PRIZE MEDAL, INTERNATIONAL EXHIBITION, 1862, 
 
 was awarded to 
 
 MESSRS. VIRTUE, 
 for 
 
 the " publication of 
 WE ALE'S .SERIES." 
 
 See JURORS' REPORTS, 
 
 CLASS XXIX. 
 
 gfft RUDIMENTARY, SCIENTIFIC, EDUCATIONAL, AND SgfcS 
 |g CLASSICAL WORKS, jgg 
 
 1^/$ FOR COLLEGES, HIGH AND ORDINARY SCHOOLS, ^fM 
 
 M 
 
 $3% 
 
 AND SELF-INSTRUCTION ; 
 
 ALSO FOB 
 
 MECHANICS' INSTITUTIONS, FREE LIBRARIES, &c. &c, oj 
 
 PUBLISHED BY '^J^' 
 
 VIRTUE BROTHERS & CO., 26, IVY LANE, 
 
 PATERNOSTER ROW. 
 
 : : tt if^ 
 
 The Public are respectfully informed that the whole of the vy~»>£^ 
 
 •** THE ENTIRE SERIES IS FREELY ILLUSTRATED ON WOOD 
 AND STONE WHERE REQUISITE. 
 
 »jT£ lhe Public are respectfully informed that the whole of the <nr»j 
 
 j^v? late Mh. Weale's Publications, contained in the folloiving Cata- 
 
 ^ X^S logue, have been purchased by Virtue Brothers & Co., and £y^3l 
 
 ^j-M^ that all future Orders will be supplied by them at 2G ; Ivy Jsf^ 
 
 p# $jH 
 
 ^C^l) *** Additional Volumes, by Popular Authors, are in Preparation. 
 
 MISCELLANEOUS. 
 
 w V§ Natural Philosophy, Introduction to the Study of, for the 
 A^t^t use of Beginners. By Charles Tomlinson, Lecturer of Natural £if <S 
 >pp Science in King's College School, London. Illustrated, Is. 
 
 Pneumatics; for the use of Beginners. By Charles 
 ijph^ Tomlinson. Illustrated, Is. r ^kt: 
 
 1£ Perspective for Beginners ; simplified for the use of Juve- 
 nile Students and Amateurs in Architecture, Painting, &c. By 
 r$j$ George Pyne. Illustrated, 2s. 
 
 
 
 2T^4- VIRTUE BROTHERS & CO., 26, Ivy Lane, Paternoster Row. 
 
VIRTUE BROTHERS & CO., 26, Ivy Lane, Paternoster Row. 
 
 ~ ^ 
 
 ^gf4£ Tne Mollusca, a Manual of; or, a Eudimentary Treatise 
 $^$£ of Eecent and Fossil Shells. By S. P. Woodward, Associate of 9^^% 
 '"^^^ tne Linnean Society, Assistant in the Department of Mineralogy 
 ^ta&t and Geology in tn © British Museum. Illustrated, 5s. 6d. Jaf 
 
 ^5§S 
 
 ^^jjj Tile Painter,s ^ rt > Rudiments of; or, a Grammar of 3$c& 
 5*/^'^ Colouring applicable to Operative Painting, Decorative Archi- I^N^H 
 tecture, and the Arts. By George Field. Illustrated, 2s. ^*v^ 
 
 Paintin £ on Glass ' or Glass Staining, the Art of; com- M0j 
 f~%*C% prisin S Dir ections for preparing the Pigments and Fluxes, for 
 ^^C^f laying them upon the Glass, and for Firing or Burning in the o^^l 
 ^ffiffi Colours. From the German of Dr. Gessert. To which is added 
 £V an A PP en dix on the Art of Enamelling, &c. Is. x v 
 
 ^2 
 
 IP: 
 
 ££4f Tlie Law of Contracts for Works and Services. By 
 ^3L^ David Gibbons. Is. Cv } | : 
 
 •^J * MS' 
 
 the S ° ils and M ° de of Cultivation — Cropping and Value of the 
 
 CIay Lands and Loam y Soils; comprising the Origin and 9^r^ : 
 ' Chemical qualities of each variety of Land— Natural Properties of 
 
 Lands ' changes effected b 7 mixing with Hot Lime. By Pro- Hp^f*^ 
 
 '^La^ fessor Donaldson. Is. Srl^^ 
 
 Music, a Eudimentary and Practical Treatise on. By 
 
 ^ jC X ' Charles Child Spencer. 2s. M 
 
 ^ A The Pianoforte, the Art of Playing, with numerous Exer- 
 
 hxtfc cises and Lessons » written and selected from the best masters. 
 ?{%> 3^ By Charles Child Spencer. Is. *Y'>€^Q 
 
 VIRTUE BKOTHEES & CO., 26, Ivy Lane, Paternoster Row. f&Stf; 
 
A 
 
 RUDIMENTARY A 1ST) PRACTICAL TREATISE 
 
 ON 
 
 PERSPECTIVE FOR BEGINNERS; 
 
 SIMPLIFIED 
 
 YOR THE USE OF JUVENILE STUDENTS AND AMATEURS 
 IN ARCHITECTURE, PAINTING, ETC.; 
 
 ALSO 
 
 I 
 
 ADAPTED FOR SCHOOLS AND PRIVATE INSTRUCTORS. 
 
 By GEORGE PYNE, Artist. 
 
 Sixty €Vxtxon, 
 
 EIGHTY-SIX ILLUSTRATIONS. 
 
 LONDON: 
 
 VIRTUE BROTHERS & CO., 26, IVY LANE, 
 
 PATERNOSTER ROW. 
 18G6. 
 
INTRODUCTION. 
 
 In the course of twenty years' extensive practice as 
 a teacher of drawing, the Author has frequently had 
 considerable difficulty in making his juvenile pupils com- 
 prehend the necessity for and the value of a knowledge 
 of Perspective. Many works have appeared, proposing 
 to enable the student and the amateur to instruct them- 
 selves in this indispensable branch of the Art of Painting ; 
 but the Author has never yet met with one that has 
 appeared to him well calculated to accomplish so de- 
 sirable an end. To furnish amateurs, and especially 
 young ladies, with the means to acquire, by themselves, 
 a knowledge of Perspective, sufficient to enable them 
 to make agreeable sketches from nature, without sacri- 
 ficing too much of the time that must be required for 
 other occupations, has been the object of the Author. 
 In the little work he now puts before the public, his 
 principal endeavour has been to avoid every possible diffi- 
 culty — every superfluous line. It is addressed to those 
 who require a simple and comprehensive knowledge of 
 Perspective, to enable them to avoid committing any of 
 those gross errors, so constantly to be observed in the 
 works of those entirely ignorant of it. He strongly ad- 
 vises all desirous of drawing from nature to make them- 
 selves masters of the modes here given for drawing various 
 forms, so as to be able to apply them mentally in sketch- 
 ing from nature. , It is universally admitted, that sketches 
 
ir 
 
 INTRODUCTION. 
 
 made by those who draw by their eye, having at the 
 same time a thorough knowledge of Perspective, produce 
 more agreeable paintings than those who draw entirely 
 by rule. To demonstrate to the juvenile student the 
 value of a knowledge of Perspective, let him examine 
 the cut at the end of this Introduction, as also that at 
 the end of the First Part. The first is a correct repre- 
 sentation of a double cross in perspective, drawn, as it 
 would appear, when quite new and perfect ; the latter 
 (which is drawn over the same outline) is intended to re- 
 present a similar cross in an ancient and dilapidated state. 
 The student will perceive that the perspective drawing 
 looks formal and uninteresting, while the other has an 
 agreeable and picturesque appearance, though perfectly 
 correct. The art of painting is to represent objects in 
 nature as they appear to the eye ; but if any lines, either 
 from time or accident, have lost their perpendicular or 
 horizontal direction, great care should be taken in the 
 representation of them, that they are so drawn as not to 
 appear like faulty Perspective, but as the result of time or 
 some other cause. It is the absence of formality that 
 constitutes picturesque form. 
 
 The Second Part, which is entirely new, and written 
 for this Third Edition, carries the student still further, 
 and opens to view all the requisite acquirements for a 
 perfect knowledge of the art of Perspective. This edition 
 will be found to comprehend all the principles, with simple 
 representations, to enable the learner by ordinary appli- 
 cation to execute perspective drawings with facility. 
 
CONTENTS. 
 
 PART L 
 
 Page 
 
 Chap. I. — Explanatory Introduction 1 
 II. — Representation of Plane Surfaces bounded oy straight 
 
 Lines .. .. .. .. ..14 
 
 III. — Representation of Plane Surfaces bounded by curved 
 
 Lines . . . . . . . . . . . . 34 
 
 IV. — Representation of Solids . . .. .. ,.55 
 
 PART II. 
 
 I. — I j near Perspective .. .. ..73 
 
 it. — Rules of Perspective deduced from the Science of Optics 81 
 
 III. — Number of Objects combined ... .. ..90 
 
 IV. — Objects changing require a fresh Vanishing Point . . 99 
 V. — How Solid Figures arc drawn in Perspective .. .. 107 
 
 VI. — In the projection of Roofs, Pediments, Cornices, 
 
 Moulding, &c 139 
 
 VII. — On the choice of Position for the Spectator, relative to 
 
 the Objects to be represented in Perspective. . . . 156 
 
UST OF PLATES. 
 
 Plate I. to face page 
 
 IT. 
 
 ft 
 
 III. 
 
 * * 
 
 IV. 
 
 
 V. 
 
 ft 
 
 VI. 
 
 1* 
 
 VII. 
 
 It 
 
 VIII. 
 
 ft 
 
 IX. 
 
 It 
 
 x. 
 
 
 XI. 
 
 tt 
 
 XII. 
 
 tt 
 
 XIII. 
 
 »• 
 
 10 
 
 .. S3 
 .. 2i 
 .. 23 
 
 .. 28 
 .. 35 
 .. 38 
 .. 43 
 .. 45 
 51 
 
 .. 5*? 
 .. 59 
 .. 64 
 
PERSPECTIVE FOR STUDENTS. 
 
 CHAPTER L 
 
 Students, from the first commencement of drawing, 
 should never neglect an opportunity of submitting their pro- 
 ductions to the inspection of those who, from their superior 
 knowledge, may point out defects, and suggest alterations 
 extremely useful. But in criticising their works, those who 
 have attained some proficiency may frequently make use of 
 terms which, though perfectly correct, may by possibility 
 not be understood by very young pupils, and hence they may 
 lose much valuable assistance. 
 
 Before commencing Perspective, the pupil will therefore 
 find it to his advantage to make himself acquainted with the 
 following preliminary matters, which more properly belong 
 to practical geometry. Many young persons, in copying a 
 drawing, if they draw a line that is out of the perpendicular, 
 or not horizontal, are apt to say, " That line is not straight." 
 The first thing to comprehend is, that all lines lying evenly 
 between their two extremities (which are called points) are 
 straight lines, whatever direction they may take (Fig. 1). 
 The line A b is a straight line, and each 
 of the lines that run from it, and through 
 it are also straight lines, although they 
 vary in their direction. 
 
 Lines that run in the same direction, 
 and continue always at the same distance 
 
 Perspective, B 
 
 A 
 
 
 
 
2 
 
 PERSPECTIVE FOR STUDENTS. 
 
 from each other, are called parallel lines (Fig. 2). Lines 
 
 which incline towards each other 
 ^'S' 2 - and meet in a point, are said to form 
 
 ~ angles. Angles have three different 
 ~ names, according to the space con- 
 
 tained between the two lines at an 
 equal distance from their point of meeting (Fig. 3). The 
 
 lines a E and c e meet together at the 
 3 ' point E ; the lines b e and c e also 
 
 meet together at the point e : the space 
 between a c and the space between 
 B c will be found to be exactly equal. 
 Whenever one line stands upon an 
 other line, and, upon drawing a semi- 
 circle from the point of contact (as the semicircle A c B, 
 drawn from the point e), the line divides the semicircle into 
 two equal parts, it is said to be perpendicular* to the line on 
 which it rests, and the angle on either side is called a right 
 angle. If the space contained between two lines forming an 
 angle be less than that contained between the lines forming a 
 right angle, the two lines are said to form an acute angle. 
 The angle formed by the lines D e and a e is less than the 
 right angle, because the space contained between a d is less 
 than the space contained between c a : for the same reason, 
 the lines c e and d e also form an acute angle. If the space 
 contained between the two lines be greater than the space 
 contained between the two lines that form the right angle, 
 
 * It is a common error to confound the terms vertical and perpendi- 
 cular. One line is always said to be perpendicular to another line when 
 the angle formed by the two lines is a right angle. Vertical lines are 
 those lines perpendicular to the horizon, or to the surface of the globe. 
 If a vessel lie on the surface of the water in a dead calm, having her 
 masts perpendicular to her deck, the masts may be said to be vertical ; 
 but if the water were agitated so as to throw the vessel at an angle with 
 the horizon, though the masts would still be perpendicular to the deck 
 they would no longer be vertical lines. 
 
PERSPECTIVE FOR STUDENTS. 
 
 3 
 
 the two lines are said to form an obtuse angle. The angle 
 formed by the lines b e and d e is greater than the right- 
 angle, because the space contained between b d is larger than 
 the space contained between b c. If the learner open a pair 
 of compasses exactly half way, the legs of the compasses will 
 form a right angle ; if they are shut-to a little, they then 
 form an acute angle ; if opened a little wider, they form an 
 obtuse angle. If the extremities of the two lines forming an 
 angle are joined by a third line, the figure formed by the 
 three lines is called a triangle, from its containing 
 three angles (Fig. 4). 
 
 In making perspective drawings, certain instru- 
 ments are indispensable ; and one of the most essen- 
 tial is a proper drawing-board, in the choice of 
 which great care should be taken that the edge at the bottom 
 be perfectly straight, and that at all events one of the sides 
 be perfectly at a right angle with the line of the bottom — or, 
 in other words, that the side of the board be perpendicular 
 to the bottom : if not, and the pupil should make use of the 
 T square,* his drawing can never be correct ; because all lines 
 drawn with the T square are parallel : consequently, what- 
 ever error may exist in the drawing-board will be multiplied 
 by your ruler. To be certain that you commence with a 
 perpendicular line, draw, as in the following example 
 
 * The tee, or, as it is commonly written, from its form, T square, is 
 a ruler to which is attached at one end a cross piece of wood ; and this 
 cross piece, being made thicker than the ruler itself, enables the drafts- 
 man to slide it backwards and forwards on the edge of his board. The 
 ruler attached to this cross piece is exactly at right angles with it ; and 
 consequently, in moving it along the bottom edge of the board, and 
 drawing lines from it, the lines must all be parallel to each other, and 
 perpendicular to the bottom line of the board. Now if the drawing- 
 board have one of its sides at a right angle with the bottom edge, by 
 shifting the T square from the bottom to the side of the board, and 
 sliding it on this edge, all the lines ruled from it will be parallel to each 
 other, and at right angles with the lines drawn from the bottom. The T 
 square is the most convenient and quickest ruler for drawing all perpen- 
 dicular and horizontal lines. 
 
 B 2 
 
4 
 
 PERSPECTIVE FOB STUDENTS. 
 
 4*- 
 
 Fig. 5. (Fig. 5), with a ruler a straight lino, 
 
 which is to form the bottom or base 
 ( line of your picture. From the point 
 
 on this line from which your per- 
 pendicular line is to be raised, as at 
 
 9^ A, mark off an equal space on each 
 
 side, as the spaces a b and a c ; 
 from the extremity of each of these spaces, at the points b 
 and c, with a pair of compasses, at an extension of not less 
 than once-and-a-half the length of a b or a c, describe two 
 portions or arcs of a circle immediately over the point A; 
 from the point D, where these two arcs intersect each other, 
 draw the line d a, which will always be perpendicular to the 
 line a b, and may be continued to any length. The learner 
 must be aware that in a work of this kind, illustrated by wood- 
 cuts, the space for the insertion of the examples is extremely 
 limited ; he is therefore recommended, in drawing them for 
 his own practice and improvement, to enlarge them very 
 considerably — say from four to six times the size. 
 
 There are various other rules in practical geometry that 
 the author has found useful to his pupils ; but as this is not 
 a treatise on practical geometry, they are not given. The 
 foregoing are introduced from a conviction that with the 
 very young, they are nearly, if not quite, indispensable. 
 
 In introducing my young readers to an elementary know- 
 ledge of perspective, as the most simple definition, I should 
 say that perspective is the art of representing objects at 
 various distances, and is of two kinds — Aerial Perspective, 
 and Linear Perspective. Aerial Perspective is the art of 
 giving the appearance of distance, independent of lines. 
 Claude de Lorraine is celebrated for his exquisite manner of 
 representing aerial perspective : many English painters are 
 also highly and deservedly celebrated for this portion of the 
 art of painting, more particularly the painters in water- 
 colours ; among whom, perhaps, Glover and Copley Field- 
 ing have been the most successful. It is of the latter, Linear 
 
PERSPECTIVE FOR STUDENTS. 
 
 5 
 
 Perspective, that we have to treat : of this it may be said, 
 that it is the art of drawing outlines of objects from nature, 
 of their relative sizes according to their distance, and of their 
 apparent variety of form according to their position, as they 
 would appear in looking through a sheet of glass placed be- 
 tween them and the spectator. The reader is doubtless aware 
 that all objects of the same magnitude apparently diminish as 
 they recede from the eye of the spectator. In walking in a 
 long street at night, the reader must have noticed the appear- 
 ance of the gas lamps as they gradually recede from him : if 
 the street be very long, they will appear to come closer and 
 closer together, till they apparently meet in a point ; * yet 
 the more distant lamps are as far apart from each other as 
 those close to the spectator. The same appearance is observ- 
 able in a long avenue of trees. In a long series of arches, 
 the first few will show their curves wide and distinct : as they 
 recede from the eye they appear gradually narrower and nar- 
 rower, till in the extreme distance they assume the appearance 
 of mere straight lines. To demonstrate clearly to the young 
 reader that objects at a great distance seem very small, 
 let him look through a pane of glass, and imagine that this 
 pane of glass were a sheet of paper, on which he had to re- 
 present all the objects he sees through it : though this pane 
 of glass may only be a foot square, he may see houses, ships, 
 tracts of country, mountains, rivers, &c. &c. represented on 
 this small space, though perfectly aware of their actual size. 
 
 Most of my readers must have heard the term horizon 
 frequently used in conversation — in such cases as " the sun 
 is above the horizon," or, " the sun has sunk below the hori- 
 zon," 'Ssc Every perspective drawing has a line running 
 across it, parallel to the bottom of the picture, to designate 
 the line of the horizon, which line is called the horizontal 
 line. In drawing from nature, this line is at a height exactly 
 level with the eye of the draftsman ; and its position, or dis- 
 
 * This point is termed the vanishing point, and is most important, 
 as will be seen in our progress. 
 
6 
 
 PERSPECTIVE FOR STUDENTS. 
 
 tance from the base of the picture, which is called the ground 
 line, depends entirely on the position in which the artist 
 places himself to take his sketch. In the following example 
 (Fig. 6), we will suppose the lines 1, 2, 3, 4, to form the 
 j,. g boundary lines of the pic- 
 
 ture. If the draftsman 
 is placed in a sitting pos- 
 ture, as at a, the hori- 
 zontal line will be at the 
 height of the line 5, even 
 with the painter's eye, 
 and parallel to the ground 
 line 1. If the draftsman 
 stand up to take his 
 sketch, as at B, the horizontal line will be higher, in conse- 
 quence of his eyes being in a more elevated situation, and 
 will be at the line 6. If, to get into his picture some more 
 distant object, the artist should find it necessary to raise him- 
 self still higher, as at c, upon the bank, the horizontal line 
 will also be raised, as seen by the line 7 ; or, as I have before 
 stated, the height of the horizontal line depends on the raised 
 or lowered position of the eye of the artist. 
 
 In making a picture, the choice of height of the horizontal 
 line is of considerable importance. To make the horizontal 
 line exactly half- way between the top and bottom of the pic- 
 ture, has generally a bad effect ; it appears to cut the picture 
 in half, and the perspective is not pleasing to the eye. It is 
 generally considered that the most agreeable perspective is 
 produced by placing the horizontal line at about one-third 
 the height of the picture from the ground-line : to place it 
 lower than this is generally preferable to placing it higher. 
 There are painters, however, of great celebrity, who in some 
 of their finest productions have placed their horizon so high 
 as to be removed only one-third from the top of the picture. 
 Gaspard Poussin, Francesca Mola, Domenichino, &c. have 
 frequently painted pictures with these high horizons ; but the 
 
PERSPECTIVE FOR STUDENTS. 
 
 7 
 
 subjects are peculiar, and the painters so talented, that any- 
 thing emanating from their pencils cannot fail to be good. 
 All those views that come under the denomination of bird's- 
 eye views must necessarily have the horizontal line very 
 high, being taken always from some high window, tower, or 
 eminence of some sort, such as the views of London from 
 St. Paul's, of Paris from the Pantheon, &c. &c. ; but such 
 views are intended more for topographical curiosities than 
 for pictorial representations. 
 
 In order to give the reader an idea of the use of perspec- 
 tive, we will commence with some object of the most simple 
 form, a square, or oblong (figures which are technically 
 called rectangular parallelograms, from their opposite sides 
 being parallel to each other, and the angles all right angles). 
 Let the student take any rectangular object — a workbox, for 
 instance ; let him place it in front of him, close to his feet, 
 then bend his head slightly forward till his eyes come imme- 
 diately over the centre of the box (Fig. 7) : so 
 placed, he will be able to see nothing but the 
 simple form of the lid, it being impossible in 
 this position to see either the front, back, or 
 sides. Let the student now place the box on 
 the chimney-piece, the front towards him, and place himself 
 about two yards from it, and in such a position that his eyes 
 shall come on a level with the middle of the front of the 
 box, and exactly midway between its two sides 
 (Fig. 8) : thus placed, the student will see 
 nothing but the front of the box, it being impos- 
 sible in this position to see either the top or sides. 
 The student must now place the box on a chair, 
 or other support, so as to be in height about halfway between 
 his head and feet, placing himself at two or three Fig. 9. 
 yards' distance from the object, but still in such 
 a position as to stand exactly opposite the key- 
 hole of the box (Fig. 9) : he now, from the 
 changed position, sees the top and front of the 
 
 Fig. 7. 
 
 Fig, 8. 
 
8 
 
 PERSPECTIVE FOR STUDENTS. 
 
 box. Let the student now shift his position about one yard 
 to the left, leaving the box in the same situation ; he will 
 here find that he sees the front, the top, and one side of the 
 Fig 10 k° X (Fig* The student will here ob- 
 
 serve, that according to the variation of the 
 position from which he regards the object, it 
 changes its apparent form. In the first two 
 figures he will see that the lines are all 
 
 parallel to their opposites, or, as it is commonly called, are 
 in geometrical drawing ; but in the third figure he will per- 
 ceive that the lines of the sides of the top converge, and that 
 the line of the top of the box at the back is shorter than the 
 line of the top in front. Perspective teaches how to find the 
 proper directions for these converging lines, and also shows 
 now to regulate the length of the line at the back of the box, 
 so as to make it agree with its apparent diminution of size 
 to the visual organs. The same remarks apply equally to 
 the last figure. 
 
 As another example of the use of perspective, let the 
 student procure a common bowl, and place it at his feet, 
 looking at it in a similar manner as at the workbox in its first 
 situation. In looking at it in this position, the student will 
 Fig. 11. see nothing to draw but a plain circle (Fig. 11). 
 
 If the bowl be placed on a chair, as the work- 
 box in its third situation, the spectator being in 
 the same relative position, the circular opening of 
 the bowl appears of only half its width, and a 
 Fig. 12. portion of its outer part is seen (Fig. 12). If the 
 s \ bowl be now placed on the chimney-piece, and 
 
 v" y the eye of the spectator brought to a level with 
 
 the upper edge of the bowl, none of the inside 
 Fig 13 °^ ^ e ^owl is perceptible, the circle from this 
 point of view appearing as a straight line (Fig. 
 13). The student will here observe that, accord- 
 ing to the position in which the spectator is 
 placed relatively to a circular object, it takes the 
 
PERSPECTIVE FOR STUDENTS. 
 
 9 
 
 form of a circle, an ellipse, or a straight line. Perspective 
 teaches how to delineate the form the circle apparently as- 
 sumes, according to the point of view from which it is seen. 
 
 In the preceding pages we have introduced four diagrams, 
 representing the change of appearance a work-box, or any 
 similar object, assumes, as viewed from four different posi- 
 tions. In the first and second figures, the upper and lower 
 lines of the box are parallel, as are the upright lines repre- 
 senting the sides ; they are in fact of precisely the same form 
 as that they are intended to represent, the position in which 
 they are viewed presenting the simple geometrical figure. 
 The third position of the box presents the front, similar to 
 the second, but being below the eye, the top as well as the 
 front of it is seen. Now, as objects appear smaller as they 
 are further removed from the spectator, the back of the box 
 will appear less than the front, and Fig. 14. 
 
 must necessarily be represented by 
 a shorter line ; hence it must be 
 obvious that to draw the lines re- 
 presenting the sides of the top, they 
 must incline towards each other, 
 and if continued, would meet in a 
 point, as in the annexed figure. 
 
 In the fourth diagram, the front of the box is still drawn 
 geometrically, but from its position being again changed 
 relative to the spectator, both the top and one side of the box, 
 as well as the front, are vissible ; and as the lines representing 
 the back of the top and the further angle are both drawn 
 shorter than the front edge Fig. 15. 
 
 and nearer angle of the box, 
 the lines drawn to represent 
 the sides of the top and the 
 side of the bottom must in- 
 cline towards each other, and 
 the three lines would, if con- 
 tinued, meet in the same 
 point. Now these three lines, which in Figure 15 incline 
 
 b 3 
 
10 
 
 PERSPECTIVE FOR STUDENTS. 
 
 towards each other so as to meet in the same point, in the 
 original object (the workbox) are parallel lines; and herein 
 consists the difference between what is called Geometrical or 
 Elevation drawing, and Perspective drawing. In the former, 
 all lines that are parallel in the original object, are drawn 
 parallel in the representation ; whereas in perspective drawing 
 all representations of parallel lines incline towards each other, 
 and tend to the same point. This point is always placed on 
 the horizontal line, and is called the vanishing point. Thus, 
 D in the foregoing figure is the vanishing point for the lines 
 abc, and would be the point to which all lines which in the 
 original object are parallel to those they represent (the side 
 edges of the box) would be drawn, however numerous ; this 
 is exemplified by the line e, showing where the lid of the box 
 shuts on. 
 
 It is to be presumed, that before commencing the study 
 of perspective, the student has already dabbled a little in 
 drawing ; in which case he must now make an attempt to 
 draw a little perspective for himself. Let him place himself 
 in a chair, immediately opposite a closed door, and at a 
 distance of six to eight feet, and in that position let him 
 draw the door, and the cornice if any ; if not let him sketch 
 a little of the pattern of the papering above the door, as in 
 fig. 1, Plate I., which is a geometrical drawing of a door, to 
 be put in perspective. 
 
 Let the student now imagine a straight line passing di- 
 rectly from his eye to the door, always at the same height 
 from the floor — or, more correctly speaking, parallel to the 
 floor : this line would touch the door at the point A ; and 
 this point fixes the height of the horizontal line, and is called 
 the point of sight.* But we must here proceed with the 
 second figure, Plate I. 
 
 The student must first draw the four outer lines of the 
 
 * The point exactly opposite the eye of the spectator is always termed 
 the point of sight, and forms the perspective centre of a picture : when 
 used as a vanishing point, it is for those lines only that are parallel to the 
 imaginary straight line passing from the eye to it. 
 
PERSPECTIVE FOR STUDENTS. 
 
 11 
 
 door, b c D K, as in the geometrical drawing, and then 
 through the point A (the height of his eye from the ground) 
 draw a line across his picture parallel to the ground line, or 
 bottom line of the drawing ; this is the horizontal line. In 
 looking at the geometrical drawing (fig. 1), it will be seen 
 that the two lines b c, which represent the two sides of the 
 door, from each of them being at the same distance from the 
 eye of the spectator, are of an equal length ; that the lines 
 D and k, representing the top and bottom lines of the door, 
 are parallel to each other ; and that the lines representing the 
 top and bottom of the panels are parallel to each other, and 
 to the lines d and K also. Let the student now open the 
 door about one foot : here he will observe an extraordinary 
 difference ; — the directions of all the horizontal lines,* as 
 seen in the geometrical drawing, are now changed. Observe 
 that the upper and lower corners of the door, 1 and 2, the 
 side where the hinges are fixed, remain the same as in the 
 geometrical drawing: they have not changed their situation, 
 but the corner 3 appears raised, and the corner 4 lowered, 
 making the side c of the door consequently appear longer; 
 the side of the door c, from its being approached nearer the 
 eye, becomes apparently larger ; but the side b, as it remains 
 in precisely the same position, remains of the same size as in 
 the geometrical drawing. The student must now carefully 
 notice at what particular spot on the cornice, or at what 
 particular mark on the pattern of the papering, the point 3, 
 marking the top of the door, appears to touch, and mark the 
 spot on his drawing, as at A : from this point, through the 
 point 1, marking the other corner of the top of the door, the 
 student must draw a line till it touch the horizontal line ; and 
 the point l, where it touches, is its vanishing point. Now 
 the student must bear in mind, that this vanishing point is 
 
 * All lines in a drawing that are parallel to the horizontal line are 
 called horizontal. The student must understand that the line drawn 
 through the point a is the Horizontal Line, or line representing the 
 horizon ; and that those lines parallel to it are only called horizontal in 
 reference to their being parallel to it. 
 
12 
 
 PERSPECTIVE FOR STUDENTS. 
 
 the point to which every line of the door, parallel to che line 
 of the top of the door in the geometrical drawing, must be 
 drawn in his perspective drawing, whether above or below 
 the horizontal line. In order to get the perspective line of 
 the bottom of the door, the student must place his ruler to 
 the vanishing point L, and draw a line through the point 2 
 till it passes nearly under the right-hand side of the door : to 
 determine the length of this line, the student must draw a 
 perpendicular line from the point 3 till it meets it at the 
 point 4. The student should now, with a firm hand draw 
 over the lines b, c, d, k, to make them stronger than the 
 other lines ; and he will then have the external lines of the 
 door in perspective, as it appears to him from the position in 
 which he is placed. The next thing necessary is to find the 
 perspective inclinations of the lines forming the top and the 
 bottom of the panels of the door — the lines e, f, g, h, i, j, 
 of the geometrical drawing. To accomplish this, the student 
 must mark upon the line B the relative distances of these 
 lines, as at the points 5, 6, 7, 8, 9, 10; and from the vanish- 
 ing point l through each of these points he must draw a line 
 till it touch the line c. Here, then, are all the horizontal 
 lines of the panels of the door in their perspective directions : 
 and the student will observe that the panels of the door, as 
 also the framework of the panels, gradually widen as they 
 approach the eye of the spectator, or, in other words, they 
 diminish as they recede from it. Having obtained the lines 
 which will regulate the height of the panels, it is now neces- 
 sary to determine their width. It must be obvious to the 
 readei, from what has already been said, that the framework 
 surrounding the panels must be wider on the side nearest 
 to him than on the side at the greater distance. To find the 
 width of the panels, the student must draw a line parallel to 
 the horizontal line from the point 3 of the geometrical length 
 of the top of the door, and measure off with his compasses 
 from each extremity, 3 and 11, a space equal to the width 
 of the framework of the panels, as at 12 and 13, the space 
 between being obviously the width of the panel. From the 
 
PERSPECTIVE FOR STUDENTS. 
 
 13 
 
 point 11, passing through the point 1, a line must be drawn 
 till it touch the horizontal line, as at m ; and this point is 
 called the point of distance, by which the perspective width 
 of all the spaces between the perpendicular lines upon the 
 door may be ascertained. From the points 12 and 13 two 
 lines must be drawn to the point of distance, m ; and where 
 these lines intersect the line d, at 14 and 15, they mark the 
 perspective width of the framework or panels on the top of 
 the door: from these points, 14 and 15, two perpendicular 
 lines must be drawn till they touch the line K ; and where 
 these perpendicular lines pass between the lines E and f, 
 g and H, and i and J, they form the perpendicular boun- 
 daries of the panels. The student must now strengthen all 
 the lines of the panels, as in the example ; and he has com- 
 pleted his task, — he has drawn the door in perspective. 
 
 In order to make the foregoing example simple enough 
 to be comprehensible to the most inexperienced, the drawing 
 is confined to the fewest possible quantity of lines. The 
 thickness of the door and the projection of the framework 
 round the panels has been purposely omitted, — a multiplicity 
 of lines tending always to perplex the learner; but the rules 
 for drawing these are the same as those already explained. 
 That the student may satisfy himself that he has clearly 
 understood what he has just accomplished, let him open the 
 door so wide as to bring the handle of the door within a 
 foot of the wall, and reseat himself in the same position. 
 He now loses sight entirely of the side of the door he has 
 just drawn, and the outer side becomes visible. The point 
 of sight, and consequently the horizontal line, is precisely 
 the same, but the vanishing point of the door changes sides : 
 instead of being to the left of the artist, it is now to his right 
 hand ; the whole drawing of the door is reversed, but the 
 process of putting it in perspective is precisely similar to 
 that of the last example. It is strongly recommended to 
 the student to proceed carefully and steadily to draw it in 
 this altered position. 
 
14 
 
 PERSPECTIVE FOR STUDENTS. 
 
 CHAPTER II. 
 
 The Author, when very young, on being strongly recom- 
 mended by an artist, now an R. A., to draw from nature, 
 replied that he had no possibility of getting into the country. 
 
 " My young friend," said Mr. C , " you have got a 
 
 notion, like many other foolish people, that to draw from 
 nature it is necessary to go into the country. Let me advise 
 you, if you cannot find a tree to draw from, to draw the 
 plants in your mothers flower-pots ; if you cannot get to 
 draw the outside of a house, draw the inside of a room ; if 
 you are unable to find a wheelbarrow, take a coal-scuttle ; if 
 cows and sheep are not to be found, draw the family cat ; — 
 you will find it equally improving, and it will give you the 
 power ultimately of representing every object you desire on 
 paper." The advice was most excellent ; and the Author 
 most strongly recommends it to his juvenile readers. He is 
 about to lead them step by step to draw various objects in 
 perspective ; and the forms selected will be the most familiar 
 and the best adapted to the purpose : but in the limits of a 
 small work like this the principles on which certain objects 
 may be represented in drawing is all that is attempted. If 
 an example of a square object is given, the rules for drawing 
 that square object will apply to everything of a similar form 
 seen from a similar point of view. If an example is given 
 for drawing a circular, octagonal, or any other form, all 
 similar forms may be drawn by the same rules. Once 
 clearly comprehend how to draw a circle in perspective, and 
 it is immaterial what circular object is to be represented : the 
 same rules apply to all, whether a plate, a tumbler, a column, 
 or a dial, &c. 
 
 One of the great difficulties experienced by teachers is that 
 of making their pupils understand the manner of finding the 
 Vanishing Points and Points of Distance. For architectural 
 
PERSPECTIVE FOR STUDENTS. 
 
 15 
 
 draftsmen, and those who go deeply into perspective, there 
 are rules by which all the various points are to be found ; 
 but they are perplexing and tedious, unfitted for an elemen- 
 tary work like this, and unnecessary for those whose object 
 is simply to acquire that knowledge of perspective which will 
 enable them to make correct and agreeable sketches from 
 nature. In order to find the Vanishing Points, some teachers 
 recommend their pupils to make use of an instrument called a 
 moveable angle, or guiding-rule. It is an instrument of this 
 form. (Fig. 16.) It is made simply of three 
 straight pieces of wood, the two outer pieces 
 of which, by means of a moveable screw, open 
 and shut like a pair of compasses. The use of it 
 is, to hold it at arm's length, between the spec- 
 tator and the object to be represented — as, for 
 instance, the two top lines of a church tower — and, by means 
 of the screw, move the legs of the guiding-rule till they follow 
 the direction of the inclination of these two upper lines ; then, 
 laying the guiding-rule on your paper, and placing the point 
 formed by the angle over the point representing the highest 
 point of the nearest corner of the tower, rule the lines in the 
 direction of the two- sides of the guiding-rule, and continue 
 them till they touch the horizontal line. The points where 
 these lines would touch would form the vanishing points for 
 the horizontal lines on the respective sides of the tower. 
 
 Presuming that the reader draws a little before attempting 
 to draw from nature (and if not, he is strongly recommended 
 so to do), the author considers it far preferable for the drafts- 
 man to depend rather on his eye and judgment than to make 
 use of a guiding-rule or other mechanical instrument ; that 
 he make his first sketch by eye, and correct it afterwards by 
 the rules of perspective. 
 
 Problem I. — Let the student imagine himself placed 
 before a cottage, having a gable at one end and four win- 
 dows in front, and let him further imagine that he is so 
 
16 
 
 PERSPECTIVE FOR STUDENTS. 
 
 situated as to see both sides nearly equal — that he stands, in 
 fact, nearly in a line running from the angle formed by the 
 Fig. 17. lines J K to the corresponding corner, 
 
 ■~ which is hidden. (Fig. 17.) Suppose 
 A to be the plan of the house, and B 
 the position of the draftsman, c would 
 represent the line drawn from the 
 spectator's eye to the point of sight : 
 and the student will perceive that the lines r> and e, the two 
 sides of the house visible, are neither of them in the direction 
 of this line c ; consequently, that the point of sight cannot 
 form the vanishing point for any lines running parallel to 
 either D or E ; and that as these two lines are also at an angle, 
 each of them must have its respective vanishing point : the 
 line D will have its vanishing point to the right, and the line 
 e to the left. 
 
 The student, if sketching from nature, must first draw, 
 according to the best of his judgment, the first upright line, 
 a, of the building, and set a mark upon it at the height of 
 his eye, in order to get the horizontal line. To make this 
 perfectly simple, we will suppose the real height of this line 
 to be twenty feet, and that the spectator is so situated as to 
 have his eye at five feet from the ground ; he must then 
 measure off from the bottom of the line one-fourth of its 
 length, which will give the height of his e^e at five feet from 
 the ground ; and through this point he must draw a line, 
 B, across the picture, which will form the horizontal line.* 
 
 * The student must here bear in mind that the height of the hori- 
 zontal line depends entirely on the situation in which he is placed. If 
 the building from which he is drawing stood on a rising ground, say a 
 rise only of five feet, the horizontal line would be exactly on a line with 
 the base of the building, the spectator's eye being supposed five feet from 
 the ground on which he stands. If, on the contrary, the spectator stood 
 on a rise of five feet, the horizontal line would cut the line a in half, 
 because, the house being twenty feet, the spectator's eye, being five feet 
 above the spot on which he stands, would bring it to ten feet high. 
 If the spectator stoof on a rise of fifteen feet, the horizontal line would 
 
PERSPECTIVE FOR STUDENTS. 
 
 IT 
 
 From the top of the line A the student must now sketch the 
 lines c and D, marking their inclination towards the hori- 
 zontal line as carefully as possible, and he must then sketch 
 the lines e and F, to determine the width of the two sides of 
 the building. This is all that is necessary for the student to 
 draw by eye, and he must now correct his sketch by rule. 
 He must first, with his T square, the use of which has been 
 already described, make the line A perpendicular, so as to be 
 at right angles with the horizontal line on each side, both above 
 and below it : he must then, placing his rule upon the top of 
 the line a, marked 1, in the direction he has sketched the line 
 c, rule a line till it meet the horizontal line at g, which will 
 be the vanishing point for all the horizontal lines on the left 
 side of the house. From the same point 1, the top of the 
 line a, following the direction of the sketched line D, another 
 line must be drawn till it meet the horizontal line at the 
 point H, which will be the vanishing point for all horizontal 
 lines on the right side of the house. The rule must now be 
 placed at the point 2, the bottom of the line a, and from it 
 to the vanishing points, g and h, the lines J and k must be 
 drawn, which lines represent the perspective inclination of 
 the bottom lines of the house, as the lines c and d represent 
 the perspective inclinations of the top lines. The lines e 
 and F, determining the width of the two sides of the house, 
 must now be corrected by the T square, taking care to draw 
 the line e so as exactly to meet the lines c and J at the points 
 3 and 4, and the line f so as exactly to touch the lines d and 
 K at the points 5 and 6. Here let the student well notice 
 these three lines, a, e, and f, which, though really of the 
 same height in nature, are all dissimilar in the perspective 
 drawing. The line A, from being the nearest to him, appears 
 the longest ; the line E, from the left side of the house being 
 
 be on a level with the top of the house. Practice, and attentive exami- 
 nation of the works of clever artists, will gradually teach the amateur 
 a good choice of position, upon which the agreeableness of his drawing 
 greatly depends. 
 
18 
 
 PERSPECTIVE FOR STUDENTS. 
 
 narrower than the right, is nearer to the spectator than the line 
 f, and is consequently, though considerably shorter than the 
 line a, much longer than f, the farthest removed from the eye. 
 
 The upper part of the left side of the house is terminated 
 by a pointed roof, or what is called a gable, and the point 
 of this gable in nature is perpendicularly over a point mid- 
 way between the lines a and e. The student must be aware 
 that the perspective centre of the side of the building cannot 
 be exactly half-way between the lines a and e in the drawing, - 
 because that half of the building nearest to him must appear 
 wider than the half that is farther off. If the centre is 
 required of any rectangular parallelogram, it is found by 
 Fig. 18. ruling two lines from its opposite angles, which are 
 \ a called diagonal lines (Fig. 18), the intersection of 
 
 X. which denotes the centre of the figure. So in per- 
 
 ^ spective, — the space contained by the lines a, c, e, j, 
 
 is a rectangular parallelogram in perspective ; and if from 
 the opposite points, where these lines join, as from 4 to 1 and 
 from 3 to 2, the diagonal lines l and m are drawn, the point 
 where they intersect at 7 is its perspective centre,* and the 
 point of the gable must be drawn directly over it ; to do which 
 the student must draw a perpendicular line n through this 
 point 7 above the line c ; and at some point on this line the 
 lines forming the sides of the gable must meet. In order to 
 determine the height of the point of the gable, the student 
 must continue the line a above the point 1. This line being 
 the nearest perpendicular line, is the most convenient for 
 finding the height of all objects on either side of the house. 
 Let us suppose the height of the point of the gable to be five 
 feet above the line c ; this five feet must be set upon the line 
 A, above the point 1. The student must therefore put on 
 this line one-fourth of its length, as at 8, and from it (the 
 
 * This mode of finding the perspective centre of a parallelogram by 
 diagonal lines is eminently useful in sketching from nature ; it often 
 obviates the necessity for a great many points and lines that would other- 
 wise be needed. The student will do well to bear it in mind. 
 
PERSPECTIVE FOR STUDENTS. 
 
 19 
 
 point 8) rule a line o to the vanishing point where this 
 line intersects the line n is the perspective position of the 
 point of the gable, to which, from the points 1 and 3, draw 
 the lines p and q, which complete the drawing of the left 
 side of the building. 
 
 The student is here shown the method of finding the exact 
 perspective height of the point of the gable ; but in sketching 
 from nature it is quite sufficient to choose the point on the 
 line N by the eye, and from it rule the lines p and q, — as 
 whether it is a trifle higher or lower is of little importance. 
 
 The mode used for finding the position and width of the 
 windows, is similar to that for drawing the panels and frame- 
 work of the door, in Fig. 2, Plate I. From the point 1 a 
 horizontal line R must be drawn, to represent the geometrical 
 length of the line d in the perspective drawing ; * and on this 
 line must be measured off at each end the distance of each 
 window from the side of the house, as at 10 and 13, and 
 from each of these points the width of each window, as at 
 1 1 and 1 2. From 9, the extremity of this line R, a line must 
 be drawn through the point 5, till it meet the horizonial line 
 at s ; which point forms the point of distance, by which the 
 width of all objects on the right side of the house is deter- 
 mined. From each of the points on the line R, viz. 10, 11, 
 12, 13, a line must be drawn to the point of distance, s; and 
 where these lines intersect the line d (which represents r in 
 perspective) they designate the perspective positions of these 
 points, from each of which a perpendicular line, as 14, 15, 
 16, 17, must be drawn, till it touch the bottom line, K, of the 
 building. The space between a and 14 represents the per- 
 
 * It is immaterial to what length the line r is drawn, so that it 
 be longer than the line d. The student must be aware that r, being 
 the geometrical line represented in perspective by the line d, must 
 necessarily be the longest. If the line r were lengthened so as to bring 
 the point 9 further to the right, but keeping the distances and width 
 of the windows in their relative proportions, the point of distance would 
 be further to the left, but the intersections on the line d would be the 
 same. 
 
20 
 
 PERSPECTIVE FOR STUDENTS. 
 
 epective distance between the side of the house and the first 
 window; that between 14 and 15, the perspective width of 
 the first window; from 15 to 16 is the perspective width of 
 the space between the two windows ; from 16 to 17 the per- 
 spective width of the second window ; and from 17 to the 
 line F the perspective width of the space between the last 
 window and the farther side of the house. It now only re- 
 mains to determine the height of the windows, and their 
 respective distances from the top and bottom lines of the 
 building. Let us suppose that the upper window is one foot 
 below the line d, and that the window is four feet high ; a 
 twentieth part (one foot) must be marked off on the line a 
 below 1, as at 18, which will be the geometrical distance of 
 the top of the window from the roof, and below this one-fifth 
 of the line A (four feet), as at 19, which will be the geome- 
 trical height of the windows, and from each of these points 
 a line must be drawn to the vanishing point H. Where the 
 line drawn from 18 passes between the lines 14 and 15, and 
 16 and 17, it gives the perspective drawing of the top of each 
 of the upper windows ; and where the line drawn from 19 
 passes between the same lines, 14, 15, and 16, 17, it gives 
 the perspective drawing of the bottom lines of the upper 
 windows. Supposing the lower windows to be of the same 
 height as the upper ones, and that they are three feet from 
 the ground, these distances must be placed on the line A ; 
 that is to say, from the bottom, 2, of the line A, must be set 
 up three-twentieths of its length (three feet), as at 20, and 
 above that one-fifth of the length of a (four feet), as at 21. 
 From each of these points, 20, 21, a line must be drawn to 
 the vanishing point h ; and where these lines pass between 
 the lines 14, 15, and 16, 17, they give the perspective draw- 
 ing of the top and bottom lines of the lower windows.* 
 
 * The student should now draw in with a pen the strong lines, leaving 
 the remaining lines, as well as the letters and figures, in pencil, and care- 
 fully preserve his drawings, as he will find them always useful, and towards 
 tLe end of the work they may save him much time and trouble. 
 
PERSPECTIVE FOR STUDENTS. 
 
 21 
 
 It is hardly necessary to tell the student, that the dark 
 lines in the plates represent only the object to be drawn, and 
 that the faint lines are those used for finding the correct per- 
 spective. In the foregoing example, on the right side of the 
 drawing, the student is made to comprehend a mode for 
 finding the perspective distance and size of any object on the 
 face of a building : the forms chosen — the windows — are 
 rectangular figures, as being the most simple ; but the posi- 
 tion and size of any object, whatever may be its form, can 
 be ascertained by the same rule. In our progress we 
 shall endeavour to render intelligible the mode of putting 
 a variety of forms into perspective ; but, like everything 
 else, it is necessary to proceed step by step, and to thoroughly 
 understand one problem before proceeding to another. 
 
 On the left side of the building the student is made to 
 comprehend a mode for putting a pointed roof or gable in 
 perspective ; and, simple as it is, it is surprising the number 
 of errors constantly committed by the neglect of its use. 
 The author has seen many paintings where the artist, from 
 mere carelessness, has brought the point of the gable nearer 
 to the line represented by a than to the side represented by 
 e, which is most offensive to the eye. Many of the Dutch 
 and Flemish paintings show a great deficiency in perspective 
 drawing ; and the great Teniers, notwithstanding his beautiful 
 representations of still-life, sadly outrages perspective in some 
 of his out-of-door scenes.* 
 
 Problem II. — In the foregoing example, the mode for 
 finding a point of distance is given upon a line above the 
 horizontal line : but many instances occur in drawing per- 
 spective where all the lines are below the horizon ; as, for 
 instance, a chess-board placed on a table, where, even in a 
 sitting position, every line must be below the eye, or the 
 
 * There is an entertaining print by Hogarth, the title of which I do not 
 recollect, that would amuse, and at the same time be useful to the young 
 reader : in it he has outraged perspective as much as possible. The 
 btudent would do well to examine it and find out its errors. 
 
2? 
 
 PERSPECTIVE FOR STUDENTS. 
 
 squares on it could not be seen. The student should place a 
 chess-board before him, so as to view it in the same position 
 as that represented in the plate. He must first sketch, to 
 the best of his judgment, the square of the board a, b, c, d.* 
 The line A must be drawn over with a rule, to make it per- 
 fectly straight ; and parallel to it, at the distance the eye is 
 above the board, a long line, e, must be drawn across the 
 picture to represent the horizontal line. From the point 1 — 
 the nearest left-hand corner — in the direction of the sketched 
 line b, draw a line till it touch the horizontal line E at F, 
 which will be the vanishing point. From the point 2 — the 
 nearest right-hand corner of the board — a line must also be 
 drawn to the vanishing point F. These two lines, B and D, 
 represent the perspective inclinations towards the vanishing 
 point of the two sides of the chess-board ; and the student 
 will perceive how easily the two sketched lines are corrected. 
 At the distance of from a to c, and parallel to A, a line must 
 be drawn between b and d, to touch them at the points 3 
 and 4. The lines a, b, c, b, represent the outer lines of the 
 chess-board in perspective. In order to regulate the perspec- 
 tive widths of the squares, which gradually diminish from the 
 line A to c, it is necessary to find a point of distance. The chess- 
 board being a square, the student will understand that the 
 line b, between 1 and 3, is the perspective length of the 
 line a, between 1 and 2. If the student then rule a line 
 from the point 2, making it pass through the point 3, and 
 continue it up to the horizontal line, the point g, where it 
 touches, will be the point of distance, and will regulate the 
 perspective lengths of the squares on the line B.t The line 
 
 * The dotted lines represent a sketch of the square of the chess - 
 board, as it might be made by a beginner, to show with what facility a 
 very indifferent sketch may be corrected by rule. 
 
 f It is immaterial whether the line b or the line d, each of which 
 represents the perspective length of a, be taken for rinding the perspec- 
 tive distances of the squares. If the student measure off to the right of 
 the point 2 a space equal to the line a, between 1 and 2, and from its 
 
PERSPECTIVE FOR STUDENTS. 
 
 A mast now be divided into eight parts ; and from each oi 
 the points of division, viz. 5, 6, 7, 8, 9, 10, 11, a line must 
 be drawn to the vanishing point f. These lines represent 
 the gradually decreasing width of the squares from a to c. 
 From each of these points — 5, 6, 7, 8, 9, 10, 11 — a line 
 must be drawn to the point of distance, G ; and where these 
 lines intersect the line B, at the points c, d, e, /, g, they 
 represent the gradual decreasing length of each square from 
 A to c. From each of these points of intersection, a, b, c, &c. 
 a line parallel to the line A must be drawn till it meet the 
 line d ; and these lines, by their intersections with those drawn 
 from the points 1, 2, 3, &c, give the perspective representa- 
 tion of the whole 64 squares of the chess-board. The alter- 
 nate squares are slightly shaded, to make the figure perfectly 
 intelligible to the juvenile student. 
 
 Here let it be understood, that when the four sides of the 
 square, a, b, c, d, are put in perspective, if, in order to find a 
 point of distance, a line had been ruled from the point 1 
 through the point 4, the point at which that line would touch 
 the horizontal line would give a point of distance that would 
 have produced the same result ; observing, that in this case 
 the points of intersection, a, c, J, &c, would have come on 
 the line D instead of the line B. 
 
 Problem III. — The student, in drawing this figure, 
 must, according to the explanations given in Problem I., 
 draw the nearest house, so far as it is described, up to the 
 lines lettered to q, and figured to 8 ; observing that, with a 
 view of exercising his ingenuity, the gable end is on the 
 opposite side, — the letters and figures up to q, and 8 referring 
 to similar lines in Problem I. 
 
 In* order to determine the perspective width of the second 
 
 extremity, h, rule a line to the point of distance, it will intersect the line 
 c at 4, the point determining the length of the line d by means of the hori- 
 zontal line drawn from the point 3 of the line b. 
 
 * It must be understood that the description here commenced, and 
 continued to the end of this and the following paragraph, is not the 
 
24 
 
 PERSPECTIVE FOR STUDENTS. 
 
 and third houses, the same means might be used as employed 
 for determining the position and width of the windows in 
 Problem I. ; that is, a horizontal line might be drawn to the 
 right of the point 1, the top of the line A, and from any part 
 of the horizontal line to the left of the line e a point might 
 be chosen as a point of distance ; and from it a line drawn 
 through the point 3 till it meet this horizontal line, would 
 give the geometrical width of the house between its point of 
 contact and the point 1. If two similar spaces were mea- 
 sured off on this line to the right, to represent the geometri- 
 cal width of the second and third houses, and from each of 
 the points of division a line were drawn to the point of dis- 
 tance, where these lines intersect the line c would be the 
 perspective widths of the second and third cottages. If the 
 reader has thoroughly understood the First Problem, he would 
 now have no difficulty in putting the gables to these two 
 further houses, on the same principles as those used for 
 drawing the first : but the author, in a long experience of 
 teaching, has found so frequently that in the slightest varia- 
 tion in the application of a rule the juvenile student is apt to 
 get bewildered, that, at the risk of being thought tedious, he 
 will repeat the mode necessary for proceeding. 
 
 From each of the points of intersection on the line c, that 
 determine the perspective widths of the second and third 
 cottages, a perpendicular line should be drawn down to meet 
 the line J ; and these two lines, with the portions of the lines 
 c and j lying between them, would represent the rectangular 
 parallelograms of the second and third cottages, answering 
 to that contained by the lines a, c, e, j, of the first. In each 
 of these perspective parallelograms two diagonal lines should 
 
 description of the mode by which the gables in this representation are 
 drawn. It is given in order to impress on the mind of the reader what 
 he has already done, and to accustom him to comprehend perspective 
 drawing by general description. The student would do well, however, 
 to draw the problem on a separate sheet, according to the description 
 nere given. 
 
PERSPECTIVE FOR STUDENTS. 
 
 25 
 
 be drawn, corresponding with the lines l and M in the first ; 
 and from their points of intersection two perpendicular lines 
 should be drawn to touch the line o, similar to the line N 
 drawn from the point 7 to 9. The line o ruled from the 
 point 8 to the vanishing point G fixes the height of the first 
 gable ; and as it is supposed that all three of the gables are 
 of the same height, the line o would also determine the 
 height of the gables of the second and third cottages : so that 
 where the line o would meet the perpendicular lines just 
 drawn, would be the points where the two sides of the gable 
 must meet. From each of these points to the top of the per- 
 pendicular lines right and left (corresponding to the points 1 
 and 3 of the first gable) draw the sides of the gables, corre- 
 sponding to the lines p and q of the first ; and in a similar 
 manner any number of cottages with gables may be con- 
 tinued on. 
 
 Where many gables follow in succession, as in a long row 
 of houses with gable ends, or with garret or other windows 
 having pointed tops, there is a rule for putting them in per- 
 spective much more simple than the foregoing, the use of 
 which, with a little extra attention, the student will fully 
 comprehend. Let us suppose that on some part of the front 
 of each of these cottages was fixed a clock-dial, and let us 
 further suppose the time marked upon each dial to be a quar- 
 ter to twelve : the hour-hand of the dial would then be per- 
 pendicular, (or so nearly so, that, for the sake of our lesson, 
 we must grant it to be perpendicular,) and the minute-hand 
 in a horizontal position. To represent a series of dials with 
 the hands in this position would not require any additional 
 points, because the hour-hands, being perpendicular, would 
 be parallel to the other perpendicular lines on the face of the 
 building ; and the minute-hands, being horizontal, would be 
 drawn to the same vanishing point as the other horizontal 
 lines on the face of the building: but if, instead of the hands 
 of the dials indicating the time a quarter to twelve, they stood 
 at ten minutes to five, they would then be at an angle both 
 
 Perspective* C 
 
2G 
 
 PERSPECTIVE FOR STUDENTS. 
 
 with the horizontal and perpendicular lines of the building. 
 It has been already remarked, that all lines that are geome- 
 trically parallel are drawn in perspective to the same vanish- 
 ing point. Now if the hands of all these dials stand precisely 
 at ten minutes to five, all the minute-hands must be parallel 
 to each other, and all the hour-hands must also be parallel, 
 and certain points must be found by which the directions of 
 the lines representing these hands may be drawn. The 
 minute-hands of the dials pointing to the figure ten, the 
 lines representing them must necessarily run upwards from 
 the horizontal line, and some point must be found to repre- 
 sent them above it ; but where, on the contrary, they point to 
 the figure five, they would run downwards, and some point 
 must be found to represent them below the horizontal line. 
 These points are to be found on a line perpendicular to the 
 horizontal line, either above or below it, and passing through 
 the vanishing point. 
 
 As it would be with the hands of a series of dials just 
 described, so is it with the lines corresponding with p and q 
 in a series of gables, these lines being at an angle both with 
 the perpendicular and horizontal lines of the building and 
 with each other. By finding the respective vanishing points 
 for these two lines, the student will not only be enabled to 
 find the perspective directions for an infinite number of 
 gables, but in drawing them they determine the perspective 
 width of each building. 
 
 To proceed with the drawing, which we left with the first 
 house completed, as in Problem I. to the letter q and figure 8. 
 Through the vanishing point G a long line R must be drawn 
 perpendicular to the horizontal line, above and below it ; and 
 the line P of the first gable must be continued upwards till it 
 meet the line R at s, which will be the vanishing point for 
 all the lines forming the left sides of the gables ; all of which 
 lines the student is aware are geometrically parallel. The 
 line q, the second line of the first gable, must then be con- 
 tinued downwards till it meet the line r at t, which will be 
 
PERSPECTIVE FOR STUDENTS. 27 
 
 the vanishing point for all the lines forming the right sides 
 of the gables. From the point 3 a line must be drawn to 
 the vanishing point s, which will give the perspective direc- 
 tion of the first line of the second gable : and where this line 
 at 10 intersects the line o (which drawn from the point 8 
 regulates the height of each gable), it determines the point 
 where the two lines of the second gable meet ; and from it 
 a line must be drawn to the vanishing point t, which gives 
 the perspective direction of the second line of the second 
 gable. Where this line intersects the line c, which gives the 
 perspective height of all the lines from which the lines of the 
 gables are drawn, it determines the perspective width of the 
 second cottage, and from it the third gable is drawn pre- 
 cisely as was the second from the point 3. By the same 
 process a fourth, fifth, or more gables may be drawn, at the 
 will of the artist ; the three given are quite sufficient to 
 enable the student to comprehend the rule. But one of the 
 most important features of this mode of representing the 
 gables, is the facility and accuracy with which the perspec- 
 tive direction of the sloping line of the roof from the point 5 
 on f is drawn. It is a common error to draw this further 
 line v parallel to the line p ; but the student will readily 
 perceive, from the example before him, as also by looking at 
 nature, the inaccuracy of so doing — the further line v sloping 
 much more than the line p. From the point 9, the point of 
 the first gable, draw the line u to the vanishing point h ; 
 this gives the perspective direction of the upper line of the 
 roof : then from the point 5 draw the line v to the vanishing 
 point s; and where this line intersects the line u at 11 is a 
 point corresponding to the point 9 on the line p. From each 
 of the points of the second and third gables a line must be 
 drawn to the vanishing point H, to give the direction of the 
 upper lines of the roofs of the respective cottages, which 
 completes the drawing. These additional points, s and t, 
 are found to be valuable in various ways, as will be shown 
 in our progress onward : they greatly facilitate the findiug, 
 
 C 2 
 
28 
 
 PERSPECTIVE FOR STUDENTS. 
 
 the positions of chimneys or windows on sloping roofs 
 of houses, of towers or soires on the sloping roofs of 
 churches, &c. 
 
 The student will perceive that diagonal lines are put on 
 the gable end of each cottage, and that perpendicular lines 
 have been drawn from their points of intersection (the per- 
 spective centres of each gable end). This is done to demon- 
 strate to the student that the mode of finding the points of 
 the gables by means of the two vanishing points s and t 
 produces the same result as that of finding them by means 
 of the diagonal lines ; the perpendicular lines drawn from 
 the intersections of the diagonals passing directly through 
 the points of the gables found by the vanishing points s 
 and t. 
 
 Problem IV. — In a note in a former part of this work 
 we drew the attention of the student to the advantage he 
 would find from using the diagonal lines. In sketching from 
 nature, it is rarely possible — neither is it necessary — to have 
 the actual measurement of the objects to be represented ; 
 most of the relative proportions of one object with another 
 must depend on the eye of the artist ; but if the position and 
 form of any one object be carefully drawn on one part of the 
 face of a building, the position and form of any similar object 
 in a corresponding part may be found by means of the diago- 
 nal lines. The skeleton of the house is drawn in the same 
 manner as in the last problem and Prob. I. For the advan- 
 tage of having the references distinct, the figure is drawn 
 rather larger ; in consequence of which the vanishing points 
 are out of the picture, but they are referred to in the first 
 and third problems as G and H ; and the student in making 
 his drawing must necessarily have them. The points to 
 which the figures referred in the former problems, being un- 
 necessary for our present purpose, are not marked ; and the 
 references by figures here given relate only to the new rule 
 about to be explained. 
 
 The student must first, as before described (Prob. I.), 
 
PERSPECTIVE FOR STUDENTS. 
 
 29 
 
 draw all the lines of the house, with their letters A b, &c. 
 for reference, up to the letter q, marking the respective 
 vanishing points of each side, G and h. This done, he 
 must sketch the position and size of the first window on the 
 gahle end of the house, and then with his T square draw 
 correctly the lines 1 and 2, carrying them a little above and 
 below the lines he has sketched for the top and bottom of 
 the window. Now in order to get the relative distance of 
 the second window from the line E that the first window is 
 from the line a, it is necessary, from the point 3, where the 
 line 1 intersects the diagonal line L, to draw a line to the 
 vanishing point g. This line intersects the other diagonal 
 line M at 4 : and through this point of intersection 4 draw a 
 perpendicular line 5. The point 4 on the diagonal m cor- 
 responds with the point 3 on the diagonal l, and the line 5 
 drawn through it is at the same relative distance from the 
 line E that the line 1 of the first window is from the line A. 
 To find the relative perspective width of the second window, 
 from the point 6, where the line 2 of the first window inter- 
 sects the diagonal l, another line must be drawn to the 
 vanishing point G : and the point where it intersects the dia- 
 gonal m at 7 corresponds with the point 6 on the diagonal l ; 
 through this point 7 another perpendicular line (8) must be 
 drawn, which corresponds with the line 2 of the first window, 
 and the space between the lines 5 and 8 represents, relatively 
 to its perspective distance, the same as that between the lines 
 1 and 2. The ruler must now be placed on the line 1, at that 
 point denoting the top line of the window, as a 9, and from 
 it a line must be ruled to the vanishing point G : this will 
 correct the original sketched line of the first window; and 
 when it passes between the lines 8 and 5 it will represent the 
 top line of the second window. The ruler must now be 
 placed at the point on the line 1, that denotes the position 
 of the bottom line of the window ; and from that point a line 
 drawn to the vanishing point g will give, where it passes be- 
 tween the lines 1 and 2, the bottom line of the first window, 
 
30 
 
 PERSPECTIVE FOR STUDENTS. 
 
 and where it passes between the lines 8 and 5, the bottom 
 line of the second. 
 
 In the first window, just drawn, the perpendicular lines 
 forming the sides intersect the diagonal line L, as at 3 and 
 (i ; and consequently the corresponding points on the diagonal 
 line M are found easily, by ruling at once from these points 
 to the vanishing point G. But it happens sometimes that 
 the windows are so situated on the face of a building, that 
 their sides neither intersect nor touch the diagonal lines. In 
 order to point out the mode of proceeding when the windows 
 are so situated, we will take the other side of the building. 
 We will suppose a window to be in the situation of that 
 represented in the engraving near the line a, between that 
 line and f : this being sketched, the diagonal lines w and 
 x must be drawn. The student will here perceive that neither 
 of the upright lines of this window touch the diagonal lines ; 
 the student must therefore with his T square, continue them 
 upwards till they meet the diagonal line x at the points 11 
 and 12, and from each of these points draw a line to the 
 vanishing point H. Where the upper line intersects the 
 diagonal line w at 13, is a point corresponding with the point 
 11 on the diagonal line x ; and where the lower line intersects 
 the diagonal line w at 14, is the point corresponding with the 
 point 12 on the diagonal line x. From each of these points 
 (13 and 14; a perpendicular line must be drawn downwards; 
 and the space between these two lines represents the perspec- 
 tive width of the second window, at its perspective distance 
 from the line F, corresponding with the distance of the first 
 window, from the line A. The upper and lower lines of the 
 second window are found, as on the other side of the house, 
 by continuing the lines of the top and bottom of the first 
 window to the vanishing point H. 
 
 Let us now suppose that on the roof there are two garret 
 windows, situated immediately over the two windows just 
 drawn, of the same width, and each window having a pointed 
 roof. To find their width and position, the upright lines of 
 
PERSPECTIVE FOR STUDENTS. 
 
 31 
 
 the windows just drawn must be continued up through the 
 line d, which will form their sides. Let any point on the 
 nearest of these upright lines be chosen, as at 15, to fix their 
 height (the mode for getting a fixed height would be the 
 same as that explained foi getting the height of the gables, 
 Problem I. 8, o), and from it rule a line to the vanishing 
 point H ; this, crossing the upright lines already drawn, will 
 give the rectangular parallelograms of the garret windows 
 in perspective : and as there are only two garret windows, 
 and consequently only two pointed roofs, to be drawn, the 
 readiest way will be to find the situation of the points by 
 raising perpendicular lines from the intersection of the dia- 
 gonal lines of each parallelogram. The pointed roofs of these 
 two windows are here drawn, and the lines used for finding 
 them left ; but it would be quite superfluous again to go 
 over the explanation of drawing them. In order to find the 
 side of the first garret window, it is necessary first to draw a 
 line from the point of the gable to the vanishing point G, as 
 also from the point 1 5 to the same vanishing point, which 
 lines will represent the perspective direction of the upper and 
 lower lines of the roof of the garret window, — and which, the 
 student must understand, in the real object are parallel to 
 the horizontal lines on the gable side of the house. To find 
 the points where these two lines terminate on the roof of the 
 house, will reouire a little attention : the rule is similar to 
 that employed for finding the directions of the gable in 
 Problem III. The student must first find the vanishing 
 point for the line p of the gable of the house. The lines of 
 each of the sides of these windows, where they touch the 
 roof, are in reality parallel to the line P of the gable (because 
 the whole side of the roof is a uniform slope), and must con- 
 sequently vanish to the same point ; therefore, from the point 
 16*, where the upright line of the window touches the lower 
 line of the roof of the house, a line must be drawn to the 
 vanishing point s ; and where this line intersects that drawn 
 from the point 15 to the vanishing point g at 17, is the point 
 
32 
 
 PERSPECTIVE FOR STUDENTS. 
 
 marking the spot where the lower line of the roof of the garret 
 window touches the sloping roof of the house. To find the 
 point where the upper line of the roof of the garret window 
 touches the sloping roof of the house, is a little complicated ; 
 and to render it quite clear, an additional figure is introduced. 
 Fig. 2 is drawn up to the point marked 17 of Fig. 1. The 
 window here drawn contains the lines of both sides, as if it 
 were transparent. The student will observe that the point of 
 the front of the gable comes directly on a line, exactly mid- 
 way (perspectively) between the two sides ; consequently, the 
 point at the back must come on a line midway between the 
 sloping lines on the roof forming the two sides ; from the 
 points 3 and 1, two lines have been drawn towards the van- 
 ishing point s. Where the line drawn from the point 3 
 meets the line drawn from the point of the gable 5 to the 
 vanishing point G at 6, is the point where the two sides of 
 the roof join ; and a line drawn from the point 6 to 4 will 
 complete the drawing of the first garret window. The student 
 will observe, that where the line drawn from the point 1 to 
 the vanishing point s intersects the line drawn from the 
 point 7 to the vanishing point g at 8, the lines forming the 
 triangle 1, 7, 8, represent the farther side of the window, and 
 correspond with the lines forming the triangle 2, 9, 4, the 
 near side ; the lines forming the triangle 4, 8, 6, represent 
 the form of the gable on the sloping roof of the house, and 
 correspond with the lines forming the triangle 7, 5, 9. The 
 garret windows in the drawing (Fig. 1) must now be com- 
 pleted, in the manner described in Fig. 2 ; and the highest 
 line of the roof of the house, u, with the extreme line of the 
 slope, v, drawn to their respective points, as in the preceding 
 problem (III.) ; and this figure will be finished. 
 
 The rules given in this and the preceding plate will be 
 found useful for drawing the divisions of tiles or slates on 
 the roof. In Fig. 3, that portion only of the drawing of the 
 house is introduced necessary for the purpose. The lines 
 a, c, D, f, p ? u, v, are drawn as before described. From 
 
PERSPECTIVE FOR STUDENTS. 
 
 33 
 
 the point of the gable a horizontal line must be drawn to the 
 left, to represent the geometrical length of the perspective 
 line u ; this geometrical line must be divided into as many 
 equal parts as there are tiles in each row, and a point of dis- 
 tance found, to give the perspective positions of these several 
 divisions on the line u. These being found, a line must be 
 drawn through each from the vanishing point s to the line D, 
 which will give the correct perspective direction of the divi- 
 sions of the tiles or slates. From the point a, a horizontal 
 line must be drawn to the right, to represent the geometrical 
 length of the near half of the line c ; and this geometrical 
 line must be divided into as many equal parts as there are 
 rows of tiles on the roof, and a point of distance found to get 
 the perspective positions of these points on the near half of 
 the line c. These divisions, however, are required on the 
 line P, and from each point of intersection on c a perpendi- 
 cular must be drawn till it touch the line p, and from each 
 point of contact a line must be drawn to the vanishing point 
 H, which, by their intersections with the lines drawn between 
 u and d, give the relative forms and positions of the different 
 tiles ; as the lines crossing each other in Prob. II. represent 
 the 64 squares of the chess-board. The tiles may be of various 
 forms ; but we do not attempt to do more at present than 
 point out the mode of rinding the perspective distances. The 
 student may easily, on these, draw any form of tile that may 
 happen to have been used, as in the example just given. 
 
 The rules employed in this problem will be found ex- 
 tremely useful ; the positions of all the objects in Fig. 1 are 
 found without the necessity for using a point of distance, the 
 diagonal lines answering for that purpose : they produce 
 equal correctness, and save time and labour. The rule for 
 finding the triangle 8, 6, 4, in Fig. 2, will be found useful 
 in drawing roofs of buildings, where the pointed ends slope 
 back as well as the sides ; a mode of construction very com- 
 mon in old buildings, especially abroad, and not unfrequently 
 met with in the roofs of country churches. 
 
 c 3 
 
34 
 
 PERSPECTIVE FOR STUDENTS. 
 
 Before proceeding to the following pages, the author 
 strongly recommends the student to choose certain familiar 
 articles composed of straight lines, and endeavour to put 
 them in perspective, according to the rules already explained. 
 
 CHAPTER III. 
 
 It is now necessary to advance a step farther. We trust the 
 directions for drawing the foregoing problems will be found 
 sufficiently clear to enable the young student to draw the 
 superficies of any object of simple form represented by 
 straight lines : thicknesses, such as the width of objects, 
 like windows', doors, &c, have been purposely omitted in 
 the preceding problems. The rules for drawing these 
 thicknesses are the same as those employed in drawing the 
 superficial forms, but demand a considerable number of 
 additional lines ; these would tend seriously to embarrass 
 the student, from their complication : it is therefore thought 
 advisable to postpone this portion of our work, till the reader, 
 by gradually accustoming himself to this mechanical draw- 
 ing, will be less liable to become perplexed with a multi- 
 plicity of lines. We will therefore proceed with some rules 
 necessary for drawing curves in perspective ; and, as the 
 most simple, we will commence with the circle. 
 
 Let us suppose that a series of semicircular arches were to 
 be drawn in perspective. We trust that the reader has so 
 far profited by the foregoing examples, that he would have 
 no difficulty in finding the width of each arch, the width of 
 each column, pilaster, or pier between the arches, and their 
 gradations of height. All this can be accomplished by the 
 use of a common ruler, because it can be done by means of 
 straight lines : but no ruler has yet been invented that will 
 enable the student to draw the changes of forms taken by 
 curves in perspective. The mode of proceeding is, first to 
 draw geometrically the curve intended to be represented in 
 
page, 2>5 
 
PERSPECTIVE FOR STUDENTS. 
 
 35 
 
 perspective, and through this geometrical figure to draw in 
 certain directions various straight lines, that shall intersect 
 or touch one another in certain points of the curve ; to put 
 these straight lines in perspective which will change the 
 relative positions of the various points, and through them, 
 by hand, to draw the curve in perspective. In this Fig. 19. 
 diagram (Fig. 19) for instance, we have a circle 
 drawn ; and in order to find certain points in this 
 circle, that will enable us to put it in perspective, 
 it is enclosed in a square; and the student will perceive that 
 the circle touches at four points of this square, exactly at the 
 points of contact of the two cross lines. Nothing can be 
 more simple than to put the six straight lines of this diagram 
 in perspective ; which, when done, would furnish four points 
 through which the curve line forming the circle in perspec- 
 tive must be drawn. But four points are not found sufficient 
 for the representation of a circle in perspective ; we must 
 therefore find some additional points, by adding Fig. 20. 
 to the straight lines already drawn, two diagonal 
 lines (Fig. 20). The student will here observe that 
 these diagonal lines intersect the line of the circle 
 at four other points, exactly midway between those in the 
 former diagram. Let us now proceed to construct the figure. 
 
 Problem V. — The student ought now, without assistance, 
 to be able to put in perspective the square, the diagonal lines, 
 and the perpendicular and horizontal lines that pass through 
 the centre ; but that no error may by possibility occur, we 
 will give him a little aid. First, below the ground line, of 
 any size that may be required, he must construct a geome- 
 trical figure similar to the second diagram given in the 
 preceding paragraph, and taking the upper line of the square 
 of this diagram for his first line, draw the square in per- 
 spective;* then from the opposite corners draw the two 
 diagonal lines : from the point 1 draw a line to the vanishing 
 
 * By referring to the drawing of the chess-board, Problem II., the 
 manner of drawing the square in perspective will be found. 
 
36 
 
 PERSPECTIVE FOR STUDENTS. 
 
 point, and through the centre of the square (where the 
 diagonal lines intersect) draw a horizonual line across, from 
 the line d to the line e, to the points 4 and 2. The student 
 will here perceive that he has put in perspective the straight 
 lines contained in the second diagram above, and found the 
 four points contained in the first : viz. the points marked 
 1, 2, 3, 4 of the geometrical drawing here given. It was 
 observed, in the foregoing paragraph, that certain straight 
 lines must be drawn, that shall intersect or touch one another 
 at certain points of the curve, &c. Now the student will 
 perceive that the diagonal lines drawn in the second diagram, 
 though they intersect the line of the circle, have no points of 
 intersection with any other straight lines, and that therefore 
 these diagonal lines in the perspective drawing in this stage 
 are quite useless : in order, therefore, to find the points 
 where the diagonal lines intersect the circle, we must have 
 two additional straight lines. In the square of the geome- 
 trical drawing on each side, through the points where the 
 diagonals intersect the circle, draw a line running from the 
 top to the bottom line of the square, as the line A passing 
 through the points 6 and 7, and touching the bottom line of 
 the perspective square at 9, and the line B passing through 
 the points 5 and 8, and touching the bottom line of the per- 
 spective square at 10. From each of these points 9 and 10 
 a line must be drawn to the vanishing point ; and where the 
 line drawn from the point 9 intersects the first diagonal, it 
 gives a point corresponding with the point 6 in the geome- 
 trical drawing ; where it intersects the second diagonal line, 
 it gives a point corresponding with the point 7. In like 
 manner the line drawn to the vanishing point from the point 
 10, at its intersections with the diagonal lines, gives two 
 points corresponding with the points 5 and 8 of the geome- 
 trical drawing. 
 
 The perspective positions of the whole of the eight points 
 being thus found, the student must carefully draw the curve 
 to represent the circle, touching the points 1, 2, 3, 4, and 
 
PERSPECTIVE FOR STUDENTS. 
 
 37 
 
 passing through the points 5, 6, 7, 8. This mode is generally 
 found sufficient for all ordinary purposes ; but where circles 
 are required to be drawn in perspective of very large di- 
 mensions, more points of intersection may be found in the 
 geometrical drawing : these do not at all increase the diffi- 
 culty, on the contrary, the curve line is drawn with more 
 ease and accuracy ; but the multiplicity of lines would be apt 
 to puzzle the student, and, as we before remarked, the fore- 
 going is quite sufficient for all ordinary purposes. 
 
 Let us suppose that the circle just drawn represents the 
 spot on which a column is to be erected, and that a row of 
 these columns is to be built; that the columns are to be 
 distant from each other exactly their own width, and that 
 the circle is marked on each spot where a column is to be 
 erected. In order to represent this in perspective, it is first 
 necessary to find a point of distance : this must be done by 
 the same rule employed in Problem II. (the finding the 
 point g). The student must first find the proper distance for, 
 and afterwards draw, the perspective square in which the 
 circle is to be drawn. To find the distance, he must measure 
 off on the ground line, and on the opposite side to where he 
 has fixed his point of distance, two spaces of the width of the 
 geometrical square ; and from each point of division, 1 1 
 and 12, a line must be drawn to the point of distance c. 
 Where the line drawn from the point 1 1 intersects the line 
 D at 13, it gives the perspective distance between the two 
 circles; and where the line drawn from the point 12 inter- 
 sects the line d at 14, the space between that point and the 
 point 13 represents the left side of the square in perspective 
 in which the second circle is to be drawn. From the points 
 13 and 14 two horizontal lines must be drawn to touch the 
 line E at the points 15 and 16. These two lines, with 
 the portions of the lines d and e between their extremi- 
 ties, form the four sides of the. square in perspective in which 
 the second circle is to be drawn. From the points 13 to 16 
 and 14 to 15 draw two diagonal lines, and through their 
 
38 
 
 PERSPECTIVE FOR STUDENTS. 
 
 points of intersection draw a horizontal line between the 
 lines D and E. The line running from the point 1 of the 
 first square, in passing through the bottom and top lines of 
 the second, gives the points corresponding to the points 
 1 and 3 in the first. The line running from the point 9 of 
 the first square, where it intersects the diagonals of the 
 second, gives the points corresponding with the points 6 and 
 7 of the first : in like manner, the line running from the 
 point 10, at its intersections with the diagonal lines of the 
 second square, gives the points corresponding with those 
 marked 5 and 8 in the first ; and the horizontal line passing 
 through the centre of the second square, gives, at its points 
 of contact with the lines D and e, points corresponding with 
 the points 4 and 2 in the first. The whole of the eight 
 points being thus found in the second square, it remains only, 
 as before described, to draw the curve line through them, 
 which will represent the perspective position and form where 
 the base of the second column is to be placed. By continuing 
 in this manner, a third, fourth, or any number of circles may 
 be drawn at their perspective distances : the two given are 
 quite sufficient to illustrate the rule. 
 
 It may here be well to remark, that every circle correctly 
 drawn in perspective forms a perfect ellipse, whether, from 
 the position from which it is viewed, it appear broad or 
 narrow. By those who understand perspective but imper- 
 fectly, this is frequently denied : and their disbelief arises 
 from their mistaking the middle horizontal line for the axis 
 of the ellipse, whereas it simply divides the circle into its 
 perspective halves. If all the lines serving to draw the curve 
 were to be erased, and the curve left ; if its proper axis (a 
 long straight line, that divides it longitudinally into two equal 
 parts,) were to be found, it would show that the curve forms 
 a true ellipse. 
 
 Problem VI. — By the applipation of the same rule as that 
 explained in the foregoing problem, with a little variation 
 in the manner of employing it, an arcade or succession of 
 
I 
 
 if 
 
 4 
 
PERSPECTIVE FOR STUDENTS. 
 
 39 
 
 arches may be put in perspective ; a small geometrical draw- 
 ing (g d) or elevation of which is placed at the side of the 
 problem, and is drawn to a scale of one-fourth of the per- 
 spective drawing. 
 
 Let the student first draw an elevation similar to the one 
 in the plate, of one-fourth the size he intends to make his 
 perspective drawing; and then let him draw the perpendi- 
 cular line a by measurement from it, and at the supposed 
 height of a figure (or his eye from the ground line) draw the 
 horizontal line across his picture. To the best of the judg- 
 ment of the artist, from the point 1 (the top of the line a) let 
 him sketch the perspective inclination of the line B, and con- 
 tinue it till it meet the horizontal line : the point at which it 
 touches will be the vanishing point. From the point 2 (the 
 bottom of the line a) the line c must also be drawn to the 
 vanishing point. These two lines B and c represent the per- 
 spective directions of the upper and lower lines of the struc- 
 ture. The student will find, by reference to the geometrical 
 drawing, that the height of each arch is three-quarters the 
 height of the whole structure ; the width of each arch, one- 
 fourth ; and the width of each pier, between the arches, one- 
 eighth of the height of the stiucture. If the student mark 
 off, on the line a, three-fourths of its length from the point 2, 
 as at the point 3, it will mark the real geometrical height 
 of the arch ; and from this point 3, if he rule a line, D, to the 
 vanishing point, it will determine the height of the respective 
 arches as they recede from him The perspective distances — 
 that is, the perspective width of the piers and arches — may be 
 found on the line B by the sume rule as that employed for 
 finding the position and width of the windows in Problem L* 
 A long horizontal line (e) must be drawn from, and to the 
 
 * They might be found with equal correctness on the line c, from the 
 ground line, by employing the rule given in Problem II. for finding the 
 squares of the chess-board. Two lines are drawn from similar distances 
 on the ground line to those on the line e, to show that the points of inter- 
 section are the same. 
 
40 
 
 PERSPECTIVE FOR STUDENTS. 
 
 left of the point 1, to represent the geometrical line of the top 
 of the structure, and on it must be marked the geometrical 
 width of the several piers and arches ; as from 1 to 4, the 
 geometrical width of the first pier (one-eighth of the line a) ; 
 from 4 to 5, the width of the first arch (one-fourth of the 
 line a). The student should now, if he were sketching from 
 nature, draw lightly with his pencil the first arch by eye, or 
 mark, as at the point 6 (on b), the distance of the nearest 
 side of the arch to the line A ; and through this point rule a 
 line from the point 4 till it meet the horizontal line ; its point 
 of contact will be the point of distance. From the point 5 a 
 line must also be drawn to the point of distance, intersecting 
 the line b at 7. These two points, 6 and 7, corresponding 
 on the perspective line B with the points 4 and 5 on the 
 geometrical line E, give the perspective width of the first 
 pier and arch ; and from each of them a perpendicular line 
 must be drawn till it meet the line c. These two lines cor- 
 respond with the lines a and b in the elevation g d. 
 The student will observe that the arches are all formed of 
 semicircles ; consequently, he will only have to construct 
 semicircles for finding the points for the curve on the geo- 
 metrical line E ; therefore on this line, placing one point of 
 the compasses at 8, from the points 4 to 5 describe a half- 
 circle ; from the points 4 and 5 draw upwards two perpendi- 
 cular lines, and parallel to the line E, so as just to touch the 
 top of the semicircle,* a line meeting the two perpendiculars 
 at the points 9 and 10. The semicircle will thus be enclosed 
 in a half- square. From the point 8 a line must be drawn to 
 each of the corners 9 and 10 ; and through the points where 
 these lines (which represent the upper halves of two diagonal 
 lines) intersect the semicircle, two perpendicular lines must 
 be drawn to touch the line e at 11 and 12 : if the student now 
 draw a perpendicular line from the point 8 till it meet the top 
 of the semicircle, he will perceive, by comparing it with the 
 
 * Straight lines touching a curve in this manner are called tangents. 
 
PERSPECTIVE FOR STUDENTS. 
 
 41 
 
 preceding problem, that lie has drawn the upper half of the 
 geometrical figure there represented for drawing a whole circle. 
 
 The line d being that which regulates the height of the 
 several arches, the points 13 and 14, given by the intersec- 
 tions on it from the perpendicular lines drawn from 6 and 7, 
 represent the perspective position of the points 9 and 10 of 
 the geometrical drawing : the points 4 and 5 must now be 
 found, for which purpose the geometrical height of the half- 
 circle must be set on the line A below the point 3 (respresent- 
 ing the height of the top of the arch), as at 15 ; and from this 
 point rule a line to the vanishing point. Where this line 
 intersects the perpendicular lines at 16 and IT, drawn from 
 the points 6 and 7, are the points corresponding with 4 and 
 5 of the goometrical drawing on e. To find the points 11, 
 8, 12 on e in the perspective drawing, a line must be drawn 
 from each of them to the distance point ; and from their 
 points of intersection on the line b, perpendicular lines must 
 he drawn to the line F at 18, 19, 20, which are the perspec- 
 tive positions of the points 11, 8, and 12. From the point 19 
 to each of the corners 13 and 14 a line must be drawn, which 
 will complete the perspective drawing of the straight lines 
 in the geometrical figure erected on the line e. The curve 
 must be drawn as in the former problem, through the points 
 corresponding with those of the geometrical elevation. 
 
 To continue the line of arches, another space of one-eighth 
 of the line a must be measured off on the line e, for the 
 width of the second pier, and beyond that a space of one- 
 fourth of a, for the width of the second arch, as from 5 to 
 21, and from 21 to 22. From these points, lines must be 
 ruled to the point of distance, and the sides of the second 
 arch (c d of the geometrical drawing) must be drawn on the 
 perspective drawing, in the same manner as the sides a b of 
 the first arch. To find the points between 21 and 22 (23, 
 24, 25), corresponding with the points 11, 8, and 12, between 
 4 and 5, it is not necessary again to construct a geometrical 
 figure, similar to the one for finding the points for the first 
 
42 
 
 PERSPECTIVE FOR STUDENTS. 
 
 arch, because the space between 21 and 22 being exactly the 
 same as that between 4 and 5, these corresponding points 
 must come at precisely the same distance from each other, 
 and may therefore be measured off with a pair of compasses, 
 the points of the geometrical distances for the second arch 
 (on e) corresponding with the points of the first, thus : 
 T2-> ih ih ?V ^he mode of drawing the second and fol- 
 lowing arches in perspective is precisely similar to that em- 
 ployed for drawing the first: in the example given, all the lines 
 necessary for drawing the second arch are introduced, but with- 
 out the references. The three remaining arches are drawn, 
 but the lines used for finding them are purposely omitted. 
 
 There is another mode of applying this rule, equally correct, 
 which it is desirable for the student to understand. The 
 lines A, B, D, E, and F (Fig. 2) must be drawn as in the 
 preceding example (Fig. 1). The lines d and f being drawn, 
 the points 3 and 15 must have been found ; let that portion 
 of the line A between the points 3 and 15 form the left side 
 of a half-square, similar to the side 5, 10, of the one erected 
 on the line e, Fig. 1, and upon this line construct a half- 
 square of the same dimensions as that on E, and describe 
 within it a semicircle. From the centre point draw to the 
 two upper corners lines corresponding with the lines 8, 9, 
 and 8, 10 ; and from the same centre point draw a perpen- 
 dicular line to touch the top of the half-circle, and you will 
 then have a geometrical figure similar to the upper half of 
 the second diagram in the introduction to Part III. 
 
 The variation in applying this rule consists in the mode of 
 finding the points of intersection of the diagonal lines with 
 the curve. In placing the geometrical drawing at the side 
 of the line a, instead of on the line e, it is necessary for 
 finding these points in perspective (marked a and b in Fig. 2), 
 to draw through them a horizontal line to touch the side 
 (as at c), instead of two perpendicular lines to give the points 
 at the bottom of the half-square ; and from this point c, a 
 line (g) must be drawn to the vanishing point. The points 
 
PERSPECTIVE FOR STUDENTS. 
 
 43 
 
 13 and 14, and 16 and 17, are found as in Fig. 1 ; and the 
 student must now find, by drawing a perpendicular line 
 dividing the arch into its perspective halves,* the point cor- 
 responding with the point 19 of Fig. 1, and from it draw a 
 line to each of the points 13 and 14. The line g, where it 
 intersects the diagonal lines of the half-square in perspective, 
 will give the points corresponding to the points a b in the 
 plan at the side, through which to draw the curve, and in its 
 possage towards the vanishing point would give the corre- 
 sponding point for every arch. In the example here given, 
 those lines only are used that are absolutely necessary for 
 the explanation of the rule ; the student will do well to draw 
 the whole figure with the five arches on this plan. 
 
 Problem VII. — The reader has, doubtless, at one period 
 or another, been in some place where he has seen a row of 
 arches straight before him, such as the Burlington Arcade, 
 the side aisle of a church, &c. Let him suppose, then, that 
 he is standing before a row of arches, and in such a position 
 as that the point of sight (in this case, also the vanishing 
 point,) be exactly in the middle, between the two sides of 
 the arch. 
 
 Fig. 1. The student must first draw the elevation of the 
 first archway : this is so extremely simple, that it scarcely 
 needs any directions. Having drawn the external lines, 
 
 6, £?, the ground line, and the horizontal line, draw the 
 sides of the archway a and b, up to the points 1 and 3 (from 
 where the curve springs), and draw a horizontal line between 
 these two points : from the centre of this line, at the point 5, 
 with a pair of compasses describe a semicircle from the points 
 
 * The student, it is to be hoped, recollects that the centre of any 
 rectangular parallelogram in perspective is found by the intersection of 
 its diagonal lines. The figures 6, 7, 13, 14, represent the four corners of 
 a rectangular parallelogram in perspective, as do also the figures 13, 14, 
 16, 17, and 6, 7, 16, 17. If the student, in any one of these, draw two 
 diagonal lines, as from 7 to 13, and from 6 to 14, and through this point 
 of intersection draw a perpendicular line to r, it will give the point (19) 
 required. 
 
44 
 
 PERSPECTIVE FOR STUDENTS. 
 
 1 and 3. From each of the points 1, 2, 3, 4, draw a line to the 
 vanishing point.* The student must now on the line c (drawn 
 from the point 1 to the vanishing point), mark the distance 
 of the second arch from the first, as at 6, and from this point 
 draw a perpendicular line till it touch the line d, at 7. From 
 the line a, through the point 7, draw a horizontal line, till it 
 touch the line b ; and from the point 8, where it intersects the 
 line F, draw a perpendicular line to meet the line e at 9. The 
 points 6, 7, 8, 9, are the points of the second archway, cor- 
 responding with the points 1, 2, 3, 4 of the first. Draw a 
 horizontal line between the points 6 and 9 ; and from the 
 point 5 draw a line to the vanishing point : where this line 
 intersects the line 6 to 9 just drawn, at 10, with a pair of 
 compasses open to the distance of from 10 to 6, or 10 to 9, 
 which are equal, describe another half-circle ; this completes 
 the second archway. 
 
 We will suppose the archways to be equidistant from each 
 other, and that the distance between each is a space equal to 
 from a to b. In order to find the relative distance the third 
 arch appears from the second that the second appears from 
 the first, we must find the point of distance, which may be 
 found in a similar manner to that employed in the Problems 
 II. and Y. From the point 4, passing through the point 7, 
 a line must be drawn till it meet the horizontal line ; the 
 point of contact is the point of distance. To the right hand, 
 from the point 4, two spaces must be measured off on the 
 ground line, each equal to the space between A and b, as at 1 1 
 and 12 ; from each of these points a line must be ruled to the 
 point of distance, and where they intersect the line D they 
 give the points determining the perspective distances of the 
 third from the second arch at 13, and the fourth from the 
 third at 14. The mode of drawing the third archway is 
 similar to that employed for the second, with this difference 
 only, that the perpendicular line must be drawn up from the 
 
 * This point has already been settled to be placed midway between 
 the two sides of the arch. 
 
page 45. 
 
PERSPECTIVE FOR STUDENTS. 
 
 45 
 
 point 13 on d to the line c ; whereas the second arch was 
 commenced by drawing a perpendicular line down from the 
 point 6 on c to the line D. To draw the third archway, 
 raise a perpendicular line from the point 13 till it meet the 
 line c at 15 ; from 15 draw a horizontal line till it meet the 
 line E at 16; from 16 draw down a perpendicular line to 
 meet the line f at 17; draw a horizontal line between 15 and 
 16, and from the point midway between them describe a 
 third semicircle ; this completes the third archway. The 
 fourth is drawn in a similar manner, commencing at the 
 point 14 ; and a fifth, sixth, or indefinite number, may be con- 
 tinued by the same rule. 
 
 If, however, instead of being placed exactly in the middle, 
 between the two sides A B of the archway, the spectator had 
 placed himself a little on one side — as opposite the point 
 marked g on the horizontal line, the arches would have had a 
 very different appearance. Viewed from this position, more 
 of the left side of the inner archways would be visible, 
 and the right side of the first archway would entirely exclude 
 the view of the right side of the inner ones. The appearance 
 of the archways, as seen from this position, would be as 
 represented in Fig. 2. 
 
 In this example there are neither letters nor figures of 
 reference : the mode of drawing it is exactly similar to that 
 employed in Fig. 1, the position of the vanishing point only 
 being altered. The student will observe that the right side 
 of all the inner arches is hidden by the first, and that a por- 
 tion only of their semicircles is seen ; nevertheless, it is best 
 to complete each archway in the drawing, to insure correct- 
 ness ; and this would be more necessary if it were drawn in 
 oblique instead of parallel perspective. A few of the lines 
 are left, to assist the student in his drawing the figure. 
 Fig. 3 we shall have to consider in a more advanced part 
 of the work. 
 
 Problem VIII. — In the introductory portion of this 
 work, it will be recollected that some observations were 
 
4G 
 
 PERSPECTIVE FOR STUDENTS. 
 
 made respecting circular objects ; and the various forms tbey 
 took, according to the positions from which they were seen, 
 was familiarly explained by the example of a common bowl. 
 It has already been remarked, that all perspective represen- 
 tations of circles form perfect ellipses ; * but the width of 
 these ellipses varies according to the distance the circular 
 form is placed above or below the eye of the spectator. To 
 illustrate this, a column, composed of five distinct pieces, is 
 chosen, and the spectator is supposed to view it from such a 
 position as to bring the vanishing point (f) exactly in the 
 centre of the column, and midway between the top and bot- 
 tom. The student will understand that the joints of a circu- 
 lar column are circular, like the top and bottom, and that the 
 forms of the curves of the joints vary in appearance accord- 
 ing to their distance above or below the eye of the spectator. 
 The student will observe that the curve or ellipse a, the base 
 of the column, from being at the greatest .distance below the 
 eye, is much broader than the curve B, representing the first 
 joint ; that the second joint of the column c, from its being 
 exactly level with the eye, forms only a straight line ; for if 
 the student imagine, that instead of this joint he had before 
 him a thin circular plate, from its being exactly level with 
 his eye, it would be impossible to see either its upper or 
 lower surface, and consequently could only be represented 
 by a straight line. The curve D, from being the same 
 height above the eye that the curve b is below it, forms 
 a precisely similar ellipse ; and, for the same reason, the 
 ellipses a and e are also similar : this the student will readily 
 perceive by turning the example upside down. The curves 
 d and - e, from being above the eye, show the upper half of 
 the curve; the curves B and A, from being below the eye, 
 show the lower half. 
 
 In order that the student may fully comprehend the draw- 
 
 * The only exception is where the curve comes, as in the joint c of 
 this figure, exactly on a level with the eye of the spectator, in which case 
 st is represented by a straight line. 
 
PERSPECTIVE FOR STUDENTS. 
 
 4? 
 
 ing of this problem, let him cut five square pieces of card, 
 and draw on the face of each of them a figure similar to the 
 diagram represented in the plate ; and through each card, at 
 the points tf, 5, c, and 1, 2, 3, 4, 5 } 6, 7, 8, pierce a hole, 
 as also through the point at the centre. This done, let him 
 place the cards one over another, so that the various points of 
 each card shall lie under and over the corresponding points 
 of the others. In this position of the cards, let him put a 
 piece of stick or wire, the length of the column, through the 
 centre holes, and separate the cards on this stick to an equal 
 distance one from another, always keeping the corresponding 
 points on the cards in the same relative position. If the 
 student were now to put a straight piece of wire (a common 
 knitting-needle will answer the purpose) through any one of 
 the points of the top cards, and holding it perpendicular, 
 were to push it downwards, it would pass through the 
 corresponding holes in the four cards underneath. By 
 clearly understanding this, the student will find his progress 
 through the problem greatly facilitated. 
 
 The student must first draw the square, and find the points 
 in perspective, corresponding to the points in the diagram, 
 for drawing the curve of the base of the column. This is 
 fully explained in Problem V., with this slight variation, 
 that the vanishing point in this drawing is placed in the 
 middle of the object to be represented, instead of at the side : 
 this done, he must letter and figure the several points to 
 correspond with the letters and figures on the points in the 
 diagram, as the points a, c, 6?, 1, 2, 3, 4, 5, 6, 7, 8. 
 
 From the points d and c draw two perpendicular lines, g 
 and h, each four times the height of the side of the square 
 of the diagram, marking on each line the divisions of each 
 fourth, as at the several points marked c and d* and draw 
 
 * The student must understand that the same figures and small 
 letters are employed as references for the corresponding points in each 
 of the squares, and that the several squares, with their first line and the 
 
48 
 
 PERSPECTIVE FOR STUDENTS. 
 
 a horizontal line between the lines g and H, from each point, 
 c to up to the toi). The first three of these horizontal 
 lines, b, c, and d, represent the first line of the three squares 
 in which the curves to represent the joints are to be drawn : 
 the fourth line, E, represents the first line of the square in 
 which the curve to represent the top of the column is to be 
 drawn. From each of the points c d, the extremities of the 
 lines b, d, e, a line must be drawn to the vanishing point F, 
 as the lines J, K, l, m, n, o.* 
 
 By reference to the cards, in the position before described, 
 the student will perceive that the circles drawn on them are 
 in a similar position to that of the top and bottom, and the 
 three joints of the column ; and that a perpendicular line 
 passing through any one of the points, representing certain 
 points in the circle on the top card, would pass through the 
 corresponding points in the cards beneath. So is it in the 
 perspective drawing ; the corresponding points in the several 
 perspective squares lie exactly one over the other, and will 
 be found by means of perpendicular lines. 
 
 From the point a of the first square a, a perpendicular 
 line P must be drawn till it meet the line J of the square e 
 at a ; and from the point b of the square A, a perpendicular 
 line r must be drawn till it meet the line k of the square e at 
 b ; by drawing a horizontal line between these two points 
 a and b (square e), the square E will be completed. The 
 student must here understand that the several lines J, L, and 
 N, drawn from the points d to the vanishing point, represent 
 the perspective directions in each of the squares B, D, and 
 e, of the line d — a of the diagram ; and that the line P is 
 the perspective representation of the wire passing through 
 
 curve drawn within them, are alike designated by their respective letters, 
 a, b, c, D, E. 
 
 * The middle joint c being represented by a straight line, no points 
 can be required on it. It will only be necessary to determine its per- 
 spective width, which will be done in determining the width of the oth«r 
 curves. 
 
PERSPECTIVE FOR STUDENTS. 
 
 49 
 
 the points a of the cards : consequently, where the line p 
 intersects the lines n, l, and J, it gives the perspective posi- 
 tions of the points a in the several squares b, d, and e. In 
 like manner the lines K, m, o, running from the point c to 
 the vanishing point, represent the perspective directions in 
 each of the squares of the line c — b of the diagram ; and the 
 line R is the perspective representation of the wire passing 
 through the several points b of the cards : consequently, the 
 points at which this line intersects the lines K, m, and o, are 
 the perspective positions of the several points b in the respec- 
 tive squares b, d, and e. The student must now draw a 
 horizontal line from the points a to 6, in each of the squares 
 B and D ; and then, in each of the three squares b, d, e, he 
 must draw the diagonal lines from the points d to and from 
 a to c. 
 
 From the point 7 (square a), the point designating the 
 extremity of the perspective circle to the left, a perpendicular 
 line s must be drawn till it meet the line J (square e) ; and 
 from the point 3, the extremity of the perspective circle on 
 the right, a perpendicular line t must be drawn to meet the 
 line K (square e). These lines s and t, represent the two 
 sides of the column, and are the perspective delineation of the 
 wires passing through the holes 7 and 3 of the cards : conse- 
 quently, where the line s intersects the lines representing in 
 perspective the line d — a of the diagram (those portions of the 
 lines J, L, and n, between the letters d and a), it gives the 
 perspective positions of the points 7 in the respective squares 
 B, D, and e. In like manner, where the line t intersects 
 those portions of the lines k, m, o, between the letters c and b 
 (representing the line of the diagram c — b in perspective), it 
 gives the perspective positions of the point 3 in the respective 
 squares B, D, and E. 
 
 The student must now, from the point 5 of the square A, 
 draw a perpendicular line u, till it touch the top line of the 
 square e ; and this line, from the spectator being placed 
 exactly in the centre of the column, will answer for a per- 
 
 . Perspective* D 
 
50 
 
 PERSPECTIVE FOR STUDENTS. 
 
 pendicular line that should be drawn from the point 1, square 
 A, till it touch the further line of the square E. This line u, 
 then, represents in perspective the wires passing through the 
 holes 5 and 1 of the cards, and designates at its intersections 
 on the first line of each square the point corresponding with 
 the point 5 of the diagram ; and at its intersections with the 
 further line of each square it designates the positions of 
 the points corresponding with the points 1 of the diagram. 
 The student will perceive that he has now, in each of 
 the perspective squares, points for drawing the curves cor- 
 responding with the points 1, 3, 5, ? of the diagram ; he 
 must now find the remaining points, 2, 4, 6, 8, in each of the 
 squares. 
 
 From the point 6 on the diagonal line d — 5, square A, a 
 perpendicular line v must be drawn, till it touch the diago- 
 nal line d — 5, square e. This line v represents in perspec- 
 tive the wire passing through the holes 6 of the five cards ; 
 and where it intersects the several diagonal lines d — in 
 the respective squares b, d, e, will be a point corresponding 
 to the point 6 in the diagram. In like manner, from the point 
 4 on the diagonal line c — square A, a perpendicular line x 
 must be raised, till it meet the diagonal line c — «, square e. 
 This line x representing the wire passing through the points 
 4 of the cards, will give, at its intersections on each of the 
 diagonal lines c — a point corresponding with the point 4 
 on the diagram. Similar to the preceding perpendicular 
 lines v and x, the two others, Y and z, must be drawn from 
 the points 8 and 2 on the diagonal lines c — a and d — 
 square A, to meet the corresponding diagonal lines c — a 
 and d — 5, square e ; these will give the perspective repre- 
 sentations of the wires passing through the holes 8 and 2 of 
 the five cards. Where the line Y intersects the diagonal lines 
 d — b of the several squares, will be found the points corre- 
 sponding with the point 2 of the diagram ; and where the 
 line z intersects the diagonal lines c — a of the several squares, 
 will be the points corresponding with the point 8 of the 
 
PERSPECTIVE FOR STUDENTS. 51 
 
 diagram. The whole of the perspective positions of the points 
 of the diagram being found in each perspective square, the 
 student, as in Prob. V., must draw the curve in each, through 
 the several points 1, 2, 3, 4, 5, 6, 7, 8, which will complete 
 the perspective drawing. 
 
 This problem, though not difficult, requires attention, 
 from its being a little more intricate than any preceding ; 
 but, with the assistance of the cards, the author thinks it 
 barely possible for the intelligent student to fail understand- 
 ing it. 
 
 Problem IX. — In sketching from nature, the artist has 
 frequently to represent, and in old buildings especially, doors, 
 windows, arches, &c. of a pointed form, commonly called 
 gothic arches, gothic windows, &c. The curves of these 
 pointed arches are formed in geometrical drawing by the 
 intersections of segments of circles, Fig. 
 21. The pointed arch here drawn is Fig. 21. 
 
 formed by the intersection of two semi- 
 circles ; and by the rules already explained, 
 the student would be able easily to put it 
 in perspective ; as by putting the two semicircles in perspec- 
 tive, their intersection would give the pointed arch, which 
 would be in perspective also. But these pointed arches vary 
 so much in their proportions, some being extremely obtuse, 
 whilst others are very pointed, that it is better to give a 
 general rule by which pointed arches of any form may be 
 put in perspective. At the side of the problem, an elevation 
 of the building to be put in perspective is drawn, represent- 
 ing a gothic arched doorway, with a gothic window on each 
 side ; and, like the elevation in Problem VI., it is drawn to 
 the scale of one-fourth of the perspective drawing : the 
 student will therefore bear in mind, that in making for his 
 perspective drawing geometrical measurements, they are 
 understood to mean four times the size of that given, and 
 marked g d. 
 
 The whole of this figure, with the exception of the curves 
 
 D 2 
 
52 
 
 PERSPECTIVE FOR STUDENTS. 
 
 of the doorway and windows, must be drawn in a similar 
 manner to Prob. VI, The line A must first be drawn ; and 
 across the picture, at the height of the spectators eye, the 
 horizontal line ; then the line B, the perspective top line of 
 the building, to the horizontal line, to fix the vanishing point ; 
 and from the bottom of the line A, a line c to the van- 
 ishing point, for the perspective base-line of the building. 
 The point 3 on a, the geometrical height of the pointed top 
 of the windows, must next be marked, and from it the line 
 D drawn to the vanishing point, to regulate their perspective 
 height. The line e must next be drawn, and on it the geo- 
 metrical widths of the doorway, windows, and spaces be- 
 tween must be marked, as at a, 6, c, 0,/, g ; a distance- 
 point must next be found, to get the perspective positions of 
 these points on the line b.* The distance-point having been 
 found, and the perspective positions of the points a, c, d, 
 0, /, g marked on the line b, draw from each of these points 
 a perpendicular line till it touch the line c. 
 
 On the line e, between the points a and 5, construct a 
 geometrical figure of the arch of the first window. This is 
 done by fixing one point of the compasses at the point a, 
 and opening them till the other point touches the point b; 
 describe upward the segment of a circle, and then changing 
 sides, and fixing the point of the compasses (at the same 
 extension), at the point 6, describe another segment of a 
 circle from the point a, till it join the one before drawn : this 
 will give the geometrical drawing of the arch. From the 
 points a and b draw up two perpendicular lines, and, just 
 touching the point of intersection of the arch, draw a hori- 
 zontal line to meet them at the points 4 and 5. From the 
 
 * Throughout this problem the student must constantly refer to 
 Prob. VI. Although the figures and letters vary, in this, from those used 
 in the sixth problem, still the principle is the same; and it would occupy 
 too much space to go over the same ground in every problem : moreover, 
 by changing the references, and giving more general explanations, it will 
 oblige the young student to exert his faculties. 
 
PERSPECTIVE FOR STUDENTS. 
 
 53 
 
 point of the arch draw down a perpendicular line to touch 
 the line e at 6, and from the point 6 draw two lines to the 
 corners 4 and 5. Through each of the points 7 and 8, where 
 these lines intersect the curve lines, draw a perpendicular 
 line to the line e, at 9 and 10, and a geometrical figure will 
 be constructed, containing points that, put in perspective, 
 will be a guide for drawing the curves. The perspective 
 positions of the points 9, 6, 10 (on e) must be found on the 
 line B, in a similar manner to the points 11, 8, 12, in Prob. 
 VI. The line d, drawn from the point 3, regulates the per- 
 spective height of the points of the arches ; and to find the 
 perspective positions of the points from which the curves 
 commence, their geometrical height must be marked on the 
 line a, as at the point 11 ; and from this a line f must be 
 drawn to the vanishing point : this line, where it intersects 
 the perpendicular lines drawn from the points a and b (line 
 b), gives the points corresponding to the points a and b of 
 the geometrical drawing ; and where the line d intersects the 
 same lines (drawn from a and 5, line b), it gives the points 
 corresponding to the points 4 and 5 of the geometrical draw- 
 ing. The perspective points 9, 6, 10, on the line b, must 
 now be carried down by perpendicular lines to the line F, 
 and from the point 6 (on f) two lines must be drawn to the 
 corners 4 and 5 : this completes the perspective drawing of 
 the straight lines of the geometrical figure, and through the 
 corresponding points the curve must be drawn. The point 
 12 (the height of the bottom of the windows) must now be 
 marked on a, and from it a line g drawn to the vanishing 
 point : where this line o passes between the two perpendi- 
 cular lines drawn from a and b (line b), representing the 
 sides of the window, it gives the perspective line of the 
 bottom of the window. The farther window is drawn in a 
 precisely similar manner to the nearer one, finding the geo- 
 metrical points on the line E, between e and /. All the lines 
 necessary for drawing this second window are shown, but 
 without references. 
 
54 
 
 PERSPECTIVE FOR STUDENTS. 
 
 The mode of drawing the doorway is similar to that used 
 for representing the windows. The point 13, the geometrical 
 height of the point of the arch, must be marked on a, and a 
 line H drawn from it to the vanishing point, to fix its per- 
 spective height, as the line D does that of the windows : 
 and the point 14, the geometrical height of the points 
 from which the curve lines commence, must also be marked 
 on the line A, and from it a line J drawn to the vanish- 
 ing point, to determine their perspective height, as the line 
 f does those for the points where the curves of the windows 
 commence. 
 
 The geometrical form of the arch of the doorway must 
 now be drawn on the line E, between c and <f, in a precisely 
 similar manner to that of the window between a and 5, and 
 the corresponding points found on the perspective drawing, 
 between the lines h and J, as the points for the arch of the 
 window were found between the lines d and f. 
 
 The student must understand that these pointed arches, 
 like the semicircular ones in Fig. 2, Prob. YL, might be 
 drawn quite as correctly in perspective, by making the 
 geometrical drawing of the arch at the side of the line a, 
 between the points 3 and 1 1 ; observing, that the points 7 
 and 8 (like the points a Fig. 2, Prob. VI.) must be found 
 by a horizontal line passing through them, instead of two 
 perpendicular lines. The point that would thus be found, is 
 represented at the point 15 on the line A ; and from it a line 
 is drawn to the vanishing point, to show the student that the 
 intersections through the points 7 and 8 are similar to those 
 given by the intersections of the perpendicular lines. 
 
 The points here given are usually found sufficient for 
 drawing the curves of pointed arches ; but where great 
 nicety is required, or the arches are very long and pointed, 
 additional points of intersection may be chosen. Fig. 2 
 represents a narrow pointed or lanciform arch, in which 
 the points 1, 2, 3, 4, and 5 represent the points found in 
 the foregoing example ; but other lines are here drawn, by 
 
PERSPECTIVE FOR STUDENTS. 
 
 55 
 
 giving intersecting points* on the curve lines, as 6, 7, 8 5 9. 
 These additional lines the student would find no difficulty in 
 putting in perspective. 
 
 CHAPTER IV. 
 
 Having led the student by degrees to represent in per 
 spective various superficial forms, and we trust in a manner 
 sufficiently explicit to enable him to clearly comprehend 
 what he has gone through, we must now proceed to the 
 representation of solids — or, in other words, to draw the 
 objects with their thicknesses. It was observed at p. 13, 
 in reference to the first object put in perspective, the door, 
 that its edge or thickness would have been seen, but that it 
 was purposely omitted, to avoid embarrassing the student, 
 at the commencement of his studies, with a complicity of 
 lines. In Prob. I. (Chap. II.), four windows are drawn on 
 the side of the house ; but they represent only the superficial 
 form, or outer line, of the space to contain the window. We 
 will therefore proceed, with this example, to point out the 
 mode by which the width of the recesses of the windows is 
 found ; and, to make the reference more clear, the figure is 
 represented on a larger scale. The lines are drawn in the 
 direction of the vanishing points ; but from our limited space 
 the points are out of the picture. 
 
 Problem X. — The student must now take his drawing 
 of the first problem ; and if he has followed the directions 
 given in a note, in drawing it, he will have put in ink all 
 the lines representing the object to be drawn. If this is 
 done, he must rub out all the pencil lines, letters^ and figures, 
 
 * Horizontal lines, for getting the points of intersection, are here 
 employed, as, from the narrow form of the arch, perpendicular lines would 
 be more confused, especially in a perspective representation. 
 
56 
 
 PERSPECTIVE FOR STUDENTS. 
 
 excepting the horizontal line, the vanishing points, and the 
 point of distance. To proceed : — 
 
 First, on the left side of the line 1 of the upper and nearer 
 window, take a point to mark the width of the recess by 
 eye,* and through this point draw a perpendicular line 2 
 from the top of the upper window to the bottom of the lower 
 one. The upper window being considerably above the eye 
 of the spectator, would cause the width of the recess to be 
 seen below the top line, 3. To find this perspective width, 
 a line must be drawn from the point 4 to the vanishing point 
 G, to intersect the line 2 at 5 ; and through the point 5, in 
 the direction from the vanishing point h, a line A must be 
 drawn to meet the near line 6 of the window, at the point 7. 
 This represents the width of the recess for the window at 
 the top and right side, where only it could be visible to the 
 spectator. 
 
 In the first lower window, a line must also be drawn from 
 the upper right-hand corner to the vanishing point G ; and 
 through the point where it intersects the line 2, a line must 
 be drawn in the direction from the vanishing point H, to 
 meet the line forming the left side of the window : this gives 
 the perspective width of the recess at the top. From the 
 upper part of this window being so much nearer to the 
 horizontal line than that of the window above, the student 
 will perceive how much narrower the recess appears. The 
 bottom line of the lower window being below the eye, the 
 width of the recess will be seen below, as well as above : from 
 the lower right-hand corner of the under window, therefore, 
 a line must also be drawn to the vanishing point G ; and 
 through its point of intersection with the line 2, a line must 
 be drawn from the vanishing point H, to meet the left-hand 
 line of the window ; which will complete the representation 
 
 * It would be easy to find a line for getting the perspective width of 
 the recesses corresponding with the line by which the width of the 
 outer line of the windows was found ; but the mode here given is equally 
 correct, and more simple. 
 
PERSPECTIVE FOR STUDENTS. 
 
 57 
 
 of the thicknesses of the top, bottom, and side of the recess 
 of the first lower window. The student will observe, that 
 the upper and lower line of the under window being nearly 
 equidistant from the horizontal line, the perspective width of 
 the recess is nearly equal, above and below. 
 
 To find the perspective width of the recess of the farther 
 upper window, the application of the rule is changed. In 
 the first window, the width of the top of the recess is found 
 from the line giving the width of the side ; in the farther 
 window, the width of the side is found bv the line desig- 
 nating the width of the top. 
 
 The line A being continued through the farther window, 
 denoting the perspective width of the recess above, a line 
 must be drawn from the upper right-hand corner to the 
 vanishing point G, and, from its point of intersection with 
 the line A, a perpendicular line must be drawn down to the 
 bottom line, which should be continued down through the 
 lower window. This perpendicular line gives the width of 
 the recess at the side, and the line A gives it at the top. 
 The width of the top, bottom, and side of the recess of 
 the second lower window is found as in the first lower 
 window. 
 
 The rule here given for finding the width of these 
 recesses, is applicable to all other objects represented by 
 straight lines. Suppose the object required to be drawn in 
 perspective was a church tower with battlements at the top, 
 the perspective width of the battlements, and their distances 
 one from another, would be found in a similar manner to 
 the position and width of the windows in Problem I., from 
 the lines R to D, and the width or thickness of them by the 
 rule just given. Also the perspective width of doorways, 
 &c. with their thicknesses, may be found in the same 
 manner. 
 
 Let us suppose that each of these windows contains nine 
 panes of glass : they would, in addition to what is already 
 drawn, require two perpendicular and two horizontal lines 
 
 D 3 
 
58 
 
 PERSPECTIVE FOR STUDENTS. 
 
 to complete the frame-work (Fig. 22), which would be' 
 Fig. 22. within the inner lines of the recess of the window. 
 
 To find the perspective position of these lines, 
 
 it will be necessary to have all the four inner 
 
 LJ - J - J lines of the recess drawn in perspective: we must 
 therefore continue our drawing as if the building were trans- 
 parent. From each corner of the window a line must be 
 drawn to the vanishing point G (that from the corner 4 1. 
 already been drawn, to find the point 5 and line a) ; the 
 line A must now be continued upwards till it meet the line B 
 at 8 ; from the point 8 a perpendicular line must be drawn 
 till it meet the line c at 9 ; from the point 9 draw a line to 
 the vanishing point h, which will intersect the line D at 10, 
 the same point at which the line D intersects the line 2. The 
 lines between the points 8 and 9, 9 and 10, 10 and 5, and 5 
 and 8, represent in perspective the inner parallelogram of 
 the recess of the window, in which the framework is to be 
 drawn. 
 
 To avoid a confusion of lines and references, this figure 
 of the window is drawn at the side, and marked Fig. 2. 
 It is drawn of the same size, and the points are in the same 
 relative position. 
 
 The most simple and quickest mode of putting the lines 
 of this framework in perspective, is, first, to draw a hori- 
 zontal line to the right of the point 8 (the top of the nearest 
 line of the figure), and from the distance-point, through the 
 point 5, draw a line to intersect it at 11. The space between 
 8 and 11 here represents the geometrical length of the line 
 8 — 5. Divide the line 8 — 11 into three equal parts (to 
 represent the width of the three panes of glass), and get the 
 perspective positions of their points of division on the line 
 8 — 5, from each of which points draw a perpendicular line 
 to the line 9 — 10. These will give the perspective positions 
 of the perpendicular lines of the window-frame. The line 
 8 — 9 must now also be divided into three equal parts (to 
 give the height of the panes of glass) ; and from each point 
 
PERSPECTIVE FOR STUDENTS. 
 
 59 
 
 of division a line must be drawn to the vanishing point h. 
 These lines, where they pass between the lines 8 — 9 and 
 5 — 10, represent in perspective the horizontal lines of the 
 framework of the window. 
 
 The student will observe, that a portion both of the hori- 
 zontal and transverse lines of the framework would not 
 be visible. The dark lines represent the parts that are 
 seen. It is most essential to attend to this rule, as any 
 inaccuracy in representing windows of a similar form is 
 most offensive to the educated eye. Any number of lines 
 forming the framework of a window may be drawn by the 
 same rule ; and it would be quite as easy, though a little 
 more complicated, to make the lines representing the frame- 
 work double ; the object here, however, is solely to exemplify 
 the rules sufficiently to enable the student to draw other 
 similar and more intricate forms by them. 
 
 The framework of the other windows may be drawn in 
 the same manner, but, to save time, the perpendicular lines 
 of tbe upper windows may be drawn through to the lower 
 ones. 
 
 Problem XI. — Of the double arch, here given, the front 
 one is drawn in the same manner as the first arch in the plate, 
 Prob. VI. It varies in its proportion, but the mode of 
 drawing it is the same. 
 
 We will endeavour so to figure and describe the representa- 
 tion here given, as to enable the student to draw it with the 
 least possible difficulty. He must first imagine the object 
 he is representing to be transparent; most perspective 
 drawings are made as if such were the case ; and he must 
 understand, that in order to get the lines that represent the 
 thickness or width of the object, the whole perspective draw- 
 ing of the front is repeated a little way behind it, at its 
 perspective distance. To render this problem perfectly intel- 
 ligible, the lines of the front of the structure will have their 
 references in capital letters ; and the corresponding lines in 
 the drawing behind it in small letters. The references by 
 
60 
 
 PERSPECTIVE FOR STUDENTS. 
 
 figures will be the same in the corresponding points of each 
 archway. 
 
 The mode of drawing the front face of the structure, it is 
 quite superfluous to go over ; the student can draw it from 
 the description given for drawing the first archway in Prob. 
 VI. We will therefore consider the whole of the front or 
 face of the structure as drawn and lettered, and proceed at 
 once to the farther one, representing the thickness eft* width 
 of the arch and side. From the points 1 and 2, the top and 
 bottom of the line a, draw two lines to the vanishing point 
 H, and between them draw the perpendicular line a, to deter- 
 mine the width of the structure. It is here made very 
 narrow, for the purpose of showing a greater portion of the 
 inner curve. From the points 1 and 2 of the line a, draw 
 two lines, b and £, to the vanishing point J ; and from the 
 points 21 and 22, the upper and lower points of the line g, 
 draw two lines to the vanishing point H ; from the point 21, 
 where the upper line intersects the line 5, draw a perpendi- 
 cular line g, to meet the point 22, where the lower line 
 intersects the line c. This will complete the external lines 
 of the second archway ; the lines a, g, c of the second cor- 
 responding with the lines a, b, g, c of the first archway. 
 
 To find the line determining the top of the second arch, 
 draw a line from the point 3 on A to the vanishing point H ; 
 and from its point of intersection on the line a at 3, draw a 
 line to the vanishing point J, which is the line d required. 
 To find the line determining the height of the points from 
 which the second arch rises (4 and 5 of the geometrical 
 drawing), from the point 15 on A draw a line to the vanish- 
 ing point h; and from its point of intersection on a at 15, 
 draw a line to the vanishing point J, which is the line / 
 required; the line d of the second archway corresponding with 
 the line D of the first, and the line / of the second corre- 
 sponding with the line F of the first. 
 
 From the points 13 and 14 on the line d, draw two lines 
 to the vanishing point h; and from their points of inter- 
 
PERSPECTIVE FOR STUDENTS. 
 
 61 
 
 section at 13 and 14 on the line J, draw two perpendicular 
 lines to meet the line c : these lines represent the two sides 
 of the arch ; and where they intersect the line / will be the 
 points 16 and 17, corresponding with the points 16 and 17 
 of the line f. The corresponding points to 13 and 14 on 
 D, and 16 and 17 on f, of the first archway, being found on 
 the lines d and / of the second, there remain only the points 
 18, 19, 20 on f of the first, to find on / of the second. 
 The student will recollect that these points were found in the 
 first archway by getting the perspective distances on the 
 line b of the points 11, 8, 12 of the elevation on e, and from 
 these points on b drawing perpendicular lines that meet the 
 line F at 18, 19, and 20. Now, from each of the points 
 where these lines intersect the line D, draw a line to the 
 vanishing point h, and from each of the points where these 
 lines drawn to h intersect the line d, draw a perpendicular 
 line to meet the line / at the points 18, 19, 20. From the 
 point 19 / draw two lines upwards, one to the point 13 on 
 
 the other to the point 14 on d ; and the straight lines of 
 the elevation on the line E are represented in perspective 
 between the lines / and d of the second archway, corre- 
 sponding with those of the first between f and d. The second 
 curve must be drawn through the points corresponding with 
 those of the first. The student must here understand, that 
 the line b corresponding with the line B, the line d with d, 
 and /with F, the perspective positions of the points 11, 8, 12 
 of the elevations might with equal correctness have been 
 transferred from either of the three lines of the first (b, d, or 
 f), to its corresponding line of the second d y or /), by 
 ruling from each point on either line to the vanishing point 
 H, to give the intersections on the corresponding line : the 
 lines d d were chosen as the most convenient, for, as they 
 form the upper line of the figure for constructing the arch, 
 the taking them obviates any unnecessary length of line, 
 which is always desirable. 
 
 It would be quite superfluous to give a plate representing 
 
62 
 
 PERSPECTIVE FOR STUDENTS. 
 
 the gothic arch and windows, with their thickness or width, 
 as the mode of drawing them is line for line with the 
 example just given ; but the student is strongly recom- 
 mended to draw them, and, for the sake of practice, to 
 draw another double structure of them, some little distance 
 behind the first — say, at the distance of the point 1 from the 
 point d on the line E, Prob. IX. He will find this extremely 
 simple, inasmuch as the points (refer here to the problem 
 just gone over) for the perspective distances of the points 
 4, 11, 8, 12, 5, and 9 and 10, are already found. We will 
 explain this by continuing the problem before us. Continue 
 the line c (the base-line of the farther side of the first struc- 
 ture) to the ground line, at the point 21 ; the space between 
 21 and the bottom of the line a (2) will represent the 
 geometrical width of the structure; the vanishing point J 
 serving as a point of distance for the objects on the right- 
 hand side. From the point 21 measure off on the ground 
 line, to the right, a space the length of the line E, to repre- 
 sent the geometrical distance of one structure from the other 
 (any other distance might be chosen) ; and farther to the 
 right, measure off the space between the points 2, the bottom 
 of the line A, and 21, to represent the width of the second 
 structure. From each of the points of division, rule a line 
 to the vanishing point J (as a point of distance), and from 
 the intersections of these lines with that drawn from the 
 lower point (2) of the line A to the vanishing point H, draw 
 two perpendicular lines to meet the line drawn from the top 
 of the line A to the vanishing point H. This will give four 
 points in the second double archway, corresponding with the 
 points A 1, a 1, and a 2, a 2, and represents the sides of the 
 second archway. The corresponding lines of this archway, 
 as far as we go, are lettered and figured the same as in the 
 first, to make them perfectly intelligible. The line already 
 drawn from the point 8 on A of the first structure, gives the 
 points 3 — 3 on the lines A a of the second ; and from these 
 points the lines D d are ruled to the vanishing point J. The 
 
PERSPECTIVE FOR STUDENTS. 
 
 63 
 
 line ruled from the point 13 on the line d of the first struc- 
 ture, to get the point 13 on d of the same, gives the points 
 13 — 13 on the lines d — d of the second ; and all the lines 
 drawn from the points on D, to find the corresponding points 
 on d of the first, in their passage to the vanishing point H, 
 give the corresponding points on the lines d d of the second 
 structure, and would to any number it were necessary to 
 draw. The line drawn from 15 A of the first, gives in like 
 manner the points 15 a and 15 a on the second ; and the 
 lines F f are ruled from these points to the vanishing point J. 
 The above is quite sufficient to enable the student to complete 
 the drawing of the second double archway ; and the gothic 
 archway and windows may be represented in precisely the 
 same manner. 
 
 The student must now refer back to Plate VIII., Prob. 
 VII. Fig. 3, here given, is a fac-simile of Fig. 2, with the 
 width of the arches added. Nothing can be more simple 
 than to make this addition, as it is merely repeating what 
 has been done before. The second figure is chosen on 
 account of its showing more of the thickness of the arches . 
 but the mode of adding the width to the arches in either of 
 the figures is the same. The student had better, perhaps, 
 commence with Fig. 1, on account of the references. In 
 this case, from his being situated exactly midway between the 
 two sides of the arch, he must necessarily see the thickness 
 on both sides. The width of each arch is supposed to be one- 
 third of the space between the two sides : take, then, one- 
 third of the distance from a to B, and put it on the ground 
 line to the right of the point 2, as at d, and from it rule a 
 line to the point of distance ; from the point where this line 
 intersects the line d draw up a perpendicular line till it meet 
 the line c ; from the point where the perpendicular line 
 touches c draw a horizontal line to the line e, and from that 
 point again draw a perpendicular line to the line f. From 
 the middle point of the horizontal line drawn from c to e 
 (given by the intersection of the line drawn from the point 
 
64 
 
 PERSPECTIVE FOR STUDENTS. 
 
 5 to the vanishing point) describe a semicircle from its two 
 extremities, and the drawing of the thickness of the first 
 archway will be completed. The student will doubtless per- 
 ceive that the mode here pursued for representing the width 
 of the archways, is precisely the same as that before de- 
 scribed for drawing the second archway. To get the thick- 
 ness of the second archway, set the geometrical width (one- 
 third of the distance from a to b) on the ground line, to the 
 right of the point 4, find its perspective width on the line d, 
 and proceed, as with the first, and so on, from the geometrical 
 width beyond the points 11 and 12, for the third and fourth 
 archways. The widths of the archways in Fig. 3 were drawn 
 in the same way. 
 
 "We have, in Problem IV., pointed out the utility of 
 diagonal lines for finding the perspective centres for gables, 
 &c, as also for determining certain perspective distances. 
 There is another and most essential use of this rule ; which 
 is, the finding the points of spires, turrets, &c, whether their 
 basis be of a square, circular, or other form. If a pyramidal 
 figure, the base of which is a square (Fig. 1, Plate XIII.), 
 the square must first be put in perspective, its centre found 
 by diagonal lines, and from the centre a perpendicular line 
 D drawn up to the height of the top of the pyramid e ; and 
 to this point, from each of the angles a, b, c, a line must be 
 drawn, which will give the true representation of a pyramid 
 m perspective. Where this rule is applied to the drawing of 
 spires, from the height of the tower, the sides of the square 
 generally incline downwards to the vanishing point ; but it is 
 immaterial whether the perpendicular line D, to find the point 
 of the spire, be drawn up from the centre of the upper or 
 lower square of the tower. In Fig. 2, a drawing is given of 
 the outer lines of a tower with a square spire, and the per- 
 spective squares are drawn both at the top and bottom. The 
 student will perceive from it, that the perpendicular line D, to 
 find the point of the spire, drawn from the centre of the lower 
 square, passes directly through the centre of the upper one. 
 
PERSPECTIVE FOR STUDENTS. 
 
 65 
 
 In representing conical forms, the same rule is to be fol- 
 lowed. Fig. 3 represents a cone in perspective. The per- 
 spective circle being drawn as described in the preceding 
 problems, and the points for drawing the curve being found 
 by means of a square, the perspective centre of the square 
 will also be the perspective centre of the circle, from which 
 a perpendicular line must be drawn up for the point of the 
 cone ; and from each of the lateral extremities of the circle a 
 line must be drawn to it, which will give the perspective 
 appearance of a cone. 
 
 Fig. 4 is a geometrical figure of eight sides, called an 
 octagon, with the mode of constructing it. It is a similar 
 figure to that given in Problem V. for constructing a circle ; 
 and if from point to point, through which the circle is 
 drawn, straight lines are ruled, it will produce a regular 
 eight-sided figure, or octagon, of which all the sides and all 
 the angles are equal. By putting in perspective the same 
 figure of straight lines as that given for describing a circle 
 in Problem V., these eight points will be in their perspec- 
 tive positions, and eight lines ruled from one point to another 
 will give the octagon in perspective : a perpendicular line 
 must then be drawn up from the centre of the square (which 
 is also the centre of the octagon), and lines drawn to it from 
 the angles of the octagon. Fig. 5 is the representation of 
 an octangular pyramid : in the position from which this is 
 seen, four sides of the octagon are visible to the spectator: 
 but it is very commonly the case, that only three sides of an 
 octagonal tower or pyramid are visible. This depends en- 
 tirely on the position in which the spectator places himself to 
 view it. 
 
 Figs. 6 and 7. It frequently occurs that spires of churches 
 do not commence from quite the top of a tower, and that the 
 base of the spire is less than the square of the tower. In 
 going to the summit of a church tower, it is very common 
 to find, between the base of the spire and the battlements, a 
 sort of terrace, to walk round, of from three to four feet 
 
66 
 
 PERSPECTIVE FOR STUDENTS. 
 
 wide, and that the walls of the tower rise three or four feet 
 from this platform. Fig. 7 represents the upper portion of 
 a structure of this kind. 
 
 Fig. 6 is a diagram representing the plan of the square of 
 the tower, with the square of the base of the spire within it : 
 such a figure is technically called a drawing of concentric 
 squares. The student will observe, that in this diagram the 
 diagonals drawn from the corners of the outer square form 
 the diagonals of the inner one, and would form the diagonals 
 of any number of squares, when their sides are always at an 
 equal distance from the sides of the outer one. Concentric 
 squares, then, may with great facility be represented in per- 
 spective ; for the outer square being put in perspective, and 
 the diagonal lines drawn, any one point of an inner square 
 being found on either diagonal line, the three remaining 
 points can readily be found from it. In Fig. 7, suppose A 
 to be the distance of the platform below the top of the tower ; 
 at this height a square must be drawn, with its diagonal 
 lines. Let b represent the distance of the base of the spire 
 from the outer wall of the tower ; from it rule a line to the 
 vanishing point, and where it intersects the diagonal line at 
 cr, will be a point for drawing the square of the base of the 
 spire corresponding with the point a of the diagram. From 
 the point a draw a horizontal line to the opposite diagonal at 
 
 which will answer to the point b of the diagram. From 
 the point b a line must be drawn to the vanishing point, 
 which will intersect the diagonal beyond at d. The line 
 already drawn from b has given the point c ; a straight line 
 drawn from d to c will be found to be parallel to the upper 
 line from a to 6, and completes the drawing of the inner 
 square in perspective. A perpendicular line must now be 
 drawn up from the centre of the square to the height of the 
 spire ; and to this point lines must be drawn from the points 
 a b and d of the inner square, or base of the spire. The 
 dark lines represent the perspective appearance of such a 
 tower and spire as were proposed to be drawn. Conical, 
 
PERSPECTIVE FOR STUDENTS. 
 
 67 
 
 octagonal, and other spires, similarly placed, may be repre- 
 sented in the same manner. 
 
 It may be well here to make a few observations regarding 
 the reflections of objects in water. Reflections in water 
 require the same attention to be paid to the rules of 
 perspective as when drawing the representations of the 
 objects themselves 1 this must be understood to apply to 
 reflections in still water, where their appearance is similar 
 to the reflections in a large looking-glass. It is highly 
 necessary that this should be attended to in sketching from 
 nature. The introduction of water in landscape scenery 
 greatly adds to its interest ; and though it would have a bad 
 effect generally to reflect every object on a sheet of water 
 with the precision the objects themselves are represented, 
 still it is well to know that such might be the case, and to be 
 able to do it when requisite. In some of the works of the 
 Dutch and Flemish painters, who executed their pictures for 
 the most part on the spot — and hence their deserved reputa- 
 tion for truth — an occasional bit of reflection may be met 
 with, that, if the painting were turned upside down, the 
 reflection might be taken for the reality, and the reality for 
 the reflection. There are a variety of circumstances that 
 have their influence on the appearance of reflections ; such 
 as ripples, currents, &c, all of which, fleeting as they may 
 be, are turned to advantage by the observant and skilful 
 artist. A judicious observation on the influence twilight pro- 
 duces on reflections, frequently has a fine effect on a picture ; 
 much of positive form is lost in the mysterious appear- 
 ance of objects between the spectator and an evening sky, 
 though the forms of these objects are often vigorously 
 reflected in the water. These remarks, though somewhat 
 foreign to our purpose, we are led to make to caution the 
 young aspirant from adhering too much to rule. Though a 
 bit of real reflection, judiciously introduced here and there, 
 has an excellent effect, to reflect every object as if it were 
 standing on a magnified mirror would have a very bad one. 
 
68 
 
 PERSPECTIVE FOR STUDENTS. 
 
 It very commonly occurs that parts of a building, or 
 other object, that are not visible to the spectator, are 
 shown in the reflection, and also much is visible to the 
 spectator that in reflections cannot be shown : this applies 
 principally to horizontal surfaces ; as, for instance, if a 
 square post were represented standing in the water, the 
 flat of the top would be visible, but there could not by any 
 possibility be any reflection of it; but if a succession of 
 posts, with railings running from them, as is constantly 
 met with across country streams, for preventing cattle stray- 
 ing, the upper side of the rail would be visible, but the 
 under part would be shown in the reflection. These 
 apparent trifles should be carefully attended to ; the intro- 
 duction of objects of this kind gives a local truth to a land- 
 scape, and they are frequently invaluable as bits of foreground. 
 When this occurs, they form necessarily a large feature, and 
 any want of accuracy is easily detected. Thus, in Fig. 23, 
 
 a bit of brick wall, with the 
 coping, the top of the coping is 
 seen in the object, but in the re- 
 flection the under part of it is 
 shown. All the horizontal lines of 
 this diagram, both of the object 
 itself and of the reflection of it, 
 tend to the same vanishing point, 
 as is shown very distinctly in the 
 lines dividing the rows of bricks 
 forming the wall, which are drawn 
 in precisely the same manner as if the wall and its re- 
 flection were one plain surface. It will be well to bear this 
 in mind, as want of thought sometimes causes some to 
 draw the reflections in the water exactly the reverse of 
 what the objects appear, which in many instances is a 
 gross error, as is shown in the example (Fig. 24), where the 
 difference between the object and its reflection is very con- 
 siderable. 
 
 Fig. 23. 
 
PERSPECTIVE FOR STUDENTS. 
 
 69 
 
 Fig. 24. 
 
 As it is with reflections, so is it with shadows ; they must 
 also follow the same rules of perspective as the various objects 
 that cast them. In all ordinary perspective drawings, the 
 shadows are projected as in the geometrical drawings, — 
 which for the most part are at an angle of forty-five degrees. 
 This is an extremely convenient way of casting the shadows, 
 as in an elevation drawing the depth of the shadow ia the 
 same as the extent of the projection. Though, for the style 
 of drawing to which this first part of the work is expressly 
 devoted, it would not be proper that the light and shade 
 should be put in according to rule, yet it is well to bear 
 in mind, that to represent correctly the forms of shadows 
 thrown from any projection, as the projecting roof of a 
 house, a cornice, window-sill, &c, or shadows cast on the 
 ground, requires as much attention to the rules of per- 
 spective, as for drawing the outlines of the objects that cast 
 them. If the shadow thrown from a projecting roof were 
 to be made of a greater depth at the further angle of a 
 building than at the nearer one, it would produce the effect 
 of a wider projection at the further end, which, in repre- 
 sentations of new buildings, would have an extremely awk- 
 ward effect; on the contrary, in representations of cottages 
 and ruins, this same knowledge is of advantage, as the 
 inequalities in the depth of the shadows give an excellent help 
 to the effect of dilapidations. We offer these few observations 
 for the advantage of the amateur without illustrating them by 
 examples, the compass of this treatise not admitting of our 
 entering into the subject of light and shade. 
 
70 
 
 PERSPECTIVE FOR STUDENTS. 
 
 There is a problem in perspective, one extremely useful in 
 drawing very large objects — viz. that of finding the perspec- 
 tive inclinations of parallel lines by means of arcs of circles. 
 It appears curious, that in order to draw the representations 
 of curves in perspective, we are compelled to use straight 
 lines, and that in this instance we find the proper directions 
 of straight lines by means of curves. It occasionally happens 
 that some object in a picture is so large, and the vanishing 
 point for finding the proper inclination of the parallel lines 
 so far out of the picture, that it would be difficult to find a 
 means of drawing lines with sufficient accuracy to a vanish- 
 ing point, if one were fixed ; as, for instance, some high object 
 towering above everything in the picture. This ingenious 
 mode of inclining lines towards a vanishing point, without 
 having occasion to make use of one, is as follows : — Let 
 A and b* represent the two sides of the front of a tower ? 
 
 D 
 
 G 
 
 * From our limited space, we are compelled to give this diagram on a 
 very small scale, but its peculiar advantage is when applied on a large 
 scale. It was pointed out by a deceased artist of great celebrity, to the 
 author, when making a very large outline of one of the towers of Rochester 
 Castle. He has found it on many occasions of great service. 
 
PERSPECTIVE FOR STUDENTS. 
 
 71 
 
 D E being the perspective inclination of any horizontal line 
 on the face of it, and c the horizontal line. From e draw 
 an indefinite line e f, parallel to the horizontal line c ; 
 from the point G, at an extension of the compasses g d, 
 describe an arc of a circle till it meet the line E f at f ; 
 join f G. If the arc of a circle be drawn from any point 
 on the line G D through the line G f, the point of intersec- 
 tion on it will furnish a point that will regulate the inclina- 
 tion a horizontal line should take on the face of the tower, 
 drawn from the point on the line G d the arc springs from. 
 Suppose the perspective inclination of a horizontal line 
 required across the face of the tower from the point h ; 
 in order to find this inclination, from the point g, with an 
 extension of the compasses g h, describe the arc of a circle 
 till it intersects the line F G at K, and from k draw a line 
 parallel to the horizontal line c to meet the line b at ar ; 
 then join H and m, and the line H M will be the perspective 
 inclination of a horizontal line on the face of the tower at 
 the height H, tending towards the same vanishing point as 
 the line d e. Take any other point on the line a, as at j ; 
 from g, with an extension of the compasses g j, describe the 
 arc j l ; draw a horizontal line from l to the line B at n, and 
 a line drawn from j to n will be in its perspective inclination 
 towards the same vanishing point as the lines h m and d e. 
 By extending the horizontal line c to the left, and continuing 
 the line d e till it intersects it, the point of intersection would 
 of course be the vanishing point, and it will be found that 
 both the lines h m and J n would terminate in the same 
 point. The perspective inclinations of any number of hori- 
 zontal lines may be drawn in a similar manner, and the per- 
 spective divisions of stones, battlements, windows, &c. may 
 be found either by means of a point of distance found as de- 
 scribed pp. 12 and 13, Plate I. Fig. 2, or by means of diagonal 
 lines as described p. 29, Prob. IV. Plate 5. Thus the per- 
 spective divisions and inclinations of lines may be drawn on 
 the face of a building without the necessity for a vanishing 
 
72 
 
 PERSPECTIVE FOR STUDENTS. 
 
 point, which in certain cases, such as being deficient of large 
 drawing-boards, long rulers, &c, will be found of very con- 
 siderable advantage. 
 
 The foregoing examples, with their descriptions, the author 
 has commonly found ample for the generality of his pupils, 
 mostly young ladies, whose numerous avocations leave them 
 but little leisure for the study of what they mostly consider 
 so dry a subject as perspective. In sketching from nature, 
 even by those well acquainted with the more complicated 
 branches of perspective, it is generally found sufficient to 
 mark the positions of the various points by the eye, without 
 resorting to drawing ground plans, &c. and finding them by 
 rule : nevertheless it is always desirable, if time permit, to 
 thoroughly understand the principles upon which perspective 
 drawing is founded, and more especially for those who pro- 
 pose to become teachers themselves. One principal object 
 with the author has been so to arrange his material, that 
 any intelligent youth, without the assistance of a master, 
 might proceed by easy stages, from chapter to chapter, and 
 render himself, after a diligent perusal of the whole work, 
 competent to instruct others. By an attentive perusal of 
 this First Part of Practical Perspective, the Second will be 
 rendered easy of comprehension to the reader ; and we shall 
 be enabled, we trust, to take the problems used for illus- 
 trating the descriptions of the First Part, and show how the 
 same things may be constructed from plans drawn from 
 actual measurement in the Second. 
 
PERSPECTIVE FOR STUDENTS, 
 
 PART II. 
 
 CHAPTER I. 
 
 Before entering into the matter of the second part of this 
 treatise, it cannot be too strongly impressed on the mind of 
 the student, that Perspective — that is to say, our portion of 
 it, Linear Perspective — is the art of representing in outline 
 the forms of objects on plane surfaces, as they appear to us 
 when viewed in different positions ; we say as they appear, 
 because it must be fully understood that an object viewed 
 perspectively never appears of its positive form. There is, 
 however, one exception to this rule, that of a sphere; this 
 figure, viewed from whatever position it may, near or far 
 from it, above or below it, appears always of the same 
 form. There are certain positions in which objects are 
 viewed, that, whatever their distance may be from the spec- 
 tator, they retain their geometrical figure, the apparent 
 size only changing according to the distance ; such would 
 be the case in any plane figure viewed in the direction of 
 Figs. 7 and 8, Part I. p. 7, and Fig. 11, p. 8, id.; which, 
 at whatever distance they may be removed from the spec- 
 tator, always appear of their geometrical form : objects 
 viewed in such directions cannot be said to be in perspective. 
 We may then really define Linear Perspective to be the art 
 
 Perspective. E 
 
74 
 
 PERSPECTIVE FOR STUDENTS. 
 
 Fig. 1 
 
 of representing certain forms by other forms dissimilar to 
 those they are intended to represent, and yet conveying to 
 the mind a perfect idea of the object intended. Thus a 
 circle, unless perpendicularly opposite the eye, never appears 
 a circle, but always as an ellipse ;* and a cube, which every- 
 body knows to be a figure of six equal sides, each face being 
 a perfect square, may be represented by a figure in which no 
 two lines form a right angle, and yet convey the perfect idea 
 of a cube, as in the following example, which represents the 
 form the outline of a cube would assume, 
 as viewed from a certain position. This 
 figure filled up with the intermediate lines, 
 shaded, and certain round marks on it, 
 will convey at a glance the idea of an 
 object familiar to most individuals. 
 
 Let us first inquire how it is that objects 
 viewed perspectively, apparently change 
 their figure. A very slight observation of 
 the appearance of real objects must con- 
 vince any thinking person, that any object 
 whatever, when seen from a distance, ap- 
 pears smaller than if it were close to him, 
 and that the farther it is removed the 
 smaller it appears. Were this not the case, 
 how, when sitting in a room, could we account for seeing 
 the houses on the opposite side of the street ? If the houses, 
 from their distance, did not appear reduced in magnitude, it 
 would be impossible to see even one of them through so 
 comparatively confined a space as that occupied by a window. 
 To illustrate this, we introduce a light sketch made from the 
 spot at which we are now pursuing our labours, seated about 
 three feet from the window. 
 
 In this cut we see that the whole of a house on the opposite 
 
 * When placed horizontally at exactly the height of the eye, it appears 
 as a straight line, See Fig. 13, Parti, p. 8. 
 
PERSPECTIVE FOR STUDENTS. 
 
 75 
 
 side of the street, from its greater distance from the spec- 
 tator than the window through which it is seen, appears only 
 
 Fig. 3. 
 
 of the height of about two panes of glass ; we perceive that 
 a large projecting lamp from the corner of the opposite shop, 
 appears much about the size of the bowl of a wine-glass 
 situated on the window ledge; that the full height of a 
 
 e 2 
 
76 
 
 PERSPECTIVE FOR STUDENTS. 
 
 figure under the lamp, from its distance, is scarcely higher 
 than the crown of a mans hat close to the railings in front 
 of the window, and much less than a bottle standing by the 
 side of the wine-glass within the room. Though we have 
 every reason to believe the first-floor window of the house 
 opposite to be fully as large as that through which we are 
 looking at it, yet from its distance it occupies but a small 
 portion of a single pane of glass of that so close to us ; 
 in fact, this cut must in every part clearly demonstrate 
 the fact of objects appearing smaller as their distance is in- 
 creased. 
 
 The following very simple experiment will perfectly satisfy 
 the most incredulous as to the fact of objects of an equal size 
 appearing less and less as they recede from the spectator, 
 and the diagram will prove very serviceable in our progress. 
 On the wall, or on a large piece of board, draw a long straight 
 line, similar to the line a b in the following diagram ; 
 
 Fig. 4. 
 
 and on it draw four perpendicular lines at equal distances 
 one from the other, say each line eighteen inches in height, 
 with an interval of one foot between each, similar to the 
 lines c d, E f, g h, and J k. Drive a nail into the top 
 and bottom of each of the lines G H and J k, and to each 
 nail attach a piece of twine ; then taking the four strings 
 together, draw them tight, and bring them to a point any- 
 where on the line c D, as shown in the diagram at L. If the 
 eye be now placed at the point where the four strings meet 
 on the line c d (as shown at l), and the line E F taken as a 
 
PERSPECTIVE FOR STUDENTS. 
 
 77 
 
 line for the measurement of the two more distant ones, the 
 line G h will appear of the length g and the line J k will 
 appear of the length j k ; clearly showing that lines of the 
 same length appear less and less as their distance is increased. 
 It may be here noticed that this apparent diminution in size 
 according to distance, is in a regular progression ; that a line 
 placed at a certain distance from the spectator, at double that 
 distance will appear exactly half the length it appears in its 
 first position : thus, g h, viewed from the point l, appears 
 exactly half its length, g h measuring precisely one- half of 
 E f.* At three times the distance, it will appear exactly 
 one-third the length it appears in its original position ; thus, 
 j k, three times the distance of E F from c d, viewed from 
 the point L appears one-third the length of e f, j k mea- 
 suring precisely one-third of this line. At four times the 
 distance, it would appear one-fourth ; at fives times the dis- 
 tance, one-fifth, and so on progressively. 
 
 We may venture to presume, then, that no doubt remains 
 in the mind of the reader that objects do really appear to 
 diminish as their distance from the spectator is increased, as 
 on this fact we propose to show how it is that objects viewed 
 at an angle in perspective apparently change their form. 
 Let us, for example, take the simple form of a square (any 
 rectangular piece of board will answer the purpose), and 
 place it upright before us ; the relative positions of the spec- 
 tator and board being as a b (Fig. 5) ; A re- 
 presenting the position of the spectator, and 5 * 
 B the base of the board, the board itself 
 standing perpendicularly over this line ; the 
 line from a to b representing the direction in 
 which the board is viewed. In such a position, 
 the simple geometrical form of the square (or 
 whatever form it may be) would be apparent, a 
 
 * If the strings were brought to a point on any other part of the Ike 
 c d, the result would be the same. 
 
78 
 
 PERSPECTIVE FOR STUDENTS. 
 
 Fi°r. 6, 
 
 and this would be the same at whatever distance it may be 
 removed, so long as the board stands at a right angle with the 
 direction of the line from A to b in which it is viewed ; the 
 change of distance affecting only the apparent size of the 
 object, but the form always remaining the same : but if the 
 angle at which the board stands with reference to the line from 
 A to B is in the slightest degree changed, an apparent change 
 of form is the immediate consequence, as we will endeavour to 
 point out by the following diagram (Fig. 6). In this figure, let 
 
 the square abed represent a closed 
 shutter, opening by hinges on the 
 side a b, viewed from the point A 
 in the direction a e b. If this 
 shutter were opened, so as to lay 
 
 /, %. ^ & against the wall to the left, 
 
 the edge d c must necessarily de- 
 scribe a semicircle in its passage 
 from d to /, as shown by the line 
 d e f. In the passage of this 
 shutter from its first position, 
 closed over the line d 6, to its 
 full opening over the line b /, it 
 assumes an infinite variety of forms. In its first position we 
 see the outside, and in the last the inside, of the shutter ; and 
 as in both these positions the shutter stands at a right angle 
 with the direction of the line (a e b) at which it is viewed, it 
 would appear of its simple geometrical form either shut or 
 open. Now it must be quite evident that as in the first 
 position the outside is visible, and in the latter the inside, 
 there must be in its passage from dtof one position in which 
 neither side could be seen, but merely the edge of the shut- 
 ter ; and this would occur when the line d c comes imme- 
 diately over the point 0, the point immediately opposite the 
 eye of the spectator at a. Such being the case, it follows 
 tli at in the passage of the edge d c towards e, the 
 space between the two sides d c and a b must gradually 
 
 a 
 b 
 
 
 r 
 
 
 
 
PERSPECTIVE FOR STUDENTS. 
 
 79 
 
 appear to become narrower and narrower, till it is at last 
 entirely lost to view : and by the same reasoning, the 
 shutter having passed the point e towards /, the inside 
 when it first becomes visible appears extremely narrow, 
 and gradually appears wider and wider till it comes to its 
 place against the wall over e /, where it resumes its real 
 geometrical form. 
 
 Let us open the shutter as far as the point marked g, and 
 examine what form the square shutter would assume in this 
 position. The side represented by the line a b remains 
 stationary in all the variety of positions, but that represented 
 by d c, when brought to the point g, from its being brought 
 nearer to the spectator, would appear longer ; and as we have 
 shown that in the passage of the shutter from the point d 
 towards 0, the space between the two sides becomes appa- 
 rently narrower, it would, when standing over the point g, 
 require to be drawn closer to a b than in the diagram (Fig. 6). 
 The appearance then of the square shutter a b c d, opened as 
 far as the point g, would be as shown in Fig. 7 ; the line a b, 
 from its remaining in the same position, is drawn ^ 
 of the same length as a b, Fig. 6 ; but the line c d, 
 from its being brought nearer to the spectator, is 
 drawn longer : and as the space between a b and 
 c d, for the reasons before given, is apparently less 
 than in Fig. 6, it is drawn nearer to a b than in that 
 diagram. The top and bottom of the shutter must 
 be drawn by joining the points c a and d b ; and 
 this figure, a b c d, notwithstanding the difference of its 
 appearance from the original form, is nevertheless a correct 
 representation of what a square would appear in the position 
 we have described, and with the addition of light and 
 shade would convey a perfect idea of the object it proposes 
 to represent. 
 
 The following diagram (Fig. 8) will enable us to under- 
 stand how it is that a circle must necessarily change its appa- 
 rent form according to the point from which it is viewed. 
 
80 
 
 PERSPECTIVE FOR STUDENTS. 
 
 Fig. 8, 
 c 
 
 E li 
 
 
 G \B 
 
 
 
 
 1) 
 
 Suppose the circle A B c d to represent a hoop standing 
 
 upright, and the eye of 
 the spectator exactly op- 
 posite the point g ; the 
 line E f an axis on which 
 the hoop may be made to 
 revolve : the hoop might 
 be turned upon this axis 
 so as to bring the point 
 d before the point G. In 
 this position, the curve of 
 the circle would appear 
 a part of the straight line e f. Now on the same principle 
 that the shutter appears to become narrower and narrower 
 in its passage from d to e (Fig. 6), the space from d to G 
 must appear gradually to lessen in width in the change of 
 the hoop from a perpendicular to a horizontal position, and 
 the line d g consequently appear to shorten till it becomes 
 apparently a mere point. The centre of a circle is a point 
 w r ithin the circumference equidistant from all parts of it, 
 and all lines drawn from this centre to any part of the cir- 
 cumference are of equal length, as is the case in the 
 geometrical figure before us, g a, g b, g c, and g d 
 being all equal ; but the instant the hoop is turned upon 
 the axis, the figure becomes totally changed ; the point d 
 is brought forward, and the point c thrown behind ; 
 D G appears shorter, c g, from being further removed, 
 shorter still, and A G and b g remain the same. If the 
 hoop be turned completely round upon the axis, the dia- 
 meter a b is always the same, but the diameter c d is 
 always apparently changing, producing a constant apparent 
 variety of forms. Hence it is clear that the apparent 
 change in the form of an object arises from the increased 
 or diminished distance of one part from another ; so that 
 what is required to enable us to represent any object, 
 or combination of objects, in perspective, is to find the 
 
PERSPECTIVE FOR STUDENTS. 
 
 81 
 
 relative heights and distances of the different parts accord- 
 ing to the angle at which they stand in reference to the 
 spectator. 
 
 CHAPTER II. 
 
 The rules of perspective are deduced from the science of 
 optics, depending on the fact of vision being caused by rays 
 of light passing from the object seen into the eye, and form- 
 ing an image at the back of it, similar to the reflection of 
 an object in a mirror. These rays of light forming the 
 image are supposed to proceed from the object in straight 
 lines, and in every possible direction, as shown by the lines 
 proceeding from the line A b (Fig. 9). It will be clear from 
 this diagram, that the 
 
 same rays that give the Fig. 9. 
 
 representation of the A c 
 line A b to a spectator 
 situated at c, cannot be 
 the same as those that 
 give the representation 
 to a spectator situated b 
 at D. It is by means 
 
 of lines drawn from the object to the eye of the spectator, 
 that the perspective positions of the various points are 
 found ; these lines are called visual rays, and in their trans- 
 mission from the object to the spectator pass through an 
 intermediate plane,* called the plane of delineation (to be 
 
 * Many are deterred from commencing, and more from persevering in 
 the study of Perspective, from the multiplicity of definitions too frequently 
 put at the very commencement of most works on this subject, as if it were 
 purposely done to frighten the young student from his pursuit ; still, 
 certain technicalities are unavoidable, such as the point of sight, the 
 vanishing points, &c. : the ground plane, horizontal planes, perpendicular 
 
 E 3 
 
82 
 
 PERSPECTIVE FOR STUDENTS. 
 
 hereafter described), and there determine the size and form 
 the object must take in the representation. 
 
 Much difficulty is always avoided by clearly comprehend- 
 ing every step we take in the pursuit of knowledge ; let us 
 then understand that the cause of vision is the rays of light 
 coming from the object to which our eyes are directed, pass- 
 ing through the pupil of the eye to the back part of it, and 
 there forming an image representing every object on that 
 small space in the most exquisite perfection.* The rays of 
 light are supposed to proceed from all objects in straight 
 lines, and in every possible direction. Now if our object 
 be only the representation of one single straight line, not- 
 withstanding the rays of light proceed from every part of 
 the line we are about to draw, and in an infinite variety of 
 directions, all that we require for finding the position of this 
 
 planes, parallel planes, &c. are terms so commonly in use that an expla- 
 nation of them is indispensable. Some of these technicalities it is to be 
 hoped are already understood from the perusal of Part I. of this treatise ; 
 others we will endeavour to explain as the necessity for using them 
 occurs ; for the present we will only observe, that the word plane is used 
 to designate a surface ; thus the ground on which objects stand is called a 
 plane (the ground plane) ; the surface of glass interposed between the 
 spectator and the subject is also called a plane (the plane of delineation) 
 &c. &c. 
 
 * The optic nerve, the supposed medium by which the image is con- 
 veyed to the brain, is expanded at the back of the eye into a beautiful 
 transparent membrane called the retina, at the back of which nature has 
 spread a dark pigment, forming a most perfect concave mirror, and on 
 this minute surface a most perfect representation is formed of everything 
 within the scope of vision. To understand clearly how this wonderful 
 miniature image of all that we see is represented on the retina, would 
 require a knowledge of the anatomical structure of the eye, as well as 
 some proficiency in the science of optics ; the form of the eye, of the lens, 
 or crystalline humour within it, and the density of the humours both in 
 front and behind the lens, all have an influence on the direction of the 
 rays of light, so as to bring them to such a focus as will produce the 
 image. All that is required for our purpose is to understand that the 
 rays of light do proceed in straight lines from every object we see, to the 
 eye. 
 
PERSPECTIVE FOR STUDENTS. 
 
 83 
 
 Fig. 10. 
 
 line in the picture are the two rays from the extremities of the 
 line ; by these alone we are enabled to determine the perspec- 
 tive position of the two points from which the rays proceed 
 on the paper or canvas, when a line drawn from point to 
 point must be the representation of the straight line. 
 
 In Fig. 10, let A B represent the line to be drawn, and c the 
 position of the spectator's eye ; 
 from A a number of straight 
 lines are drawn, to represent the 
 rays of light proceeding in va- 
 rious directions from this point: 
 and it must be evident that none 
 of the rays proceeding in the 
 direction of the lines A 1, A 1, 
 could possibly reach the eye of 
 
 a spectator situated at c, neither could any of the rays pro- 
 ceeding in the directions A 2, a 2. It is sufficient to draw 
 a line from a to c and from b to c to determine the position 
 of the two points on any intermediate line, as D E (which 
 would represent a perpendicular section of the plane of deli- 
 neation) ; and from A to b, being a straight line, no inter- 
 mediate points would be required, as all straight lines are 
 represented by straight lines ; thus, supposing D e to be the 
 distance from the spectator at which the representation is to 
 be made, or, if we express ourselves technically, suppose D E 
 to be a perpendicular section of the plane of delineation, the 
 intersections on it, a b, would represent the position of a b ; 
 and though from every portion of the line a b it is under- 
 stood that rays are proceeding to the eye, it is unuecessary 
 for perspective drawing to introduce them. The visual rays 
 F c and g c, which give the points f and g on the section of 
 the plane of delineation, are perfectly unnecessary if the line 
 A b only is required ; for A B being a perpendicular line, and 
 D e the section of the plane of delineation, also perpendi- 
 cular, the line A b must necessarily come through these 
 points / and g. It must, therefore, be understood that m 
 
84 
 
 PERSPECTIVE FOR STUDENTS. 
 
 drawing visual rays from any object composed of straight 
 lines, it is only necessary to draw them from their extremi- 
 ties : thus, in drawing the visual rays to find the perspective 
 position and form of a square, it will only be necessary to 
 draw a visual ray from each corner ; the positions of the four 
 corners being found, the lines may be ruled from point to 
 point, as we shall presently demonstrate. 
 
 Now perspective is said to be the art of representing an 
 object or combination of objects on a plane, such as a sheet 
 of paper or canvas, as they would appear to the spectator 
 looking through a sheet of glass, or window, interposed 
 between himself and the objects to be delineated. This sheet 
 of glass, or window, is a most important feature in perspec- 
 tive drawing ; and though views are more frequently taken 
 in the open air than from a room, an intervening sheet of 
 glass, or transparent plane, is always supposed to exist 
 between the spectator and the original objects, and this sup- 
 posed intermediate plane is called the plane of delineation.* 
 Thus, supposing a view to have been selected for delineation, 
 with the correct measurement of all the various parts of 
 which it consists ; before we can proceed a single step to 
 prepare the points necessary for making a perspective draw- 
 ing, the position of the spectator and the plane of delineation 
 must be determined ; the size of the objects in the repre- 
 sentation depending on the nearer or more distant position 
 of the plane of delineation from the spectator. Let us refer 
 to Fig. 4, p. 76, and suppose the line J K to be the object 
 to be represented, and l the position of the spectator's eye ; 
 either e f or g h may represent a section of the plane of 
 
 * The cut, Fig. 3, p. 75, is an excellent example of this. The house, 
 the windows, the lamp, the figures, railings, and everything, in fact, 
 outside the window from which the sketch is taken, are the objects to be 
 drawn, or, as they are commonly termed, the original objects ; we have 
 the window through which they are seen (which may be called the plane of 
 delineation), on which the form of all the objects might be traced — and the 
 position of the spectator three feet from it. 
 
PERSPECTIVE FOR STUDENTS. 85 
 
 delineation, but it makes a vast difference in the relative 
 space the object will occupy in the picture, whether it is 
 placed in the one position or the other. The lines J l and 
 K l represent the visual rays ; and it will be seen by the 
 diagram how much longer the representation of the line 
 would be were g h the position of the plane of delineation, 
 than if it were placed at E f, by the different length between 
 the intersections of the visual rays on the two lines.* 
 
 We will endeavour to illustrate the preceding observa- 
 tions by the following diagram. Let A (Fig. 11) represent 
 
 Fig. 11. 
 
 a square to be drawn (the original object), B the position of 
 the spectator, and c the plane of delineation ; the square, the 
 spectator, and the plane of delineation, all standing on the 
 same level plane. The appearance of the square on the plane 
 
 * By reference to the cut, p. 75, this may be easily understood, as, if 
 the reader station himself first at about three feet from a window, and 
 notice exactly how much of the opposite side of the way he sees through 
 it, he may easily satisfy himself of the wonderful difference consequent on 
 the slightest change of position. If the student retire three yards from the 
 window instead of three feet, he will perceive that a much less portion of 
 the opposite side of the way is now visible, though the window, which we 
 will suppose to be the picture, is of the same area. Again, if the 
 student comes quite close to the window, he will find a much greater 
 portion of the opposite side of the way is seen. This experiment is merely 
 to show how much a perspective representation depends on the relative 
 positions of the spectator and plane of delineation with the original 
 objects. 
 
86 
 
 PERSPECTIVE FOR STUDENTS. 
 
 of delineation would be as d, and its form and position are 
 there ascertained by means of visual rays drawn from the 
 four corners of the original object, e f g A, to the eye of the 
 spectator at e, perpendicularly over b. We have in this 
 diagram the original object (the square), the position of the 
 spectator, and the position of the plane of delineation, with 
 the visual rays drawn through it ; and if we could imagine 
 these visual rays to be pieces of thread, or wire, passing 
 through the plane of delineation, we should be able to mark 
 the points of intersection a b c d, and at once draw the 
 lines representing the square from point to point ; but though 
 we know them to pass through, we know also that they are 
 not tangible, and we must therefore find some means to 
 ascertain the points of intersection of the visual rays with the 
 plane of delineation ; to explain which we must have a fresh 
 diagram. 
 
 In this cut (Fig. 12), A B is the object to be represented, 
 
 c D a perpendicular section of the 
 12# plane of delineation, and the eye 
 
 E of the spectator anywhere on the 
 line e f ; for, as affects the length 
 G the line a b will appear on c D, 
 it is immaterial at what point on 
 this line the eye may be situated, 
 F as may be seen by the intersections 
 made by the visual rays on c d, 
 a b, c d, and e f being of an equal length. As it is with the 
 height of an object, so it is with the width : the visual rays 
 drawn from e and /, and g and A, to e (Fig. 11), would give 
 the same width on the plane of delineation if drawn to any 
 other point on the line B e. We have stated that the base 
 of the square, the base of the plane of delineation, and the 
 spectator (Fig. 11), all stand on the same level plane ; there- 
 fore, if we suppose the eye of the spectator to be on the 
 ground, the visual rays drawn from g and 7i to that 
 point (b) would intersect the ground line of the plane of 
 
PERSPECTIVE FOR STUDENTS. 
 
 87 
 
 delineation, and these points of intersection determine the 
 width of the square, at whatever height the representation 
 of it may be on the plane of delineation. If from each of 
 these points a perpendicular line is drawn from the base to 
 the top of the plane of delineation, the space between the two 
 lines determines the width of the square, at whatever point on 
 the line b e the eye of the spectator may be placed ; the 
 perpendicular from either point of intersection representing 
 a section of the plane of delineation, on which the height 
 of e g or / h may be found in a manner precisely similar to 
 that of finding the height of a b on the section c D, Fig. 12. 
 
 In Fig. 11, then, we have the relative positions of the 
 spectator and object, with the interposition of the plane of 
 delineation : the visual rays are drawn, and the points of inter- 
 section were found, in the following manner. In this diagram 
 (Fig. 13), the relative positions of the spectator and original 
 
 Fig. 13. 
 
 
 
 C 
 
 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 IE 
 
 A 
 
 
 
 
 
 
 h 
 
 
 IE 
 
 
 
 
 object are the same as in Fig. 11, visual rays being drawn 
 to a point on the ground plane, perpendicularly under the 
 eye of the spectator ; and the intersections of these lines with 
 the base of the plane of delineation at j Jc determine the 
 width the square would appear. The combination of the 
 visual rays drawn in these two Figs. 11 and 13, will enable 
 us to give a perfect representation of what the square would 
 appear in its relative position with the other parts described. 
 Fig. 14 is a combination of the two (the relative positions 
 of the parts being similar to Figs. 11 and 13), with the per- 
 
88 
 
 PERSPECTIVE FOR STUDENTS. 
 
 pendicular lines drawn from the points j and k : the line drawn 
 fromy, where it intersects the visual ray c E, gives a point a, 
 
 Fig. 14. 
 
 />— ■ 
 
 f 
 
 C 
 
 
 h 
 
 
 
 
 
 
 a 
 
 
 
 
 
 
 
 
 D 
 
 
 
 
 
 A 
 
 
 
 
 
 1 
 
 ¥ 
 
 
 
 
 
 
 
 
 J n 
 
 
 
 
 the perspective position of the point e of the original object ; 
 where it intersects the visual ray g e it gives a point c, the 
 perspective position of the point g in the original object. 
 Where the perpendicular line from k intersects the visual ray 
 / e, it gives the perspective position b of the point / of the 
 original ; and the point of intersection with the visual ray h e 
 gives a point d corresponding with h in the original object. 
 Thus we have in the points abed the perspective positions 
 of the four corners of the square A. Consequently, the 
 perpendicular drawn from where it passes between the 
 visual rays e e and g e, represents on the plane of deline- 
 ation the line e g of the original object A ; that drawn from 
 
 where it passes between the visual rays / e and h e, 
 represents on the plane of delineation the line f h of the 
 original object. By ruling lines from the points a to b and 
 c to d, we have on the plane of delineation a form such as 
 the object would take on a sheet of glass similarly placed, 
 if the spectator were to trace on it the form he sees through 
 it ; the relative positions of the three essentials for perspec- 
 tive drawing being as described, — viz. the positions of the 
 original object, the spectator, and the plane of delineation. 
 
 Trusting that the matter in the preceding pages, which 
 may be said to contain the groundwork from which the 
 
PERSPECTIVE FOR STUDENTS, 
 
 89 
 
 various modes of representing objects in perspective are 
 deduced, is tolerably understood, we will first offer a few 
 observations, and then proceed to practice. The foregoing 
 may be summed up in very few words : — The rays of light, 
 which are the cause of vision, are understood to proceed 
 from every object in straight lines to the eye of the spec- 
 tator, and in their transmission are supposed to pass through 
 an imaginary plane, situated somewhere between the spec- 
 tator and his subject. In order to make a perspective draw- 
 ing, the points where the rays of light (visual rays) pass 
 through this imaginary plane must be ascertained, these 
 points transferred to the canvas or paper, and by means of 
 these the form and size of the objects accurately delineated. 
 The diagrams requisite have been made with as much atten- 
 tion to simplicity as the subject admitted, and are intended 
 solely to illustrate the principle, that correct representations 
 of form are to be drawn by finding the above-mentioned 
 points : various modes are employed for finding them, which 
 it will be our endeavour to make thoroughly understood. 
 
 The horizontal line on the plane of delineation selected for 
 finding the perspective width of the square a (Figs. 11, 13, 14), 
 is the base line, though the same result might be obtained 
 by taking any other horizontal line. The base line is that 
 usually chosen for this purpose ; and as it is unquestionably 
 the most convenient, from the circumstance of draftsmen 
 being furnished with a ground plan and elevations from 
 which they are to execute the perspective drawing, it would 
 only be perplexing the student to furnish him with examples 
 and rules for what he may most probably never require. To 
 avoid any errors, it may be well to remark that the ground 
 plane, which is but another name for the surface on which the 
 object to be drawn rests, is supposed always to be a flat even 
 surface of great extent, and that the plane of delineation is 
 always supposed to stand perpendicularly on it. Any devia- 
 tion from this might materially affect the representation. 
 
90 
 
 PERSPECTIVE FOR STUDENTS. 
 
 CHAPTER III. 
 
 Where a number of objects combined are required to be 
 represented in perspective, to proceed by finding every point 
 by means of intersecting points on the plane of delineation, 
 would be an extremely tedious mode of procedure; and 
 these points are used as sparingly as possible, the main object 
 being to produce accuracy with the smallest possible number 
 of lines. In order to make our illustrations clearly intel- 
 ligible, and in the comparison of one mode with another, to 
 show that we may arrive at the same result by various 
 means, we are necessarily compelled to employ more lines 
 than would be requisite for an ordinary drawing; but in 
 doing so, we will endeavour to point out the best and readiest 
 modes of proceeding. Vanishing points are what are com- 
 monly used for determining the perspective heights of per- 
 pendicular lines, as well as for regulating the length and 
 direction of horizontal ones. It has already been explained 
 in Part I. pp. 9, 10, Figs. 14, 15, that those lines in an ori- 
 ginal object that are parallel, in the perspective representation 
 incline towards each other and meet in a point, which is called 
 the vanishing point : keeping this in mind, let us proceed to 
 draw a plane figure in perspective from the following plan. 
 
 Fig. 15, Plan. 
 
 Let A represent the top of a table, perpendicularly over the 
 
PERSPECTIVE FOR STUDENTS. 
 
 91 
 
 edge of which, b, stands a sheet of glass ; let c represent 
 the position of the spectator, and d a square figure lying 
 flat on the table. We should here call A the ground plane, 
 the upright sheet of glass over B the plane of delineation, 
 and D the square figure to be represented, the original object 
 viewed from the point c : * from these data it is required to 
 draw the figure d in perspective ; the first step towards which 
 will be to draw a parallelogram (Fig. 15, Rep. l)to represent the 
 picture (efg h), and across the 
 
 picture, parallel to the ground Fi S- 15 > Representation 1. 
 line e F, at the height above H 
 the ground line the spectators 
 eye is above the ground plane, 
 draw the horizontal line J (see 
 Part I. pp. 5, 6, Fig. 6). 
 
 Before proceeding with this 
 
 drawing, it must be thoroughly e j, a j ' F 
 
 understood that the line e f, 
 
 the ground line in the representation, is in all respects similar 
 to the line B of the plan, and that all points of intersection 
 found on b are to be carried to ef; and moreover, that the 
 parallelogram e f g h is the representation of the plane of 
 delineation standing upon the line b. As in this case we 
 propose to find the perspective positions of the points b c h j 
 by means of a vanishing point, the position of this vanishing 
 point must first be ascertained; and the vanishing point in this 
 position of the square relative to the plane of delineation, 
 will be the point of sight, which is always perpendicularly 
 
 * Nothing can be more readily imagined than that the whole of this 
 figure might be drawn from a description of the absolute measurements 
 of the different objects ; as, for instance, we might say d to be twelve 
 inches square, situated one foot from b ; b four feet long ; and c four feet 
 perpendicularly distant from the point a on b ; the eye of the spectator 
 two feet above the ground plane a. From such a description, a perspec- 
 tive drawing may be made of any size, either large or small, by working 
 from a scale of so much to a foot — either the eighth of an inch, or half a 
 dozen yards. 
 
 — jr— 
 
 
 K 
 
 
 
 
 
 1 
 
 \ 
 
 \ 
 
 
 
 "/ 
 
 
 
 
 c 
 
 1 
 
 
 
 
 
 \ 
 
92 PERSPECTIVE FOR STUDENTS. 
 
 opposite the spectators eye on the plane of delineation : * 
 we must therefore mark on the base line of the plane of 
 delineation, the point a perpendicularly opposite c ; carry 
 this point to the ground line e f, and perpendicularly over 
 it on the horizontal line, mark the point of sight at k. In 
 order to find the length the line b c will appear, it is 
 necessary to draw the visual rays b c and c c ; carry the 
 points of intersection they make with the base of the plane 
 of delineation d and e to the ground line E F, and from each 
 draw a perpendicular line on the picture. From d to e will 
 be the length the line b c would appear ; the point b will 
 be found somewhere on the perpendicular drawn from d, 
 and the point c on some part of that drawn from e. 
 
 The lines b h and c j of the plan are parallel lines ; and 
 though if they were continued to any extent they would never 
 meet, yet in the representation they incline towards each 
 other and meet in a point ; and being at a right angle with 
 the plane of delineation, this point would be the point of 
 sight. Continue the lines h b and j c up to the plane of 
 delineation at / and g, and carry these points to the ground 
 line E F, and from each point draw a line to the point K, the 
 point to which the parallel lines f b h and g c j of the plan 
 must incline (or any others that might be parallel with them) : 
 the intersection of the line / K with the perpendicular from d, 
 gives the perspective position of the point b of the plan ; 
 the intersection of the line g K with the perpendicular from 0, 
 gives the perspective position of the point c of the plan ; 
 
 * The point of sight is always used as the vanishing point for lines 
 that are situated at a right angle with the plane of delineation ; those 
 lines that lie in a direction parallel to it have no vanishing point, but 
 are represented by lines parallel to the ground line of the picture. This 
 will be understood when we come to the rule for finding the positions 
 of vanishing points in general ; it is necessary here to understand that 
 all lines at a right angle with the plane of delineation do have their 
 vanishing point in the point of sight, and that this point is on the plane of 
 delineation perpendicularly opposite the spectator's eye (Part I. Plate I. 
 A, Fig. 2). 
 
PERSPECTIVE FOR STUDENTS. 
 
 93 
 
 and a straight line drawn from one to the other, the repre- 
 sentation of the line b c. 
 
 The direction of the sides b h and c j are represented by 
 the lines b k and c K ; and to determine their length, it is 
 necessary to find the positions on them of the points h and 
 j ; which is extremely simple. From the points h andy, draw 
 the visual rays h c and j c ; carry the points of intersection 
 Hon B to the ground line E F, and from each point draw 
 a perpendicular line : where that drawn from k intersects 
 the line / K is the perspective position of the point h ; and 
 where that drawn from the point I intersects the line g k is 
 the perspective position of the point / A line drawn from 
 h to j completes the drawing, the figure b c j h being the 
 perspective representation of the square d, viewed in the 
 positions described. 
 
 In the foregoing example more lines have been used 
 than are absolutely required for drawing this figure, but they 
 have been introduced for the more clearly exemplifying the 
 mode for finding the positions of points, by means of a 
 vanishing point and the visual rays; but the visual rays^ c 
 and c c (in the plan) might have been omitted, and con- 
 sequently the points and lines derived from them on the 
 picture ; for, as is shown in the Second Representation, 
 Fig. 15, having the lines / K and g k drawn, and the points 
 b and h found by means of the 
 
 intersections d and h of the Fig. 15, Representation 2. 
 visual rays b c and h c, the H 
 points c and j are found from 
 the points b and h. The lines 
 b c and h i in the plan, being 
 parallel to the base of the plane 
 of delineation, are represented 
 in the drawing by lines parallel e 
 to the ground line ; hence lines 
 
 drawn parallel to the ground line from the points b and h to 
 the line j K, give an eoualb* accurate reioresentation of the 
 
 — dp 
 
 K 
 
 I 
 
 1 
 
 
 
 
 
 
94 
 
 PERSPECTIVE FOR STUDENTS. 
 
 figure in a more simple manner, and at a considerable saving 
 of labour. 
 
 In geometry, a point is defined to be that which has 
 position but not magnitude, and a line to be length without 
 breadth or thickness ; the extremities of lines are points, and 
 the intersections of one line with another are also called 
 points. Any straight line may be represented in perspective 
 by finding the positions of its extremities, and any combi- 
 nation of straight lines by finding the positions of the 
 extremities of each separate line. Curves may be drawn in 
 perspective by finding intersecting points in the original 
 figure (Part I. pp. 35, 36), and finding the positions of 
 these points on the plane of delineation. If then a drafts- 
 man has the knowledge how to find any single given point, 
 he will be able by the same means to find a second, third, 
 or any quantity ; hence, straight lines may be drawn in per- 
 spective by finding the position of their extremities, and 
 curve lines by finding points of intersection, by understand- 
 ing clearly how any single point is to be found. We will 
 therefore proceed to show, first, how the perspective position 
 of any single point is to be ascertained ; and afterwards, 
 a combination of them : so that any plane figure may be 
 put in perspective, upon the same principle employed for 
 
 Fig. 16, Plan. 
 
 
 A 
 
 
 C 
 
 
 
 B 
 
 a 
 
 c 
 
 drawing the square, Fig. 15 ; as in the above-given plan, 
 Fig. 16, from which it is required to find on the picture 
 
PERSPECTIVE FOR STUDENTS. 
 
 95 
 
 Fig. 16, Representation. 
 
 E F 
 
 
 
 
 \ 
 
 the position of the point a, situated on b the ground plane, 
 the relative positions of the plane of delineation and the 
 spectator being as c and d, the eye of the spectator situated 
 above the ground plane, the height shown by the horizontal 
 line E (Fig. 16, Rep.). First, find the position of the point of 
 sight (as in Fig. 15), by drawing 
 the line a d perpendicular to c; 
 draw the visual ray a d, inter- 
 secting the base of the plane of 
 delineation at b ; bring the point 
 a perpendicularly forward to the 
 plane of delineation at c,* and 
 then carry these three points 
 a b c to the ground line of the 
 picture. From a draw a line 
 
 perpendicular to the ground line, to the horizontal line at f ; 
 this will be the point of sight ; also a perpendicular line 
 across the picture from 5, on some part of which the point 
 A will be found. The line c a of the plan, being at a right 
 angle with the plane of delineation, will be represented by a 
 line drawn to the point of sight, the vanishing point for all 
 lines lying in that direction ; draw therefore the line c f, and 
 where this intersects the perpendicular drawn from b y will 
 be the position in the picture of the point A required, the 
 line c A being the correct perspective length and direction 
 in the representation, Fig. 1 6, of the line c A of the plan. 
 
 Simple as the above diagram appears, it is important that 
 it be most clearly understood. In making a perspective 
 drawing consisting of a great variety of objects, the intricacy 
 
 * The drawing a straight line from the point required to be found in the 
 picture up to the plane of delineation, is called bringing the point up to the 
 plane of delineation, and the position of it in the picture may be found by- 
 bringing it forward in any direction ; but in this instance it must be 
 brought forward perpendicularly, otherwise we could not find the position 
 of the point a by means of the point of sight, that being the vanishing 
 point only for lines at a right angle with the plane of delineation. Tin* 
 will be fully explained in the ensuing chapter. 
 
96 
 
 PERSPECTIVE FOR STUDENTS. 
 
 of lines may at the outset perplex the student, which will 
 pass away with practice ; but with a thorough comprehension 
 of the principle on which the foregoing diagrams (Figs. 16) 
 are drawn, he can never be at a loss to find the position of 
 any given point in a picture, whether situated on the ground 
 plane or above it, — as we shall presently show. By this 
 mode any plane figure, however complicated, may be repre- 
 sented in perspective by actual measurement of the original 
 objects: for instance, suppose A, Fig. 16, to be the spot on 
 which any object stands (a man, a tree, or a post, is im- 
 material for our present purpose)*, distant perpendicularly 
 from the plane of delineation four feet, the spectator being 
 four feet in front of this plane, his eye situated three feet 
 above the ground plane ; from these premises the position of 
 this point may be ascertained with the greatest ease on a sheet 
 of paper or other plane surface, and of any size required, 
 in the following manner : — First, draw a horizontal line, on 
 which mark a series of equal divisions, to serve as a scale to 
 work from, similar to the diagram (Fig. 1?) here given,* 
 
 Fig. 17. 
 
 . 2 2 3 U 5 ____ *P 
 
 which represents a scale of ten feet in length ; then draw 
 a line c, equal in length to that of your picture (Fig. 16, 
 Rep. 1), to represent the base line of your plane of delinea- 
 tion ; and behind it, at the (perpendicular) distance of four 
 feet,t mark the position of the spot on which the object 
 
 * All geometrical drawings furnished to artists are worked on a scale of 
 so much to a foot, yard, &c. according to the size required. Having 
 the dimensions of the various parts to be represented, measurements 
 are taken from the scale and applied to the drawing, by which the rela- 
 tive proportion of the parts is preserved, however minute the repre- 
 sentation. 
 
 f The plan, Fig. 16, is worked to the scale here given, for the advantage 
 of L h* student ; but in making his own drawing, it will be better to con- 
 struct a much larger scale to work from. 
 
PERSPECTIVE FOR STUDENTS. 
 
 97 
 
 stands at A, to the right hand of the spectator, then on the 
 picture, three feet of the scale above the ground line, draw 
 the horizontal line, and from these find the position of the 
 point as described, Fig. 16, Representation. 
 
 Referring back to the diagrams, Figs. 16, at the risk of 
 being accused of prolixity, we will once again go over the 
 several parts; and we will first observe that the space be- 
 tween the ground line 3 — 4 and the horizontal line e repre- 
 sents a very great extent of flat surface, the whole extent of 
 space between the situation of the plane of delineation to the 
 greatest distance our vision extends ; that the point f is the 
 point of sight, that point perpendicularly opposite the eye of 
 the spectator, and the vanishing point for all lines at a right 
 angle with the plane of delineation ; the perpendicular line 
 over b shows that on some part of that line the point A will 
 come, and the line c F determines at what point. 
 
 We will now, by the same means employed for finding the 
 position of the point a, find the position of another point on 
 the same plane, b ; and that we may not interfere with the 
 simplicity of Figs. 16, we will take a fresh diagram (Fig. 18), 
 premising that the references marked A, B, c, D, e, and F, 
 and a, c, are precisely similar to those in Fig. 16. 
 
 Fig. 18, Plan. 
 
 Let us place another point on the plane B, as at G, the 
 position of which is to be found in the picture by the same 
 process as we found the position of the point A, viz% a visual 
 
 Perspective. F 
 
98 
 
 PERSPECTIVE FOR STUDENTS. 
 
 ray G d, must be drawn, and the point of intersection, d, 
 
 carried to the ground line of the picture at d (Fig. 18, 
 
 Representation), and a perpendicu- 
 Fig. 18, Representation. fep y me dfawn ^ ;| . then bring 
 
 the point G perpendicularly to the 
 plane of delineation at e, carry this 
 point to the ground line of the 
 picture at 0, and from it draw a line 
 to the point F ; the intersection, g, 
 of this line with the perpendicular 
 drawn from d will be the perspec- 
 tive position of the point G of the 
 Plan. If we suppose these two points a and g to be the 
 extremities of a straight line, as shown by the dotted line a g 
 in the Plan, then the dotted line A g in the Representation 
 would be the perspective appearance of it. It must then be 
 evident that as by this means we can ascertain the perspective 
 direction and length of one straight line, the direction and 
 length of any other may be found in the same manner, and 
 consequently, any plane figure of straight lines may be drawn 
 in perspective by means of the point of sight only as a 
 vanishing point ; thus, if we place another point at H on the 
 Plan, and draw the (dotted) lines h g, g a, and a h, we have 
 the plan of a triangle ; and by rinding the position of h in 
 the picture in the same manner as the points A and g were 
 found (shown in the diagrams), by drawing the lines G A, 
 A H, and g h, as is done by dotted lines in the Representation, 
 it presents the appearance the triangle would assume viewed 
 as described. Although the mode described in Figs. 16 and 
 18 for finding the perspective form of any plane figure would 
 produce a correct representation were it ever so complicated, 
 it would be found but a round-about method of proceeding 
 where the figure contains a quantity of lines. The usual and 
 most convenient manner of finding the perspective directions 
 of lines that are at an angle with the plane of delineation is 
 by means of their respective vanishing points ; and we will 
 
PERSPECTIVE FOR STUDENTS. 
 
 09 
 
 therefore proceed in the next chapter to point out the manner 
 of finding the vanishing points for straight lines at whatever 
 angle they may lie with the plane of delineation, and how- 
 ever numerous the variety of their directions. 
 
 CHAPTER IV. 
 
 We have already more than once stated that all lines that 
 in the original object are parallel, in their perspective repre- 
 sentations incline towards each other and meet in the same 
 point, somewhere on the horizontal line (see Part I. pp. 9, 
 10, Figs. 14, 15 ; pp. 11, 12, Plate 1, Fig. 2). It must be 
 further understood, that in drawing a variety of objects, every 
 change of angle a line makes in its inclination towards the 
 plane of delineation requires a fresh vanishing point. Thus 
 in the diagram (Fig. 1 9) the rectangular figures a, b, c, and d 
 
 Fig. 19. 
 
 represent the plans of four objects standing on the ground 
 plane F, the situation of the plane of delineation being at E ; 
 so situated, each figure would require two distinct vanishing 
 points, the sides of each figure lying at different angles with 
 the plane of delineation. The same would be the case in 
 representing any polygon in perspective ; the sides of the 
 
 F 2 
 
100 
 
 PERSPECTIVE FOR STUDENTS. 
 
 polygon being at different angles with the plane of deline- 
 ation, would require distinct vanishing points. 
 
 In order to find the vanishing points requisite for putting 
 any object or combination of objects in perspective, it is 
 indispensable to have the plan of them, the position of the 
 plane of delineation (its base line drawn), and the position of 
 the spectator. Whatever may be the direction of any Jne 
 on the plan for which a vanishing point is required, it is 
 found by ruling a line (parallel to the line on the plan) from 
 the point marking the position of the spectator, and con- 
 tinuing the same till it intersects the plane of delineation ; the 
 point perpendicularly over this on the horizontal line will be 
 the vanishing point not only for the line selected on the plan, 
 but for every line parallel to it. Thus in Fig. 20, let A be the 
 
 Fig. 20, Plan. 
 d 
 
 c 
 
 position of a square, b the plane of delineation, and c the 
 position of the spectator. The lines a b and c d being 
 parallel lines in the geometrical plan, in the perspective 
 representation must tend to the same point; to find this 
 point, the edge of a parallel rule must be placed against 
 either a h or c d, and brought down over c, from which 
 
PERSPECTIVE FOR STUDENTS. 
 
 101 
 
 point a line must be ruled till it intersect the line b,* as at e. 
 The lines a c and b d are also parallel, but at a different angle 
 with the plane of delineation to a b and c d, and will require 
 a fresh vanishing point, which is found in the same manner ; 
 that is to say, from the position of the spectator, c, a line pa- 
 rallel to a c, or (what is the same thing) b must be drawn 
 to the plane of delineation at /, and the point perpendicularly 
 over this, on the horizontal line will be the vanishing point 
 for the lines a c and b and for all lines in the plan parallel 
 to them. Let us now carry these points from the plane of 
 delineation to the ground line of the picture, as in the 
 previous examples, and proceed to put the square figure 
 in perspective, first drawing the horizontal line D in the 
 Representation (the height of which depends on the height 
 
 Fig. 20, Representation. 
 
 E F 
 
 V 
 
 
 
 
 h 
 
 
 
 
 
 
 
 
 e j k g h f 
 
 of the spectators eye above the ground plane), and placing 
 the vanishing points E and f on it respectively perpendicular 
 
 * The plane of delineation being an imaginary plane, may be sup- 
 posed to extend to any distance. The visual rays must come within 
 that portion of it where the picture is supposed to be, and, as we have 
 shown, the whole might be accomplished by using the point of sight ; 
 in such case, no more than that space would be required : but it rarely 
 happens when more than one vanishing point is required that they fall 
 within the picture. Where objects stand in a direction nearly parallel 
 with the plane of delineation, the vanishing points are at a great dis- 
 tance. It is therefore necessary, in finding the positions of the vanish- 
 ing points, to extend the line representing the plane of delineation 
 some distance both to the right and to the left, and the same will be 
 required with the horizontal line in order to mark the vanishing points 
 on it. 
 
102 
 
 PERSPECTIVE FOR STUDENTS. 
 
 over the points e and /. On the plan, draw the visual rays 
 a c and c c, carry their points of intersection, h, on the 
 plane of delineation to the ground line of the picture, and 
 draw perpendicular lines from each. As in the preceding 
 examples, the point a will come somewhere on the perpendi- 
 cular drawn from g, and the point c somewhere on that 
 drawn from h. Now in order to find the position of a on 
 the perpendicular drawn from gr, let us refer back to the 
 square we put in perspective in a different position (Figs. 1 5, 
 Plan and Representation), and we shall there find that in 
 order to get the position of the point corresponding with a 
 of our present figure, we first drew the visual ray b c, carried 
 the point of intersection d to the ground line of the picture, 
 and, as in the present case, drew a perpendicular line from 
 it ; then in order to find the point of intersection on it, the 
 point b was brought perpendicularly to the plane of deli- 
 neation at /, / carried to the ground line of the picture, and 
 a line ruled from it to the point of sight k, which gave the 
 point of intersection b. In the present figure (20), the 
 points a, 6, d are found in the same way as the corre- 
 sponding points 6, A, c, j (Fig. 15) ; only as the vanishing point 
 for the line a c in this case is not the point of sight, but the 
 point F, instead of bringing a to the plane of delineation 
 by a perpendicular line, it must be brought forward in the 
 direction c a, as at j, and j carried to the ground line of the 
 picture ; if a line be ruled from this to the vanishing point 
 p, we have the perspective position of the point a on the 
 perpendicular over g, and the perspective position of the 
 point c on the perpendicular over h ; the line j 1 a of the Repre- 
 sentation being the perspective length and direction of the 
 line j a of the Plan, the line a c the perspective representa- 
 tion of the line a c oi the Plan, and the portion c f of the 
 line j a c F, the continuation of the line^ a c in the direction 
 of the dotted line in the Plan, as far as it is possible for it 
 to be seen. The positions of the points b and d might be 
 found in the same way as that employed for finding the posi- 
 
PERSPECTIVE FOR STUDENTS. 
 
 103 
 
 tions of the points a and c, i. e. by drawing visual rays from 
 b to c, and from d to c, carrying their points of intersection 
 to the plane of delineation, and drawing perpendiculars from 
 them, then bringing the point b to the plane of delineation 
 in the direction d b, and carrying the point of contact to the 
 ground line of the Representation. If from this point a line 
 were drawn to the vanishing point f, its intersections with 
 the perpendicular lines found by the visual rays b c and d c 
 would fix the perspective positions of the points b and as 
 the intersections of the line j f fixed the points a, c on the 
 perpendiculars over g and h. This manner of proceeding, 
 however, though perfectly accurate, is not the readiest way 
 to determine the positions of these points ; it is quite suf- 
 ficient to draw the visual ray b c, and carry the point of 
 intersection, to the ground line of the representation, and 
 draw a perpendicular from it, on some part of which the 
 point b will come. The position of the point a is already 
 ascertained, and the vanishing point for the line a b ; there- 
 fore, if a line is drawn from a to the vanishing point E, the 
 point of intersection with it and the perpendicular drawn 
 from is the position of the point b. F being the vanishing 
 point for the line b d (this line being parallel to a c), from b 
 draw the line b F ; E being the vanishing point for the line 
 c d (c d being parallel to a from c draw the line c e ; the 
 point of intersection of the two lines b f and c e is the per- 
 spective position of the point d of the Plan, and completes 
 the perspective drawing of the square a of the Plan by means 
 of the vanishing points. 
 
 Having shown by the diagrams of Figs. 16 and 18 how 
 a triangle (or any other rectilinear figure) may be drawn in 
 perspective by finding the positions of the points at the 
 extremities of the lines, using the visual rays and point of 
 sight only as a vanishing point, we will now take a similar 
 figure, and point out the manner of drawing a perspective 
 representation of it by means of the respective vanishing 
 joints for each line of the triangle, and comparing the former 
 
104 
 
 PERSPECTIVE FOR STUDENTS. 
 
 mode, that of Fig. 18, with the one we are about to describe, 
 show that the result is the same. 
 
 Fig. 21, Plan. 
 
 c 
 
 C 
 
 Let A in the plan, Fig. 21, represent the triangle, b the plane 
 of delineation, and c the position of the spectator. Find the 
 respective vanishing points on the Representation (after draw- 
 ing the horizontal line) for the lines a b, a c, and b c, the 
 sides of the triangle a at D, e, and f ; * draw the visual rays 
 a c and b c, and carry the points of intersection g^ h to the 
 ground line of the picture, from each of which draw a per- 
 pendicular line ; bring the point in the direction c 5, to the 
 plane of delineation at /, carry j to the ground line of the 
 picture, and to the vanishing point for the line b c (f) draw 
 
 * This being a different figure to the last, and knowing from experience 
 as a teacher how trifling a variation will sometimes confuse beginners in 
 these matters, we repeat for their assistance the proceedings of the last 
 figure ; from c, the position of the spectator, parallel to b c, draw the 
 line c ft the point on the horizontal line perpendicularly over f will be 
 the vanishing point for the line b c, or any lines that may be parallel to it. 
 From c, parallel to a b, draw the line c d; the point d on the horizontal 
 line perpendicularly over d, will be the vanishing point for the line a b 
 and all lines parallel to it. From c, parallel to a c, draw the line c e,- the 
 point e on the horizontal line perpendicularly over the point e will be 
 the vanishing point for the line a c and all lines parallel to it. If a 
 figure consist of twenty or more sides, each side in a different angle, their 
 respective vanishing points may be found in the same manner. 
 
PERSPECTIVE FOR STUDENTS. 
 
 105 
 
 the line j f ; the intersection of this with the perpendicular 
 from h is the perspective position of the point b. From the 
 vanishing point D (the vanishing point for the line a b), 
 through the point draw a line till it meet the perpendicular 
 drawn from gr, this will give the position of the point ay* from 
 a draw the line a e, the intersection of which with the line 
 b f gives the perspective position of the point e, which com- 
 pletes the figure of the triangle, a b c, in perspective, and by 
 a mode much more simple than that described in Fig. 18. 
 
 To prove, however, that the result would be the same 
 whichever mode of proceeding were adopted, we have drawn 
 the visual ray c c by a dotted line, and carried the point of 
 intersection, to the ground line of the picture, from which 
 we have also drawn a perpendicular (dotted) line. We have 
 also by a perpendicular from c to the plane of delineation 
 (c I) determined the position of the point of sight, which we 
 have placed perpendicularly over I on the horizontal line at 
 G ; the point c we have brought perpendicularly to the plane 
 of delineation at m ; we have placed m on the ground line of 
 the Representation, and from it drawn a line to the point of 
 
 Fig. 21, Representation. 
 F G E D 
 
 \ 
 
 
 / 
 
 
 
 
 / 
 
 
 
 
 1 a 
 
 ! ! 
 i • i 
 
 i * \ 
 
 
 \ 
 
 
 fglkmhej d 
 
 sight, g. It will be seen that the intersection of this line 
 with the perpendicular drawn from k is in the same point (c) 
 as that where the lines a r> and b f intersect each other, 
 
 * This is a much readier way of finding the position of a than if we 
 had brought, as was done, with the point b, the point a of the Plan, in the 
 direction b a, to the plane of delineation, carrying that point to the ground 
 
 F 3 
 
106 
 
 PERSPECTIVE FOR STUDENTS. 
 
 proving that the same result may be arrived at by different 
 means. 
 
 After perusing the descriptions given, and examining the 
 diagrams illustrating them, for rinding the positions of van- 
 ishing points, a little attention ought to make the student 
 comprehend why it is that the lines of original objects that 
 are situated at a right angle with the plane of delineation have 
 their vanishing point in the point of sight, and that those lines 
 which in an original object are parallel to the plane of deline- 
 ation have no vanishing point, but are drawn parallel to it, or 
 rather to the ground line of the picture. This will be fully 
 understood by comparing the plans of Figs. 20 and 21 with 
 that of Fig. 15 ; for if we require to find the vanishing point 
 for the sides b c j of the square d, Fig. 15, both of which 
 are at a right angle with the plane of delineation, it is clear 
 that a line drawn from c, the position of the spectator parallel 
 to them, to the plane of delineation at a, must be perpendicu- 
 lar to it, which is always the position of the point of sight. 
 Again, according to this rule for finding the vanishing points 
 of lines from a plan, if we endeavour to find one to which 
 the sides b c and h j should be drawn, a line through the 
 point c parallel to them must also be parallel to the plane of 
 delineation, and consequently could never meet ; hence it is 
 that all lines that in original objects are parallel to the plane 
 of delineation, are drawn parallel in the perspective repre- 
 sentations. This manner of representing objects is frequently 
 called Parallel Perspective, and that where all the sides of an 
 object are at an angle with the plane of delineation, Oblique 
 Perspective. 
 
 The object of this work not professing more than to 
 make the student thoroughly comprehend the system on 
 which he is to proceed in making perspective drawings, 
 
 line of the picture, and from it ruling a line to the vanishing point d. The 
 result is obviously the same, as it can make no difference whether the line 
 a b d is commenced from one point or the other, the direction must be the 
 same. 
 
PERSPECTIVE FOR STUDENTS. 
 
 107 
 
 the examples already given are sufficient, it is hoped, to 
 enable him to draw any plane rectilinear figure in perspec- 
 tive that may be put before him. No example has been 
 given for drawing curves, but as we have in Part I. pp. 35, 
 36, already stated that in order to draw curves in perspec- 
 tive, rectilinear figures must first be made, to get intersecting 
 points through which the curves are to be drawn, the ex- 
 amples already given are considered ample for the present. 
 Nevertheless, before proceeding farther, the student, if so dis- 
 posed, will find it to his advantage to take the plar of the 
 circle (Part I. Prob. V. Plate VI.) with the right lines about 
 it, and marking a point for the position of a spectator and a 
 line for that of the plane of delineation, draw it in perspec- 
 tive, as also Figs. 4 aud 6, Part I. Plate XIII. 
 
 CHAPTER V. 
 
 We have hitherto confined our examples entirely to the 
 representations of plane figures, without taking into considera- 
 tion at all the height or thickness of objects. We will now 
 proceed to show how solid figures are to be drawn in perspec- 
 tive upon the principles we have already laid down. In order 
 to be able to give a correct representation of any solid figure 
 in perspective, it is not only requisite that we should have the 
 form of the base of the object on a plan with its relative posi- 
 tion with the plane of delineation and position of the spectator, 
 but we also require, either by drawing or description, the form 
 and dimensions of its different parts. If the solid to be repre- 
 sented be simply a cube, it is unnecessary to have more than 
 the plan furnished, as, one face of a cube being given, the 
 remaining five are known to be similar. It must be under- 
 stood, that in drawing plans from which perspective draw- 
 ings are to be executed, a mere ground plan will not be 
 found adequate to the purpose, for as the base line only of 
 
108 
 
 PERSPECTIVE FOR STUDENTS. 
 
 the plane of delineation is used for finding the perspective 
 positions of the different parts of a structure, projections, 
 recesses, &c., though they may occur twenty or thirty feet 
 above the base of a building, must be first drawn on the 
 geometrical plan, their perspective positions found, and then 
 carried up to the height required ; in fact, the plan of a 
 building required for making a perspective drawing must 
 consist not only of the ground plan, but of a series of hori- 
 zontal sections to the very top, wherever any change of form 
 occurs. This will be better understood as we proceed, and 
 when we have occasion for such a plan, we will recur to the 
 foregoing remarks. 
 
 The cube being the most simple form of solid to put into 
 perspective, we will select that as our first example. In Fig. 
 15, pp. 90, 91, 93, we have drawn in perspective the plan of a 
 cube as seen in a certain position — in what is termed parallel 
 perspective. In referring to this figure (in which it will be 
 remembered that the sides b h and c j\ from their being at a 
 right angle with the plane of delineation, have their vanish- 
 ing point in the point of sight), the points b and c are said 
 to be brought to the plane of delineation at / and which 
 makes the distance between /and g exactly the same as that 
 between b c and h j. To make this perfectly clear, let us 
 suppose the dotted lines g c j and / b h to be grooves, on 
 which the face of a cube perpendicularly over b c could be 
 slid backwards and forwards ; if we were by means of these 
 grooves to slide the square forward, we should positively 
 bring the points b and c up to the plane of delineation. The 
 pupil can therefore have no difficulty in understanding that 
 in Fig. \b,fg on the ground line is the width of the square 
 right up to the plane of delineation, and that the line b c in 
 the Representation is the width the same line would appear 
 at the distance of fboi the Plan from the plane of delineation. 
 Now let us imagine the square d (Fig. 15) to be a base on 
 which a cube stands, and that this cube could be slid forward 
 on the grooves, up to the plane of delineation. We should then 
 
PERSPECTIVE FOR STUDENTS. 
 
 109 
 
 have the front face of a square absolutely on the plane of deli- 
 neation ; we should have a perpendicular line over /the length 
 of b c, one of equal length over and a line parallel to / g 
 passing from the top of one to the top of the other ; in fact, 
 the form of the square on the plane of delineation (see Fig. 22, 
 Rep.). It is unnecessary in constructing this cube to go over 
 
 Fig. 22, Plan. 
 
 the ground a second time for rinding the position of the 
 square D on the picture, as in Fig. 15, the plan of which we 
 have given again in Fig. 22. The square in this plan is 
 placed distant from the plane of delineation exactly its own 
 width, and for the advantage of showing the lines more dis- 
 
 Fig. 22, Representation 1. 
 
 i g 
 
 tinctly, the position of the spectator a little to the left. Those 
 points on the plane of delineation only are introduced that 
 
110 
 
 PERSPECTIVE FOR STUDENTS. 
 
 are requisite for drawing the square in perspective, as de- 
 scribed for Fig. 15, Representation 2 ; and in the diagram of 
 the Representation, Fig. 22, all lines are erased but those 
 necessary for our present purpose. In case the student may 
 have forgotten any portion of the directions for drawing 
 Fig. 15, Representation 2, Jie can refer back to p. 93 to refresh 
 his memory. 
 
 We have above stated, that in order to find the points for 
 constructing a cube in perspective standing over the square 
 b c j A, we must suppose the face of it opposite to the 
 spectator, to be brought forward to the plane of delineation, 
 and according to that description, we must construct a square 
 on the line / (7, as / g k I, which is really the geometrical 
 elevation of the cube of its full size. It is very evident that 
 the perspective position of the point k must come some- 
 where on a perpendicular line over the point 6, and the 
 perspective position of the point I somewhere on a perpen- 
 dicular line drawn from c ; draw then from each of these 
 points, b and c, an indefinite perpendicular line, and from 
 each of the points k and I draw a line to the vanishing point 
 (the point of sight) k ; where the line k K intersects the 
 perpendicular drawn from b at m, it gives the perspective 
 position of the point k ; where the line I K intersects the 
 perpendicular drawn from c at w, it gives the perspective 
 position of the point I ; by joining the points m and w, we 
 have in the figure b c n m the appearance of the figure 
 f g Ik at the distance of / b from the plane of delineation 
 shown in the Plan, and the plane b c m n* being parallel to 
 
 * The student must here understand, that as we speak of the lines in 
 the plane figure of the square in the Plan being parallel to or at an 
 angle with another line, so do we speak of the planes in a solid figure ; 
 thus the square surface perpendicularly over the line b c we call a plane 
 parallel to the plane of delineation, as we also call the perpendicular sur- 
 face over the line h j, the surfaces of the back and front of a cube neces- 
 sarily being parallel. In speaking of the sides of the cube perpendicularly 
 over the lines b h and c j, we say that the planes of the sides are at right 
 angles with the plane of delineation. The top and bottom of the cube are 
 
PERSPECTIVE FOR STUDENTS. 
 
 Ill 
 
 the plane of delineation, does not change its form, but only 
 decreases in magnitude according to its distance. The angles 
 of the cube standing over the points h and j of the Plan, 
 must now be drawn perpendicularly from the points h and j 
 of the Representation, that from h up to the line k k at o, 
 and that from j up to the line I K at p ; by joining the points 
 
 and jp, h j p o will be the representation of the square plane 
 at the back of the cube ; this also being in a plane parallel 
 with the two former, still preserves its geometrical form, 
 though smaller from its increased distance. Thus, b c n m 
 being the position and form of the front of the cube, and 
 h j- p o of the back, m n p o must be the appearance of the 
 top, and b m o h of the side that is visible, it being unne- 
 cessary to point out that the sides of the top and the side of 
 the bottom that is seen, must have been drawn in determining 
 the positions of the points A, o, and p. If another cube stood 
 over the square in the Plan f g cb, the square f g Ik would 
 represent the front, b c n m the back,* and consequently 
 
 1 n m k the top, and / k m b the side visible to the spectator. 
 
 In this figure, as in Fig. 15, Representation 2, in order 
 that the principle on which the different planes of the cube 
 are drawn should be well understood, more lines are intro- 
 duced than are absolutely required to represent the figure it 
 assumes ; a more simple manner of drawing the Representa- 
 tion would be as shown in Fig. 22, Representation 2. Having 
 drawn (as described, Fig. 15) the square b c j h m per- 
 spective, draw from each of the points, 6, £, and an 
 indefinite perpendicular line ; from the point g, set up the 
 perpendicular height of the cube at Z, and from / draw the 
 
 also in planes at right angles with the plane of delineation, but they are in 
 horizontal planes, and the sides in perpendicular planes. It is as easy, 
 after a little practice, to comprehend the terra " plane," as the word 
 " surface." 
 
 * In order to find the positions of certain points, it is frequently 
 necessary to draw parts that cannot by possibility be seen, and in order 
 to make this figure perfectly intelligible, several lines are drawn as if the 
 cube were transparent. 
 
112 
 
 PERSPECTIVE FOR STUDENTS. 
 
 line I K ; where I K intersects the perpendicular from c, it 
 gives the point n of the preceding figure. The top of the 
 cube being parallel with the bottom, n m must be parallel 
 
 Fig. 22, Representation 2. 
 K 
 
 to c b ; draw then from n a line parallel to c 6, and where 
 this intersects the perpendicular from 6, it gives the point m ; 
 from m draw the line m K ; the intersection of this with the 
 perpendicular from h gives the point o ; o p being parallel 
 to m n, draw from o, parallel to m n, a line to meet the line 
 I k, which gives the point p : the intersection of these two 
 lines obviates the necessity for drawing a perpendicular 
 from j. 
 
 Suppose another cube to stand upon the square e, and it 
 is required to put this cube also in perspective behind the 
 cube D. To do this, we must draw the visual rays 1 c and 
 3 c, and carry the points of intersection 5 and 6 to the 
 ground line of the Representation, and from each point draw 
 a perpendicular line between / K and m K ; from the points 
 of intersection on / k draw horizontal lines to meet the line 
 g k ; from the points of contact on g K draw perpendicular 
 lines to meet the line I K ; from the points of contact on I k 
 draw horizontal lines to m K ; the points of contact with the 
 line m K will be the same as with those from the perpen- 
 diculars drawn from the points 5 and 6, and will complete 
 the perspective drawing of a cube standing over the square 
 E of the Plan. 
 
PERSPECTIVE FOR STUDENTS. 
 
 113 
 
 Let us place a third square (p) on the plan, over which we 
 will suppose another cube to stand, and we shall perceive 
 with how much facility this third cube may be drawn in 
 perspective. Bring the points 7 and 8 to the plane of 
 delineation at 9 and 10 (in the same way as the points b c 
 of the square D were brought to /, g) ; bring these points to 
 the ground line, and from each of them draw a line to the 
 vanishing point K. The square F being (as is seen in the 
 Plan) at the same distance from the plane of delineation as 
 the square e, all the points for drawing the square F and 
 the cube standing on it may be found from the cube e 
 already drawn, as follows : continue the horizontal lines 
 2 — 1 and 4 — 3 till they intersect the lines 10 K and 9 k; 
 from the points of intersection 7 and 8, 11 and 12, draw up 
 indefinite perpendicular lines ; continue the horizontal line 
 of the top of the front face of the cube e to intersect the 
 perpendiculars from 7 and 8 at the points 13 and 14, and 
 from each of these draw a line to the vanishing point K. If 
 a line be now ruled from the points where the lines 13 k and 
 14 k intersect the perpendiculars drawn from 11 and 12, it 
 will complete the perspective drawing of the cube standing 
 over the square f of the Plan, and show that having by 
 means of the visual rays fixed the positions of certain points 
 in one object, the positions of the points required for drawing 
 another object may be found from the first without the 
 necessity of additional visual rays, and the result will be the 
 same with less labour. To illustrate this, we have intro- 
 duced the visual rays 8 c and 7 c, and placed the points of 
 intersection r on the ground line ; it will be seen by the 
 perpendicular (dotted) lines drawn from these points, that 
 they pass directly through the points 7 and 8, found by a 
 different mode. 
 
 The principle on which a solid is drawn in perspective, 
 where all the planes are at an angle with the plane of 
 delineation, is as simple as the one just given, Fig. 22, 
 where some of the planes are parallel to it. In the example 
 
114 
 
 PERSPECTIVE FOR STUDENTS. 
 
 given, p. 100, Fig. 20, we have put in perspective a square 
 having all its sides at an angle with the plane of delinea- 
 tion. Let us refer to these diagrams, Figs. 20, supposing a 
 cube to stand over the square A, and proceed to draw a 
 
 Fig. 23, Plan. 
 
 perspective representation of it, according to the Plan.* In 
 this representation, the square a is put in perspective in the 
 manner described for the Representation, Fig. 20, and the 
 
 Fig. 23, Representation. 
 
 
 
 
 
 E 
 
 
 
 P 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 
 d> 
 
 
 
 
 
 
 c 
 
 
 ? 
 
 > 
 
 % 
 
 
 
 
 
 a 
 
 
 
 
 
 
 f q h g jh 8 e 
 
 * The same plan we have given, Fig. 20, would fully answer the pur- 
 pose for constructing a cube on the square a, but a new one is considered 
 requisite, in order to introduce some additional lines, which would have 
 caused considerable confusion if put into the former one; the principle for 
 drawing it is the same, although the figure is reversed. 
 
PERSPECTIVE FOR STUDENTS. 
 
 115 
 
 point a is brought to the plane of delineation at j\ as the 
 point c of Fig. 22 was brought to the plane of delineation 
 at only that as the side c j is at a right angle with the 
 plane of delineation, the point c is brought (following the 
 direction of j c) perpendicularly to it ; whereas the line a c 
 of Fig. 23, not being at a right angle with the plane of 
 delineation, the point a is brought to it in the direction of 
 c a. From this point j, set up the perpendicular height oi 
 the cube (the length of any side of the square A of the Plan) 
 at I ; this point will determine the height of the four angles 
 of the, perspective square over the points a, 6, c, d ; we have 
 already got the perpendiculars from a, 6, and £, in the lines 
 drawn from g, and A, and may therefore proceed at once 
 to determine the height of them. From I draw a line to the 
 vanishing point F, which will determine the height of the 
 perpendicular lines over a and c at m and n, as the line I K 
 (Fig. 22, Representation 2) determined the positions of the 
 points n and p. The point m, in Fig. 22, Representation 2, 
 was found by drawing a line parallel to the ground line from 
 a to intersect the line drawn up from 6, on account of the plane 
 c n m b being parallel to the plane of delineation ; but the 
 plane standing over the line a b, Fig. 23, being at an angle 
 with the plane of delineation, all the horizontal lines on that 
 plane must be drawn to the same vanishing point ; therefore 
 from the point m, a line must be drawn to the vanishing 
 point E, the intersection of which w r ith the perpendicular 
 from 6, determines the height of it at o : if a line be now 
 drawn from o to the vanishing point f, and from n to the 
 vanishing point E, the point of intersection p of the two lines 
 will be found perpendicularly over and completes the 
 perspective drawing of a cube standing in the position de- 
 scribed in the Plan, Fig. 23. 
 
 In order to determine the heights of the different angles 
 of the cube a m, c n, d p, and b o, it is immaterial whether 
 the point a is brought to the plane of delineation in the 
 direction c a to j, or whether it is brought forward in the 
 
116 
 
 PERSPECTIVE FOR STUDENTS. 
 
 direction b a ; in fact, either of the three points a, 6, or c, 
 would answer equally well to find the heights of these per- 
 pendicular lines ; for if the point c were brought forward to 
 the plane of delineation in the direction of the line d 
 as at q (in the Plan), and this point carried to the ground 
 line of the Representation, a perpendicular drawn from it the 
 geometrical height of the cube, will be found to produce the 
 same result as that produced by the point /, as is shown in 
 the Representation. The point q is placed on the ground 
 line, and the geometrical height of the cube set over it at 
 r ; from r a line is drawn to the vanishing point E, and it 
 will be seen that the point of intersection with the perpen- 
 dicular from £, is in the same point with that drawn from I 
 to the vanishing point F ; and the intersection giving the 
 point p must necessarily be the same. Again, if the point 
 b was brought forward to the plane of delineation in the 
 direction d 6, as at s (Plan), s carried to the ground line, 
 and the geometrical height of the cube set over it at £, and 
 this point chosen for determining the height of the per- 
 pendicular angles, the same result would ensue as in the two 
 preceding cases ; the line t f intersecting the perpendicular 
 from b in the same point as the line m e. 
 
 If the perspective height of the cube is to be determined 
 from the line q r on c it determines by the same line the 
 height of dp ; the point m would be found by ruling a line 
 from the vanishing point f through n to intersect the per- 
 pendicular from a ; the intersection will be found in the same 
 point as found by the line I F ; the point o, in this case, 
 would be found on the perpendicular line from 6, either by 
 the intersection of a line drawn from the point m to the 
 vanishing point E, or of a line drawn from the vanishing 
 point F, passing through the point p. It has then been 
 clearly shown that any point may be selected, and that the 
 perspective height of the cube being found at any one angle, 
 the height of the remaining three may be determined from it. 
 
 The student, if he has paid attention to the directions 
 
PERSPECTIVE FOR STUDENTS. 
 
 117 
 
 given tor drawing all the preceding figures, ought now to be 
 able to draw in perspective with tolerable facility a more com- 
 plicated one than any we have yet used for our illustrations, 
 and that by simple description, By a mere servile copying, 
 in his progress through the work, the diagrams we have 
 introduced, he may be liable to forget some of the earlier 
 ones in his anxiety to get forward, and he is strorgly recom- 
 mended, as he proceeds, not only to draw with care and more 
 than once each separate figure as it occurs, but to vary the 
 positions of the objects or the spectator in his plans, and then 
 put them in perspective. To draw an octagonal tower from a 
 plan in perspective, is not more difficult than to represent the 
 cubes in Figs. 22 and 23, it only requires a greater number 
 of lines ; but by a careful attention to the rules we have 
 already given, a tower of any number of sides may be drawn 
 without the necessity for any extra directions, as it ought to 
 be quite superfluous to inform the student, that if the hori- 
 zontal lines that in the perspective representation tend to any 
 vanishing point are situated above the spectator's eye, they 
 incline downwards instead of upwards. In order, therefore, 
 that the student may satisfy himself that he has fully com- 
 prehended the foregoing rules, we will furnish him with the 
 plan of an octagon tower, with the relative situations of the 
 spectator and tlie plane of delineation, of which he must 
 draw the perspective representation from description. We 
 have, therefore, in the diagram (Fig. 24) given the plan of 
 the tower, a, the position of the spectator at c, his eye 
 situated five feet above the ground plane, and the plane of 
 delineation at B, marking each angle of the tower at 1, 2, 3, 
 4, 5, 6, 7, 8. According to the position in which the spec- 
 tator is placed in looking at an octagon tower, he may see 
 either three or four of its sides ;* in the position in which it is 
 
 * It is possible to place the spectator opposite either angle of the 
 octagon, so that only two sides would be visible ; indeed, if he were close 
 to such a tower, perpendicularly opposite either of the sides, that side only 
 would be seen. But we shall be able to point out in our remarks on the 
 
118 PERSPECTIVE FOR STUDENTS. 
 
 here placed, four of the sides are visible, 8 — 1, 1 — 2, 2 — 3, 
 
 and 3 — 4. The width 
 these sides would appear 
 in the picture is deter- 
 mined by the visual rays 
 drawn from the points 
 8, 1, 2, 3, 4, precisely 
 similar to the manner of 
 finding the width of the 
 sides of the cube, Figs. 
 22 and 23 ; the points of 
 intersection, a, 6, c, 0, 
 on the plane of delinea- 
 tion, must be carried to 
 the ground line of the 
 picture, from each of 
 which a long perpendi- 
 cular line must be drawn 
 (the top of the tower be- 
 ing considerably above 
 the eye of the spectator). 
 In the position of this 
 figure relative to the 
 plane of delineation, we 
 see that the lines 1 — 2 
 and 6 — 5 are parallel 
 to it, and consequently 
 have no vanishing point ; that the lines 3 — 4 and 8 — 7 
 are at right angles with it, and will , consequently have 
 their vanishing point in the point of sight ; that the pa- 
 rallel lines 1 — 8 and 4 — 5, being at an angle, not a right 
 angle, with the plane of delineation, require a distinct vanish- 
 ing point, as will also, for the same reason, the parallel lines 
 
 positions to be chosen for making perspective drawings, that those posi- 
 tions where only one or two sides are seen are such as are inadmissible for 
 making perspective drawings. 
 
PERSPECTIVE FOR STUDENTS. 119 
 
 2 — 3 and 7 — 6. The position of these vanishing points then 
 must; be found on the plane of delineation, as described in 
 Figs. 20 and 21, carried to the ground line, and perpendicu- 
 larly over them set on the horizontal line. 
 
 Having set these points on the horizontal line, that is, the 
 point of sight, which should be marked D,* the vanishing 
 point for lines parallel to the sides 1 — 8 and 4 — 5, which 
 mark e, and the vanishing point for the sides 2 — 3 and 7 — 6, 
 which mark f, proceed to draw the figure in perspective. 
 It is immaterial, in determining the positions in the picture 
 of the points 1 and 2, which of them is brought to the plane 
 of delineation, as the position of one being ascertained, the 
 other may be got from it (see the points £, and Fig. 23). 
 Let us choose the point 1, and we shall find that it is imma- 
 terial also whether this point be brought perpendicularly to 
 the plane of delineation at /, or in the direction of the side 
 8 — 1, at g. If the former, f would be carried to the ground 
 line, and a line ruled to the point of sight d, the intersection 
 of which with the perpendicular drawn from would be the 
 perspective position of the point 1 ; if the latter, g would be 
 carried to the ground line, and a line ruled from it to the 
 vanishing point E, which would intersect the perpendicular 
 from b in the same point as the line / d (Fig. 21). The 
 position of the point 1 being ascertained, the remaining 
 corners may be found from it ; as, from 1 draw a line parallel 
 to the ground line, where this intersects the perpendicular 
 from c is the point 2 ; from this point draw a line to the 
 vanishing point f, the point of intersection with the perpen- 
 dicular from d is the position of the point 3 ; from 3 draw a 
 line to the vanishing point d, the point of intersection with 
 the perpendicular from e is the position of the point 4. If g 
 be the point chosen from which the position of the point 1 
 is determined by g e, the same line will give the point 8 
 
 * The student must be careful to mark the references as they are de- 
 scribed, as he proceeds ; he will by this means get on without difficulty ; 
 by neglecting to do so, he will get sadly perplexed. 
 
120 
 
 PERSPECTIVE FOR STUDENTS. 
 
 at its intersection with the perpendicular from a; if the 
 point 1 be determined by the line / D, then a line mus 
 be drawn from the point 1 to the vanishing point E to deter- 
 mine the point 8. These four lines, 8 — 1, 1 — 2, 2 — 3, and 
 3 — 4 of the base of the octagon tower, are all that can be 
 seen from the station c, and we have now to determine the 
 heights of the perpendicular angles over these points 8, 1, 2, 
 3, 4. The perpendicular lines from these points are already 
 drawn, and the height of one being determined, the heights 
 of the remainder may be ascertained from it. Supposing the 
 height of the tower to be five times the length of from 1 to 2, 
 this height must be set up on a perpendicular line from the 
 point g (on the ground line), and marked 7i, from which 
 point a line drawn to the vanishing point E will deter- 
 mine the height of the angles of the tower over 1 at j and 
 8 at k; the heights of the three remaining angles standing 
 over 2, 3, and 4, may be determined in the same way that 
 the points 2, 3, and 4 were found ; that is, from^ draw a line, 
 parallel to the ground line, to the perpendicular from c; from 
 the point of intersection draw a line to the vanishing point f 
 till it meets the perpendicular from and from this point of 
 intersection, a line to the point of sight d to meet the per- 
 pendicular from 0, which will complete the drawing of an 
 octagon tower, viewed from the position shown in the plan, 
 Fig. 24.* 
 
 * In giving the description for putting this figure in perspective, we 
 stopped at finding the positions of the points 8 and 4, four sides of the 
 octagon only being seen from the position at c. It may be as well to point 
 out how the whole plan of the octagon might be completed from the 
 points already found, without drawing any additional visual rays. It 
 will be seen that the lines joining the points 2 and 3 with 7 and 6, are 
 parallel with the lines 1 — 8 and 4 — 5, and consequently, to represent 
 them in perspective they must be drawn to the same vanishing point e : 
 therefore from 8 draw a line to the point of sight d, and from 2 draw a 
 line to the vanishing point e ; the intersection of these lines gives the 
 point 7 ; from 7 draw a line to the vanishing point f, and from 3 to the 
 vanishing point e ; the intersection of the lines gives the point 6 ; 
 
PERSPECTIVE FOR STUDENTS. 
 
 121 
 
 Fig. 25. 
 
 An experiment practically proving the accuracy of a 
 series of diagrams, not only tends to fix certain principles in 
 the mind of the student, but frequently, if the principles are 
 but imperfectly comprehended, is an inducement to him to 
 retread his ground, in order to render himself capable to 
 become the exhibitor of the same in his turn ; and moreover, 
 the satisfaction derived from witnessing the perfection of an 
 experiment gives great encouragement for perseverance in 
 the continuation of his studies. To this end we propose 
 describing a simple but most satisfactory experiment, that 
 will afford a convincing proof of the correctness of the 
 principles on which the preceding figures have been drawn. 
 
 In the annexed diagram, Fig. 25 (the whole of which is 
 drawn to a scale from the ob- 
 jects described), the parallelogram B 
 A B c D represents the top of a 
 common table, upon which over 
 the square E stands a cube ; * on 
 the line c D (which represents the 
 edge of the table), take any point 
 as the position of the spectator, 
 which we have here fixed at f, 
 and across the table parallel to the 
 edge c I), draw a line G h, to re- 
 present the base of the plane of 
 delineation, over which the plane 
 of delineation is supposed to d~ 
 stand; then from the corners of 
 
 the object on the table, draw the visual rays ; find the 
 
 from 4 draw a line to the vanishing point e, and from 6 a line parallel 
 to the ground line to meet it ; the point of intersection will give the 
 point 5, and complete the figure of the octagon in perspective. The same 
 result might be arrived at in various ways, but the principle would be the 
 
 same. 
 
 * We are obliged to specify some particular figure, but a work-box, 
 desk, book, or other object, no matter what, may be chosen* 
 Perspective, q 
 
122 
 
 PERSPECTIVE FOR STUDENTS. 
 
 Fig. 26. 
 
 c b a f 
 
 position of the point of sight by a perpendicular from p, 
 and bring one of the angies of the square on which the 
 cube stands perpendicularly to the plane of delineation, as 
 described in Figs. 15 and 16. Construct a parallelogram 1, 2, 
 3, 4, to contain the representation ; carry all those points to 
 the ground line, Fig. 26, and then draw the cube in per- 
 spective as described in Fig. 
 22. The representation, to 
 answer the purpose of our 
 experiment, must be drawn 
 on a piece of stiff paste- 
 board ; the height of the 
 horizontal line being placed 
 above the ground line, the 
 same height the eye is 
 situated above the edge of 
 the table c d, and should be 
 sufficiently elevated to enable the top of the object to be 
 distinctly seen. A strip of card must now be cut similar to 
 K, Fig. 26, a straight line 5 6 drawn across it, and from 
 this a perpendicular line must be drawn the length of the 
 space from the ground line to the horizontal line, to at 
 which point drill a small hole with a pin. Place this strip 
 of card upright on the edge of the table c D, the point 5 at 
 the point 7, and the point 6 at 8, which will bring the point 
 g exactly opposite the point of sight. 
 
 Let the form of the cube (the whole of the tinted figure) 
 in Fig. 26, be carefully cut out, and the piece of pasteboard 
 on which it was drawn set perpendicularly on the table in 
 the place where the plane of delineation is supposed to be 
 situated ; the point 3 standing on g, and the point 4 on n, 
 the points/, a, 6, c, d must necessarily come over their cor- 
 responding points on g h, and the hole g in the card per- 
 pendicularly opposite the point of sight J. If in this 
 position of the different parts, the student place his eye close 
 to the hole g in the strip of card k, he will find the cube, 
 
PERSPECTIVE FOR STUDENTS. 
 
 123 
 
 standing over the square e, to fit exactly to the hole cut out 
 of the pasteboard. In order to be quite successful in this 
 experiment, a perfect adjustment of all the parts is indis- 
 pensable ; the drawing must be made with great accuracy, 
 and the greatest care must be taken that the piece of paste- 
 board containing the figure of the cube, and the slip of card 
 through which the hole is pierced, stand perfectly perpendi- 
 cular to the plane of the top of the table. 
 
 This experiment may be repeated with advantage in a 
 variety of ways, all of which variations will illustrate some 
 portion of the text in the preceding chapters. It will bo 
 found, that the slightest change in the position of any of the 
 parts will destroy the effect : the position of the eye must 
 neither be moved to the right nor left, neither higher nor 
 lower ; if the line G h were drawn closer to c d, the hole would 
 be too large for the cube to appear to fit it ; if it were dr^wn 
 farther back, the hole would appear too small ; any change 
 in the position of the cube itself would alike destroy the effect. 
 
 If only one face of the cube were to be drawn, the figure 
 would be similar to D, the representation of A, Fig. 11 ; in 
 such a figure, whether the square stood parallel to or at an 
 angle with the plane of delineation, if pieces of twine were 
 attached to the four corners of .the real square, the four 
 strings first passed through the hole cut in the pasteboard 
 standing over g h, then through the hole g in the strip of 
 cardboard, and the strings pulled tight so as to form straight 
 lines from the original square to the point they would be 
 found to touch the four corners of the hole representing the 
 form of the square in perspective ; thus referring to Fig. 11, 
 and supposing a to be the front of the cube, D the hole cut 
 in the pasteboard, and the point E the hole in the sirip of 
 cardboard, the strings from the corners of the original 
 square drawn tight to the point w^ould touch thr* four 
 corners of the hole cut in the board, as the lines drawn from 
 the corners of the square A to E, touch the corners a y 6, 
 
 d y of the souare D. 
 
 a 2 
 
124 
 
 PERSPECTIVE FOR STUDENTS. 
 
 For teachers at schools, or for those who receive pupils 
 in classes, a small apparatus for this experiment would be 
 attended with a very trifling expense, and would prove as 
 advantageous to the teacher as to the pupil : to illustrate by 
 experiment being easier to the master than by description, 
 and much less difficult of comprehension to the scholar. 
 Any figure may be selected for this experiment, and the 
 representations of circular figures, to those not conversant 
 with perspective drawings, cause considerable astonishment. 
 
 After an attentive examination of the rules contained in 
 the preceding chapters, the student ought to experience no 
 difficulty in finding the situation on the horizontal line of all 
 vanishing points that are requisite for drawing in perspective 
 any plane figure, however complicated ; nor do we think he 
 ought to be at fault in constructing a solid figure upon it ; it 
 is frequently, however, necessary to have on the same plan a 
 variety of figures, drawn one within the other, representing 
 projecting and receding parts situated over the plan. To 
 represent these in perspective with accuracy, requires great 
 attention and considerable nicety ; and as we have given no 
 figure of this kind either in this or the first part of our work, 
 we will introduce a plan and perspective view of one of the 
 buttresses of Magdalene Bridge, Oxford, which affords an 
 excellent example for illustration. Before, however, pro- 
 ceeding to any more complicated representation, we propose 
 to make a few general observations, and compare the pro- 
 cesses described for drawing perspective in the First and 
 Second Parts. 
 
 In the various diagrams we have already given, it must 
 be quite evident that the same result in finding the perspec- 
 tive positions of points in a picture is to be attained in a 
 variety of ways ; and though, in the first instance, in order to 
 determine the position of some leading point or points from 
 which others may subsequently be drawn, it is requisite that 
 the relative positions of the plane of delineation, &c, must 
 be fixed, so that the general outline of the subject shall be 
 
PERSPECTIVE FOR STUDENTS. 
 
 Fig. 27. 
 
 Station, 
 ® 
 
126 
 
 PERSPECTIVE FOR STUDENTS. 
 
 arranged by the visual rays and vanishing points, much of 
 the detail may be accomplished by more simple means. This 
 is clearly shown in the manner of drawing the perspective 
 cube f, in Fig. 22, Representation 2, the whole of which 
 may be drawn without the necessity for introducing any 
 visual ray at all. The same may be observed by referring 
 to the Representation, Fig. 21, where the point c of the tri- 
 angle is formed by the intersections of the lines a D and b F, 
 without the necessity of a visual ray, as is also the point d in 
 the Representation, Fig. 20. 
 
 Those modes for making perspective drawings that are 
 attended with the fewest number of lines are always to be 
 preferred ; and it would surprise many who are not accus- 
 tomed to execute drawings in perspective, to witness the rapid 
 and very simple manner in which intricate drawings are made 
 by those who make it their business. In our endeavours to 
 explain with sufficient clearness the manner of finding the 
 perspective positions of certain points by means of drawing 
 the visual rays through the plane of delineation, we have in 
 every instance made the plan quite distinct from the repre- 
 sentation, which is really the fact, as it must never be lost 
 sight of that the picture you are making is to represent the 
 original objects as they would appear if traced on a sheet of 
 glass (-the plane of delineation) placed between the spectator 
 and the objects to be drawn. The manner most commonly 
 in use, however, is to make the ground line of the picture 
 and the ground line of the plane of delineation the same 
 line ; to place the position of the spectator above this line 
 according to his distance from the plane of delineation and 
 the plan of the original objects below it, the points of inter- 
 section of the visual rays on the ground line of the plane of 
 delineation thus come at once on the ground line of the pic- 
 ture;* those points required to be brought forward to the 
 
 * The student must understand, that in the diagrams given in this Part 
 of the work, as well as in the figures of the Problems in Part I. for the 
 purpose of instruction, lines of all kinds, whether to vanishing points, to 
 
PERSPECTIVE FOR STUDENTS. 
 
 127 
 
 plane of delineation are brought at the same time to the 
 ground line of the picture. This process is a much readier 
 mode than making the base of the plane of delineation ana 
 the ground line of the picture two separate lines, as we will 
 show in our next figure. 
 
 The original object of the First Part of this treatise on per- 
 spective was to furnish information just sufficient to enable 
 the amateur to make sketches from nature without violently 
 outraging perspective. The forms chosen and the directions 
 given for drawing them were as simple as the subject would 
 admit, and the Author trusts that it is impossible for any 
 intelligent person to go steadily through the pages without 
 comprehending the matter. Although the present Part goes 
 much farther into the art of perspective than the First, and 
 the mode pointed out for representing the perspective forms 
 of objects is different, yet there is nothing in the First Part 
 of the work to unlearn; an attentive perusal of the two parts, 
 with careful drawings made on a larger scale from the illus- 
 trations contained in both, we may venture to say, would 
 enable the student to draw in perspective any geometrical 
 figure set before him. It may appear to some that in giving 
 rules for drawing a number of figures in perspective in the 
 First Part, and leaving the explanation of the principles on 
 which perspective drawing is founded for the Second, is, to 
 use a homely adage, putting the cart before the horse ; but 
 perspective is generally allowed to be an extremely difficult 
 subject to write on, as it is necessary before we can enter 
 into the principles on which perspective drawing is founded, 
 
 distance points, or the visual rays that are requisite fo»* rinding points of 
 intersection,— in all cases the whole length of the line is drawn from point 
 to point ; but in the execution of a perspective drawing, where all lines 
 for finding the form required are erased, this is not required. All that 
 is necessary is to place the rule from point to point, and mark delicately, 
 but distinctly, only the point of intersection required. By this proceeding 
 a vast confusion of lines is obviated, and the progress of the drawing 
 rendered more simple. 
 
128 
 
 PERSPECTIVE FOR STUDENTS. 
 
 first to understand what perspective really is. To those who 
 are ignorant of drawing, the geometrical elevation of a 
 building appears more correct than a perspective representa- 
 tion, yet to those who understand the principles of drawing 
 it must be quite evident that a geometrical elevation, how- 
 ever useful it may be, cannot be a correct representation of 
 what we see, let the position of the spectator be where it 
 may ; as a simple geometrical drawing does not represent the 
 thicknesses either of projections or recesses, though they may 
 be ascertained generally by the depth of the shadows. The 
 frontage of a rectangular building may be so situated with 
 reference to the position of the spectator, as to present a rec- 
 tangular figure, but all recesses or projections on the face of 
 it must be drawn according to the rules of perspective if the 
 representation be really as it appears, which is not done in a 
 geometrical elevation. Hence we have preferred the system 
 of showing practically by the most familiar examples, in the 
 First Part, in the various figures from 7 to 15, that objects 
 vary considerably in their form according to their change of 
 position with reference to the spectator ; that parallel lines 
 viewed in certain directions appear to meet in a point called 
 the vanishing point ; how the position of these vanishing 
 points may be found on the horizontal line with sufficient 
 accuracy for ordinary sketches, with directions for determin- 
 ing the height of this line ; and then proceeding from these 
 premises to put a variety of figures in perspective ; showing 
 in the First Part what is meant by perspective drawing, and 
 leaving it to the Second to point out the principles on which 
 it is grounded. The First Part, in fact, being but an intro- 
 duction to the Second, the proper understanding of which is 
 greatly facilitated by an acquaintance with the First, some of 
 the diagrams of the Second being difficult to comprehend 
 without this knowledge. 
 
 In referring back to the first problem, Part I. p. 15, if we 
 were to proceed to draw a similar figure on the principles 
 we have described in the Second Part, which would be on the 
 
PERSPECTIVE FOR STUDENTS. 
 
 129 
 
 same plan as the Representation, Fig. 23, we should find 
 that after determining the positions of the lines a, e, f by 
 means of visual rays, and finding the positions of the vanish- 
 ing points G and H, as shown in the diagrams Figs. 20 and 
 21, by which the inclinations of the lines D, c, H, and J are 
 drawn, all the remaining parts may be as accurately deli- 
 neated by the system described for drawing this problem, 
 pp. 15 — 21, Part I., as if done by the rules given for draw- 
 ing the diagrams Figs. 22 and 23, as it is immaterial as 
 affects the accuracy of the drawing, whether the positions of 
 the points necessary for finding the width of the windows, 
 which were determined by means of a distance point, or the 
 point necessary for drawing the point of the gable, which 
 was determined by the use of diagonal lines, be found by 
 the means used in drawing Prob. I., or whether the positions 
 of all the points are found by means of the visual rays; 
 whichever mode is employed the result will be the same. 
 The point of distance is most valuable in perspective draw- 
 ing, and a variety of ways are shown by different writers on 
 perspective by which the position of this point may be deter- 
 mined. In the directions given for making perspective 
 representations of objects embraced in the diagrams from 
 Fig. 15 to Fig. 24, it must be very evident that the whole of 
 a perspective drawing may be executed without the necessity 
 for employing a distance point at all, the visual rays answer- 
 ing the same purpose ; but it is frequently the case that the 
 introduction of a distance point saves considerable labour, 
 and the directions given for fixing its position in Prob. I. 
 and II. Part I., and elsewhere, are sufficient for any pur- 
 pose, as they will produce intersecting points on any line 
 with as much accuracy as is to be obtained by visual rays. 
 
 If a drawing of the plan of the house, the original object 
 of the representation, Prob. I. Part I., were given to the 
 student, with the position of the spectator and situation of 
 the plane of delineation, it is barely possible he could have 
 any difficulty in putting in perspective the parallelograms of 
 
 G 3 
 
130 
 
 PERSPECTIVE FOR STUDENTS. 
 
 the two sides of the house, formed by the lines a, d, f, k and 
 a, c, E, J ; the heights of all the different parts being the 
 same as described for drawing the first problem, Part L 
 In the cut (Fig. 27) we have here introduced, we have placed 
 the plan of the building below the ground line, so as to bring 
 the intersections of the visual rays direct to it ; this will 
 prove a great saving of labour ; but it is necessary to point 
 out, that in all cases where this is done, the plan must be 
 drawn the reverse way to that where the base line of the 
 plane of delineation and the ground line of the picture are 
 distinct lines, which has been the case in all our previous 
 examples. This will be understood, by turning the figure 
 27, upside down ; in looking at it in this position, taking the 
 ground line of the representation as the base of the plane of 
 delineation, if the points of intersection on it were carried to 
 another line as the ground line of the picture (similar to the 
 diagrams, Figs. 22, 23), the gable end would be to the right 
 hand instead of to the left, the way we absolutely see it. 
 
 It is not our province to enter into any theories ; all that 
 is required is to make the student understand as a practical 
 fact, that where the plan is placed underneath the picture, so 
 that the base line of it (the ground line) is used at the same 
 time as the base of the plane of delineation, the plan must be 
 drawn reversed ; by making the ground line serve both 
 purposes, we gain a saving of labour, which is always an 
 object. The annexed diagram shows that the result will be 
 the same, whether we use a distinct line for the base of the 
 plane of delineation, carrying the intersecting points to the 
 ground line of the picture, drawing the plan as it stands 
 before us, or whether we make one line serve the purpose of 
 both, by drawing the plan reversed. Let A represent a plan 
 similar to that in Fig. 27, only drawn as it stands before us, 
 B representing the plane of delineation, from which the 
 points are to be carried to the ground line of the picture, c 
 the position of the spectator. The plan a is a reversed, and 
 the point c is in the same relative position to a as c to A ; the 
 
PERSPECTIVE FOR STUDENTS. 
 
 131 
 
 Fig. 28, 
 
 line B serving for the plan a and station point c, both as ground 
 line of the picture and base of the plane of delineation ; the 
 visual rays b c and d c, it will be seen, intersect the line B in 
 the same points as the visual rays b c 
 and d c. In comparing figure 27 with 
 Plate II. of the First Part, it will be 
 seen that the lines K and J in the plan, 
 are the lines by which the positions of 
 the vanishing points H and J are deter- 
 mined ; the angle a of the house touch- 
 ing the plane of delineation, is of course 
 drawn perpendicularly up from the 
 point 2 ; the perpendiculars E and F 
 are determined by the visual rays drawn 
 from a and b to the station point of the 
 spectator © ; the lines c, J, D, and k, are 
 drawn as in Fig. 23, Representation.* The 
 points by which the perspective width 
 of the windows was determined in drawing the house in 
 Prob. I., was by means of a distance point, full directions 
 for the manner of determining the position of which, were 
 given in the directions for drawing that problem ; and we 
 here propose to show that the result of finding them by this 
 means is precisely the same as if the visual rays had been 
 used. The distance from 1 to 9 in this diagram (Fig. 27), 
 will be found to be really the geometrical length of the 
 perspective line d, by comparing it with the line K of the 
 plan, and the divisions 10, 11, 12, 13, correspond on the 
 two lines R and K. From each of the points 10, 11, 12, 13, 
 on K of the plan, draw a visual ray to the station of the 
 spectator, intersecting the ground line (also the plane of 
 delineation) at c, d, e 9 /t From the point 9 through the 
 
 * As regards the height of the line a and the height of the windows, the 
 manner of determining them was fully explained at p. 20, Part I. 
 
 f In this figure, to avoid confusion of lines, the visual rays are not 
 drawn to the station point through the ground line, but only up to it 9 
 
132 
 
 PERSPECTIVE FOR STUDENTS. 
 
 corner of the house, 5, draw a line to the horizontal line, to 
 determine the point of distance, s;* then from each of the 
 points 10, 11, 12, 13, on r, draw a line to the distance 
 point s, to determine the perspective positions of these points. 
 If we now from each of the points on the ground line, c, d, 0,/, 
 draw a perpendicular line up to d, they will be found to 
 intersect that line in the same points, as the lines drawn from 
 the points 10, 11, 12, 13, on r, to the point of distance s, 
 proving that the positions of these points are determined 
 with as much accuracy by means of a point of distance, as 
 by drawing the visual rays. 
 
 In Prob. I. Part I., the perpendicular line dividing the 
 parallelogram A, c, e, j into its perspective halves, was found 
 by means of the intersection of the diagonal lines l and m 
 at the point 7 ; we shall find this mode for drawing the 
 perpendicular line N at its perspective distance from a and e, 
 equally correct as employing either a visual ray or a point 
 of distance. The point g on the plan is exactly midway 
 between 2 and a on the line J ; if from this point a visual 
 ray is drawn to the ground line at a perpendicular line 
 drawn from it will pass directly through the point 7 found 
 by the intersection of the diagonals l and m. Again, from 
 the point 1 on a, a horizontal line 1 j is drawn to represent 
 the geometrical length of the line c (equal to j of the plan) ; 
 if from j through the point 3 a line is ruled to the horizontal 
 line, it will give a point of distance T, by which the width of 
 any details on the gable end of the house may be determined 
 on the line c ; thus, if from the point the half of the 
 geometrical representation of c, a line is drawn to the point 
 of distance T, the intersection of it with the line c determines 
 the perspective centre, and is in the same point with the 
 
 were it not a figure for instruction, even this would be unnecessary, 
 as simply marking the points of contact at c, d, e, and f, would be 
 sufficient. 
 
 * All the references in this figure that occur in Proh. I. Part I., are 
 lettered and figured the same. 
 
PERSPECTIVE FOR STUDENTS, 
 
 133 
 
 intersection produced by the perpendicular line from 7 and L 
 Hence it must be clear, that to determine the perspective 
 distance of any perpendicular line between a e or A F, it is 
 immaterial, so far as correctness is to be obtained, whether 
 these distances are determined by a distance point, by visual 
 rays, or by means of diagonal lines.* 
 
 It is difficult to say of these three modes, which is the best; 
 in some cases one is to be preferred, in others another. 
 The distance point and the diagonals are the most used in 
 sketching from nature, as it is seldom found necessary to 
 construct a plan, the general outline and position of the 
 vanishing points being taken at discretion. In minute parts 
 of a drawing, the use of the point of distance is valuable, as 
 correctness is more readily attained, and it saves much 
 trouble where a perspective line is required to be divided 
 into a number of equal parts, to use the point of distance 
 instead of visual rays, as will be seen on the line drawn 
 from I (on a), to the vanishing point H. Suppose that portion 
 of the line from I to m, required to be divided (perspectively) 
 into four equal parts : draw an indefinite horizontal line 
 from and set off four equal parts at w, 0, p, q; from q 
 draw a line to the horizontal line at u ; these lines drawn 
 from 0, p, to u, will give the perspective positions of 
 these points on Z, m. The reason why the distance point 
 is more convenient for this purpose than the visual rays 
 from a line on the plan, is, first, that you may place your 
 distances on the horizontal line at discretion, and secondly, 
 that by being able to get them wider apart, correctness is 
 more easily attained. Suppose the distance chosen to have 
 been a trifle more than from I to n y as from I to r, the horizontal 
 line would extend to u, and the point of distance must be 
 found by drawing a line from u through m to the horizontal 
 line at v, and lines ruled from the intermediate points r, s, t 
 
 * The diagonal lines may be used for finding the positions of other 
 perpendicular lines besides the middle line n, as shown in the First Part, 
 Prob. IV. Plate 5. 
 
134 
 
 PERSPECTIVE FOR STUDENTS. 
 
 to the point of distance v,* will intersect the line I m in the 
 same points as those drawn from the points w, 0, to the 
 point of distance u. It may be well here to notice, that from 
 any point on a from which a line is drawn to the vanishing 
 point, the point of distance may as readily be determined as 
 from the point (1) chosen ; or that the point of distance 
 being fixed from the real geometrical length of the line R, 
 measurements may be made on any other line, and the 
 same point of distance made use of ; as if we required to 
 divide the perspective line Jc in half, we have only to measure 
 off on the ground line to the right of the point 2, tlie geome- 
 trical half of the line k of the plan, and a line ruled from 
 it to the point of distance s, would intersect the line K in the 
 middle ; this is clearly shown in the geometrical width of 
 the first arch on the line E and the ground line, Fig. 1, 
 Prob. VI. Part I. In the second problem, Part I., the 
 student could have no difficulty in finding the positions 
 of the points 3 and 4 from a plan ; and as the near edge of 
 the chess-board is on the ground line, and therefore up to the 
 plane of delineation, the line 1 2 has only to be divided into 
 eight equal parts to get the divisions of the squares. It 
 must be evident in this figure, that the point of distance 
 is the readiest way of determining the perspective positions 
 of the points from a to g; if these were to be found by means 
 of visual rays, it would be necessary to divide one of the 
 sides of the plan into eight equal parts, to draw a visual ray 
 from each to the ground line, and from each intersection draw 
 a perpendicular line to intersect either the line 1 — 3 or 
 2 — 4 ; whereas in the manner adopted in Prob. II. (the 
 point 3 having been found by a visual ray), the point of 
 distance is determined by a line from 2 through 3 to the hori- 
 zontal line, and the intermediate points between 1 and 3 are 
 found by one operation. 
 
 * This point is out of the picture as well as the vanishing points, but it 
 would be found on the horizontal line by continuing the line u m v up 
 to it. 
 
PERSPECTIVE FOR STUDENTS. 
 
 135 
 
 In Prob. X. Part I. we pointed out the mode (pp. 56, 
 57) for finding the perspective width of recesses, leaving 
 the first step, the distance of the line 2 from 1 to be 
 determined by eye ; in this figure (27) we shall show with 
 what readiness the depth of the recess may be determined 
 by rule. Let v on the plan represent the plan of the recess ; 
 we shall then require the perspective width of from w to 11; 
 to ascertain which, draw a visual ray from w intersecting 
 the ground line at and from x draw a perpendicular line 
 across the front of the house ; the perpendicular from d is 
 the line 1 of Prob. X. Part I., and the perpendicular from 
 x the line 2 of the same. In this figure (27) we have not 
 introduced the top and bottom lines of the windows, as the 
 manner of proceeding for drawing these is fully described 
 in the directions given for drawing Prob. X. Part I. We 
 have merely shown how the line 2 (Prob. X.) is to be 
 found by a visual ray from a plan ; that being determined, 
 proceed as described, p. 56, Part I., from the corner of the 
 window 4. 
 
 The same remarks we have made in comparing Fig. 27 
 with the first problem in Part I., will apply to other plates 
 in the same part ; thus, in Fig. 1 , Prob. VI., if the plan 
 were given so as to fix the position of the vanishing point 
 and determine the distance of the line d from a, the width 
 of the piers and arches, and the intermediate points required 
 for drawing the curves, are as readily found by means of a 
 point of distance as by drawing visual rays to the plane of 
 delineation. The readiest way of drawing this figure, would 
 be to find the width of the arches and the middle line 
 between their sides, by the visual rays, and find the points 
 for the curves as described in Fig. 2, Plate 7. If in addi- 
 tion to what is represented in Prob. VI., the thickness of the 
 arches was required to be drawn as in Prob. XI. Part I., 
 the thickness shown from A to a (Prob. XI.) would be 
 determined from a plan, in a way similar to finding the 
 perspective depth of the recess 11 to w in Fig. 27. 
 
13G 
 
 PEftSPECTIVE FOR STUDENTS. 
 
 In the diagrams introduced in Part II., none have been 
 given for the representation of curves in perspective ; ample 
 information has been given to enable the student to construct 
 any plane rectilinear figure in perspective; and as rectilinear 
 figures must first be constructed in order to find intersecting 
 points through which the curves are to be drawn, it was 
 considered that the introductory observations and subsequent 
 examples in Chap. III. Part I., were quite sufficient to 
 enable the student from a plan to draw a circle in per- 
 spective ; the geometrical figure required for so doing, being 
 only a few straight lines intersecting one another at certain 
 points, and all within a square. There are a variety of 
 geometric curves, such as the ellipse, parabola, &c. &c, 
 that, if the mode for geometrically constructing them accord- 
 ing to the rules laid down in works on practical geometry 
 are known, are as easily put in perspective as the circle, 
 although a greater number of lines may be required. These 
 curves are met with in arches, roofs, mouldings, &c, and as 
 we before said, if the manner of constructing these figures 
 geometrically is understood, they are readily put in per- 
 spective. We will introduce one example by putting a 
 semi-elliptic arch in perspective. 
 
 Fig. 29. 
 
 The rule for drawing this geometric curve is to be found 
 in Nicholsons " Practical Geometry it is very simple, and 
 produces a good line. It must be understood that to draw 
 
PERSPECTIVE FOR STUDENTS. 
 
 137 
 
 this figure on a large scale, it would be necessary to take, in 
 proportion to the size it is to be drawn, an increased number 
 of points ; four points are very few to get a perfect line, but 
 are quite sufficient to illustrate the principle on which this 
 or any other curve may be represented in perspective. It 
 is superfluous to draw any plan for this figure, the paral- 
 lelogram containing it being drawn in perspective as any 
 other parallelogram would be ; we therefore premise all that 
 preparatory work to have been gone through, and proceed 
 at once to the representation of the curve. The geometrical 
 divisions on a must be carried to the line B, and their per- 
 spective positions found on c by means of the point of 
 distance d on the horizontal line, and from c, these points 
 must be brought to the line a* the representation of A in 
 perspective. The divisions on the line e must be put on 
 the line 0, by ruling to the vanishing point from e through e 
 to it ; this will give all the points requisite for drawing the 
 rectilinear figure in perspective ; and by drawing from / to 
 the points on the perpendiculars on each side of the paral- 
 lelogram, and from g through the divisions on lines up to 
 those from / to the divisions on the sides, similar to the 
 geometrical figure to the left, all the points requisite for 
 drawing the curve will be evident ; the rectilinear figure to 
 the right being the perspective representation of the rectilinear 
 figure to the left. 
 
 Our limits do not admit of introducing any great variety 
 of figures ; indeed the object is only to point out the princi 
 pies on which perspective drawings are to be made from 
 details of individual parts. In Prob. VIII. Part I. we have 
 given a figure by which the shaft of a column, and conse- 
 
 * It would be a shorter process to continue the line a, and there place 
 the geometrical divisions (as on b) ; by which proceeding, the divisions got 
 by the point of distance would come at once on a. We have taken the 
 Line b, because, being obliged to leave the lines required for finding the 
 positions of the points, the finding them from a continuation of the line a 
 would have created a confusion with the other lines. 
 
138 
 
 PERSPECTIVE FOR STUDENTS. 
 
 quently a series of shafts of columns, may be drawn ; fre- 
 quently the shafts of columns are of a less diameter at the 
 top than at the bottom ; to represent such a shaft in perspec- 
 tive, it would require only to construct a perspective square 
 within the top square e (see Figs. 6 and 7, Plate 13, Part I.), 
 the same width as the diameter of the top of the shaft, draw 
 the circle within it, and from the extremities of the bottom 
 draw lines to the extremities of the top, similar to the lines 
 s and t, Prob. VIII. In the forms of the roofs of interiors, 
 we constantly meet with arches crossing each other in a 
 variety of ways. It is as easy to draw an arch in perspective 
 in one direction as another : it is only necessary therefore to 
 fix the points from which the arches spring, ample imforma- 
 tion for doing which has already been given, and the arches, 
 whatever may be their geometrical form, are as easily drawn 
 as in Prob. VI. and IX. The annexed diagram (Fig. 30) 
 Fig. 30. is an example of the effect pro- 
 
 duced by the intersection of 
 arches. It would be super- 
 fluous to show how these per- 
 spective arches were construct- 
 ed, as it would be only a repe- 
 tition of preceding examples ; 
 the points from which the curves 
 spring are similar to 1, 3, 6, 9 
 of Prob. VII. Part I., the arches springing from 1 to 9 
 and 3 to 6, instead of from 1 to 3 and 6 to 9. Domes vary con- 
 siderably in their apparent forms, according to the change of 
 position from which they are seen, and to the experienced 
 eye a want of perspective knowledge is easily detected in 
 their representation. There is little difficulty in drawing in 
 perspective any form of dome, provided the student under- 
 stand thoroughly how to draw the geometrical figure, as we 
 will show by a reference to Prob. V. Part I., and Fig. 29, 
 just given. Let us suppose the form of the dome to be that 
 given in Fig. 29, springing from eight points, and we had 
 
PERSPECTIVE FOR STUDENTS. 
 
 139 
 
 drawn the plan of it similar to the nearest perspective circle, 
 Prob. Y. First draw a perpendicular line from the perspective 
 centre of the circle (the intersection of the diagonal lines), 
 the height of A F, and on the line 4 — 2 construct a geo- 
 metrical figure similar to Fig. 29 ; then, as described in Fig. 
 29, draw the geometrical figure in perspective over each of 
 the lines 1 — 3, 5 — 7, and 6 — 8, as bases representing the line 
 a, and the result will be an accurate representation of such 
 a dome in perspective, as was described. Irregular curves 
 may be represented in perspective in the same manner as 
 described for Fig. 29, constructing your own rectilinear 
 figure about the irregular curve to get a number of inter- 
 secting points in it, and putting this figure in perspective; 
 this would be required in drawing the leading lines of leaves 
 in a Corinthian capital. Any spiral from the volute of an 
 Ionic capital would prove an excellent example for the 
 student's exercising himself on the principles for representing 
 curves in perspective. 
 
 CHAPTER VI. 
 
 At the commencement of Chap. Y. p. 107, we remarked 
 that the mere outline of the ground plan of a building 
 would be insufficient as a plan from which to make a per- 
 spective drawing, and that we require for this purpose a 
 series of plans made from horizontal sections of the various 
 parts wherever any change of form occurs either as a pro- 
 jection or recess ; thus in the projections of roofs, pediments, 
 cornices, mouldings, &c, the form and extent of the pro- 
 jection must be drawn on the plan before we can put it in 
 perspective. Fig. 31, p. 141, represents the plan (a), eleva- 
 
140 
 
 PERSPECTIVE FOR STUDENTS 
 
 tion (bJ, arid perspective representation (c), of a square pillar 
 btandmg on a cube, with a projecting top, or capital ; the 
 station of the spectator is marked at ©, to which point visual 
 rays have been drawn from the three corners of each of the 
 squares or the plan that are visible, the inner square being 
 the plan of the shaft of the pillar, and the outer square the 
 plan of the projection (the projection of the base and the 
 capital being the same), and from each of the points of inter- 
 section a, 5, c, d, e on the ground line of the picture, a perpen- 
 dicular line has been drawn ; the point 2 has been brought 
 to the ground line at /, and a line drawn from / to the 
 vanishing point D, gives the perspective position of the point 
 2 of the plan at 2 ; the cube forming the base of this figure is 
 drawn as Fig. 23, only as the top of it is above the eye of the 
 spectator, only the outer edges of it are visible. The perpen- 
 dicular line from /, by which the perspective height of the 
 base was determined at 7^, must be continued up, and the 
 geometrical heights j k marked on it from the elevation, from 
 each of which points a line must be ruled to the vanishing 
 point d ; the intersections of these lines with the perpendicu- 
 lars drawn from the points c and a, will give the perspective 
 positions of the points Z, and m of the elevation, and per- 
 spectively perpendicular over the points 1 and 2 of the plan ; 
 from the points^' and k lines must be drawn to the vanishing 
 point E, their intersections with the perpendicular drawn from 
 e determine the perspective positions of the points n and o 
 perpendicularly over the point 3. Draw a line from the 
 point m to the vanishing point E, and from o to the vanish- 
 ing point D, intersecting each other in the point p, then 
 m, />, o would be the representation of the under surface 
 of the- slab or capital on the top of the square pillar, per- 
 pendicularly over the points 2 — 1 — 4 — 3 of the plan. On 
 this perspective square we have now to represent the inner 
 square of the plan in perspective, the top of the pillar on 
 which the square slab just drawn stands ; continue the line 
 
142 
 
 PERSPECTIVE FOR STUDENTS. 
 
 5 — 6 up to the ground line at q, from which draw a perpendi- 
 cular line, and from the elevation set on it the geometrical 
 height of the pillar at r, from which point draw a line to the 
 vanishing point d, the intersection of this with the perpen- 
 dicular from c gives a point, s 9 perpendicularly over the 
 point 8 of the plan ; its intersection with the perpendicular 
 from b, gives a point, perpendicularly over the point 5 of 
 plan ; from s draw a line to the vanishing point E, its inter- 
 section with the perpendicular line from d gives a point, 
 perpendicularly over the point 7 of the plan, and completes 
 the perspective representation of the elevation b from the 
 plan a required. 
 
 The mode employed for finding the positions of the points 
 s and t by setting up the geometrical height of the pillar at 
 r, is perfectly correct, and serves excellently as an exercise 
 on our previous examples ; but a much readier and equally 
 correct mode of proceeding would be to find their position 
 by means of a diagonal line between m and o; the points t 
 and u being perpendicularly (that is, perspectively speaking) 
 over 5 and 7, the diagonal m — o must give these points in its 
 intersections with the perpendicular lines from b and d. If 
 the points t and u had then been determined by the diagonal 
 m — o, the lines s — t and s — u would have been drawn on a 
 different system ; from the vanishing point D a line must 
 have been ruled upwards through £, and from the vanishing 
 point e a line ruled through u to intersect the last drawn, 
 which would be in the same point on the perpendicular 
 drawn from c, founol by the visual ray from the point 6.* 
 Any additional projection may be put in perspective by 
 
 * The visual rays drawn from all the points, 2, 6, 8, 4, are in the same 
 line, and consequently the perpendicular line drawn from c answers for 
 the perpendicular angles of any number of parallel squares when the visual 
 ray is in a direct line with a diagonal ; had the station of the spectator 
 been either to the right or left of the position in which we have placed it, 
 the angle j Jc of the slab would not have been perpendicularly over the 
 angle s of the pillar. 
 
PERSPECTIVE FOR STUDENTS. 
 
 143 
 
 drawing it iirst on the elevation, and afterwards tlie horizon- 
 tal section of it on the plan ; thus narrow lillets. such as are 
 constantly placed on pillars a little below the capital, may be 
 represented in perspective with very little trouble, on the 
 same system as was employed for drawing the base and 
 capital : — such as the one we have drawn on the elevation 
 and introduced on the plan with dotted lines.* 
 
 It is best generally in drawing the plans from which per- 
 spective drawings are to be made, to draw the largest sur- 
 face first, whether it stands on the ground, or is the section 
 of some projecting part over it ; we have, however, in the 
 figure before us, departed from this rule, in adding to the 
 representation the projecting piece round the lower part of 
 the cube on which the pillar stands. This is done to show 
 the student that he must occasionally add to his plan from 
 the elevation as he proceeds : thus the width of this pro- 
 jection is taken from the elevation and set on the plan out- 
 side the outer square, 1, 2, 3, 4, already drawn, and the pro- 
 jection is put in perspective by exactly the same process as 
 the cube standing over the square 1, 2, 3, 4. In the plan of 
 this outer square only two sides have been drawn, as they 
 give the three points that are required : the other two sides it 
 would be a waste of labour to introduce. 
 
 Let us suppose that instead of the square pillar standing 
 on the cube of the base, there stood a circular column, the 
 diameter of which is the same as the width of the square 
 5, 6, 7, 8 ; from the centre of the plan describe a circle to 
 represent the plan of the column within the square 5, 6, 7, 8 ; 
 from the point u in the perspective representation draw a line 
 to the vanishing point d, and from the point t draw a line to 
 
 * This fillet we have not introduced in the perspective representation, 
 as the lines necessary for so doing would interfere with those we shall 
 require for another illustration ; and, as we have observed before, we have 
 not the advantage of being able to erase our lines as we proceed. It may- 
 be done in the same way as a projection we are about to draw round the 
 base. 
 
144 
 
 PERSPECTIVE FOR STUDENTS. 
 
 the vanishing point e ; the intersection v of these two lines 
 gives the tourtb point in the perspective representation of the 
 sguare 5, 6, 7, 8 at the height r of the elevation. The student 
 must now turn back to Prob. VIII. (facing p. 45, Part I.), 
 and refer from this figure (31) to that problem in the fol- 
 lowing description. From the point the intersection of the 
 circle with the diagonal 5 — 7, draw a line parallel to 5 — 8 
 up to the line 5 — 6 ; from this point draw a visual ray inter- 
 secting the ground line at x ; from x draw a perpendicular 
 up to the line t — v at y. From the vanishing point D, 
 through y, draw a line to intersect the diagonals t — u and 
 s — vj where this line intersects t — u it gives a point corre- 
 sponding with the point 2 of the diagram, Prob. VIII. ; 
 where it intersects s — v it gives a point corresponding with 
 8 of the diagram ; from each of these points draw a line to 
 the vanishing point e ; the intersection on t — u will give a 
 point corresponding with 6 of the diagram, that with s — v a 
 point corresponding with 4. From each of the vanishing 
 points d and e draw a line through the intersection of the 
 diagonals t — u and s — vj where the line drawn from e inter- 
 sects the line u — it gives a point corresponding with the 
 point 5 of the diagram, Prob. VIII. ; where it intersects 
 the line s — t it gives the point 1 of the diagram. Where 
 the line drawn from d intersects the line t — v it gives a 
 point corresponding with the point 3 of the diagram ; where 
 it intersects the line s — u it gives the point 7 of the diagram. 
 Here it will be seen that we have the eight points required 
 through which to draw the curve of the top of the circular 
 column in perspective, without the necessity for drawing 
 the whole figure of the diagram, Prob. VIII., on the plan : 
 the points 6, 8, 2, 4 being all determined from the single 
 point and the points 3, 5, 7, 1 from the intersection of 
 the diagonal lines t — u and s — v; it was even superfluous to 
 draw in the plan the whole of the circle ; marking from the 
 intersection of the diagonals 5 — 7 and 6 — 8 the length of a 
 radius on the diagonal 5 — 7 at w would have been quite 
 
PERSPECTIVE FOR STUDENTS. 
 
 J 45 
 
 sufficient. The curve may be drawn through these points, 
 and the sides of the column drawn down similar to the lines 
 s and T from the curve E, Prob. VIII. Part I. 
 
 In the foregoing illustration (Fig. 31) we have shown by 
 a very simple figure how the projections of any original 
 object are represented in perspective from a plan introducing 
 the sections of the projections where they occur, all of which, 
 save the circular column, are of the same form, viz. squares ; 
 we now present the reader with an example admirably 
 adapted not only to illustrate the same, but also as an excel- 
 lent exercise for the student on all the examples given in the 
 two parts of this work. The student on looking at this 
 figure must not be alarmed at its apparent intricacy : there 
 is not a line in it but what has been explained in preceding 
 drawings, and if each portion of the figure be examined 
 separately, and looked at as a distinct figure, all intricacy 
 will vanish : the square block forming the base is drawn by 
 the same rules given for the cube in previous examples ; the 
 triangular solid that stands on it is equally simple ; the 
 directions for drawing the half-circle on the top of this last 
 figure were explained in the First Part (Prob. V. &c.) ; 
 the blocks forming semicubes are as simple as the earliest 
 examples we have given in this Part ; and the mode for 
 drawing the semicircular forms made by the shaft passing 
 through these blocks was amply illustrated in Prob. VIII. 
 The projecting parapet and pier standing on it is but a 
 variation of the last figure, and if we only hide with a piece 
 of paper all the under part, it will appear equally simple. 
 It is the imperative necessity for our leaving all the lines 
 required for constructing our figure that makes them appear 
 complicated ; whereas the student in his progress will be 
 able to erase the lines that are used in the construction of 
 one figure before he proceeds to the next, by which his pro- 
 ceedings will be divested of all confusion of lines. Before 
 commencing this figure, the student must recollect that in 
 our progress we have frequently observed, that for the pur- 
 
 Perspective, rj 
 
146 
 
 PERSPECTIVE FOR STUDENTS. 
 
 pose of instruction we have been compelled to introduce 
 more lines than were absolutely required for the construction 
 of the figure, and also we have in numerous cases pointed 
 out that a result equally correct is to be obtained by a variety 
 of modes of proceeding ; therefore in the construction of a 
 figure like the present, consisting of a combination of parts, 
 dissimilar in form, one part may be drawn by means of 
 visual rays for finding the positions of the various points, 
 and the points requisite for drawing the next may be deter- 
 mined by that previously drawn ; the mode that produces a 
 correct result with the fewest possible number of lines is 
 always the most desirable. 
 
 In this figure, for the advantage of more clearly under- 
 standing the references, we have placed a plan both under 
 the elevation and perspective representation, the manner of 
 drawing which we are about to describe ; and in order to 
 afford every facility to a perfect understanding of the figure, 
 every alternate form is drawn with dotted lines. The origi- 
 nal object, the buttress of a bridge, is viewed from a recess 
 on the bridge looking over the parapet, which necessarily 
 causes the horizontal line to be placed very high. This 
 position is chosen, as it presents a larger surface on which to 
 represent the several figures. 
 
 We will in this representation take each part of the but- 
 tress separately, treating each as a distinct figure, and before 
 the student commences his perspective drawing, he should 
 on a large scale copy the elevation and plan below it, care- 
 fully marking all the letters and figures for reference, draw- 
 ing on the plan for constructing his perspective figure each 
 part as it is described ; he will also find his progress through 
 die drawing much simplified by putting in each figure when 
 completed with ink, and rubbing out all the superfluous 
 pencil lines. We commence then with the square base 
 marked A. It is quite unnecessary to give any directions for 
 drawing this figure, as it would be but a repetition of what 
 has been already fully explained ; all the lines, however, 
 
PERSPECTIVE "FOR STUDENTS. 147 
 Fig. 32. 
 
 H 2 
 
148 PERSPECTIVE FOR STUDENTS. 
 
 that were used for drawing the perspective form are left on 
 the representation for the advantage of the student.* The 
 second, which is a triangular figure, might have been drawn 
 by finding the vanishing points from the lines 1 a and b a 
 of the Plan ; but as the perpendicular planes over these lines 
 are the only planes standing in these directions, the triangu- 
 lar figure may be drawn by a much shorter process. From 
 the point c t draw a visual ray to find the position of c % on 
 the top of the perspective square a, from which draw a line 
 parallel to the ground line across the square, which will give 
 the point d on the line 2b; from d draw a long per- 
 pendicular line, as the position of this point will be required 
 in every portion of the figure ; from a draw a perpendicular 
 line to e (the geometrical projection of this triangular figure), 
 and from e a line to the vanishing point, the intersection of 
 this with the line c d gives the position of the point a on the 
 plane a ; draw the lines 2 a and a 5, which will give the 
 perspective form of b of the Plan on the plane A. From each 
 of the points 2, a, 5, draw an indefinite perpendicular line, 
 bring from the elevation the point 3 to the line of projection 
 j k, and from it draw a line to the vanishing point ; from 
 the point of intersection d on this line draw a horizontal line 
 to the perpendicular from a ; join the points 3 a and a b, 
 
 * The student will bear in mind, that the plan being drawn below the 
 ground line, the figure is reversed. 
 
 f In order to curtail as far as possible the quantity of lines that are 
 unavoidable in making a perspective drawing, we have drawn this figure 
 in parallel perspective, and the plane of delineation right up to the 
 nearest face of the base figure; by so doing, we are enabled to get the 
 measurements of the heights on the line J k (which may be called the line 
 of projection) of the representation direct from the same line in the 
 elevation. 
 
 X The letters for reference are made on the plan of the elevation, but 
 the lines representing the visual rays are drawn from the corresponding 
 points on the plan under the perspective representation. The letters in 
 the perspective representation indicate points that are perpendicular over 
 the points, with similar references in the plan at whatever height they may 
 occur. 
 
PERSPECTIVE. FOR STUDENTS. 
 
 149 
 
 which will complete the perspective drawing of the second 
 figure. To draw the form of the third figure on the plane b, 
 we must first find the positions of the points f and h on 
 the line 3 b. This is done by visual rays drawn from these 
 points from the plan to the ground line, and the perpendiculars 
 from the points of intersection will give the points / and h on 
 the line 3 b. Continue the horizontal line to the right of the 
 point 3, the length of the radius of the circle (d f of the 
 Plan), and from its extremity draw a line to the vanishing 
 point ; then from each oi the points / and h draw a hori- 
 zontal line to meet the last drawn at g and j ; this will give 
 the perspective half-square in which to describe the semi- 
 circle. Draw the semi- diagonals d g and d j j the semi- 
 diameter is already drawn by the line da; it remains only 
 to find the perspective positions of the points of intersection 
 made by the curve on the semi-diagonals to enable us to 
 draw the figure ; from k (on the Plan) draw a visual ray to 
 the ground line ; a perpendicular drawn from the point of 
 intersection will give the perspective position of the point k 
 on the semi-diagonal d g ; a line from this point k to the 
 vanishing point will give a corresponding point on the semi- 
 diagonal d j. The curve should now be drawn through the 
 several points, which will be the perspective form of the 
 semicircle c of the Plan on the plane B. 
 
 The fourth, sixth, and eighth figures are all of the same 
 form, being projections of half-cubes. It will be seen, on 
 referring to the plan and elevation, that the perpendicular 
 angles of these figures are over the points/, g, k,j; the posi- 
 tions of these points having been determined on any one 
 plane, their positions on any other will be perpendicularly 
 over (or under) those already found ; therefore from each of 
 the four points /, g, h, j on the plane b draw up an indefinite 
 perpendicular line, then bring each of the points 4, 5, 6, 7, 
 8, 9, to the line of projection for the perspective representa- 
 tion, and from each of the points draw a line to the vanishing 
 point ; the lines drawn from 4 and 5 at their intersections 
 
150 
 
 PERSPECTIVE FOR. STUDENTS. 
 
 with the perpendicular from /, determine the height of the 
 perpendicular angle of the half-cube d standing over f ; 
 horizontal lines from these points of intersection to the per- 
 pendicular line from g determine the height of the angle over 
 g ; from the points of intersection on the perpendicular from 
 g, draw lines to the vanishing point, their intersections with 
 the perpendicular from j determine the height of the angle 
 standing over j ; from the highest intersecting point on the 
 perpendicular from j draw a horizontal line to meet that 
 drawn from 5 to the vanishing point, which will complete 
 the perspective drawing of the half-cube d ; the drawing of 
 the two other half-cubes is a mere repetition, substituting in 
 the directions above given the figures 6 and 7, or 8 and 9, for 
 4 and 5.* The process for drawing the semicircles on the 
 planes D, E, and f is the same as that for drawing the semi- 
 circle on the plane b ; the positions of the points I and m (of 
 the plan) must first be found on each of the lines drawn 
 from 5, 7, and 9 to the vanishing point ; we must therefore 
 draw visual rays from these points I and m on the plan to 
 the ground line, and draw up perpendicular lines from the 
 points of intersection ; these will give on the lines drawn 
 from 5, 7, and 9 to the vanishing point the perspective posi- 
 tions of the points I and m on each ; the position of the point 
 d is already determined by the perpendicular drawn from d 
 on the plane a ; draw then from each of the points I and m 
 a horizontal line across the several planes. From n on the 
 plan draw a visual ray, and from the point of intersection on 
 the ground line a perpendicular, which will give on the hori- 
 zontal lines drawn from m points on each of the planes D, e, 
 and F, corresponding with the point n on the plan ; from the 
 vanishing point through the points n\ draw lines to meet the 
 
 * The three half- cubes are treated as one figure, also the shaft 
 passing through them, all the points required for drawing the semicircles 
 on the upper planes of the three half-cubes being determined by one 
 process. 
 
 f From either of the points n or o, a visual ray might have been 
 
PERSPECTIVE FOR STUDENTS. 
 
 151 
 
 horizontal line from Z, which will give the points on each 
 plane corresponding with the point o of the plan ; draw in 
 each figure the lines d n and d o, on which we have to find 
 the points through which to draw the curve. From^p (on 
 the plan) draw a visual raj, and from the point of inter- 
 section on the ground line a perpendicular, which will give 
 the perspective positions of the points p on each of the semi- 
 diagonals do; from each of the points p draw a line to the 
 vanishing point, which will give a corresponding point on 
 each of the semi-diagonal lines d n. All the points being 
 determined through which the curve passes, the curve should 
 be drawn on the respective planes, and a perpendicular line 
 from the extremity of each curve to the right to meet th 
 projection above it. 
 
 We will pass by for the present the projecting parapet, and 
 proceed to the pier standing on it ; the perpendicular angles 
 of the base of this figure, as may be seen by the plan and 
 elevation, are perpendicularly over the angles of the three 
 half-cubes, and must consequently come on the perpendicu- 
 lars drawn from the points /, ^, and j y on the plane b ; 
 therefore bring the points 11 and 12 of the elevation to the 
 line of projection, and put the base of the pier in perspective, 
 according to the directions given for the half-cube D. The 
 perpendicular angles of the portion L of the pier not project- 
 ing so far as the angles of the base, the lines representing 
 them must be found by visual rays, which must be drawn 
 from the points q, r, and s of the plan to the ground line at 
 
 drawn to have determined their perspective positions ; the point n was 
 chosen, and the point o found from it by the vanishing lines n o, to 
 avoid confusion. These points n and o might have been determined in a 
 different manner, by drawing semi-diagonal lines from the points d to the 
 corners of each of the planes d, e, and f ; the intersections of these 
 semi-diagonal lines with the horizontal ones drawn from / and m would 
 be in the same points n and o ; this indeed would prove the readiest 
 way of determining these points, as it would at the same time have given 
 the semi -diagonals d o and d n, which are required for drawing the 
 curve. 
 
152 PERSPECTIVE FOR STUDENTS. 
 
 ?«, and 0, and perpendicular lines drawn from each ; carry 
 the point 13 to the line of projection, and from it draw a line 
 to the vanishing point, the intersection of this with the line 
 drawn from t will be the point q (perpendicularly over the 
 point q of the plan at the height marked 13 of the elevation); 
 from q draw a horizontal line to the perpendicular from 
 this will give the perspective position of the point r of the 
 plan at the height marked 13 ; from r to the vanishing point 
 rule a line to the perpendicular from «?, the intersection will 
 give the perspective position of the point s of the plan at the 
 height 13. The sloping lines from the points r, and s must 
 be ruled to the top of the angles of the base of the pier. The 
 line from the base of the projection 14 must be drawn pre- 
 cisely in the same way as the lines of the base of the pier from 
 11 and 12, the points of this projection being perpendicularly 
 over the points /, g y h,j; the lines got from 15 and 16 similar 
 to the line from 13, and the projecting angles from the ex- 
 treme points of projection. 
 
 The plan of the parapet on which the pier stands, from 
 our limited space, we cannot introduce so as to get our points 
 from it by visual rays ; we have, however, in very faint 
 dotted lines on the plan under the elevation, shown what the 
 projection is, sufficiently to understand our manner of drawing 
 the perspective representation ; the points 10 and 11 are 
 placed on the line of projection, and the horizontal lines from 
 them continued to the right to w and their geometrical 
 length ; through each of these points, in a direction towards 
 the vanishing point, we have ruled lines to the right and left ; 
 from the point d on the plane of the top of the parapet, 
 through the angles of the base of the pier, draw lines to meet 
 the line drawn through x 9 to determine the corners of the 
 projection in front of the pier (see the dotted lines in the 
 plan) ; from each of the points of intersection on the line 
 through x draw a parallel line to meet the continuation of the 
 base line of the pier ; the intersections determine the points 
 from which the projection commences. The under line of 
 
PERSPECTIVE FOR STUDENTS. 
 
 153 
 
 the projecting parapet showing its thickness, it is unnecessary 
 to describe, the several points lying perpendicularly under 
 those of the upper one. 
 
 We have taken considerable pains to make the directions 
 for drawing this figure so clear as to enable the student to 
 draw any other figure composed of a variety of parts of dis- 
 similar forms by a reference to it. The small space afforded 
 by a page of this work precludes the possibility of introducing 
 more than one buttress in the representation ; in the single 
 one given, the lines are necessarily so close together that great 
 care is required in attending to the references ; we wish, 
 however, to point out how any number of these buttresses, 
 forming the piers of a bridge, may be drawn in succession, 
 with the arches between them, with the fewest possible quan- 
 tity of lines ; this we will explain by referring to Prob. VI. 
 Plate VII., of which we will suppose from a to b the width of 
 the buttress (from 1 to b of the Plan, Fig. 32). On the per- 
 spective representation of any horizontal line crossing these 
 spaces between the arches, the perspective positions of the 
 points /, <7, Z, <i, m, r, and h must be found on each, whatever 
 may be the number of piers ; the positions of these points 
 may be determined either by visual rays from a plan, as in the 
 figure just drawn, or from geometrical measurements on a 
 line of projection by a point of distance, as from the line E to 
 the line b, Prob. VI. The positions of these points being 
 determined on a (perspective) horizontal line on each pier, a 
 perpendicular line should be drawn through each, the whole 
 length of the structure from top to bottom, and the several 
 points 1,/, ^, Z, cZ, m, ^, lettered on the base or top line 
 of each pier; then continue across the whole face of the 
 structure the lines to the vanishing point from the points 
 1 to 1 6 inclusive ; by these two operations the perspective 
 positions of all the points are determined that are requisite 
 for drawing the different projections of the buttress from each 
 pier. Suppose, as in Prob. VI., the structure consists of five 
 
 h 3 
 
154 
 
 PERSPECTIVE FOR STUDENTS. 
 
 arches* and six buttresses, the nine points from 1 to b found, 
 and perpendicular lines drawn through them on each pier, 
 with the references put on the base line of the structure. t 
 The lines drawn to the vanishing point (which for brevity 
 we will call m) from the points 1 to 16 on the line of projec- 
 tion, j K, will give on the perpendicular lines over the points 
 1 on each pier, points corresponding with those on the line 
 of projection ; as the lines drawn from 2, 3, 5, 7, and 9, will 
 give the corresponding positions on the perpendicular line 
 from d on each of the more distant piers, the points d re- 
 quired for drawing the different figures on the first pier. 1 f 
 from the point d on the line 2 m of each pier a horizontal 
 line is drawn to meet the line e M, it will give the position of 
 the point a on the plane A of each buttress ; a perpendicular 
 from each of these points a intersected by horizontal lines 
 from the points d on the line 3 M, will give the position of 
 the point a on each of the planes b ; the lines 2 a and 3 
 and the line a b of the plane B, may now be drawn on the 
 buttresses, the points required for drawing them being deter- 
 mined on each pier. Horizontal lines drawn from all the 
 points /and h to meet the line g M, will determine the points 
 for drawing the half-square f g h j\ in which to describe 
 the semicircle c of the plan, projecting from each pier. In 
 each of these half-squares the diagonals d g and dj may be 
 drawn, the points required, c£, g, and j 9 being determined in 
 each, the semi-diameters being already drawn in finding the 
 points a ; the line k m gives the intersecting points on each 
 of the semi-diagonals d g and d j, through which the curve 
 is drawn. 
 
 It is unnecessary to recapitulate the process for drawing 
 
 * It is needless to introduce any further observations on drawing the 
 arches between the piers. Reference may be made to Prob. VI. IX. 
 and XI. Part L, and to Fig. 29, p. 136. 
 
 f The student should make his drawing from this description, marking 
 the references as he proceeds. 
 
PERSPECTIVE FOR STUDENTS. 
 
 155 
 
 each separate figure on the further piers ; the perpendicular 
 line standing over the point 1 on each being a representation 
 of the line of projection j k, with its divisions from 1 to 16; 
 the rectangular figures are drawn exactly as described for 
 the first buttress ; and the semicircles on the planes d, e, and f, 
 as the semicircle c on the plane b just described, substituting 
 the letters /, m, w, o, for /, k,j\ g, and after drawing the semi- 
 diagonals d o and d the letter p for k. 
 
 A portico, composed of a series of columns supporting a 
 pediment, and standing on a base to which we ascend by a 
 flight of steps, being furnished with a plan and elevation of 
 the same, would be as easy to draw in perspective as the 
 example we have just given. Although it is necessary to be 
 acquainted with rules by which even the most minute part of 
 a structure can be drawn in perspective, it would be a most 
 tedious process to find all the points required for drawing the 
 curves of mouldings and the intricacies of ornament intro- 
 duced in architectural representations ; it is generally found 
 sufficient to mark a certain number of leading points, and 
 draw the intermediate lines by hand ; practice enables archi- 
 tectural draftsmen to draw capitals of columns, ornamental 
 friezes, &c. with great accuracy, from determining a com- 
 paratively few points. Whatever may be the forms of the 
 component parts of a building, such as the base, shaft, and 
 capital of its columns, or the extent of the projections of any 
 parts of a structure, the principles on which they may be 
 represented in perspective are contained in the directions 
 given for drawing our last figure (32). 
 
156 
 
 PERSPECTIVE FOR STUDENTS. 
 
 CHAPTER VII. 
 
 Having given to the full extent of our limits practical 
 examples for drawing in perspective the forms of various 
 figures and their combinations from different points of view, 
 we will make a few observations on the choice of position 
 for the spectator, relative to the objects to be represented in 
 perspective. The important place the plane of delineation 
 holds, as a feature in perspective drawing, must by this time 
 be fully appreciated by the student ; it is an imaginary plane, 
 but in the preparatory steps towards making a perspective 
 drawing it is treated as a reality, being made to intercept the 
 rays of light in their passage from the original object to the 
 eye. The rays of light being understood to proceed from 
 every part of an object in straight lines to a point, the rays 
 proceeding from any rectangular plane to this point would 
 form a pyramid of rays ; and from any circular plane they 
 would form a cone ; any section of this pyramid made 
 parallel to the plane from whence the rays proceed, would 
 present the same form as the plane itself, larger or smaller 
 according to the distance from it the section is made, and a 
 section of the cone of rays, parallel to the base of the cone, 
 would present the figure of a circle larger or smaller accord- 
 ing to its distance from it. If the section of either the 
 pyramid or cone of rays is made in a direction not parallel 
 to the planes from which they proceed, the sections would 
 present a different form. This may be familiarly illustrated 
 by the figure of a sugar-loaf, which is made in the form of a 
 cone ; if this is cut in any part in a direction parallel to the 
 base, the section will be a circle, no matter how near to or far 
 from the base ; but if cut through in any direction not parallel 
 to the base, the section would not present the form of a circle. 
 In Fig. 15, if a cube stood over the square D, the rays of 
 light proceeding from the front face of it to the eye of the 
 spectator would form a pyramid of rays, and the section of 
 
PERSPECTIVE FOR STUDENTS. 
 
 157 
 
 this pyramid made by the plane of delineation, this being 
 parallel to the plane of the base of the pyramid, the square 
 figure is shown by the section; but if the direction of the 
 plane of delineation were changed, the figure produced by 
 the section of the pyramid would not be square ; this would 
 be equally the case were the direction of the base of the platx* 
 of delineation changed, or if the base remain on the same 
 line, if it were inclined so as to be at any angle but a right 
 angle with the ground plane.* It must be evident, then, 
 that the position of this plane is of great importance, and in 
 consequence some rule for determining it necessary. 
 
 If the plan of an object, as a (Fig. 33), were given, and 
 the position of the spectator at B only, Fig 33 
 the student, notwithstanding he may have f — -, A 
 
 paid great attention to all the preceding | | 
 
 diagrams, would find himself in some 
 
 difficulty to draw the object in perspective 
 
 by rule, the mere position of the object and 
 
 spectator not affording sufficient data on 
 
 which to commence operations ; there is 
 
 required in addition to these, either the 
 
 direction in which the object is viewed, or 
 
 the position of the plane of delineation, 
 
 the one depending entirely on the other, 
 
 A spectator placed in any open situation, 
 
 by turning himself about, can see objects in all directions ; 
 
 but when looking at any object with a view to making 
 
 a perspective representation of it, the direction of the eye 
 
 must not be changed ; any change in the direction of vision 
 
 * In many works, figures are introduced, showing the change of form 
 in the representation of objects by varying the direction of the plane of 
 delineation, placing several planes of delineation between the object and 
 spectator in different directions, and showing the variety of forms they 
 assume on the different planes. Such figures here would be irrelevant 
 to our purpose, which is only to point out how the position of this plane is 
 to be determined, so as to produce proper perspective drawings, and not 
 distortions. 
 
158 
 
 PERSPECTIVE FOR STUDENTS. 
 
 producing an apparent change of form in the object at which 
 we are looking. In Fig. 34 we have, in addition to a 
 
 and b in Fig. 33, drawn a line to 
 
 Flg * 34 ' show the direction in which we are 
 
 | A | looking at the object, viz. from b to 
 
 a. In all our preceding diagrams 
 
 ^7 c we have given the object, the station 
 
 \, of the spectator, and the position 
 
 %x £ of the plane of delineation, and 
 
 / ' e wherever the position of the point 
 of sight has been required, it has 
 / been found by drawing a line from 
 
 / the station of the spectator perpen- 
 
 / dicularly to the plane of delinea- 
 
 tion ; but here the positions of the 
 object and spectator are given 
 with the direction ol vision, and the position of the 
 plane of delineation is to be determined from these. Know- 
 ing that the forms of objects perpendicularly opposite the 
 eye always present their real form, and that a section of 
 the rays conveying the image only presents the real form 
 when made parallel to the plane from which they proceed, it 
 follows that the plane of delineation which makes the section 
 of the rays on which the representation depends, must be 
 parallel to those planes that present their original form to 
 the eye, in order to arrive at a correct representation of the 
 object. The plane of delineation then, it will be understood, 
 is always placed perpendicular to an imaginary straight line 
 proceeding from the eye of the spectator to the original 
 objects, wrfich is what we term the direction of vision ; 
 instead, therefore, of drawing, as in preceding examples, 
 the direction of vision from the station of the spectator per- 
 pendicularly to the plane of delineation, we must, in drawing 
 from any object, first mark the situation of it, as at a (Fig. 
 34), then the position of the spectator at b, with the direction 
 of vision b a, and at right angles with the line K a the base 
 
PERSPECTIVE FOR STUDENTS. 
 
 159 
 
 line of the plane of delineation c d ; the distance at which 
 the plane of delineation is placed between the spectator and 
 object not having any influence on the form presented to the 
 eye, but only affecting its size. The figure a, viewed from b, 
 in the direction b a, would be represented in parallel per- 
 spective, the vanishing point being the point of sight. 
 
 On the same figure (34), the positions of the spectator 
 and original object remain the same, but the direction of vision 
 is changed from b to a to from b to 5, shown by dotted lines, 
 and the base of the plane of delineation E f drawn at right 
 angles with it. The importance of determining the direction 
 of vision must here be instantly apparent ; the original 
 object viewed in this direction (b b) must necessarily be 
 represented in oblique perspective. The representation of 
 the figure A, viewed in the direction b would be similar to 
 the square D, Fig. 22 ; the representation viewed in the 
 direction b would be as the square a, Fig. 23. These 
 observations are intended to impress on the mind of the 
 student the necessity for keeping, when he has once fixed 
 his position, and the direction in which he has determined 
 to take his view, this position always the same, never turn- 
 ing his head either to the right or to the left ; which brings 
 us to another important question, viz. in order to draw a 
 more or less extended representation, what the distance of the 
 spectator should be from the object to be delineated. 
 
 In looking at any object, however limited the distance 
 from the eye, we not only see the point that is immediately 
 opposite, but some distance from this point both above, 
 below, and on either side ; and the further the object is 
 removed from the eye, the greater extent of surface becomes 
 visible. This may be made manifest by standing before the 
 door of a house, and looking in a direction perpendicular to 
 the plane of the door, placing the eye quite close to it ; a very 
 small portion of it will be visible ; but keeping the eye in the 
 same direction, and taking a step or two backward, the wholo 
 of the door becomes visible ; by retreating a few steps further. 
 
160 PERSPECTIVE FOR STUDENTS. 
 
 we see the windows situated over and on the sides of tlie 
 door ; and by going still further back, if in a wide street, the 
 whole of the house, or two or three, may be visible : but as 
 we cannot retire further back than the houses on the opposite 
 side of the road, to represent the side of a street in parallel 
 perspective, a very limited portion only could be drawn, and 
 the extent that could be seen on either side of the point 
 opposite the eye would depend on the width of the street. 
 In an open situation we are enabled, by turning round, to see 
 every object for miles distant ; but remaining stationary, and 
 looking in one steady direction, there must be some limit to 
 the extent we see, both to the right and left ; for if we turn 
 the head to the right, we see an additional extent of country 
 on that side, and lose sight of a portion on the other; and 
 the reverse will be the case if we turn the head to the left. 
 It is therefore requisite to determine the extent we may 
 represent to the right and left of the direction of vision, what- 
 ever may be the original objects of the perspective drawing 
 we may have to execute. It is difficult, in fixing this limit, 
 to say precisely what it should be — writers differing much 
 on the subject — but the most agreeable perspective repre- 
 sentations are generally considered to be produced by fixing 
 the angle of vision at from forty-five to fifty degrees ; some 
 extend it ten degrees beyond this, and in some cases this is 
 admissible ; but as a general criterion, from forty-five to fifty 
 will be found most advantageous. In taking views from 
 nature, and more particularly street views, the position of the 
 point of sight is rarely chosen in the centre of the paper or 
 canvas, but on one side, and for the most part nearer the 
 ground line than the top of the picture ; the student must 
 understand that the centre of the picture, that is, the centre of 
 the canvas or paper on whick a picture is drawn, is only the 
 perspective centre when the point of sight comes on this 
 point ; the point of sight, wherever it falls, being the perspec- 
 tive centre of the picture. In looking at any large repre- 
 sentation, either landscape or architectural, the general effect 
 
PERSPECTIVE FOR STUDENTS. 
 
 161 
 
 is greatly enhanced by standing from the picture the relative 
 distance the artist was supposed to stand from his subject, 
 with the eye opposite the point of sight ; in very large sub- 
 jects, the perspective does not appear satisfactory to the eye 
 if this is not attended to. A striking example of this may be 
 seen at any time at a theatre ; for as it is clear from what has 
 been said, that there can only be one point of sight in a 
 picture, there can be but one situation in the theatre where 
 the representation can be perfectly satisfactory, which situ- 
 ation must be opposite the spot where the painter has fixed 
 his point of sight. 
 
 In order to get the full extent of view in a picture em- 
 braced in the angle of vision, whatever that angle may be, 
 it is necessary that the point of sight be exactly midway 
 between the two sides of the picture. Though this position 
 is perfectly admissible, so far as regards correctness, it does 
 not produce so agreeable an impression as when placed on 
 one side ; when nearer to one side than the other, the whole 
 extent embraced in the angle of vision cannot be introduced 
 the picture; this will be understood by the annexed 
 
 m 
 
 Fig. 35. 
 
 diagram, in which suppose a b to be the base of the plane 01 
 delineation, which will also represent 
 the width of the picture, and c the 
 position the angle of vision would be, 
 as B c a and the picture would em- 
 brace the whole extent that could be 
 seen ; but if D were the position of the 
 spectator, a b remaining the same, 
 only a part of the extent visible on 
 one side of the direction of vision 
 would come into the representation. 
 
 The student is intended to understand from the foregoing 
 observations in this chapter, that in his preparations for com- 
 neacmg a perspective drawing, either from nature or from 
 plans and elevations, he must not imagine he can fix his 
 
162 
 
 PERSPECTIVE FOR STUDENTS. 
 
 station-point at random, at any distance from the objects, 
 draw a line anywhere to represent the plane of delineation, 
 and then proceed according to the rules given for making 
 perspective representations in the diagrams from Figs. 15 to 
 32, and produce an effect that will be either pleasing or 
 accurate ; but that in the relative positions of the spectator, 
 plane of delineation, and original objects, the arrangement 
 should be such as will produce a representation similar to 
 what we really can see from some fixed point, if drawing 
 from nature, or that we know might be seen, if drawing 
 from plans and elevations, or from description, which the 
 directions here given will enable him to do. Thus, sup- 
 posing a range of objects occupying a lateral extent of from 
 A to B, Fig. 35, and the direction of vision 
 as the line a b y the station of the spectator 
 A- must be somewhere on that line. Sup- 
 pose any one not acquainted with the 
 limit of vision on each side of the line a b y 
 Fig. 36, they might fix the station at c, 
 which would be a position where it would 
 be necessary to turn the head from side to 
 side, it being impossible in one view to 
 see so large an extent as this angle em- 
 braces ; the lines, it is true, may be 
 drawn according to the directions contained in the various 
 problems and diagrams we have introduced, but the re- 
 presentation would have an extremely bad effect, no per- 
 spective representation ever being satisfactory to the eye 
 but such as the original objects would present in one single 
 view, without shifting the direction in which we look at 
 them. Whatever the angle of vision determined on, — sayi, for 
 instance, an angle of fifty degrees, — a point D on the line a b 
 must be found, from which a line drawn to A will form an 
 angle of twenty-five degrees with the line a b ; a line any- 
 where across this line a b, at right angles with it, may be 
 
PERSPECTIVE FOR STUDENTS. 
 
 163 
 
 drawn for the base of the plane of delineation, the distance 
 from the spectator to be regulated according to the size of 
 the representation. 
 
 The knowledge of how to place the relative positions of 
 original objects, plane of delineation, and station of the 
 spectator, from a plan or description, is particularly service- 
 able in drawing architectural views in confined situations, 
 such as small quadrangles, interiors of rooms, &c. By 
 taking an imaginary position further back than the confined 
 space a small quadrangle would allow you to take, a 
 representation may be made that shall be perfectly satis- 
 factory to the eye, give a faithful idea of the place it repre- 
 sents, and yet no position on the spot exist from which such 
 a view can be seen. In crowded cities the effect intended 
 to be produced by architects in looking at large buildings is 
 completely lost, no situation existing so as to get a general 
 view of the whole structure. St. Paul's Cathedral is an 
 instance of this ; the houses being crowded so thickly around 
 it, no position is to be found by which the grand effect so 
 imposing a building must present as a whole can be seen ; 
 the situation where the grandest impression this magnificent 
 structure produces, is perhaps from Watling-street. A 
 knowledge of perspective, the student ought now to under- 
 stand, would enable a draftsman, with a plan and elevations of 
 St. Paul's, to fix an imaginary station from which a per- 
 spective representation might be made, giving a just idea of 
 how it would look from such a position, though no real 
 position can be found that affords such an uninterrupted 
 view in the vicinity of the building itself. In narrow streets, 
 the general effect of large buildings is lost, and in making 
 topographical drawings, unless the station of the spectator is 
 assumed, the perspective is disagreeably sudden ; in many 
 instances, no position can be chosen from which the whole 
 extent of the building can be seen without shifting the 
 position of the eye. Artists are frequently excessively 
 worried by the demands of their employers, who, completely 
 
164 
 
 PERSPECTIVE FOR STUDENTS 
 
 Fig. 37. 
 
 ignorant of the principles of perspective, are frequently 
 requesting the draftsman to furnish them with topographical 
 views that it is impossible to excute. We will here intro- 
 duce one more diagram, in order to show how an imaginary 
 station is to be taken, that shall produce a satisfactory 
 representation, though no position actually exists from which 
 the original objects can be so seen in one direct view. Let 
 abed represent the plan of some 
 small quadrangle, such as is fre- 
 quently met with in cloisters of old 
 monastic buildings, and suppose the 
 spectator at the point marked a, his 
 back to the end c d, the greatest dis- 
 tance he can possibly get from the end 
 a b. It will be understood how very 
 little of the sides could be seen from 
 this position, only from a to e and from 
 b to /, though we have made the angle 
 of vision to the extent of sixty degrees ; but supposing we had 
 a, plan of the three sides of the quadrangle with their elevations, 
 we could readily imagine the spectator to be situated at b, and 
 proceed as if the quadrangle were viewed from that point, by 
 which as much of the sides as from a to g and b to h would 
 be represented; and by taking an imaginary station still 
 further back, the whole of the sides of the quadrangle might 
 be represented, and still produce an effect as if drawn from 
 nature. Frequently, in subjects of this kind, one side con- 
 tains much more interest in its architectural detail than the 
 other; in such case the artist should take his station nearer 
 to the side with the less interest, by which means he wiD 
 have the opportunity of displaying to greater advantage thj 
 beauties of the other. 
 
 We have now, we sincerely trust, succeeded in carrying 
 out the intention of this treatise, by leading the pupil by 
 almost imperceptible degrees to understand the principles on 
 
PERSPECTIVE FOR STUDENTS. 
 
 165 
 
 which perspective representations are made, and have fur- 
 nished ample directions to enable him to execute perspective 
 drawings himself. In our earnest endeavour to make the 
 whole proceedings perfectly intelligible, we have deviated 
 from the general course of works on this subject ; and 
 that of which the knowledge is essential before even a 
 perspective drawing can be commenced, meaning the deter- 
 mination of the positions of the spectator and plane of 
 delineation, we have left to the last chapter, under the con- 
 viction that it would be more perfectly understood at that 
 stage. Throughout the whole work the endeavour has been 
 to make one part bear upon another, without attending to any 
 consecutive arrangement, so as at the conclusion the student 
 should feel himself master of the whole. In taking leave of 
 the reader, to make use of the simile in Mr. Weale's 
 prospectus, the author trusts that the boat which it has 
 been his province to provide for conveying the student to the 
 ship of science will carry him safely on board. 
 
INDEX 
 
 Acute angle, 2. 
 Aerial perspective defined, 4. 
 Aisles, to draw, perspectively, 43. 
 Angle, every change of, in an object 
 
 requires a fresh vanishing point, 99. 
 Angle, movable, 15. 
 Angles defined, 2 ; acute, 2 ; obtuse, 
 
 3 ; right, 2. 
 Arcade, to draw an, in perspective, 
 
 43, 38. 
 Arch, to draw a double, 59. 
 Arches, to put in perspective, 38, 43 
 
 to find the width of, 63 ; to find 
 
 the width, various methods, 135. 
 Arches, pointed, rules for drawing, 
 
 54 ; to draw the thickness, &c, of, 
 
 62. 
 
 Arcs of circles, to find the perspective 
 inclination of parallel lines by, 70. 
 
 Bird's-eye views, position of the 
 
 horizontal line in, 7- 
 Buttress of a bridge with projections, 
 
 method of drawing described, 146. 
 Buttresses, to draw a number of, in 
 
 succession, 153. 
 
 Capitals, to represent, 140. 
 
 Centre, perspective, to find the, of 
 any parallelogram, 18. 
 
 Chess board, to represent a, per- 
 spectively, 21. 
 
 Circle, perspective, 34 ; perspective 
 of every, an ellipse, 38 ; perspec- 
 tive of the, in different positions, 
 45 ; forms assumed by a, in dif- 
 ferent positions, illustrated, 79. 
 
 Circles, to find the perspective in- 
 clination of parallel lines by 
 means of, 70. 
 
 Column, to draw a circular, on a 
 square base, 143. 
 
 Concentric squares, to draw, 66. 
 
 Conical forms, representation of, 65. 
 Cottage with gable, to draw a, 1 5 ; 
 
 to draw the gable, 18; to draw 
 
 the windows, 19. 
 Cubes, to draw, in perspective, 108 ; 
 
 proof of the accuracy of the rule, 
 
 121. 
 
 Curved lines, representation of plane 
 surfaces bounded by, 34. 
 
 Curves, to represent, in perspective, 
 94 ; rules for drawing, in per- 
 spective, 136. 
 
 Delineation, plan of, defined, 82^., 84. 
 
 Diagonal lines, use of, explained and 
 illustrated, 18, 28. 
 
 Dials, to represent a series of, in 
 perspective, 25. 
 
 Distance, effect of increasing or de- 
 creasing the, between the object 
 and the student, 160. 
 
 Distance, point of, value of, 129; 
 point of, to find a, upon a line 
 above the horizontal line, 15 ; to 
 find a, all the lines being below 
 the horizontal line, 21. 
 
 Dome, to draw a, in perspective, 138. 
 
 Doorways, pointed, to draw, 54 ; to 
 draw the recesses of pointed, 62. 
 
 Double Arch, to draw, 39. 
 
 Drawing, difference between geo- 
 metrical and perspective, ex- 
 plained, 10. 
 
 Drawing from nature, a substitute 
 for, 14. 
 
 Drawing-board, 3. 
 
 Ellipse, perspective drawing of every 
 
 circle an, 38. 
 Eye, structure of the, 82 n. 
 
 Framework of windows, to draw 
 the, 57. 
 
INDEX. 
 
 167 
 
 Gable, to represent a, 18. 
 
 Gables, representation of a row of 
 houses with, 25. 
 
 Garret windows with pointed roofs, 
 directions for representing, 30. 
 
 Geometrical drawing defined, 10 ; 
 compared with perspective draw- 
 ing, 128. 
 
 Geometrical drawing, 96 n. 
 
 Gothic arches, to draw, 63. 
 
 Ground line, 6 ; to be used as the 
 base of the plane of delineation 
 where the plan is under the pic- 
 ture, 130. 
 
 Ground plan defined, 82 n.\ position 
 of, in relation to the plane of de- 
 lineation, 89. 
 
 Guiding angle, 15. 
 
 Horizontal line defined, 5 ; position 
 of, 6 and 16^.; position of in 
 bird's-eye views, 7- 
 
 Houses, representation of, with 
 gables in succession, 23. 
 
 Line, ground, defined, 5 ; horizontal, 
 
 defined, 6. 
 Linear perspective defined, 4. 
 Lines, parallel, 2 ; perpendicular, 2; 
 
 straight, defined, 1 ; vertical, 2 n. 
 
 Mouldings, to draw, 155. 
 Movable angle, 15. 
 
 Nature, position of the artist in 
 sketching from, 160. 
 
 Objects, diminution of, in size when 
 viewed perspectively, 76 ; diminu- 
 tion in regular progression, 77 ; 
 diminution illustrated, 77 ; method 
 of forming views of, from imagi- 
 nary stations, 163; on the dis- 
 tance of the spectator from the 
 object 1 to be represented, 159. 
 
 Oblique perspective, 106. 
 
 Obtuse angle defined, 3. 
 
 Octogen, to draw an, 65. 
 
 Optic nerve described, 82 n. 
 
 Optics, rules of perspective deduced 
 from, 81. 
 
 Parallel lines, 2; to find the per- 
 spective inclination of, by means 
 of arcs of circles, 70. 
 
 Parallel perspective, 106. 
 
 Parallelogram, to find the centre of 
 a rectaugular, 18. 
 
 Perpendicular defined, 2. 
 
 Perpendicular lines, three ways of 
 finding the perspective distance of, 
 130; comparison of the various 
 methods, 133. 
 
 Perspective, aerial, defined, 4. 
 
 Perspective, linear, defined, 4 ; de- 
 scribed, 73 ; cause of objects 
 changing their figure when viewed 
 perspectively, 74. 
 
 Perspective centre, to find the, 
 18; distance of perpendicular 
 lines, three modes of determining, 
 130; ditto, comparison of the 
 methods, 133 ; drawing described, 
 10; example of drawing, 11; 
 compared with geometrical draw- 
 ing, 128; modes of making draw- 
 ing to be preferred, 1 26 ; rules of, 
 deduced from the science of 
 optics, 81 ; use of, illustrated, 7 ; 
 oblique, 106; parallel, 106. 
 
 Piers, to find the width of, various 
 methods, 135. 
 
 Pillar, to represent a square, with 
 capital in perspective, 140. 
 
 Plane defined, 82 n. 
 
 Plane of delineation defined, 82 n. • 
 described, 84 ; influence of the 
 position of, on the object to be 
 drawn, 84 ; object of the, illus- 
 trated, 86, 89 ; position of, in rela- 
 tion to the ground plan, 89 ; 
 bringing the point up to the, 
 defined, 95 n. 
 
 Plane figure, method of drawing in 
 prospective, described, 90. 
 
 Plane surfaces, representation of, 
 bounded by curved lines, 34. 
 
 Plane surfaces, representation of, 
 bounded by right lines, 14. 
 
 Plans, preparation of, for perspective 
 drawing, 143. 
 
 Point of distance, value of the, 129. 
 
 Point of sight, to draw any rectili- 
 near figure in perspective by 
 means of the, as a vanishing 
 point, 98. 
 
168 
 
 INDEX. 
 
 Pointed arches, rules for drawiug, 54; 
 to draw the thickness, &c, of, 62. 
 
 Pointed doorways, method of draw- 
 ing, 54. 
 
 Pointed windows, to draw, 52. 
 
 Points defined, 94 ; to find the per- 
 spective position of single, 94 ; 
 various methods for finding the 
 perspective positions of, in a 
 picture, 1 24 ; to find the position 
 of, in combination, 97. 
 
 Points, vanishing, defined, 10 ; de- 
 scribed, 90; to find, 15. 
 
 Portico, to draw a, 155. 
 
 Position, necessity of a student keep- 
 ing the same, in sketching, 159. 
 
 Pyramid, octangular, 65. 
 
 Recesses, rule for determining the 
 depth of, 135. 
 
 Recesses of windows, to draw the 
 width of, 56. 
 
 Rectilinear figure, to draw any, in 
 perspective, by means of the point 
 of sight as a vanishing point, 98. 
 
 Reflections in water, 67 ; circum- 
 stances influencing, 67; influence 
 of twilight on, 67 ; of parts of an 
 object not seen by the spectator, 69. 
 
 Retiua of the eye described, 82 n. 
 
 Right angle defined, 2. 
 
 Right lines, representation of plane 
 surfaces bounded by, 14. 
 
 Rule, guiding, 15. 
 
 Shadows, projection of, 69 ; of geo- 
 metrical drawings, 69 ; cases in 
 which it is of importance to cor- 
 rectly represent, 69. 
 
 Slate roofs, directions for drawing, 
 32. 
 
 Solid figures, to draw, in perspective, 
 107. 
 
 Solids, representation of, 55. 
 
 Spectator, choice of position for the, 
 relative to the object to be repre- 
 sented, 156; circumstances in- 
 fluencing, 157 ; necessity of keep- 
 ing the position when taken, 159 ; 
 distance of the, from the object, 
 
 159; effect of increasing or decreas- 
 ing the distance, 160 ; position of 
 the, in sketching from nature, 160. 
 
 Spires, to draw, 66 ; use of diagonal 
 lines for finding the points of, 64. 
 
 Square, T, described, 3. 
 
 Square pillar with capital, to repre- 
 sent, in perspective, 140. 
 
 Squares, to draw concentric, 66. 
 
 Straight line defined, 1. 
 
 Straight lines, to represent, in per- 
 spective, 94. 
 
 T square described, 3 n. 
 Tangents defined, 40. 
 Tile-roofing, directions for drawing, 
 32. 
 
 Triangle, to draw a, in perspective 
 by the vanishing points for each 
 line, 103 ; to draw a, in perspec- 
 tive by means of the point of 
 sight as a vanishing point, 98. 
 
 Turret, to find the points of, by 
 diagonal lines, 64. 
 
 Twilight, influence of, on water, 67. 
 
 Vanishing point defined., 10. 
 
 Vanishing points described, 90 ; to 
 find, 1 5, 100 ; instrument for, 1 5 ; 
 preferable to depend on the eye 
 than on the instrument, 15 ; every 
 change of angle in an object re- 
 quires a fresh vanishing point, 99. 
 
 Vaulting, to draw, 138. 
 
 Vertical lines defined, 2 n. 
 
 Visual rays described, 81 ; trans- 
 mission of, 83, 89. 
 
 Water, reflections in, 67 ; circum- 
 stances influencing, 67 ; influence 
 of twilight on, 67 ; parts of an 
 object reflected but not visible to 
 the spectator, 68. 
 
 Windows, to draw, 1 9 ; to draw the 
 framework of, 57 ; to represent 
 the width of the recesses, 56. 
 
 Windows, garret, with pointed roofs, 
 to draw, 30. 
 
 Windows, pointed, to draw in per- 
 spective, 52 ; to draw the re- 
 cesses of, 62. 
 
 VIRTUE AND CO., PRINTERS, CITY ROAD, LONDON. 
 
PEIZE MEDAL, INTERNATIONAL EXHIBITION, 1862, was 
 awarded to Messrs. VIRTUE for the *' publication of 
 "Weale's Series." 
 
 See JURORS' REPORTS, 
 
 CLASS XXIX. 
 
 CATALOGUE 
 
 OF 
 
 RUDIMENTARY, SCIENTIFIC, EDUCATIONAL, AND 
 CLASSICAL WORKS, 
 
 FOR COLLEGES, HIGH AND ORDINARY SCHOOLS 
 AND SELF-INSTRUCTION ; 
 
 ALSO FOR 
 
 MECHANICS' INSTITUTIONS, FEEE LIBEAEIES, &c. &c, 
 
 PUBLISHED BY 
 
 VIRTUE & CO., 26, IVY LANE, 
 
 PATEENOSTEE EOW, LONDON. 
 
 *** THE ENTIRE SERIES IS FREELY ILLUSTRATED ON WOOD 
 AND STONE WHERE REQUISITE. 
 
 The Public are respectfully informed that the whole of the late 
 Mb. Weale's Publications, contained in the following Catalogue, 
 have been p%erchased by Virtue & Co., and that all future 
 Orders ivill be siqyplied by them at 26, Ivy Lane. 
 
 *** Additional Volumes, by Popular Authors, are in Preparation. 
 
 RUDIMENTARY SERIES. 
 
 2. NATUEAL PHILOSOPHY, by Charles Tomlinson. Is. 
 12. PNEUMATICS, by Charles Tomlinson. New Edition. Is. 6cl 
 20. PEESPECTIVE, by George Pyne. 2s. 
 
 27. PAINTING ; or, A GEAMMAE OF COLOUEING, by G. 
 Field. 25. 
 
 40. GLASS STAINING, by Dr. M. A. Gessert, with an Appendix 
 on the Art of Enamelling. Is. 
 
2 
 
 SCIENTIFIC AND MECHANICAL WORKS. 
 
 41. PAINTING ON GLASS, from the German of Fromberg. Is. 
 
 50. LAW OF CONTRACTS FOR WORKS AND SERVICES, 
 by David Gibbons. Is. 
 
 66. CLAY LANDS AND LOAMY SOILS, by J. Donaldson. Is. 
 
 69. MUSIC, Treatise on, by C. C. Spencer. 2s. 
 
 71. THE PIANOFORTE INSTRUCTIONS, by C. C. Spencer, la. 
 
 72. RECENT FOSSIL SHELLS (A Manual of the Mollusca), 
 
 by S. P. Woodward. 5s. 0>d. 
 
 In cloth boards, 65. 6^. ; half morocco, 7s. 6d. 
 79**. PHOTOGRAPHY, a Popular Treatise on, from the French 
 
 of D. Van Monckhoven, by W. H. Thornthwaite. Is. 6d. 
 96. ASTRONOMY, by the Rev. R. Main. Is. 
 
 107. METROPOLITAN BUILDINGS ACT, and THE METRO- 
 
 POLITAN ACT FOR REGULATING THE SUPPLY 
 OF GAS, with Notes, by D. Gibbons and R. Hesketh. 2s. 6d. 
 
 108. METROPOLITAN LOCAL MANAGEMENT ACTS. Is. 6d. 
 
 108*. METROPOLIS LOCAD MANAGEMENT AMENDMENT 
 ACT, 1862; with Notes and Index. Is. 
 
 109. NUISANCES REMOVAL AND DISEASE PREVENTION 
 
 ACT. Is. 
 
 110. RECENT LEGISLATIVE ACTS applying to Contractors, 
 
 Merchants, and Tradesmen. Is. 
 113. USE OF FIELD ARTILLERY ON SERVICE, by Jaubert, 
 translated by Lieut.-Col. H. H. Maxwell. Is. 6d. 
 
 113* MEMOIR ON SWORDS, by Marey, translated by Lieut.-CoL 
 H. H. Maxwell. Is. 
 
 140. OUTLINES OF MODERN FARMING, by R. Scott Burn. 
 
 Vol. I. Soils, Manures, and Crops. 2s. 
 
 141. Vol. II. Farming 
 
 Economy, Historical and Practical. 3s. 
 
 142. Vol. III. Stock- 
 Cattle, Sheep, and Horses. 2s. 6d. 
 
 145. Vol. IV. Manage- 
 ment of the Dairy — Pigs — Poultry. 2s. 
 
 14G. Vol. V. Utilisation of 
 
 Town Sewage — Irrigation — Reclamation of Waste Land. 2s. 6d. 
 The above 5 vols, hound in 2, cloth boards, 14s. 
 
 150. A TREATISE ON LOGIC, PURE AND APPLIED. By 
 S. H. Emmens, Esq. Is. 6d. 
 
 VIRTUE & CO., 26, IVY LANE. 
 
SCIENTIFIC AND MECHANICAL WORKS. 
 
 3 
 
 151. A HANDY BOOK ON THE LAW OP FRIENDLY, IN- 
 
 DUSTRIAL AND PEOVIDENT, BUILDING AND LOAN 
 SOCIETIES, by N. White. Is. 
 
 152. PRACTICAL HINTS FOR INVESTING- MONEY: with 
 
 an Explanation of the Mode of Transacting Business on the 
 Stock Exchange, by Francis Playford, Sworn Broker. Is. 
 
 153. LOCKE ON THE CONDUCT OF THE HUMAN UNDER- 
 
 STANDING, Selections from, by S. H. Emrnens. 2s. 
 
 154. GENERAL HINTS TO EMIGRANTS. 2s. 
 
 PHYSICAL SCIENCE. 
 
 1. CHEMISTRY, by Prof. Fownes, including Agricultural Che- 
 mistry, for the use of Farmers. Is. 
 
 3. GEOLOGY, by Major-Gen. Portlock. Is. 6d. 
 
 4. MINERALOGY, with a Treatise on Mineral Rocks or Aggre- 
 
 gates, by Dana. 2s. 
 
 7. ELECTRICITY, by Sir W. S. Harris. Is. 6d. 
 
 7*. GALVANISM, ANIMAL AND VOLTAIC ELECTRICITY, 
 by Sir W. S. Karris. Is. 6d. 
 
 8. MAGNETISM, Exposition of, by Sir W. S. Harris. 3s. 6d. 
 11. ELECTRIC TELEGRAPH, History of, by E. Highton. 2s. 
 
 133. METALLURG Y OF COPPER, by R. H. Lamborn. 2s. 
 
 134. METALLURGY OF SILVER AND LEAD, by R. H. Lam- 
 
 born. 2s. 
 
 135. ELECTRO-METALLURGY, by A. Watt. Is. 6d. 
 
 138. HANDBOOK OF THE TELEGRAPH, by R. Bond. Is. 
 143. EXPERIMENTAL ESSAYS— On the Motion of Camphor 
 and Modern Theory of Dew, by C. Tomlinson. Is. 
 
 BUILDING AND ARCHITECTURE. 
 
 16. ARCHITECTURE, Orders of, by W. H. Leeds. Is. Gd. 
 
 17. Styles of, by T. Bury. Is. 6d. 
 
 18. Principles of Design, by E. L. Garbett. 2s. 
 
 22. BUILDING, the Art of, by E. Dobson. Is. 
 
 23. BRICK AND TILE MAKING, by E. Dobson. 2s. 
 
 25. MASONRY AND STONE-CUTTING, by E. Dobson. 2s. 
 30. DRAINING AND SEWAGE OF TOWNS AND BUILD- 
 INGS, by G. D. Dempsey. 2s. 
 (With No. 29, Drainage of Land, 2 vols, in 1, 3s.) 
 
 VIRTUE & CO., 26, IVY LANE. 
 
4 SCIENTIFIC AND MECHANICAL WORKS. 
 
 35. BLASTING AND QUARRYING OF STONE, AND BLOW- 
 
 ING UP OF BRIDGES, by Lt.-Gen. Sir J. Burgoyne. Is. Qd. 
 
 36. DICTIONARY OF TERMS used by Architects, Builders, 
 
 Engineers, Surveyors, &c. 45. 
 
 In cloth boards, 5s. ; half morocco, 6s. 
 
 42. COTTAGE BUILDING, by C. B. Allen. 1*. 
 
 44. FOUNDATIONS and CONCRETE WORKS, by E.Dobson. Is. 
 
 45. LIMES, CEMENTS, MORTARS, CONCRETE, MASTICS, 
 
 &c, by G. R. Burnell. Is. Qd. 
 
 57. WARMING AND VENTILATION, by C. Tomlinson. 3s. 
 
 83**, CONSTRUCTION OF DOOR LOCKS, by C, Tomlinson. 
 Is. U. 
 
 111. ARCHES, PIERS, AND BUTTRESSES, by W.Bland. Is. U. 
 
 116. ACOUSTICS OF PUBLIC BUILDINGS, by T. R. Smith. 
 Is. 6d. 
 
 123. CARPENTRY AND JOINERY, founded on Robison and 
 
 Tredgold. Is. 6d. 
 123*. ILLUSTRATIVE PLATES to the preceding. 4to. 4s. 6d. 
 
 124. EOOFS FOR PUBLIC AND PRIVATE BUILDINGS, 
 
 founded on Robison, Price, and Tredgold. Is. 6d. 
 
 124*. IRON ROOFS of Recent Construction— Descriptive Plates. 
 4to. 4s. 6d. 
 
 127. ARCHITECTURAL MODELLING, Practical Instructions, 
 
 by T. A. Richardson. Is. 6d. 
 
 128. VITRUVIUS'S ARCHITECTURE, translated by J. Gwilt, 
 
 with Plates. 55. 
 
 130. GRECIAN ARCHITECTURE, Principles of Beauty in, by 
 the Earl of Aberdeen. Is. 
 
 132. ERECTION OF DWELLING-HOUSES, with Specifications, 
 Quantities of Materials, &c., by S. H. Brooks, 27 Plates. 25. Qd. 
 
 MACHINERY AND ENGINEERING. 
 
 33. CRANES AND MACHINERY FOR RAISING HEAVY 
 
 BODIES, the Art of Constructing, by J. Glynn. Is. 
 
 34. STEAM ENGINE, by Dr. Lardner. Is. 
 
 43. TUBULAR AND IRON GIRDER BRIDGES, including the 
 Britannia and Conway Bridges, by G. D. Dempsey. Is. 6d. 
 
 VIRTUE & CO., 26, IVY LANE. 
 
SCIENTIFIC AND MECHANICAL WORKS. 
 
 5 
 
 47. LIGHTHOUSES, their Construction and Illumination, by Allan 
 Stevenson. 3s. 
 
 59. STEAM BOILERS, their Construction and Management, by 
 R. Armstrong. Is. 6d. 
 
 62. RAILWAYS, Construction, by Sir M. Stephenson. Is. 6d. 
 
 62* RAILWAY CAPITAL AND DIVIDENDS, with Statistics of 
 Working, by E. D. Chattaway. Is. 
 
 (Vols. 62 and 62* bound in 1, 2s. 6d.) 
 
 67. CLOCK AND WATCH MAKING, and Church Clocks and 
 Bells, by E. B. Denison. 3s. Qd. 
 
 78. STEAM AND LOCOMOTION, on the Principle of connecting 
 Science with Practice, by J. Sewell. 2s. 
 
 78*. LOCOMOTIVE ENGINES, by G. D. Dempsey. Is. <5d. 
 
 79*. ILLUSTRATIONS TO THE ABOVE. 4to. 4s. 6d. 
 
 98. MECHANISM AND CONSTRUCTION OF MACHINES, 
 by T. Baker ; and TOOLS AND MACHINES, by J. Nasmyth, 
 with 220 Woodcuts. 2s. 6d. 
 
 114. MACHINERY, Construction and Working, by C. D. Abel. 
 
 Is. Qd. 
 
 115. PLATES TO THE ABOVE. 4to. 7s. 6d. 
 
 139. STEAM ENGINE, Mathematical Theory of, by T. Baker. Is. 
 
 CIVIL ENGINEERING, &c. 
 
 13. CIVIL ENGINEERING, by H. Law and G. R. Burnell. 4s. 6d. 
 29. DRAINING DISTRICTS AND LANDS, by G.D. Dempsey. Is. 
 (With No. 30, Drainage and Sewage of Towns, 2 vols, in 1, 3s.) 
 
 31. WELL-SINKING, BORING, AND PUMP WORK, by J. G. 
 
 Swindell, revised by G. R. Burnell. Is. 
 
 46. ROAD-MAKING AND MAINTENANCE OP MACADA- 
 MISED ROADS, by Gen. Sir J. Burgoyne. Is. 6d. 
 
 60. LAND AND ENGINEERING SURVEYING, by T. Baker. 2s> 
 
 VIRTUE & CO., 26, IVY LANE. 
 
6 
 
 SCIENTIFIC AND MECHANICAL WORKS. 
 
 63. AGRICULTURAL ENGINEERING, BUILDINGS, MOTIVE 
 POWERS, FIELD ENGINES, MACHINERY, AND 
 IMPLEMENTS, by G. H. Andrews. 35. 
 
 77*. ECONOMY OF FUEL, byT. S. Prideaux. Is. 
 
 80*. EMBANKING LANDS FROM THE SEA, by J.Wiggins. 2s. 
 
 82. WATER POWER, as applied to Mills, &c., by J. Glynn. 2s. 
 
 82**. A TREATISE ON % GAS WORKS, AND THE PRACTICE 
 OF MANUFACTURING AND DISTRIBUTING COAL 
 GAS, by S. Hughes, C.E. Ss. 
 
 82***. WATER-WORKS FOR CITIES AND TOWNS, by S. 
 Hughes. 3s. 
 
 117. SUBTERRANEOUS SURVEYING, AND EANGING THE 
 
 LINE without the Magnet, by T. Fenwick, with Additions 
 by T.Baker. 2s. Qd. 
 
 118. CIVIL ENGINEERING OF NORTH AMERICA, by D. 
 
 Stevenson. 3s. 
 
 120. HYDRAULIC ENGINEERING, by G. R. Burnell. 3s. 
 
 121. RIVERS AND TORRENTS, and a Treatise on NAVI- 
 
 GABLE CANALS AND RIVERS THAT CARRY SAND 
 AND MUD, from the Italian of Paul Frisi. 2s. 6d. 
 
 125. COMBUSTION OF COAL, AND THE PREVENTION 
 OF SMOKE, by C. Wye Williams, M.I.C.E. 3s. 
 
 SHIP-BUILDING AND NAVIGATION. 
 
 51. NAVAL ARCHITECTURE, by J. Peake. 35. 
 
 53*. SHIPS FOR OCEAN AND RIVER SERVICE, Construction 
 of, by Captain H. A. Sommerfeldt. Is. 
 
 53**. ATLAS OF 15 PLATES TO THE ABOVE, Drawn for 
 Practice. 4to. 7s. 6d. 
 
 54. MASTING, MAST-MAKING, and RIGGING OF SHIPS, 
 by R. Kipping. Is. 6d. 
 
 VIRTUE & CO., 26, IVY LANE. 
 
SCIENTIFIC AND MECHANICAL WORKS. 
 
 54*. IRON SHIP-BUILDING, by J. Grantham. 2s. Gd. 
 
 54**. ATLAS OF 24 PLATES to the preceding. 4to. 22s. Gd. 
 
 55. NAVIGATION ; the Sailor's Sea Book : How to Keep the Log 
 and Work it off, &c.; Law of Storms, and Explanation of 
 Terms, by J. Greenwood. 2s. 
 
 80. MARINE ENGINES, AND STEAM VESSELS, AND THE 
 SCREW, by R. Murray. 2s. Gd. 
 
 83 bis. SHIPS AND BOATS, Form of, by W. Bland. Is. Gd. 
 
 99. NAUTICAL ASTRONOMY AND NAVIGATION, by J. R. 
 Young. 2s. 
 
 100* NAVIGATION TABLES, for Use with the above. Is. Gd. 
 
 106. SHIPS' ANCHORS for all SERVICES, by G. Cotsell. Is. Gd. 
 
 149. SAILS AND SAIL-MAKING, by R. Kipping, N.A. 2s. Gd. 
 
 155. ENGINEER'S GUIDE TO THE ROYAL AND MER- 
 CANTILE NAVIES. 35. 
 
 ARITHMETIC AND MATHEMATICS. 
 
 6. MECHANICS, by Charles Tomlinson. Is. Gd. 
 
 32. MATHEMATICAL INSTRUMENTS, THEIR CONSTRUC- 
 TION, USE, &c, by J. F. Heather. New Edition. Is. Gd. 
 
 61*. READY RECKONER for the Measurement of Land, Tables 
 of Work at from 2s. Gd. to 20s. per acre, and valuation of 
 Land from £1 to £1,000 per acre, by A. Arman. Is. Gd. 
 
 76. GEOMETRY, DESCRIPTIVE, with a Theory of Shadows and 
 Perspective, and a Description of the Principles and Practice 
 of Isometrical Projection, by J. F. Heather. 2s. 
 
 83. BOOK-KEEPING AND COMMERCIAL PHRASEOLOGY, 
 
 by James Haddon. Is. 
 
 84. ARITHMETIC, with numerous Examples, by J. R.Young. 1,9. Gd. 
 84*. KEY TO THE ABOVE, by J. R. Young. Is. Gd. 
 
 VIRTUE & CO., 26, IVY LANE. 
 
8 
 
 SCIENTIFIC AND MECHANICAL WORKS. 
 
 85. EQUATIONAL ARITHMETIC : Tables for the Calculation 
 
 of Simple Interest, with Logarithms for Compound Interest, 
 and Annuities, by W. Hipsley. 2s. 
 
 86. ALGEBRA, by J. Haddon. 2s. 
 
 86*. KEY AND COMPANION TO THE ABOVE, by J. R. 
 Young. Is. 6d. 
 
 88. EUCLID'S GEOMETRY, with Essay on Logic, by H. Law. 2s. 
 
 90. GEOMETRY, ANALYTICAL AJSTD CONIC SECTIONS, by 
 
 J. Hann. Is. 
 
 91. PLANE & SPHERICAL TRIGONOMETRY, by J. Hann. 2s. 
 
 93. MENSURATION, by T. Baker. Is. Gd. 
 
 94. LOGARITHMS, Tables of ; with Tables of Natural Sines, Co- 
 
 sines, and Tangents, by H. Law. 2s. 6d. 
 
 97. STATICS AND DYNAMICS, by T.Baker. Is. 
 
 101. DIFFERENTIAL CALCULUS, by W. S. B. Woolhouse. Is. 
 
 101*. WEIGHTS AND MEASURES OF ALL NATIONS; 
 
 Weights of Coin3, and Divisions of Time ; with the Principles 
 which determine the Rate of Exchange, by W. S. B. Wool- 
 house. Is. 6d. 
 
 102. INTEGRAL CALCULUS, by H. Cos. Is. 
 
 103. INTEGRAL CALCULUS, Examples of, by J. Hann. Is. 
 
 104. DIFFERENTIAL CALCULUS, Examples of, with Solutions, 
 
 by J. Haddon. Is. 
 
 105. ALGEBRA, GEOMETRY, and TRIGONOMETRY, First 
 
 Mnemonical Lessons in, by the Rev. T. P. Kirkman. Is. 6d. 
 
 131. KEADY-RECKONER FOR MILLERS, FARMERS, AND 
 MERCHANTS, showing the Value of any Quantity of Corn, 
 with the Approximate Value of Mill-stones and Mill Work. Is. 
 
 136. EUDIMENTARY ARITHMETIC, by J. Haddon, edited by 
 
 A. Arman. Is. 6d. 
 
 137. KEY TO THE ABOVE, by A. Arman. Is. 6d. 
 
 147. STEPPING STONE TO ARITHMETIC, by Abraham 
 
 Arman, Schoolmaster, Thur'leigh, Beds. Is. 
 
 148. KEY TO THE ABOVE, by A. Arman. Is. 
 
 VIRTUE & CO., 26, IVY LANE. 
 
NEW SERIES OP EDUCATIONAL WORKS, 
 
 [This Series is kept in three styles of binding — the prices of each 
 are given in columns at the end of the lines.'] 
 
 HISTORIES, GRAMMARS, AND DICTIONARIES. 
 
 Limp. 
 
 1 Cloth 
 Boards. 
 
 Half 1 
 Murucco.j 
 
 
 s. 
 
 d. 
 
 s. d. 
 
 s. d. 
 
 1. ENGLAND, History of, by W. D. Hamilton 
 
 A 
 
 4 
 
 V 
 
 O U 
 
 o b 
 
 5 GREECE Historv of bv W T) Hamilton 
 
 
 O 
 
 ^ ft 
 
 4. n 
 
 and E. Levien 
 
 9 
 ~i 
 
 7. ROME, History of, by E. Levien 
 
 9 
 
 a 
 u 
 
 
 4 O 
 
 9. CHRONOLOGY OF CIVIL AND ECCLE- 
 
 
 
 
 
 siastical History, Literature, Art, and 
 Civilisation, from the earliest period to 
 
 
 
 
 
 
 6 
 
 3 G 
 
 4 
 
 
 2 
 
 11. ENGLISH GRAMMAR, by Hyde Clarke . 
 
 1 
 
 
 
 
 
 11*. HANDBOOK OF COMPARATIVE PHI- 
 
 
 
 
 
 lology, by Hyde Clarke .... 
 
 i 
 
 
 
 
 
 12. ENGLISH DICTIONARY, above 100,000 
 
 
 
 
 
 words, or 50,000 more than in any existing 
 
 
 
 4 6 
 
 5 
 
 work. By Hyde Clarke .... 
 
 3 
 
 6 
 
 , YV 1 L 1 J. \-A I Ciilllilcii 
 
 
 
 5 G 
 
 
 
 14. GREEK GRAMMAR, by H. C. Hamilton . 
 
 1 
 
 
 
 
 
 15. DICTIONARY, by H. R. Hamil- 
 
 
 
 
 
 ton. Vol. 1. Greek — English . 
 
 9 
 
 
 
 
 
 17. Vol. 2. English — Greek 
 
 2 
 
 
 
 
 
 ■ Complete in 1 vol. 
 
 4 
 
 
 
 5 
 
 5 6 
 
 , with Grammar 
 
 
 
 G 
 
 G 6 
 
 19. LATIN GRAMMAR, by T. Goodwin . 
 
 1 
 
 
 
 
 
 20. DICTIONARY, by T. Goodwin. 
 
 
 
 
 
 Vol. 1. Latin — English .... 
 
 2 
 
 
 
 
 
 22. Vol. 2. English— Latin 
 
 1 
 
 6 
 
 
 
 Complete in 1 vol. 
 
 o 
 
 o 
 
 G 
 
 4 G 
 
 5 O 
 
 ■ , with Grammar 
 
 
 
 5 G 
 
 6 O 
 
 M. FRENCH GRAMMAR, by G. L. Strauss . 
 
 1 
 
 
 
 
 I 
 
 VIRTUE & CO., 2G, IVY LANE. 
 
NEW SERIES OF EDUCATIONAL WORKS. 
 
 HISTORIES, GRAMMARS, AND DICTIONARIES. 
 
 25. 
 26. 
 
 27. 
 28. 
 
 30. 
 32. 
 
 34. 
 35. 
 
 39. 
 40. 
 41. 
 
 42. 
 43. 
 
 FRENCH DICTIONARY, by A. Elwes. 
 Yol. L French — English . 
 
 Yol. 2. English— French 
 
 Complete in 1 vol. 
 
 f with Grammar 
 
 ITALIAN GRAMMAR, by A. Elwes . 
 
 TRIGLOT DICTIONARY, by 
 
 A. Elwes. Yol. 1. Italian — English — 
 
 French 
 
 Yol. 2. English — Italian — French 
 Yol. 3. French — Italian — English 
 
 Complete in 1 vol. 
 
 , with Grammar 
 
 SPANISH GRAMMAR, by A. Elwes . 
 ENGLISH AND ENGLISH- 
 SPANISH DICTIONARY, by A. Elwes . 
 . with Grammar 
 
 GERMAN GRAMMAR, by G. L. Strauss . 
 
 READER, from best Authors . 
 
 TRIGLOT DICTIONARY, by 
 
 N. E. Hamilton. Yol. 1. English— Ger- 
 man — French 
 
 Yol. 2. German — English — French 
 
 Yol. 3. French — English — German 
 
 Complete in 1 vol. . 
 
 , with Grammar 
 
 44. 
 
 46. 
 
 46* 
 47. 
 48. 
 
 49. 
 
 HEBREW DICTIONARY, by Dr. Breslau. 
 
 Yol. 1. Hebrew — English 
 , with Grammar 
 
 Yol. 2. English — Hebrew 
 
 -Complete, with Grammar, in 2 vols. 
 - GRAMMAR, by Dr. Breslau . 
 
 FRENCH AND ENGLISH PHRASE BOOK 
 
 COMPOSITION AND PUNCTUATION, 
 by J. Brenan 
 
 DERIYATIYE SPELLING BOOK, by J. 
 Rowbotham 
 
 s. d. 
 
 1 
 
 1 6 
 
 2 6 
 
 1 
 
 2 
 2 
 2 
 
 1 
 
 4 
 
 1 
 1 
 
 1 
 
 1 
 1 
 3 
 
 6 
 
 7 
 3 
 
 1 
 1 6 
 
 VIRTUE & CO., 26, IVY LANE. 
 
GREEK AND LATIN CLASSICS. 
 
 GREEK AND IiATIN CLASSICS, 
 
 With Explanatory Notes in English, principally selected from the 
 best German Commentators. 
 
 LATIN SERIES. 
 
 1. LATIN DELECTUS, with Vocabularies and Notes, by 
 
 H. Young Is, 
 
 2. OESAE'S GALLIC WAE ; Notes by H. Young . . 2s, 
 
 3. COKNELIUS NEPOS; Notes by H.Young . . . Is. 
 
 4. VIEGIL. The Georgics, Bucolics ; Notes by W. Eushton 
 
 and H. Young ... • Is. Gd. 
 
 5. VIEGIL'S -ZENEID ; Notes by H. Young • 2^r. 
 
 6. HORACE. Odes and Epodes ; Notes, Analysis and Ex- 
 
 planation of Metres Is. 
 
 7. HOEACE. Satires and Epistles ; Notes by W. B. Smith Is. 6d. 
 
 8. SALLUST. Catiline, Jugurtha ; Notes by W. M. Donne Is. 6d. 
 
 9. TERENCE. Andria and Heautontimorumenos ; Notes by 
 
 J. Davies Is. 6d. 
 
 10. TERENCE. Adelphi, Hecyra, and Phormio; Notes by J. 
 
 Davies 2s. 
 
 14. CICEEO. Do Amicitia, de Senectute, and Brutus ; Notes 
 
 by W. B. Smith 2s. 
 
 16. LIYY. Part I. Books i., ii., by H. Young . . Is. Gd. 
 16*. Part II. Books iii., iv., v., by H. Young Is. 6d. 
 
 1 7. — Part III. Books xxi., xxii., by W. B. Smith Is. 6d. 
 
 19. CATULLUS, TIBTJLLUS, OYID, and PEOPERTIUS, 
 
 Selections from, by W. M. Donne . . . • *' • 2s. 
 
 m. SUETONIUS and the later Latin Writers, Selections from, 
 
 by W. M. Donne . 2s. 
 
 VIRTUE & CO., 2G, IVY LANE. 
 
12 
 
 GREEK AND LATIN CLASSICS. 
 
 GREEK SERIES, 
 
 ON A SIMILAR PLAN TO THE LATIN SERIES. 
 
 1. GEEEK INTRODUCTORY READER, by H. Young. 
 
 On the same plan as the Latin Eeader . . . .15. 
 
 2. XENOPIION. Anabasis, i. ii. iii., by H. Young . . Is. 
 
 3. XENOPHON. Anabasis, iv. v. vi. vii., by H. Young . Is. 
 
 4. LUCIAN. Select Dialogues, by H. Young • • .15. 
 
 5. HOMER. Iliad, i. to vi., by T. H. L. Leary . . Is. Gd. 
 G. HOMER. Iliad, vii. to xii., by T. H. L. Leary . Is. Gd. 
 
 7. HOMEE. Iliad, xiii. to xviii., by T. H. L. Leary . Is. Gd. 
 
 8. HOMEE. Iliad, xix. to xxiv., by T. H. L. Leary . Is. Gd. 
 
 9. HOMEE. Odyssey, i. to vi., by T. H. L. Leary . Is. Gd. 
 
 10. HOMEE. Odyssey, vii. to xii., by T. H. L. Leary . Is. Gd. 
 
 11. HOMEE. Odyssey, xiii. to xviii., by T. H. L. Leary Is. Gd. 
 
 12. HOMEE. Odyssey, xix. to xxiv. ; and Hymns, by T. II. 
 
 L. Leary ......... 2s. 
 
 13. PLATO. Apology, Crito, and Phsedo, by J. Davies . . 2s. 
 
 14. HERODOTUS, i. ii., by T. H. L. Leary . . . Is. Gd. 
 
 15. HEEODOTUS, iii. iv., by T. H. L. Leary . . Is. Gd. 
 
 16. HEEODOTUS, v. vi. vii., by T. H. L. Leary , . Is. Gd. 
 
 17. HEEODOTUS, viii. ix., and Index, by T. H. L. Leary Is. Gd. 
 
 18. SOPHOCLES; (Edipus Tyrranus, by H. Young • . Is. 
 20. SOPHOCLES ; Antigone, by J. Milner . . . . 2s. 
 23. EURIPIDES ; Hecuba and Medea, by W. B. Smith . Is. Gd. 
 26- EURIPIDES; Alcestis, by J. Milner .... Is. 
 30. ZISCHYLUS; Prometheus Vinctus, by J. Davies . . 1*, 
 32/JlSCHYLUS ; Septem contra Thebas, by J. Davies , , Is. 
 
 40. ARISTOPHANES ; Acharnians, by C. S. D. Townshend Is. Gd. 
 
 41. THUCYDIDES, i., by H. Young U. 
 
 "VIRTUE & CO., 26, IVY LANE. 
 
GET 
 
 TY RESEARCH INST 
 
 TUTE 
 
 
 
 
 
 
 3 3125 01360 4497 
 
•on 5^ 
 
 VIRTUE BROTHERS & CO., 26, Ivy Lane, Paternoster Row. 
 
 ^'05 Photography, a Popular Treatise on. Also a Description 
 ^ "^o °^ an( ^ ^ emar ^ 8 on » ^he Stereoscopic and Photographic Optics, &c. 
 
 By Monckhoven. Translated by W. H. Thornthwaite, Ph.D., 
 F.C.S. Illustrated, Is. 6d. 
 
 Astronomy, Eudimentary. By the Rev. Eobeet Main, i^'jc 
 M.A., F.R.A.S., First Assistant at the Royal Observatory, Green- jfif sj; 
 
 ^m^")T wich. Illustrated, Is. p^fc 
 
 ^2 The Metropolitan Building Act, 18th and 19th Victoria, 
 ■^tS cap. 122. Followed by Appendices of the Unrepeated Sections of 
 
 the former Building Acts, 14th Geo. III., c. 78, and 7th and 8th r&J& 
 
 'sV^.^ Vic, c. 84, and the Sections of the City of London Sewers Acts, 2$* * 
 
 1848 and 1851, relating to Buildings. With Notes and Index by i^f^ 
 
 David Gibbons, Esq., of the Middle Temple, Special Pleader, and v^VJ 
 R. Hesketh, Esq., Architect and District Surveyor. Illustrated, 
 
 ^£ The Metropolis Local Management Act, 18th and 19th 
 
 Vic, c. 120; 19th and 20th Vic, c. 112; 21st and 22nd Vic, 
 
 % c. 104; 24th and 25th Vic, c 61 ; also the last Pauper Removal '<Y^ 
 
 Act, and the Parochial Assessment Act, comprehending all that is fVy^ 
 
 essential for the use Of Members of Parish Vestries, and the Local ^£o, 
 
 Managers of the Metropolis. Is. 6d. vV^Wl 
 
 £5 The Metropolis Local Management Amendment Act, 
 
 1862, 25th and 26th Vic, c. 120. With Notes and an Index. Is. 
 
 3li 
 
 The Nuisances Removal and Diseases Prevention 
 
 Amendment Act, 18th and 19th Vic, c. 121. To which are 
 added The Smoke Nuisance Act and the Common Lodging House *&\^V 
 Act. With Notes and an Index. Is. 
 
 oi^| Field Artillery on Service, on the Use of; with especial 
 reference to that of an Army Corps. For Officers of all Arms. 
 By Taubert, translated by Lieut.-Col. Henry Hamilton Maxwell, ^ ^ 
 
 m 
 
 Bengal Artillery. Is. 6d. 
 
 ; — W 
 
 VIRTUE BROTHERS & CO., 26, Ivy Lane, Paternoster Row. - 
 
flc VIBTUE BROTHERS & CO., 26, Ivy Lane, Paternoster Row. . 2 
 
 Jfy 
 
 £j$ t Six Legislative Enactments for the Guidance of Con- 
 $k tractors, Merchants, and Tradesmen. I. An Act to 
 
 ✓ ~. j facilitate the remedies on Bills of Exchange and Promissory Notes. 
 <jC 4 II. An Act to amend the Laws relating to Stamp Duties, and on 
 3; Bonds and Securities. III. The Newspaper Stamp Act, and the 
 ^ Transmission by Post of Periodicals, &c. IV. An Act to con- 
 <§t solidate the Excise and Stamps and Taxes into one Board. V. An 
 '3\t Act for facilitating arrangements between Debtors and Creditors. 
 VI. An Act to amend the Law of Insolvency, Bankruptcy, and 
 Execution. With an Index. Is. 
 
 57" 
 
 [gp Swords, &c, Memoir on. By Col. Maeet, translated by 
 A, J Lieut.-Col. H. H. Maxwell. With Notes. Illustrated, Is. 
 
 Modern Farming, Outlines of. By Eobert Scott Burist. 
 
 C^j Vol. I. — Soils, Manures, and Crops. Illustrated, 2s. 
 
 Vol. II. — Notes, Historical and Practical, on Farming and Farm- 
 
 ing Economy. Illustrated, 3s. 
 ^ o1, HI— Stock— Cattle, Sheep, and Horses. Illustrated, 2s. 6d. 
 ^£jt Vol. IV. — The Management of the Dairy — Pigs — Poultry. With 
 ^Jfc^ Notes on the Diseases of Stock, by a Veterinary Surgeon. 
 It0c> Illustrated, 2s. 
 
 VoL V. — Utilisation of Town Sewage — Irrigation — Reclamation 
 of Waste Land. Illustrated, 2s. 6d. 
 
 Complete in 2 vols., cloth boards, 14s. 
 
 The Law of Friendly, Industrial and Provident, Building, 
 
 and Loan Societies, a Handy Book on. With Copious Notes. 
 
 By Nathaniel White, Esq. 12mo., Is. o^c^ 
 
 Logic, Pure and Applied, a Treatise on. By S. H. Emmens, 
 Esq. 12mo., Is. 6d. 
 
 . 
 
 Xjt* Investing Money, Practical Hints for: with an Expla- 
 
 nation of the Mode of Transacting Business on the Stock Exchange. ^^'^ 
 r 5 ^* By Francis Playford, Sworn Broker. Third Edition, revised and ^ 
 t^l corrected. 12mo., Is. 
 
 LONDON: VIRTUE BROTHERS & CO., 26, IVY LANE. 7^2 
 Q/" wAdfli ^#<# « complete Catalogue of the Weale Series. <§^< ^