquare S, and afterwards divide it into as many fmall fquares, perfpeétively, as are contained in the geometrical fquare, Fig. 2 : As thus, take the meafures of the fmali fquares l, 2, 3, 4, 5, 6, on the fide of the geometrical fquare, Fig % and transfer them on the ground line each way, from X, as indicated by the figures 1, 2, 3, 4, &c.; and from thofe points draw lines to the points d ι and d 2, which are the diftances of the vaniihing points for the fides of the fquare, which lines will produce the inter- feétions a, b, e, d, e, upon the fides of the reprefentation of the large fquare S ; from which interfeclions draw lines to the vaniihing points V l, V 2, and the great fquare S will be divided into the fame number of fmall fquares perfpeétively, with the geometrical fquare. The perfpeclive appearance of the circle muft then be finiihed by
hand, which can be completed by no other method than comparing the reticulations of the perfpeclive fquare S, with thofe of the geome- trical one Fig. 2, and then drawing the appearance of the circle through thofe points in the fquare S, which correfpond with thofe in Fig. 2. Thus the points e, f, g, h, k, in both figures correfpond with Β f each
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PERSPECTIVE.
each otherj confequently, thofe m S are the guides for the required
reprefentation. This procefs is fo plain, that a ileady examination of the figure will
render further explanation unneceffary. It muft be obferved, that if the circle is circumfcribed by a fquare, whofe fides are parallel, and perpendicular to the picture, the procefs is nearly the fame; the center of the picture being the vaniiliing point for the fides of the fquare; in which cafe the diftances of the vaniiliing points are noti required. Tedefialtmth Steps inclined to the Picture.
Fig. J, Plate XVI. In this example there are three fteps, each of which is 6 inches in
the rife, and é foot in the tread *. The face F of the pedeftal and fteps are inclined to the picture in
an angle of 50 degrees. The height of the eye, or horizontal line is 6 feet.
The center of the picture and place of the eye are marked.
The diftance of the picture is about 11 feet.
FI II is the horizontal line.
G G is the interfection or ground line of the picture.
Ñ Ñ is the parallel of the picture, to which the line or radial R in-
clines 50% producing the vaniiliing point V 1. V I, V 2, are the vaniiliing points.
d í 1, d í 2 are the diftances of thofe vaniiliing points. V D, is
the vaniiliing point of the diagonal. * By rife is meant the thicknefs of the ftep, or its height from the ground, which is
feldom more than 6 inches, generally lefs. By the tread is meant the flat part, on which the foot falls, which is feldom more that»
12 inches, unlefs the building to which they approach is large. 4 The
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PERSPECTIVE.
The vanilliing points and their diftances are found by the methods-
already explained.
The point X is the neareft angle of the loweft ftep, which touches the picture.
Therefore from X draw lines to the vaniihing points V I, V 2, as
the lines A and B, and at the point X draw the line × Õ perpendi- cular to the hafe line G G 2. Upon the line × Õ fet up the height of the neareft ftep, as from X
to o, fix inches by the fcale; and from the point ï draw lines alfo to the vaniihing points V l and V 2. Then will the lines which are drawn from the points X and ï to
the vaniihing points, reprefent the indefinite appearance of the firft or loweft ftep. Proceed to find the length of the firft or loweft ftep, as follows :
From X fet off on the bafe line each way to the points 7 *, the
length of the loweft ftep 8 feet 10 inches by the fcale, and fromthofe points draw lines to the points d í é and d í 2, which are the dif- tances of the vaniihing points ; and the interfeclions of thofe lines with the lines A and  of the fteps, as at Ê E, will determine the ap- parent length each way of the loweft and longeft ftep. Then proceed to reprefent the tread or furfaee of each ftep, as
follows : From the angle X of the loweft ftep, fet off each way the dimen-
ftons of the horizontal furfaee or tread of the ftep, which in the example is one foot f; and repeat that meafure ns often as there are fteps required : as from X to the points 1 2 3 each way; outward from the point X for the neareft angles, and for the farther extreme * Obferve that the marks of reference are the fame on both fides of the fleps, beeaufe
the procefs is the fame on both fides. f That is, 1 foot from the point g to r.
Of
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PERSPECTIVE.
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of each ftep, from the point 7, the fame meafure inward; that is,
one foot from the point 7 to 6', to 5 and 4. From thofe points draw right lines to the points d í 1, d í 2,
which will interfeeT; the line A in the points a, b, e, and d, e, f; at which draw lines perpendicular to the horizon, as the lines a g, b h, e m, at the near angle of the ftep ; and alfo lines in the fame direc- tion from the points d, e, f, at the fartheft angle of the ftep; as in the example. Then determine the neareft angle or corner of the fecond ftep, as
follows: - V D, is the vaniihing point of the diagonal of the object; there-
fore from o, which is the upper angle of the loweft ftep, draw a line to V D, a-nd at a, draw the vertical line ag; then from g draw a line to V 1, and the interfection of the lines ï V D, and g V ], at q, will give the feat of the angle (or as vulgarly called, the corner) of the fecond ftep at q. Having found the point q, which, as before obferved, is the loweft
point of the angle of the fecond ftep, draw the vertical line q r: Then upon the line × Õ fet up the height of the fecond ftep, as from ï to P, and transfer that meafure to the line q r by a line drawn from P, to the vaniihing point V D, of the diagonal; and the height of the fecond ftep will be obtained by its interfecTion at r with the line q r. From q and r draw lines to the vaniihing points V 1, V 2, as was
done in the firftftep, which will exprefs the fides of the fecond ftep; then cut off the length of the fecond ftep, as follows : From the point 6 in the bafe line, which is the meafure of the length
of the fecond ftep, draw a line to the vaniihing point V 1, which will interfeeT; the bafe line of the loweft ftep in the point d, at which point draw a vertical line that may interfeeT; the upper line of the loweft ftep;
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PERSPECTIVE.
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ftep ; from which interfeciion draw a right line to the vaniiliing·
point V i, and it will interfeft the loweft line of the fecond ftep in the point w, confeqnently the point w is the extreme length of the fecond ftep. If the inftructions given for obtaining the proportions and forms
of the two lower fteps are well underftood, there will be little dif- ficulty in obtaining the proportion of the third, and of the bale of the dado of the pedeftal; for a repetition of the foregoing procefs is ajl that is neceifary, even if there were many more fteps to be defcribed. There wants nothing to complete the figure but to find the height
of the pedeftal, for its thicknefs is determined by the tread of the third ftep; becaufe that ftep furrounds the bafe of the pedeftal, which is defined by the angles marked a b d. Therefore to find the height, draw the vertical angles of the dado
or ihaft of the pedeftal, from the points a b d, which are found in the upper ftep, by the fame procefs that determined the lower fteps. Then upon the vertical line × Õ fet up from X in the bafe line, 9
feet 6 inches to Y, and from that point draw a line to the diagonal vaniiliing point D V; and the interfeciion of that line with the angle of the pedeftal will give the height required, as at the point z. From the point æ draw lines to the vanifhing points V 1 and V 2, which will determine the form of the ihaft or dado of the pedeftal. The method for determining the crown or covering of the pedeftal is given in the Addenda.—See Index. In the courfe of the foregoing operation, let it be obferved, that in
marking the meafures for the treads of the fteps, which determine their fituations within each other, thofe meafures are applied out- wardly on the bafe line from X, and inwardly on the fame line from 7; and the fpace between the points 3 and 4 is the breadth of the dado
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PERSPECTIVE.
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dado or ihaft of the pedeftal. It miift alfo be particularly noticed, that
the proportions of the treads of the fteps are]obtained by lines drawn to the difiances of the xmnifhmg points; but the lines which produce the forms of the fteps muft be drawn to the vanifiing points. A Garden Seat.
In Plate XVII. is an example for a building with a pediment,
which may be confidered as adapted to contain feats or other acco- modations in a garden. The center of the picTuie is C, the diftance or eye is D.
Ç Ç is the horizontal line.
Ñ Ñ is the parallel of the picture, as written.
L l is the radial or parallel of the front Ê of the building, which
is inclined to the picture in an angle of 38°. . L 2 is the radial or parallel of the fides of the building, which are
perpendicular to the front. * Â Â is the bafe line.
The vaniihing points are V l and V 2, determined by the radials
L l, L 2 (as before taught, Page 95) ; and d l, d 2 are the diftances of thofe vaniiliing points. • After having ftudied the inftrucYions and praétifed the rules which
are given in the preceding part of this Section, particularly thofe of the Example of a houfe inclined to the picture ; the ftudent will not require to have them repeated here, for the conftruction of * In all architectural reprefentations, it will ever be the moft convenient praftice, to
place the objeft immediately upon the bafe line; as in this example, in which the point A is the interfection of the pifture, or feat of the angle upon the bafe line. And after the reprefentation of the building is obtained, to add as much fpace below as may make the whole appear pleafar.t to the eye, as the line X X was added at pleafure, after the the object was finifhed.—This method will fave a great deal of trouble, yet be equally true, as if the building had been drawn at the fame apparent diftance from the line X X as it now appears. |
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PERSPECTIVE,
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129
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this figure is produced by the fame procefs which was employed in SECT.
, J. 1.1·
that example, with the addition of the pediment; for the conftruc- v—v—j
Plate
tion of which, inftruétions are given in the next Seétion. XVIL
Yet it may be proper to obfe-rve, that the heights of the funk pan-
nels, and of the fluted fafciae, are determined by fetting up their dimenfions from the point A, upon the angle of the building A f, as at the points a, b, d, h, and from thofe points drawing lines to the vaniihing points V ι and Va ; which lines determine the heights of the required parts. The breadths of thofe parts are found by fetting their meafures
upon the bafe line Β Β, from the angle A to the points l, 2, 3, 4, and e, on the left fide, and from the fame angle to the points 7, 8, and III, on the right fide, and from every one of thofe points drawing lines to the diflances of the vaniiliing points; as thofe on the left fide are drawn to the point d ], which is the diftance of the vani firing point V l, and thofe on the right are drawn to the point d 2, which is the diftance of the vaniihing point V 2. The rails or palifadoes, marked M, at the back of the building, are
parallel to the front and back ; confequently have the fame vaniih- ing point with the front, which is V 1. The line of pofts and chains is parallel to the ends or fides of the
building; confequently has the fame vaniiliing point with the ends, which is V 2. The figures of the man and Avoman are fuppofed of equal heights
with the fpeélator who views the building; therefore the horizontal line of this example paifes through the eyes of thofe figures. The fcale by which the building is drawn, is given at the bottom
of the print; therefore the ftudent who confiders this example may eafily obtain the meafures of all the parts ; and it will be a good practice for him to purfue the following method : S Take
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PERSPECTIVE.
Take with the compaffes the different fpaces which are figured upon
the bafe line, and then apply the compaffes, fo extended, to the fcale, and the real dimenfions will be known. Thus   is the bafe line, upon which the fpace between 4 and 5 is equal to the real width of the door or aperture: therefore take that fpace in the compaffes, and while fo extended apply them to the fcale, and it will give 9 feet, which is the roeafure. The fame procefs will determine the heights of the different parts,
taking the dimenfions at the angle of the building upwards from A, as from thence to the top is 24 feet by the fcale. It muil be obferved, that no drawing of an object of this kind can
be made with truth but by meafurement.—- If it is a view of a build- ing already conftructed, the meafures muil be obtained from the original object: but if it is only a defign for fome future edifice, the geometrical iketch muft firft be made, and the meafures figured, by which the perfpective drawing muft be regulated.. |
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Plate
XVII. |
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A Church inclined to the Picture.
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• Plate XVIII. Fig. 3.
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Is a view of a country church, of which the fide A is inclined to
the picture in an angle of 46 degrees. All the fides of the building are at right angles with each other.
The center of the picture is C, which is in the middle of the hori-
zontal line, reckoning from the fides of the picture. D is the diftance of the picture, on the prime vertical line.
The vaniihing points are V é and V 2; and d l, d ä, are the dif-
tances of thofe vaniihing points. é The
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PLATE
XVIII. |
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PERSPECTIVE.
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The roof is divided into two parts, and the line Ô Æ is in the fame
plane with the fide S of the bell tower; while the fide U of the tower is parallel to the end × Ô of the church, as are alfo the fides of the porch. D 2 is the diftance of the picture, brought down to the horizontal
line: it is employed to find the point A, which is the feat of the neareft angle of the building upon the ground; which point A is determined by the procefs which is employed in example Fig. 2, Plate XI. Page 99. The point A being found, from that point, draw the lines which
form the bafe of the building, to their different vaniihiiig points, as Á í é, Á í .2. Then find the height of the wall, as at the point W, which is
in the angle of the walls of the fide and end. For which purpofe, continue the line which is drawn from V 1
through A, till it cuts or interfecls the bafe line, as at the point b, at which point, draw the line b ï perpendicular to the horizon ; and upon that line fet up the known or fuppofed height of the wall, as from b to 2, which is 18 feet by the fcale ; and from 2 draw a line to V 1, and it will mark the apparent height of the wall by its in- terfeclion at W. It is fcarcely neceflary to give a minute detail of the procefs for
obtaining every line in the reprefentation of the object, as fuch in- ilructions would be no more than a repetition of the foregoing leifons. Yet, as the general form of the building is rather complex, it
might by fome be thought unintelligible without further explana- tion, particularly concerning the receding or remote parts of the dhjecl. It muft then be obferved, that the building is compofed of two
parts or divifions, the further and longer past of which may be called the chancel, and is connected with the bell tower, S £ * To
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PERSPECTIVE.
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* To find the appearance of what may be called th& chancel, that
is from Ô to S, including the tower, proceed as follows : Suppofe the line Ô ç already determined by the rules before given,
and confider Ô as dividing the building, at the part equal the chan- cel's width, and in the fame plane with the face Ì and S; then from ç draw a line to the vaniiliing point V 2, as ç ×, Then through the point ç draw a line from d 2, which is the dis-
tance of the vaniiliing point D 2, that may cut or interfeér. the bafe line at 9 ; and from 9 fet upon the bafe line the meafure of the whole length of the church 6o feet, and from that point draw a line to the fame diftance d 2 of the vaniiliing point, which will determine the apparent length of the building. The height of the fpire is deter- mined as follows: -eithe four angles of the tower draw diagonals, and at their interfecüon draw a line perpendicular to the horizon, which will be the center, and find its height upon that line by the foregoing rules. In the example the roof of the church is divided into two parts>
each of which has the fame inclination to the horizon ; which in- clination is determined without vaniiliing points, by the following- method :- Suppofe the ridge R of the roof to be fix feet higher than the
angle W of the walk The line o b is the interfèftion of the picture by the plane of the
* As in this example the plate is not long enough to admit the whole meafure of the
length of the church, the half may beufed, with the half diftance, as follows: V 2 is the vaniihing point of the fide A A of the church, and d 2. is the diftance of that vaniiliing point; therefore divide the fpace between V 2 and d 2 in half or two equal parts, as at j2i. then from d 2 f draw a right line through n, which may interfect the ground linej as at æ: then from æ fet on half the meafure of the length of the Church, 28 feet to y, and thence draw a right line to d í f, and its interferon with a line a drawn from ç to V 2 will give the apparent length of the church with equal' truth as if the whole meafure had been employed. end
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PERSPECTIVE. 133
end Ô Xlcontinued, and the point 2 in that line is the height of the SECT.
Ë „ ■ é e ■ r. Éз
wall upon which the roof refts ; therefore, from 2 to 3 fet up the w-v—»
Plate
height 6 feet, and from 3 to 0, 6 feet alfo; which is double the XVlli. height of the roof from 2 to 0; at each of thofe points draw lines
to the vaniihing point V l; then continue or produce the line A W,
which is the neareft angle of the wall to k, and the line ç Ô, torn, and
the farther angle to F. Then draw the diagonals W m, Æ k, Í m, and
Æ F; and the pitch or form of the roof will be determined by the
lines ø R Æ and Æ L N.
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Demonfiration.
The points W k Æ m reprefent the four angles of a parallelogram,
feen obliquely, the bafe of which is W Z, equal the chord or fpan of the roof required; and R, which is the center of the parallelogram, marks the apex or fummit of the roof, the perfpeélive appearance of which is produced by the fcalene triangle W, R, Z. The procefs for obtaining the reprefentation will be better under-
llood by confidering the diagrams marked Fig. l and Fig. 2 in the fame plate. The firft ihews the geometrical form of the roof, the line 45 being
the chord or fpan, and the point 1 the apex or fummit of the roof. Upon the line 34, as a bafe, a parallelogram is conftruéted, whofe
height is twice that of the roof; confequently the diagonals 25* S4, of the parallelogram coincide with the inclined furfaces of the roof. It muft be obferved, that the vaniihing point V 4, by which Fig. 2
is drawn, is to be confidered as in fome horizontal line, and cor- refponds with the point V l in Fig. 3.* * All the lines and points in Fig. é, and Fig. a, are marked by fimilar letters and figures,
by comparing which the fludent will eafily comprehend the procefs. This
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PERSPECTIVE.
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134
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SECT. This procefs is more eafy in theory than what is employed in the
\^s^s firft example of the next Section. Yet both produce the fame effect;
Plate
XVIII. and as it may frequently happen that the vaniihing point for the in-
clined planes of the roof may lie above the limits of the canvas or tablet, the method here given will, in general practice, be found the moft convenient A Chair inclined to the Picture.
PLATE Plate XIX. contains examples for drawing a chair in Perfpe&ive, ãôã
when the front is inclined to the picture.
The form of chair reprefented in the example is by workmen
called a balufter back, parlour chair; and the ftudent muil obferve that in general, all modern chairs are conftracled upon the princi- pies of this example; that is, the back of the feat is narrower than its front; confequently the feat and general plan of the chair are trapeziums, of which two fides are parallel to each other, as a and b, Fig. 1 and 2; while the other two are inclined to each other, as e and d. By this form of the original object, it is abfolutely neceffary to find three vaniihing points, for the delineation of the chair feat and rails; a fourth mufi alfo be employed to facilitate the operation. The meafures of the chair are as follows :
The height of the feat is ] 8 inches.
The width of the front of the feat, 25 inches.
The width of the back of the feat, 19 inches.
The depth of the feat from the front to the back, as from a to b,
19 inches. The height of the back above the feat, 18 inches.
The width of the top of the back is the fame with the front of the
feat.
The
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PERSPECTIVE.
The height of the eye or horizontal line is 5 feet.
The diftance of the picture is 8 feet 5 inches. Ç Ç is the horizontal line, G G is the interfe&ion or ground line, and the center of the pióture is marked by the word. Procefs, Fig. I.
Determine the center of the picture, and likewife the diftance or
place of the eye, as E, and draw the parallel of the picture Ñ Ñ, as in the foregoing examples. At the eye E draw the plan a b e d of the feat of the chair, by the
fcale; difpofing the front a with the fame inclination to the parallel of the picture Ñ Ñ, as the original is fuppofed or known to incline to the picture.
Continue the line a, which expreifes the front, till it cut the
horizontal line atYl; the point V l will be the vaniihing point for the front and back of the chair, for they are parallel to each other. Then continue the line e of the plan, till it interfects the horizon- tal line at V 2; then will V & be the vaniihing point of the fide e of the plan of the chair. Through the point E draw another line parallel to the fide d of the
plan, till it interfect the horizontal line, as at V 3, and V 3 will be the vaniihing point of the fide d of the chair. Thus are the three vaniiliing points obtained, V é for the front
and back of the chair, V 2 for the fide e, and V 3 for the fide d» Find the vaniiliing point V 4, by drawing a line at E perpendicu-
lar to the line E V l, and its interfecüon at V 4 will be the vaniih- ing point for lines perpendicular to the front of the chair. After having determined thefe Vaniiliing points, find their diftances
by the foregoing rules ^ as d 1 is the diftance of V l, d 2 the dif- tance |
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is6 PERSPECTIVE.
SECT, tance of V 2, d 3 the diftance of V 3, and d 4 the diftance
é—^j of V 4.
Plate XtX. Having thus obtained the vaniihing points and their diftances,
proceed as follows.:
Suppofe the point l, Fig. l, in the ground line, to be the point at
which the angle of one of the front legs touches the picture. At é draw the perpendicular é e, and upon that line fet up the
height of the feat 18 inches by the fcale ; and from the bottom and top of the leg, that is from é and e, draw lines to the vaniihing point V 1. Then on the bafe line fet off from l to 2 the width of the front of the chair 2 feet l inch, and from 2 draw a line to d l, which is the diftance of the vaniihing point V l ; and the interfeclion at 9 will be the apparent width of the front of the chair. Draw the front with the rail and breadth of the legs by the foregoing· rules, as in the Example of the table, Plate XIV. Page 114. Then from the upper angles of the front draw the fide d to the
vaniihing point V 3, and the fide e to the point V 2. This will be hetter understood by infpeéting the Targe example
Fig. 2; where the front and back, a and b, are both drawn to one* Vaniihing point, they being both parallel to each other; while the fide e is drawn to the point V 2, and the fide d to the point V 3 ; becaufe thofe fides incline to each other. Determine the depth of the feat as follow*;:
The vaniihing point of the fide e of the chair feat, Fig. 1, is V 2.
Therefore, from the angle l draw a line to V 2, and cut off a por- tion of that line equal the depth of the feat; for which purpofe fet the meafure of the fide d or e (being both equal in dimenfions) upon the ground line, from 1 to 12, and from 12 draw a line to the point * The vaniihing point for the front and back a and b, Fig, 2, are out of the plate, for
want of fpace. d fi;
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PERSPECTIVE.
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137
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&% which will give the interferon x, and determine the apparent SECT.
depth of the feat upon the ground; which meafure maft be trans- t-^^-j Plate
ferred to the point 10, by a line drawn perpendicular to the horizon, ÷é÷,
which will produce the interfeftion 10. Having obtained the point 10, the reprefentation of the feat of the
chair is formed by drawing a line from that point to the vaniihing point V l ; a portion of which line is b, which reprefents the back of the feat. Then find the reprefentation of the back, as follows : Continue upwards the angle of the front and neareft leg, as from
é to 4; and from the point l fet up the whole height of the chair back, 3 feet 2 inches by the fcale; and from the point 4 draw-a line to the vaniihing point V 1, which will interfect a line drawn upwards through the angle of the other front leg, in the point 3 ; then from the points 3 and 4 draw lines to the vaniihing point V 4, and find the point 6, which is the upper angle of the back, as follows: From the point l draw a line to the vaniihing point V 4, and fet
upon the bafe line from ú to 8 the meafure of the depth of the back, with its * inclination ; and from the point 8 draw a line to d 4, which will give the interferon 5 ; from which point draw a line perpendi- cular to the horizon, to interfecl the line drawn from the point 4 to the vaniihing point V 4, as at 6; from which point draw a line to the vaniihing point V 1, which will produce the interfeclion 7 : then will the line b, which is the back edge of the feat of thé chair, together * By inclination muil be underftood the bend or flope of the back; which, in the
kind of chair reprefented by the example, declines backward from the feat; by which circumftance the top of the back is further from the front rail a of the feat, than the back rail b. The beft way to obtain the meafure of tire depth of the inclination of the back, is to
place the chair againft the fide of a room, and meafure from the front of the feat to the' wall. In the example, the difference is about 3 £ inches; the depth of the feat being 19 inches, and fpring of the back, or back feet, 23 f inches, cxprefled by the points 12 and 8 in the ground line G, Fig. 1. Ô with
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Ñ E E S Ñ E C Ô ß V E,
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with the points ï and 7, be the principal guides for the delineation;
of the back of the chair; which, with the back feet, muft be drawn· by hand : for in ail complex forms compofed of curved lines, like the example here given, the rules of perfpeóHve can only give points of direction, by which the curves muft be completed by hand. In this example, many of the lines that are neceifary for the com-
pletion of the object are omitted, as their number would appear fo complex as to render the figure unintelligible; but the whole will be clearly underftood, if thofe which are given are well examined, for they are the effential, and govern the reft. The Student muft pay particular attention to this figure, becaufe
it is the fir ft of this Treatife in which the fides of the object are not at right angles with each other. But of this further notice will be taken hereafter. He muft alfo obferve, that the whole form of the object is contained in a parallelepiped, the angles of which are mark- ed as follows: l, x, 9, o, confidered as the bafe, and 3, 4, 6, 7, as the top. This muft ever be the procefs when objects are compofed of curvi-
linear forms, fuch as the back and balufter marked A, Fig. 2, toge- ther with the back feet; for thofe parts muft be drawn by hand, through points which determine the extreme angles, or principal parts of the required forms. The Example, Fig. 2, is given to ihew the object at large, more
particularly to demonftrate the direction of the fides of the object to their vaniihing points. The horizontal line for this figure is marked Ç 2, and the vaniihing
points for the fides of the chair are V 2 and V 3, which correfpond with the points V 2 and V 3 in Fig. l. The vaniihing points for the front and back are beyond the limits,
of the plate, as before obferved. The
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"PERSPECTIVE.
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1~3§
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The thicknefs of the legs, the place and dimenfions of the fide SECT
' in.
rails, together with the ftretcher, are determined by the fame pro- c-v*—»
P.Jatc
cefs, which was employed in the example of the Table, Plate XIV. xix.
The balufter A of the back muil be drawn by hand; to aflift
which, find the perfpeclive reprefentation of the middle of the back of the chair, together with as many points as may ferve as guides to direót the delineation of the fmaller forms; after which the whole inuft be finiihed by hand, as already obferved. Door inclined to the Picture.
Plate XX.
Is an example for determining the apparent width of an open PLATE
door, when inclined to thepiclure. This figure is intended to ihew the procefs by which a door may
be reprefented in true proportion to the aperture, after the other parts of the room are finiihed; and therefore may be confidered as extremely ufeful to the artiil, who wiihes to reprefent an open door, the dimen- fions of which may correfpond with a given aperture in any pofi- tion fuited with the compofition of his pióiure. Let it be fuppofed, that the fides of the room and the aperture
for the door are already drawn, and * that R R, R R, reprefent the upper and lower lines of the fides of the apartment, and that the points V, W, mark the apparent width of the opening of the door—g is the vaniihing point of the fide C Υ of the room, and d is the diitance of that vaniihing point. Obferve that this point is in the iliade, therefore rather obfcure. Firft find the real width of the aperture, by the inverfe procefs, as
follows: * In this problem it is afliimed that the center and diflance of the pifture are already
determined : C is the center, and E is the eye or diftance of the picture. Τ 3 Confider
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PERSPE C Ô É V E;
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140
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SE C Ô. Confider the width of the aperture of the door as a given portion·
V—■í—' of the* line TWV, the vaniihing point of which, as before obferved} XX, is g, and its diftance d. From d draw right lines through W and V, till they cut the ground
line G, G l, at a and b : then will the fpace between a and b repre- fent the real width of the aperture. Having thus obtained the real width of the aperture, proceed as
follows: At the point V, which is in the line of the door upon which it is
hinged, draw a right line at pleafure, as may beft fuit the circum- ftances required in the picture; and continue that line, till it cut or interfect both the ground line and the horizontal line, as the line V S, which interfefe the ground line at the point m, and the horizontal line in the point o : confequently the line m.SVois the indefinite reprefentation of the face of the door, and o is its vaniihing point. * From o draw a line to the eye or diftance E, and bring down the
length of that line to the horizontal line, as at d 2, which will be the diftance of the vaniihing point o. Thus is o the vaniihing point of the loweft line of the face V X
of the door, and d 2 is its diftance : therefore, from d 2 draw a right line through V, till it cut the ground line or interfe&ion of the pic- ture, as at G l; then take the width of the door, which has already been found at a b, and fet it on from G é to a; and from a draw a right line to d 2 ; and its interfection S with the line of the face of the door, which is, drawn from m to o, will give the apparent width of the door. It is prefumed that the ftudent has already made fuch progrefs
before he attempts this problem, that after having obtained the point S, he will be able to complete the reprefentation of the door without * The line O E. is the radial parallel to the original of face S X V of the door; and
the length of every radial is the diftance of the vaniihing point produced by that radial; therefore d æ is the diftance of the vaniihing point o. further
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PERSPECTIVE.
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141
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further inftruótions ; becaufe, all that follows has been already ex- SEC T.
plained. Therefore it will be fufficient to obferve, that the vaniiliing c-v-^-r
. - Plate
point for the thicknefs of the door at S, is h, the interferon of its ÷÷
parallel radial with the picture : fo that to find the true dimenfions
of the thicknefs of the door, the diftance of the vaniiliing point h
.muft be brought down to the horizontal line, the dimenfion of the
thicknefs marked on the * ground line, and transferred to the door,
as before.
In this example the procefs, which may be confidered as the
natural order of operation, is reverfed; for inftead of finding the points V W, by their places a b firft given, on the ground line or interfection of the picture,- the reprefentation of the opening of the door, as V, W, are firft given, or fuppofed, and afterwards their places found upon the interfection of the picture, as at b, a. Again, the line S V of the face of the door being drawn out at
pleafure, as may beft fuit the defign of the artift, the vaniihing point of that line, and the diftance of that vaniihing point muft afterwards be found. Therefore, inftead of firfc drawing the radial E O, making the fame angle with the parallel of the picture, which the door is required or known to> make with the picture; the radial muft be drawn from O to E, becaufe O is the vaniiliing point already deter- mined by the pofition of the line m S V. Here it will be proper to remark, that if nothing more were required than to cut off a portion of the line m S V, the radial O E need not be drawn; becaufe the diftance from O to E, taken by the compaffes, is fufficient to deter- mine the diftance of the vaniiliing point O : but, as it is neceifary to find the vaniihing points for the thicknefs of the door, the line O E * The diftance of the vaniiliing point h is not marked, and other lines are omitted,
that the figure might not be too much confufed. The perpendicular line m ç is the line upon which the real height of the door is marked 3
which being transferred by the vaniiliing point 0, marks the apparent height of the door at X. muft
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PERSPECTIVE.
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.142
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SECT, muft be drawn, that the line E h may be drawn perpendicular to ϸ,
thereby to obtain the vaniihing point h, which is the vaniibing point |
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Plate
XX. |
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for the thicknefs of the door at S X.
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As the inftruclions and obfervations given in this Example may be
confidered as a fcholium upon the preceding part of the Section, it will be very advantageous to the ftudent to underftand it clearly, and alfo to confider the following: Corollary. The line V Õ is the line upon which the door is hinged; therefore, the point V, which is upon the floor, may be confidered as the center, round which the lower line S V of the door revolves, producing a circle, all the radii of which are equal the original of the line S V. Hence it follows, that this problem may alfo be confidered in effect the fame with that which is given by Dr. Brook Taylor in his firft edition, Prob. 12, Example 3, Page 24, which teaches to complete the reprefentation of a circle from the reprefentation of one given radius. For fuppofe the line V Wthe given reprefentation of a radius, and
V the center of the required circle; then will the line V S repre · fent a fecond radius ; confequently, all the other radii may be deter- mined by a repetition of the fame procefs, by which the line V S was obtained ; and as the circle paifes through the extreme of its radii, the points S and W may be confidered as two points in the perfpeclive reprefentation of the periphery of a circle, the reft of which remain to be found. |
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Of the Polygon and Triangle.
Plate XXI. In this plate are two examples, one of which may be called au
oblong hexagon, the other a triangle, in perfpe&ive, both of which have all their fides inclined to the picture. é In
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PLATE
XXI. |
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PERSPECTIVE.
In drawing the perfpective appearances of objects fimilar to thefe
examples, the readieft way of finding their vaniihing points, is to draw their geometrical forms in fome part of the canvas or paper ; difpofing the fides with the fame inclination to the parallel of the piclure, as thofe of the original object Let the hexagon, Fig. l, be fuppofed fo fituated that one of its
longeft fides inclines, to the picture in an angle of 37 degrees. After having determined the center * of the picture, and its dis-
tance, as at the words center and diftance, draw the horizontal line H, and the parallel of the picture Ñ Ñ, as in the foregoing examples. This being done, draw the geometrical figure of the hexagon, No. l, in fuch difpofition that the fides R i, Ri, ihall incline to the parallel of the picture Ñ Ñ, in an angle of 37 degrees, which is the inclination that the original object makes with the picture. Through the eye draw the radial R l X, parallel to the fides R i, R i, of the geometri- cal plan No. é;. and the point where that line interfeéts the hori- zon, as at V i, will be the vaniihing point for the projection of the reprefentation of the lines Ri,Ri, in the geometrical plan, No. l. Again, through the eye draw the radial R 2 X parallel to the ori-
ginal lines R % R 2, in the geometrical plan No. 1; and the interfec- tion of the radial R 2 X with the horizontal line H, will be V 2, which will be the vaniihing point for two of the fides of what may be called the ends of the hexagon. The other radial R 3 X is drawn in the fame manner through the
eye, parallel to the fides R 3, R 3, of the geometrical figure No. 1, and wrould, if continued, cut the horizontal line, and produce a third vaniihing' point; but the angle of the radial R 3 being but little * The center of the picture is marked by the word, and the mark X, but V 2, clofe
to it is a vaniihing point, produced by the radial RzX. inclined
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PERSPECTIVE,
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inclined to the picture, the interfeclion will not be in the paper;
therefore, it is neceffary to find * the vaniihing point V 4, which is for lines perpendicular to the radial R é X, or which is the fame thing, to lines perpendicular to the lines R i, R i, in the geometrical plan No. 1. Having thus obtained the vaniihing points V J, V 2, V 4, and
the diftances d V 1, d V 4, proceed as follows : Let the point a be found in the perfpe&ive reprefentation, by
the rules given in example, Fig, 2, Plate XI, page 99; from the point a draw a line to the vaniihing point V l, which line may be confidered as the indefinite reprefentation of that longeft fide of the hexagon which is neareft to the picture. From the point d V l, which is thediftance of the vaniihing point
V 1, draw a right line through a, till it cuts the bafe line at é; and then fet off the meafure of the fide, as from l to % in the bafe line5 which is equal to one of the longeft fides of the hexagon, No. é; and from 2 draw a right line to d V i, which will interfeci the line a b at b: confequently, the portion of line between a and b will be the per- ipecrive reprefentation of one of the longeft fides of the hexagon neareft the .picture. Having thus obtained the neareft fide of the hexagon, find the per-
fpecüve reprefentation of the oblong or parallelogram which contains the whole figure, as 5, 9, 6, 12. For this purpofe, on the geometrical figure, No. ], draw the lines
b b through the interfecrions which the ihorteft fides of the hexagon * In drawing this kind of figures, the great bufinefs of the artift is to contrive that
they may be executed with the feweft vaniihing points poffible. In this figure three vaniihing points are employed, which are V I, V 2, V 4.—But
V 4 is not a vaniihing point of any one of the fides of the figure, but is a vaniihing point of lines at right angles with the fides R i, of No. I; for as the vaniihing point of the fides R 3 would lie at a great diftance beyond the limits of the paper, it is neceflary to find fome other point which may fupply the want of it. |
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make
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PERSPECTIVE.
make with the longeft. Then take the projection of the angle from
b to a, and apply that meafure on the ground line G G from 2 to 3, and from é to 4 ; from thefe points draw lines to d V l, and they will interfeet the line b a, in the points 5 and 6; from which points draw lines to the vaniihing point V 4. Having drawn thefe lines, find the breadth of the parallelogram : for which purpofe draw a right line from d V 4 through the point 5, till it interfeet the bafe line at ÷; and let off from ÷ the whole breadth of the hexagon by two equal divifions, as at 7 and 8 ; and from thole points draw right lines to d í 4, the diftance of the vaniihing point í 4 ; and they will produce the interfections 5, ï, g : from which interfections draw lines to the vaniihing point V J, which will give the fides and longeft diameter of the parallelogram which circumfcribes the hexagon. Complete the hexagon, as follows—From the point b, * draw a
right line to the vaniihing point V 4, and its interfeétion with the oppofite fide of the hexagon will give the point 12. Draw right lines from o and b, to the vaniihing point V 2, and
join the points a, o, and n, 12, by right lines; and the hexagon will be complete. Triangle inclined to fhe Picture.
Plate XXI. Fig. % is an example of a triangle, the fides of which
are inclined to the picture. Having determined the center of the picture, with its diftance, draw
the horizontal line, and the parallel of the picture Ñ P. Then draw the triangle, No. 3, in any part of the paper or canvas,
fo that the fides may make the fame inclination with the parallel of the picture Ñ Ñ, which the fides of the original is known to make with the picture. • This line is not marked, to aroid confufio'n,
U Through
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PERSPECTIVE.
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146
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SECT.
III. Plate
XXI. |
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Through the eye draw lines parallel to the original lines d 1, é %
Ü 3, in No. 3, which parallel lines will be R 1, R 2, R 3, producing the vaniiliing points V 1, V 2 : but the vaniiliing point which would be produced by R 3, falls beyond the limits of the paper, but is not required in this diagram. Having found the vaniihing points V 1 and V 2, Math their dif-
tances d í i, and d í 2, determine the place of the angle A, as already directed in the foregoing example; the point d Ñ being the dif- tance of the picture, brought down to the horizontal line, then from ti**t point draw right lines to V l and V 2, which M'ill be the in- definite reprefentation of two fides of the triangle. Through the point A, draw right lines from d V 1 and d V 2,
which ihall cut or interfecl; the ground line at a and b ; from which points fetofFthe meafures of the fides of the triangle, as from a to e, equal the fide d 2 in No. 3, and from b to d, equal the fide d' l in the geometrical triangle, No. 3. From e draw a line to d V l, which will interfecl; the line that is
drawn from A to V ], in the point P; and from d draw a line to the point d V 2, which will interfecl; the line drawn from A to V 2, in the point Ï: confequently, the line A P reprefents the fide of the triangle equal d 2, in the geometrical figure No. 3; and Á Ï the fide equal d l, in the fame geometrical figure. Complete the triangle by drawing the line Ï P. It is almoft needlefs to obferve, that having obtained two fides of
any triangle, the third is found of courfe ; therefore two vaniihing points are fufficient In all folid figures, of which the plans or bafes are polygons or
triangles, the elevations are conftrufted by the fame vaniiliing points which produce the plans : therefore, if the foregoing examples con- cerning |
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PERSPECTIVE. 14?
cerning folids, the fides of which are at right angles with each other, SECT.
. v HI.
-are well underftood, there need no fnrther inftructions concerning >w-vO
the conilruótion of buildings, the bafes of which are polygons or Xxj.
trapeziums, for the fame methods are employed in both cafes. Thus, if a folid were required to be reprefented upon the triangle Fig 2, nothing more is neceffary than to draw lines at the different angles Α Ο Ρ, perpendicular to the horizon; then to determine the heights of the neareft line in the object, and from the point fo found draw- ing lines to the vaniihiag points of the fides ; which points are V 1, and V 2. Octagon and Pentagon, Plate XXII.
Fig. 1, and Fig. 2, are examples for drawing the oétagon and pen- PLATE XXII
tagon in perfpeétive, when their fides are all inclined to the picture.
Before any inftruclions are given for finding the vaniihing points,
it will be neceifary to inform the ftudent (particularly if unikilled in Geometry) of the readieil methods of drawing thofe forms geome- trically. * The oétagon being the fimpleft, ihall be firft given.
To draw a regular oétagon in a fquare of any given dimenfions,
the following is the procefs : Draw the fquare, Fig. 1, No. 1, of the given fize; and at the
points 1, 2, 3, 4, draw the diagonals AC, Β D. Then with a radius equal to half the diagonal, that is with com-
paffes on the point C or D extended to E, which is the center of the fquare, draw the femicircles a d, g b, cutting the fides of the fquare at the points a e, and g b. Repeat the fame at the angles A and B, producing the points e, h, and f, e ; draw right lines from b to e, from a to e, from f to g, and h to a; and the oólagon will be completed. * Thefe problems do properly belong to the firft feéHon; but they are placed here, that
the ftudent might not have the trouble of turning back, before he could proceed with the perfpe&ive part of the diagrams. U 2 - For
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PERSPECTIVE.
For tlie regular pentagon, the fides of which are of given dimen--
fions, the following are the methods of operation. Draw the line A B, Fig. 2, No.. l, equal to the given dimenfions off
one of the fides of the pentagon.. Upon the points A and  draw two circles, the radii of which are
equal to the line A B, and they will interfect each other at the points a and b; through a and b draw a right line, as GD. With the radius equal to the given line A B, on either of thofe
points, mark the point a on the line C D ; and upon that point de- fcribe the circle l 2, A.B, producing the point E.. Through the points 1 2 and E draw right lines, till they cut the
upper circles in the points s and 4 ; on the points 3 and 4, with the radius Á Â defcribe the arcs D, and join the points A 3, Â 4, D a, and. D 4, by. right lines; and the pentagon will be complete.. Plate XXII. Fig. i*. Procefs for. the Perfpeeüve reprefentation,
of the oftagon, the fides of which are inclined to the piélure. Having drawn the horizontal line. Ç Ç, and the parallel of the pic-
ture Ñ Ñ, as before directed ; draw the radial E Vl, making the, fame inclination with the parallel of the picture Ñ Ñ, that the fide of the original object makes with the picture... Thus the line E V l, makes an angle of 35" with the parallel of
the picture Ñ Ñ, and produces the vaniihing point V é. Perpendicular to the line E V i, draw the line E V-Q, producing
by its interfection with the horizontal line, the vaniihing point V 2. Bring down the diftances of thofe vaniihing points to the horizon--
tal line, as at d V l, and dV2, Thefe points being determined, draw the perfpective reprefentation £
of the circumfcribing fquare, as follows: Let é be fuppofed the angle of the fquare, touching the interfection ;
or bafe line of the piclure G G. From é draw lines to the vaniihing points
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PERSPECTIVE. 149
points V1 and Vs, and fet off on the bafe line the dimenfions of the SECT.
circumfcribing fquare; that is, from 1, to 2 and 3 on each fide, equal «—-v—J
Plate
to the fide A B or C D of the original·geometrical fquare, No. 1. xxn. From thofe points draw lines to the diftances of the vaniihing
points d í 1 and d í 2, and the interferons 4 and 5 will be the perfpeóiive appearances of the angles of the circumfcribing fquare. Complete the perfpecüve fquare 1 4 5 6, by a line drawn from
the point 4, to the vaniihing point V l, and from the point 5 to the vaniihing point V 2. Find the vaniihing point V 3, of lines producing the fides of the
oêtagon, which lines are parallel to the diagonals of the given cir- cumfcribing fquare, as b e, g f, and d e, h a, in Fig. 1, No. 1. Bifeét the angle Ì E Í, and draw a right line from the eye E,
through X, which by its interferon with the horizontal line, will give the vaniihing point V 3, the vaniihing point for one of the diagonals of the perfpecüve fquare, Fig. I, which circumfcribes the reprefentation of the octagon. From the point 1 in the bafe line (which is the neareft angle of the
circumfcribing fquare) fet off the meafures which may determine the interfections of the fides of the octagon with that fquare, as follows ; ' From either of the angles of the fquare No.- 1 as from C, take, with
the compaffes, the fpace e b, or C e, and apply that meafure from the point 1 in the bafe line, Fig. 1, on each fide, as to A and C; from A draw a right line to dV2, and from C to d V 1, which will interfeél' the perfpective fquare at the points 7 and X ; alfo fet the fame mea- fures from the points 2. to B, and from 3 to D. From·. draw a line to d í 2, and from D draw alfo a line to d. í é, .
which lines will produce the interfeftions 8 and 9. Then from the points 8 and 9, draw lines to the vaniihing point
V 3, which lines will form the reprefentations of two fides of the octagon as 811 and go. From:.
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PERSPECTIVE,
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From the points 7 and X draw lines alfo to the vaniihing point V 3,
which will give the interfections 12 and 13. Join the points 7 and X, as alfo 12 and 13, by right lines, and. the perfpective appearance of the octagon will be completed.* It is needlefs to obferve, that the fides ÷ 7, and 11,12, are parallel
to each other, confequently, have the fame vaniihing point; which point would be produced by the radial R, if it were continued till it interfered the horizontal line; but as this point falls out of the limits of the paper, it cannot be conveniently employed. The Pentagon. Fig. 2, Plate XXII.
Suppofe the center of the pióture C, with the eye or diftance E,
the horizon Ç Ç, and the parallel of the picture PEP, as before directed. At the eye E, draw the regular pentagon No. 2, geometrically,
with one of its fides at the fame inclination to the parallel of the pic- ture PEP, that the original inclines with the picture. Continue the fides E l and E 4, till they interfect the horizontal
line; alfo draw right lines from E through the angles 2 and 3, till they interfect the horizon; all which interferons will give the vaniihing points V l, V 2, V 3, and V 4, the vaniihing points for four of the fides of the pentagon. After having found thofe vaniihing points, bring down their dif
tances, as d í é, d í 2, &c. Let a in the ground line G G, be fuppofedthe angle of the penta-
gon, which touches the picture. From a, draw lines to V l, and V 2, * When a regular octagon is fo fituated that one of its fides is parallel to the picture,
there will be no occafion for any other vaniihing points than the center and diftance of the picture, as commonly employed in the fecond feftion: for in fuch a fituation, the fides of the octagon, which are inclined to the picture, may be confidered as parallels to the diagonals of a circumfcribing fquare; confequently, are inclined to the pifture in an angle of 45 degrees. Therefore, the points of diftance placed on the horizontal line will be the vaniihing points of thofe fides of the octagon, which are inclined to the picture. which
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PERSPECTIVE.
which will be the indefinite reprefentations of two fides of the penta-
gon. From the point a, fet oft* on the ground line the length of one fide, as from a to b, equal A B or A 3 in the geometrical figure No. 2, and draw a line from b to d í1; and the interfecüon d will give the apparent length of one of the fides of the pentagon, as from a to d. Repeat the fame procefs from a to e equal the meafure of a fide,
and from e draw a line to d í l, which will give the interferon e, and determine a fecond fide of the figure: from d draw a line to the vaniiliing point í 3, and from e to í 4, which will produce the in- definite reprefentation of two other fides of the pentagon. Then determine the length of thofe fides, as follows :
Draw a right line from d í 3, through d, till it interfecls the ground
line, as at h; and from h fet off the length of a fide to k, and draw a right line from k to d í 3 ; and its interfeclion m will give the length of the fide d m. For the fide e n, repeat the fame procefs, by means of the vaniiliing
p©int d í 4. Having obtained the points m, n, join them by a right line, which
will form the fifth fide, and complete the pentagon. In the foregoing problems, it is directed to draw the form of the
polygon required at the eye or diftance; but it muft be obferved, that the reprefentation may be obtained, if one fide only is drawn in the given inclination to the picture ; after which lines muft be drawn at the eye> making fimilar angles with the firft given line, which the different fides of the polygon make with each other ; which lines will produce the vaniiliing points, by their interfeétions with the hori- zontal line. But this procefs, though elegant and fcientific* is not fo ready and ufeful to the artift as the one here given. But iliould any one defire to purfue this part of the fubje<3, he will find it amply treated by Hamilton and Malton : the latter in his firft vo- lume, Book iii. Plate XIII, As
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152
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SECT. As edifices are very feldom conftrucled -upon the plan of any poly-
e—v-1»» gon, except that of the octagon, the foregoing inftructions will be Plate
XXII. fufficient for the artiftin all cafes whatever, for the vaniihing points
of the fides, as alfo the dimenflons of thoie fides are all determined by the fame procefs which is employed in the four preceding .examples.
Of Circles and Semicircles in Perfpectwe, Plate XXIII.
As it may he prefumed that the undent has already made himfelf
mailer of the rules, by which the vaniihing points of horizontal lines inclined to the picture are found ; no directions are given for that part of the procefs in the fucceeding examples of this Section, but the center of the picture and vaniihing points are indicated, as already determined. For it muft be evident to the flighteft obfervation, that the lines which are called the parallel of the piélure, and the parallel of the eye, are of no ufe after the vaniihing points are found; there- fore, thofe lines are omitted in the following examples, to prevent the confufion which would otherwife neceflarily refult from their number. x In the following examples and inftructions, the application of cir- cles and femicircles in the reprefentations of columns and arcades is
ihewn, particularly when they are inclined to the picture. In a foregoing leffon, Plate XV, Page 122, the manner of obtaining
the reprefentation of a circle was ihewn ; which was done by reticu- lating the geometrical figure by a number of right lines, and after- wards throwing that reticulation into perfpective : which method is very eafily underftood, and is perhaps the befY when the circle to be re-- prefented is large ; but when required of moderate fize, it will be more convenient to employ fewer lines ; at moft, no more than will produce by their interfections eight given points, as in Plate XXIII. Fig. 2 ; which figure is obtained by the following procefs. 1 Firft
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PERSPECTIVE. i$3
Firft draw the circle, No. 2, at full fize j after which inclofe it in SECT.
in.
a # fquare. *— -v-wJ *P ate
At the angles of the fquare draw the diagonals a e, b d, and through XXIn„
the center draw the diameters, 2, 8, and 5, 7, parallel to the fides of the fquare; which diagonals and diameters interfeét the circle in the points 2, 8, 5, 6, and a, b, e, d, producing eight regularly difpofed points in the periphery of the circle. Parallel to either diameter, draw two right lines, that may pafs
through the points, in which the circle is interfeaed by the dia- gonals, as the lines l d, 3 e. It muit be obferved, that it is indifferent to which diameter thefe
lines are parallel; becaufe they are only required to indicate the feat of the points a, b, and e, d, upon one fide of the fquare, as the points l, and 3, which would be j lift as convenient in operation, if they were upon the bafe of the fquare, as upon the fide. Having thus completed the geometrical figure, No. 2, proceed to
the perfpeftive reprefentation at No. 3. The line G G is the bafe or ground line; and the line Ç Ç is the
vaniihing line of the plane in which the circle is fituated, or in other words, it is the horizontal line. V é and V 2 are the vaniihing points of the fides of the fquare ;
and d í 1, d í 2 are the diftances of thofe vaniihing points. V 3 is the vaniihing point for the diagonal of the fquare a e; which may or may not be employed by the artift, as he lhall find convenient. Determine the appearance of the fquare, No. 3, in perfpe&ive;
which is done by applying the meafures equal to thofe of the geome- trical fquare, No. 2, on each fide of the point 4, upon the bafe line; as to 5 and 6; and then determine the fquare as in the former leifons; * Euc. B. iv. Prop. 7,
X by
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154
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SECT, by imes drawn'to. the vanifliing points, and to the diftances of thofe
irr.
Plate
XXIII. After the reprefentation of the fquare is obtained, draw the dia-
gonals a e, b d; which muft be drawn through the points or angles of the fquare. At the point X, where thofe diagonals interfecl; each other, draw
right lines from the vanifliing points V l and V 2 through that inter- feclion, and thofé lines will reprefent diameters of the circle, touch- ing the fquare in the points % T, S, 9. Having thus prepared the reprefentation of the fquare, No. l>,
with the diagonals and diameters; find the points a, b, e, d, in thofe diagonals, which may indicate four points in the periphery of the ellipfis ; which is to be the reprefentation of the circle required. For this purpofe the following procefs muft be employed: At the angle 4 of the reprefentation of the fquare, fet upon the bafe line the fpace 4 i, equal 4 é in the geometrical fquare No. 2; and from 5 to 3 the^fame fpace; and transfer thofe points to the re- prefentation of the fide of the fquare- as at the points é and 3, by the iiflance of the vanifliing point d í é · and from é and 3 draw lines to the vanifliing point V 2, and thofe lines will interfeét the diagonals in the points a, d, b, e; which are the points required. Through thofè four points, and the points 2, f, 8, 9, making eight points, draw the reprefentation of the circle by hand, in the beft manner poflible. It muft be obferved, that the method here given, together with
that of the Example in Plate XV, are the beft adapted for the ufe of the artift: but of' the two th"e latter is the moft elegant. There is alfo a method, which is extremely elegant, given By
Dr. Brook Taylor, in page 24 of the firft edition of his Linear Perfpeclive; but the' diagram, Fig. 22, is not fufficiently clear, nor is the procefs adapted to the painter's ufe. ø
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PERSPECTIVE.
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Of the Shafts of Columns.
Plate XXIII. The foregoing diagrams being clearly underdood, the ftudentmay
proceed to the following leflbn, which ihews the manner of repre- fenting the ihafts of columns when {landing in a row. Fig. l, reprefents the rudiments of two columns; the centers of
which are in a right line; the vaniihing point of which is V 1. The other vaniihing point is V 2; for as the plinth and abacus of
every column are perfect fquares, fo the vaniihing points of their fides arb the vaniihing points of lines at right angles with each other. Therefore Vl is the vaniihing point for the fronts, and V2 the
vaniihing point for all the fides óf the plinths ; and as every abacus is parallel to the plinth, fo V 1 and V 2 are the vaniihing points for the fronts and fides of thofe abaci. The line   is the bafe line.
Ç Ç is the horizontal line.
€ is the center of the picture.
And the eye is marked by the word.
V 3 is the vaniihing point of one of the diagonals of the plinths,
as alfo for one diagonal in each of the abaci. Let ï be fuppofed the feat upon the ground of the neareft angle of
the plinth of the firft column, and let the line X 1, X 2, be confidered as the interferon of the pifture by a plane paffing through the dia- gonal of the plinth and abacus of that column; therefore, it will be the line upon which the height of the column muft be deter- mined. From the point ï draw a right line to V 1, which line will exprefs
the indefinite reprefentation of the fronts of all the plinths that may X 2 be
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156 PERSPECTIVE.
be required; from o draw alfo a right line to V % which will produce
the indefinite reprefentation of the fide of the neareft plinth. ××ÉÐ Then determine the apparent breadths of thofe plinths as follows: From the point d í l, which is the diftance of the vaniihing point V l, draw a right line through o, till it interfecls the bafe line of the
picture, as at 4; and repeat the fame procefs from d í 2, which will give the interfe&ion A. From thofe two points determine the forms of all the plinths that
may be required, as follows : Upon a feparate paper, or remote part of the picture, draw two
circles, the diameters of which iliall be equal to the lower and upper diameters of theihafts of the intended columns, as at No. l and No. 2 ; and circunrfcribe thofe circles each with a fquare, and draw the dia- gonals and ordinates, as, was dcme in the cafe of the circle Plate XXII. page 153. Then take with the compaffes the dimenfions of the lower diame-
ter of the ihaft, equal to the line from 4 to 6 in No. % and apply it upon the bafe line B, Fig. l, from the point 4 to the point 6, and alfo from A to the point Z. From the point 6 draw a right line to the point d í J, which is
the diftance of the vaniihing point V 1; and its interfecüon of the line which is drawn from o to V l, as at n, will give the apparent breadth of the front of the plinth of the foil column. Then from the point Æ draw a right line to the point d í 2, which
is the diftance of the vaniihing point V 2 ; and its interfection with the line that is drawn from the point ï of the plinth to the point V 2, will determine the depth of the plinth of the fame column, as
at k. Then fet up the height of the plinth upon the line X 1, X 2; and
transfer that height by the vaniihing point V 3 of the diagonal, |
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157
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as has been already taught in the example of the fteps, Plate XVI. S EC T.
Pasce 125, and complete the form of the plinth. »-—«^—f ó Plate
Then upon its upper furface find the reprefentation of a circle, the ÷÷éá
diameter of which ihall be equal to the diameter of the loweft part of the ihaft of the required column; which muft be done by the fame procefs which was employed to obtain the reprefentation of the circle Plate XXII.
■JV. B. The diameters are found by means of the points 2 and 6 in
the bafe line; from which points lines are drawn to the points d í l and d í % as before direded; and the ordinates which determine the interfeaions of the circle by the diagonals of the fquare, are found by means of the points é and s in the bafe line, by the fame procefs with the former.
Then determine the height of the neareft column, and alfo the
reprefentation of its abacus; as follows: The line X l, X % as before obferved, is the interfeaion of the
piaure by the diagonal plane, fuppofed to pais through the angles of the plinth and abacus of the neareft column. Therefore, upon that line fet up the whole height of the column, as from × é to X 2 ; and at X 2 draw the right line Ì W, parallel to the horizontal line. Then from the point X 2 draw a right line to the diagonal vaniih-
ino· point V 3 ; and in that, line find the point S, the neareft angle of the abacus of the neareft column. But here it will be neceftary to obferve, particularly to thofe who
do not underftand architeaure, that the perfpeftive arrangement of columns in rows muft be conduaed by the diftance between their centers or axes, rather than between their iliafts; which diftances muft be determined as follows : The points 4 and 6, as alfo u and i6, in the bafe line B, Fig. é,
mark the breadths of the plinths of the two columns; and the points
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PERSPECTIVE,
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points 5 and 15 mark the middle of thofe plinths: and as it is
necefiary to find the middle of the abaci of both columns, the points 5 and 15 in the bafe line B, muil be transferred to the line Ì W, as the points b f, which ftand exactly over the points 5 arid 15. Thefe points may be obtained by drawing lines perpendicular to
the horizon, from the points 5 and 15 in the bafe line, till they in- terfeér. the line Ì W in the points b and f; or they may be obtained by the following procefs : Take with the compafles, the fpace between the points X l and 5 in
the bafe line, and fet it on upon the line Ì W, from the point X % %o b ; and alfo from b, fet on to f the dimenfions, equal to the fpace from 5 to 15 in the bafe line B; which laft meafure is equal to the diftance between the centers or axes of the columns. Then find the reprefentation of the abacus of the neareft column,
as follows: From the plan of the capital No. 1, take with the compaifes half
the dimenfions of either of its fides, as from b to d, or from b to X; and apply that meafure upon the line Ì W, from the point b each way, as to d and P, and from the point Ñ draw a line to the point d í l; and its interferon with the line that is drawn from X 2, to the diagonal vaniihing point V 3, will determine the angle of the abacus of the neareft column, as at S. Then from the point S draw a right line to the vaniihing point V 1, as the line S Ô ; which line may be confidered as the indefinite reprefentation of the upper edges of the fronts of the abaci for all the columns that may be required. Then determine the reprefentation of the abacus of the fecond
column.
At the point f, in the line Ì W, fet off half the breadth of the
abacus each way to a and d; and from a and d draw right lines to the point
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PERSPECTIVE.
point d í 1, which will produce the interfedions Ô and k in the line
S V l; which points determine the apparent breadth of the abacus of the fartheft column. Then determine the fide of each abacus, for which purpofe pro-
ceed as follows : Through the point S, which is the angle of the neareft, draw a right
line from the point d í % which is the diftance of the vaniihing point V % that may interfed the line Ì W, as at the point h; and. from the point k, fet on to m the depth of the abacus, equal to the fpace between d and X in the plan No. 1 ; and from m draw a right line to d í % which will give the interfedion P, in the line S í 2 : confequently, the fpace S P, is the apparent depth of the neareft abacus. After what has been taught in the former part of this fedion, it is
unneceffary to continue inftrudions for obtaining the reprefenta- tion of the abacus of thefecond column; it will be fufficient to ob- ferve, that a right line drawn from the angle Ñ to the ■vaniihing point V 1 will determine the apparent breadth of the fecond abacus, as at F and of as many more as might be required. After having obtained the reprefentations of the plinths and abaci}
find the ellipfes or reprefentations of circles; one upon the upper furface of each plinth, and alfo one on the under furface of each aba- cus. And by the affiftance of thofe ellipfes defcribe the ihafts of the columns, as follows : From the points which reprefent the centers of' the ellipfes in the
plinths, draw right lines to the centers of the ellipfes in the abaci, as the lines A A ; which lines will reprefent the axes of the columns. Then obferve where the ellipfes in the plinths are interfeded by the
diagonals that are leafl inclined to the picture; and from thofé points draw right lines to the points in the abaci, m which the up- per ellipfis is interfeded by the diagonal that is neareft to parallel ' with.
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PERSPECTIVE.
with the picture: and thofe lines which in the example are marked
y y, will be the apparent boundaries of the ihaft; which will have the full eftecl, if ihadowed with ikill. Obferve, that the line A, which indicates the axis of the column,
is of ufe to direcT; the artift in drawing the lines y y, which ex- prefs the outlines of the ihafts; for the latter lines muft incline equally on both fides to the line A, which reprefents the axis of the column. In re-confidering this example, the ftudent muft obferve, that the
chief purpofe of the foregoing operation is to obtain the reprefenta- tion of two circles, in two planes parallel to each other; which planes are the upper furface of the plinth, and the under furface of the aba- cus. Thefe circles reprefent the lower and upper extremes of the lhaft of the column; confequently, both thofe circles, though of dif- ferent dimenfions in their diameters, muft have both their centers in one right line ; which right line is the axis of the ihaft of the co- lumn. And it muft be particularly noticed, that the meafures which determine the lower diameters and the intercolumnations, muft be arranged by firft finding the reprefentations of the feats upon the ground of thofe centers or axes ; which muft ever be at equal dif- tances from each other, both at their lower and upper extremes; while the fpace between each abacus is greater than that between each plinth : therefore, when the point 5, in the bafe line, is deter- mined, the point b, in the line Μ W, muft ftand exactly over it, and thofe two points muft govern all the reft in the difpolition of the columns. It cannot be improper to obferve, that the foregoing inftruótions
direót a procefs which the author confiders as attended with the leaft trouble, and as being the moft convenient for the artift; becaufe it does not require a plan to be drawn out upon the ground plane : a method which is always inconvenient, and frequently impracticable to
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PERSPECTIVE.
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to the painter. It muft alfo be noticed, that the proportions em-
ployed in the example do not accord with thofe given by Vignola and other architects, but are more conformable to the ancient co- lumns of Pasftum, which were purpofely felecled that the example might be more explicit, as the procefs muft ever be the fame, fup- pofmg the height of the column to confift of ten, or only of two diameters. The Tufcan Baft and Capital*.
Plate XXIV,
Fig. 1, No. 2, is for the bafe, when the plinth is difpofed with its
fides parallel, and perpendicular to the piclure. In drawing objects like the given examples, it will be the iliorteft
procefs to draw the geometrical elevations, and alfo the plans of the parts, as in Fig. l, No. l, which reprefents the Tufcan bafe with its plan beneath. Having drawn thefe at their proper dimenfions (which may be
done upon any ffeparate paper) draw upon the plan the diagonals "a d, and b e; after which draw the ordinates or right lines f g, e h, pafling through the points where the diagonals interfeél the circle of the torus, as at 2, 4, 6, 8. Again draw the right lines ï ñ, and q r, which muft pafs through
the fmall circle, in the points where it is interfefiled by the diagonals. Draw the diameters l 5, 3 7, and the lines 2 8 and 4 6, which are tangents to the fmall circle, afcertaining its dimenfions or breadth. • Of the-examples, one is parallel to the pi&ure, and the other oblique or inclined:
and to thofe who have ftudied the former part of this fe&ion, there can be no difficulty in
the 2d figure, for it is evident that the procefs for defcribing the parts is the fame in both
■ figures ; obferving that in the firft the center and diilance of the pifture are the points
employed; whereas in the fecond oblique vanifhing points and their diftances are ufed.
Õ The
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iöa PERSPECTIVE.
S ^n T' ^ie ê'eornetr'ca^ elevation of the bafe, together with it» plan, beins*
*-~-v>«~> thus prepared, let the line a b d, Fig. i, No. % be confidered as the k i3tC
XXIV. bafe line, C the center of the picture, and V the diftance.—d b is the
breadth of the plinth, which being always fquare, fet off the fame breadth from b to a, and determine the perfpecrive appearance of it* form, by the rules given in the fecond Section. Having completed the plinth, upon its upper furface draw the
diagonals, as from the angle 8 to 9, and from 10 to 7. Then interfecl thofe diagonals by lines correfponding with the
lines f g, e h, in the plan Fig. l, No. l; which muft he done as follows: From the angles 10 and 8 of the plinth, Fig. 1, No. £, mark the
points 2 and 4, equal to the fpacesfrom b to f and a to e', in Fig. l, No. 1; and from the points 2 and 4 draw lines to the center of the jpióture C; and thofe lines will interfeet the diagonal lines in the points §, k, and m, n. Divide the upper line of the plinth at the point a in half, and
perfpecfively on the receding fide at b, and draw the diameters b d and a e. By which procefs eight points will be obtained upon the furface of the plinth, which will ferve as guides for the delineation ©f the ellipfis, which includes the general form of the perfpecrive ap- pearance of the feat of the Torus.. After, having drawn the lower ellipfis, proceed to form the upper
©ne, which conftitutes the lower part of the ihaftof the column. It muft be obferved, that in this example the torus- and the liftel:
united, are equal in their heights to the height of the plinth ; con- fequently, they may be contained,in.a fquare of equal thicknefs with that which forms the plinth. Therefore draw the upper fquare perfpe&ively, as exprefled, by
^ Xili*"
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PERSPECTIVE.
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the upper lines w, x, y, æ; and upon its upper furface defcribe the
ellipfis, which forms the lower part of the fliaft of the column. It is not neceifary to give further infiru6tions for finding the
upper ellipfis,* as it is fufficientto obferve, that the fame procefs, which was employed to defcribe the lower, muft be repeated to produce the upper; and all that can be done by the rules of perfpeétive, is to find as many ellipfes as will aifift the artift in drawing the members of the bafe by hand, whether their numbers be equal, or more than are contained in the given example. It muft be obferved, that when the upper ellipfis is determined, the lines which mark the fliaft of the column, as the lines A and B, muft be drawn perpendicular to the horizontal line, from the points k, m, in the diagonal line W Z. See obfervations in the Addenda.f Fig. % No. 2a is an example for the Tufcan capital, inclined ta
the piclure. Fig. 2, No. l, is the geometrical elevation and plan of the capital,
which, as in the foregoing example, muft be fo drawn before the per- fpeétive reprefentation can be attempted. In the plan Fig. 2, No. l, the outer circle is the extreme of the
ovalo, and the inner circle G % the dimenfions of the fliaft at the neck of the column, • The letters o andq in the bafe iine Fig. é,Íï. 2, correfpond with the letters ï, q: ia
the bafe of the plan, they mark the dimenfions of the upper ellipfis. f The examples here given are of the Tufcan bafe and capital, which are compofed of
few members, abftrafted from the plinth, while the other orders are compofed of many : but as all their members are difpofed parallel to each other, they can only be determined by the reprefentations of a greater number of parallel circles, the largeit of which, upon the upper furface of the plinth, Ihould reprefent the extreme dimenfions of the torus, and the fmalleft, the thickeft and loweit part of the fliaft. In the capital, the largeft circle ihould reprefent the upper part of the ovalo, and the leaft circle the higheft and fmalleft part of the lhaft, juft above the aftragaL J The lines for the liftel and the aftragal arc omitted, both in the plan Fig. 2, No. ô9
and alfo in Fig. 2, No. 2, to prevent a confufion of lines in the example. Õ 2 ' The
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PERSPECTIVE.
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The geometrical plan and elevation being drawn, let the line V I,
V 2, Fig. 2, No. 2, be the horizontal line, and the points Vi, V% the vaniihing points for the inclined fides of the abacus, and (hi, d í % the diftances of thofe vaniihing points.. V s is the vaniihing' point for the diagonal. The point A is the angle of the abacus, touching the picture \ there-
fore at A draw the right line ABC parallel to the horizontal line. From A draw lines to the vaniihing points V é, V 2, which will be
the indefinite reprefeiitations of the upper edges of the abacas. Upon the line  C fet off from the point A the lengths  and C of the
inclined faces of the abacus, equal BC in Fig. 2, No. 2; and draw lines; tod V é and d V 2, the diftances ©f the vaniihing points; which lines- will produce the interfeétions a, b, the upper angles of the abacus. The thicknefs of the abacus is· AD,.equal BD in Fig. 2, No. 2;
from which dimenfions complete the repreientation of the abacus, by* the methods which are given in the preceding inftructions. Then proceed to find the reprefentation of the circle, which ex-
preifes the upper part of the ovalo, upon the abacus, Obferve,, that the right lines l 4, 2 3,. Fig. 2, Íá i, are ordi-
nates palling through the points, where the circle of the ovalo is. interfered by the diagonals, and ï ñ, r s are the diameters marking, the extent of the ovalo upon the abacus. Therefore, upon the line BAG, Fig. 2, No. 2, fet the meafures of
the fpaces l, o, 2, from A, either way equal l, o, 2, in the lower line of the plan, Fig. 2, No. %; at the points D, E, and F, draw the dia- gonals of the under furface of the abacus, as the lines-D G and E F.. From the points 1, 0, 2, Fig. 2, No. 2, draw lines to the diftance
of the vaniiiiing point d.V l, and they will interfeót the abacus at the points k, in, n; which points, muft be transferred to the lower line of the abacus; and from thofé points draw lines to the vaniihing point V 2, which lines will represent two ordinates interfering the diagonals
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PERSPECTIVE.
diagonals and alfo the edges of the abacus, producing points through
which the ellipfis muft be drawn, which ferves as a guide for the re- prefentation of the upper part or periphery of the ovalo, which muft be drawn by hand. After having obtained the reprefentation of the large or upper
circle which reprefents the ovalo, find the reprefentation of the fniall one, which is equal to the diameter of the neck of the column. For this purpofe obferve, that the lines 5 6 and 7 8 in the plan
Fi°\ 2, No. 1, mark the dimenfions of the full extent of the fniall circle; and the dotted lines X, X, are the * ordinates which mark the points in which the diagonals are interfected by the fame circle,, which, as before obferved, is the upper diameter of the ihaft. Therefore, below the fquare of the abacus A D, Fig. 2, No. 2, con-
ftruól the reprefentation of another fquare ρ q s v, equal in thicknefs to the height of the ovalo and liftel in the geometrical capital Fig. 2, No. 1; and upon the under part of that fquare produce the reprefen- tation F of the circle, which indicates the upper diameter of the ihaft; which reprefentation muft be obtained by the fame methods which were employed for the delineation of the greater ellipfjs or upper circular appearance of the ovalo. When thefe two ellipfes are obtained, they muft be confidered as the
boundaries of the ovalo and liftel of the capital, and confequently the guides to direét the artift in completing the delineation of the capital, which muft be done by hand. After the foregoing inftruftions have been confidered, it will be
evident to the painter, that no methods can be employed by which thebafes and capitals of columns may be delineated with mathematical • In the line A B, Fig. 2, No. 2, the points 6, 7, o, correfpond with the points 6, 7, ©>
in the geometrical plan Fig. 2, No. 1 ; which points are transferred from the line A C, by the point d V 2, fo as to produce the ordinates- for the conftru&ien of the ellipfis F. certaintyj
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PERSPECTIVE.
certainty; particularly tliofe of the Ionic, Corinthian, and Com-
pofite Orders: therefore, they who are engaged in that line of art, in which the reprefentations of fuch objects are required, will do well to provide themfelves with fmall models of fuch parts of the principal orders, about four or fix inches diameter, and work from them. Yet the ftudent fliould confider and copy the examples here given;
for by fo doing he will acquire the principles which muft guide him in the reprefentation of fuch objects, which are fo intricate in their forms, that, although they cannot be decidedly drawn by the rules of perfpeclive, yet they can never be elegantly delineated by thofe who poffefs not a competent ikill in that fcience. Left it mould be objeéted, that the foregoing inftruaions are too
concife, it muft be obferved, that they ought not to be ftudied with- out a previous acquaintance with the foregoing parts of this Section, as well as of the fcience of architecture itfelf: for although the prin- ciples of conftruótion be as Ample as any problem in the fcience ; yet the figures are compofed of fo many repetitions of fimilar forms, that to offer more explicit examples than are here given, would produce fuch confufion as to render them unintelligible ; and what are given will be better underftood by attending to the following obferva- tions. All the members of theTufcan and Doric capitals, as alfo their bafes,
together with the bafes of the other orders, are compofed of circles : therefore, all that can be done is to obtain the perfpe&ive reprefenta- tions of as many fquares as may contain ellipfes expreffive of the dif- ferent members required; which fquares ihould be difpofed parallel or obliquely to the picture, as may be neceffary; each of thofe fquares jnuft then be divided by two diagonals, two diameters, and two ordi- nates,
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PERSPECTIVE.
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16?
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nates, the latter interfecring the diagonals in the fame points in which SECT.
Hi.
the diagonals are interfeéted by the circle; by which procefs eight v-v—* . Plats
points are obtained, through which an ellipfis is drawn by hand, part χχ,ιγ,
of which will reprefent the line of moulding required. For the reprefentations of the capitals of the other orders, as of the
Ionic, the Corinthian, and Compofite, the author forbears to attempt any examples ; becaufe their forms are fo extremely complex, that no intelligible or fatisfaclory inftrucuons can be given upon fuch intricate fubjeéts. This is acknowledged by the fenior Malton in his firft volume, who yet lias given examples of the Ionic and Corinthian capitals in his XXII Plate: but as if refolved to render difficulties totally unintelligible, he has fo entangled the lines of conftruóiion,; that no patience can unravel the web of his diagrams ; confequently, no artift can receive much, if any affiftance from his inftruclions upon this fubjeét The only procefs that can afford any affiftance to the artift, is to
defcribe a fquare in perfpeélive, that may contain the abacus, and through the angles of fuch fquare to draw the diagonals that ihall direct the pofition of the volutes. Then below the abacus find the reprefentations of two circles, that may ferve as guides for the difpo- fition of the leaves or acanthi, if the capital be Corinthian or Compofite: and this is the moft that can be done by way of correal; arrangement; the reft muil depend upon the eye and hand of the painter, who ihould procure models of fuch capitals, and difpofe them in proper fituations, efpecially if the work be large in which the reprefentations of fuch ornaments are required. As it would be improper to give finiihed examples of the objecls,
entangled with all the lines that are neceffary for their conftruclion, the finiihed reprefentations are given at the commencement of this workj
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PERSPECTIVE.
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work, drawn to the fame dirnenfions with thole given in Plate XXIV ;
the bafe, as before obferved, having the plinth parallel, and the capital having the abacus inclined to the picture. Of Arches which are in Planes inclined to the Picture.
Plate XXV.
The reprefentations of arches which are inclined to the picture are
eafily obtained, by thofe who underftand the manner of reprefenting the circle, for the procefs is nearly the fame in both cafes, but with this additional circumftance, that it is neceffary to reprefent the thick- nefs or foffit of the arch; and therefore the exterior proportions by which the femi-ellipfis, that reprefents the arch in the front, are ob- tained, muft be transferred to the interior of the piers, thereby to obtain the reprefentation of that femicircle which lies within, and determines the thicknefs of the arch. Let Fig. l, Plate XXV. reprefent a fmall building, fitted to contain
a garden feat, with an arch in front; which front is inclined to the picture. Ç Ç is the horizontal line.
C is the center of the picture.
G G is the bafe line.
V 1 and V 2, are the vaniihing points for the front and fides of the
building ; and d í 2 is the diftance of the vaniihing point V 2. V 3 is the vaniihing point of the diagonal or mitre line.
Ô is the angle of the ftep or plinth, which touches or interfecls
the picture. Although the general form of this example differs but little from
many of the foregoing, yet part of the procefs for obtaining the prin- /cipal lines ihall be repeated. The
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PERSPECTIVE.
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The point T, as before obferved, is the interferon of the ftep with
the piclure : therefore, at the point Ô draw the line Ô Ñ perpendi- cular to the bafe line G G ; and upon that line fet up the height of the building, as from Ô to d; and from T, upon the fame line, the height of the ftep, as at n. At the point ç draw the line ç S parallel to the horizontal line. Then the line S n, may be confidered as the bafe line, becaufe it is
in the plane of the horizontal furface of the ftep or plinth : therefore, all the meafures for the width of the building, together with that of the plinth, may be applied upon that line. Having completed the reprefentation of the ftep or plinth R,
which müft be done by the method already taught in the conftruo tion of fteps, page 124, proceed to the reprefentation of the body of the building; for which purpofe determine the point o, which is the loweft angle of the neareft pier, upon the ftep. This point is obtained by drawing a line from the angle ç of the
ftep, to the point V 3, which is the diagonal vaniihing point; and then cutting off the portion ç ï, by the fame procefs that was em- ployed to obtain the point S, the neareft angle of the reprefentation of the parallelogram, Fig. 2, Plate XI. Having obtained the point 0, draw a line from the point d V 2
through 0, till it interfeéts the line S f, as at Ñ; and from Ñ fet on to f the meafure of the width of the front of the building, and from f re j turn a line to the point d V 2, which will interfeót the front in m ; confequently the fpace m o is the width required. To thofe who have confidered the inftruclions contained in the
foregoing part of this Section, it will be unneceffary to continue a repetition of what has been already taught; therefore let it be fup- pofed that the general form of the objeót is completed, except the arch, for the reprefentation of which proceed as follows : Æ Upon
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PERSPECTIVE.
Upon· any part of the paper, or upon a detached piece, draw a
femicircle, whofe diameter is equal to the chord of the required arch^ as at No. I, and circumfcribe that femicircle with the parallelogram Then draw the diagonals 3 X and 6 y> and through the centec
W, draw the diameters a e and e 0. Through the points in which the diagonal lines cut the femicircle,
as b and d, as. alfo through the center W, draw right lines parallel to; the longeft fides of the parallelogram, as the lines b d and a e. Then through the points in which the femicircle is interfered by
the longeft diameter a e, draw the right lines a 4 and e 5, parallel to the fliorteft fides of the parallelogram; and the lines fo drawn wil interfeót the fides of the parallelogram in the points 12 4 5-; which points, when transferred to Fig. 1, will be the guides for the delineation of the perspective reprefentation of the arch. To complete which, proceed as follows : At the point d, Fig. l, which, as before obferved, marks the ex-
treme height of the arch at the interfection of the diagonal of the building with the picture, draw the line d L parallel to the horizon- tal line ; which line muft be confidered as the interfeetion of the plane of the top of the building with the piciure. The-line 0 3,. is the angle of the building, the apparent height of
which is found by drawing a right line from the point d to the dia- gonal vaniihing point V s, which gives the interfeclion 3 ; from which point a line drawn to the vaniihing point V % produces the indefinite reprefentation of the top of the front. The lines which reprefent the apparent breadths, of the piers in
Fi°·. 1, when continued upwards, interfect the line 3 y in the points \ 1 ;·' confequently the fpace h 1 is the reprefentation of a line inclined to
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PERSPECTIVE.
to the piaure, equal the line 3 6 in No. 1, which line is equal the
chord of the arch, the reprefentation of which is required. Therefore, through the points h, 1, draw lines from the point d V %
which may interfeft the line d L in the points 6 and 7. Then with compaffes take the fpace from 3 to 4, in the line 3 6
No. 1, and fet that meafure from 6 to 4, and from 7 to 5, in the line L d, Fig. 1 ; and from the points 4 and 5 draw lines to the point d V 2, and they will interfea the upper line of the building in the points i and k ; draw alfo a line from the point L to the point d V 2 ; which will give the interfection g, the middle of the crown of the arch. Then determine the height of the arch as follows :
Take with the compaifes the radius or femidiameter of the arch; or,
which is the fame thing, the meafure of the Ihorteft fides of the pa- rallelogram No. l, and fet it downwards on the line d T, Fig.. 1, from the point d to a ; and divide that fpace into as many fmall parts as there are divifions in the line 3 y, /No. é; that is, transfer the fpaces marked 3, % l, in No. l, to the line d Ô in Fig. l, as from d to e, equal 3 %, and from e to b, equal 21 ; and from the points a, b, e, draw lines to the diagonal vaniihing point V 3; and they will intevfect the ano'le of the building in the points x, 1, 1; and from thofe points draw lines to the point V 2, the vaniihing point of the front of the building. Then from the points i, k, draw lines parallel to the vertical angles
of the building, or, which is the fame in effect, perpendicular to the horizontal line; and thofe lines will interfeö; the line é 8 in the points t and u; which two points are in the femi-ellipiis that forms the reprefentation of the arch. At the points h and 1 draw the diagonals hz, and J r; and the
interfeéfions of thofe lines with the line 2 9 will give two additional points for the delineation of the arch; which, together with the Æ 2 points
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PERSPECTIVE.
points g r and æ produce feven points; through which a curve line
drawn by hand will be the reprefentation of the outer line of the arch. It next follows to determine the thicknefs or foffit of the arch;
which requiring many lines, is given at greater magnitude in Fig. 2, and No. 2, the better to explain the procefe. Example, Fig. 2, in the fame Plate,
Of the Method by which the Soffit or thicknefs of an Arch is-
determined.
The center of the piaure, together with the vaniïhing points and
their diftances, are the fame with thofe employed in the foregoing Example. The geometrical figure, No. 2, ihews the arch at full fize, which
being inclofed in a parallelogram, the diagonals m and ç are drawn, together with the diameters é 8 and e s, as was done in No. é of the foregoing Example. The diagonals interfe6t the femicircle in the points b and d, and
the diameters in the points a, e, and e. Through the points b, d, right lines are drawn parallel to the longeft
fides of the parallelogram, as the lines l 8 and 27; and through the points a and e lines are drawn parallel to the ihorteft fides of the parallelogram, as the lines 5 f and 6 g. It mu.il- be obferved, that thofe lines are drawn for the purpofe of
markino· the projections or feats of the points a, b, e, d, e, in the femicircle, upon the different fides of the parallelogram which con- tains the femicircle. Fig. 2 is the perfpeétive reprefentation of No. 2.
The line o 3 being fimilar to the line h r in Fig. l, muil be con»
ôé iidered |
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PERSPECTIVE.
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173
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fidered as the inner line in the front of the neareft pier continued SECT,
upwards, the height of which is equal the femidiameter of the arch, t_,-v-„_j the reprefentation of which is required. XXV; At the point 3 draw the line A, parallel to the horizontal line, and
from the point 3 fet off to 4 the meafure of the chord or breadth of the arch, equal the line 3 4, in No. Q. From the points o and 3 draw lines to the vaniiliing point V 2,
and from the point 4 in the line A draw a line to d V £, which is the diftance of the vaniihing point V 2; and the interferon of the latter line with the line that is drawn from 3 to V 2, as at the point a,, will determine the breadth of the arch. At the point a draw the line a W parallel to the line 3 0, and the
trapezium, which muft contain the reprefentation of the arch, will be completed ; the angles of which are 3 O a W. It is unneceffary to continue the inftruétions for the reprefentation
of the front of the arch, the procefs being the fame with that before employed in Fig. 1, and which will be eafily underilood by com- paring the latter Example with the former; the only difference to be obferved is, that the arch in the firft Example is diftant beyond the picture; whereas that in the fecond Example has the angle o 3 touch- ing the piclure; which circumftance occafions a flight variety in the operation, which hereafter will be noticed. In the mean time it will be proper to proceed to the conftruclion of the thicknefs or foffit of the arch, which is determined by the following procefs : From the point 3 draw a right line to the vaniiliing point V 1, as
the line 3 9. Then continue the line A, which is drawn parallel to the horizon-
tal line, from the point 3 towards B, as far as may be thought neceifary. Then from the point s, which marks the height of the arch, fet
off
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174 PERSPECTIVE.
SECT, off by the compaffes the meafure of the foffit or depth of the arch',
>—v-~~/ as from the point 3 to D; and from D draw a line to the point Plate
XXV. d V l, the diftance of the vaniihing point V i, which will interfect
the line that is drawn from 3 to V ι in the point 9 : then is the line 3 9 equal, the apparent thicknefs of the foffit. At the point o draw a line to the vaniiliing point V l, and at the
point 9 draw the line 9 X parallel to the line 3 0. Then from the point 9 X draw lines to the vaniiliing point V 2,
and from the points a and W draw lines to the vaniihing point V 1, which will produce the reprefentation of a fecond trapezium, which muft inclofe the femi-ellipfis, reprefenting the inner outline, which determines the breadth or foffit of the arch. The angles of the inner trapezium are at the points 8, 9, y, x;
, therefore, through thofe points draw the diagonals 8 X and 9 y. Then transfer the points which are in the top and neareft fide of
the firft or front trapezium, to the top and neareft fide of the fecond or inner one.—Thus, from the points 1 and 2 draw lines to the vaniili- ing points V 1, which will interfeót the neareft fide of the inner tra- pezium, in the points 10 and 12 ; and from the points e S b in the top of the neareft, draw lines to V 1, which will interfeól the top of the inner trapezium in the points f and k.—The third interfeclion, which is near the point 8, is not marked, to prevent confuiion. From the points 10 and 12 draw lines to the vaniiliing point V 2,
and at the points f and 8 draw lines parallel to the line 9 X, which is the fide of the remote trapezium; and the mutual interfeétions of thofe lines will produce the points by which the femi-ellipfis muft be drawn, that will exprefs the depth or inner edge of the foffit of the arch. As it is not pofiible to mark or figure every point neceffary to indi-
cate the curvature of both the femi-elliptical lines, by which the arch is reprefented, without producing confufion; the points neceflary for
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PERSPECTIVE. *?5
for the inner one only are marked ; as at Ñ, 13, k, h, g, X, and y, in S;EC Ô
Fio·. é: and in the Example Fig. l, the outer femi-ellipfis only is *~-v->. ° é «J ^ pjate
marked by the points neceffary for its conftru&ion, as the points XXV.
r, Ñ, u, g.
As it impoffible to avoid the appearance of intricacy and labour in
the conftruclion of the Examples, the ftudent will do well to con- sider them attentively, fo that he may perfectly comprehend the principles of their conftruaion and thereby be enabled to employ the leadino- lines when required, without regard to the minuter divi- fions, which may be omitted. By the preceding mftruótions, together with the Examples of Plate
XXIII. are ihewn the methods for obtaining the reprefentations of circles and femicircles, whether the planes in which they are feated are inclined or perpendicular to the picture; and the whole of the pro- cefs confifts in determining the perfpeótive reprefentations of certain given points in the original circle or femicircle, and then drawing the required reprefentation through thofe acquired points. The fame metbod muit alfo be employed to obtain the reprefenta-
tions of all other kinds of arches, as the Gothic, the Elliptic, and the Catinarian; as alfo for the reprefentations öf all objects compofed of curved lines, fuch as femicircular. and compound pediments, with every other variety that can be required in curvilinear figures : and although thefe rules cannot enfure juft or pleafing reprefentations, without affiftance from the hand of a ikilful artift; yet if thefe gene- ral hints are not underftood and applied in a certain degree, no one muft hope to produce the reprefentations of curved lines in architec- ture, but what will be diftorted and unpleafant to the eye, |
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PERSPECTIVE;
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Of Mouldings inclined to the Picture.
In Plate XXVI. is an example of a pedeftal #with its cornice?
which is inclined to the picture. C is the center of the picture. Ç is the horizontal line. G is the ground line or bafe line. The eye or diftance is marked Eye. Ñ Ñ is the parallel of the piclure. V l and V 2 are the vaniiliing points, found as before directed ;
And d í l, d í 2, are the diftances of thofe vatiiihing points. V 3 is the vanifhing point of the diagonal of the dado of the
pedeftal; confequently, it is the vaniiliing point for the mitres of the mouldings, and d V 3 is the diftance of that vaniiliing point. The lines f, at the bottom, and f i, at the top of the pedeftal, are both
parts of the fame line, which is the interfeclion of the piclure by the angle of the loweft ftep and cornice; or in other words, it is the line in which thofe parts touch the piclure. Firft determine the reprefentation of the fteps and dado of the
pedeftal, as follows: X is the feat upon the ground of the angle of the loweft ftep ; therefore, fet up from that point the height of the firft ftep to e, and complete the general form of the whole ftep, by draw- ing lines to the vanilhing points V l and V 2. At the point e draw the line'e g parallel to the ground line.
Then from e to g fet on the depth of the mitre or diagonal of the
• To this, as alfo to fome of the following Examples, there is no annexed fcale ; be-
caufe the mouldings are drawn at their given fize upon the Plate: therefore, the fteps and dado are drawn in proportion to thofe mouldings. The diftance of the piilure is rather too ihort in this Example, which is fo difpofed, the
better to demonftrate the procefs. ftep
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PERSPECTIVE,
Hep *, and from g draw a right line to d V 3 (the diftance of the
diagonal vaniiliing point) and the interferon at h will give the feat of the angle of tne fecond ftep. Complete the fecond ftep by the methods given in Example Plate
XVI. page 126, as alfo the dado of the pedeftal; which dado is ter- minated at the top in the points a, b, e, d. Through thofe points drawthe diagonals marked Diagonal é and d 2,
Obferve, that a, is the apparent height of the dado of the pedeftalt
including the fteps and cornice'; which height is obtained by fetting up the real meafurefrom X in the bafe line to 1 upon the line f, and from 1 drawing a line to V 5, the vaniihing point of the diagonal, which line interfeéts the angle of the dado at a, and gives the required height. After having thus prepared the pedeftal, proceed to the cornice, as
follows : Upon any remote part of the drawing, or upon any feparate paper,
draw the geometrical profile of the mouldings, at their true dimen- fions, by afcale, as at Fig. l; above which draw the fquare X 1 8 7, each fide equal to the projection of the upper line 7 8 of the cornice : at the point 7 draw the line 7 Ì é, which will be the diagonal or mitre line of the cornice. Parallel to the line e 7, which is marked dado line (and correfponds
with the line fo marked in Fig. 2.) draw lines from every projection of the members, as from g, h, l, k, m, that may interfeét. the mitre line Ì in the points 2, 3, 4, 5, 6; which points will exprefs the pro- jections of the various members at their joints or mitres. Having thus prepared the geometrical profile, Fig. l, which is the
guide for the perfpeétive reprefentation, proceed as follows : At the point l, Fig. 2, which projects equally with the lower ftep,
• That is, the fpace between the upper angle of the firft ftep and the lower angle of the
fecond ftep; the former marked e, the latter h. A a draw
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PERSPECTIVE.
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draw the line é Ì parallel to the horizontal line, upon which fet oa
from the point é the following fpaces, l, 2, 3, 4, 5, 6, 7, equal to the fimilar points in the mitre line Ì of the geometrical profile Fig. é ; then from thofe points draw lines to the diilance of the vaniihing* point d V 3 ; which lines will interfecT; the fpace upon the diagonal line that lies between the points é and a; which interfecTions mark the projections of the different mouldings, as at the points k, m, n, a. Then upon the line f 1, Fig. 2, fet downwards the heights of all:
thofe members, as at the points b, f, e, d, e, equal the fame points in the dado line of the geometrical profile Fig. i. From the points a 2, b, e, d, e, draw lines to the diagonal vaniih-
ing point V S-. Then from the points k, in, n, a, in the diagonal line, draw lines
perpendicular to the horizon, which will interfecT the lines that are drawn from the points b, f, e, d, e, to the diagonal vaniihing point V 3 in the points ; which points determine the angles of the
profiles of the mouldings, the curved parts of which muil be drawn by hand, and the right lined parts by the ruler. From the points o, p, q, draw lines to the vaniihing points V l}
V 2, which will determine the appearances of their different faces.
Then find their terminations or projections, at the angles S and T, as follows :
Transfer the points k, m, n, a, to the diagonal line S T, by lines
drawn from all the points marked k, m, n, a, to the vaniihing points V 1, V 2, till they interfecT; the diagonal line S Ô in the points b,
u, w; from which points draw lines perpendicular to the horizon, that may interfecT the lines that tend to the vaniihing points, as at m> P, and n, u; which points are the prqjecTions of the different members at thofe mitre angles, and which muft be completed by the fame pro- cefs employed at the angle Ï 1. In examining this figure it will eaiily be feen, that the projecTion
or
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PERSPECTIVE. ^
©f the mouldings are firft marked upon the jnitre line, by trans- S EC T.
ferring their meafures, which are marked upon the line M, by v-~v--^ means of the point d V S, which is the diftance of the mitre line. XXVI. Itmuft be obferved, that in this diagram, as in the others, the occult
or dotted lines are no more than acceflary lines, «which, though ab- folutely neceifary for the production of the figure, are ufelefs after it is finiihed. Of thefe fome are wanting. They are omitted to avoid confufion: but the ftudent will eafily comprehend the whole, if he carefully examines thofe which are given, and above all compar- ing them with fome real mouldings, which he would do well to pro- cure for the purpofe; for, without being well acquainted with their real conftruétion, it is impoffible to delineate them with accuracy. It is needlefs to obferve, that if mouldings were required at the
bafe of the pedeftal, they muft be produced by the foregoing procefs reverfed; which will eafily be underftood by turning up the example, and fuppofing a b to be the loweft line of the bafe of the dado. In clofing this Section it will be proper to obferve, that it contains
a feleclion of Examples, which the author confiders as being the moil ufeful to the artift, at the fame time that they illuftrate the moil, ele- gant and fimple principles of the fcience, as far as relates to objecÏs which have their fides difpofed obliquely or inclined to the picture. But the figures, which are particularly recommended to the con-
sideration of the ftudent, are the following : The fquare and parallelogram, Plate XI. the reprefentations of the
houfes, Plate XII. and XIII. the table, Plate XIV. together with the chair, and pentagon, Plates XIX. and XXII. all of which are of the utmoft confequence to the artift who wiihes to attain perfection in his defigns and pictures : therefore, thofe Examples are recommended to his particular attention and confideration. To the foregoing obfervations it is neceifary to add the following
caution.
A a 2 Let
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PERSPECTIVE.
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Let it ever be remembered that the reprefentations of thofe horizon*
tal lines, which produce the appearances of the fides or faces of right lined figures, are conftantly drawn to the vaniihkig points; but the meafures, or proportionate divifions of thofe lines are determined by drawing lines from the points which mark the real meafures on the bafe line, to the diftances of thofe vanifiiing points, to which the reprefentative lines are drawn. Thus, in the parallelogram Plate XL Fig. 2, Page 100, the line S 8, which expreifes the indefinite repre- fentation of the fide D, is drawn to the vaniiliing point V 2; but the length or required portion of that line is determined by drawing a line from the point 6 in the bafe line, to the point d V 2 in the hori- zontal line, which gives the interfeótion 8 ; after which the fide A is formed by drawing a line from the point 8 to the point V i, which is the vaniiliing point of lines, the originals of which are at right angles with thofe reprefented by lines drawn to the vaniiliing point V 2. The above caution is the more earneftly recommended to the ftu-
dent, becaufe the author has found by long experience, that the young practitioner is more apt to forget, or miftake this part of the procefs, than any other in the operation. It cannot be improper to obferve, by way of eonclufion, that as
the old writers on the fcience of Perfpective knew nothing of oblique vaniiliing points, fo were they confequently ignorant of their dif- tances : from which circumftance it followed, that they never could proportion the reprefentations of objects inclined to the pióhire by juft meafures: which defect may 'be feen in the Jefuit's Perfpective, in which the reprefentation of a houfe inclined to the picture, given in the upper Example of page 111 is falfe, owing to the want of thofe points. End of the third Section.
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• PERSPECTIVE.
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181
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SECTION THE FOURTH.
Containing Examples, the Lines and Planes of which are, inclined
both to the Picture and Horizon. É Í this Seótion inftructions are given for the reprefentations of
objects, the component lines or planes of which are inclined both to the picture, and to the horizon. Before the ftudent proceeds to in veftigate the fubject of this Sec-
tion, it will not be improper for him to re-confider the principles laid down in the fecorid and third preceding Sections. The fecond treats of objects, the lines and planes of which are
parallel, and perpendicular to the picture. The third, of objects the lines and planes of which are inclined to
the picture. , · And then the following, or fourth Section, which treats of objects
eompofed of lines and planes which are inclined to the picture, and alfo to the horizon. ·, i : Ë . ' : : .· By confidering the principles contained in this and the foregoing
Sections, the ftudent will eafily perceive, that lines and planes may be difpofed in:all the following,directions ; ... Firft, Parallel, - - .-*,_ . , , Secondly, Perpendicular, i to the picture ;.
Thirdly, Inclined, J
Fourthly, Inclined, both to the picture, and to the horizon.
The
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PERSPECTIVE.
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Ιβί
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SECT. Thefe are all the varieties in which lines oPplanes can be difpofed
IV. ir-y-^yj to the picture, or in other words, to the eye of the fpeélator.
To familiarize thefe diftinctions, let it be fuppofed, 'that "a houfe,
whofe plan is * oblong or fquare, be fo difpofed to the eye of the fpeétator, that the loweft line of the bafe and the upper line of the blocking courfe of one of its fides, appear perfectly parallel to each other; then is that front parallel to the piclure; confequently, the fides are perpendicular to it, combining the firft and fecond pofition of lines and planes. Or, let it be fuppofed that the fame building be feen by the fpec-
tator from fuch a ilation, that the lower lines of the bafe, and the tipper lines of the blocking courfe, on both the fides, appear to ap- proach each other: then are the fides of that building inclined to the picture ; which is the third pofition of lines and planes. Again, fuppofe the fame building, as feen in the foregoing fitua-
tion, be covered with a Hoping or inclined roof; then will the lines and planes of fuch roof be inclined, not only to the piéture, but alfo to the horizon. Example, Plate III. Fig. l and 2.
PLATE ^n ^ig· *s tfte fr°nts Β ar,<* E °f ^e buildings are parallel to the
picture, and the fides A and G are perpendicular to it, uniting the firft and fecond pofitions of lines and planes. In Fig. 2, the fides X l, X 2, are inclined to the piéture, and the
roof R 2 is inclined to the horizon ; confequently, the line R 2 of the roof t has a double inclination, being inclined both to the pic- ture, and alfo to the horizon. * An oblong hath all its angles, right angles, but has not all its fides equal.
Simpfon, Defin. 31ft.
la
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PERSPECTIVE.
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In Fig. l, the building A B has a roof, the planes of which are
all inclined to the horizon: but it muft obferved, that two of thofe planes, although inclined to the horizon, are yet perpendicular to the picture, as the plane R, and its oppofite; while the front plane T, and its oppofite are inclined to the picture as well as to the horizon. As the inftructions contained in this Section cannot be intelligible
to thofe who do not underftand the nature and conftruotion of va- niihing lines, it will be neceffary to beftow particular attention on this abftrufe part of the fcience, and to confider the fubject in a more theoretic view, than has been hitherto employed in this Treatife. By the general conftruction of objects of art, the lines and planes
of which they are compofed are feldom inclined to the horizon, though they are frequently fo fituated to the picture ; therefore it is feldom neceffary to employ any other vaniihing line than the hori- zontal line. But when the line or plane to be reprefented is inclined to the
horizon as well as to the piéture, then other vaniihing points, not in the horizontal line, muft be found; and it muft be remembered, that in moil cafes the vaniihing line of an inclined plane cannot be determined, unlefs the vaniihing points of two right lines in that plane are fifft found. Example»
In Plate III. Fig. 2, the houfe Τ has its fides or planes inclined to
the picture, while the roof is inclined to the horizon. The tops and bottoms of the windows and doors are horizontal;
therefore they vaniih in the points V 1 and V 2, which points are in the horizontal line, or in what is called the vaniihing line of horizon- tal planes. But
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I
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PERSPECTIVE.
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18
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But the lines of the roof in the fide X % which are inclined to the
horizon, vaniih in the point R ; which point is in the vaniiliing line of the plane or fide X 2 of the building. When a vertical plane is inclined to the picture, there is no great
difficulty in finding its vaniiliing line; but when an original plane is inclined both to the picture and to the horizon, the procefs for de- termining its vaniihing line is more intricate and abftrufe. In the firft cafe, nothing more is neceffary than to find the vanijh-
ing point of any horizontal line in the inclined plane, and through that point draw a right line perpendicular to the horizontal line; which will be the vaniihing line fought *. |
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Example, Plate III. Fig. 2.
V 2 is the vaniiliing point of all horizontal lines in the vertical
plane or fide of the houfe X 2, found by the methods taught in the preceding. Section; therefore, through V 2 draw the line V 2 R, perpendicular to the horizontal line H, and it will be the vaniiliing line fought, and it is fo marked : vaniihing line of fide X 2. Again, in Plate XXVII. Fig. 1, C is the center of the picture,
XXVII. and the vaniiliing point of all the horizontal lines in the planes or
fronts of the houfes A, B; therefore, through the point C draw the
right line V l, V 2; which will be the vaniihing line of the fronts
A and B.
In Fig. 2, of the fame Plate, V 2 is the vaniihing point of all
horizontal lines in the fronts of the houfes A and Β; which fronts are inclined to the picture; and the line V 3, V 4, which is drawn * Every vaniihing line is parallel to its original plane; therefore the horizontal line in
the piélure is parallel to the plaue of the earth. And as every vertical plane is perpendi- cular to the horizon, fo the vaniihing lines of vertical original planes are, in the piélure, perpendicular to the horizontal line. 1 * through
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PERSPECTIVE.
through the point V 2 perpendicular to the horizontal line H, is the
raniihing line of the fronts A and B. Rudiments of Ρ lams inclined to the Horizon.
In Plate XXVII. which is the firft Plate of the fourth Section, are
two Examples, in which lines are inclined to the horizon, as in the roofs of the buildings A, B, Fig. l and 2, and in the Hoping edges of the walls F, G, Fig. i. It muft be obferved, that in Fig. 1, the lines a 2 and b 2 are in
the plane of the fronts of the houfes A and B; which plane is ver- tical and perpendicular to the piclure. In Fig. % the lines a b, e d, are, like the former, in the plane of the
fronts A, B; but the vertical plane in which they are feated is inclined to the piéture : confequently, the line V 3, V 4, is the vaniihing line of that plane- The fronts A and B, in Fig. 1, being perpendicular to the piclure,
the vanijhing line of thofe fronts is the prime vertical line, which is drawn through C, the center of the piéture, perpendicular to the horizon, as the line V l, V 2*. The prime vertical line being drawn, and the lower parts of the
houfes completed by the rules which are given in the firft Section, proceed as follows; At the diftance of the picture (which is fo marked) upon the ho-
rizontal line conftruét an angle -f, equal to the known pitch of the roof, as the angle Η Ρ Κ, Fig. ι, equal to e b d, Fig. i, No. 2; and continue the line Ρ Κ till it cuts the prime vertical line at V 1: * As the center of the piéture is the vaniihing point for all lines perpendicular to
the piéture, fo the prime vertical line, by paffing through the center of the pi&ure, is the vaniihing line of all planes perpendicular to the piéture. f For the conftruétion of the angle, fee the inftruéiïons page g.
Β b then
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PERSPECTIVE.
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then will V l be the vaniiliing point for the lines a and a of the·
roofs. To find theother vaniihing point V 2, below the horizontal line*
conftruét the angle HPN, equal to the angle e d e in No. 2, and continue the line Ñ Í till it cuts the prime vertical line at V 2,. which will be the vaniiliing point for the lines in the roofs marked 2 and e. The Hoping edges or tops of the walls, or blocks G F, having
edges, I E, with the fame 'inclination to the horizon with the pitch of the roofs, and their fides G and F being perpendicular to the pióture, have the fame vaniihing point V 1, with the lines a and a of the roofs. The vaniiliing points of all lines are in. the vaniihing line of the
plane in which thofe lines' are feated. Thus the lines of the roofs, windows, and doors in the fronts of the houfes A and B, Fig. 2, have their vaniihing points in the line V 3, V 4; that being the line in which the plane Á Â vaniihes. To complete the forms of the roofs, draw lines from the angles of
the buildings to the vaniihing points V l, V'2, alternately ; and thofe lines will produce the perfpeétive reprefentations of the roofs. The vaniihing points V i, V a, may be found by another procefs,,
as follows : In any part of the paper or canvas, as may be moil convenient,
'draw the form of the roof or pediment geometrically, as Fig. 1,. No. 2, with its bafe G parallel to the horizon. Then through the point ÃË which is the diftance of the picture,
draw the lines E and. X parallel to the lines a, a, in Fig. i, No. 2 ,· and the lines K, and X, will produce, by their interfections with the prime vertical line, the vaniihing points required, as at V é and V 2 ia Fig.. L
In
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PERSPECTIVE. m
In Fig. 2, the fronts and fides of the buildings A and  are in- §E c T*
clined to the piclure; therefore their forms, except the roof, are e—í—t Plate
produced by the rules given in the loregoing Section. XXVtf, C is the center of the piclure, D is the eye or diftance, V 2 is the
vaniihing point for all the horizontal lines in the fronts A and  of the houfes, the lower parts of which are drawn by the rules given in the third or foregoing Section. Through the vaniihing point V 2 draw the line V 3, V 4, perpen-
dicular to the horizontal line, and at d V 2, which is the diftance of the vaniihing point V 2, conftru<5t the angle b a e, equal to the angle of the pitch of the roof in Fig, 2, No 2 ; draw the line from d V % through e, till it cuts or interfeóls the vaniihing line V 2 in the point V 3 : then will V 3 be the vaniihing point for the lines f and g of the roofs. To obtain the vaniihing point V 4, at the point d í 2 conftruét the
angle bad, below the horizontal line, fimilar to the angle bac; and draw the right line ad, continuing it tillit interfeéts the line Vs, V-2, in the point V 4 : then will. V 4 be the vaniihing point for the lines b, b. Having found the vaniihing points V 3, V 4, finiih the roofs by
drawing lines from the angles of the buildings k, 1, m, n, to the vaniihing points V 2, V 4 j and the objects will be completed. Roof inclined to the Horizon,
In Plate III. Fig. 2, is an Example of a building, the fides of which P^jjJE
incline to the piclure, with a ■* roof inclined alfo to the horizon. It is not neceffary to give inftructions for the delineation of the
body of the building, fuch having been repeatedly given in many * The fpecies of roof given in this Example is common in Italy.
 b 2 parts
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PERSPECTIVE.
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parts of the foregoing Seélion; therefore the following inflruélions
will be confined to the roof, that being the only part of the Example which belongs to this Section. The points V l and V 2 are the vaniihing points of the fides X 1
and X 2 of the building ; which fides are vertical or perpendicular to the horizon : therefore, through the point V 2 draw the line V 2 R, perpendicular to the horizon; which line will be the vaniiliing line of the fide of the houfe X 2, as it is marked in the Example. At the point d 2 in the horizontal line (which is the diftance of
the vaniihing point V 2) conilruét an angle equal to the known in- clination of the roof to the horizon, and draw a line from b 2 till it interfeéts the vaniihing line of the fide X 2, as at R ; then will R be the vaniihing point for the inclined lines of the roof R 2. The lines of the roof on the fide X l are all horizontal; confe-
quently, they vaniih in the point V 1. Having determined the vaniiliing point for the inclination of the
roof, draw lines from the point V, and determine its form by the rules which are given in the foregoing Seétion, and in the next Example of this Section. It may perhaps be objected, by thofe who are not ikilled in the
fcience, that the lines R 2 of the roof, do not rife, but, on the con- trary, fink towards the horizontal line, and therefore do not produce the defired effeól. To this it muft be replied, that when original lines are very much elevated above the eye, their reprefentations will ever appear to approach the horizon, whether they are inclined or parallel to it;—and thofe who erroneoufly fuppofe the contrary, have only to correét their opinions by examining objeéls of forms fimilar to that which is reprefented in the given Example.—Some additional obfervations will be made upon this circumftance in the confidera- tion of views of acclivity and declivity. Block
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PERSPECTIVE.
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Block with double Inclination.
In Plate XXVIII. is an Example of a block, which is inclined
both to the picture and alfo to the horizon. It is inclined to the picture in an angle of 39°, and to the horizon
in an angle of 36 degrees. No. l, ihews the geometrical plan of the block, as fituated to the
picture;
G é being the interfeótion of the picture with * the orginal plane.
The line from 0 to X 2 indicates its diftance beyond the picture. X 2 is the feat of the neareft angle of the block on the original
plane, and o the feat of the fame angle upon the picture. No. 2, is the geometrical elevation of the block, ihewing its fitua-
tion to the horizon; the line G 2 indicates the original plane, or ground, upon which the block is placed ; X 2 is the angle upon which it refts, expremve of the fame angle X 2 in No. i, and of a in the perfpective reprefentation. In the plan No. i,'the lines b and e exprefs the full length of the
block; but the line 3 a marks the feat of the line e on the ground, or original plane, when the end e of the block is elevated. In the Example Fig. é, Ç Ç is the horizontal line.
Ñ Ñ is the parallel of the picture.
The center of the picture is marked by the words.
The point D Ñ is the diftance of the picture, brought down on the
horizontal line, andVi, V2 are the vaniihing points for the fides, agreeably to their inclination to the piQure; and d V l, d V 2 are the diftances of thofe vaniihing points.
The vaniihing points V 1, V 2, as alfo the figure of the block C, on
• By original plane is meant, the ground or plane upon which the block refts.
which
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PERSPECTIVE.
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190
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SE^T* which the inclined one refts, are found by the procefs which has
J. V · -
v-—/"**>» been taught in the foregoing Section. The feats of the angle a is
Plate
XXVIII. alfo found by the methods which have been iliewn by Fig. 2, Plate
XI. See inftructions, page 100. V 1 is the point in which all the horizontal lines vaniih, which are
inclined to thepiSiure in an angle of 39 degrees; confequently, the vani/hing line of a vertical plane, which is inclined to the picture in a fimilar angle, will pafs through that point *. Therefore, through V é draw the line V 3, V 4, perpendicular to
the horizontal line Ç Ç. Then find the vaniihing point V 3, as follows :
At the point dV é, which is the diftance of the vaniihing point
V 1, conftruót an angle with the horizontal line, equal to the incli-
nation of the block to the horizon, 36 degrees, as the angle S F P. Continue the line F S till it interfeds the vaniihing line V l,* in
the point V 3 : then will V 3 be the vaniihing point for all the lines which are inclined to the horizon, both in the upper face  of the block, and alfo in its parallel face beneath. The block being fquared (fo called .by workmen) the fides A and
0, are confequently perpendicular to the face  ; therefore find the vaniihing point of lines perpendicular to the face B, as follows : At the point d V l draw the right line M, perpendicular to the
line S F, and it will interfeét the vaniihing line V é, V 3, in the point V 4 : confequently, V 4 will be the vaniihing point for the angles of
the fides o and A of the block. « It muft follow, that the vaniihiag line of all vertical planes, which are equally
inclined, will pafs through the fame point, perpendicular to the horizontal line,; feeing that every vertical plane is perpendicular to the horizon. The face A of the inclined block, and of C, the under block, are both in one plane, whole vaniihing line is V 3, V 4. Now the angles or edges of the inclined block muft be confidered as lines inclined to the
harixon, in a vertical plane, which is inclined to the picture : therefore, the vaniihing points muft be fomewhere in the vaniihing line of the plane in which thofe lines are feated. 1 Having
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PERSPECTIVE. 191
Having thus found the vaniiliing points, complete the figure, as SECT.
follows: e—í-—» From the point a draw lines to the vaniihing points V 1, V 2, V 3, ÷÷íðé.
and V 4, as in the Example. Then from the point d V 1 draw a right line through the point a,
till it interfects the ground line at 5. Then from the point 5 fet off to 3 the meafure of the feat of the
line a 8, equal to the fpace ÷ 2, ÷ 3, in the orthography No. 2, and from 3 draw a line to the point dVi; and its interferon 6 with the line which is drawn from a to the vaniihing point V 1, will be the perfpective feat, upon the ground, of the elevated angle 8 of the block. From 6 draw a right line perpendicular to the horizon, and it will
produce the interfection '8, which will give the apparent length of the fide A of the block. Through the points a and 8 draw right lines to the vaniihing
points V 4, which will produce the angles m 8, and k a. Then determine the apparent thicknefs of the block, as follows :
—Continue the line which is drawn from V 1, through a, till it inter- feas the ground line at 4, and at that point draw the line 4 9 per- pendicular to the ground line. The line b 4, in No. 2, gives the height of the angle 4 of the'
block above the ground; therefore, take the length of the line b 4, and transfer it to 4 9 in Fig. 1, and from the point 9 draw a right line to the vaniihing point V I, and it will interfeét the line which is drawn from V 4 through a, and produce the point k, the neareft angle in the face  of the block A B.—From k draw a line to the vaniiliing point V 2. Then determine the breadth of the block, as follows:
¥ 2 is the vaniiliing point for the end k y, and d V 2 is the
diftance
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192 PERSPECTIVE.
SECT, diftanee of that vanifliing point: therefore, draw a right line from
^IX^ d V 2, through the angle a, till it interfefts the ground line at w. XYvfrr Then take the meafure of the width of the block, 3 feet 6 inches, by the fcale, or equal to the length of the lines a or b, in the plan No. 1, and fet it on the ground line from w to ÷; and from ÷ draw a right line to d V 2, which will interfeéfc the line that is drawn from the point a to the vanifliing point V 2, in the point 0, making the line a È equal the apparent width of the block. Through 0 draw a right line from the vanifliing point V 4, pro-
ducing the interfeétion y. Having thus obtained the points a, k, and y, from thofe points
draw lines to the vanifliing point V 3, and from k m draw alfo lines to the vanifliing point V 2 ; and the reprefentation of the inclined block will be completed. That the ftudent may with more certainty underftand the con-
ilracHon of thefe figures, it will not be impropef to add fome obfer- vations to the foregoing initructions. The point X 2 in No. l, is the fame with X 2 in No. 2; while
the point X 3 in No. l, may be confidered as the point 8 in No, 2; at the fame time obferving, that from X 2 to X 3 in No. 1, is the fame length as from X 2 to X 3 in No. 2. The real length of the block is from the line b to e in No. é; but
as it xeits upon the line b, having the oppofite fide elevated equal to the height, the line which is drawn from the point X 3 to 8 in No. 2, it follows, that the whole length of the block, from the line b to e, covers no more of the ground than is exprefled by the line drawn from X 2 to X 3 in both figures,. Therefore X 3 in No. 2, is called the feat of the point 8, and the
line from X 3 to 8, is called the fupport of the point 8 ; while the line drawn
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PERSPECTIVE, 193
drawn from X 2 to X 3 in No. 2, is called the feat upon the original SECT,
plane of the line X 2, 8, in the original object. u-v^—» As in Figure 1, there are fome points marked which are not allud- ÷÷ãðé,
ed to in the foregoing inftructions, it will be proper to explain them in this place. Thepointa, Fig. 1, which is the angle of the block  upon the ground,
is the fame point which is expreffed by the points X 2, both in the plan and elevation No. 1 and No. 2. It is obtained by the procefs which hath already been given, but which fliall be here repeated. Fig. 1. The feat upon the picture, of the point a, is at the point
1, and its diftancebeyond the picture is equal to the length of the line from 0 to X 2 in No. 1. · . Therefore, from the point 1 draw a right line to the center of the
picture; then take the diftance of that point from the picture, 1 foot by the fcale, or equal to the line 0 ÷ 2, in No. 1, and fet it on the bafe line from 1 to 2, and from 2 draw a line to the diftance of the picture D P, which will interfect the line that is drawn from 1 to the center of the picture in the point a; confequently a is the perfpective reprefen- tation of theneareft angle of the block, which touches the ground. Obferve, that D Ñ is the diftance of the picture brought down to
the horizontal line.; by the fcale it is 6 feet; it is the fame meafure from the, center with the point marked Eye. In this Example, there is a line marked vanifhing line, of face  of
block. This vanifhing line is found as follows : V 2 is the vaniihing point for the lines k y and m n, which are
inclined to the picture only ; and V 3 is the vaniihing point for the lines k m and ç y, which are inclined to the horizon and alfo to the picture : now all thefe lines are in one plane ; confequently, a line drawn through the vaniihing points of thofe lines, will be the vaniihing line of the face B. C e Box
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m PERSPECTIVE.
SECT.
IV. ñß^Ã"^ - Box with open Lid.
XXIX.
Plate XXIX. Fig. 2, contains an Example of a box with the lid
open, every fide of which is inclined to the pidure, and the lid in-
clined alfo to the horizon. The lower part of the figure is given in Fig. l, Plate XV. in the
third or foregoing Seótion, that this part of the Example might not be incumbered with an excels of lines .'therefore, having drawn the lower {part A E by. the inftruétions before given, proceed to com- plete the lid as follows : Fig. é, reprefents the geometrical profile of the box with the lid
open—A, being the body of the box, ancl.D, the lid. In Fis:. 2, the horizontal line is marked Ç Ç.
C is the center of the picture, V é and V 2 are the vaniihing points
for the fides of the box, as found in Fig. l, Plate XV. Through the vaniihing point. V 2, draw the line V 3, V 4, perpen-
dicular to the horizontal line, which will be the vaniihing line for the fides A and D of the. box and lid. At the point o, which is the interfecf ion of the fide of the box with
the ground line, draw the vertical line X, X l, which mufi be con- fidered as the interfe&ion of the fide A of the box, with the piciure. Find the vaniihing points for the fide D, and alfo for the front Â
of the lid, as follows : Bring down the diflance of the vaniihing point V 2 to the hori-
zontal line, as at d 2, as was done for finding the meafure of the , lower part of the box in Plate XV. At the point d 2, conilruft an angle with the horizontal line, equal
to the angle of inclination of the lid of the box with the horizon ; or, which is the fame thing, draw a line through the point d 2, parallel 7 tO
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PERSPECTIVE. 195
to the lid of the box, Fig. 1, till it cuts the vaniihing line V 2 at SEC T..
V 3 : then will V 3 be the vaniihing point of the lid of the box, u^^j
which is inclined to the horizon. XXIX. At the point d 2 draw another line perpendicular to the line d 2,
V 3, till it interfecls the line V 2, V 3, at V 4 ; then will V 4 be the
vaniihing point for the angles  S, a, and e of the lid of the box. Having found the vaniihing points V 3, V 4, bring down the
diftances of thofe points into the vaniihing line· V % as is marked at
their points.
Thus, with the cornpaffes upon the point V 4, take the length of
the line or radial R R to the point d 2, and mark it upon the vaniihing
line V 2, as at the point marked diilance of V~4.
Then repeat the fame at the point V 3, which will give the point
marked diilance of V 3. Having thus found the vaniihing points and their diftances, pro-
ceed to form the lid as follows : Through the angle a of the box draw a right line from the vaniih-
ing point V 3, as the line a B, which will reprefent the lower line of the fide of the lid. From the diilance of the vaniihing point V 3, draw a line through the
angle of the box, at which the lid is hinged, as at a, and continue the line till it interfecls the line X, X 2, as at m;—then from m fetup on the line X, X 1, the meafure of the fide of the lid to n, equal the mea- fure a e in Fig. 1 ; and from ç draw a right line to the diilance of V 3 ; and the interfeclion of that line with the line a B will give the
apparent dimenfion of the fide of the lid of the box, as at B. Having thus found the point B, which is the outer angle of the
lid, from  draw a line to the vaniihing point V 1, as the line  e. Then from the point V 3 draw a line through the further angle b
of the box, till it cuts the line which is dra\vn from  to V 1; and C e 2 the
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PERSPECTIVE.
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the 'interfeéUon of thofe lines at e will produce the general form of
the infide of the lid. * From the points B, e, and a, draw lines to the vaniihing point
V 4, which will give the angles of the thicknefs of the lid. To obtain the thicknefs of the lid at thofe angles, proceed as follows :
From the diftance of the vaniiliing point V 4, draw a right line
through the point  till it cuts the line X, X l, in the point l; then from é fet up to 2 the meafure of the thicknefs of the lid equal b e, Fig. 1 ; and from the point 2 draw a line to the diflance of V 4 ;— and its interfeftion with the line which is drawn from  to the vaniili- ing point V 4, as at S, will give the apparent thicknefs of the lid. Complete the general form of the lid, by drawing lines from the
points  and S to the vaniiliing point V 3, and alfo from the angle S to the vaniiliing point V 1; after which the interior thicknefs of the lid muft be found. The thicknefs of the border of the lid correfponds with the thick-
nefs of the lower part of the box : therefore, having found the thick- nefs of the lower part, as direcled in the explanation of Fig. l, Plate XV. the upper part muft alfo be conftrucled by the aihftance of that thicknefs which is already found. For which purpofe continue the interior lines of the fide A, and its
oppofite, till it cuts the back line of the box a, b, and through thofe points draw lines from the vaniihing point V 3, which lines will give the thicknefs of the fides of the lid; after which find the thicknefs of the front and back of the lid, as follows : f At the points m and ç on the lineX, X 1, fet on the meafure of the
thicknefs
* It is almoft needlefs to obferve, that the angles a, b, e, B, of the lid coincide with
the angles of the lower part of the box, when it is Unit down. f Neither the points which are propofed from m and n, nor their interfe&ions on the
lid, are marked on the figure: fome others are alfo omitted, that the diagram might not be too
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PERSPECTIVE. 197
thicknefs of the lid, as from m towards 0, and from η towards E ; SECT,
and from thofe points draw lines to the diftance of the vaniihing ^^j point V 3 ; and thofe lines will interfeét. the line of the lid a B, and χχ'χ mark the thicknefs both of the front and the back of the lid. From thofe points of interferon draw lines to the vaniiliing point
V 1, and they will give the appearance of the interior dimenfions of the thicknefs of the fides of the lid. . . It muft be obferved, that there are three vaniihing points abfolute-
ly neceffary for the conftruclion of the lid, they are V 1, V 3, V 4. V 1 is the vaniihing point for all thofe lines in the lid which are
parallel to the horizon, and V 3, V 4, for thofe which are inclined to the horizon. The vaniiliing line of the fide of the box, which is marked in the
Plate by its appellation, is the vaniiliing line in which all the vaniih- ing points for the lines in the fidé D of the lid, and alfo of the fide A of the box, muft be found ; for the lines a B and D S muft be con- fidered as being all in the fame plane with the fide A. Thus V 2 is the vaniihing point for the" lines in the fide A, which
are parallel to the horizon, and V 3 and V 4 for thofe which are in- clined to it. It is found thus: The radial or line that paifes from the eye to V 2
is parallel to the original of the fide A, and cuts the picture at V 2. Now this radial muft be confidered as a line in a vertical plane ; con- fequently, fuch vertical plane will cut or interfeét the picture in the line that is drawn from V 3 to V 4. But this plane cannot be ap- plied in operation ; but as it is known to be a vertical plane, its in- terfeétion will be perpendicular to the horizon; therefore, the line too much confufed. For the author is convinced that the reft will be eafily difcover-
ed, provided thofe which are given are properly underftood by the ftudent: but fliould there be any difficulty, let him apply a ruler or ftraight edge to the different points and lines and he will foon find out their direction and ufe. ■ V 3, V 4,
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PERSPECTIVE.
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V 3, V 4, is drawn through the point V 2, perpendicular to the hori-
zontal line Ç Ç. By the term interjection As commonly meant the ground line, or
bafe line of the piclure ; but the the term is equally applicable to other lines in the diagram ; for if any original plane, whether verti- cal, inclined, or horizontal, be produced or continued forward to the piclure, the line in which fuch plane cuts it will be the interfection ; and it is upon fuch line that all meafures muft be applied, to deter- mine the proportionate magnitudes of the parts in the plane pro- ducing that interfection. Thus in Fig. 2, Plate XXIX. the fide A of the box, if continued
or brought forward, cuts the piclure in the line X, X 2 ; therefore, that line is the interfection of the pi&ure by the fide A of the box, equally fo with that which is produced by the ground or plane upon which it ftands, and which in the Example is marked interfec- tion or bafe line. It muit be remembered, that the interfeétion of every plane is
parallel to its vaniihing line. Thus the line commonly called the ground line or bafe line, is parallel to the horizontal line : therefore the vaniihing line of any plane may be confidered as the horizontal line, and the interfection of that plane as the ground line. Demonfiration.
Turn the Example Plate XXIX. fo that the words vani/hing line
of fide A of box, may ftand in the regular order ; then confider that line as the horizontal line, and the line X, X 1, which is parallel to it, as the Bafe line : the point V 2 is the center of that vaniihing line, and the points V 3 and V 4 are vaniihing points for lines in the plane A or fide of the box; and the length of the line which is ^| drawn |
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PERSPECTIVE.
drawn from V 2 to the point marked Eye, is the difiance of the
vanifhing line of the fide A of the box. Of the Pediment on a Building, the Tront of which is inclined to the
PiBure.
Example, Plate XXX.
This figure» although not abftrufe in theory, is difficult in practice.
It is extremely ufeful to the painter and architect, yet the perfpeótive arrangement cannot poflibly be underftood without a previous ac- quaintance with the principles of arehiteéture ; nor ihould it be at- tempted until the ftudent has well confidered the Examples already given of mouldings fituated in the different directions of parallel and inclined to the picture, as demonftrated in Examples, Plates IX. X. and XXVI. As it is neceffary that the ftudent ihould have a clear and accurate
knowledge of the architectural form of the pediment, before he at- tempts to draw it in perfpe&ive, it will be neceffary for him to mnderftand its geometrical conftruélion; for which purpofe the Exam- ple Fig. l, Plate XXXVI. is given. This Figure reprefents half the cornice and pediment, which fur-
mounts a building, and is exactly fimilar in conftruélion to the Example in Plate XXX.—It is drawn by the following procefs : Firft determine the breadth of the erection or building, upon which
the cornice is placed, and then draw the mouldings or members with their projections or profiles, as A B ; which in the Example eonfifts of a cima recla A, the fafcia or corona B, with the cavetto K, beneath. The
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200 PERSPECTIVE.
SECT. The iludent muft obferve, that when there is a pediment, the cima
IV.
j^^^j re<5ta is omitted in the execution of the horizontal part of the' cor- Y YY nice, and is only admitted upon the inclined or floping part, called
the pediment; yet it muft be marked in the drawing, as is done in
the Example by occult lines; becaufe, if this member be not defcrib-
ed, the inclined parts cannot be determined *.
After having drawn the cornice, with the projection of its mem-
bers, divide the whole length from C to the other extreme, into nine equal parts; and having drawn the line a b, perpendicular to the line 0 X C, and in the center of the pediment; fet up from the point O, two of the nine parts, as from O to b : then is b the fummit or apex of the pediment: therefore from b draw a right line to the extreme of the cornice C, and the line b C will be the upper edge of the inclined part of the pediment. Then determine the breadth of the members of the inclined part,
as follows: Draw a right line perpendicular to the upper raking line b C of
the pediment, as the line X, X 2, and continue it through the cor- nice, as at 6, 7 ; then take by compaffes the depths of the horizontal mouldings of the cornice, as they appear upon the line X, X 2, from 7 to b, to l, and to 6; and transfer thofe meafures from the upper raking line b C of the pediment downwards upon the line X, X 2, from 9 to 8, which gives the whole depth of the inclined members of the pediment; which muft be again divided by the fame procefs for the fmall members : through which diviiions, draw lines parallel * In the Example no more than half the pediment is given ; but the intire general form
öf a pediment may be feen in Plate V. Fig. 3, where the whole length of the upper hori- zontal member a b, is divided into nine equal parts, two of which are fet up for the height from 3 to e. It may be confidered as more geometrical to obferve, that the inclination of the raking members of a pediment form angles of 240 with the horizontal part; but this proportion is fometimes varied, as circumitances may require. to
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PERSPECTIVE.
to the line b e, and the geometrical form of the pediment will be
completed. Procefsfor the Perfpectwe Reprefenfathn.
Draw the geometical form of the pediment upon a feparate paper,
as Fig. i, Plate XXXVI. Then determine the general form of the building perfpecüvely, as
A B, Plate XXX. the vaniihmg lines and points for which are as follows : The bafe line is marked A 'G-.
¹ Ç is the horizontal line.
*C is the center of the picture; which is marked Center.
V é is the vaniihing point for the front A B; and d V é is its diftance.
The vaniihing point of the fide F is out of the plate ; but its dif-
tance is marked d V 2. Í is the vaniihing point for the mitre line; but its diftance is out
•of the plate on the left hand. The vaniihing line of the front A B is marked by its title ; it paffes
through the vaniihing point V l, perpendicular to the horizon; in which line the point V 3 is the vaniihing point for the raking or in- clined members D of the pediment; which point is determined by Ihe rules given for the Example Fig. 2, Plate XXVÏI. The other vaniihing point for the members, which may be con*
fidered as the declining part P, is out of the plate, equally diftant from the point V l, with the point V % but below the horizontal line. Thele vaniihing points being determined, and the front and
fides being drawn according to the rules given for Figs, é in Plates XII. and XIII. Pages 102 and 106, confider the points S and Q as the upper angles in the 'front of the edifice, round which the mould* ings aredifpofed; the profile of which in the neareft angle T, muft D d be
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PERSPECTIVE.
be drawn by the procefs given in Page 177, for the Example in Plate
XXVI. therefore, by the fame methods complete the horizontal mouldings Ì and the angle T. The line A S being the angle of the building, and S the point
which terminates its height without the pediment, continue the line A S to any convenient height, and fet up from S to b the whole height of the pediment, equal the height from Ï × tob, in Fig. 1, Plate XXXVI. Then from b fet downwards towards S the depths of the different
members e d e, equal to the geometrical meafures b, e, d, e, in Fig. j, Plate XXXVI. Find the point m, which is the middle of the bafe line of the
building, and through m draw the line w o, tending to the vaniih- ing. point of the fide F, which is out of the plate. Then from d í 2, which is the diftance of the fore-mentioned
vaniihing point, draw a right line through the point m, and con- tinue it till it interfeóis the ground line at the point 6; from which point fet on upon the bafe line the projection of the mouldings, as to 7, 8, 9, equal to the projections of the geometrical profile, mark- ed with fimilar letters, Fig. I, Plate XXXVI. and from the points '7, 8, 9, draw lines to the point d V 2, that.may interfecr, the line w Ï in the points w, x, y; from which points draw lines perpendicular to the horizon, as W K, XI, y v, and m n. Note, the line m ç reprefents the middle of the front A of the
building; confequently, the point f is the fummit of the pediment upon the face of that front. From the points b, e, d, e, In the lineAB, draw right lines to the
vaniihing point V l; which lines will interfecr. the line mn, in the
points f, g, h, and L; through which points draw right lines from
the vaniihing,point of the fide F of the building (but which, as be-
* fore
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PERSPECTIVE.
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fore ohferved, lies out of the plate) that may interfeefc the perpen-
dicular line WK, and the other perpendicular lines in the points E, I, r, L, wHich points determine the projection of the mouldings of what may be called the apex of the pediment. Therefore, from the vaniihing point V 3, draw lines through the
points E, I, r, L, to the various angles of the mouldings in the profile T, which will exprefs the members of the inclined part D of the pediment. Again, from the fame points K, I, r, L, draw lines to the vaniih-
ing point * of the declining part P. of the pediment, which will complete its form. It mull be ohferved, that the contour of the curved mouldings at
the angle of the pediment, muft be drawn by hand ; for any attempt to delineate them by rule would be vain and ufelefs. This Example is given as the laft which can be ufeful to the artift;
therefore this Seclion (hall be clofed with it, together with fome obfervations upon up-hill and down-hill views; their conftracüons being founded upon the principles contained in this Section. It is unneceiTary to apologize for omitting the regular fblids,
feeing that they are of no ufe to the artift, and are at beft no more than curious inquiry. Yet if any one ihould chufe to ftudy their * This vaniihing point lies below the horizontal line, in the vaniihing line of the front
E, at an equal diftance from the point V é with the point V 3 ; how thefe points are ob- tained, has been already {hewn ; yet the procefs fhall be here repeated. V'i is the vaniihing point of all the horizontal lines in the front E of the building,
and the line X pafles through it perpendicular to the horizon : confequently, it is the vaniihing line of the front E of the building. The point d V 1 is the diftance of that va- niihing line ; therefore at the point d V é conftruét an angle on the line Ç Ç, equal to the inclination of the pediment to the horizon, as the angle UWIZ, and the line W 15 continued, will interfeft the line X in the point V 3, which is the vaniihing point required. . The lower vaniihing point for the declining part of the pediment P, may be found by-
making the fame angle at the point d V 1, but below the horizontal line; though it will be fufficient to take the meafure ftom V 1 to V 3, and fet it downwards from V 1 on tha line X, continued below the ground line. Dda conftruc-»
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£04 , Ñ E R S Ñ E C Ô É V E.
SECT, coiiftruótion, lie will find ample inftruclion in the elaborate Work of
IV.
%^-v-~i Mr. Hamilton,- and fome good diagrams in the elder Malton's Trea- "Plate
XXX. tife, which will gratify the curiofity of tho'fe 'who chufe to purfue
the ftudy of Perfpeclive through all its intricacies. Yet they wilt by no means improve the artift in the ufeful part of the fcience, be- yond what may be acquired by the problems of this and the fore- going Sections. Of the Reprefeniations of Up-hill and Down-hill Views.
In terminating the inftruCtions for the laft Example, which treats
of a pediment, whofe front is inclined to the picture; it was ob- ferved, that all which is neceifary to the artift in the fcience of Per- fpeéVive, was concluded by that problem; but as fome hints con- cerning up-hill and down-hill views were alfo promifed in that con- clusion, it will be proper to offer fome inftruftions, which muft be considered as elementary principles for the conduct of the artift in the delineations of up-hill and down-hill views in landfcape. At the fame time it muft be remarked, that this fubjeét is among the many which Dr. Brook Taylor fays, muft be affifted " by a good judg- " ment, founded on much obfervation; it being difficult to bring " every thing to exact mathematical conftruction, at leaft fo as to " be convenient for practice*." On this account nothing more than general hints can be offered, which the artift muft apply to his ufe as occafion may require; at all times paying attention to the following circumftances: Firft, The pofitions of plants, whether inclined to the horizon or
declining from it, cannot eafily be diftinguiihed from each other by * See his firft edition, é ?é |, page 31, of Linear PerfpeiUvf,
the
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PERSPECTIVE.
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the fpe&ator, unlefs fome vertical object or horizontal plane inter-
vene, by which a comparifon of their pofitions can be made; for if it were poffible that a defcending plane or declivity could be feen without the horizon, it would not ftrike the eye of the fpeclator with any other effect than that of being perfectly horizontal; the fame equivocal appearance takes place alfo in the cafe of the afcend- ing plane or acclivity. It is not neceifary to offer a geometrical demonftration of this
fa<5t, as common obfervation will prove the truth of the affertions, however paradoxical they may at firft fight appear. Secondly, All buildings and edifices of every kind muft have their
elementary parts perfectly vertical and horizontal, otherwife they will fall, or be ufe.lefs by their conftruclion *. Thus a houfe built upon the declivity of a hill, muft have its walls
vertical, and the floors perfectly horizontal, together with the tops and fills of the windows, and the ornamental fafcias of the front; yet the ground or plane upon which the building ftands, may be in- clined to the horizon, either afcending or defcending. Thirdly, Although declining or defcending planes have their re-
mote parts below thofe which are neareft to the fpeótator's eye, yet do- thofe remote parts, when reprefented upon the picture, appear higher, or above thofe which are neareft to the picture. Thus the defcending part of a flight of ftairs, when reprefented upon the picture, will have the loweft ftep marked above the reprefentation of that which is the neareft and higheft f, and this will ever be the appearance when the eye of the fpeétator is above the declining plane. On the contrary,, when an inclined plane is above the eye of the fpeétator, then the * There are fome buildings which are pyramidical, others conical in their exterior
forms; yet fuch muft have their bafes perfectly horizontal. f See an excellent Example of a ftaircafe, in Malton's Treatife on Perfpeclive*.
Plate XXXI. where this effect is well illuftrated. fartheft/
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PERSPECTIVE;
fartheft or higheft part of the original plane will in the reprefentation
often fall below the part which expreffes the neareft and loweft parts of fuch inclining plane. It is thus in Example, Fig, 2, Plate III. in which the houfe C, X 2, has the roof R inclined to the hori- zon, the fartheft part of which, at the point W, is in the original objecl much higher than the neareft part marked Ζ ; yet in the reprefenta- tion upon the picture, the higheft part w is below the angle Z, which in the original objeót is the loweft. iitie author has given thefe remarks, becaufe it is neceffary that
the ftudent and artift fhould imprefs their minds with thefe im- portant facts, though they may not perfectly underftand the theory 'by which fuch phenomena^ are produced. In the procefs for drawing views, where the general face of the
•country -is nearly horizontal, the ground line or bale line iliould be •foil of all determined, and then the horizontal line ihould be drawn at a certain height, equal to the height of the eye of the fpeclator, above the ground upon which he ftands; but in drawing down-hill views, it will be proper to draw the horizontal line firit, and then determine the bafe or ground line below it; and this courfe of opera- tion is founded on the following circumftances : When a fpeétator ftands upon the declivity of a hill, and looks
directly forward, he cannot fee any part of the declining plane, but fuch as will be at a much greater diftance from his ftation, than if the plane upon which he ftood were perfeftly horizontal *■; confe- quently the portion of the picture will be greater below the horizontal line in the down-hill view, than in that which reprefents the level country. * It is for this reafon that the difference of procefs is recommended : for in all other
views, it fliould be the invariable practice of the artift firft to mark the ground line or bafe line of the picture, and then the horizontal line, at the proper height above it. Operation
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PERSPECTIVE.
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Operation f or the down-hill View.
Plate XXXI. Fig. 1.
Firil draw the horizontal line Ç H, fo difpofed upon the paper or
canvas, as may beft correfpond with the apparent height of the horizon, in the natural view; and determine the center of the pic- ture, C. Then find the vaniihing points of the fides of the various buildings'
which may be required in the view, by the procefs directed in the Third Section, Page 97, as the points V é and V 2 ; which are the vaniihing points of the fides A and â of the houfe ; obferving, that all its fides or faces are inclined to the picture. Through the vaniihing point V l, draw the vertical line V l, 3,
which will be the vaniihing line of the fides A A of the building. Bring down the diftance of the vaniihing point V l, to the hori-
zontal line, as at d V 1, and at that point conftrucl an angle with the horizontal line, but below it, equal to the known or fuppofed decli- nation of the ground, and draw a right line, as the line M, which; being continued will cut the vertical vaniihing line V 1, at the: point V 3; eonfequently, V 3 is the vaniihing point for the decli- vity of the hilL Thefe points being obtained, let all the conftmclive lines of the-
houfe, and barn, be determined by'the vaniihing points V i, V%i which being in the horizontal line, are the vaniihing points for all the horizontal lines in the fides of thofe buildings which are inclined- to the picture : but the fences or Avails b and e? have their upper* edges parallel to' the hill upon which they Hand; therefore, they- vaniih in the point V 3. And as the three trees are in a line parallel to the fence, and are all fuppofed of equal heights with each other;, their·
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PERSPECTIVE.
their heights are determiued by drawing a line from the fummit of
the neareftj to the vanifliing point ¥ 3. . . .■ < Of'the afcending or up-hill View.
Plate XXXI. Fig. 2. This Example contains two buildings, a barn, marked A, and a
■iioufe, marked Â; the former hath the fide A, perpendicular, and the latter hath its fides  E, inclined to the piclure. ïProcefs.
Draw the ^horizontal line at the common height above the bale or
jground line, and let C be the center of the .picture. As the hill in this Example is fuppofed to be a regular afcending
plane, the bafe or lower line of which is parallel to the picture, pro- ceed as follows : Through the center of the piclure C, draw the prime vertical line
E Cj and let D, upon the horizontal line, be the diftance of the jpiclure. At the point D -draw a line, which iliall make the fame inclination
with the horizontal line, that the hill or afcending plane is known -or fuppofed to make by its afcent with the natural horizon, and the interfection of fueli inclined line with the prime vertical line, as at V, determines the vaniiliing point for lines that exprefs the direclt afcent of the hill.—Therefore, through V draw the line a b parallel >to the horizontal line, then will the line ab be the vaniiliing line for the up-hill or afcending plane *. The houfe or building B, inclines to the piclure, and V 1, Vaare
* The angle V D C is equal to the angle which the afcent of the hill makes with the
tiodzon.; confequenfly-the line D V is parallel to the acclivity of the hill. the
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PERSPECTIVE. 209
the vaniiliing points of all the horizontal lines in the fides of that SECT.
. IV.
building; which vaniihing points are found as directed in the fore- u-„-!_>
Pla*·*
going Seaion. See Plate XIII. Page i-ofiL ÷÷÷À.
Through the point V % draw the vertical line X F, and its inter-
feétion with the line a b (which is the vaniiliing line of the afcend- ing plane) will give the vaniihing· point for all thofe lines which are in the afcent of the hill, parallel to the front  of the houfe, which is inclined to the picture. It would be an endlefs tafk to attempt further directions for deli-
neating the finaller parts of the foregoing figures, nor would it be iufeful, fmce thofe who underftand the preceding, together with the former part of this Section, will eafily comprehend the principles upon which afcending and defcending views are delineated : and it mull be again repeated, that without fuch preparatory knowledge, the lnoft minute and accurate inftructions cannot be underftood by the iludent. Yet the following obfervations will be found ufeful to thofe who are but little informed in the fcience. In all defcending views, although there may be no rifing ground
after the hill terminates upon which the fpectator Hands; yet the level· plane that lies below will appear to afcend or rife towards the Iiorizontal line, which paffes through the eye of the fpectator. This .phenomenon hath fometimes deceived intelligent perfons, who, viewing diftant objects from elevated itations, have fuppofed that thofe objects flood upon elevated ground, when in fact they were npon the loweft that could be feen in the view. On the contrary, when a fpectator approaches afcending ground,
•he is often itruck with the idea of its being lefs elevated than it -really is; but this deception is caufed by tire plane of the afcent pairing through the eye of the fpectator, by which circumftance the elevation cannot eafily be perceived. E e Thefe
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PERSPECTIVE.
Thefe laft obfervations may be confidercd as theoretic, and th&
ftudent will do well to confider them with attention, and to com- pare thefe precepts with the appearances of nature, v/hich in this cafe, as in all others, will furniih him with the beft Examples for his ftudy and confederation, |
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End of the fourth Section»
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Of Shadows.
J.T was not originally the author's intention to give any inftruólions
in this work relative to ihadows, becaufe it feldom happens that the painter is required to reprefent the determined ihadows of the build- ings, which form the back ground of his picture : for this reafon fome writers on the fcience of Perfpeótive have omitted to give any srule concerning ihadows.—* One in particular has the following obfervation: " The geometrical or perfpeclive knowledge of ihadows " is of very little confequence to a painter: it is eafily underftood, " when we have learned that of objecls." Although there is much truth in this remark, yet, in the painting
of fcenes, and in executing perfpeftive drawings of architecture, the correét reprefentations of the ihadows will ever be required ; it was therefore thought neceffary to add this Section, that the work might not be considered as imperfect by thofe who ihould require infrac- tions upon the fubjecl; of ihadows. * The Pra&ice of Painting and Perfpe&ive made eafy, by Thomas Bardweil, Fainter,
.quarto. The patent for this work is dated in January 1756. There was a fecond edition ï'773· |
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PERSPECTIVE.
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It muil be obferved, that however eafy the conftruclion of ihadows
may appear to thofe who underftand the fcience of Perfpeclive, ac- cording to * Dr. Brook Taylor's principles, yet they will be found extremely difficult by thofe who do not clearly comprehend the nature of vaniihiug lines, particularly thofe of planes inclined to the horizon. Fortius reafonit is abfolutely neceffary, before the ftudent engages
in the ftudy of the problems contained in Plates XXXIV. and XXXV. that he ihould well confider the problem given in Fig. 3, Plate XXXVII. which teaches the conftruclion of the vaniihing line of an inclined plane, together with the methods of determining its center and diftance ; all which, ihould be clearly understood before any attempts are made to reprefent ihadows, as caft or projected upon planes inclined to the horizon. In this Seclion no inftruclions are given for the reprefentation of
the objecls whofe ihadows are required, as fuch would only be a repetition of what has been already given in the foregoing Sections, and would alfo produce great confufion in the Examples. Of Shadows.
Shadows may be considered as of two kinds; thofe of art, and
tliofe of nature. 1. The ihadows of art are thofe which are produced by the torch,
lamp, or candle. 2. Thofe of nature are produced by the fun.
* It is worthy of obfervation, that the advantage of the Doctor's principles over all ,
others is in no inftance fo clearly demonftrated as in the conftrucYion of ihadows, parti- cularly thofe which are caft upon inclined planes; for no other methods can determine fuch reprefentations with truth and certainty ; an inconteftible proof of that great writer's fuperiority in the fcience of Perfpeitive. |
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PERSPECTIVE. ai»
In the firft, the luminary mull always be * reprefented, or its SEC T.
place indicated on the picture ; and as the rays of light diverge u-¥-*«J every way from the luminary, the ihadows of all objeóts, which are thus illuminated, muft be reprefented as diverging from a point ia all directions upon the piéture. In the fecond, the rays of light are parallel \o each other, in our
fyftem, in confequence of the immenfe diftance of the luminary; therefore the ihadows of objects produced by the fun's rays are in nature projected or caff in a parallel direction only; yet in the pic- ture they muff be reprefented' as diverging from, or converging to a point, unlefs the fun's rays be parallel to the picture, in which cafe the ihadows of vertical lines will be parallel to the picture. In reprefenting ihadows-as-produced by the fun's rays, it is ab-
folutely neceifary to confider with attention the direction of thofe rays; for there are three different relations in which they may be difpofed to the picture, or, in other words, to the eye of the ipectator. The firft is when the fun's rays are parallel to the picture, or, as fome-
times called, in the plane of the picture ; in which cafe the ihadows of all right lines that are vertical or perpendicular to the horizon, will be parallel to the bafe line or interfeótion of the picture. Ex- ample, Plate VIII. Fig. i, and Plate XL. Fig. 6. The fecond is when the fun is behind the picture, or its body he-
fore the fpectator ; in which fituation the ihadows of all right lines that are vertical will be caft or projected forward towards the fpec- tator. Example, Plate XXXIII. Fig. r, and Plate XXXIV. Fig. I. The third is when the funis before the picture, or its body behind
the fpectator; for then the ihadows of all' right lines that are verti- * In reprefentations of candle-light fubjeils, a fine effecT: may be produced by con-
cealing the luminary; but to fuppofe it out.of the picture will render it unintelligible, and! therefore fliould never be attempted. cai
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«é* Ñ E R S Ñ E C Ô I V E.
SECT, cal will he projeaed from the fpeaator. Example, Plate XXXIII.
^^ Fig. 2, and Plate XXXIV. Fig, 2. To the foregoing obferyations the following may be added, which
may be confidered as Theorems. Theorem lit
The ihadow of every right line, when call or projeaed by the fun upon a plane, is a right line, unlefs the line is parallel to the ray of light, and perpendicular * to the plane of projeaion ; for in that cafe the ihadow will be no more than a point. Theorem 2d.
The ihadow of a plane is alio a right line, whenever the original plane producing that ihadow is parallel to the fun's rays, and per- pendicular to the plane of projeaion; but if oblique to either, the ihadow will then be a trapezium. Theorem 3d.
The ihadows of all right lines, when projeaed upon planes that are parallel to them, have the fame vaniihing points with the lines them- felves. Thus the ihadow lines e f and 8 9, vaniih into the fame points with the lines g, h and k 1 which are the original lines producing thofe ihadows. Plate XL. Fig. 6. To determine the Reprefeniations of Shadows, when the Sun's Rays
are parallel to the Picture.
Plate Plate XL.
XL
Suppofe the block a, to be already drawn, with the face or fide a,
' parallel to the piaure.
In this and the following Example the ihadows are projeaed to-
wards the right fide of the fpeftator; therefore continue the lower * By the plane of proje&ion is meant any plane upon which the ihadow is caft or
projeiled. line
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PERSPECTIVE.
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line of the block from 3 to e, and at the upper angle g draw the line
í 2, making the fame inclination with the horizontal line that the fun's rays are known to make with the natural horizon; and the interfection at e will determine the ihadow of the face a, upon the ground. From e draw a line to C, which is the vaniihing point of the fide of the block; and then draw the line í l parallel to í 2, which will determine the ihadow of the other angle. The line í 1 may be omitted, provided a line be drawn from the farther angle d parallel 'to the line 3 e, till it interfeóts the line e f, at f. Block b, Fig. 6.
The block b has all its fides inclined, while the fun's rays are-
parallel to the picture; therefore at the neareft angle 5, draw the line 5 6 parallel to the bafe line, and through the upper angle d, draw the the line í 4 with the fame inclination to the horizontal line that the fun's rays incline to the natural horizon; and its inter- feétion at 6 will determine the length of the ihadow of the neareft angle of the block.—From the point -6 draw a line to the vaniihing point V % which is the vaniihing point of the ihadowed fide of the block. Then draw í 3 parallel to í 4, which will give the interfection, S, and from 8 draw a line to the vaniihing point V é; and from the point 7 in the indicated * plan of the block, draw a line parallel to the bafe line, which will give the interfection 9. Thus will the lines be obtained, which form the boundaries of the required ihadow. In the Example the inclined line í 4, which reprefents a ray of
the fun, paffes through both the points k and d of the block, and thereby marks the fhadows of both thofe points on the ground by one line only, as at 9· But this is an accidental circumftance, for had the ©bjeét been more or lefs oblique to the piéluïe, it might then have been * It is frequently necefiary to find the perfpeitive reprefeïitatian of the plan of the
öbjeét, without which'the ihadow caBaot be coftipleted. - ■ neceffarv.
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Ñ E II S Ñ E C Ô I V-E;
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neceffary to draw another ray through k ; but as the line producing
the required ihadow is vertical, a line drawn parallel to the bafe of the picture from the point 7, which is the remote angle ®f the plan of the block, will be the ihadow required, as the line 7 9; and therefore a third ray would be fuperflous. In Plate VIIL Fig, l,
Is the Example of a ftool, the ihadow of which is proje&ed upon
the floor, by rays parallel to the picture. , The line L a, indicates one ray of light, to which the others are all
parallel, | |||||