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PERSPECTIVE 
 
 A 
 
 Series of Elementary Lectures 
 
 BY 
 
 ADA CONE, 
 
 Late Instructor in Massachusetts State Art Schools ; Super¬ 
 visor OF Drawing in Concord, N. H., Public Schools, 
 
 AND Lecturer on Industrial Art to Teachers’ 
 Institutes ; Pupil Museum of Fine 
 Arts, Boston. 
 
 NEW YORK : 
 
 WILLIAM T. COMSTOCK, 
 
 23 Warren Street. 
 
 1889 
 
COPYRIGHT, 
 
 A. L. CONE, 
 
 1889. 
 
PREFACE. 
 
 rriHOUSA^^DS of people are giving attention to 
 art nowadays, who would be glad of some 
 knowledge of perspective if they could have it pre¬ 
 sented with sufficient clearness and simplicity to make 
 it come within easy comprehension, and the work of 
 numberless artisans suffers for lack of a concise and 
 simple manual on this subject. These lessons are ad¬ 
 dressed especially to these and other adults who 
 have either no time or no inclination to go into the 
 details of the science or to follow it to a far analysis. 
 I have developed with a care which I have not seen 
 heretofore in any elementary work those points 
 which experience tells me are the greatest stumbling 
 block to beginners, and some of the illustrations 
 are novel. 
 
 THE AUTHOR. 
 
INTRODUCTION. 
 
 Perspective lias been considered one of the most 
 difficult subjects which come within the special range 
 of the art student. The more books have accumu¬ 
 lated purporting to explain its elements the more com¬ 
 plicated it has become, until the luckless pupil is 
 involved in a labyrinth of lines whose directions he 
 does not understand and whose endings are goals at 
 which his mind never arrives. “ What awful perspec¬ 
 tive!” exclaimed Wordsworth as he looked down the 
 aisles of King’s College Chapel, and ‘‘What awful 
 perspective ! ” echoes the teacher by methods in vogue 
 as he walks down the school-room aisles on a perspec¬ 
 tive recitation day and observes retreating lines pro¬ 
 duced impartially to M. Ps, S. Ps, and other Ps not 
 laid down on the ordinary diagram, a drawing here 
 and there recalling the line of a late newspaper poet: 
 
 “And see where lines diverging meet.” 
 
 The best treatises on the subject are too abstruse for 
 elementary pupils, and those which have attempted 
 simplification are, for the most part, too vague. Ele¬ 
 mentary works have consisted of sets of drawings with 
 explanatory text rather than of analyses of principles 
 with diagrams by way of illustration, and have had 
 for this reason little educational value, the mind instead 
 of being concentrated upon important truths being 
 distracted by particular problems without general appli- 
 
8 
 
 PEE8PECTIVE. 
 
 cation. As a result many persons whose work calls for 
 practical application of this science remain ignorant of 
 its primary principles. 
 
 Perspective refers to the appearance of objects as 
 influenced by position and distance from the eye. 
 Facts which are local and ascertained by actual meas¬ 
 urement, make when delineated what are called geo¬ 
 metric or working drawings. Such are used by the 
 artisan for patterns. Perspective drawings give the 
 appearance from one point of view, as is necessary in 
 pictures. If we see three sides of a cube and draw the 
 geometric facts of what we see, the drawing will look 
 like Fig. 1, but it does not appear thus; it appears. 
 
 
 Fig. 1 
 
 
 
 instead, like Fig. 2. By this we know that there is a 
 science of appearances; and the fact that every object 
 which occupies space, be it a house or a flower, an 
 avenue of trees, or a row of preserves on a shelf, in¬ 
 volves the aid of this science to represent makes knowl¬ 
 edge of it imperative to the draughtsman. 
 
PERSPECTIVE. 
 
 9 
 
 The following lessons state simply the important 
 truths and make several practical applications, going 
 very little into the mathematical part of the subject, 
 but giving information in such a form as to be of easy 
 application to the artisan and the amateur. 
 
CHAPTEE I. 
 
 Pkin. 1 .—Pamllel retreating lines converge. 
 
 For proof of this stand in the street and observe the 
 rows of buildings on either side; as they retreat they 
 tend toward each other: or stand on a railroad track 
 and observe the same phenomenon; the rails as they 
 retreat converge. Place yourself under an elevated 
 railroad and observe the lines above and the lines of 
 the surface track below, the lines on cither side of 
 pillars and spandrels, all converging towards a common 
 
 point in the distance. An 
 open door is a convenient 
 illustration; the top and bot¬ 
 tom do not appear parallel, as 
 they really are, but conver¬ 
 gent. 
 
 Parallel lines which do not 
 retreat do not appear to con¬ 
 verge but to remain parallel. 
 Take for example the nearest 
 side of the cube. Fig. 2. It 
 is a square and has parallel 
 lines—two vertical and two 
 horizontal—but these lines do 
 not retreat and therefore re- 
 Fis .3 main parallel in appearance. 
 
 As soon, however, as they 
 
PERSPECTIVE. 
 
 11 
 
 were placed in a retreating position, they would appear 
 to converge. 
 
 This convergence causes the farther side of an object 
 to appear smaller than the nearer side. For if the 
 retreating lines of an object, as the top and bottom of 
 a door, being parallel, appear to converge, then the 
 farther upright line, being of the same length as the 
 nearer one, must appear shorter, in order to meet the 
 ends of the apparently converging lines. 
 
 If an object, as a stick, is moved into the distance 
 its ends will describe two parallel retreating lines. 
 
 which lines, according to the law of perspective, 
 appear to converge. In other words the stick grows 
 smaller constantly as the distance increases. (See 
 Fig. 4.) Of a row of houses equal in size, the house 
 nearest the spectator will appear largest, and they will 
 decrease in size as the distance increases (Fig. 5). 
 
 The statement of the principle may be varied thus : 
 
 The apparent size of objects is according to the 
 square of the distance. 
 
 There is a common point where parallel retreating 
 lines meet if produced. Continue for illustration the 
 
12 
 
 PEESPECTIVE. 
 
 retreating lines in any one of the drawings given and 
 it will be seen that they have a common centre ; note 
 that the rails on the track come together in the dis¬ 
 tance ; observe also that the lines seem to rise as they 
 retreat; that the lines of the elevated track above seem 
 to tend downwards; that the left side of the street 
 
 tends towards the right, the right towards the left, 
 and all towards a point directly opposite the eye. 
 
 Ketreating lines whether above or below the eye, 
 tend towards the level of the eye. Parallel retreating 
 lines meet at the level of the eye. The point where 
 parallel retreating lines meet is called the Yanishing 
 Point. 
 
PEESPECTIVE. 
 
 13 
 
 Lines which run directly back from the spectator, 
 as do all the retreating lines in the drawings to this 
 lecture, retreat at an angle of 90°. When an object 
 is so placed that its retreating lines are at this angle 
 the object is said to be in parallel perspective. 
 
 Lines which retreat at an angle of 90° find their 
 Vanishing Point directly opposite the eye. 
 
 To make a perspective diagram, draw first a hori¬ 
 zontal line of indefinite length. This is called the 
 Horizontal Line and represents the level of the eye. 
 Objects which are above or below the level of the eye 
 are drawn in a corresponding position with regard to 
 this line. Make a dot at, or near, the centre of the 
 line. This represents the point directly opposite the 
 eye, and is called the Centre of Vision (C. Y.), or, 
 sometimes, Point of Sight (P. S.) It is the Vanishing 
 Point for all lines which retreat at an angle of 90° 
 and should therefore be marked, also, Y. P. Draw a 
 line from this point at right angles to the Horizontal 
 Line, indefinite in length. This represents the prin¬ 
 cipal visual ray, called also the Line of Direction. 
 The end of this line is the position of the eye of the 
 spectator, and is called the Station Point (S. P.) 
 
 Obtain two other points in the following manner: 
 Measure the distance from the Y. P. to the S. P. and 
 mark points the same distance from the Y. P. each 
 side on the Horizontal Line. These are called Meas¬ 
 uring Points (M. Ps). They may be most easily 
 located by means of compasses, as seen at Fig. 12, 
 thus: Take the Y. P. as a centre, and the distance to 
 the S. P. as a radius, and draw an arc to cut the Hor¬ 
 izontal Line on each side; the intersections will be the 
 M. Ps. 
 
14 
 
 PERSPECTIVE. 
 
 Having the diagram made proceed to draw the ob¬ 
 ject. Begin always by drawing its nearest side, or 
 edge, the part which is unchanged by perspective and 
 which requires only to be placed on the diagram in its 
 requisite geometric relation to the lines of the diagram. 
 Draw next the retreating sides towards their Y. P. 
 Fig. 6 shows the nearest side of a cube. The cube in 
 this position is below the level of the eye because it is 
 below the Horizontal Line which represents the level 
 
 S.P. 
 
 of the eye on the diagram. It is also to left of the 
 spectator because it is to left of the Principal Yisual 
 Bay. Being in this position we are able to see the 
 top and one upright retreating side. These retreat at 
 an angle of 90°, their direction is, therefore, towards 
 the C. V., which is their Vanishing Point. 
 
 How that the retreating lines are drawn, w^e come 
 to Prin. 2, for the question arises,—where shall we 
 place the farther vertical and horizontal lines to com¬ 
 plete the object ? For we have seen by Figs. 1 and 2 
 
PERSPECTIVE. 
 
 15 
 
 that it will not do to draw these lines as far off as they 
 really are. 
 
 Hold a sheet of paper directly facing you and it 
 looks as wide as it is, turn it a little from you and it 
 appears narrower, and if ‘it is turned as far round as 
 is possible it is narrowed to a straight line. From 
 this we get 
 
 Prin. 2.— Objects seen obliquely are foreshortened. 
 
 The degree to which the retreating sides are fore¬ 
 shortened depends upon the position of the object. 
 The nearer the object approaches the Principal Visual 
 
 M.P.l 
 
 V.P. 
 
 C.Y. 
 
 H.L. M.P.2 
 
 
 G.S. 
 
 Fig. T 
 
 S.P. 
 
 Pay the narrower the vertical retreating side becomes, 
 until on this line it is a single line and top and hot. 
 tom retreating lines are one. The same phenomenon 
 occurs as the horizontal retreating side approaches the 
 level of the horizontal line. 
 
 Perspective measures the foreshortening in this 
 way ; 
 
 Produce, as in Fig. 7, the ground line A from the 
 nearest ground angle B indefinitely. Measure from 
 
16 
 
 PEKSPECTIVE. 
 
 the angle B out on the produced ground line a dis¬ 
 tance equal to the actual width of the adjacent side, 
 whose perspective width it is desired to find. This 
 measurement is called a Geometric Scale (G. S.) From 
 the end of this scale, C, draw a line to the M. P.,which 
 is on the same side of the diagram as the cube. The 
 place where this line crosses the retreating ground 
 line, D, is the farther ground angle, E. This deter- 
 
 v.p. 
 
 G.S. 
 
 Fig. 8 
 
 SP. 
 
 mines the perspective width of the side in the posi¬ 
 tion in which it is placed. Baise the vertical line to 
 meet the upper retreating line. This intersection, H, 
 gives one of the farther angles for the top. From H 
 draw a horizontal line and complete the cube. 
 
 The extremes of foreshortening are shown in 
 Fig. 8. 
 
 Copy Figs. 7 and 8, and draw the cube in other 
 positions on the diagrams. 
 
CHAPTER IL 
 
 In a perspective diagram there are two lines which 
 invariably bear to each other the same geometric rela¬ 
 tion. They are always present and the other lines and 
 the points are dependent upon them. These are the 
 Horizontal Line and the Principal Yisnal Ray. The 
 first indicates the level of the eye; the second is the 
 axis of tlie cone of visual rays, and extends from the 
 eye of the spectator to the Horizontal Line, at right 
 angles to that line. (A further explanation of the 
 cone of visual rays is not necessary to the present pur¬ 
 pose. It may be found under Vision in any cyclo¬ 
 pedia.) The length of these lines is arbitrary, dej^end- 
 ing upon convenience or special necessity. Wlien the 
 length of the P. Y. R. is determined upon, however, 
 it decides the limits of the drawing, for all the points 
 on the diagram are determined by the position of the 
 Station Point, which is at the outer end of this line. 
 
 Of points the Station Point, which represents the 
 position of the eye in the diagram, is fixed, and the 
 Centre of Yision, representing the point opposite the 
 eye in the horizon, is always at the juncture of the 
 P. Y. R. with the Horizontal Line. 
 
 The Yanishing Point is any point in the diagram 
 where two or more lines meet which are really par¬ 
 allel, but which on account of retreating appear to 
 converge. The position of this point is variable, 
 
18 
 
 rEEa’ECTI v^E. 
 
 depending on the direction of the retreating lines of 
 the object to he drawn. 
 
 All horizontal retreating lines have their Vanishing 
 Points in the Horizontal Line. 
 
 The Principal Yisnal Eaj, although drawn verti¬ 
 cally in order to represent it on the diagram, represents 
 a horizontal line retreating from the eye at an angle 
 of 90°. Its Vanishing Point is in the Centre of 
 Vision. As all parallel retreating lines converge to a 
 common point, it follows that all lines parallel to the 
 
 Principal Visual Kay: i. e., which retreat at an angle 
 of 90°, find their common point of convergence in the 
 Centre of Vision, and hence the Centre of Vision-is 
 the Vanishing Point for all such lines. 
 
 As the apparent size of objects varies according to 
 distance it is necessary when proportions are to be 
 measured that they all be found in some one plane, at 
 a chosen distance from the eye. PerspecLve assumes 
 a plane, an imaginary, vertical plane, in which to 
 measure them. It is called the Plane of Measuies. A 
 
PEKSPECTIVE. 
 
 19 
 
 line indicating the lower edge of tins plane is usually 
 drawn through the lower line or angle of the nearest 
 object. This line is called the Ground Line and is 
 used for horizontal measures. 
 
 All proportions and all distances, however remote, 
 can be measured in this plane. The measurements are 
 called Geometric Scales, because they give the actual 
 proportions of the parts to each other. The only 
 indication of this plane to be seen in the diagram is 
 the position of the scales. 
 
 There is still another imaginary plane, called the 
 Picture Plane. It represents the surface on which 
 the drawing is supposed to be made, and its position 
 marks the foreground limits of the picture. The Pict¬ 
 ure Plane and Plane of Measures are parallel and may 
 coincide or not. In the diagrams here given they are 
 regarded, for simplicity, as one and the same, so that 
 
20 
 
 PERSPECTIVE. 
 
 Plane of Measures ” and “ Foreground ’’ are synon- 
 omous terms. 
 
 The Picture Plane is indicated at Fig. 10, where 
 the objects rest on the ground with their nearest sides 
 against it. When objects touch this plane they are in 
 the extreme foregrounds This shows the Picture 
 Plane and Plane of Measures as coincident, the lower 
 line of the Picture Plane being used as the ground 
 Line or horizontal scale of measures. The Plane of 
 Measures may be assumed at any distance beyond the 
 Picture Plane, as it is only necessary that wherever it 
 is all the scales be made in it, in order to be propor¬ 
 tional ; it is simpler, however, to have it in front of 
 the objects, and, as before stated, the nearest object is 
 usually assumed to touch it. When objects do not 
 touch the Plane of Measures their scales are still 
 drawn in it and transferred to the distance, as will be 
 shown in the next lecture. 
 
 Lines parallel to the Picture Plane, whatever their 
 direction—whether vertical, horizontal or oblique—re¬ 
 main parallel to it when put into perspective, but the 
 length of such lines decreases as the distance increases. 
 This may be seen by practical observation and by re¬ 
 ference to the drawings here given. 
 
 Lines not parallel to the Picture Plane do not re¬ 
 main parallel to themselves, but appear, instead, to 
 converge. 
 
 When an object is so placed that all its retreating 
 lines are parallel to the Principal Yisual Ray the ob¬ 
 ject is said to be in Parallel Perspective. 
 
 The Vanishing Point for all retreating lines in Par¬ 
 allel Perspective is the Centre of Vision. 
 
PEKSPECTIVE. 
 
 21 
 
 Lines which are parallel to the Picture Plane remain 
 parallel in aj)pearance, but grow shorter as they retire 
 into the distance. It is easy to find the perspective 
 of such lines ; as, for example, take the nearest line 
 of the square at A, Fig. 11. If lines are drawn par¬ 
 allel to it at any distance back between the retreating 
 lines its perspective length at those distances will be 
 accurately represented, though what the distances are 
 is not determined, except in case of the farther line 
 of the square, which is definitely placed. 
 
 The common method for finding perspective dis¬ 
 tances was described in the last lecture; following is 
 an explanation which, if thoroughly understood, will 
 make clear the whole ground of elementary perspec¬ 
 tive. 
 
 If, as we have seen, a line drawn from the Station 
 Point at 90°, as the P. Y. R., in Fig. 9 :—finds at its 
 junction with the Horizontal Line a Vanishing Point 
 for itself and all lines parallel to it, so a line drawn 
 from the S. P. at any angle, as say 45°, will, in cut¬ 
 ting the Horizontal Line, determine a Vanishing Point 
 for all lines running in that direction. fSTow it will be 
 found that horizontal lines directed from the Station 
 Point at an angle of 45° will strike the Measuring 
 Points. These, therefore, are the Vanishing Points 
 for all horizontal lines which retreat at an angle of 
 45°. 
 
 If the sides of a square retreat at an angle of 90°, 
 the diagonals of the square will retreat at an angle of 
 45°, therefore the Measuring Points are the Vanish¬ 
 ing Points for the diagonals of the square. This may 
 be tested by drawing the diagonals of the horizontal 
 
22 
 
 PERSPECTIVE. 
 
 retreating squares which have been obtained by the 
 scale as described in the last lecture, if correctly drawn 
 their diagonals will run tov^ards the M. P’s. 
 
 In Fig. 10 (A) the nearest side of an oblong block 
 being drawn, parallel with the Picture Plane and 
 touching it, draw next the retreating lines to the Y. 
 P. It is now necessary to find the perspective wddth 
 of the retreating side. Measure out from the nearest 
 ground angle a distance equal to the actual width of 
 the foreshortened side. This measurement is the Geo¬ 
 metric Scale. From the end of this scale draw a line 
 towards the M. P., to cross the adjacent retreating line 
 of the object. The intersection will be one of the 
 farther angles. This determines the distance back 
 from the Picture Plane of the two remaining lines of 
 the object. As these lines do not retreat draw them 
 parallel to the corresponding lines in the foreground, 
 between the retreating lines which mark their lengths 
 at this distance. The proportions of the object are dif¬ 
 ferent but there is no difference in procedure between 
 the drawing of these and the cubes at Figs. T and 8. 
 
 Suppose the point, C, to remain stationary and the 
 block to be turned on that point round to the right, 
 till the retreating line, D, is coincident with the scale. 
 It will be readily seen that the angle E will coincide 
 with the end of the scale, G. This may be easily de¬ 
 monstrated by experiment. 
 
 This process of finding perspective distances is based 
 on a simple geometric problem. If we have one side 
 of an isosceles triangle given, and wish to determine 
 the other side on an indefinite line we may do so by 
 drawing the base to cut the indefinite line ; the inter- 
 
PERSPECTIVE. 
 
 23 
 
 section will determine the length of the side on the 
 indefinite line. The Geometric Scale is one side of 
 an isosceles triangle, the retreating line from C to 
 the Vanishing Point is another side whose length is 
 not determined, the line from the end of the scale 
 towards the Measuring Point is the base of the triangle, 
 whose direction was found by geometric measurement 
 at the Station Point. 
 
 The lines from the Vanishing Point to the Station 
 Point and from the Vanishing Point to the Measur¬ 
 ing Point are two legs of an insosceles triangle, of 
 which a line from the Station Point to the Measuring 
 Point (shown at Fig. 10) is the base, and are parallel, 
 respectively, to the retreating line of the object, the 
 Geometric Scale, and the line from the Geometric 
 Scale to the Measuring Point. For the geometric re¬ 
 lation of the last named lines see (C) Fig. 10. 
 
 The lines from the S. P. to the M. P’s must be 
 regarded as being in the same horizontal plane as the 
 P. V. K., and as having the M. P. for its V. P. 
 
 In parallel perspective the base of the triangle is at 
 an angle of 45° with its legs. In consequence the 
 diagonals of horizontal squares tend towards the 
 Measuring Points, and in many instances it is more 
 convenient to dispense with the scale and find the 
 perspective of such squares by drawing their diag¬ 
 onals, as is done in the figures which follow. 
 
 Fig. 11 (A) is a square plane in a horizontal position. 
 The farther angles are determined by drawing the 
 diagonals. In this case the diagonal is the base, and 
 the adjacent sides are the legs, of the triangle. Fig. 11 
 (B) is a cube whose perspective has been determined 
 
24 
 
 PEESPEOTIYE. 
 
 by drawing all its retreating lines—tliougli in an opaque 
 object but three would be seen—and the diagonals of 
 the base. The diagonals of the upper side would have 
 obtained the same result and obviated the necessity of 
 making a transparent figure, but the ground surface 
 was used because it afforded a larger angle, being 
 farther from the TI. L., and the intersections could be 
 more easily seen. 
 
 S.P. 
 
 It should be explained that in making scales the 
 nearest ground corner is used only because it gives the 
 largest angle and thereby secures the greatest accuracy. 
 If the object is above the Horizontal Line the nearest 
 U23per corner is the best jDlace for the scale, and if the 
 ground line of the object rests on the Horizontal Line 
 a scale at the top is necessary. Either of the other 
 corners might be used, with the other Measuring 
 Point, but they give a slighter angle and so offer more 
 difficulty in determining the point of intersection, and 
 
PERSPECTIVE. 
 
 25 
 
 ill case of one of tliem it would be necessary to make 
 a transparent drawing, as that at (B) Fig. 11. If a 
 scale for the side C, of this figure, were to be drawn 
 out to the right from the angle D, and a line drawn 
 from it to M. P. 1, it would pass through the angle 
 E, already obtained in another way; this proves both 
 processes. 
 
 Fig. 12 is a square floor with square tiles whose 
 edges are parallel to the edges of the floor. Having 
 
 S.P, 
 
 drawn a ground line set ofi upon it the nearest edge 
 of the floor, which is one-half on each side of the P. 
 Y. P. From each end of the measured edge draw 
 retreating lines to the Y. P. Draw diagonals and 
 complete the square. Divide the nearest edge into 
 four equal parts and draw lines from the points so 
 found to the Y. P. Thus we have the retreating edges 
 for all the tiles. The diagonals of the large square 
 are diagonals also of the small squares they cross. 
 
26 
 
 PEKSPECTIVE. 
 
 Draw horizontal lines at the intersections and com¬ 
 plete. 
 
 Fig. 13 is a rectangular pyramid. Draw the square 
 base as in Fig. 12. To draw the inclined sides it is 
 necessary first to find their upper terminus, the apex 
 of the pyramid. The apex is at the upper end of the 
 axis. The position of the axis is already determined 
 by the intersection of the diagonals, which occurs at 
 the perspective centre of the base, but its length is 
 still to be found because, being removed from the 
 
 foreground half the width of the base, it appears 
 shorter than it really is. According to the rule laid 
 down, its actual height must be measured in. the Plane 
 of Measures, in which the nearest edge of the base is 
 drawn. If the axis should be moved to the fore¬ 
 ground, keeping it parallel to the retreating edges of 
 the base, it would touch the nearest edge, A. E., at C. 
 In this position it would show its true height and pro¬ 
 portion to the base. Draw therefore a line at C the 
 
PEESPECTIVE. 
 
 2T 
 
 actual height it is desired to make the axis. From its 
 upper end draw a line to the Y. P., because the axis, 
 which is now in the foreground, retreats at an angle 
 of 90°, and stops at the centre of the base. Raise a 
 line from the centre of the base to meet the retreating 
 line. Finish by drawing the inclined lines from the 
 base angles to the apex. It is obvious that the per¬ 
 spective height of the axis could have been found as 
 well by erecting the scale at either foreground angle 
 of the base, and drawing the retreating line to an M. 
 P. In this case the axis would have been moved to 
 its position along a plane at an angle of 45°, which is 
 the plane of the diagonal. 
 
 The student is recommended to draw the diagrams, 
 placing the objects in different positions from those 
 given, until he is sure that he understands the princi¬ 
 ples involved. Drawings should be made larger than 
 the plates here given ; not less than five inches for the 
 length of the P. Y. R. He is urged also to prove the 
 statements made herein by actual observation. A 
 great German savant has said that Nothing which 
 comes through the eyes into the head ever goes out,” 
 and this is true at least to the extent that things are 
 better remembered when seen than when merely heard 
 of. It is comparatively easy to comprehend a subject 
 in the abstract, but abstract knowledge of Perspective 
 is not enough for the artist. He must, in homely 
 metaphor, eat, drink and sleep with it; it must inform 
 his pencil as magic informed the sword of Orlando, 
 for he can never make a pictorial line in which it is 
 not involved. 
 
CHAPTEE III. 
 
 It lias been shown that the appearance of objects in 
 any position and at any distance can be accurately 
 measured and drawn, and a simj)le method has been 
 explained for accomplishing this result. What we 
 have learned may be summed up as follows. 
 
 The direction of retreating lines in Perspective is 
 found by determining their real or geometric direc¬ 
 tion at the Station Point, as see Figs. 9 and 10, and in 
 case of horizontal retreating lines, which only w^e have 
 discussed, producing these lines from the Station 
 Point to cut the Horizontal Line, the intersections deter¬ 
 mining the direction of all lines in the object which 
 are parallel to these lines. 
 
 The foreshortening of retreating lines is deter¬ 
 mined by drawing their real length in the Plane of 
 Measures in the position they would occupy if pro¬ 
 jected forwai’d to that plane. These lines are called 
 Geometric Scales. 
 
 It \vas shown also that the perspective direction of 
 the diagonals of horizontal squares in Parallel Per¬ 
 spective is towards the Measuring Points, and that, in 
 consequence, the diagonals may be used to determine 
 the farther angles, instead of dravdng a scale. 
 
 It may be remarked, thougli obvious, that the Yan- 
 ishing Points have reference only to the direction of 
 lines and planes, and not to their position. Objects 
 
V.P. 
 
 WP.1_c.v,__ m.p .2 
 
PEESPECTIVE. 
 
 31 
 
 however placed have the same Yaiiishing points when 
 their lines and planes are parallel. 
 
 It has seemed more logicahto explain at the outset 
 tlie means by which the Yanishing Point of any line 
 may be found though in fact it is not necessary in 
 Parallel Perspective to measure the angles of the re- 
 treating lines at the Station Point, because their Yan¬ 
 ishing Point is known beforehand, and the order of 
 drawing the lines of the diagram is, practically, as was 
 described and illustrated in Chapter I. 
 
 It remains now to make a little further application 
 of what has been learned of parallel perspective and 
 to speak of some alternative means of obtaining the 
 same results. 
 
 Fig. 14 shows objects adjacent to the plane of 
 Measures, and others removed from it, and is intended 
 to illustrate the fact that all objects in the picture 
 must be measured in the same plane in order to be 
 imoportional. 
 
 A and B are the same size as C and D but, being 
 distant, appear smaller. To draw A draw first the 
 real length of its vertical edge in the plane of Meas¬ 
 ures, G H. This line is a vertical scale. As the ob¬ 
 ject is parallel to the P. Y. B., draw the lines from the 
 scale to the C. Y. which is their Y. P. Mark off on 
 the Ground Line a scale equal to the distance of the 
 first upright of the object from the foreground, G K. 
 Draw a line from the end of the horizontal scale, K, 
 to the M. P. and the intersection with the retreating 
 ground line at M will be the perspective distance of 
 the nearest edge of the object, which is an upright 
 line, M M, between the two vanishing lines. Deter- 
 
32 
 
 PERSPECTIVE. 
 
 mine upon the width of the object and add a corre¬ 
 sponding width, K O, to the horizontal scale. Obtain 
 the farther angle in the same manner as the first one. 
 Draw the vertical edges between the retreating lines 
 and complete the object. Proceed in same way for 
 the horizontal plane, B. If it is desired to draw 
 planes still more distant, add on to the horizontal scale 
 the required distance, always measuring from the 
 point in the plane of Measures which would coincide 
 with the point sought if the point were projected 
 forward. 
 
 There is another method of finding perspective dis¬ 
 tances used sometimes in place of the one we have 
 described. It is called the method of Diagonals and 
 is based on the following propositions: 
 
 1. A line drawn through the intersection of the 
 diagonals of a parallelogram bisects the parallelogram. 
 This is the common way of dividing a perspective sur¬ 
 face into halves, and in Fig. 16 (A) determines the 
 centre line of the gables. 
 
 2. If a line drawn from the corner of a parallelo¬ 
 gram to the middle of one of the opposite sides, as in 
 (A), Fig. 15, and continued to meet the other side 
 produced, the intersection with the last line will be a 
 farther angle for another equal and similar parallelo¬ 
 gram adjacent to the first. 
 
 Suppose an object, as the nearest block in the row 
 in Fig. 15, A, to be drawn by means of the scale, and 
 it is desired to draw another, or a row of similar blocks, 
 adjacent. Divide the nearest upright line of the object 
 in the middle, B, and draw a line from the point thus 
 made to the Y. P. This line w/ 1 bisect the side of the 
 
M'.P.I _ C.V. H.L. M.p.2 
 
PERSPECTIVE. 
 
 35 
 
 block, and of every block which may be drawn beyond. 
 Draw a line from one of the nearest angles, C, of the 
 block to cut the middle of the opposite side, D, and 
 produce to cut the lower retreating line, which will 
 give E. E will be the farther ground angle of the 
 next block. This is a shorter method than drawing a 
 continuous number of horizontal scales. But if the 
 objects are not adjacent it is most convenient to use 
 the scales. The square planes on the right are obtained 
 by drawing their diagonals to the M. P., but they 
 could as well have been made by drawing a dividing 
 line to the Y. P., and proceeding as for the blocks at 
 (A). If, instead of squares, it were desired to draw 
 oblong planes, either this last method or the use of 
 the scale would be necessary, unless a Y. P. were found 
 for the diagonals, which would be unnecessary labor. 
 
 In Fig. 16 we have two rows of gable-roofed houses, 
 showing different methods of obtaining the same result. 
 
 For (A) draw first the nearest side of the nearest 
 house, A B C D. Draw lines from A and B to the 
 Y. P. Draw a scale, B, O, for the width of the 
 house through the nearest ground corner. By means 
 of a line from the scale to the M. P. locate the farther 
 ground-angle of the front, S. Draw the farther up¬ 
 right, S. T. Draw the diagonals of the front, and 
 from their intersection raise a vertical line indefinite 
 in length. The apex of the gable will be found in 
 this line but its height must be measured by a verti¬ 
 cal scale in the plane of measures, as in case of the 
 axis of the pyramid in the last chapter. A. E. is such a 
 scale. A line from E to the Y. P. will give, at its in¬ 
 tersection with the line bisecting the gable, the height 
 
36 
 
 PEKSPECTIVE. 
 
 of the apex. This line, from the vertical scale to the 
 y. P., will give also the height of the other gables 
 beyond. Draw the inclined lines, and complete the 
 front of the house. The upper line of the roof, or 
 ridge-pole, is parallel to the plane of measures, but as 
 it is distant from it half the width of the house it 
 appears shorter, and its perspective length at this dis¬ 
 tance must be measured. Erect a scale, G, from the 
 corner, D, and draw a line from it to the Y. P. Draw 
 the ridge-pole to cut this line and the intersection 
 will determine its perspective length. From the point 
 of intersection, H, draw the inclined line to D. 
 Measure off horizontal scales on the ground line for 
 the widths of the remaining houses, and proceed as for 
 the first. 
 
 For (B), draw the nearest side of the first house, 
 A B C D. Draw lines from A and B to the Y. P. 
 Place the further upright line of the front N. P. by 
 means of a scale, B E. Draw diagonals of the front, 
 and at their intersection draw a vertical line upwards, 
 indefinite in length. Make a vertical scale for the 
 height of the roof, AG. A line from this scale to 
 the Y. P. gives, at its intersection with the line from 
 the diagonals, the height of the apex. Complete the 
 front end of the roof. All this is the same as was 
 done at (A). The slope of the other roof lines is ob¬ 
 tained in a different manner. 
 
 If a sheet of paper is held in the position of the 
 slope of the roof, A D K L, it will be seen that it is 
 not parallel to the picture plane but inclines upwards 
 from it, and that as the upper line is farther away 
 than the lower one it appears shorter and the retreating 
 
.GR.L, 
 
i 
 
PERSPECTIVE. 
 
 39 
 
 ends converge. These converging ends, A K and D L, 
 must have a Y. P., and if it is found we will have 
 only to draw the slopes towards it to give their appar¬ 
 ent inclinations. It will be observed that these lines 
 lie in the same plane as the back and front of the 
 house, which retreats at an angle of 90°, and, therefore, 
 if they were horizontal lines, like H, their Y. P. 
 would be in the C. Y., but they are not horizontal 
 but incline upwards, instead, and so their Y. P. must 
 be found by looking directly ahead and raising the 
 eyes above the C. Y. If a line is drawn upwards 
 from the C. Y., the Y. P. for the inclined lines will be 
 found somewhere in it. Where depends on the de¬ 
 gree of inclination of the roof, which must be deter¬ 
 mined beforehand. In this case the inclination of the 
 line, A K, was found by means of the vertical scale 
 A G, and the diagonals of the front. If this line, 
 A K, is produced to cut the vertical line from the 
 C. Y. the intersection will be the Y. P. sought, and 
 all lines parallel to A K must be drawn towards this 
 Point. Having located this Y. P. draw the inclined 
 line from D and finish the first house. The width of 
 the farther houses is found in the same way as that of 
 the blocks in Fig. 15. The line from the vertical scale, 
 at G, to the Y. P. gives the height of each roof, and its 
 intersection with the inclined line from the nearest 
 corner of each house to its Y. P. determines the height 
 of the roof and position of the apex. Finish by draw¬ 
 ing the farther slopes of the roofs, which incline equally 
 in the opposite direction. A Y. P. for these last 
 slopes would be found below the level of the eye in 
 the P. Y. P., but its use was not here necessary. 
 
40 
 
 PEESPECTIVE. 
 
 Fig. 17 is an application of what we have already 
 learned to some details of a honse. In (A) the position 
 of the windows is found by fixing their position on the 
 scale for that side of the house and transferring the 
 points to the retreating ground line ; their height was 
 marked on the near end of the house, O, and a line 
 from thence to the Y. P. crossed by vertical lines from 
 the points on the ground line. The door of the house 
 (B) was found in the same way. 
 
 To draw the chimney on house (A), mark its width, 
 height and position, ABC, on the end of the near¬ 
 est gable, which is in the Plane of Measures. Draw 
 lines from these points to the Y. P. Mark its dis¬ 
 tance from the Plane of Measures and the width of 
 its retreating side on the Geometrical Scale for the 
 retreating side of the house at E F. Transfer these 
 
PERSPECTIVE. 
 
 41 
 
 points, by lines to tlie M. P., to the retreating ground 
 line. Carry lines thence to the roof and across it, 
 keeping them parallel to the Plane of Measures. 
 Where these lines cut the line from B to the Y. P. 
 we have the position of two lower angles of the 
 chimney. Erect two vertical lines to meet the retreat¬ 
 ing line from C. This completes one side of the 
 chimney. Draw a horizontal line from G to cut a re¬ 
 treating line from A at H. This gives the width of 
 the front, H G. From H draw a vertical line to meet 
 a horizontal line from K. The oblique line, G N, 
 showing the insertion of the chimney into the roof 
 is parallel to the other oblique lines of the roof. 
 
 Several alternative ways will suggest themselves for 
 drawing the chimney on house (B). As the process 
 shown involves only what has been already explained, 
 a description is unnecessary. 
 
 When mechanical accuracy is required perspective 
 work is reduced from actual measurements to a defi¬ 
 nite scale of inches or fractions of an inch. These are 
 set off in the Plane of Measures, and thus the smallest 
 details are drawn in true proportion. Perspective 
 problems usually state the scale of reduction. Work¬ 
 ing to a scale is not introduced here because it would 
 cumber the student while being no help towards a 
 knowledge of principles. 
 
CHAPTEE lY. 
 
 To draw any curve in perspective enclose it in a rec¬ 
 tangle ; draw tlie perspective of the rectangle; find 
 the points of contact of the curve with the rectangle; 
 draw the curve through these points, and its true per¬ 
 spective will be found. 
 
 If the curve is a circle the rectangle will be a square, 
 as A B C D, Fig. 18, and the points of contact will 
 be the ends of the diameters, E F Gr H. 
 
 If it is desired to find other points through which to 
 draw the perspective curve not in the rectangle, find 
 these points first in the geometric circle, and transfer 
 them to the perspective circle by the rule already given 
 for placing a point in any given perspective position. 
 
PERSPECTIVE. 
 
 4a 
 
 viz : Fix its position in the plane of measures, as A., 
 Fig. 19, and draw a line thence towards its Y. P. 
 Cross this line by a line towards the M. P., from a 
 scale which represents the actual distance of the point 
 from the foreground or plane of measures, and the 
 intersection B will be the perspective position of the 
 point. 
 
 Fig. 20 shows the circle in perspective. A descrip¬ 
 tion of the process is as follows: 
 
 Having drawn the H. L., P. Y. B. and G. L., fix the 
 point where the circle is to come in contact with the 
 Plane of Measures, as at A. Mark off on the G. L. the 
 width of the diameter of the circle, B C, one-half on 
 each side of A, and draw lines from thence to the Y. 
 P. Draw the diagonals and complete a perspective 
 square. Draw the diameters. The ends of the diame¬ 
 ters, A F G H, are the four points of contact of the 
 circle with the square. 
 
 It is desirable for greater accuracy to find points 
 where the curve crosses the diagonals. These are 
 
PEKSPECTIYE. 
 
 U 
 
 found by constructing the square, or half of it, in the 
 plane of measures, and having found the points in 
 this, transferring them to their corresponding places 
 in the perspective plan. Construct half the square 
 with B C as one side. Draw the semi-diameters and 
 semi-diagonal. Find the points on the diagonals 
 through which the circle passes by measuring out on 
 them, from the centre, the length of the semi-diame¬ 
 ters or by inscribing the half circle. This gives points 
 
 Fiy;. so 
 
 D and E. Transfer these to the ground line at I and 
 K by vertical lines. From I and K draw lines to the 
 Y. P., and where these cross the diagonals of the per¬ 
 spective square will be the points corresponding to D 
 and E. Draw the curve freehand through the points 
 obtained. 
 
 If it is desired to draw on the diagram a circle 
 removed from the foreground, construct a square at 
 the desired distance and proceed as for the first. 
 
PEKSPECTIVE. 
 
 45 
 
 A circle appears as such only when the eye is oppo¬ 
 site its centre. When the side only is seen it is a 
 straight line. In all other positions it is an ellipse. 
 As these other positions are practically unlimited, it 
 oftenest appears as an ellipse. 
 
 Fig. 21 shows circles in an upright position parallel 
 to the plane of measures. In this position they may 
 be drawn without the aid of the enclosing rectangle, 
 in the following manner. 
 
 v.p. 
 
 Draw the nearest circle in the plane of measures, 
 touching the G. L. Draw line A B from the cen¬ 
 tre to the point of contact with the ground. This 
 is the radius of the circle. Draw lines from A and B 
 to the Y. P. A will pass through the centre of all 
 circles which may be drawn beyond and B will pass 
 through their point of contact with the ground. Make 
 a scale from B the actual distance of the farther circle 
 from its position in the foreground. A line from the 
 end of this scale to the M. P. will, where it crosses 
 
46 
 
 PEESPECTIVE. 
 
 the retreating line from B, at O, give the position for 
 the farther circle. Draw a vertical line from O to D. 
 With D as a centre and the distance to O as a radius 
 draw the circle. 
 
 Circles in any other position than parallel to the 
 picture Plane must be enclosed in squares. Draw the 
 square in any position required for the circle and then 
 proceed as described for Fig. 20. 
 
 Any number of points may be found on the per¬ 
 spective circle by first placing them on the geometric 
 circle in the plane of measures, and transferring them, 
 as was done with points D and E. 
 
 The profile lines of circular solids, as cylinders, 
 must be drawn tangent to the circles, as are the lines 
 E and H in Fig. 21. 
 
 The forms of all bodies are continuous deviations 
 from geometric forms which compose their bases, and 
 which, when imagined round them, can be called their 
 general outline. To determine this general outline is 
 the first process in making a correct drawing. It is 
 found by circumscribing the object as closely as pos¬ 
 sible with straight lines or geometric curves, in such a 
 way that we complete some of its parts, and perhaps 
 cut off some of its ^protuberances.” Such an outline 
 will represent the essential form and character of the 
 object. 
 
 After the geometric form of the general outline is 
 decided upon, it must be drawn in the perspective 
 position in which the object is to be represented, and 
 afterwards modified into the object. 
 
 Only by this means, by reference to the intentional 
 structure, can disordered forms be represented truly ; 
 
PEESPECTIVE. 
 
 47 
 
 for if the normal structure is recognized the departure 
 from it will seem an accidental deviation, but if the 
 irregularities are drawn without this guide the result 
 will have neither strength nor character. The one 
 process will show the operation of mind while the 
 other will be chaos. 
 
 The perspective of flowers which are based on the 
 circle will serve to illustrate this point. The choice 
 suggests itself from the fact that flowers, while being 
 popular objects of imitation are commonly repre¬ 
 sented as though they neither occupied solid space nor 
 possessed essential form. It would seem, moreover, 
 from the ordinary portrayal, that they were exempt 
 from the visual laws which govern the appearance of 
 other objects, and if their scientiflc classiflcation 
 were to be determined from these efflgies, the most 
 expert botanist might well be perplexed. But not¬ 
 withstanding that ignorance is satisfied by variety 
 without order the flower has a perfect plan, and no 
 one can hope to draw it correctly by scoring down 
 irregularities as essential beauties, without reference to 
 the law which is behind. 
 
 I hold this rose in my hand. I perceive above tlie 
 straggling leaves, like a halo, its structural plan, the 
 circle. In the winged life of each petal there is a his¬ 
 tory. I note how the rain beat on this one and tlie 
 sun came out and confirmed its downward growth ; I 
 see the trail of the worm on that, and how the heat 
 has withered, the next, and the countless graceful vol¬ 
 untary movements of the parts among themselves. 
 Scarcely one is in its normal place, yet it is by refer¬ 
 ence to the normal that the deviations are interesting. 
 
48 
 
 PERSPECTIVE. 
 
 The irregularities are special truths and mark the in¬ 
 dividuality of each leaf, but art is concerned with 
 types more than with special truths, and the artist who 
 rightly depicts a flower must indicate its essential 
 form, and must show that the variations are not mere 
 chance without organic connection, but that they are 
 temporary escapings from a law to which, while re¬ 
 taining their individuality, they yet submit.” 
 
 Flowers have either bi-lateral symmetry, like the 
 pansy, or the stellar symmetry of the daisy. The plan 
 or general outline of the latter class is the circle. The 
 petals radiate from the centre and are balanced and 
 equal. Fig. 22 A gives the construction as seen from 
 above or below, the radiating lines being balancing 
 lines for the petals. The number of radiating lines 
 
PEESPECTIVE. 
 
 49 
 
 differs according to the number of petals, but the plan 
 is the same for all radiating flowers. 
 
 When viewed from the side the flower is balanced 
 on a straight line, as shown at B, tho line at the 
 top, G. H., being always at right angles to the balanc¬ 
 ing line, E C. 
 
 These two positions, A and B, require but two posi¬ 
 tions for the eye with relation to the circle, but be¬ 
 tween these two there are an inflnite number of 
 appearances, and all are ellipses of varying widths, for 
 
50 
 
 PERSPECTIVE. 
 
 an ellipse may be any width between a straight line 
 and a circle, these two being the perspective poles of 
 the curve. 
 
 JTig. S4: 
 
 There needs yet a little further construction, for ij 
 is necessary to fix not only the outside limit for the 
 petals, but also the point from which they start, E. 
 When the outline appears a circle this point is, of 
 course, the center of the circle, as shown at Fig. 22, 
 
 and when the view is a side one, as B, Fig. 22, the 
 point is found on the balancing line at its actual dis¬ 
 tance from the outside limit, but in all elliptical views 
 the perspective position of this point must be especi¬ 
 ally determined. It will approach the centre of the 
 
PEKSPECTIVE. 
 
 51 
 
 curve or retire from it according as the ellipse grows 
 wider or narrower, as may be seen at Fig. 23. The 
 construction is shown by a cone at Fig. 24. Let the 
 base of the cone be the outline of the flower and its 
 centre will be found in the axis; not necessarily at 
 the apex but if not there a cross section must be drawn 
 at the desired distance along the axis. This cross sec¬ 
 
 tion will be another ellipse as C D, and will enclose 
 the centre or heart of the flower. 
 
 To draw parallel ellipses, freehand, draw their 
 diameters parallel. If the centre of one ellipse is op¬ 
 posite the centre of the other, as occurs in the flower, 
 their short diameters will coincide. Two lines con¬ 
 necting these curves, as A B, B C at Fig. 25, will be 
 the sides or proflles of the flower. If these sides are 
 
52 
 
 PERSPECTIVE. 
 
 produced to meet, it will be seen that the point of 
 meeting is opposite the centre of the circles, and will 
 fall in the line of the short diameters produced. In 
 the flower this intersection occurs in the stem, which 
 is always a prolongation of the short diameters. 
 
 If it is desired to draw ellipses a deflnite distance 
 apart, it may be done freehand by enclosing them in 
 squares, as seen at Fig. 27, the distance between them 
 being measured by the distance between the nearest 
 edges of the squares, as from A to B. If one ellipse 
 
 is smaller, its size and position are set off in the nearest 
 edge of the square in whose plane it is to be drawn, 
 as C. D. Vanishing lines from the points thus made, 
 crossing the diagonals of the square will give its per¬ 
 spective position and size. 
 
 Construction lines show the fundamental form of 
 the flower or other object which they circumscribe, 
 and its perspective position. If the draughtsman is 
 expert they need not be drawn but they must be kept 
 always in mind. With this precaution the common 
 mistakes in form may be avoided. 
 
CHAPTEE Y. 
 
 We have thus far considered objects so placed that 
 the retreating lines are at an angle of 90°, and which 
 being thus parallel to the P. Y. K. find their Y. P. in 
 the C. Y. 
 
 But when the object is turned to left or right so 
 that its retreating lines are no longer parallel to the P. 
 Y. E., then the C. Y. is no longer their Y. P. Under 
 this change the object is said to be in angular per¬ 
 spective. 
 
 As the retreating lines remain horizontal under this 
 change, the Y. Ps remain in the Horizontal Line, but 
 are removed to a distance from the C. Y. depending 
 upon how much the object is turned from the plane of 
 the picture. 
 
 To fix the Y. P. in a definite place it is necessary 
 to know the angle at which the lines of the object are 
 turned from the P. P. Distances of this kind are 
 measured by degrees of a circle. For example: in a 
 quarter circle there are 90°. If one side of a square 
 is parallel with the line C D (A), Fig. 28, then the 
 retreating side is at an angle of 90° with C D. This 
 is the relative position of the object to the picture 
 plane in Parallel Perspective. But if it is turned fur¬ 
 ther round as in B, the angle is smaller than 90°; 
 and this is the position of the object to the picture 
 plane in angular perspective. How much smaller it 
 
64: 
 
 PERSPECTIVE. 
 
 is may be ascertained by measuring the number of 
 degrees between F and E. 
 
 If it is desired to measure the angle at which the 
 object is to retreat it must be done at the Station 
 Pointj as explained in Chapter II. The rule is as fol¬ 
 lows : Draw a line from the 8. P. at any angle and 
 produce to cut the II. Z., and the intersection will he 
 the Y. P. for all lines running, in that direction. 
 
 Therefore, if the angle at which the lines of the object 
 retire is known the Y. P. may be fixed as follows. 
 
 Suppose the angle to be Y5°, as in Fig. 29. Having 
 drawn the H. L. and P. Y. H., draw a horizontal line 
 through the S, P. Draw a half circle on this last line 
 with the S. P. as a centre. Measure ofi on the curve 
 75°. Draw a line from the S. P. through the point 
 thus obtained, C, and produce to cut the H. L., and 
 the intersection with the H. L. will be tlie Y. P. 
 
PERSPECTIVE. 
 
 55 
 
 sought. Draw the nearest upright line of the object, 
 A B, resting on the ground line. From each end 
 draw lines to the Y. P. and they will be at an angle of 
 75° with the Gr. L. or Picture Plane. 
 
 When the angle is not known, the direction of the 
 retreating line may be determined by judgment of 
 the eye, thus: Draw the nearest upright line of the 
 object, resting on the Gr. L., as A. B. in Fig. 30. 
 Draw a retreating line from A to meet the H. L., 
 
 H.L. C.V. \l.?. 
 
 judging its direction by the eye. This gives the 
 Y. P. for this line and all others parallel to it. Draw 
 a line from B to the same Y. P. But now, in com¬ 
 pleting the cube, it will be seen that whereas in 
 parallel perspective all the retreating lines converged 
 to one point, here they do not; for the lines which 
 formerly were parallel to the Picture Plane now re¬ 
 treat, and in a different direction from the first set. 
 As they retreat they must converge, whereas formerly 
 they did not; as they converge they have a Y. P., and 
 
56 
 
 PEESPECTIVE. 
 
 as they are horizontal their Y. P. will be in the H. L. 
 The Y. P. for this second set of lines is as far from 
 the first Y. P. as the actual difierence in direction of 
 the two sets of lines, the angle to be measured at the 
 S. P. To find the second Y. P. draw a line from the 
 first Y. P. to the S. P., and make with it there an 
 angle equal to the difierence in direction of the two 
 sets of lines. In case of the cube, the lines are at 
 right angles, therefore make a right angle at the S. P., 
 and produce the line to cut the H. L., and the inter¬ 
 
 section will be the Y. P. for all horizontal lines at 
 right angles to those running towards the first Y. P. 
 
 Draw retreating lines from A and B to the second 
 
 Y. P. 
 
 It is necessary now to find the M. Ps., in order to 
 measure the foreshortening of the sides. The rule for 
 finding the M. Ps., given in Chapter I, is: Measure the 
 distance from the Y, P. to the S. P.^ and marh a 
 jpoint the same distance from the P. P. on the II.L.; 
 the point thus found is the M. P. 
 
PERSPECTIVE. 
 
 5r 
 
 A Y. P. needs but one M. P. In Parallel Per¬ 
 spective, however, the Y. P. occupies the centre of 
 the diagram, and it is convenient to have the M. P. 
 sometimes on one side and sometimes on the other, 
 depending on the position of the object, whether to 
 right or left of the P. Y. K. It occupies on either 
 side the same relative position, being used to right or 
 left as convenience dictates. Where there is more 
 
 S.P. 
 
 than one Y. P. each one has its own M. P., used ex¬ 
 clusively to measure the foreshortening of the lines 
 which vanish towards it. The M. P. for any Y. P. is 
 found when an isosceles triangle is formed with the 
 Y. P. as the apex, the line to the S. P. as one -side, 
 and an equal side laid oil on the H. L. 
 
 To find the M. Ps. in angular perspective take 
 each Y. P. as a centre and the distance to the S. P. as 
 
58 
 
 PEESPECTIYE. 
 
 a radius, and strike an arc to cut the H. L.; the inter¬ 
 sections will be the respective M. Ps. 
 
 To continue the drawing of Fig. 30, measure off on 
 the Gr. L. a scale for each side of the cube. Draw a 
 line from each to the M. P. for that set of lines which 
 it is to measure. At the intersections with the lower 
 retreating lines of the object raise vertical lines to 
 meet the upper retreating lines. This completes the 
 vertical sides. It now remains to complete the top 
 
 S.P. 
 
 which is done by drawing lines from the upper angles 
 just found, one to each Y. P., the intersection of these 
 two lines being the farther angle of the top. 
 
 It will be seen that there is nothing here different in 
 procedure from parallel perspective. Another Y. P. is 
 added because the Tines which formerly were drawn 
 parallel now converge, but the Y. Ps. are used in pre¬ 
 cisely the same way, and the M. P. for each is found 
 
PERSPECTIVE. 
 
 61 
 
 by precisely the same rule as in Parallel Perspective. 
 It may be observed that in Par. Per. the isosceles tri¬ 
 angle is a right triangle, because the Y. P., which is 
 always the apex, is in the C. Y., making the P. Y. P. 
 one of its sides, but when the Y. P. is removed from 
 the C. Y. the P. Y. P. is no longer one of its sides, 
 and the angle is therefore smaller. 
 
 Fig. 31 shows planes in angular perspective. But 
 one Y. P. is needed. The first three are measured by 
 scales, and the others by the method of Diagonals 
 explained in Chapter III. 
 
 Fig. 32 shows an object whose sides are not at right 
 angles to each other, and, in consequence, whose Y. 
 Ps. are not 90° apart. 
 
 Draw the nearest upright line, A B. Draw one of 
 the lower retreating lines from A to the H. L., deter¬ 
 mining its direction by the eye. Draw a line from 
 the Y. P. so obtained to the S. P. Make an angle with 
 this line equal to the difference in direction of the 
 sides C and D of the object. In this case the sides 
 are 60° apart. Produce the line so obtained to meet 
 the H. L. This gives the Y. P. for the side D. Draw 
 the remaining lines from A. B. to the Y. P’s. Find 
 the M. P. for each Y. P. Draw the scale for each 
 side, making them equal, as the object is a triangular 
 prism. By lines from the scales to the respective M. 
 P’s. find the foreshortening of each side. Paise the 
 vertical lines and complete the sides. As there are 
 but three sides to the object finish by drawing a line 
 connecting the upper side angles. 
 
 The method of finding the Y. P. for inclined lines 
 was explained in Chapter III, the illustration being the 
 
62 
 
 PEESPECTIVE. 
 
 lines of a roof. The Y. P. for such lines is found in 
 a vertical line running from the Y. P. of the plane or 
 side in which the lines lie. In Pig. 16 they were in a 
 plane which had its Y. P. in the C. Y., and therefore 
 their Y. P. was in a line drawn vertically from the C. 
 Y. In Fig. 33 the inclined lines are in the plane which 
 tends towards Y. P. 1. The inclination of the first 
 line, A, may be assumed or it may be measured accu- 
 jately by drawing a scale for the height of the roof, 
 B, and drawing a line thence to Y. P. 1; drawing the 
 diagonals of the front and erecting a vertical line from 
 their centre to meet the line from the scale to Y. P. 1. 
 Prolong the line A to meet the vertical line from Y. 
 P. 1, and the third Y. P. will be found. 
 
 When an object is so placed that its retreating lines 
 incline up or down from the horizontal, as does the 
 roof of Fig. 33, the object is said to be in oblique 
 perspective. 
 
 The process of measuring angles as at the S. Ps. in 
 Figs. 29 and 32 is geometrical and for its explanation 
 the student is referred to the simple problems of plane 
 geometry. 
 
A Practical Book on PerspectiYO. 
 
 Arclitectural PersiifictiYe for Bepners. 
 
 -- 
 
 F. A. WRIGHT, Architect. 
 
 -- 
 
 Containing’ 11 large Plates and full descriptive 
 letter-press. One large quarto handsomely bound 
 in cloth. Price, $3,00, 
 
frtJ&V FrTB3LiXS3EZ3£SX3 
 
 KIKXM E»IXIO?« 
 
 Of I^^z-gLOtioetl Xjossozxs Ino. 
 
 ArcMtoctural Drawing 
 
 OR 
 
 How to Make tke Workiig Drawiags for BnlMings. 
 
 44 pages descriptive letter press, illustrated by %% full-page plates 
 {one in colors'), and 33 woodcuts, showing methods of construction and 
 representation. 
 
 The work embraces Scale Drawings of Plans, Elevations, Sections 
 and Details of Frame, Brick and Stone Buildings, with full descrip¬ 
 tions and a form opSpecifications adapted to the same. 
 
 Suited to the wants of Architectural Students, Carpenters, 
 Builders, and all desirous of acquiring a thorough knowledge of 
 Architectural Drawing and Construction. 
 
 COIVTONTS. 
 
 Chap. I—Introduction. Chap. II.—A Small Frame House. Chap. 
 III. A Frame Building. Chap. IV.—A Brick Building. Chap. V.— 
 A Stone Building. Chap. 'Sfl.— The Specifcations. Chap. VII.— 
 Color. By 
 
 WILLIAM B. TUTHILL, A. M., ArcMtect. 
 
 One large 8vo volume, oblong. Cloth. Price, post-paid, $2.50. 
 
 WM. T. COMSTOCK. Publisher, 
 
 23 WARREN ST., NEW YORK. 
 
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