ERS 
 
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 ~^' Goraafi University 
 
 PH-2- 
 
 CSfatttcU Uttiueraitg Ethrarg 
 
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 BOUGHT WITH THE INCOME OF THE 
 
 SAGE ENDOWMENT FUND 
 
 THE GIFT OF 
 
 HENRY W. SAGE 
 
 1891 
 
''^iiffinl'imSnlli'.f "'•' '"" '"e "ew method. 
 
 3 1924 020 548 479 
 
The original of tiiis book is in 
 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924020548479 
 
PERSPECTIVE 
 
 The Old and the New Method 
 
 By 
 
 ARCHIBALD' STANLEY PERCIVAL I §^2 
 
 M.A., M.B., B.C.Cantab 
 Author of "Geometrical Optics," "Practical Integration," etc 
 
 WITH DIAGRAMS 
 
 LONGMANS, GREEN AND CO. 
 39, PATERNOSTER ROW, LONDON, E.G. 
 New Vork, Bombay, Calcutta, and Madras. 
 1921 
 
PREFACE 
 
 Several books have been written on Perspective, but while some 
 are too voluminous for the amateur, others are obscure or inaccurate. 
 
 In this little book the Art School rules have been concisely and, 
 it is hoped, clearly explained ; the illustrations given should remove 
 every difficulty in their application. 
 
 It will be found however that in Oblique Perspective the 
 Vanishing and Measuring Points are frequently five or six feet away, 
 and ridiculous distortion results if they are brought so close that they 
 all can be marked on the same sheet of paper. Figs. 6 and 7 show 
 •such distortion, and Figs. 10 and 18 show the same objects drawn 
 by Plan in correct Perspective without the use of Vanishing Points. 
 
 Several useful devices have been given of which some are original, 
 such as the method of reducing or enlarging any pictures ; while for 
 others I am indebted to Mr. Storey's " Theory and Practice of 
 Perspective." 
 
 My brother (Mr. A. H. Percival) has supplied me with the 
 illustrations on the perspective of shadows whether cast on horizontal, 
 vertical or oblique planes, and has given me invaluable help in 
 ■explaining clearly this somewhat perplexing subject. 
 
 I think this little handbook should be of use in all drawing 
 ■classes at schools and to all junior art students. 
 
 A. S. Percival. 
 " Westward," 
 
 Newcastle-upon-Tyne . 
 
 2()ih April, 1921. 
 
CONTENTS. 
 
 Introduction — 
 
 Field of View .... 
 
 Cardinal Planes 
 Direct Perspective . 
 
 Tesselated Pavement 
 
 Fundamental Rules and Distance Points 
 
 Block and Box .... 
 
 Oblique Perspective — 
 
 Oblique lines (i) in H.PL. 
 , (2) in S.Pl. . 
 
 (i) Box and Open Lid 
 
 (2) Tilted Brick . . . . 
 
 Use of the Tables 
 
 Oblique Perspective by Plan and Side elevation — 
 
 Box 
 
 Useful Devices — 
 
 Division of Oblique- Line into Proportional Parts 
 
 Similar Oblique Squares . 
 
 Direct Square ... 
 
 Oblique Square .... 
 
 Direct Circle . ... 
 
 Oblique Circle . 
 
 Polygon . . ... 
 
 Oblique Perspective by Plan and Elevation — 
 
 Tilted Brick 
 
 Octa-hedron enclosed in a Cube . 
 Diminution or Enlargement 
 Shadows 
 Summary and Tables . ; . . 
 
 PAGE 
 
 6,7 
 8 
 
 9-13 
 
 9-11 
 
 9, 10 
 
 11-13 
 
 13. H 
 
 1+ 
 
 15-17 
 18-19 
 
 19-21 
 21-24. 
 
 24-25 
 
 25 
 
 26 
 
 27 
 
 28-29 
 
 30' 
 ■ 31 
 
 31-32- 
 32-33 
 34 
 35-41 
 41,42 
 
 
 
 
 
 FOLDING 
 
 I. 
 
 Figs 
 
 4, 
 
 5,6 
 
 
 n. 
 
 Figs. 
 
 7> 
 
 II 
 
 
 III. 
 
 Figs. 
 
 21 
 
 , 22, 
 
 23 
 
 PLATES 
 
 at end of book 
 
PERSPECTIVE 
 
 Perspective is the art of drawing solid objects as seen with one 
 eye on a plane surface. 
 
 A true stereoscopic view of a solid object cannot be represented 
 by one picture, as stereoscopic vision necessitates the fusion of two 
 ■dissimilar pictures corresponding to the two views of the object 
 seen by the two eyes. For instance, on looking at a cube, one inch 
 wide, held 8 inches in front of the eyes, the front surface of the cube 
 is seen by both eyes : the right side is foreshortened by the right 
 ■eye, and the left side is similarly foreshortened by the left eye. As 
 these different views depend entirely on the different positions of 
 the two eyes we see that the position of the observing eye with 
 regard to the object is an all-important consideration. 
 
 The simplest way to make a perspective drawing of any object 
 in the field of view is to trace out its outline as seen through a glass 
 screen that is held in the vertical plane. It is necessary to close 
 one eye, and to keep the other at a stated distance from the centre 
 ■of the screen. If the distance of 12 inches be chosen, then the 
 finished drawing must always be viewed from this distance of 12 
 inches. In the following pages is given the simplest njethod of 
 making a similar perspective drawing on paper of any object bounded 
 by straight lines. 
 
 Large pictures are usually intended to be viewed at a distance 
 ■of 13 feet, and the artist's eye when painting is usually about 5 feet 
 from the ground. In Fig. i the picture plane (P. PI.) is represented 
 by PP' and the eye of the artist (and of the observer) S at a distance 
 SC or 13 feet from the centre C of the picture Both S and C 
 are supposed to be 5' feet from the level of the ground, so that CO 
 would represent 5 feet. HCL represents the horizon line (H.L.) 
 and GOL' the ground line (G.L.) in the picture. 
 
 5 
 
Fig. 1. 
 Small pictures are usually viewed from a shorter distance sa 
 that SC may be 2 feet or only 10 inches. The " ground line " 
 G.L. in the picture need not represent the level of the ground, it. 
 may indicate the level of a table as in Fig. 11. 
 
 FIELD OF VIEW 
 
 Now as the eye when fixed on the point of sight C cannot 
 see at all distinctly more than 30° on either side of the line of sight 
 SC, by making CSE = 30° we can determine the upper limit of the 
 
 6 
 
picture, and by drawing a circle of radius CE on the picture plane 
 PP' we define the boundary of all that can be legitimately represented 
 for an observer placed at S. The extreme distortion shown in 
 Fig. 6 and in Fig. 7 is due to the field of view exceeding this 
 limit and to the fact that the distance SC, from which they should 
 be viewed, is far too short. 
 
 In Fig. 2 the fine SC is supposed to be folded down into the 
 picture plane. This is the standard diagram from which all the 
 perspective drawings in the following pages have been made. 
 
 Better pictures will often be obtained by making SC three or 
 four times greater than CL, i.e. by making the angle CSL about i8|° 
 or even 14°, but in order to keep the diagrams small the usual 
 English value (30°) has been adopted. 
 
 It should be noted that in this little book the horizon fine HCL 
 will always be denoted by H.L. ; the ground line GOLI by G.L. ; 
 the point of sight by C ; and the line of sight SC by C.S.L. 
 (central sagittal line). 
 
 7 
 
CARDINAL PLANES 
 
 It is important to assign names to the three cardinal planes of 
 space that correspond to the ordinary terms— length,, breadth and 
 height. On referring to Fig. 3 it will be seen that the horizontal 
 ground plane (G.Pl.) is represented by the plane AEMN which 
 contains in the foreground the ground line (G.L.), whereas NMKF 
 represents a vertical plane (V.Pl ). Every plane that is parallel to 
 G.Pl. is a horizontal plane (H.Pl.), and in this book every plane 
 that is parallel to the picture plane (P.Pl.) is called a vertical plane 
 (V.P1.). The other planes, such as AF, BH, OI, DJ, and EK, 
 that are at right angles to each of the previous planes are called, 
 sagittal planes (S.Pl.). The central sagittal plane (C.S.Pl.) is the 
 plane that contains the point of sight (C), and in this diagram is 
 
 C 
 
 represented by OI, and occupies the middle of the diagram. As 
 the edge only of C.S.Pl. can be seen, it is represented only by a 
 vertical line. The horizontal line drawn through C is the horizon 
 line (H.L.). 
 
 When viewing a long street that extends horizontally in front 
 of one, the line of roofs, if uniform, either above or below H.L., 
 may be seen to converge towards the right and left limits of H.L. ; 
 in such a case, however, the field of view extends beyond 30° on 
 either side of the C.S.P1., and a horizontal line appears slightly 
 
 8 
 
curved towards H.L. As pictures only include what can be 
 distinctly seen, z.e.,'not more than the prescribed 30° on either side, 
 the very slight concavity of the horizontal lines is inappreciable, 
 and, as we know that the line is straight, we expect to see it straight. 
 Note that, as AEMN is a horizontal plane, the line MN is the 
 same distance below the H.L. that G.L. is, and that, although MK 
 is actually so much shorter than EV, perspectively the same height 
 is represented by each, for EV is touching P.Pl., but MK is some 
 distance behind it and further from the observer. 
 
 The H.L. should strictly be at the same height from the ground 
 as the observer's eye, and consequently pictures with a low H.L. 
 should be hung rather high on the wall, and vice versa. It is 
 interesting to note that many pictures of the old masters have been 
 painted with a low horizon line, as they were evidently intended to 
 be hung rather high. Nearly all pictures of the present day are 
 painted so that they can be only seen to advantage by being hung 
 " on the hne " as it is called. 
 
 TESSELLATED PAVEMENT IN PERSPECTIVE 
 
 The diagram Fig. 4 represents tessellated pavements of two 
 kinds : that on the right consisting of 2-inch squares, while that on 
 the left is formed of smaller squares arranged diagonally, and are 
 such that each diagonal is 2-inches in length. Successive pairs of 
 these squares are shaded so as to form a familiar pattern. It will be 
 noticed that the pavement does not look flat ; this is because the 
 distance SC. is so short ; had it been greater, it would have required 
 a much larger sheet of paper to contain the indicated points Dj and Dj 
 The drawing would look better if it could be viewed from the 
 distance SC. The lines SH and SL, 30° on either side of C.S.Pl. 
 show that the diagram lies wholly within the prescribed field of view. 
 
 The first three fundamental rules of Perspective will now be 
 
 given : — 
 
 (1) All Lines in a Vertical Plane are Drawn in their 
 Real Direction 
 
 Or every line in a plane that is parallel to P.Pl. is drawn in 
 its real direction ; in other words, there is no vanishing point 
 (V.P.) for lines in a.V.Pl. For instance, in Fig. 4, the lines that 
 are really horizontal are drawn horizontal, and in Fig. 5 the lines 
 that are vertical or oblique, which are in a V.Pl., are drawn vertical 
 or with their real obliquity. 
 
(2) All Sagittal Lines are Drawn Converging towards C 
 
 In other words, C is the V.P. for all sagittal lines. For instance, 
 in Fig. 4 three parallel sagittal lines from the points 2-4-6 are drawn 
 converging towards C. 
 
 (3) All Measurements are Made in P. PI. 
 
 In Fig. 4 all the measurenjents are made on G.L. As 2-4-6 
 indicate points 2-ins., 4-ins. and 6-ins. respectively to the right of O, 
 the three lines 2C, 4C, 6C represent three parallel sagittal Hnes 
 distant from C.S.Pl. 2-ins., 4-ins. and 6-ins. respectively. These 
 three sagittal Hnes, therefore, indicate the lateral boundaries of the 
 two-inch squares. 
 
 We have now to measure on 2C lengths to represent 2", 4", 6" 
 and 8" behind P.Pl. The two dashes (") represent inches. First 
 we must find the measuring points (M.Ps.) for sagittal lines ; they 
 are commonly called the distance points. 
 
 DISTANCE POINTS 
 
 On H.L. mark off D^ and Dj on either side of C, so that CD^ 
 and CDj are each equal to SO. Then Dx and Dj are the distance 
 points. To mark off 2C into the four lengths stated above ; from 
 the point 2 we mark the point 2" to the left of 2 {i.e., the point O), 
 then the points indicated by 2', 4' and 6', which are 4'', 6" and 8" 
 respectively to the left of the point 2. On now joining 6', 4', 2', O 
 and Dj, the points where these slanting lines cut 2C will indicate 
 the points on this line that are respectively 8", 6", 4" and 2" behind 
 P.Pl. So a is 2" and d is 8" behind P.Pl. We have only to draw 
 horizontal lines from these points of intersection to meet the sagittal 
 line 6C to represent the six 2'' squares of pavement on the right 
 side — see (i) above. 
 
 On the left side of ad is another tessellated pavement 
 that reaches from GL to 8" behind P.Pl. as it is bounded by the 
 horizontal line through d. Join SDj, 60^, ^D^, 2Dx, OD^, 2'Di 
 and 4ID1 ; also join i6'D2, I4'D2, i2'D2, lo'Dj and 8ID2. As the 
 remaining oblique lines from 6', 4 1, 2' and O have already been 
 drawn, we have now a series of oblique squares between SH and 
 2d seen in perspective. It only remains to shade the successive 
 pairs of squares. 
 
 It will be noticed that Dj 4 and D^ O intersect at a, and it will 
 be seen that any point, 2" or 4" for instance, behind P.Pl. in the 
 
sagittal line 2C can be found by using either D, or Dj according 
 as measurements are made to the right or to the left of the point 2. 
 
 The sides of the oblique squares on the left are parallel to the 
 diagonals of the direct squares on the right, so they must represent 
 lines at an angle of 45° with P. PI. It may be inferred as they converge 
 either to D^ or Dg, that Dg is the V.P. of Unes at 45° with P.Pl. which 
 tend to the right, and that Dj^ is the V.P. of the similar lines that 
 tend to the left. This will be explained when obUque perspective 
 is dealt with (p. 17). 
 
 (A.) DIRECT PERSPECTIVE, or Perspective of Lines in 
 the Cardinal Directions 
 
 If we consider a cube or a brick or any object ot similar shape, 
 we se^ that its six surfaces consist of three pairs, each pair being 
 parallel to a plane that is at right angles to the planes of the other 
 pairs. Consequently if two of these planes coincide with two of 
 the cardinal planes, the remaining plane must also coincide with the 
 remaining cardinal plane. When such an object is drawn in this 
 position, the principles of direct perspective must be used. 
 
 In the figure on the left of Fig. 5 a rectangular block 3"X 2 X 6" 
 has been drawn with its base (the 3" X 6" surface) resting on a table 
 2" to the left of C.S.Pl. and 2" behind P.Pl. It is unnecessary now 
 to indicate the position of the observer by S., as we know that he 
 is assumed to be at a distance from the picture represented by CD. 
 
 As the block is placed 2" to the left of C.S.PL, and as it is 3" 
 wide, the points 2 and 5 are marked to the left of O on G.L., and 
 lines are drawn from 2 and 5 to C which will give the directions 
 of the lateral edges of the block. It will be found that 5 C need 
 only be drawn as far as b, as the block will obscure part of this line. 
 In order to find the corner a which is 2" behind P.Pl. we must 
 measure 2" to the right of 2 (i.e., O), and a is the intersection of 
 2C and OD, so ab drawn horizontally to meet 5C in 6 will define 
 the near edge of the base of the block. The side ad is 6" long ; 
 as a is 2" behiild P.Pl. d must be 6" further in than a, so the point 6 
 (6" to the right of O, which corresponds to a) must be joined to D, 
 and its intersection with 2C will determine d. 
 
 Note that whenever D (or a measuring point of any kind on 
 H.L.) is used to determine the length of a line in a horizontal 
 plane which is situated some distance behind P.PL, such as 
 
ad, D<2 must be joined and produced to meet P.Pl. in that 
 horizontal plane at some point m : similarly join Dd and 
 produce to meet P.Pl. at mK Then the length of the 
 horizontal line m m' in P.Pl. is equal to the real length 
 required of ad according to the scale used. 
 
 In the present case the horizontal plane considered is G.Pl. 
 so we have only to produce Da. to O, and produce Dd to 
 meet G.L. at 6 ; as 06 represents 6 ins., ad is 6 ins. 
 
 We have now to find the position of c when be represents a height 
 of 2". On 5 erect a vertical 2" high, and join its summit to C ; the 
 vertical from b which meets this slanting line will define be the height 
 of the block. This procedure is conveniently called the transference of 
 the vertical from 5 to 6. The figure is completed by drawing ce 
 horizontally to meet the vertical on a at e, by joining eC, and from 
 the intersection of eC with the vertical on d drawing a horizontal 
 line to meet cC 
 
 All this is simple enough, but there will be a difficulty if there 
 be not sufficient paper to receive the point D. It will be found 
 that an equally correct drawing may be made if any arbitrary fraction 
 of CD be taken, as long as the same fraction of the corresponding 
 measurement in P.Pl. is used. In order to illustrate this, a box 
 4" X 2" X 8" has been drawn 2" to the right of C.S.Pl. and 2" behind 
 P.P1., when CJD is one-fourth of CD. The lid of this box is 
 represented as open 30°. 
 
 To draw the box the points 2' and 6 are taken 2" and 6" to the 
 right of O, and lines drawn from these points would indicate the 
 lateral boundaries of the box. (The dashes (I) are used to distinguish 
 the points on G.L. that refer only to the box.) 
 
 As we are now using JD. we must measure one-fourth of 2" 
 or Y to the left of 2' (i.e., i|l to the right of O). The intersection 
 of ij' JD. and 2'C. will mark the inner corner of the box, and the 
 near edge will be given by a horizontal line from this corner to 6C. 
 As the length of the box is 8" we must measure J of 8' or 2" to the 
 left of i^", or the point Ji on the left of O. On now joining J' and 
 ^D. its intersection with 2'C. will give the far corner of the box. 
 
 The drawing can be completed in the same way as the block 
 on the left. It will be noticed that the near end of the box is as 
 
far behind P.Pl. as the near end of the block, and that the whole 
 drawing is more compact by using JD. 
 
 As the box is 2" high and 4" wide, a vertical is erected at 6, 
 6" high with a small mark at the height of 2". When this vertical 
 is transferred to the edge of the box, the dotted line would represent 
 the margin of the hd if it were open 90°. But as already intimated 
 (p. 9) all lines in a vertical plane are drawn in their real direction, 
 so it is only necessary to draw a Une 30° with the top of the box and 
 of length exactly equal to the length of the dotted line. The end 
 of this line is joined by a sagittal line to C, and the further edge 
 of the lid is defined by drawing a line from the far corner parallel 
 to the edge just drawn to meet this sagittal line. 
 
 Fig. 5 completes the illustrations of direct perspective, which 
 includes no obUque lines except those that lie in a vertical plane. 
 
 We may now tabulate for convenience of reference some of the 
 points that we have so far illustrated-, dealing only with lines drawn 
 in the cardinal directions. 
 
 Lines 
 
 Direction 
 
 Vanishing Point 
 
 Measuring Point 
 
 V 
 
 Vertical 
 
 Absent 
 
 C 
 
 H 
 
 Horizontal 
 
 Absent 
 
 C 
 
 S 
 
 Towards C 
 
 C 
 
 DonH.L.;CD = SC 
 
 We have already shown how oblique lines in a vertical plane 
 should be drawn, and we must now describe how oblique lines in 
 the other cardinal planes should be represented. This involves the 
 principles of oblique perspective. 
 
 (B.) OBLIQUE PERSPECTIVE 
 Oblique Lines in a Horizontal Plane 
 
 In Fig. 6 a box is drawn, the end of which recedes 
 30° to the right, while the long side recedes 60° to the left. 
 Through the observer's station point S a horizontal line BSB' is 
 drawn, then the angle BSV^ is made equal to 60°, and SV^ meets 
 H.L. in Vi. Similarly B'SYg = 30° and SV^ meets H.L. in V^. 
 Then V^ is the V.P. for all lines in any horizontal plane that are at 
 . 60° to P.P1. and tend to the left, and Vg is the V.P. for all lines in 
 any horizontal plane that are at 30° to P.Pl. and tend to the right. 
 
 13 
 
The corresponding measuring points are also on H.L., and their 
 situation is determined by making V^ Mj^ ^ V^ S, and Y^ M.^ = Vj S. 
 It will be found that the measuring point and its corresponding 
 vanishing point will always be on opposite sides of C.S.Pl. 
 
 Oblique Lines in a Sagittal Plane 
 
 In Fig. 5 the side aed of the block is in a sagittal plane and C 
 is the V.P. of its sagittal edges. But if the further end of the block 
 were raised 30°, while the edge ab still rested on the table, it is clear 
 that the V.P. of the raised edges would no longer be C, but would 
 be some point immediately above C. Consequently the V.P. of all 
 oblique Unes in a sagittal plane will lie somewhere in OC produced 
 (see Fig. 7). As D is the M.P. of all sagittal lines we make CDV = 6 
 where ^ is the angle that the oblique sagittal line makes with H.PL, 
 and DV meets OC. produced, at V. When the oblique line 
 recedes upwards, the V.P. is above H.L. and vice versa. 
 In Fig. 7 Vj is the V.P. for lines receding 30° upwards, and Vj for 
 those receding 60° downwards. The corresponding M.P.'s are also 
 on OC. produced, and are determined by making VM = VD. The 
 M.P. and its corresponding V.P. will always be on opposite sides 
 of H.L. 
 
 We can now tabulate the rules for drawing oblique lines in the 
 three cardinal planes. The angle of obliquity is always measured 
 from a horizontal or from a vertical plane. For instance, the 
 obliquity of lines in a V.Pl. or a S.Pl. is denoted by ^, the angle 
 which these lines make with a horizontal plane ; while 6' is the 
 angle which oblique lines in a H.Pl. make with a V.Pl. (or the P.Pl.). 
 
 The explanation of the expressions in brackets is given on p. 19. 
 OBLIQUE LINES AT ANGLE 6 OR 9' 
 
 
 Direction 
 
 Vanishing Point 
 
 Measuring Point 
 
 InV.Pl. 
 
 t) 
 
 Absent 
 
 C 
 
 In S.P1. 
 (Fig. 7) 
 
 Towards V 
 
 V in OC ; CDV = 
 
 (CV = SC tan 6) 
 
 M in OC ; MV - DV 
 (MV = SC sec B) 
 
 In H.P1. 
 (Fig. 6) 
 
 Towards V 
 
 VinH.L.;BSV = fli 
 (CV = SC cot ei) 
 
 M in H.L.; MV = SV 
 (MV = SC cosec fli) 
 
 14 
 
(1) OBLIQUE LINES IN H.Pl. 
 
 We will first take the case of oblique lines in the horizontal 
 ground plane. Fig. 6 shows a box 4" x 2" X 8" with its nearest 
 corner h 3" to the right of C.S.Pl. touching P.Pl., and with its sides 
 ba and hd forming angles of 30° and 60° with P.Pl. 
 
 According to the rule given, BSB' is drawn horizontally through 
 S, B'SVj is made equal to 30°, and SVg meets H.L. in Vg. Similarly 
 BSVj = 60°, and SV^ meets H.L. in V^. The corresponding 
 measuring points are easily found on H.L. for Vg Mj = Vj S, and 
 Vi Ml = Vi S. 
 
 The point h is given 3" to the right of O, and from b the lines 
 iVj and 6V1 are drawh to represent the receding edges of the box. 
 As ba is 4" long, and as it lies in the direction of Vg, we must use Mjj 
 as its measuring point. The point on G.L. 4" to the right of b is 
 marked by 4 ; join 4M2 and its intersection with bW^ defines the 
 corner a. Similarly as bd is 8" long and lies in the direction of V^, 
 the intersection of SM^ and fiV^ defines the far corner d of the box. 
 
 The drawing of the box may be completed in the usual way 
 by erecting verticals at the corners a, b and d; be is 2" high as it 
 is in P.Pl., and the intersections of eVj and eV^ with the verticals 
 above a and d will define the sides of the box. If the summits of the 
 verticals on a and d were joined to Vj^ and Vg, the remaining edges 
 of the box would be defined, as the intersection of these Hnes would 
 give the most distant corner of the box. However, as a Ud open 30° 
 is to be added to the drawing, these edges will be obscured, except 
 a part of that which leads to V^. 
 
 Now the lid, as the roof of a house seen at a similar angle, is 
 incHned to each of the cardinal planes, and the table previously 
 given for oblique lines in one of these planes will give no help. If 
 the lid were closed, the V.P. of its short edges would be Vj, but as 
 it is open 30° its line recedes upwards to the right, so its V.P. 
 must be somewhere immediately above Vj. It will be found that if 
 from Mg a line Mg V3 be drawn at 30° with H.L. meeting the vertical 
 through Vj at V3 this point V3 will be the required V.P. for the short 
 edges of the lid. So from e and from the corresponding corner g 
 lines are drawn to V3 in which the edges of the lid must lie. 
 
 Now it was stated on p. 14 that when a V.P. is in the vertical 
 line OC produced, the corresponding measuring point M is found 
 
 It 
 
also in that line on the opposite side of H.L. In this case the 
 corresponding measuring point M3 is found by making 
 
 V3 M3 = V3 M„ 
 when M3 is below H.L. on the vertical just drawn through V3. 
 
 As be is in P.Pl., erect a vertical 4" high on e, then 64' would 
 represent the edge if the lid were open 90°. The intersection of eVg 
 and 4'M3 will define the length ef of the lid at tbs angle 30°. The 
 long edge of the lid is obviously parallel to any one of the other long 
 edges such as bd, so it is only necessary to join Vj/, and the intersection 
 of this line with the line from the corner g to V3 will determine the 
 far corner /I of the lid. 
 
 As a sketch may be required of an oblique house or something 
 similar, that has not one corner touching P.Pl. it will be well to 
 describe the procedure ah initio. It is required to erect a vertical 
 dg at d 2" high and from g to draw a line 4" long inclined at 30° with 
 the horizontal. 
 
 From any convenient point on H.L. draw a line through the 
 base {d) to G.L. In the figure we already have the Une Vj^rffi to 
 G.L. so that will serve. Erect he vertical 2" high, and above that 
 64' four inches high ; transfer Je4' to dgi^' by means of V^. The 
 process may be regarded as one of sliding the vertical be^^ between 
 the guiding lines V^ft Y-^e and V14' to the position dgi\!' . The 
 points g and 4'' will occupy the, same position, whatever point is 
 taken on H.L. as the origin of the guiding lines. Try for instance 
 with M2 join Mj d and produce to G.L., meeting it at »z ; erect a 
 vertical at m, 2" + 4" high, and join Mg with each of these points 
 to form the guiding lines, and slide this vertical between these 
 guiding lines to d, and the distances dg and ^4" will be found as 
 before. 
 
 However found dg represents a vertical 2" high and ^4" a vertital 
 on g, 4" high in perspective. 
 
 The line gV^ is drawn to give the direction of the inclined line, 
 and its intersection with 4"M3 will define the point /', where gp is 
 4" long. (It is an accident that ^/'Vg happens to pass the top of the 
 original vertical S4I.) 
 
 Two vertical lines are shown on the near end of the box, if we 
 wish to find the actual distance between them, we use the method 
 described on p. 11. The base of the line ha is in G.Pl. ; and since 
 
 16 
 
ba is directed towards Vg we join Mg to the base of each line and 
 produce to meet G.L. at points indicated by dots. The distance 
 between these dots is ij", so the distance between the two lines 
 is i^" or one-third of ab. If Mg be joined to the upper extremities 
 of the two lines and produced to G.L., a totally erroneous value 
 will be given, for the upper edge of the box is not in G.Pl. A 
 correct result however will be obtained if the lines from M.^ are 
 drawn only as far as to meet a horizontal line through e, for then the 
 measurement is made in P.Pl. in the same horizontal plane as that 
 m which the upper edge of the box lies. This is a special instance 
 of the first of the two following rules : — 
 
 (i) Whenever the V.P. and the M^P. of a line are on a horizontal 
 line, the line on which the measurement is made is the intersection 
 of the P.Pl. with the horizontal plane of the line to be measured. 
 
 (2) Whenever the V.P. and M.P. of a line are on a vertical line, 
 the measurement is made on the line formed by the intersection of 
 the P.Pl. with the 'vertical plane '(p. 35) of the line to be measured. 
 
 If it be required merely to divide an oblique line into any number 
 of proportional parts the M.P. need not be used as will b^ explained 
 on p. 24. 
 
 If each side of the box in Fig. 6 were so placed that they formed 
 angles of 45° with P.PL, of course BSV^ and B'SVg would each be 
 45°, and then it would be found that V^ would coincide with D^ 
 and V2 with Dg. This explains why the diagonal lines on the left 
 of Fig. 4 all tend either towards Dj^ or Dj. 
 
 The box and its lid have now been fully drawn, but the result 
 is not satisfactory ; the lid seems distorted. The reason is twofold. 
 
 (i) In order to show all the vanishing and measuring points, etc., 
 on the drawing, the distance SC has been made exceedingly short, 
 i.e., the observer's eye must be at this distance from the picture. 
 Whenever a photograph of a high building has been taken with a 
 wide-angle lens at too close a distance, a similar distortion may be 
 seen. Such a photograph can only be seen in true perspective by 
 using strong convex glasses, so that it can be viewed at a distance 
 of 3 or 4 inches. 
 
 (2) The box on the right side exceeds the limit of the field of 
 view allowed 30". (See Fig. 2, p. 7, when CSL = 30°) 
 
 A method will be presently given that will obviate this defect. 
 
 17 
 
(2) OBLIQUE LINES IN S.PL. 
 
 An example is now given of a brick 4^" X 3" X 9", whose sides 
 are in sagittal planes, but the other surfaces are inclined both to 
 H.P1. and to V.Pl., the far end being raised 30° from the table or 
 ground plane. 
 
 The edge A A' rests on the table 3" to the right of C.S.Pl. 
 and 4" behind P.Pl., while the edge BBl is in the P.Pl. (Fig. 7). 
 
 Here again distortion is seen as, in order to show all the necessary 
 V.P.'s and M.P.'s in the figure, the distance of the observer is supposed 
 to be merely SC or CD. The distortion will be less if the drawing 
 be viewed from this distance, but it will not disappear entirely, as 
 the right part of the brick is beyond the field of view allowed, which 
 is limited by the line SL where CSL = 30° (p. 7). 
 
 In Fig. 18 the same object is drawn in exactly the same position 
 by another method without any distortion. 
 
 On referring to the table (p. 14) it is seen that as both the long 
 and the short edges of the brick are oblique lines in a S.PL., both 
 the V.P.'s. and the M.P.'s must be in OC produced. As AE is 
 inclined 30° up from H.PL., CDVj is made 30° and V^ is the inter- 
 section of DVi and OC produced. Similarly as the short edge BA 
 incUnes down at an angle of 60° with H.PL., CDVg = 60°, and Vg 
 is the intersection of DVg and CO produced downwards. The 
 points Ml and Mg are found by making V^ Mj^ = V^ D, and by 
 making Vg Mg = Vg D. 
 
 The points P and Q are situated 3'' and 7I" from O on G.L., 
 CP and CQ are joined, and the intersection of i|D and CP will 
 denote the point A, and AA', horizontal through A meeting CQ 
 in A', will define the lower edge of the brick on H.PL. 4^" wide 
 and ij" behind P.Pl. Join V^ A. and Vg A', and produce them 
 upwards to give the direction of the short edges, and join AV^ to 
 give that of the long edge of the brick. 
 
 In order to determine the length of the edge AE. a scale of the 
 height 9" should be made on OCV, which lies conveniently on the 
 left of the drawing. Transfer A to P on G.L. by means of C ; 
 mark e, 9" vertically above P ; transfer Pe to Ael by means of C ; 
 join Ml e\ and its intersection with AV^ will define the point E. 
 We need not go through all this process in order to define B. or B', 
 for we are told in this special case that the BB' line is in P.PL, so 
 
 18 
 
it is only necessary to transfer A' to Q by means of C, to erect 
 at Q a vertical in P. PL, and B' will be the intersection of Vg A' 
 produced and this vertical. Through B' draw BIB. horizontally 
 meeting Vj A produced in B, and the surface AA'B'B tilting towards 
 P.P1. is defined. 
 
 The intersection of BV^ and Vg E produced defines the point D ; 
 •draw DD' horizontal meeting BIB^ in D', and the outline of the brick 
 is completed. 
 
 The above is the method of drawing in perspective that is usually 
 taught in the art schools ; but, as has been shown, it is impossible 
 to draw correctly by this method if one's space is limited. The 
 vanishing and measuring point may be, perhaps, six feet away from 
 one's picture, and in such cases the use of a centrolinead is recom- 
 mended. This is a simple instrument that enables one to draw lines 
 to any vanishing point, however distant, and is very useful when a 
 ■number of parallel lines have to be drawn to a distant V.P. A 
 centrolinead is not, however, allowed in the examination room, so 
 it is necessary to give some other method when this instrument is 
 not available. 
 
 *Now on p. 14 there are some trigonometrical expressions given 
 in brackets in the table, and it will not. require any knowledge of 
 trigonometry to explain their use. In Fig. 6 the line bd is in H.Pl. 
 •drawn in the direction bV^, and it represents a line at an angle 60° 
 Avith P.Pl. It will save trouble as well as paper if V^ can be found 
 vsithout marking S on the paper, for in large pictures SC may be 
 12 feet or even more. It will be convenient to regard SC, the 
 •distance at' which the picture is to be viewed, as either 10 inches, 
 10 feet, or 100 c. m. (i.e., about 40-ins.), for in the little table on 
 p. 42 is given the value of the trigonometrical ratios, so that 
 •one can mark the V.P.'s and M.P.'s. at once upon H.L. in Fig. 6 
 "without drawing the angles 60° and 30°. 
 
 Suppose for instance that SC is 10 inches or 10 feet, it is seen 
 from p. 41 (where the table just given is repeated) that 
 CV = SC cot 6' foi- any oblique line in H.PL, so that V^ in Fig. 6 
 is distant from Ci 10 cot 60° = 10 X -5774 = S,-774 inches or feet; 
 whereas CVg = SC cot 30° = 10 X i'732i = 17-321 inches or feet, 
 ■as the case may be. 
 
 *This need not be read by those who are unfamiliar with mathematical 
 formulae. 
 
 ^9 
 
Also (Fig. 6) MjVi = SVi = SC cosec CViS, 
 
 = SC cosec 60° = 10 X 1. 1547 = 11.547 units, 
 and M2V2 = SC cosec 30° = 10 X 2 = 20 units. 
 
 Similarly in Fig. 7 where V^ is the V.P. for 30° in a sagittal 
 plane, if SC be 10 units, 
 
 as CD = SC, 
 
 CVi = CD tan 30° = SC tan 30° = 10 X '5774 = 5-774 
 units, 
 
 and CV2 = SC tan 60° = 10 x i'732i = 17-321 units. 
 Again MiVi = DYj = CD sec 30° = SC sec 30°, 
 .•. MiVi = 10 X 1-1547 = 1 1 -547 units, 
 and M2V2 = SC sec 60° = 20 units. 
 
 (In the diagrams the value of SC is 4.1 cm. in Fig. 6, and 7.7 cm. 
 in Fig. 7, and the accuracy of the V.P.'s and M.P.'s can be 
 tested by substituting these values for SC in the above) 
 
 As we know that in Fig: 6 V^ must be to the left of C, and M^ 
 to the right, and as we know roughly the direction of every other 
 V.P. and M.P., there can be no ambiguity about placing them in 
 their proper positions. 
 
 This table may be also used for planes which are oblique to all 
 the three cardinal planes, as the lid of the box in Fig. 6. If the lid 
 were closed the line ef would be in the direction eVg, but as the lid 
 is open d, it will be found that 
 
 Vg V3 = M2 V2 tan 30° = 20 X 5-774 = 11.548 
 and Mg ¥3= Mg V3 = MgYj sec 30° = 20 X 1.1547 = 23.094. 
 The geometrical method by drawing the line M3V3 at the 
 appropriate angle will however, usually be more convenient. 
 
 The ordinary drawing board is 16" x 23", and it will be useful 
 to draw the horizon line upon it 10" from its base, and to bisect it 
 by a vertical Une cutting it at C. If Dj^ be marked 10" to the left 
 of C. on this H.L., and D2 be marked 10" to the right of C, one has 
 the measuring points for sagittal lines marked permanently, and also 
 the V.P.'s for all lines in a horizontal plane that are at an angle of 
 45° with P.Pl. A small sheet of drawing paper may be fixed to the 
 drawing-board with stamp paper, so that the H.L. and the C.S.L, 
 
 20 
 
of the purposed drawing will coincide with the horizontal and vertical 
 lines on the drawing-board. This will be suitable for all drawings 
 that are intended to be viewed at a distance of lo". If the drawing 
 is going to involve lines in a horizontal plane at an angle of 45° with 
 P.Pl., their measuring points can be at once taken from the table. 
 
 MD = MV = 10 cosec 45° = i4.i42. 
 
 As Di is 10" to the left of C, M^ is 4-142" to the right of Q 
 and D2 being to the right of C, Mg is 4.142" to the left of C, so that 
 both these measuring points can be marked on the paper. 
 
 If other V.P. s or M.P. s are required that fall outside the paper, 
 but not beyond the limits of the drawing-board, drawing-pins may 
 be inserted at their appropriate places ; against these drawing-pins 
 rules may be set to draw the necessary lines on the paper. It will 
 be found, however, that for many subjects even with such a short 
 distance point as 10", the drawing-board is not large enough to have 
 the necessary V.P.'s represented upon it. 
 
 This table will enable one to make perspective drawings from 
 plans (as described in the following pages) miich more speedily and 
 accurately. Whenever a line AB. is inclined ^ to a horizontal line, 
 its projection on the horizontal is AB cos 0, and its projection on 
 the vertical is AB sin 0. 
 
 Thus in Fig. 8 (p. 22) hg = ef cos 30° = 4 X -866 = 3-464 ; 
 in Fig. g Pb = bd cos 60° = 8 X -5=4; 
 while ?d = hd sin 60° = 8 X -866 = 6-928 ; 
 and dS = dk sin 30° = 4 X -5 = 2. 
 
 THE NEW METHOD 
 
 The method now described is really an old method that has 
 been used by architects for many years, but it is not usually described 
 in books on perspective. Architects frequently make perspective 
 drawings of buildings from plans and elevations of the building 
 without the use of vanishing points at all. 
 
 We will draw the two last figures (Figs. 6 and 7) by this method, 
 and it will be seen that all the distortion has disappeared that was 
 evident when the vanishing and measuring points were so near as to 
 be all marked on the paper. 
 
Fig. 8. 
 
 Fig. 8 represents an end view, or a side elevation, of the box 
 shown in Fig. 6 when the observer is opposite the end of the box ea^ 
 and his eye is directed towards Vj. The box is 4" x 2" X 8" with 
 its lid ej open 30° ; from /, jg is drawn perpendicular to the base 
 of the box ha. 
 
 A plan of the box must now be drawn, i.e., a view of the- 
 box as seen from above or a projection of the box on H.Pl. 
 Fig. 9 shows such a plan enclosed in a parallelogram PQRS. where 
 the end ba of the box is 30° and the long side bd is 60° from the 
 horizontal line PQ. With compasses or dividers bg is marked off in 
 the plan equal to bg in the side elevation, and from the point g 
 in the plan a dotted line ^^1 is drawn parallel to the long side ah to- 
 indicate the position of the edge of the lid in the plan. Perpendiculars 
 are now dropped from the extremities of this dotted line to the base 
 PQ at G and G', and another perpendicular from h meets the base 
 
 h R 
 
 I \ \ 
 
 \ ^ \ 
 
 I \ ^ \ 
 
 a 
 
 M g' H b 
 Fig. 9. 
 
 N 
 
in H. We have still to devise some means of indicating the positions 
 of d and a on PQ. The simplest way is to draw a diagonal QS and 
 let horizontals from d and a meet this diagonal ; from these meeting 
 points perpendiculars are drawn to PQ cutting it at M and N. 
 
 We have now determined all the necessary points on the base 
 line and we have only to transfer them to the ground line in 
 Fig. 10, which represents on the ground plane Fig. 9 in 
 perspective ; PQ in Fig. 10 is an exact repetition of PQ 
 in Fig. 9, with the point b 3" to the right of O. We take 
 C^D to be \ of the distance of the observer, so PK is 
 made \ of PS in Fig. 9, and S in the perspective plan is the 
 intersection of PC and K ^D. Hence the enclosing parallelogram 
 PQRS is easily drawn. On joining HC. the point h is given as the 
 intersection of this Une with RS ; similarly on joining MC and NC 
 and drawing horizontals from their points of intersection with the 
 diagonal QS we get the points d and a. On joining now ha, ah, 
 hd and db we have the base of the box drawn in true perspective. 
 
 O M a'H h N G Q 
 
 Fig. 10, 
 On Q is erected the vertical QF equal to gf in Fig. 8, and on 
 it the point marked 2'' indicates the height of 2-ins. for the sides of 
 the box. Slide Qa." between the guiding lines ,CQ and Ca", and so 
 transfer the vertical to a and to R ; transfer the vertical at R to A 
 along the horizontal line RA. The summit need only be indicated 
 by a dot as this edge of the box will not be seen. Erect a vertical 
 be 2" high at b, and indicate by a dot a height of 2" above P., and 
 transfer this height to d by means of C. The .three vertical corners 
 have now been drawn, and it only remains to join their summits to 
 define two sides of the box. From the summit of the vertical on a 
 a line is drawn in the direction of the dot that represents the obscured 
 corner of the box to indicate that part of the long edge that is seen. 
 
 23 
 
(As the vertical edge at d is indefinitely marked by the nearly vertical 
 dotted line from C it will be better 'iio join JD and d and produce 
 to rfi ; on rfl to erect a vertical z"" high and from its summit drawr a 
 line in the direction of ^D. ; th^ intersection of this line with the 
 vertical on d will be a better indication of the height of the edge 
 at d.) 
 
 We have now to draw the hd : a dot is placed over G at the 
 same height as QF, from this point a dotted line is drawn towards C. 
 Now on ba a dot indicates the position of the point g in the plan 
 Fig. 9, being the intersection of CG and ba. On this point (g) a 
 vertical is raised, meeting the above-mentioned dotted line from 
 C in/. The line ef is the near edge of the lid. Similarly the height 
 of QF is marked above G' and a dotted hne is drawn from it to C. 
 The intersection of CG' and dh is marked to represent g^ (in Fig. 9), 
 and a vertical from this point {g') meets the dotted line from C in/' 
 which gives the far corner of the lid. The figure is completed by 
 joining ff^ and by drawing the far edge of the Ud. Although it 
 takes so long to describe, it will be found that the actual drawing 
 can be made in a much shorter time than would be imagined, and the 
 resulting figure is seen to be much more satisfactory than that in 
 Fig. 6, which was drawn with vanishing and measuring points. 
 
 In Fig. II the same box is represented lying on an inlaid table, 
 and its near end has been divided into three equal parts by two 
 vertical lines as in Fig. 6. These details have been added since the 
 method employed will be of frequent use. I lay no claim to 
 originality in the methods suggested in the next four pages, 
 although I discovered them for myself. I afterwards found them 
 described in Mr. Storey's excellent book on the " Theory and 
 Practice of Perspective." 
 
 TO DIVIDE AN OBLIQUE LINE INTO PROPORTIONAL PARTS 
 
 (1) When the Oblique Line Touches G.L. at One Point 
 
 The line ba touches G.L. at b, and it is required to divide ba 
 into three equal parts perspectively. 
 
 Take any three equal lengths on G.L. measured from b, and 
 number them i, 2 and 3. Join 3 a, and produce to K. on H.L. ; 
 join I K. and 2 K., and the intersections of these lines with ha will 
 divide ba into three equal parts perspectively. 
 
 24 
 
It will be noticed that the divisions of ba are not geometrically 
 equal, and that the distance 6 i on the ground line is of arbitrary 
 length. Whenever a specific distance is required to be indicated on 
 an obhque hne, the M.P. for this oblique line must be found, 
 or if that is inaccessible, some other method must be used : such as 
 constructing a plan of the perspective drawing. 
 
 (2) When the Oblique Line does not Touch G.L. 
 
 On the left of Fig. ii, suppose that it is required to divide k'4' 
 into foiir equal parts perspectively. 
 
 Take any convenient point in H.L. : in this case K' has been 
 taken vertically above k' ; join K' k' and produce it to meet G.L. in 
 k ; join K'4' and produce it to meet G.L. in 4. Divide Iq. into 
 four equal parts at i, 2, 3. Join K' i, K' 2, K' 3, and the intersections 
 of these lines with the given oblique line will divide it as required 
 at i', 2' and 3'. 
 
 This procedure will be required in the next problem. 
 
 WHEN ONE OBLIQUE SQUARE HAS BEEN GIVEN, TO ADD ANY 
 NUMBER OF SIMILAR OBLIQUE SQUARES TO IT 
 
 In Fig. II it is supposed that the oblique square k^d has been 
 given. It is immaterial whether the V.P. of either pair of sides be 
 accessible or not ; we will suppose that both V.P.'s are inaccessible. 
 
 Produce ^'i' and the corresponding side, and cut off say three 
 lengths equal to A'l' at 2', 3' and 4' as just described. 
 
 Draw the diagonal k^d of the given square and produce it to 
 meet H.L. in L. Join L i ', L 2' and L 3 ' by dotted lines, intersecting 
 the produced side from d at 2", 3" and 4". Join 4' 4", 3' 3", 2' 2", 
 and three more squares have been added to the original square. 
 
 If a lateral extension is required, it is only necessary to produce 
 4' 4'' to meet 2'L in 4, and to produce 3' 3" to meet i' L in 3, and 
 so on. 
 
 The value of the diagonal in perspective can hardly be over- 
 estimated, some other uses will be mentioned presently, but an 
 incidental use will be seen from Fig. 11. It will be noticed that 
 L is the V.P. of all the diagonals, when the sides of the oblique 
 squares have inaccessible V.P.'s ; this is occasionally of inestimable 
 service. 
 
 25 
 
As the given square corresponds exactly with the square 
 under the further half of the box, by producing 4 a to G.L. part 
 of another square beyond the box is represented. 
 
 As it is frequently necessary to enclose oblique rectangles in 
 direct rectangles, ouch as PQRS, it is well to carefully study Fig. 9. 
 It will be noted that P6 = HQ or ^R ; PM = NQ and GIH = GQ. 
 These relations hold good universally. 
 
 Hence if two sides of an oblique rectajngle be given, it is a very 
 simple matter to construct such a diagram as Fig. 9 or Fig. 10. 
 
 Should it be required to enclose an oblique square in a direct 
 square, the procedure is still simpler, as will be seen from Fig. 12, 
 where H may coincide with M or N according to the position 
 that the inscribed square occupies. In Fig. 12 the corner h is near 
 the left side of the enclosing square, but on holding the figure facing 
 the Hght and viewing the back surface of the paper, another position 
 of the inscribed square will be seen in which b is near the right side 
 of the enclosing square. There will be no ambiguity if it be 
 remembered that when a point {e.g., b) lies in a line PQ near to P 
 the next point in the adjoining line PS will lie far from P and vice 
 
 TO DRAW A DIRECT SQUARE IN PERSPECTIVE 
 
 If one side and the point of distance be also given 
 
 (As in Fig. 13, only one-fourth of the distance CD is represented, 
 it will be well for the reader to make a diagram for himself in which 
 CD2 is four times the length of CJDj.) 
 
 26 
 
If PO be given and CD. 
 
 Join CP and CQ and either PDg or QDi, then R is the inter- 
 section of PDj and QC, and a horizontal through R completes the 
 direct square ; or, if QD^ be drawn, S is the intersection of QDj 
 and PC. 
 
 If CJD be given as in Fig. 13, join PC and QC, mark off pQ 
 equal to one-fourth of PQ, and join pJOj, and its intersection with 
 QC will define the point R and the horizontal through R meeting 
 PC in S will define the direct square. 
 
 If PS be given and CD. 
 
 Join PD2 and draw horizontals PQ and SR through P and S ; 
 then R is the intersection of PDj and SR, and Q is the intersection 
 of PQ and CR produced. 
 
 If CJD be given, draw the horizontals PQ and SR ; join P5D2 
 cutting SR in r (not shown) ; make SR = 4 S*-, then R is defined. 
 Join CR and produce to meet PQ in Q. 
 
 TO DRAW IN PERSPECTIVE AN OBLIQUE SQUARE FROM ANY 
 
 GIVEN CORNER THAT TOUCHES THE ENCLOSING 
 
 DIRECT SQUARE. 
 
 ^ T L HP Q 
 
 Fig 13. 
 Let b be the given point (Fig. 13), and let PQRS be the "direct" 
 square. Make HQ equal to Pb, join PR, CH and C6, let the 
 
 27 
 
diagonal be cut by CH in m and by Ch in n. Then md drawn 
 horizontally will define d on the left, and the horizontal na will 
 define a on the right, and h will be the intersection of HC and SR. 
 Then ahdb will be the required oblique square in perspective. 
 
 Let d be the given point. 
 
 Draw dm horizontal, cutting the diagonal PR in m ; join Cm, 
 cutting SR in h, and produce to H. Make Pi equal to HQ. Join 
 Cb, cutting the diagonal PR in n ; draw na horizontal meeting QR 
 in a, and ahdb is defined. 
 
 When h is the given point, the first steps are these : join Ch and 
 produce to H.. Make Vb equal to HQ, join Cb, cutting the diagonal 
 in n, and continue as before. 
 
 When a is the given point, draw an horizontally meeting PR. 
 in w ; Cre produced gives b, and the other two points are found as 
 in the previous cases. 
 
 TO DRAW IN PERSPECTIVE A CIRCLE THAT IS ENCLOSED 
 
 BY A SQUARE 
 
 C 
 
 \^^^"^* 
 
 & 
 
 Fig. 14. 
 Let PQRS be the "direct" square (Fig. 14), where PQ may be 
 ■on G.L. or on any base line parallel to G.L. The method here 
 recommended is one that is based on that of the architect. The 
 
 28 
 
architect first draws a plan of the square with its inscribed circle ; 
 he then draws the diagonals of the square, and from the points of 
 intersection of the diagonals "with the circle he raises verticals to meet 
 PQ in h and AL On the base PQ he draws the direct square in 
 perspective by the method just given. He then joins CP, Ch, Ch\ 
 CQ and he joins C to the point where the circle touches PQ. 
 Diagonals are then drawn in the perspective square which give by 
 their intersections with Ch and C/jl four points through which the 
 perspective circle must pass, and the two diameters through the 
 intersection of the diagonals give four other points where the circle 
 touches the sides of the "direct" square. 
 
 As circles have so frequently to be drawn in perspective, it is 
 well to give a method which will avoid the trouble of drawing a plan 
 in each case. I am indebted to Mr. Storey's book on Perspective 
 for the following ingenious device : — 
 
 On joining the intersections of the diagonals with the circle it 
 is seen that a square is inscribed in the circle. On turning this 
 inscribed square through 45° we get Fig. 15, from which it is clear 
 incidentally that the area of any square inscribed in a circle is half 
 that of the circumscribed square ; further that the side AB = QO or 
 one-half the diagonal, and Ac = Qc. 
 
 Fig. 15, 
 Let A^ be made J PQ (or from c drop the perpendicular eg-), 
 and from Q drop the perpendicular QE equal to A^, then E^ must 
 be equal to Qc or Ac. If then Ah and AA' be taken on PQ each 
 equal to Eg, the distance hh^ must be equal to AB, the side of the 
 inscribed square in Fig. 15 or Fig. 14. The left side of Fig. 16 
 illustrates this ready way of dealing with the problem. The 
 
 29 
 
perspective circle is really an ellipse, but the geometrical centre of 
 the ellipse is slightly below the intersection of the diagonals. 
 
 The method may be also used in drawing a circle in oblique 
 perspective as will be shown in the next section. 
 
 Fig. 16. 
 {After Storey) 
 
 TO DRAW IN PERSPECTIVE A CIRCLE ENCLOSED IN AN 
 OBLIQUE SQUARE 
 
 Let ABDF in Fig. i6 be the oblique square with one corner 
 B touching a base line PB that is either in G.L. or parallel to G.L. 
 Let two of the sides • BD and AF converge towards an accessible 
 vanishing point V in H.L. Draw the diagonals AD and BF, and 
 join V to their point of intersection cutting FD in c, and AB in a\ 
 and produce to meet the base hne PB in a, and produce VA to meet 
 PB in P. 
 
 Make ^B one-fourth of the base, say one-half of aB ; drop 
 the perpendicular BE equal to ^B ; join ^E ; make ah and M 
 each equal to ^E ; join Yh and V/j'. 
 
 Now one diameter of the perspective circle is given by ca' and 
 some device must be employed to give the diameter at right angles 
 to this, as we are presuming that the V.P. for it and the sides BA 
 and DF are inaccessible. Draw the diagonals of the half-square Ac 
 or Be, a line then drawn through their intersection and the intersection 
 of BF and AD will give the other diameter. We have now eight 
 points through which the oblique circle must pass, so it can be easily 
 drawn as in Fig. i6. 
 
 3° 
 
TO DRAW IN PERSPECTIVE ANY POLYGON 
 
 By far the simplest way is to enclose the polygon in a square or 
 a rectangle, and so forming a plan of the figure ; by dropping 
 perpendiculars from the angular points and by drawing intersecting 
 diagonals of the enclosing square the figure can be easily put into 
 perspective as in Fig. 13. 
 
 THE PROBLEM OF FIG. 7 
 
 We will now draw the inclined brick 4^" Xj 3" X 9" at an angle 
 of 30° with H PI. with one edge parallel to G.L. resting on a table. 
 
 S R K E 
 
 -^ J. ^D 
 
 /a / 
 
 P M 
 
 N 
 
 Q H 
 
 Fig. 17. 
 
 B 
 
 A side elevation is first drawn as on the right of Fig. 17 where 
 HK represents a sagittal line in H.Pl, the long side of the brick AE 
 being 30° from HK. The side }id of the enclosing rectangle 
 represents the height in V.Pl. Adjoining the figure on the left is a 
 plan of the brick 4I" wide, the edges B and D of the brick being 
 immediately above PQ and LL' while the edge A coincides with 
 A in the plan, and the edge E is above the line SR in the plan. 
 If the paper be folded along HK so that the side elevation is 
 vertical in a sagittal plane, the figure would represent the brick in 
 position. We indicate the position of the lines LL' and A on the 
 base line PQ by means of the diagonal QS ; M and N mark the 
 foot of the perpendiculars from the intersections of this diagonal 
 with the lines LL' and A, 
 
 31 
 
In Fig. 18 the plan PQRS, is put in perspective 3" to the right 
 of C.S.P1. by means of Cj and CJD which is one-third that of the 
 distance point ; hence PJ is made one-third of PS in Fig. 17. 
 (J. is just beyond the foot of the perpendicular from C.) 
 
 Now it must be remembered that when using a side elevation, 
 the lateral distances HB, He and Ud represent heights ; in Fig. 18 
 the scale of heights b, e and d are represented on C.S.L., and 
 horizontals are drawn through b, e and d. The A edge on the table 
 has already been drawn, the edge BB' is defined by the intersections 
 of the verticals on P and on Q vnth the horizontal through b ; the 
 points d and d' are given by the intersections of the same verticals, 
 and the horizontal through d, and similarly the point e' is the 
 intersection of the vertical from P and the horizontal ee'. The 
 vertical Pe' is now transferred to SE, and similarly the verticals Pd 
 and Qdi are transferred to LD and LID'. On joining AB, BD, 
 DE, EA and the corresponding edges on the other side of the brick 
 when visible, the drawing is complete. 
 
 It will be found that Fig. 18 can be drawn in half the time 
 required for drawing Fig. 7, and the result is evidently more satis- 
 factory than the distorted diagram of Fig. 7 
 
 TO DRAW IN PERSPECTIVE AN OCTA-HEDRON 
 RESTING ON ONE OF ITS POINTS 
 
 The octa-hedron Fig. 19 is supposed to be enclosed in a cube 
 that is oblique to P.Pl. The base AEDB of the oblique cube with 
 
 32 
 
B H Q 
 
 Fig. 19. 
 its enclosing square PQRS is first drawn in perspective as in Fig. 13. 
 At P a vertical is drawn equal to PQ and a dotted horizontal line 
 indicates the height of the cube. The vertical heights above P, H 
 and Q are transferred by means of C to the heights above D, E and 
 A and a vertical from B to the dotted line is drawn. On joining 
 the summits of these verticals the oblique cube is completed. 
 The centres of the six surfaces are found by intersecting diagonals, 
 as indicated on the two lateral surfaces. It only remains to join 
 these six points as in the figure, and the diagram is complete. 
 
 33 
 
It is hoped that these illustrations of the method will make true 
 perspective drawing much simpler and easier than the methods in 
 vogue at the art schools. 
 
 DIMINUTION OR ENLARGEMENT 
 
 Artists frequently wish to make copies of other pictures of a 
 different size from that of the original. Suppose that we wish to 
 make a copy one-third the size of the original, it is clear that if each 
 of the dimensions is reduced one-third, a copy not one-third, but 
 one-ninth of the original size is obtained. Mathematicians will tell 
 one that each of the dimensions must be divided by ^3, which is a 
 troublesome business. 
 
 Fig. 20 shows a very simple way of doing this graphically that 
 is due to Euclid (vi. 13 and 19). 
 
 Let AB be the width (or the height) of the original picture, 
 then if a copy one-third of the size is required, AB is produced 
 to K making BK one-third of AB. Now bisect AK at O, and 
 on AK describe a semicircle of radius OA, and erect a vertical 
 at B meeting the semicircle at D. Then BD is the required width 
 of the copy. Every length and height in the original must be reduced 
 in the same proportion, and to do this graphically is extremely 
 simple. 
 
 Fig. 20. 
 
 Join AD and mark off from B any dimension that must be 
 copied. Suppose that there is a lake in the original, we mark off 
 its width, say it is BC, draw CE parallel to AD and the width in 
 the copy is BE. 
 
 Of course an enlargement is made in precisely the same way, 
 only BK is taken to represent the original size and BD will be the 
 enlarged copy. 
 
 34 
 
SHADOWS 
 
 So far we have only dealt with outline perspective, but in order 
 to make the illustrations that have been given stand out vividly, it 
 will be necessary to shade the sides of the object that are not exposed 
 to the light, and to indicate by shading the shadow cast by the object 
 on the surface upon which it is resting. 
 
 In the following pages we shall always consider the sun as the 
 luminary, it may be to the right, to the left, in front of the observer, 
 or behind him. The shadow will, of course, be always in the opposite 
 direction to that of the sun. 
 
 The sun is so far off that his rays .falling upon any object on 
 the earth are practically parallel. If a plane object (such as a square 
 or a triangular board) and a screen be parallel to each other, and 
 if both be at right angles to the direction of the incident sun's rays, 
 the shadow cast on the screen will be exactly the same size and shape 
 as that of the object, whatever the distance between the object and 
 the screen. Of course, owing to reflection of light from dust in the 
 air and from other objects, the greater the distance between the 
 object and the screen the less intense and less defined will be the 
 shadow. If the screen be tilted either towards or from the object 
 the length of the shadow on the screen will be increased ; if the 
 object be tilted similarly while the screen remains in its original 
 position the shadow on the screen will be shortened, but in either 
 case the width of the shadow will remain the width of the object. 
 
 Usually the above conditions are not satisfied, so we must 
 consider the case when the object and the screen are not parallel 
 to each other, and when the sun's rays are no longer at right angles 
 to either. 
 
 Now if a perpendicular be dropped from any point in the course 
 of a sun's ray to the horizontal plane, the plane that contains the 
 ray and the perpendicular will be called the ' vertical plane ' of the 
 ray. As the rays of the sun that strike any visible object on the 
 earth are practically parallel, the ' vertical planes ' of the whole sheaf 
 of rays that illumine such an object are all parallel to each other, 
 and so the ' vertical plane ' of incidence may be used to express 
 any plane that is at right angles to the horizontal plane whether in 
 a sagittal plane or at any assigned angle to it.* 
 
 *Note the distinction between the vertical plane and the ' vertical plane ' 
 (between single 'quotes') ; the former means a plane parallel to P.Pl., the 
 latter any plane at right angles to H.Pl. 
 
 35 
 
We shall first deal with shadows cast upon a horizontal plane 
 by surfaces in a ' vertical plane ' at right angles to the ' vertical 
 plane ' of incidence of the sun's rays. 
 
 When the sun is low the shadow on a horizontal surface is 
 much longer than when it is high in the heavens ; when the altitude 
 of the sun is 45° the length of the shadow is exactly equal to the 
 height of the object (or L : H = i). The following table shows the 
 ratio of the length of the shadow to the height of the object (L : H) 
 at different altitudes of the sun. 
 
 Son's Altitude 
 
 L :H 
 
 Sun's Altitude 
 
 L :H 
 
 14° 21 10" 
 
 4 
 
 26° 331 54" 
 
 2 
 
 15° 561 43" 
 
 3-5 
 
 33° 411 24" 
 
 1-5 
 
 18° 261 6" 
 
 3 
 
 45° 
 
 1 
 
 21° 481 5" 
 
 2-5 
 
 63°' 261 4" 
 
 •5 
 
 It should be noticed that when sketching in the evening with 
 the sun low the shadows will get appreciably longer in a very few 
 minutes, and hence care should be taken that all shadows are drawn 
 as seen at the same time. 
 
 In all the following illustrations the altitude of the sun is assumed 
 to be 45°. 
 
 I. — Shadows cast on H.Pl. by the sun at an altitude of 45° of an 
 object that presents a surface at right angles to the ' vertical plane ' 
 of incidence of the sun's rays. 
 
 A. — The surface is in a ' vertical plane.' 
 
 Under these circumstances the two rules that we have 
 mentioned are : — 
 
 (i) The width of the shadow is equal to the width of the surface 
 that casts it. 
 
 (a) The length of the shadow is equal to the height of the surface. 
 
 These two rules give the actual dimensions of the shadow, so it 
 is only necessary to draw the shadow of known dimensions in 
 perspective. 
 
 (a) In Fig. 5 if the sun were in the C.S.Pl. in front of the 
 observer, the surface abce of the block would be in shade, and the 
 
 36 
 
shadow cast on the table would be 3" x 2", for these are the dimensions 
 of the surface abce. Hence the cast shadow will occupy the space 
 az^h, and the sides (56 and za) of the shadow, being parallel sagittal 
 lines, must be drawn towards C. 
 
 (h) If the sun were in a vertical plane to the left of the observer, 
 the surface ed would be in shade, and the cast shadow would be 
 bounded by two horizontal hues through a and d that extend to OC. 
 
 In Figs. 21-3 the sun is in a vertical plane to the left of the 
 observer and is, as before, at an altitude of 45°. 
 
 Fig. 31 (I.) shows a triangle in a sagittal plane. 
 
 From D. drop the perpendicular D^' to the base, draw d^d 
 horizontally equal to dfiD, i.e., in the vertical plane of the ray Y}d 
 shown by a dotted line. 
 
 As D^ represents the ray that just grazes D, the point d must 
 mark the shadow of the point D ; it must define a point on the 
 boundary between light and shade. Consequently on joining Yd and 
 Ed the limits of the triangular shadow are defined. 
 
 This diagram shows how to draw the shadow of a pole, for 
 instance, that is inclined in a saeittal plane. If FD represented 
 a pole tilted away from the observer, its shadow would be Yd ; 
 if ED. represented a pole tilted towards the observer, its shadow 
 would be Yd. 
 
 B. — The surface is tilted towards H.PL, while its base is in a 
 sagittal line. 
 
 The objects II. and III. in Fig. 21 are inclined to H.PL, but 
 in each the surface is a plane that is at right angles to the ' vertical 
 plane ' of incidence. 
 
 In this case the rule (2) that we have given about the length 
 of the shadow no longer holds good. 
 
 Fig. 21 (II.) shows the same triangle as in (I.) tilted towards 
 H.P1. From G drop the perpendicular G^' to the H.PL, and draw • 
 g^g horizontally equal to g^G, i.e., in the line of intersection of the 
 H.PL and the plane of the ray that grazes G. 
 
 Then g is the shadow of the point G., and Unes drawn from g 
 to each angle at the base of the triangle define the limits of the shadow. 
 
 .37 
 
In Fig. 21 (III.) precisely the same method is employed. From 
 B. and C. drop the perpendiculars B6I and Cc' and draw b^b and c^c 
 horizontally making b^b = 6 IB and c^c = c'C ; join be, and produce 
 bbf and ccl until they meet the object. The limits of the shadow 
 are then defined. 
 
 On the left side the shadow of the point A is found in the same 
 way to be the point a, so the shadow cast by this inclined plane is 
 only the small patch indicated. 
 
 As BC and the corresponding edge on the other side are both 
 sagittal lines, the shadows of these edges must also be sagittal lines, 
 so a ready method of drawing these shadows is : to determine a and b, 
 and from these points to draw lines to the point of sight until the 
 object (on the left side) or the horizontal through H. (on the right 
 side) is met. 
 
 From the above it will be seen that the shadow of a point is 
 where the ray grazing that point meets the surface upon which the 
 shadow is cast. 
 
 We must now deal with planes that are not at right angles to 
 the ' vertical plane ' of incidence, and also with shadows that are cast 
 upon other objects. The simple rules that we have just explained are 
 not sufficient for dealing with these cases, and the general law, that 
 is universally true, must now be stated. Unfortunately it is a little 
 difficult to understand and it is difficult to state simply. It should 
 be understood that the word surface in the following statement means 
 the surface of the object (that may be tilted in any direction) upon 
 which the shadow is cast. 
 
 The shadow of a point is where the ray grazing that 
 point meets the section of the surface in the ' vertical plane ' 
 of the ray. 
 
 The meaning of this law will be clear if it be applied to one 
 of the previous cases. Consider Fig. 21 II. The point g is the 
 shadow of the point G. and it lies in the plane {Ggg^) of the ray G^ 
 that grazes the point G. As g^g is. the section of the surface in the 
 ' vertical plane ' Ggg^ of the ray, g is determined by the intersection 
 of the ray G^ with g^g. 
 
 In the next two figures the ground plane is divided into one-inch 
 squares in order to show more clearly the plane of the incident rays. 
 
 38 
 
II. — The shadow cast by a sagittal plane upon surfaces that are 
 not at right angles to the vertical plane of incidence ; and the 
 shadows of such surfaces on H.Pl. 
 
 The shadow cast by a sagittal screen upon a box with its lid 
 open 90° placed at 45° to P.Pl. and the shadow cast by the box are 
 shown in Fig. 22 I. 
 
 As before described the screen ABCD would cast the shadow 
 abCD. on the ground plane if the box were not interposed, the box 
 however covers the part FbE. of the shadow. 
 
 At E, where Cb meets the lower edge of the box, erect the 
 vertical Ee, meeting the grazing ray Bb at e. Here Ee is the 
 section of the surface in the vertical plane of the ray Beb. The 
 ground shadow line ab meets the edge of the box at F. ; join Fe, 
 and the shadow on the side and lid of the box is defined by FeE. 
 
 Some care is required in defining the shadow of the box on the 
 ground plane. Draw Gh horizontal equal to GH, and ]k horizontal 
 equal to JK, and on Gh mark off Gl equal to GL. Now if the lid 
 were clpsed GJkl would define the shadow of the end GJKL, for 
 Ik and LK both tend towards the same vanishing point Dg (not 
 shown). We have to determine where the shadow of the upper 
 edge of the lid will fall. Now as this upper edge, as well as the 
 long edges of the box tend towards the vanishing point Dj (not 
 shown), their shadows must also tend towards D^. Hence draw 
 hr towards Dj^, meeting kl at r, and draw km towards Dj^ until it 
 is obscured by the side of the box. The limits of the ground 
 shadow are then defined by Ghrkm. The shadow of the lid on the 
 inside of the box will not be visible from the position of the observer. 
 
 Note that the ray Kk is more than an inch behind the ray HA. 
 
 Fig. 22 (II.) is partly a repetition of (I). As the lid is closed 
 the shadow on the ground is Glkm of the previous diagram, and the 
 shadow of the s.creen on the ground is aFECD. At E erect the 
 vertical Ee (not shown) to meet the ray Bb at e and passing the edge 
 of the box at n ; join Fe passing the edge of the box at p. Then 
 the shadow on the side of the box is hmited by EFpw. 
 
 Now the top of the box is in a horizontal plane so the shadow of 
 the screen on it represents no difficulty. Through n draw the 
 horizontal no, meeting Bb at o, and join op. The shadow on the 
 lid is therefore nop. 
 
 39 
 
Fig. 23 (I.) shows again a box set obliquely to P.Pl., but in this 
 case the lid is partly open. 
 
 The shadow on the ground and on the side of the box is determined 
 in the, same way as in Fig. 22, n and p being the intersections of the 
 edge with Ee and Ye. 
 
 It is now required to find the point o, i.e., the shadow of the 
 point B. on the inclined lid. From the rule above, must he at the 
 intersection of Bb and the vertical section of the box through the 
 plane Ee6. The simplest way to determine this section is the 
 following : From K drop the perpendicular KL to the base, and 
 draw LM to the V.P. of the long edges meeting E& at M ; at M 
 erect the vertical Mm meeting the edge of the lid at m ; join mn. 
 Then the required section is EwwM. As the ray B6 intersects nm 
 at 0, o must be the shadow of the point B. Consequently on joining 
 on and op the shadow on the lid is defined. 
 
 Fig. 23 (II.) is an interesting study in shadows ; it represents 
 a rectangular block poised on one edge at 45° to P.PL, with the point 
 n touching the square screen in a sagittal plane, while the point S. 
 touches P.Pl., and the edge Sw is in the same H.Pl. as the opposite edge. 
 
 It will be well first to consider the shadow of the block on the 
 ground, and the shaded surfaces pf the block, in the absence of the 
 screen. The surface wT will be wholly exposed to the Ught, and 
 the near end will be in shade, it is not clear at the first glance whether 
 the inclined surface will be in shade or not. This surface should 
 be treated in a similar way to Fig. 21 (III.) in which a direct view 
 of such an inclined plane is given. From n and S perpendiculars 
 are dropped to the H.Pl at C and a, and then the horizontals Ca; and 
 as are drawn equal to nC and aS. respectively ; similarly find the 
 shadows of the points T. and U. at t and u. Join xz, xs, st and tu, 
 and from u draw a line towards the V.P., z.e., D^ (not. shown) of the 
 long edges of the block. Then zxstu will define the limit of the shadow 
 of the block cast on the ground. As the ground between xs and the 
 lower edge of the block must be in the shadow cast by the upper 
 inclined surface wT of the block, its under surface s'S must be wholly 
 in shade. 
 
 The shadow of the screen must now be considered. If there 
 were no obstacle in the way the shadow on the ground would be 
 DabC, but a corner of this is cut off by the lower edge of the block 
 which intersects the line C6 at E. In order to find the points p and 
 
 40 
 
assume for the moment that in front of the inclined shaded surface 
 there were a plane CnSa, so that the figure would resemble (I.). The 
 intersection of the Une aB and the edge bS will determine the point/). 
 To determine the point o we must find the vertical section of the 
 block in the plane of the ray Bb. Draw the vertical Em meeting 
 the upper edge at m. As the upper edge of the block is exactly over 
 the lower edge, nmE. must denote the section of the block in the 
 plane of the ray Bb, and o is the intersection of Bb and nm. The 
 shadow on the upper surface is therefore nop. 
 
 SUMMARY OF THE 
 PRINCIPLES OF PERSPECTIVE 
 
 It must be remembered that the term vertical plane is used here 
 as elsewhere in this little book to denote any plane that is parallel 
 to P.P1. :— 
 
 (i) All lines in a vertical plane are drawn in their real direction, 
 and have no V.P. 
 
 Consequently all horizontal lines, whether in the same V.Pl. 
 or in different V.Pl.'s are drawn horizontal. 
 
 (2) All parallel lines which are not in a V.Pl. converge to the 
 same point, or have the same V.P 
 
 (3) All sagittal lines are drawn converging towards C, hence C 
 is their V.P. Their M.P. is at a point D such that 
 DC = SC. 
 
 (4) All measurements are made in P. PI. 
 
 Oblique Lines in Cardinal Planes. 
 
 The obhquity of lines in a V.Pl. or a S.Pl. is denoted by &, the 
 angle which these lines make with a H.PI. (see Fig. 7) ; while 9-' is 
 the angle which oblique lines in a H.PI. make with a V.Pl. (see 
 Fig. .6). 
 
 OBLIQUE LINES AT ANGLE d OR t)^ 
 
 
 Direction 
 
 Vanishing Point 
 
 Measuring Point 
 
 In V.Pl. 
 
 $ 
 
 Absent 
 
 C 
 
 In S.P1. 
 (Fig. 7) 
 
 Towards V 
 
 V in OC ; CDV=9 
 (CV = SC tan C) 
 
 M in OC ; MV = DV 
 
 (MV = SC sec e) 
 
 In H.PI. 
 (Fig. 6) 
 
 Towards V 
 
 V. in H.L. ; BSV = ei 
 (CV = SC cot et) 
 
 M in H.L. ; MV = SV 
 (MV = SC cosec «l) 
 
 41 
 
Oblique lines in oblique planes are most easily drawn by making 
 a preliminary sketch of the plan and elevation of the object. 
 
 TRIGONOMETRICAL RATIOS. 
 
 Angle 
 
 Sine 
 
 Cosine 
 
 Tangent 
 
 Cotangent 
 
 Secant 
 
 Cosecant 
 
 
 10° 
 
 •1736 
 
 •9848 
 
 •1763 
 
 5-6713 
 
 1-0154 
 
 5^7588 
 
 80° 
 
 20° 
 
 ■3420 
 
 ■9397 
 
 •364 
 
 2-7475 
 
 1^0642 
 
 2^9238 
 
 70° 
 
 30° 
 
 •5 
 
 •8660 
 
 •5774 
 
 1-7321 
 
 M547 
 
 2-0 
 
 60° 
 
 40° 
 
 •6428 ' 
 
 •7660 
 
 •8391 
 
 1-1918 
 
 1^3054 
 
 1-5557 
 
 50° 
 
 45° 
 
 •7071 
 
 •7071 
 
 1^0 
 
 1-0 
 
 1^4142 
 
 1-4142 
 
 45° 
 
 
 Cosine 
 
 Sine 
 
 Cotangent 
 
 Tangent 
 
 Cosecant 
 
 Secant 
 
 Angle 
 
 The symbols at the top of the table refer to the angles from 
 10° to 45° ; and the symbols at the bottom of the table refer to the 
 angles from 45° to 80° which are read from below upwards. 
 
Fig. 21. 
 
 Fio. 22. 
 
 Fio. 23. 
 
 II. 
 
CROYDON : 
 
 Printed by Roffey & Clark, 12, High Street, 
 and Middle Street.