LINEiH 
 
 PERSPECTIVE 
 
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 https://archive.org/details/linearperspectivOObart 
 
LINEAR PERSPECTIVE 
 
 EXPLAINED. 
 
 BY 
 
 WM. N. BARTHOLOMEW, 
 
 AUTHOR OF BARTHOLOMEW'S SKETCH BOOK, AND SERIES OF DRAWING BOOKS, 
 
 IN SIX NUMBERS. 
 
 BOSTON: 
 
 CYRUS G^. OOOKE. 
 
 18 6 6. 
 
Entered, according to Act of Congress, in the year 1859, by 
 William N. Bartholomew, 
 In the Clerk's Office of the District Court of the District of Massachusetts, 
 
 ELECTROTYPED AT THE 
 BOSTON STEREOTYPE FOUNDRY. 
 
 MaETTYCENre 
 
PREFACE 
 
 It is a matter well known to all who have given attention to 
 the subject^ that a very general and strongly-marked increase of 
 interest has been manifested in the study of Drawing within a 
 few years past in the United States. The time has come when 
 some practical knowledge of drawing is getting to be generally 
 recognized by the public as an essential part of every thorough 
 system of education. 
 
 Drawing is now very generally taught both in our public and 
 private schools. There is^ however, no branch of the art more 
 worthy of attention than that which forms the subject of this 
 work, and there is none which receives so little. The ambi- 
 tion to have something to shoiv^ or the desire to make something 
 fretty^ misleads multitudes from a proper and systematic course 
 of instruction. Those fundamental principles which lie at the 
 basis of the art, and which teach us how we may accurately 
 represent the forms and proportions of objects, are entirely 
 overlooked. Weeks, and oftentimes months, are spent in copying, 
 line for line, and dot for dot, some old ruin, and its usual accom- 
 paniments. The whole field of Art, save that which relates to 
 color, is run over without the knowledge of a single principle 
 involved. Those who have attempted the study, have, in most 
 instances, spent their time in learning to copy the characters used 
 
4 
 
 PREFACE. 
 
 in the expression of ideas, instead of studying the principles of 
 the art. They have spent their time in pursuing the shadow, to 
 the neglect of the substance. 
 
 It has been the aim of the author, in preparing this work, to 
 furnish the young with a text-book designed to impart a prac- 
 tical knowledge of the art of making truthful pictures of objects. 
 We have aimed to place the subject within the reach of the boy 
 of twelve years, and^ in doing this, have studied to avoid the error 
 into which many have fallen in their efforts to make the subject 
 easy of comprehension — that of superficiality. 
 
 This work differs from all others with which we are acquainted 
 in one or more of the following particulars : — 
 
 It contains a full explanation of first principles. 
 
 No principle is used in the explanation of another, which has 
 not itself been explained. 
 
 The problems given are of a practical character. 
 
 The method of sketching objects is explained in connection 
 with the method of determining their perspectives by means of 
 vanishing points. 
 
 In determining the perspectives of objects, a reason is given 
 why every line is drawn as it is. 
 
 All explanations are as concise as possible, and in language 
 that all may understand. 
 
 At the close of the book may be found a series of questions. 
 
 Boston, July 27, 1859. 
 
 t 
 
CONTENTS. 
 
 t 
 
 INTRODUCTION. 
 
 PAGE 
 
 Drawing Instruments, their Use and Management 7 
 
 Paper. — Lead Pencil. — Rubber. — Rule. — Right-angled Triangle. — Compasses. 
 
 LINEAR PERSPECTIVE. 
 
 CHAPTER I. 
 
 Definition of Terms 11 
 
 Linear Perspective. — Plane. — Ground Plane. — Picture Plane. — Point of View. — 
 Centre of View. — Horizon. — Distance of Picture. — Point of Distance. — Visual 
 Rays. — Vanishing Point. — Elevation, — Plan. 
 
 CHAPTER II. 
 
 Principles of Linear Perspective developed 14 
 
 Optics the basis of perspective. — Phenomena of vision explained. — Method of deter- 
 mining the perspectives of points by means of visual rays. — On finding the perspectives 
 
 of Hnes, straight or curved The perspective of a square situated parallel to the picture 
 
 plane determined. — The perspective of the square examined, and rules framed, based 
 on principles developed. — The perspective of a square situated oblique to the picture 
 plane determined. — The perspective of the square examined, and rules framed, based on 
 
 principles developed The perspectives of the squares examined, and a rule framed 
 
 based on a principle developed. 
 
 CHAPTER III. 
 
 Preparation of the Paper for the Representation of Objects by the Use of 
 
 Vanishing Points 22 
 
 On drawing the base line. — On placing the centre of view, and drawing the horizon 
 
 On drawing plans and elevations. — On placing the point of distance. — Lines and 
 points necessary in working a drawing. 
 
 (5) 
 
6 
 
 CONTENTS. 
 
 CHAPTER IV. 
 
 The Proper Place in the Picture for the Centre of View, and the Proper 
 
 Distance to assume as the Distance of Picture 24 
 
 The location of the actual centre of view as pictured on the retina of the eye, considered. — 
 Rule for placing the centre of view. — Remarks on the application of the rule. — The 
 distance at which an object must be from the eye, to be seen distinctly, considered. Rule 
 for determining the proper distance to assume as the distance of picture. 
 
 CHAPTER V. 
 
 On the Method of de^rmining the Perspectives of Points 25 
 
 The principle on which the perspectives of points are found. 
 
 PROBLEMS. 
 
 PARALLEL PERSPECTIVE. 
 
 pROB. 1. It is required to find the perspective of a point on the ground plane 26 
 
 Prob. 2. It is required to find the perspective of a square 27 
 
 Prob. 3. It is required to find the perspective of a rectangle 27 
 
 Prob. 4. It is required to find the perspectiveof a point above the ground plane. — Method 
 of determining the perspectives of vertical lines. 28 
 
 Prob. 5. It is required to find the perspective of a cube. — The perspective of the cube ex- 
 amined, and rules framed, based on principles developed. — Drawing from objects recom- 
 mended. — Method of sketching the cube. — On determining the accuracy of a drawing. 
 
 — Suitable subjects for sketching. — The proper course, to pursue in drawing many ob- 
 jects illustrated by the drawing of a chest. — Drawing from copies recommended 29 
 
 Prob. 6. It is required to find the perspective of a cube and the perspective of a pyramid. 
 
 — The method of sketching a pyramidical roof, or any object in the form of the pyra- 
 
 mid 34 
 
 Prob. 7. It is required to find the perspective of a building 36 
 
 Prob. 8. It is required to find the perspective of a building, with a door, windows, and 
 
 chimney, the front wall of the building being parallel to the picture plane 38 
 
 Prob. 9. It is required to find the perspective of the building referred to in Prob. 8, the 
 
 end wall being in the picture plane. — Instructions relating to the sketching of buildings. 41 
 
 OBLIQUE PERSPECTIVE. 
 
 Prob. 10. It is required to find the perspective of a cube. — Method of sketching the cube. 
 — Method of sketching various objects illustrated by the drawing of a chair. — An exer- 
 cise in drawing from the copy < 44 
 
 Prob. 11. It is required to find the perspective of a building. — Method of sketching 
 buildings 48 
 
 OBJECTS CONTAINING CURVED LINES. 
 
 Prob. 12. It is required to find the perspective of a circle. — Method of determining the 
 perspective of an octagon. — The perspective of the circle examined. — Method of sketch- 
 ing the circle 54 
 
 Prob. 13. It is required to find the perspective of a cylinder. — Method of sketching cylin- 
 drical objects i . . i » 56 
 
f 
 
 INTRODUCTION. 
 
 DRAWING instruments!— THEIR USB AND MANAGEMENT. 
 
 1. Paper. Paper should have just tooth enough to admit of 
 making a clear, bright line. There is a paper known as car- 
 tridge paper y which, for ordinary purposes, will meet the wants 
 of the pupil; for sketching, there is nothing better than the 
 German cartoon. 
 
 In drawing, the paper should be placed upon a flat, hard 
 surface, for the following reasons: A straight line cannot be 
 drawn with certainty when the paper is placed upon a yielding 
 surface. It is impossible to draw an even line when the paper 
 is so placed, and the paper is sure to break or dent under the 
 touch of the pencil. 
 
 Paper should never be rolled if it can possibly be avoided, as 
 it always cockles on being unrolled. 
 
 2. Lead Pencil. The selection of a pencil is a matter of im- 
 portance. A pencil should be capable of making a clear, jetty 
 line; it should be tough, and free from grit. A. W. Faber's 
 pencils excel all others in these desirable qualities, The dif- 
 
 (7) 
 
8 
 
 INTRODUCTION. 
 
 ferent degrees of hardness are indicated either by letters or 
 numerals. Those marked H H H, and those marked 4, are 
 admirably adapted for making fine and dehcate lines. For 
 sketching, the H or 3 pencil is all that is desired. 
 
 In drawing diagrams, the pencil should have a fine, delicate 
 point, and the wood of the pencil should be scarfed back at 
 least three fourths of an inch from the point. 
 
 In sharpening the pencil, let it rest firmly against the ball of 
 the thumb of the right hand, in such a manner as to support 
 the lead of the pencil. ^ 
 
 3. Rubber. Eubber is the ordinary instrument for cleaning a 
 drawing, and for erasing marks of the pencil. To meet the 
 wants of the draughtsman, the rubber should readily remove 
 particles of lead from the paper; and it should wear away as 
 used, so as to leave the rubber perfectly clean after use. The best 
 to be had is that manufactured by the Union Rubber Company. 
 
 Rubber should never be used if it can possibly be avoided, as 
 the best of rubber materially injures the surface of the paper. 
 After it has been once employed to any extent, it is almost 
 impossible to keep the paper clean. To avoid, as much as may 
 be, the necessity of resorting to its use, no more lines should be 
 made, in working the drawing, than are absolutely required, and 
 all lines not forming a part of the outline of the drawing should 
 be made as light as is consistent with the distinctness of the work. 
 
 4. Rule. The pupil will find it convenient to have two rules; 
 one should be at least two feet long, and the other from six to 
 eight inches in length. In drawing long lines, the former is 
 
INTRODUCTION. 
 
 9 
 
 indispensable; but for common use^, the latter will be found far 
 more convenient. 
 
 In selecting a rule, great care should be taken to obtain one 
 perfectly straight, inasmuch as the accuracy of the drawing 
 depends greatly on the straightness of the lines. 
 
 5. Eight-angled Triangle. This instrument is used in drawing 
 perpendiculars and parallels. The size most convenient measures 
 from four to six inches on a side. 
 
 The value of this instrument depends upon the accuracy of 
 the right angle, and the straightness of its sides. To test the 
 accuracy of ^ the right angle, place one side of the right angle 
 against the edge of the rule, and draw a fine, delicate line along 
 the edge of the other side of the right angle; then, holding the 
 rule firmly down, turn the other side of the triangle up, and bring 
 the same edge up to either extremity of the line, and draw a 
 light line, as before; if the two lines thus drawn form one line 
 of the same width throughout, the instrument is perfect ; if other- 
 wise, it is imperfect and worthless. 
 
 The triangle is used thus: Suppose it is required to draw a 
 perpendicular from a given point in a straight line. First, place 
 the edge of the rule on the straight line; then place one side 
 of the right angle of the triangle up against the edge of the 
 rule, and sUde the triangle along to the given point, and draw 
 the line. 
 
 Suppose it is required to draw a line parallel to a given line. 
 Place the triangle against the edge of the rule, as before, and 
 bring either one of the remaining edges of the triangle to 
 coincide with the given line ; then, holding the rule firmly down, 
 2 
 
10 
 
 IN^TRODUCTION. 
 
 slide the triangle along to the point from which the line is to 
 be drawn, and draw the line. 
 
 6. CoMPASSjES. The pupil should be provided with a pair of 
 compasses so made that a pencil may be attached to one of its 
 legs. It is convenient to have two pairs of compasses — one 
 small, and the other large; but if only one pair be used, they 
 should be of such a size that, when opened as far as practi- 
 cable, the distance from point to point will be five or six inches. 
 
 This instrument is used in measuring and transferring dis- 
 tances, and in describing circles. The compasses should be held 
 gently at the joint, between the thumb and forefinger, in such 
 a manner as not to press in the least against the legs. When 
 held in this way, the distance between the points will not be 
 altered. 
 
LINEAR PERSPECTIVE. 
 
 CHAPTER I. 
 
 DEFINITION OF TERMS. 
 
 7. Linear Perspective is an art wliicli relates to the representation of 
 the outlines of objects, in such a manner that the representation, when seen 
 from a particular point, shall present to the eye the same appearance as that 
 presented by the objects themselves. The representation of an object so 
 made is called its perspective^ and any line or point in the drawing, repre- 
 senting a line or point in the object, is called the perspective of the line or 
 point. When the perspective of a line is prolonged indefinitely, it is called 
 the indefinite perspective of the line. 
 
 8. A Plane is either a real or an imaginary surface, in which, if two 
 points be assumed at pleasure, and connected by a straight line, that line will 
 lie wholly in the surface. Every plane may be supposed to be produced in 
 any or in every direction, as convenience requires. 
 
 Fig. 1. 
 
 9. The Ground Plane is either a real or an imaginary plane, upon 
 
 which the objects to be represented rest. Its position is hopizontal. In 
 
 Fis:. 1, N IJ K represents this plane. 
 ^ (11) 
 
12 
 
 DEFINITION OF TERMS. 
 
 10. The Picture Plane is an imaginary, transparent plane, on which 
 the objects are delineated. It is supposed to stand perpendicular to the 
 ground plane, fronting the observer, and between him and the objects to be 
 represented. In Fig. 1, A B L D represents this plane. The line B L, 
 formed by the intersection of this plane with the ground plane, is called 
 the base line of the picture plane. 
 
 11. The Point of Yiew is the point from which the eye is supposed to 
 view the objects to be placed in perspective. In Fig. 1, the eye of the 
 observer represents this point. 
 
 12. The Centre of View is a point to which the eye of the observer is 
 supposed to be directed, when looking in a direction perpendicular to the 
 picture plane. In Fig. 1, the point C represents this point. 
 
 13. The Horizon. If one were so placed as to command an unob- 
 structed view of a horizontal plane, stretching away as far as the eye could 
 see, the water or ground before him would appear to incline upwards from 
 the spot on which he stood, and the limit of the plane, in the extreme dis- 
 tance, would be bounded by an apparently straight, horizontal line, on a level 
 with the eye, called the horizon. The horizon always appears to be on a 
 level with the eye, whether the view be had from the low land or the moun- 
 tain top. The term horizon is also applied to the perspective of this line. 
 In Fig. 1, the line H R represents this line. 
 
 14. The Distance of Picture is the supposed distance of the picture 
 plane from the observer. In Fig. 1, the dotted line drawn from the eye to 
 C, measures the distance of picture. 
 
 15. The Point of Distance is a point taken in the perspective of the 
 horizon, on either side of the point representing the centre of view, and at 
 a distance from it equal to the supposed distance of the picture plane from 
 the observer. 
 
 16. Visual Rays are those rays of light which pass from the object to 
 the eye of the observer. Visual rays are represented by straight lines. 
 
 17. A Vanishing Point is any point on the picture plane where the 
 perspectives of parallel lines meet on being prolonged. 
 
 18. An^ Elevation is a drawing giving the actual form, and the exact 
 
DEFINITION OF TERMS. 
 
 13 
 
 proportions, of the different upright parts of any upright side of a building 
 or other object. 
 
 Pig. 2. 
 
 Fig. 3. 
 
 In Fig, 2, we have a front elevation of the building represented iii 
 Fig. 3 ; and in Fig. 4, we have a side elevation of the building. 
 
 19. A Plan is a drawing constructed from a horizontal section, or 
 from horizontal sections, of a building or other object. 
 For the purpose of perspective drawing, the plan of the object to be rep- 
 
 -I- - 
 
 E 
 
 Pig. 4. 
 
 Pig. 5. 
 
 resented must indicate every change of form that may occur by cutting the 
 object horizontally at any point. 
 
 In Fig. 5, we have a plan of the building represented in Fig. 3. This 
 plan was made from the figures formed by cutting the building at two points 
 — one of these points being just above its base, passing through the door 
 and windows, as indicated by the dotted lines D E and E F, and the other 
 through the line of the ridge H I. The double-lined, rectangular figure 
 
I.INEAR PERSPECTIVE. 
 
 A B C D, Fig-, 5, is a plan of the walls of the building; the open space E, 
 of the doorway ; the single lines F and H, of the windows ; the figure 
 niarked I, of the chimney ; and the line J K, of the ridge. 
 
 From the three geometrical drawings, the front elevation, the side elevation, and the plan, the 
 proportions of the building may be known. 
 
 The length of the building is shown in the plan, and in the front elevation. 
 
 The breadth of the building is shown in the plan, and in the side elevation. 
 
 The height of the walls is shown in the front and side elevations. 
 
 The vertical height of the roof is shown in the front and side elevations. 
 
 The pitch of the roof is shown in the side elevation. 
 
 The height of the chimney is shown in the front and side elevations. 
 
 The width of the front, and back face of the chimney, is shown in the plan, and in the front 
 elevation. 
 
 The width of the faces of the chimney parallel to the end walls of the house is shown in the 
 plan, and in the side elevation. 
 
 The width of the windows, the width of the door, and their situation, is shown in the plan, 
 and in the front elevation. 
 
 The height of the door, the height of the windows, and their height above the base of the 
 building, is shown in the front elevation. 
 
 CHAPTER II, 
 PRINCIPLES OF LINEAR PERSPECTIVE DEVELOPED. 
 
 20. The principles which underlie and form the basis of linear perspec- 
 tive are few in number, and easy of comprehension. They are deduced 
 from the science of optics, and depend upon the following truths : 
 
 When an object is exposed to the light, rays are reflected from each point 
 in the surface, in every possible direction. 
 
 Rays of light, while passing through the same medium, proceed in straight 
 lines. 
 
 When the eye is directed towards an obr 
 ject, rays from every visible point enter the 
 eye, and form upon the retina an image or 
 picture of the object, and the impression 
 ^ thus made produces the sensation of sight. 
 
 21. To illustrate these truths, let the 
 circle in Fig. 6, represent a vertical section 
 of the eye, the point E, its pupil, and let 
 A B be a straight line, towards which the eye is directed. 
 
 A 
 
 B 
 
 Pig. 6. 
 
• 
 
 PRINCIPLES DEVELOPED. 
 
 15 
 
 From the point A we may suppose rays of light to ho reflected in every 
 direction ; a part of these rays enter the eye, through the pupil, one of 
 which takes the direction of the line A D, and there is formed upon the 
 retina at D, the image of the point A. Prom the point B, rays enter the 
 eye, and there is formed upon the retina at C, the image of tlie i)oint 15. 
 In Hke manner, rays from each point in the line A B enter the eye, and lall 
 upon the retina hetween the points C and D, completing the image C D. 
 
 22. Let us now suppose one viewing an object through a transparent 
 plane, placed upright before him. From every visible point in the object a 
 ray of light proceeds in a right line to the eye, passing through a point in 
 the plane. If to each point in the plane, pierced by the visual rays, liglit 
 and shade could be given, similar to that which characterizes the points 
 from which the rays proceed, the rays from the picture thus produced would 
 form in the eye an image precisely like that formed by the object itself. 
 The picture, therefore, would be a perspective drawing of the object. 
 
 This truth may be illustrat- 
 ed. Let the circle in Fig. 7 rep- 
 resent the eye, A C a straight 
 line, towards which the eye 
 is directed, and let the figure 
 I J K L represent a transparent 
 plane. 
 
 As already shown, the visual 
 rays from the line A C form on 
 the retina the image F H. The 
 visual ray from the point A, in 
 its passage to the eye, passes through the plane at the point a, and the 
 visual ray from C passes through the plane at c. Since A C is a straight 
 line, the visual rays from the points between A and C pass through the 
 plane at points between a and c ; therefor^, if the points a and c be con- 
 nected by a straight line, the line a c includes all points in the plane 
 pierced by the visual rays. Now, suppose the line A C removed ; the vis- 
 ual rays from the line a c form the image F H, the same as that formed by 
 the line A C ; therefore a c is the perspective of A C. From this investiga- 
 tion, it is evident that 
 
 If a transparent plane be placed upright between the eye and any given 
 object.) and a straight line be drawn from any point on the object to the eye, 
 the point in the plane pierced by the line will be the perspective of the point 
 from which the line is drawn. 
 
 Pig. 7. 
 
 23. Since linear perspective deals only with the outlines of objects, it is 
 
16 
 
 LINEAR PERSPECTIVE. 
 
 • 
 
 evidently unnecessary to determine every point in the plane through which 
 the visual rays pass. If we simply secure those points pierced by the rays 
 from the prominent points of the object, as, for example, those from the 
 extremities of right lines, and those from points in curved lines, which indi- 
 cate their course, we do all that is needful to secure the outline ; for, having 
 determined these points, we have but to connect them by lines corresponding 
 to those in the object, to complete the drawing. With this understanding, 
 let us proceed to find the perspective of some simple form, and, as the 
 square is well suited to our purpose, we select this for our first trial. 
 
 24. Let E F D I, Fig. 8, 
 be the square. Suppose this 
 figure to be traced on the 
 ground plane, which may be 
 represented by the plane of 
 the paper upon which the fig- 
 ure is made. 
 
 On the side F D of the 
 square, suppose a transparent 
 plane to be erected, perpen- 
 dicular to the ground plane. 
 Let this plane be represented 
 by the rectangle M B L N. 
 As we have supposed the 
 ground plane to be repre- 
 sented by the paper on which 
 the figure is traced, the rec- 
 tangle M B L N must be un- 
 derstood as representing a 
 plane perpendicular to the 
 plane of the paper. Suppose 
 the observer standing at A, a 
 point on the same plane on 
 which the square is traced, 
 and let the eye be on a level 
 with the point C, a point on 
 tlie picture plane. This will 
 place the eye vertically over 
 A, at a height above the 
 ground plane, equal to the 
 height of C above B L, the base line. 
 
 By referring to Fig. 9, the pupil will obtain a clear perception of the 
 
PRINCIPLES DEVELOPED. 
 
 17 
 
 relation which the planes sustain to each other, to the square, and to the 
 observer. In the two drawings, corresponding points are indicated by the 
 same letters. 
 
 In Fig. 8, the eye was as- 
 sumed to be above the ground 
 plane, at a height equal to the 
 height of the point C above 
 the base line ; but the pomt A, 
 to which the right lines from 
 E and I are drawn, is on the 
 same plane with the square. 
 It may be readily shown, how- 
 ever, that if perpendiculars be 
 erected on the points K and J, 
 where these lines intersect the 
 base line of the picture plane, 
 that lines drawn from the 
 points E and I, to the eye, will 
 pass through the picture plane 
 at points somewhere in these 
 perpendiculars. Take a line 
 from E to the eye, for example. 
 Suppose a vertical plane erect- 
 ed on the line A E, equal in 
 height to the assumed height 
 of the eye, a line drawn from 
 E to the eye will lie in this 
 plane, and the point in the 
 picture plane through which it 
 will pass will be found some- 
 where in the line of intersec- 
 tion of the two planes. Since 
 both planes are vertical, the 
 line of their intersection will 
 be perpendicular to the base of 
 either plane. The line K e is 
 perpendicular to B L, the base 
 
 of the picture plane ; and therefore a line from E to the eye, will pass 
 through the plane at some point in this line. In the same way it may be 
 shown that a line from I, to the eye, will pass through the plane at some 
 point in the perpendicular erected on the point J. 
 
 To determine the points in these perpendiculars through which lines from 
 3 
 
« 
 
 18 
 
 LINEAR PERSPECTIVE. 
 
 E and I will pass, we must 
 refer to Fig, 10. In this M 
 drawing, the figure Z E A P 
 is a representation of the ver- 
 tical plane, supposed to be 
 erected on the line A E. 
 The line Y K is a line formed 
 by the intersection of this 
 plane with the picture plane 
 MBLN. 
 
 A line drawn from E to P, 
 the place of the eye, passes ^ ^ 
 through Y K at the point e. 
 
 Take the measure of the line K e, which 
 measures the height of the point e above 
 the base of the plane, and lay it off on the 
 perpendiculars from the points K and J, 
 Fig. 8, and we determine e as the point in 
 the plane through which a line from E, to 
 the eye, will pass, and i as the point 
 through which a line from I will pass. 
 
 Since the points F and D are in the base 
 line of the picture plane, as well as in the 
 front side of the square, their perspectives 
 will appear in the plane at the points F 
 and D. 
 
 E 
 
 
 Y 
 1 
 
 
 
 
 \ Fig. 8. / 
 
 
 
 \ A 
 
 
 
 \g/'' 
 
 k 
 
 
 
 
 
 
 
PRINCIPLES DEVELOPED. 
 
 19 
 
 Connect the points now determined, and we have the figure F e i D, the 
 perspective of the square. 
 
 25. In determining the perspective of the square, we have developed 
 certain general truths. 
 
 Examining the figure, we observe, first, that the sides E F and I D of the 
 square are perpendicular to the picture plane, and that their perspectives 
 F e and D on being prolonged, intersect, or vanish^ as it is termed, in the 
 point C, which point is the centre of view, it being on a level with, and 
 directly opposite the eye. It is evident that what is true of the perspectives 
 of E F and I D is also true of the perspectives of all lines perpendicular to 
 the picture plane ; and hence the rule : — 
 
 Rule 1. The perspectives of all lines perpendicular to the picture plane 
 tend to the centre of view. 
 
 Upon a further examination of the figure, we notice that the sides F D 
 and E I of the square are parallel to the picture plane, and that their per 
 spectives, F D and e i, are parallel to the lines they represent. It is evident 
 that what is true of the perspectives of F D and E I is also true of the per- 
 spectives of all lines parallel to the picture plane ; and hence the rule : — 
 
 Rule 2. The perspectives of all lines parallel to the picture plane are 
 parallel to the lines they represent. 
 
 26. Having determined the perspective of tne square in what is called 
 parallel perspective, it is now proposed to find the perspective of the same 
 figure in oblique perspective, that is, with the sides of the square oblique to 
 the picture plane. 
 
 Let the figure S U 0 T, Fig. 11, be the square, the sides of which lie at 
 an angle of forty-five degrees with the picture plane. It will be observed 
 that this diagram is similar to Fig-. 8, with the exception of the additional 
 lines. Suppose this figure seen from the same point that the square E F D I 
 was seen from, that is, from a point vertically over A, and at a height above 
 it equal to the height of C above B L, the base line. 
 
 Guided by those laws to which attention was called in Fig. 8, we may 
 determine the perspective of this figure, without resorting to the method 
 there adopted in determining the perspectives of points. 
 
 Since the points S, TJ, 0, and T, are in the centres of the sides of the 
 square E F D I, it is evident that their perspectives will be found in the 
 perspective centres of the sides of the figure e ij), the perspective of the 
 square F E I D. 
 
(20) 
 
PRINCIPLES DEVELOPED. 
 
 21 
 
 Let us first determine the perspectives of the points S and 0. The per- 
 spectives of these points may be determined by finding the perspective of a 
 straight line connecting them. Conceive a line connecting the points S and 
 0, and we have a line passing through the centre of the square E F D I, 
 parallel to the picture plane. The perspective of this line will be parallel 
 to the line supposed to be drawn from S to 0, Rule 2. To determine the 
 perspective centre of the figure ¥ eiD, draw the diagonals D e and F i ; and 
 sr, the point of their intersection, is the point required. Draw through z a 
 line, as s o, and we have the perspective of the line supposed to be drawn 
 from S to 0, and the point 5, as the perspective of S, and o, as the perspec- 
 tive of 0. 
 
 The perspective of the point T is already determined ; for since this point 
 is in the picture plane, its perspective is found where the point is placed. 
 To complete the drawing, it only remains to find the perspective of the 
 point U. Tliis point may be determined by finding the perspective of a 
 straight line connecting the points T and U. Conceive a line connecting 
 these points, and we have a line perpendicular to the picture plane. The 
 perspective of tliis line will tend to C, the centre of view. Rule 1. Draw 
 a line from T, the perspective of T, in the direction of C, to meet the line 
 e i, and we have T u, the perspective of the line, and the point u, as the 
 perspective of U. 
 
 Connect the points Uy o, and T, as shown in the figure, and we com- 
 plete the perspective of the square. 
 
 27. Before examining this figure, let us draw the horizon. Since this 
 line always appears to be on a level with the eye, it is evident that its per- 
 spective must pass through the centre of view. Through the point C, the 
 centre of view, draw a line, as H R, to represent the horizon. 
 
 Examining the figure, we observe that the square S U 0 T, is bounded 
 by horizontal lines lying at an angle of forty-five degrees with the picture 
 plane ; that the perspectives of the sides of the square, on being prolonged, 
 vanish in the horizon at the points Y and Y' ; that these points are equally 
 distant from C, the centre of view ; and that their distance from C equals 
 T A, a line measuring the distance of the picture plane from the observer, 
 that is, the distance of picture. 
 
 That the points V and V are at a distance from C, equal to the measure of the line T A, may 
 be thus shown : From the points V and V, drop perpendiculars to meet the base line, as V W 
 and Y' X, and from the points W and X draw lines parallel to S T and 0 T, to meet the perpen- 
 dicular T A, making W A and X A. The lines W A and X A form with W X angles of forty- 
 five degrees. Since the angles T W A and T X A are each forty-five degrees, the distance from 
 T to W, and from T to X, is equal to T A. But C V and C V are each equal to T W or T X» 
 and consequently to T A. 
 
22 
 
 LINEAR PERSPECTIVE. 
 
 It is evident that what is true of the perspectives of the sides of the 
 square is also true of the perspectives of all horizontal lines lying at an angle 
 of forty-five degrees with the picture plane ; and hence the rule : — ■ 
 
 Rule 3. The perspectives of all horizontal lines lying at an angle of 
 forty-five degrees with the picture plane^ tend to a point on the horizon at a 
 distance from the centre of view equal to the distance of picture. 
 
 Upon a further examination of this figure, we notice that the perspectives 
 of those lines which tend to the right, as they recede, vanish on the right of 
 the centre of view, and that the perspectives of those which tend to the left, 
 as they recede, vanish on the left of this point. What is true of these lines 
 is evidently true of all others similarly situated ; and hence the rule : — - 
 
 Rule 4. The perspectives of all horizontal lines lying at an angle of 
 forty-five degrees with the picture plane, tend to the right or left of the centre 
 of view, according as the lines themselves tend to the right or left of this 
 point. 
 
 28. From this figure, we learn that the perspectives of all lines perpen- 
 dicular to( the picture plane, tend to the same point ; that the perspectives of 
 all parallel, horizontal lines, lying at an angle of forty-five degrees with the 
 picture plane, tend to the same point. Now, what is true of the perspectives 
 of these lines is evidently true of the perspectives of all parallel lines ; and 
 hence the rule : — 
 
 Rule 5. The perspectives of all lines which o/re parallel to each other 
 tend to the same point. 
 
 CHAPTER III. 
 
 PREPARATION OF THE PAPER FOR THE REPRESENTATION 
 OF OBJECTS BY THE USE OF VANISHING POINTS. 
 
 29. In Fig. 12, let E J K I represent a sheet of paper on which it is pro- 
 posed to make a drawing. To prepare the paper for this purpose, draw 
 across it a straight, horizontal line, as B L, to represent the base line of the 
 picture plane, leaving sufficient room below this line to accommodate the 
 plan of the object, and the elevations, if any are needed. The upper part 
 of the paper, E B L I, represents the picture plane ; the lower part of the 
 paper, B J K L, the ground plane. These planes, it will be remembered, 
 are supposed to be perpendicular to each other, the picture plane being 
 
PREPARATION OP PAPER. 23 
 
 vertical, and the ground plane horizontal. The supposed relation of these 
 planes to each other being clear in the mind, it will be understood that any 
 point placed on the ground plane will be near or remote from the picture 
 plane, according to the distance at which the point is placed from the line 
 B L. The point A, for example, represents a point on the ground plane, 
 and in the picture plane, while the point F represents a point back of the 
 picture plane and at a distance from it equal to the distance of F from the 
 Ime B L. 
 
 E, ,1 
 
 30. The place of the eye determines the location of the point known as 
 the centre of view, for this point is always on a level with, and directly oppo- 
 site, the eye. It also determines the place of the horizon, for this line 
 always appears to be on a level with the eye. In the drawing to be made, let 
 us suppose the eye to be on a level with, and directly opposite, the point C. 
 This point will then be the centre of view. Having placed the point C, 
 draw through it a very light, horizontal line, as H R, to represent the hori- 
 zon. The point C, it will be observed, corresponds to the point C in Fi^. 11, 
 and is the vanishing point of all lines in the object to be represented, situ- 
 ated perpendicular to the picture plane. 
 
 31. The plan of the object may now be drawn on the ground plane. 
 Suppose the object to be represented is a building. The plan of the building 
 should be drawn on the right or left of the point C, at that distance which 
 the building is supposed to be on the right or left of the eye, and at that 
 distance from the base line which the building is supposed to be back of the 
 picture plane, and making that angle with the base line which the building 
 is supposed to make with this plane. The plan drawn, we may then draw 
 
24 LINEAR PERSPECTIVE. 
 
 the elevations of the building on the ground plane, where they can be best 
 accommodated, or they may be placed upon a separate sheet of paper. 
 
 32. A distance equal to the assumed distance of the picture plane from 
 the eye may now be laid off on the horizon from the centre of view, on 
 either side. Suppose that we assume the picture plane to be at a distance 
 from the eye equal to the distance from C to D, and place a point, as D. 
 The point D, it will be noticed, corresponds to Y', in Fig. 11, it being placed 
 on the horizon at a distance from the centre of view equal to the distance of 
 picture, and is the vanishing point of all horizontal lines inclining to the 
 right as they recede, which make an angle of forty-five degrees with the 
 picture plane. 
 
 33. The paper now contains all the lines and points required in deter- 
 mining the perspective of any object. They are the base line, B L, the hori- 
 zon, H R, the centre of view, G, and the point of distance, D. 
 
 CHAPTER IV. 
 
 THE PROPER PLACE IN THE PICTURE FOR THE CENTRE OF 
 VIEW, AND THE PROPER DISTANCE TO ASSUME AS THE 
 DISTANCE OF PICTURE. 
 
 34. When the eye is directed to any scene, the image or picture formed 
 on the retina of the eye is evidently circular in form ; and it is also evident, 
 that the image of the particular point on which the eye is fixed, is in the 
 centre of this circular picture. This being the case, it is obvious that 
 
 In assuming a point as the centre of view, it should be taken near the centre 
 of the picture, 
 
 35. This rule, however, is intended to apply to the fall and complete picture, rather than to 
 the drawing of single objects, such as are usually selected for first practice, the object of which is 
 simply to render the pupil familiar with the principles of perspective, and the method of applying 
 them. These examples seldom contain more than one object, and as it is generally desirable that 
 this should be represented as situated on one side of the observer, rather than directly before him, 
 it necessarily brings the centre of view out of the picture. 
 
 36. A proper selection of the distance of picture is a matter of the 
 first importance, inasmuch as it is upon this that the naturalness of the 
 
CENTRE OP VIEW AND POINT OP DISTANCE. 
 
 25 
 
 drawing in a great measure depends. The distance chosen is often so very 
 trifling that the objects attempted to be represented couhl not be seen with- 
 out allowing the eye to wander from a fixed point, whicli is never admissil)le. 
 No more can be truthfully represented than can be seen clearly at one view. 
 If we take any object, as a book, for example, and hold it before the eye, 
 fixing it on its centre, it will be found that in order to obtain a clear view of 
 all its parts, it must be held at a distance equal to some four or five times its 
 longest measurement taken from the point on which the eye is fixed ; that 
 is, the book must be so held as to be seen under an angle of about thirty 
 degrees. 
 
 Suppose one standing at B, Fig. 13, looking in the 
 direction indicated by the line B D ; the angle ABC, 
 an angle of thirty degrees, may be supposed to include 
 all that can be clearly seen. Take A D as a radius, 
 and describe a circle, as in Fig. 14, and we have the 
 largest size for a picture which embraces all that is 
 seen, situated at the distance of B D from the observer. 
 The distance between the points A and D is, practi- 
 cally, one fourth the length of the line B D, the dis- 
 tance of the picture from the eye. From this it is 
 evident that 
 
 The proper distance to assume as the distance of 
 picture is equal to at least four times the distance of 
 the most remote point in the picture from the centre 
 of view. 
 
 In our illustrations, want of space has prevented us from assum- 
 ing that distance of picture which our rule demands. The pupH should 
 adhere to the rule. 
 
 Pig. 14. 
 
 CHAPTER V. 
 
 ON THE METHOD OF DETERMINING THE PERSPECTIVES OF 
 
 POINTS. 
 
 37. The principle on which the perspectives of points are found, by the 
 method we are now explaining, is this : — 
 
 The jierspective of any given point is found somewhere in the indefinite 
 perspective of a straight line drawn through the point. 
 4 
 
26 
 
 LINEAR PERSPECTIVE. 
 
 From this principle, it follows that 
 
 If two lines be drawn through any given pointy and their indefinite per- 
 spectives found, the point of their intersection with each other is the perspec- 
 tive of the point. 
 
 38. To avoid repetition, we shall, in the following problems, suppose the base line, the horizon, 
 the centre of view, and the distance of picture given, as shown in the accompanying diagrams. 
 These lines and points will, in all cases, be designated as follows. The base line will be marked 
 B L, the horizon H E,, the centre of view C, and the point of distance J). 
 
 It will be observed that, in most cases, the centre of view is taken near the edge of the paper. 
 This is done so as to allow as great a distance between the points C and D as the limited size of 
 the page will admit. 
 
 PROBLEMS. 
 PARALLEL PERSPECTIVE. 
 
 Prob. 1. In Fig. 15. A is a point on the ground plane. It is required to 
 find its perspective. 
 
 If R 
 
 39. Draw through the point A a line, as I A, perpendicular to B L, the 
 base line of the picture plane. The point I of this line is in the picture 
 plane, and therefore its perspective is at I. The perspective of the line 
 tends to C. Rule 1. Connect the points I and C, and we have I C, the 
 indefinite perspective of I A. 
 
 Through the point A draw a line, as J A, making an angle of forty-five 
 degrees with B L, the base line of the picture plane. To determine the 
 inclination of this line, place a point, as J, in the base line, at a distance 
 from I equal to the distance of A from I, and connect the point with A. 
 The point J of this line is in the picture plane, and therefore its perspec- 
 tive is at J, The perspective of the line tends to D. Rules 3 and 4. Con- 
 nect the points J and D, and we have J D, the indefinite perspective of J A, 
 and the point a, tlie point of its intersection with I C, is the perspective of 
 A, the given point. 
 
 t 
 
OBJECTS IN PARALLEL PERSPECTIVE. 
 
 27 
 
 40. In this figure, more lines have been used than are absolutely required. This remark 
 applies to the line J A, which is introduced simply for the purpose of more clearly exemplifying 
 the principle on which the perspectives of points are found. The pupil will readily see that we 
 have no use for the line ; all that is needed is the point J, and therefore, in drawing, the line may 
 be in all cases omitted. 
 
 41. Pros. 2. In Fig. 16, A E F I is a square lying' on the ground plane. 
 It is required to find its perspective. 
 
 Find a and /, the perspectives of the points A and F, as in Prob. 1. 
 The sides A I and E F of the square are parallel to the picture plane ; 
 
 HC 
 
 D R 
 
 Fig. 16. 
 
 WC/ llO/VC; U/ C/j l/JlVy \j VJ. J.. o-j-v^Ajj. i/iivy j^jKj^^^^ J y ^^^^ - - , 
 
 draw a line to meet K C, parallel to E F, and we have ef, the perspective 
 of E F, which completes the drawing of the square. 
 
 42. Prob. 3. In Fig. 17, A B F I is a rectangle lying on the ground 
 plane. It is required to find its perspective. 
 
 Determine the perspectives of the points A and E, as in Prob. 1. 
 
 The sides A I and E F of the rectangle are parallel to the picture plane ; 
 their perspectives are parallel to the lines themselves. Rule 2. From the 
 point a, the perspective of A, draw a line to meet J C, parallel to A I, and 
 we have a t, the perspective of A I. From the point e, the perspective o? 
 
28 
 
 LINEAR PERSPECTIVE. 
 
 E, draw a line to meet J C, parallel to E F, and we have e /, the perspective 
 of E F, which completes the drawing of the figure. 
 
 S. C P R 
 
 Pig 17. 
 
 — — IE' 
 
 43. Prob. 4. In Fig. 18, A is a point on the groumd plane. It is 
 required to find the perspective of a point situated vertically over A, and at 
 a height above it equal to the measure of the line 0 P. 
 
 H C D R 
 
 A Pig. 18. 
 
 ■P 
 
 Find a, the perspective of A, as in Prob. 1. 
 
 On the point I erect a vertical line, as E I, equal in height to the measure 
 of 0 P. Conceive a line connecting the point E with the point to be placed 
 in perspective, and we have a line perpendicular to the picture plane. The 
 
OBJECTS IN PARALLEL PERSPECTIVE. 
 
 29 
 
 point E of this line is in the picture plane, and therefore its perspective is 
 at E. The perspective of the lino tends to C. Rule 1. Draw E G, and 
 we have the indefinite perspective of the line. 
 
 Conceive a line connecting the point A with the point to be placed in per- 
 spective, and we have a vertical line, a Ihie parallel to the picture phine ; the 
 perspective of the line is parallel to the line itself. Rule '2. On the jioint a, 
 the perspective of A, erect a vertical line to meet E C, and o, tiie point of 
 intersection, is the perspective of the given point. 
 
 44. In determining the perspective of the given point, it will he observed 
 that we have found the perspective of a vertical line of the measure of O P, 
 standing on A, o a being the perspective of the line. lu this way, the 
 perspective of any vertical line may be found. 
 
 
 
 
 
 
 D a 
 
 e 
 
 B 
 
 
 
 
 L 
 
 E K 
 A 
 
 E 
 
 Pig. 19. 
 
 AI 
 I 
 
 J 
 
 45. Prob. 5. In Fig. 19, A E J I is the plan of a cube, situated on the 
 ground plane. It is required to find its perspective. 
 
 In placing a cube in perspective, elevations are not needed. Since all the faces of the cube arc 
 known to be similar, the measurements of the upright faces may be taken from the plan. 
 
 Find aej the perspective of the plan, as in Prob. 2. 
 Find na,Y e,o i, and S j, the perspectives of the vertical edges of the 
 cube, as in Prob. 4. See sections 43 and 44. 
 
30 
 
 LINEAR PERSPECTIVE. 
 
 Connect the point n with o, and V with S, and the drawing of the cube is 
 complete. ^ 
 
 There are but two faces of the cube which are seen, viz., the front face and the left face. The 
 perspectives of the lines bounding these faces are made full. The perspectives of the edges not 
 seen are dotted. 
 
 46. Examining the perspective of the cube, (^Fig. 19,) we find that the 
 front and back face present to the eye their actual form. (We suppose 
 the cube transparent, so that each face may be seen.) These faces present 
 affront view, that is, they face the observer. From this it is evident, that 
 
 A surface presenting a front view appears of its actual form. 
 
 47. The perspectives of the edges presenting a front view are in the same 
 position as the lines they represent. The perspectives of the vertical edges 
 of the cube are vertical ; the perspectives of the horizontal edges which face 
 the observer are horizontal. From this it is evident, that 
 
 Lines presenting a front view appear in their actual position. 
 
 48. Comparing the perspective of the front face with the perspective of the 
 back face, we find that the latter is every way smaller, the lines bounding it 
 being shorter than those bounding the front face. From this it is evident, that 
 
 Lines of equal length presenting a front view, when seen at unequal dis- 
 tances, appear unequal in length, the most distant appearing the shortest line. 
 
 49. Again, we notice that those faces seen obliquely — that is, those faces 
 which recede from the eye — do not appear of their actual form ; they are 
 foreshortened, as it is termed. Notice how short the distance between the 
 lines Y e and n a, in the perspective of the left face, and between the lines 
 corresponding to them in the perspective of each receding face, as compared 
 with the distance between the lines n a and o i, in the perspective of the 
 front face. Comparing the perspective of the left face with the perspective 
 of the right, we find the former foreshortened more than the latter. The 
 left face is seen more obliquely than the right. From these facts it is evi- 
 dent, that 
 
 Surfaces seen obliquely are foreshortened, and the more obliquely they are 
 seen the more they are foreshortened. 
 
 50. There is another poinjt worthy of notice in the perspectives of the 
 receding faces, viz., the perspectives of the edges of the cube situated above 
 the eye incline downward as they recede, and the perspectives of the edges 
 below the eye incline upward as they recede. The edges referred to are 
 iiorizontal, and they are seen obliquely. From this it is evident, that 
 
OBJECTS IN PARALLEL TERSPECTIVE. 
 
 31 
 
 Horizontal lines seen obliquely, if above the level of the eye, appear to incline 
 
 downward as they recede, and if below the level of the eye, they appear to 
 fy^incUne upward. 
 
 51. In our examination of this figure, we have already observed tliat 
 horizontal lines presenting a front view, appear horizontal ; that horizontal 
 lines seen obliquely, when situated above the level of the eye, appear to 
 incline downward ; and when situated below the level of the eye, appear to 
 incline upward. From this, it is evident, that 
 
 All horizontal lines on a level with the eye appear horizontal. 
 
 52. We have now reached a point where it becomes desirable that the 
 pupil should commence drawing from the object itself. In drawing from 
 objects, the sketcher relies entirely upon the eye and the reason to guide him 
 in determining their perspectives. The ability to recognize with readiness 
 and fidelity the form* presented to the eye by an object, is the result of 
 education ; nothing but discipline can beget it ; and therefore the sooner the 
 practice of drawing from the solid is commenced, after'' the mind is made 
 acquainted with the principles of perspective, the better. 
 
 53. Let the pupil procure a cube, 
 and so place it that it may be seen, 
 as near as may be, as represented in 
 Fig. 20 ; that is, let the cube be 
 placed on the right, below the level 
 
 • of the eye, and let its distance on 
 the right be a little greater than its 
 distance below the level of the eye. 
 As a matter of convenience, we will 
 suppose the cube to be lettered as 
 the drawing of the cube in the figure 
 referred to. 
 
 In drawing from any object, the first thing to be done is to decide upon the 
 size of the drawing. This, of course, may be large or small, as best suits the 
 wish of the draughtsman. The size of the drawing fixed upon, the question 
 arises, what part of the object shall be first represented. In drawing any 
 object, always begin with that part which may, with the greatest certainty, be 
 described with accuracy. In this instance, the front face of the cube should, 
 first be drawn. Tliis face presents a front vievj, and appears of its actual 
 form. Draw a square, as A B C D, and assume ^hat this figure represents 
 the front face of the cube. 
 
 In drawing the outline of any object, it should at first be made as light as is consistent with dis- 
 tmctness, and'when the outline is complete, if found to be correct, it may be strengthened. 
 
 Fig. 20. 
 
32 
 
 LINEAR PERSPECTIVE. 
 
 The left face of the cube is seen obliquely, and it is foreshortened. The 
 edge J H appears to be on the left of A B. To make this fact apparent, lot 
 the pupil hold a rule between the eye and the cube, so that the edge of the 
 rule shall exactly coincide with the edge A B ; then let him bring the edge 
 of the rule to coincide with the edge J H, and he will find, in doing this, 
 that the rule must be moved to the left. Having recognized the fact, that 
 the edge J H appears to be on the left of A B, the point to be decided is, 
 what is the apparent distance between these edges, compared with the width 
 of the front face. Suppose it is judged to be one third the width of the front 
 face ; then place a point to mark the apparent width of the face, and draw 
 through it a line, as J H, of indefinite length, parallel to A B. 
 
 The upper face of the cube is seen more obliquely than the left face, and 
 it is foreshortened more than the latter. The edge J E appears higher than 
 the edge A D, as will be seen if the rule is held so as to bring one of its 
 edges to coincide,- first with the edge A D, and then with J E. Determine 
 the apparent distance between these edges, comparing it with the height of 
 the front face. Suppose that the edge J E is found to be above A D, one 
 fifth the height of the front face ; then place a point to mark its apparent 
 height, and through this point draw a line, as J E, to meet J H, of indefinite 
 length, parallel to A D. 
 
 Draw a line, as J A. If the apparent width of the left and the upper face 
 of the cube has been accurately determined, this line truly represents the 
 apparent length and direction of the edge J A. 
 
 The edges J H and J E being farther in the distance than A B and A D, 
 they appear shorter than A B and A D. Make the line corresponding to 
 J E a little shorter than the perspective of the edge A D, and draw a line, 
 as E D. Make the line corresponding to J H equal the perspective of the 
 edge J E, and draw a line, as H B, which completes the perspective of the 
 visible outline of the cube. 
 
 If the cube were transparent, so that the back face could be seen, this face 
 would appear of its actual form, for it would present a front view. To com- 
 plete the outline of the cube, place a point, as F, on a level witli the point 
 corresponding to H, and directly under the point corresponding to E, and 
 draw lines, as H F, E F, and F C. 
 
 The drawing should now be carefully compared with the cube, and if found 
 to be correct, its faint outline may be made stronger. 
 
 54. The accuracy of a drawing may be determined by the idea which it 
 conveys to the mind. As a matter of course, no drawing can convey the 
 idea of a cube, unless it truly represents its apparent form. 
 
 55. When the cube can be readily and accurately represented, the draw- 
 
OBJECTS IN PARALLEL PERSPECTIVE. 
 
 33 
 
 ing of simple rectilinear forms should follow. Articles of lioiisehold furni- 
 ture furnish excellent subjects for first ju-actico. In drawing any object, first 
 make its general outline ; tlien proceed with its details, taking them u}) in the 
 order of their importance. For example, suppose the pupil has selected a 
 subject, as represented in Fig-. 21. Draw, lirst, the main outline of the 
 chest, drawing the lines in the same order as they were drawn in the cube. 
 
 The general outline of the cover should next be drawn. The inclination 
 of any line is most readily and accurately determined by comparing it with 
 a horizontal line or with a vertical line. It is often the case, however, that 
 there are no lines connected with the line to be drawn, bearing either of these 
 relations to it ; when this occurs, a line must be conceived to exist. In the 
 present instance, we have the vertical line B A, and the horizontal line D 13, 
 with which to compare the inclination of B C. 
 
 Next, draw the lines representing the thickness of the boards of which the 
 chest and cover are made. Then draw the lines in the interior of the box 
 and cover formed by the meeting of the sides and ends ; then those formed 
 by the meeting of the top with the sides and ends of the cover ; then draw 
 the panels. The method adopted in drawing the chest will suggest the 
 proper course to pursue in drawing a 
 great variety of objects. 
 
 56. In connection with drawing from 
 objects, it is a most excellent practice to 
 draw occasionally from copies, after the 
 following manner : Draw the object rep- 
 resented in Fig. 22, as it would appear 
 if seen on the opposite side of the ob- 
 server, that is, on the left ; or as it would 
 appear if the eye were dropped to a level 
 5 
 
 Pig. 22. 
 
34 
 
 LINEAR PERSPECTIVE. 
 
 with the top ; or as it would appear if it were turned partly round, so as 
 to have the end, instead of the front, present a front view. Practice of this 
 kind is well adapted to test one's knowledge of principles. 
 
 57. Prob. 6. In Fig. 23, A E J I is the plan of a cube, and also of a 
 pyramid; the base of the pyramid rests on the upper face of the cube, and the 
 height of its vertex above its base is equal to the measure of one half one side 
 its base. It is required to find the perspective of the cube and pyramid. 
 
 Find the perspective of the cube as in Prob. 5. Find the perspective of 
 the pyramid as follows : Connect the point Z, the plan of the vertex, with 
 the picture plane by a perpendicular, as F Z. On the point F erect a vertical 
 line, as N F, equal in length to the height of the cube and the height of the 
 pyramid taken together. The line N P measures the actual height of the 
 
 
 
 
 
 
 T 
 
 
 
 
 p 
 
 
 
 
 
 
 
 
 
 
 
 
 H D 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 B 
 
 
 
 
 
 
 X 
 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 
 
 
 E 
 
 
 
 
 J 
 
 
 Fig. 23. 
 
 vertex above the ground plane. Conceive a line connecting the point N with 
 the vertex, and we have a line perpendicular to the picture plane ; its per- 
 spective tends to C. Rule 1. Draw N C, and we have its indefinite perspective. 
 
 Conceive a line connecting the vertex of the pyramid with the centre of 
 its base, and we have a vertical line ; its perspective is vertical. Rule 2. The 
 perspective of the lower extremity of this line is the perspective centre of the 
 
OBJECTS IN PARALLFX PEBSPECTIVE. 
 
 35 
 
 figure V o s n, the perspective of the base of the pyramid, as well as of the 
 upper face of the cube. To determiue the perspective centre of tliis fijj^ure, 
 draw the (Jiagoiials v s and o n, and the point the point of their intorsee- 
 tion, is the point required. On the point u erect a vertical line, as u x, to 
 meet N C, and we have the perspective of the line conceived to connect the 
 vertex of the pyramid with the centre of its base, and the pohit x, the point 
 of its intersection with N C, is the perspective of the vertex. 
 
 Connect the points v o s and n with x and the drawing is complete. 
 
 58. The principles of perspective are more frequently violated in the 
 representation of the pyramid than in the drawing of any other form equally 
 simple. The error usually consists in misplacing the point at the vertex. 
 Suppose we are sketching a building surmounted by a tower, the roof of 
 which is in the form of a pyramid ; such, for example, as we have repre- 
 sented in Fig". 24 ; having completed the outline of the tower, we are about 
 to draw the roof. By what method can we determine its perspective ? 
 
 Draw a straight line from A to B, and we have a diagonal of the base, a 
 line corresponding to v s. Fig. 23. The perspective of the vertex of the 
 roof is directly over the perspective centre of this line. The perspective 
 centre of this line is a little nearer B than A, (see Fig. 23,) owing to the 
 fact that the extremity B is farther in the distance than A. Determine the 
 
 c 
 
 Fig. 24. 
 
36 
 
 LINEAR PERSPECTIVE. 
 
 perspective centre of the line, as near as you can judge, and place a point, 
 as D, and on this point erect a vertical line, as D C, the apparent height of 
 the roof. Connect the point C with the points at the vertices of the angles 
 of the base, as shown in the drawing, and the outline of the roof is complete. 
 
 59. Prob. 7. In Fig. 25, we have the plan and side elevation of a build- 
 ing. It is required to find its perspective. 
 
 In the plan the double lines represent the walls of the building, and the line S P the ridge of 
 the roof. 
 
 Find au e i, the perspective of the plan, as in Prob. 3. 
 
 Find the perspective of the end wall on the left, as follows : — 
 
 On the point V erect a vertical line of indefinite length, and in this line 
 place a point, as W, at a height above Y, equal to the measure of W A', the 
 height of the vertical sides of the walls. 
 
 Conceive a line drawn across the wall, as N' 0', in the elevation, con- 
 necting the upper extremities of its vertical sides, and we have a line perpen- 
 dicular to the picture plane ; its perspective tends to the centre of view. 
 Eule 1. Draw W C, and we have its indefinite perspective. 
 
 The vertical sides of the wall are parallel to the picture plane ; their per- 
 spectives are vertical. Eule 2. On the points a and u erect vertical lines to 
 meet W C, and we have o a and n u, the perspectives of the vertical sides of 
 the wall. 
 
 Take the measure of the line K' T', the height of the peak of the gable 
 above the base of the wall, and lay it off on the vertical line erected on V, 
 making V X equal K' T'. The peak of the gable is vertically over S in the 
 plan. Conceive a line connecting the point X with the peak of the gable, 
 and we have a line perpendicular to the picture plane ; its perspective tends 
 to the centre of view. Rule 1. Draw X C, and we have its indefinite per- 
 ^ spective. 
 
 Conceive a vertical line to be dropped from the peak of the gable to the 
 base of the wall, as K' T', in the elevation ; the line passes through the 
 centre of the wall ; its perspective, therefore, will pass through the perspec- 
 tive centre of the perspective of the wall ; the line is parallel to the picture 
 plane ; its perspective is vertical. Eule 2. 
 
 Draw the diagonals n a and o u, and the point of their intersection with 
 each other is the perspective centre of the wall ; through this point draw a 
 vertical line to meet X C ; and ky the point of their intersection, is the per- 
 spective of the peak of the gable. 
 
 The perspective of the peak of the gable may be found by another method. The peak of the 
 gable is vertically over the point S ; its perspective, therefore, will be found in the perspective of 
 
38 
 
 LINEAR PERSPECTIVE. 
 
 a vertical line drawn to meet X C, erected on the perspective of S. Find s, the perspective of S, 
 as in Prob. 1, and on this point erect a vertical line to meet X C, and k, the point of its intersec- 
 tion with this line, is the point required. 
 
 Connect the points n and o with k, and the perspective of the wall is 
 complete. 
 
 Find the perspective of the end wall on the right, bv the method adopted 
 in determining the perspective of the end wall on the left. 
 
 Having found the perspectives of the end walls, connect the point o with 
 /, k with and n with c, and the perspective of the building is complete. 
 
 60. Prob. 8. In Fig. 26 we have the plan, the front elevation, and the 
 side elevation of a building-. It is required to find its perspective. 
 
 In the plan the double lines represent the walls of the building ; the open space in the side A I, 
 the door way ; the single lines in the sides A U and A I, the windows ; the line S T, the ridge of 
 the roof; and the figure M the plan of the chimney above the roof. 
 
 In this figure, for the want of room, the point of distance is not represented. This point may 
 be found by extending the line of the horizon, and the line Z i, the perspective of Z I, in the 
 direction of their convergence, until they intersect ; the point of their intersection is the point of 
 distance. 
 
 Find the perspective of the outline of the building, as in Prob. 7. 
 
 Find the perspectives of the windows in the end wall, as follows : — 
 
 Place a point, as J, in the vertical line erected on F, at a height above the 
 base line equal to the height of the lower line of the windows above the 
 base of the building ; also place a point, as P, in the same line, at a height 
 above F equal to the height of the upper line of the windows above the 
 base of the building. 
 
 The upper and lower lines of the windows are perpendicular to the picture 
 plane ; their perspectives tend to the centre of view. Rule 1. From the 
 points J and P, draw lines to C, and we have their indefinite perspectives. 
 
 Find the perspectives of the points in the side A U of the plan which 
 mark the width of the windows, as in Prob. 1. 
 
 The vertical sides of the windows are parallel to the picture plane ; their 
 perspectives are vertical. Rule 2. On the perspectives of the points which 
 mark the width of the windows erect vertical lines to meet the line P C, 
 and we have the perspectives of the vertical sides of the windows, which 
 completes their outline. 
 
 In determining the perspectives of the points which mark the width of the windows, we have 
 not drawn the perspectives of all the lines passing through them to their vanishing points. By 
 the course adopted, a confusion of lines is obviated, and the drawing is rendered more simple. 
 It may be well here to say, that lines drawn simply for the purpose of finding the perspectives of 
 points, not forming a part of the perspective of the object, need not be extended farther than 
 is necessary to secure the points required. 
 
40 
 
 LINEAR PERSPECTIVE. 
 
 Find the perspective of the door, and the perspectives of the windows, in 
 the front wall, as follows : — 
 
 The front windows being of the same size, and at the same height above 
 the base of the building, as the end windows, the perspective height of the 
 upper and lower lines of the windows is determined by the intersection of 
 the lines J C and P C with o a, the perspective of the corner of the building. 
 
 The upper and lower lines of the windows are parallel to the picture 
 plane ; their perspectives are parallel to the lines themselves. Rule 2. From 
 the points v and c, draw horizontal lines across the front wall, as c and v 6, 
 and we have the indefinite perspectives of the lines referred to, also the 
 indefinite perspective of the upper line of the door, this line being at the 
 same height above the base of the building as the upper line of the windows. 
 
 Find the perspectives of the points in the side A I of the plan which mark 
 the width of the windows, as shown in the drawing. 
 
 The vertical sides of the windows and door are parallel to the picture 
 plane ; their perspectives are parallel to the lines themselves. Rule 2. On 
 the perspectives of the points which mark the width of the windows and 
 door, erect vertical lines to meet c and we have the perspectives of the 
 vertical sides of the door and windows, which complete their outline. 
 
 * 
 
 In drawing the chimney, proceed as follows : — 
 
 Find h n and n k, the perspectives of the lines h' n' and n' k\in the plan 
 of the chimney, as in Prob. 3. 
 
 On the point P, where the perpendicular P n ' meets the base line, erect a 
 vertical line, as P N, equal the height of the chimney. 
 
 The line in the top of the chimney over h' n'^in the plan, is perpendicular 
 to the picture plane ; its perspective tends to the centre of view. Rule 1. 
 Draw N C, and we have its indefinite perspective. 
 
 The vertical edges of the chimney are parallel to the picture plane ; their 
 perspectives are vertical. Rule 2. On the points h and n erect vertical 
 lines to meet N C, and the drawing of the left face is complete. 
 
 The points n' and k'/m the plan, are equally distant from the picture 
 plane ; therefore the perspectives of the vertical edges of the chimney rest- 
 ing on these points are of equal length. On the point k erect a vertical 
 line equal to t ^, and we have w k, the perspective of the vertical edge 
 resting on A;'. 
 
 Connect the points t and and the visible outline of the chimney is 
 complete. 
 
 The perspective of the outline formed by the meeting of the roof and 
 chimney may be determined as follows : — 
 
 The point where the ridge of the roof meets the left face of the chimney, 
 
OBJECTS IN PARALLEL PERSPECTIVE. 
 
 41 
 
 is vertically over r' in the plan ; the perspective of the point will be in tho 
 perspective of a vertical line erected on the perspective of the point 
 Find r, the perspective of r\ as in Prob. 1. 
 
 The perspective of a vertical line erected on r is parallel to the line itself, 
 for the line is parallel to the picture plane. Rule 2. 
 
 On the point r erect a vertical hue to meet u e, the perspective of the 
 ridge, and ^, the point of their intersection with each other, is the point 
 required. 
 
 « 
 
 We may find the point y by another method. Draw the diagonals of the figure xhnt ; the 
 point formed by their intersection with each other is the perspective centre of the figure ; through 
 this point draw a vertical line to meet the line of the ridge, and the point of their intersection is 
 the point reqmred. 
 
 Another method of determining this point is suggested in the drawing. 
 
 The line formed by the meeting of the roof with the left face of the chim- 
 ney is parallel with the receding sides of the roof, and since the perspectives 
 of all lines parallel to each other tend to the same point, the perspective 
 of this line will tend to the vanishing point of the lines to which we have 
 referred. Rule 5. Prolong o u and j e, the perspectives of the receding 
 sides of the roof, as shown in the drawing, and we obtain C ', their vanishing 
 point. From C ' draw a line through y, to meet the line t n, and we have 
 y the perspective of the line. 
 
 The line formed by the meeting of the roof with the front face of the 
 chimney is horizontal, and parallel to the picture plane ; its perspective is 
 horizontal. Rule 2. From the point z, one extremity of its perspective, 
 draw a horizontal line to meet w k, and we have z s, the perspective of the 
 line, which completes the drawing. 
 
 61. Prob. 9. In Fig. 27 we have the plan, the front elevation, and the 
 side elevation of the building represented in Fig. 26, the building being seen 
 from a different point of view. It is required to find the perspective of the 
 parts visible. 
 
 In the previous examples, the objects represented are situated some little distance from, the 
 picture plane ; in this example, to simplify the drawing, one of the end walls of the building is 
 assumed to be in the picture plane. 
 
 The end wall of the building resting on the side A K of the plan being in 
 the picture plane, its perspective is in all respects similar to the outline of 
 the wall. Draw the perspective of the wall and the perspectives of the 
 windows, as in the elevation. 
 
 The lines formed by the base of the front wall, by the eaves, and by the 
 ridge of the roof, are perpendicular to the picture plane ; their perspectives 
 6 
 
OBJECTS IN PARALLEL PERSPECTIVE. 
 
 43 
 
 tend to the centre of view. Rule 1. Draw A C, T C, and Z C, and wo 
 have •their indefinite perspectives. 
 
 The vertical edge of the front wall resting on E is parallel to the picture 
 plane ; its perspective is vertical. Rule 2. Find e, the perspective of E, its 
 lower extremity, as in Prob. 1, and on this point erect a vertical line to meet 
 T C, and we have e V, its perspective. 
 
 The side of the roof parallel to that represented by T Z is parallel to the 
 picture plane ; its perspective is parallel to tlie line it represents. Rule 2. 
 The line to be represented is parallel to T Z, we may, therefore, take this 
 line as a guide in drawing its perspective. From the point V draw a line to 
 meet Z C, parallel to T Z, and we have V the perspective of the Hne. 
 
 Find the perspective of the door, and the perspectives of the windows, in 
 the front wall of the building, as the perspectives of the windows were 
 found in the end wall, in Prob. 8. 
 
 Find the perspective of the chimney, as in Prob. 8. 
 
 Find the perspectives of the lines formed by the meeting of the roof and 
 chimney, as follows : — 
 
 The point where the ridge of the roof meets the front face of the chimney 
 is vertically over S ; the perspective of the point will be in the perspective of 
 a vertical line erected on the perspective of S ; the perspective of a vertical 
 line is vertical. Rule 2. Find 5, the perspective of S, as in Prob. 1, and on 
 this point erect a vertical line to meet Z the perspective of the ridge ; 
 and 6?, the point of intersection, is the point required. 
 
 This point may be found by a different process. Draw the diagonals of the figure w ' w o o and 
 through the point of their intersection with each other draw a vertical line to meet Z w, and we 
 obtain the point of their intersection. 
 
 We may obtain the point by another method. Place a point, as d, in the line Z equally dis- 
 tant from the lines n ■ n and o' o. In sketching, the point is determined in this way. 
 
 The line formed by the meeting of the roof with the front face of the chim- 
 ney, is parallel to the picture plane ; its perspective is parallel to the line 
 itself. Rule 2. The line to be represented is parallel to T Z, and therefore, 
 in drawing its perspective, we may be guided by this Hne, From the point 
 d draw a line to meet n' n parallel to T Z, and we have z the perspective 
 of the line. 
 
 The line formed by the meeting of the roof with the left face of the chim- 
 ney is perpendicular to the picture plane ; its perspective tends to the centre 
 of view. Rule 1. From the point z, one extremity of the line, draw a line 
 to meet m ' m, in the direction of C, and we have the perspective of the line, 
 which completes the drawing. 
 
44 
 
 LINEAR PERSPECTIVE. 
 
 62. The pupil should now attempt a sketch of some simple building, 
 seen in parallel perspective. In sketching any object as large as a buiMing, 
 we may say, speaking in general terms, that the distance of the draughts- 
 man from the object should be, at least, fifteen or twenty rods, and of course 
 as much farther than this, as he may desire. In delineating the general 
 
 outline of a building of the usual form, draw the lines in the order indi- 
 cated by the numerals in Figs. 28 and 29. The dotted lines in these 
 figures are used as aids in determining the perspectives of the peaks of the 
 gables, and, therefore, in the drawing, they should be made very light. 
 
 If the pupil is not perfectly familiar with the truths to which attention 
 was called in Sections 46 — 51, they should be reviewed before attempting a 
 sketch. 
 
 OBLIQUE PERSPECTIVE. 
 
 68. Prob. 10. In Fig. 30, A E I F is the plan of a cube situated on 
 the ground plane. It is required to find its perspective. 
 
 The picture plane passes through the vertical edge of the cube, resting on 
 A ; the perspective of this edge is therefore a vertical line resting on A, 
 equal in length to the measure of either side of the plan. On the point A 
 erect a vertical line equal to A E, and we have k A, the perspective of 
 the edge. 
 
 Find e, z, and /, the perspectives of the points E, I, and F, as in Prob. 1. 
 
 Connect the points A, e, z, and /, as shown in the drawing, and we have 
 the perspective of the lower face of the cube. 
 
 Prolong the perspectives of the sides of the lower face of the cube until 
 they intersect the horizon, and we obtain Y and D, their vanishing points. 
 
 The edge of the cube, situated over A E, is parallel to A E ; its perspec- 
 tive tends to Y, the vanishing point of A e, the perspective of A E. Rule 5. 
 Draw k Y, and we have its indefinite perspective. The edge over A F is 
 
(45) 
 
46 LINEAR PERSPECTIVE. 
 
 parallel to A F ; its perspective tends to D, the vanishing point of A /, 
 the perspective of A F. Rule 5. Draw k D, and we have its indefinite 
 perspective. 
 
 The vertical edges of the cube are parallel to the picture plane ; their 
 perspectives are vertical. Rule 2. On the points e and /, the perspectives 
 of the lower extremities of the vertical edges resting on E and F, erect 
 vertical lines to meet the lines k D and k Y, and we have j e and m /, the 
 perspectives of these edges. 
 
 The edge situated over E I is parallel to E I ; its perspective tends to D, 
 the vanisliing point of e i, the perspective of E I. Rule 5. Draw a line 
 from 7, the perspective of one extremity of the edge, to D, and we have j D, 
 its indefinite perspective. The edge over I F is parallel to I F ; its perspec- 
 tive tends to Y, the vanishing point of i /, the perspective of I F. Rule 5. 
 Draw a line from the perspective of one extremity of the edge, to Y, and 
 we have m Y, its indefinite perspective. 
 
 Connect the point n, formed by the intersection of the lines last drawn, 
 with i, and we have n the perspective of the vertical edge resting on In, 
 which completes the perspective of the cube. 
 
 64. The pupil should now practise drawing from the cube, seen in 
 oblique perspective. Place a cube ' so as to be seen as represented in 
 Fig-. 31 ; that is, place the cube on a horizontal plane, and so arrange it that 
 the point corresponding to H, in the drawing, shall be farther in the distance 
 than the point corresponding to E. Suppose the cube lettered as shown in 
 the drawing. 
 
 c 
 
 Pig. 31. 
 
OBJECTS IN OBLIQUE PERSPECTIVE. 
 
 47 
 
 In drawing the cube, begin by making a vertical line, as A C, of any 
 desired length, and assume that this line represents the edge A C. 
 
 The faces F H C A, and AGED, are seen obliquely, and are therefore 
 foreshortened; the face F H C A is seen more obliquely than the face 
 A C 10 D, and is therefore foreshortened more than the latter. Taking the 
 edge A C as a measure, determine the apparent distance of the edge F H 
 from A C, and the apparent distance of D E from A C, measuring in the 
 direction of the line R T, and draw lines of hidefniite length, as N 0 and 
 P S, parallel to the perspective of A C. 
 
 The edge C H, being below the level of the eye, appears to incline 
 upwards. To find the perspective of the edge C H, conceive a horizontal 
 line drawn from the point H to meet the edge A C, as H B ; or hold a 
 string in this position, so as to cover the point H ; note the size of the angle 
 formed by the line or string, with the edge C H ; also note the distance 
 from B, the point in the edge, A C, where the line or string appears to 
 intersect it, to 0, as compared with the length of the edge A C, and, 
 according to the best of your judgment, place a point, as H, in the line 
 corresponding to N 0, and draw a line, as H C. 
 
 In like manner determine the perspective of the point E, and draw C E. 
 As the cube is situated, the point E does not appear to be as high above the 
 level of C as the point H, for the reason that E is not as far in the dis- 
 tance as H. 
 
 The edge F H is farther in the distance than A C, and for this reason it 
 appears shorter than A C, Determine its apparent length, and draw a line, 
 as FA. 
 
 The edge D E, being farther in the distance than A C, appears shorter than 
 this edge, and not being as far in the distance as F H, appears longer than 
 F H. Determine its apparent length, and draw a line, as A D. 
 
 The parallel edges A D and F J, being seen obliquely, appear to con- 
 verge as they recede from the eye. Bearing in mind that the perspec- 
 tives of parallel lines tend to the same point, draw a line, as F J, of indefi- 
 nite length, approaching the perspective of A D. 
 
 The parallel edges A F and D J appear to converge a^ they recede, 
 tending to the same point as A F and C H. From the point D draw a line 
 to meet the line last drawn, approaching the perspective of A F and C H, 
 and we have the perspectives of those faces which are seen. 
 
 The perspectives of the faces which are not seen may be determined as 
 follows : — 
 
 The edge J I is farther in the distance than F H, the most distant of the 
 vertical edges which are seen ; its perspective, therefore, is shorter than 
 that of F H. From the point corresponding to J draw a vertical line, as 
 J I, shorter than the perspective of the edge F H, and of such a length that 
 
48 
 
 LINEAR PERSPECTIVE. 
 
 a line connecting the points corresponding to H and I, will tend to the same 
 point as the perspectives of the edges F J, A D, and C E. 
 
 Connect the points corresponding to H and I, and the points correspond- 
 ing to I and E, and the drawing is complete. 
 
 65. When the cube can be drawn with readiness and accuracy, the draw- 
 ing of simple objects in oblique perspective should follow. A plain, old- 
 fashioned chair, such as we have represented in Fig. 32, will furnish tlie 
 pupil with a desirable subject. In drawing a chair of this description, first 
 draw an outline like that in Fig. 33, giving the general form of the chair, 
 
 Fig. 32. Fig. 33. 
 
 as presented to the eye, making the lines very faint ; guided by this out- 
 line, draw in the details. 
 
 The method pursued in drawing the chair will suggest the proper course . 
 to take in drawing a great variety of objects, such as tables, stands, desks, 
 and the like. * 
 
 66. Fig. 32 may be taken as an exercise in drawing from the copy. 
 Draw the chair as it would appear if seen from a point farther to the right, 
 or left, than that here taken ; or as it would appear if the back legs rested 
 on the points E and F ; or on the points E and I ; or on the points I and J. 
 
 6T. Prob. 11. In Fig. 31 ive have the plan and elevations of the build- 
 ing represented in Figs. 26 and 27, the building being seen from a different 
 point of view. It is required to find its perspective. 
 
50 LINEAR PERSPECTIVE. (J 
 
 The perspective of the end wall, resting on the side A U of the plan, may 
 be found as follows. 
 
 The vertical corner of the house, resting on the point A, is in the picture 
 plane ; its perspective, therefore, is a vertical line resting on A, equal in 
 length to the line itself. On the point A erect a vertical line equal to 0' U', 
 and we have F A, the perspective of the vertical side of the wall, rest- 
 ing on A. 
 
 Find the perspective of U, as in Prob. 1, and connect ^this point with 
 A, and we have A the perspective of A U, the base of the wall. 
 
 Prolong the line A to meet the horizon, and we obtain V, its vanishing 
 point. 
 
 Conceive a line drawn across the wall, as 0' F', in the elevation, connect- 
 ing the upper extremities of its vertical *sides, and we have a line parallel to 
 the base of the wall ; its perspective tends to V. Rule 5. From the point 
 F draw a line to V, and we have F Y, its indefinite perspective. 
 
 The vertical side of the wall resting on U is parallel to the picture plane ; 
 its perspective is vertical. Pule 2. On the point u erect a vertical line to 
 meet F Y, and we have o, its perspective. 
 
 Prolong the vertical line F A, as shown in the drawing, making M A equal 
 the height of the peak of the gable above the base of the wall. 
 
 Conceive a line drawn through K', the peak bf the gable, parallel to U' A', 
 the base of the wall ; its perspective tends to Y, the vanishing point of A 
 the perspective of the base of the wall. Rule 5. Draw M Y, and we have 
 its indefinite perspective. 
 
 The peak of the gable is vertically over the point S, in the plan ; its per- 
 spective is a point in the line M Y, where the perspective of a vertical line, 
 which we may conceive to be erected on S, will intersect this line. The 
 perspective of a vertical line is vertical. Rule 2. Find 5, the perspective of 
 S, as in Prob. 1, with this exception ; omit drawing the indefinite perspec- 
 tive of c S. This line may be dispensed with ; for the point S, being in the 
 line A U, its perspective must be in A the perspective of this line ; and 
 since it is in A it must be at 5, where the indefinite perspective of E' S 
 intersects this line. On the point s erect a vertical line to meet M Y, and A:, 
 the point of their intersection, is the perspective of the peak of the gable. 
 
 Connect the point o with A;, and k with F, and the perspective of the out- 
 line of the wall is complete. 
 
 To find the perspectives of the windows in this wall, proceed as follows : — 
 
 In the vertical line erected on A, place a point, as Q, at a height above A 
 equal to the height of the lower line in the windows, above the base of the 
 building ; also place a point, as G, in the same line, at a height above A 
 equal to the height of the upper line in the windows above the base of the 
 building. 
 
OBJECTS IN OBLIQUE PERSPECTIVE. 
 
 51 
 
 The upper and lower lines in the windows are parallel to the base line of 
 the wall ; their perspectives vanish at V, the vanishing point of A u. Rule 5. 
 From the points Q and G draw lines to meet o u, in the direction of V, and 
 we have the indefinite perspectives of the upper and lower lines of the 
 windows. 
 
 The perspectives of the vertical sides of the windows arc vertical. Rule 2. 
 Find the perspectives of the points in the plan which mark the width of the 
 windows, as the perspective of the point S was found, and on these points 
 erect vertical lines to meet G F, and we complete the perspectives of the 
 windows. 
 
 The perspective of the front wall may be determined by a process entirely 
 similar to that by which the perspective of the end wall was found ; but 
 since the vanishing point of the line A I is without the limits of our page, 
 we must find its perspective by another method. 
 
 Find ^, the perspective of I, as in Prob. 1. 
 
 Prolong the side I E of the plan to meet the picture plane, as shown in 
 the drawing, and on Y, the point of its intersection with the base line, erect 
 a vertical line of indefinite length, and in this line place a point, as W, at a 
 height above Y equal to the height of the wall. 
 
 Conceive a line connecting the upper extremity of the vertical side of the 
 wall resting on I, with W, and we have a line parallel to Y I ; its perspec- 
 tive tends to the vanishing point of this line. Rule 5. The line Y I being 
 parallel to the line A U, its vanishing point is V. Draw a line connecting 
 the points W and V, and we have W V, the indefinite perspective of the line 
 connecting the points referred to. 
 
 The perspective of the vertical side of the wall resting on I, is vertical. 
 Rule 2. On the point i, the perspective of I, erect a vertical line to meet 
 W Y, and we have i b, the perspective of this side of the wall. 
 
 Connect the point F with b, and A with i, and the outline of the wall is 
 complete. 
 
 To find the perspective of the door, and the perspectives of the windows, 
 proceed as follows : — 
 
 In the vertical line erected on Y place a point, as H', at a height above Y 
 equal to the height of the lower line in the windows above the base line of 
 the wall ; also place a point, as L', in this line, at a height above Y equal to 
 the height of the upper line in the windows and door above the base of the 
 building. 
 
 Conceive the lower line in the windows, and the upper Hue in the door 
 and windows, prolonged to meet the vertical sides of the wall, as shown in 
 the elevation. Conceive a line connecting the point L' with a point in the 
 
52 
 
 LINEAR PERSPECTIVE. 
 
 vertical side of the wall resting on I, corresponding to D', in the elevation ; 
 also conceive a line connecting the point H' with a point in this side of the 
 wall corresponding to B' in the elevation. These lines are parallel to that 
 represented hjWb; their perspectives vanish at V. Rule 5. From the 
 points L' and draw lines to meet b i, in the direction of V, and we obtain 
 / and /, the perspectives of points in the vertical side of the wall resting on 
 I, corresponding to B' and B', in the elevation. 
 
 The points G and Q, in the perspective of the vertical side of the wall 
 resting on A, correspond to G' and QS in the elevation. Connect the point 
 G with /, and Q with /, and we have G / and Q the indefinite perspectives 
 of the upper and lower hues of the windows, and the upper line of the 
 door. 
 
 From the points in the side A I of the plan, which mark the width of the 
 door and windows, draw lines to meet the base line of the picture plane, 
 parallel to the side A U of the plan. The perspectives of these lines tend 
 to Y. Rule 5. From the points where these lines intersect the base line, 
 draw lines to meet A i, in the direction of V, and the points of their inter- 
 section with A i are the perspectives of the points in the side A I of the 
 plan which mark the width of the door and windows. On the points thus 
 determined, erect vertical lines to meet G /, and we complete the drawing 
 of the door and windows. 
 
 The perspective of the roof may be found as follows : — 
 
 We have already found k F and F b, the perspectives of two sides of the roof, and to complete 
 its outline, we have only to findj, the perspective of the end of the ridge over T, in the plan, and 
 to connect this point with 7i and b. 
 
 In the vertical line erected on Y place a point, as X, at a height above Y 
 equal to the height of the ridge of the roof above the base of the building. 
 
 Conceive a line connecting the end of the ridge over T with the point X, 
 and we have a line parallel to Y E ; its perspective tends to Y, the vanishing 
 point of this line. Rule 5. Draw a line from X to Y, and we have the 
 indefinite perspective of the line conceived to connect the end of the ridge 
 with X. 
 
 The end of the ridge is vertically over T ; its perspective is a point in the 
 line drawn from X to Y where the perspective of a vertical line, which he 
 may conceive to be erected on T, will intersect this line. The perspective 
 of a vertical line is vertical. Rule 2. Find t, the perspective of T, as in 
 Frob. 1, and on this point erect a vertical line to meet X Y ; and/, the point 
 of their intersection, is the perspective of the end of the ridge over T. 
 
 Connect the point k with j, and / with and the outline of the roof is 
 complete. 
 
O^ECTS IN OBLIQUE PERSPECTIVE. 
 
 63 
 
 The perspective of the chimney may be found as follows : 
 
 Prolong the lines n' h' and d' x' , in the plan of the chimney, to meet the 
 base line of the picture plane. The lines N h' and J x' are parallel to A U ; 
 their perspectives tend to V. Rule 5. Draw N V and J V, and wc have 
 their indefinite perspectives. 
 
 On the points N and J erect vertical lines, as N N' and J J', equal to the 
 height of the chimney. 
 
 Conceive the upper lines of the chimney parallel to n' h' and d' x' pro- 
 longed to meet the picture plane ; they intersect the plane at N' and J' ; 
 their perspectives tend to V, the vanishing pohit of the perspectives of N A' 
 and J x'o Rule 5. Draw hues from N' and J' to V, and we have their 
 indefinite perspectives. 
 
 Find h, n, and d, the perspectives of h' n' and d', in the plan, as the per- 
 spective of the point S was found. 
 
 The perspectives of the vertical edges of the chimney are vertical. 
 Rule 2. On the points A, n, and d, erect vertical hues to meet N' V and 
 J' Y, and we have the perspectives of the vertical edges which are seen. 
 
 Connect the points a and iv, and the drawing of the chimney is complete. 
 
 The perspective of the outline formed by the meeting of the roof with 
 the chimney may be found as follows : — 
 
 The point where the ridge of the roof meets the left ^ce of the chimney 
 is vertically over r ' ; the point where the ridge of the roof meets the right 
 face of the chimney is vertically over v ' ; the perspectives of these points 
 will be in the line kj^ the perspective of the ridge, where the perspectives of 
 vertical lines, which we may conceive to bo erected on r ' and v will inter- 
 sect this line. The perspectives of vertical lines are vertical. Rule 2. To 
 find the perspectives of r' and v\ draw a line from 5, the perspective of S, 
 to the perspective of T ; the point r, where s t intersects N C, is the per- 
 spective of r and the point where it intersects J C, is the perspective of v 
 On the points r and v erect vertical lines to meet k y, and we obtain e and 
 m, the perspectives of the points referred to. 
 
 The lines formed by the meeting of the roof with the left and right faces 
 of the chimney are parallel to the sides of the roof, represented by F k and 
 b j ; their perspectives tend to the vanishing point of these lines. Rule 5. 
 Prolong the lines F k and b j in the direction of their convergence, until 
 they intersect each other, and from the point of their intersection draw a 
 line through e to meet the line a and a line through m to meet the line 
 w d, and we have e z and m the perspectives of the lines formed by the 
 meeting of the roof with the left and right faces of the chimney. 
 
 If the pupil has been accurate in his drawing, the lines F k and bj intersect at a poiiit 
 vertically over V. 
 
64 
 
 LINEAR PERSPECTIVE. .« 
 
 Connect the points z and and we have the perspective of the line formed 
 by the meeting of the roof with the front face of the chimney, which com- 
 pletes the drawing. 
 
 68. In sketching a building seen in oblique perspective, of the form rep- 
 resented in Figs. 35 and 36, draw the lines in the order indicated by the 
 
 Fig. 35. Fig. 36. 
 
 numerals. The dotted lines in these figures, which are not used in deter- 
 mining the perspectives of the peaks of the gables, represent imaginary 
 lines. The remarks made on the line H B, in Fig. 31, will suggest the 
 use of these lines. • 
 
 OBJECTS CONTAINING CURVED LINES. 
 
 69. Prob. 12. In Fig. 37, NIMK is a circle lying on the ground 
 plane. It is required to find its perspective. 
 
 The perspective of a circle is determined by finding the perspectives 
 of a number of points in its circumference, and drawing a curved line 
 through them. The number of points usually taken is eight, and they 
 are determined as follows : — 
 
 Dl-aw a square enclosing the circle, as A F E J, making the side A J par- 
 allel with the base line of the picture plane. 
 
 Draw the diagonals A E and F J, and through the point of their intersec- 
 tion with each other draw lines, as I K and N M, parallel with the sides of 
 the square. 
 
 Through the points where the diagonals intersect the circumference of the 
 circle, draw lines, as shown in the figure. 
 
 We now have eight points in the circumference of the circle, equally dis- 
 tant from each other, viz., N, 0, 1, Y, M, U, K, and S. The perspectives of 
 these points may be found as follows : — 
 
V 
 
 OBJECTS CONTAINING CURVED LINES. 65 
 
 Find afej, the perspective of the square A F E J, as in Prob. 2. 
 Draw a e and / y, and we have the perspectives of the diagonals A E 
 and F J. 
 
 Tlic line N M is parallel to the picture plane ; its perspective is parallel to 
 N M. Rule 2. Through c, the perspective of C, draw a line to meet W C 
 and P C, parallel to N M, and we have w, the perspective of N, and w, the 
 perspective of M. 
 
 Prolong the lines 0 S, I K, and V U, to meet the base of the picture 
 plane ; these lines are perpendicular to the picture plane ; their perspectives 
 tend to C. Rule 1. Draw X C, Y C, and Z C, and we have their indefinite 
 
 perspectives. The points o, s, and v, formed by the intersection of the 
 lines X C and Z C with a e and / j, the perspective? of the diagonals, are 
 the perspectives of the points 0, S, U, and Y. The points i and k, formed 
 by the intersection of the line Y C with a j and f e, the perspectives of the 
 sides A J and F E of the square, are the perspectives of the points I and K. 
 
 Through the points n, o, t, v, m, w, and s, draw a curved line, as shown 
 in the figure, and we have the perspective of the circle. 
 
 70. In Fig. 37, the eight points, N, 0, 1, &c., in the circumference of the 
 circle, are equidistant from each other ; if these points be properly connected 
 
56 LINEAR PERSPECTIVE. 
 
 by straight lines, the figure formed is an octagon, and if the perspectives of 
 these points be properly connected by straight lines, the figure formed is the 
 perspective of the octagon. 
 
 71. Examining the perspective of the circle, we observe that the per- 
 spective of the half nearest the eye is larger than that of the half most dis- 
 tant, the distance from i to c being greater than from c to k. Examining 
 the perspective of the circumference of the circle, we notice that no part of 
 the line is the arc of a circle, the figure described by it being that of an 
 ellipse. 
 
 To represent a circle seen obliquely is more difficult than the drawing of any other simple 
 form, and it will be well for the pupil to make a careful study of the perspective of this figure 
 before he attempts to represent an f>bject containing curved lines. 
 
 72. In sketching, the perspective of the circle may be determined as 
 follows : Suppose, for example, that you are about to draw the outline of a 
 wheel, seen as the circle N I M K, in Fig. 37, when the page lies horizon- 
 tally before you, and you are looking in the direction of the line K I. 
 
 Draw, first, a light, horizontal line, as 
 a 6, in Fig. 38, and assume that this line 
 represents the distance between the points 
 in the rim of the wheel corresponding to 
 N and M in the circle. 
 
 Determine the apparent distance be- 
 tween the points in the rim of the wheel, 
 corresponding to K and I in the circle, as compared with the distance 
 between the points corresponding to N and M, and place points, as i and e, 
 Fig. 38, opposite the centre of the line a b, to represent the apparent dis- 
 tance between the points referred to, making the distance from ^ to c a little 
 less than from e to c. 
 
 Guided by the line a b, draw through the points a, i, b, and e, the per- 
 spective of the circle, making the line, at first, very light. In drawing this 
 line, begin with that part which, when drawn^will not be covered by the 
 pencil or hand in drawing the remainder of the line. The perspective of a 
 circle, seen obliquely, in any position, may be drawn in a similar manner. 
 
 73. Prob. 13. In Fig. 39, the circle N I M K i5 the plan of a cylinder; 
 the lower base rests on the ground plane ; the height of the cylinder is equal 
 to C M, one half the diameter of the circle. It is required to find its per- 
 spective. 
 
 Find the perspective of the lower base of the cylinder, as in Prob. 12. 
 
OBJECTS CONTAINING CURVED LINES. 
 
 57 
 
 To find the perspective of the upper base, proceed as follows : — 
 On the points Q and P erect vertical lines equal to C M, the height of 
 the cylinder, and connect the points T and G. 
 
 In the line T G place a point, as ^V, vertically over W ; a point, as X', 
 vertically over X ; a point, as Y', vertically over Y ; and a point, as Z', ver- 
 tically over Z. 
 
 Conceive the upper base of the cylinder circumscribed by a square, and 
 lines drawn across it, as shown in the plan, and these lines extended to meet 
 the picture plane. The line corresponding to W P meets the plane at W', 
 the line corresponding to X S meets the plane at X', the line corresponding 
 to Y K meets the plane at Y', the line corresponding to Z U meets the plane 
 at Z', the line corresponding to P E meets the plane at G, and the line cor- 
 responding to Q E meets the plane at T. The lines meeting the plane at 
 W, X', Y', Z', and G, are perpendicular to the picture plane ; their perspec- 
 tives tend to C. Rule 1. Draw W C, X' C, Y' C, Z' C, and G C, and we 
 have their indefinite perspectives. The line meeting the plane at T is par- 
 allel to Q E ; its perspective tends to D. Rule 3, 
 
 S C D R 
 
 To complete the drawing of the upper base, proceed as in drawing the 
 lower base. 
 
 Connect the point r with w, and d with w, and the drawing is complete. 
 
 8 
 
58 
 
 LINEAR PERSPECTIVE. 
 
 74. In sketching any cylindrical object, as a cask, for example, to de- 
 termine its general outline, first draw a rectangle, as A B C D, in Fig. 40, 
 giving the apparent height of the cask, and its width through the chimes. 
 Guided by the line A D, draw the ellipse as in Fig. 38, forming the perspec- 
 tive outline of the upper head ; guided by the lines A B and D C, draw the 
 outlines of the upright sides ; and guided by the line B C, draw what is 
 seen of the outline of the lower head. 
 
 1 
 
 Fig. 40. 
 
QUESTIONS. 
 
 1. What instruction is given on the selection of paper? Why should the paper 
 be placed upon a hard surface in drawing? What is the objection to rolling paper? 
 
 2. What are the desirable qualities of a lead pencil ? Whose pencils are recom- 
 mended ? How are Faber's pencils marked ? What pencils are suitable for draw- 
 ing diagrams ? What pencils are suitable for sketching ? What instruction is given 
 on sharpening the pencil? 
 
 3. What are the desirable qualities of rubber ? Whose rubber is recommended ? 
 What is said on the use of rubber ? How are you to avoid the necessity of too fre- 
 quently resorting to its use ? 
 
 4. What is said of the rule ? 
 
 5. For what is the right-angled triangle used ? What size is most convenient ? 
 Upon what does the value of the instrument depend ? How may the accuracy of 
 the right angle be tested ? How is this instrument used in drawing perpendiculars ? 
 How is it used in drawing parallels ? 
 
 6. What is said of the compasses? To what use is this instrument applied? 
 When working with the compasses, how should they be held ? 
 
 CHAPTER I. 
 Of what does this chapter treat ? 
 
 7. What is linear perspective ? What is the perspective of an object ? What is 
 the perspective of a line or point ? When the perspective of a line is indefinitely 
 prolonged, what is the line called ? 
 
 8. What is a plane ? May we assume any plane to be extended indefinitely in 
 » any required direction ? 
 
 9. What is the ground plane ? What is the position of the ground plane ? What 
 part of Fig 1 represents this plane ? 
 
 10. What is the picture plane ? What is the position of the picture plane? How 
 is it situated with regard to the observer ? What part of Fig. 1 represents this plane ? 
 
 11. What is the point of view ? What represents this point in Fig. 1 ? 
 
 12. What is the centre of view ? What represents this point in Fig. 1 ? 
 
 13. What is said of the horizon? Does the horizon under all circumstances 
 
 (59) 
 
60 
 
 LINEAR PERSPECTIVE. 
 
 appear to be on a level with the eye ? What is the perspective of the horizon called ? 
 What represents the horizon in Fig. 1 ? 
 
 14. What is the distance of picture ? What line in Fig.- 1 measures the distance 
 of picture? 
 
 15. What is the point of distance ? 
 
 16. What are visual rays ? How are visual rays represented ? 
 
 17. What is a vanishing point ? 
 
 18. What is an elevation ? 
 
 19. What is a plan ? What must be indicated in the plan of an object ? At how 
 many points was the building represented in Fig. 3 cut to enable us to construct the 
 plan represented in Fig. 5 ? At what points was the building supposed to be cut ? 
 
 Turn to Figs. 2, 4, and 5. In what is the length of the building shown ? In what 
 is the breadth of the building shown ? In what is the height of the walls shown ? 
 In what is the vertical height of the roof shown ? In what is the pitch of the roof 
 shown ? In what is the height of the chimney shown ? In what is the width of the 
 front and back face of the chimney shown ? In what is the width of the faces of the 
 chimney parallel to the end walls shown ? In what is the width of the windows 
 shown ? In what is the width of the door shown ? In what is the situation of the 
 windows and door shown ? In what are the height of the door, the height of the win- 
 dows, and their height above the ground shown ? 
 
 CHAPTER II. 
 
 Of what does this chapter treat ? 
 
 20. Upon what science is the art of linear perspective based ? State the facts 
 connected with vision. 
 
 21. Draw upon the board a diagram explaining the phenomenon of vision. 
 
 22. What is said in regard to viewing an object through a transparent plane ? Go 
 to the board and illustrate this truth. What general truth is deduced from this inves- 
 tigation ? 
 
 23. What is said in regard to finding the perspectives of lines, straight or curved ? 
 
 24. What part of Fig. 8 represents the square which it is proposed to place in 
 perspective ? How is the square supposed to be situated ? What is supposed to rep- 
 resent the ground plane ? What part of the figure represents the picture plane ? 
 What relation is the plane M B L N supposed to bear to the plane of the paper upon 
 which the square is traced ? Where is the observer supposed to stand ? Is the point 
 A supposed to be on the same plane as the square ? At what height is the eye sup- 
 posed to be above the point A ? What relation would a line drawn from the eye to 
 the point C on the picture plane bear to that plane ? What is the point C called ? 
 What line in this figure measures the distance of picture ? What is the figure F e ^ D 
 called ? What are the lines F C and D C called ? Show how the perspective of the 
 square is determined. 
 
 25. What relation do the sides E F and I D of the square bear to the picture 
 plane ? To what point do their perspectives tend ? Is the truth here recognized a 
 
QUESTIONS. 
 
 Gl 
 
 general principle, or does it apply simply to the perspectives of the lines E F and 
 I D? State the rule expressing this general truth. What relation do the sides E 1 
 and F D of the square bear to the picture plane ? What relation do the perspectives 
 of these lines bear to the lines themselves ? Is the fact here noticed a general truth ? 
 State the rule expressing the principle here recognized. 
 
 26. What is the distinction between parallel and oblique perspective ? What part 
 of Fig. 11 represents the squiu-e which it is projjosed to place in perspective? From 
 what point is this figure supposed to be seen ? Since the points S, U, O, and T are 
 in the centres of the sides of the square E F D I, where will their perspectives be 
 found ? Show how the perspective of the square is determined. 
 
 27. What relation do the sides of the square bear to the picture plane? On pro 
 longing the perspectives of the sides of the square, in the direction of tlieir conver- 
 gence, where do they vanish ? What is the distance of their vanishing points from 
 the centre of view, compared with the distance of picture ? How may this be shown ? 
 Is the truth here recognized a general principle, or is it limited in its application to 
 the perspectives of the sides of the square ? State the rule expressing this law of 
 perspective. Name the sides of the square which tend to the right as they recede. 
 On which side of the centre of view is the vanishing point of their perspectives ? 
 Name the sides of the square which tend to the left as they recede. On which side of 
 the centre of view is the vanishing point of their perspectives ? Is the truth here 
 recognized a general truth ? State the rule expressing this principle. 
 
 28. State Rule 5. What are the facts upon which this rule is based ? 
 
 CHAPTER III. 
 Of what does this chapter treat ? 
 
 29. In preparing the paper, what line is first drawn? In what position, and 
 where on the paper, is this line to be drawn ? What does that part of the pai)er above 
 this line represent ? What does that part of the paper below this line represent ? 
 What relation are these planes supposed to bear to each other ? Of two points placed 
 at unequal distances below the base line, which point is nearer to the picture plane ? 
 Where is the point A in Fig. 12 ? Is the point F to be understood as bemg in front, 
 or behind the picture plane ? 
 
 30. Having drawn a line to represent the base line of the picture plane, what is 
 the next step in preparing the paper ? What determines the location of the point 
 which represents the centre of view ? Why does the place of the eye determine tliis 
 point? What determines the place of the horizon ? For what reason? In Fig. 12, 
 where is the place of the eye supposed to be ? What does the line H R in this figure 
 represent? What point in Fig. 11 corresponds to the point C in Fig. 12? Wliat 
 relation must a line bear to the picture plane to have its perspective tend to the point 
 C, in Fig. 12? 
 
 31. What is the next step in the preparation of the paper? Suppose it is 
 required to find the perspective of a cube seen on the right of the eye, and so situated 
 that one of its faces makes with the picture plane an angle of twenty degrees, and 
 
62 
 
 LINEAR PERSPECTIVE. 
 
 the point in the cube nearest the plane is distant from the plane the measure of one 
 of the edges of the cube ; how should the plan be drawn ? Draw upon the board a 
 plan of a cube thus seen, making the angle as near as you can judge by the eye, and 
 let the distance of the cube on the right of the eye be more or less, as you please. 
 When elevations are required, where are they to be placed ? 
 
 32. Having drawn a plan of the object to be placed in perspective, what is the 
 next step in the process? What is the assumed distance of picture in Fig. 12? 
 Give the exact place of the eye. What is the point D, which marks the distance of 
 picture, called? To what point in Fig. 11 does the point D correspond? What 
 must be the direction of a line, what must be its position, and what relation must it 
 bear to the picture plane, to have its perspective tend to the point D, in Fig. 12 ? 
 
 33. Have we alluded to all the lines and points required in preparing the paper ? 
 Name the lines and points. 
 
 CHAPTER IV. 
 
 Of what does this chapter treat ? 
 
 34. What is the form of the picture made on the retina of the eye ? What point 
 in the scene is pictured in the centre of the image formed in the eye ? What con- 
 clusion is drawn from these facts in regard to assuming a point as the centre of view? 
 
 35. What remarks are made touching the application of this rule ? 
 
 36. Upon what does the truthfulness of a drawing depend? What remark is 
 made on the distance often assumed in pictures ? How much of a scene can be rep- 
 resented in one picture with accuracy ? In order to obtain a clear view of all parts 
 of an object within the range of vision, how far must the object be from the eye ? 
 Have you tried the experiment ? What is Fig. 13 intended to illustrate ? Draw this 
 figure upon the board, and explain. What is the ri^le for assuming the distance of 
 picture ? 
 
 CHAPTER V. 
 
 Of what does this chapter treat ? 
 
 37. What is the principle on which the perspectives of points are found ? What 
 follows from this principle ? 
 
 38. In the problems throughout the book, certain lines and points are to be under- 
 stood as given, as shown in the diagrams to which the problems refer ; what are these 
 lines and points, and by what letters are they in all cases to be distinguished ? 
 
 39. State Prob. 1. Go to the board and work out this problem, explaining each 
 step in the process. 
 
 40. What is said respecting the line J A, in Fig. 15 ? Why is it used ? In de- 
 termining the perspective of points, are you to draw this line ? What course are you 
 to take? 
 
 41. State Prob. 2. Go to the board and work out this problem, explaining each 
 step in the process. In determining the perspective of the square, you first found the 
 perspective of the point A ; why not begin with the point E or I ? Is the drawing 
 rendered more simple by the course pursued ? 
 
QUESTIONS. 
 
 G3 
 
 42. State Prob. 3. Go to llio boai-il and work out tliis proljluin, explaining each 
 step in the process. Instead of first finding tlie perspectives of the points A and E, 
 would it have been just as well to have taken the points I and F? In Fig. IG, the 
 perspective of the point F is found in the line M D ; why is not the perspective of 
 the point F, in Fig. 17, found in the line M D ? 
 
 43. State Prob. 4. Go to the board and work out this problem, explaining each 
 step in the process. 
 
 44. What is the line o a, in Fig. 18, the perspective of? Supjiose it were required 
 to find the perspective of a vertical line standing on a given point on the ground 
 plane ; how Would you proceed ? 
 
 45. State Prob. 5. Are elevations required in placing a cube in perspective? 
 Why are they not required ? Go to the board and work out this problem, explaining 
 each step in the process. 
 
 46. Point out the perspectives of those faces of the cube which present a front 
 view ; what is their form compared with the form of the faces represented ? What is 
 evident from this ? 
 
 47. Point out the perspectives of the edges which present a front view ? What 
 is their position compared with the position of the edges represented ? What is evi- 
 dent from this ? 
 
 48. How do the perspectives of the edges nearest the eye compare with those of 
 the edges most distant? What is evident from this? 
 
 49. How are the left, right, upper and lower faces of the cube seen ? Which face 
 is seen the most obliquely ? How are these faces affected by being thus seen ? Wliat 
 is meant by the term foreshortened ? Which face is the most foreshortened ? What 
 is evident from this ? 
 
 50. Is the upper face of the cube above or below the level of the eye ? Do the 
 perspectives of the receding sideg of the upper face incline up or down ? Is the lower 
 face of the cube above or below the level of the eye ? Do the perspectives of the 
 receding sides of this face incline up or down ? What is the conclusion drawn front) 
 these facts ? 
 
 51. In what position do all horizontal lines on a level with the eye appear? From 
 what is this evident ? 
 
 52. In sketching, upon what does the draughtsman rely in determining the per- 
 spectives of objects ? Does one readily recognize the forms which objects present to 
 the eye, without discipline ? Why should one obtain a knowledge of the principles 
 of perspective before he attempts to draw from objects ? 
 
 53. You are advised to draw from a cube seen as that represented in Fig. 20. 
 How is the cube seen ? In drawing from an object, what is the first thing to be 
 considered? What part of an object should be first represented? Draw upon the 
 board the outlmes of a cube seen as in Fig. 20, and give your reasons for each step 
 in the process. 
 
 54. How may the accuracy of a drawing be determined ? 
 
 55. In sketching, what objects are recommended as suitable subjects for first prac- 
 tice ? What directions are given in regard to the method of drawing objects ? Pomt 
 out the course pursued in drawing the chest represented in Fig. 21 ? How is the 
 
64 
 
 LINEAR PERSPECTIVE. 
 
 inclination of a line most readily and accurately determined ? In case the line to be 
 represented is not connected with either a horizontal or a vertical line, what is to be 
 done ? What does the method of drawing the chest suggest ? 
 
 56. What is said in regard to Fig. 22 ? What is one of the objects of this kind 
 of practice ? 
 
 57. State Prob. 6. Go to the board and work out this problem, explaining each 
 step in the process. 
 
 58. When an error is made in the drawing of a pyramid, in what does it usually 
 consist? What is the object of Fig. 24 ? Explain the method of drawing the roof? 
 
 59. State Prob. 7. Point out the course pursued in determining the perspective 
 of the building, referring to Fig. 25, giving the reasons for every step in the process. 
 
 60. State Prob. 8. Explain the plan of the building in Fig. 26. Is the point of 
 distance represented in this figure ? Where is this point situated ? How may this 
 point be determined ? Point out the course pursued in determining the perspective 
 of the building referring to Fig. 26, giving the reasons for every step in the process. 
 
 61. State Prob. 9. Explain the plan of the building in Fig. 27. Point out the 
 course pursued in determining the perspective of the building, giving the reasons for 
 every step in the process. 
 
 62. In sketching a building, how far should the draughtsman be removed from it ? 
 Go to the board and point out the order in which the general outlines of a building 
 should be drawn, when the building is so situated that the end wall presents a front 
 view ; when so situated that the front wall presents a front view. 
 
 63. State Prob. 10. Is this cube seen in parallel or in oblique perspective? Go 
 to the board and work out this problem, explaining each step in the process. 
 
 64. How is the cube represented in Fig. 31 seen? Draw upon the boarcl the out- 
 lines of a cube thus seen, explaining each step in the process. 
 
 65. In drawing a chair such as we have represented in Fig. 32, how do you pro- 
 ceed ? What course would you pursue in drawing a *table or stand ? 
 
 66. What is said in regard to taking Fig. 32 as an exercise in drawing from the 
 copy? 
 
 67. State Prob. 11. Point out the course pursued in determining the perspective 
 of the building, giving the reasons for each step in the process. 
 
 68. In sketching a building of the usual form, in what order are you instructed to 
 draw the lines ? Explain the use of the dotted lines in Figs. 35 and 36. 
 
 69. State Prob. 12. How is the perspective of a circle determined? How many 
 points are usually taken in the circumference of the circle ? How are these points 
 determined? Point out the course pursued in determining the perspective of the 
 circle, giving the reasons for each step in the process. 
 
 70. How may the perspective of an octagon be determined ? i 
 
 71. In the examination of the perspective of the circle, to what points is attention 
 called ? i 
 
 72. In sketching, how may the perspective of the circle be determined ? 
 
 73. State Prob. 13. Point out the course pursued in determining the perspective 
 of the cylinder, giving the reasons for each step in the process. 
 
 74. In sketching an object, such as we have represented in Fig. 40, how are you 
 instructed to proceed ? 
 
GETTY RESEARCH INSTITUTE ^ 
 
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