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THE 
 
 ELEMENTS OF PEESPECTIVE. 
 
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 j-BTNTEJ BT SPOTTISWOODE ATfD CO. 
 yBW-STBEET SQTTABB 
 
THE 
 
 ELEMENTS OF PERSPECTIVE 
 
 ARRANGED FOR THE USE OF SCHOOLS 
 
 AND INTENDED TO Be" BEAD IN CONNEXION WITH THE 
 PIEST THEEE BOOKS OF EUCLID. 
 
 BY 
 
 JOHN RUSKIN, M.A. 
 
 Author of 
 "modern painters," "seven lamps of architecture," 
 
 'stones of VENICE," "LECTURES ON ARCHITECTURE AND PAINTING, 
 "elements of DRAWING," ETC. 
 
 LONDON 
 
 SMITH, ELDEE, AND CO., 65 COENHILL. 
 
 1859 
 
 The right of tramlation is reterv? 7, 
 
PEE FACE 
 
 For some time back I have felt the want, 
 among Students of Drawing, of a written code 
 of accurate Perspective Law; the modes of 
 construction in common use being various, 
 and, for some problems, insufficient. It would 
 have been desirable to draw up such a code 
 in popular language, so as to do away with 
 the most repulsive difficulties of the subject ; 
 but finding this popularization would be im- 
 possible, without elaborate figures and long 
 explanations, such as I had no leisure to pre- 
 pare, I have arranged the necessary rules in 
 
 A 4 
 
VI PKEFACE. 
 
 a short mathematical form, which any school- 
 boy may read through in a few days, after 
 he has mastered the first three and the sixth 
 books of Euchd. 
 
 Some awkward compromises have been 
 admitted between the first-attempted popu- 
 lar explanation, and the severer arrangement, 
 involving irregular lettering and redundant 
 phraseology ; but I cannot for the present do 
 more, and leave the book therefore to its 
 trial, hoping that, if it be found by masters 
 of schools to answer its pm^pose, I may here- 
 after bring it into better form.* 
 
 * Some irregularities of arrangement have been admitted 
 merely for the sake of convenient reference; the eighth problem, 
 for instance, ought to have been given as a case of the seventh, 
 but is separately enunciated on account of its importance. 
 
 Several constructions, which ought to have been given as 
 problems, are on the contrary given as corollaries, in order 
 to keep the more directly connected problems in closer se- 
 quence ;. thus the construction of rectangles and polygcms \n 
 vertical planes would appear by the Table of Contents to 
 have been omitted, being given in the corollary to Problem IX. 
 
PREFACE. VU 
 
 An account of practical methods, sufficient 
 for general purposes of sketching, might in- 
 deed have been set down in much less space : 
 but if the student reads the following pages 
 carefully, he will not only find himself 
 able, on occasion, to solve perspective pro- 
 blems of a complexity greater than the ordi- 
 nary rules will reach, but obtain a clue to 
 many important laws of pictorial efiect, no 
 less than of outline. The subject thus exa- 
 mined becomes, at least to my mind, very 
 cmious and interesting ; but, for students who 
 are unable or unwilling to take it up in 
 this abstract form, I beheve good help will 
 be soon furnished, in a series of illustrations 
 of practical perspective now in preparation by 
 Mr. Le Vengem. I have not seen this essay 
 in an advanced state, but the illustrations 
 shown to me were very clear and good ; and, 
 as the author has devoted much thought to 
 
VIU PREFACE. 
 
 their arrangement, I hope that his work will 
 be precisely what is wanted by the general 
 learner. 
 
 Students wishing to pursue the subject 
 into its more extended branches will find, I 
 beheve, Cloquet's treatise the best hitherto 
 pubhshed.* 
 
 * Xouveau Traite Elementaire de Perspective. Bachelier, 
 1823. 
 
CONTENTS. 
 
 Page 
 Preface v 
 
 Introduction 
 
 PROBLEM L 
 To FIX THE Position op a given Point . . , .18 
 
 PROBLEM IL 
 To DRAW A Right Line between two given Points . 22 
 
 PROBLEM in. 
 
 To FIND the Vanishing-Point of a given Horizontal 
 Line 27 
 
 PROBLEM IV. 
 
 To FIND THE Dividing-Points of a given Horizontal 
 Line 36 
 
X CONTENTS. 
 
 PROBLEM V. * 
 
 Page 
 
 To DRAW A Horizontal Line, given in Position and 
 
 Magnitude, by means op its Sight-Magnitude and 
 
 Dividing-Points 37 
 
 i»ROBLEM VL 
 
 To DRAW ANY TrIANGLE, GIVEN IN POSITION AND MAGNI- 
 TUDE, IN A Horizontal Plane 41 
 
 PROBLEM VIL 
 
 To draw any Rectilinear Quadrilateral Figure, given 
 IN Position and Magnitude, in a Horizontal Plane 43 
 
 PROBLEM VIH. 
 
 To DRAW A Square, given in Position and Magnitude, 
 IN A Horizontal Plane 45 
 
 PROBLEM IX. 
 
 To DRAW A Square Pillar, given in Position and 
 Magnitude, its Base and Top being in Horizontal 
 Planes 50 
 
 PROBLEM X. 
 
 To draw a Pyramid, given in Position and Magnitude, 
 ON A Square Base in a Horizontal Plane . . 53 
 
 PROBLEM XI. 
 
 To DRAW ANY CURVE IN A HORIZONTAL OR VERTICAL PlANB 55 
 
CONTENTS. XI 
 
 PEOBLEM XIL 
 
 Page 
 
 To DIVIDE A Circle drawn in Perspective into ant 
 
 GIVEN Number of Equal Parts 61 
 
 PROBLEM XIII. 
 
 To DRAW A Square given in Magnitude, within a larger 
 Square given in Position and Magnitude; the Sides 
 OF the two Squares being Parallel . . . .65 
 
 PROBLEM XIV. 
 
 To draw a Truncated Circular Cone, given in Posi- 
 tion AND Magnitude, the Truncations being in Ho- 
 rizontal Planes, and the Axis of the Cone vertical 67 
 
 PROBLEM XV. 
 
 To DRAW AN Inclined Line, given in Position and 
 Magnitude 71 
 
 PROBLEM XVL 
 
 To FIND THE Vanishing- Point of a given Inclined 
 Line 75 
 
 PROBLEM XVII. 
 To find the Dividing^Points of a given Inclined Line 78 
 
 PROBLEM XVIII. 
 To find the Sight-Line op an Inclined Plane in which 
 
 1 wo LlNliS ARE given IN POSITION , . . .81 
 
XU CONTENTS. 
 
 PROBLEM XIX. 
 
 Page 
 To FIND THE VaNISHING-PoINT OF STEEPEST LiNES IN 
 
 AN Inclined Plane whose Sight-Line is given . . 83 
 
 PROBLEM XX. 
 
 To find the Vanishing-Point of Lines perpendiculae 
 to the Surface of a given Inclined Plane . . 85 
 
 APPENDIX. 
 
 L 
 Practice ajh) Observations on the preceding Problems 97 
 
 IL 
 
 Demonstrations which could not conveniently be in- 
 cluded in the Text 135 
 
THE ELEMENTS. 
 
INTRODUCTION. 
 
 When you begin to read this book, sit down very near 
 the window, and shut the window. I hope the \dew 
 out of it is pretty; but, whatever the view may be, 
 we shall find enough in it for an illustration of the 
 first principles of perspective (or, literally, of "look- 
 ing through "). 
 
 Every pane of your window may be considered, if 
 you choose, as a glass picture ; and what you see 
 through it, as painted on its surface. 
 
 And if, holding your head still, you extend your 
 hand to the glass, you may, with a brush full of any 
 thick colour, trace, roughly, the lines of the land- 
 scape on the glass. 
 
 But, to do this, you must hold your head very still. 
 Not only you must not move it sideways, nor up 
 and down, but it must not even move backwards or 
 forwards ; for, if you move your head forwards, you 
 
 B 
 
2 THE ELEMENTS OF PERSPECTIVE. 
 
 will see more of the landscape through the pane ; 
 and, if you move it backwards, you will see less: 
 or considering the pane of glass as a picture, when 
 you hold your head near it, the objects are painted 
 small, and a great many of them go into a little 
 space ; but, when you hold your head some distance 
 back, the objects are painted lai'ger upon the pane, 
 and fewer of them go into the field of it. 
 
 But, besides holding your head still, you must, 
 when you try to trace the picture on the glass, shut 
 one of your eyes. If you do not, the point of the 
 brush appears double ; and, on farther experiment, 
 you will observe that each of your eyes sees the 
 object in a different place on the glass, so that the 
 tracing which is true to the sight of the right eye 
 is a couple of inches (or more, according to your 
 distance from the pane,) to the left of that which 
 is true to the sight of the left. 
 
 Thus, it is only possible to draw what you see 
 through the window rightly on the surface of the 
 glass, by fixing one eye at a given point, and 
 neither moving it to the right nor left, nor up nor 
 
INTRODUCTION. 3 
 
 dowTi, nor backwards nor forwards. Every picture 
 drawn in true perspective may be considered as an 
 upright piece of glass*, on which the objects seen 
 through it have been thus drawn. Perspective can, 
 therefore, only be quite right, by being calculated 
 for one fixed position of the eye of the observer; 
 nor will it ever appear deceptively right unless 
 seen precisely from the point it is calculated for. 
 Custom, however, enables us to feel the rightness 
 of the work on using both our eyes, and to be sa- 
 tisfied with it, even when we stand at some distance 
 from the point it is designed for. 
 
 Supposing that, instead of a window, an unbroken 
 plate of crystal extended itself to the right and left 
 of you, and high in front, and that you had a 
 brush as long as you wanted (a mile long, sup- 
 pose), and could paint with such a brush, then 
 the clouds high up, nearly over your head, and the 
 landscape far away to the right and left, might 
 
 * If the glass were not upright, but sloping, the objects might 
 still be drawn through it, but their perspictive would then be 
 different. Perspective, as commonly taught, is always calculated 
 for a Tcrtical plane of picture. 
 
 B 2 
 
4 THE ELEMENTS OF PERSrECTIVE. 
 
 be traced, and painted, on this enormous crystal 
 field.* But if the field were so vast (suppose a mile 
 high and a mile wide), certainly, after the picture 
 was done, you would not stand as near to it, 
 to see it, as you are now sitting near to your win- 
 dow. In order to trace the upper clouds through 
 your great glass, you would have had to stretch 
 your neck quite back, and nobody likes to bend 
 their neck back to see the top of a picture. So 
 you would walk a long way back to see the 
 great picture — a quarter of a mile, perhaps, — 
 and then all the perspective would be wrong, and 
 would look quite distorted, and you would discover 
 that you ought to have painted it from the greater 
 distance, if you meant to look at it from that distance. 
 Thus, the distance at which you intend the observer 
 to stand from a picture, and for which you calculate 
 the perspective, ought to regulate to a certain degree 
 the size of the picture. If you place the point of 
 observation near the canvass, you should not make 
 
 * Supposing it to have no thickness; otherwise the images would 
 he. distorted by refraction. 
 
INTRODUCTION. § 
 
 the picture very large : vice versa, if you place the 
 point of observation far from the canvass, you should 
 not make it very small ; the fixing, therefore, of this 
 point of observation determines, as a matter of con- 
 venience, within certain limits, the size of your 
 picture. But it does not determine this size by any 
 perspective law ; and it is a mistake made by many 
 writers on perspective, to connect some of their rules 
 definitely with the size of the picture. For, suppose 
 that you had what you now see through your 
 window painted actually upon its surface, it would 
 be quite optional to cut out any piece you chose, 
 with the piece of the landscape that was painted 
 on it. You might have only half a pane, with a 
 single tree; or a whole pane, with two trees and a 
 cottage ; or two panes, with the whole farmyard and 
 pond; or four panes, with farmyard, pond, and 
 foreground. And any of these pieces, if the land- 
 scape upon them were, as a scene, pleasantly com- 
 posed, would be agreeable pictures, though of quite 
 different sizes ; and yet they would be all calculated 
 for the same distance of observation. 
 
6 THE ELEMENTS OF PERSPECTIVE. 
 
 In the following treatise, therefore, I keep the 
 size of the picture entirely undetermined. I con- 
 sider the field of canvass as wholly unlimited, and 
 on that condition determine the perspective laws. 
 After we know how to apply those laws without 
 limitation, we shall see what limitations of the 
 size of the picture their results may render ad- 
 visable. 
 
 But although the size of the picture is thus inde- 
 pendent of the observer's distance, the size of the 
 object represented in the picture is not. On the 
 contrary, that size is fixed by absolute mathematical 
 law ; that is to say, supposing you have to draw a 
 tower a hundred feet high, and a quarter of a mile 
 distant from you, the height which you ought to 
 give that tower on your paper depends, with ma- 
 thematical precision, on the distance at which you 
 intend your paper to be placed. So, also, do all the 
 rules for drawing the form of the tower, whatever 
 it may be. 
 
 Hence, the first thing to be done in beginning a 
 drawinor is to fix, at your choice, this distance of 
 
INTRODUCTION. 7 
 
 observation, or the distance at which you mean to 
 stand from your paper. After that is determined, 
 all is determined, except only the ultimate size of 
 your picture, which you may make greater, or less, 
 not by altering the size of the things represented, 
 but by taking in more, or fewer of them. So, then, 
 before proceeding to apply any practical perspective 
 rule, we must always have our distance of observa- 
 tion marked, and the most convenient way of marking 
 it is the following. 
 
 PLACING OF THE SIGHT-POINT, SIGHT-LINE, SIATION- 
 POINT, AND STATION-LINE. 
 
 Vr 
 
 A I 
 
 N/' "\0 
 
 S 
 
 P - 
 
 o 
 
 T 
 
 R 
 
 Fig. 1. 
 b4 
 
 H 
 
 D 
 
8 THE ELEMENTS OF PEKSPECTIVE. 
 
 I. The Sight-Point. — Let abcd, Fig. 1., be your 
 sheet of paper, the larger the better, though per- 
 haps we may cut out of it at last only a small 
 piece for our picture, such as the dotted circle 
 N o p Q. This circle is not intended to limit either 
 the size or shape of our picture : you may ulti- 
 mately have it round or oval, horizontal or upright, 
 small, or large, as you choose. I only dot the line 
 to give you an idea of whereabouts you will pro- 
 bably like to have it; and, as the operations of 
 perspective are more conveniently performed upon 
 paper underneath the picture than above it, I put 
 this conjectural circle at the top of the paper, 
 about the middle of it, leaving plenty of paper on 
 both sides and at the bottom. Now, as an observer 
 generally stands near the middle of a picture to 
 look at it, we had better at first, and for simpli- 
 city's sake, fix the point of observation opposite the 
 middle of our conjectural picture. So take the point 
 s, the centre of the circle n o p Q ; — or, which will 
 be simpler for you in your own work, take the 
 point s at random near the top of your paper, and 
 
INTRODUCTION. 9 
 
 strike the circle n o p Q round it, any size you like. 
 Then the point S is to represent the point opposite 
 which you wish the observer of your picture to place 
 his eye, in looking at it. Call this point the " Sight- 
 Point." 
 
 II. The Sight-Line. — Through the Sight-point, s, 
 draw a horizontal line, G h, right across your paper 
 from side to side, and call this line the "Sight- 
 Line." 
 
 This line is of great practical use, represent- 
 ing the level of the eye of the observer all through 
 the picture. You will find hereafter that if there 
 is a horizon to be represented in your picture, as 
 of distant sea or plain, this line defines it. 
 
 TIL The Station-Line. — From s let fall a per- 
 pendicular line, s R, to the bottom of the paper, and 
 call this line the " Station-Line." 
 
 This represents the line on which the observer 
 stands, at a greater or less distance from the picture ; 
 and it ought to be imagined as drawn right out from 
 
10 THE ELEMENTS OF PERSPECTIVE. 
 
 the paper at the point s. Hold your paper upright 
 in front of you, and hold your pencil horizontally, 
 with its point against the point s, as if you wanted 
 to run it through the paper there, and the pencil 
 will represent the direction in which the line s R 
 ought to be drawn. But as all the measurements 
 which we have to set upon this line, and operations 
 which we have to perform with it, are just the same 
 when it is drawn on the paper itself, below s, as 
 they would be if it were represented by a wire in 
 the position of the levelled pencil, and as they are 
 much more easily performed when it is drawn on 
 the paper, it is always in practice, so drawn. 
 
 IV. The Station-Point. — On this Kne, mark the 
 distance s t at your pleasure, for the distance at 
 which you wish your picture to be seen, and call 
 the point T the " Station-Point." 
 
 In practice, it is generally advisable to make the 
 distance s t about as great as the diameter of your 
 intended picture ; and it should, for the most part, be 
 more rather than less ; but, as I have just stated, this is 
 
INTRODUCTION. 
 
 11 
 
 quite arbitrary. However, in this figure, as an ap- 
 proximation to a generally advisable distance, I make 
 the distance s T equal to the diameter of the circle 
 N p Q. Now, having fixed this distance, s T, all the 
 dimensions of the objects in our picture are fixed 
 likewise, and for this reason : — 
 
 Let the upright line A b. Fig. 2,, represent a 
 pane of glass placed where our picture is to be placed ; 
 
 s 
 
 Fig. 2. 
 
 but seen at the side of it, edgeways; let s be the 
 Sight-point; s T the Station-line, which, in this 
 figure, observe, is in its true position, drawn out 
 from the paper, not down upon it ; and t the 
 Station-point. 
 
12 THE ELEMENTS OF PERSPECTIVE. 
 
 Suppose the Station-line s t to be continued, or 
 in mathematical language "produced," through s, 
 far beyond the pane of glass, and let p Q be a tower 
 or other upright object situated on or above this 
 line. 
 
 Now the apparent height of the tower pq is 
 measured by the angle Q T P, between the rays of light 
 which come from the top and bottom of it to the eye 
 of the observer. But the actual height of the image 
 of the tower on the pane of glass A b, between us 
 and it, is the distance p' q', between the points where 
 the rays traverse the glass. 
 
 Evidently, the farther from the point T we place 
 the glass, making s T longer, the larger will be the 
 image ; and the nearer we place it to t, the smaller 
 the image, and that in a fixed ratio. Let the dis- 
 tance D T be the direct distance from the Station- 
 point to the foot of the object. Then, if we place 
 the glass A b at one third of that whole distance, 
 p' q' will be one third of the real height of the object; 
 if we place the glass at two thirds of the distance, as 
 
INTEODUCTION. 13 
 
 at E F, p'' q''' (the height of the image at that point) 
 will be two thirds the height * of the object, and so 
 on. Therefore the mathematical law is that p' q' 
 will be to p Q as s T to D T. I put this ratio clearly 
 by itself that you may remember it : 
 
 p' q' : p Q : : s T : D T 
 
 or in words : 
 
 p dash Q dash is to p Q as s T to d T 
 
 In which formula, recollect that p' q' is the height 
 of the appearance of the object on the picture ; p q 
 the height of the object itself; s the Sight-point: T 
 the Station-point; d a point at the direct distance 
 of the object; though the object is seldom placed 
 actually on the hue T s produced, and may be far 
 to the right or left of it, the formula is still the 
 same. 
 
 For let s. Fig. 3., be the Sight-point, and A b the 
 glass — here seen looking down on its wpjper edge, 
 
 * I say " height " instead of " magnitude," for a reason stated in 
 Appendix I., to which you will soon be referred. Read on here at 
 present. 
 
14 
 
 THE ELEMENTS OF PEESPECTIVE. 
 
 not sideways ; — then if the tower (represented now, 
 as on a map, by the dark square), instead of being 
 at D on the line s T produced, be at e, to the right 
 
 (or left) of the spectator, 
 still the apparent height 
 of the tower on A b will 
 be as s' T to e t, which 
 is the same ratio as that 
 of s T to D T. 
 
 Now in many perspective 
 problems, the position of 
 an object is more conveni- 
 ently expressed by the two 
 measurements n t and d e, 
 
 Fig. 3. 
 
 than by the single oblique measurement e t. 
 
 I shall call d t the " direct distance " of the object 
 at E, and d e its '' lateral distance." It is rather a 
 license to call d t its " direct " distance, for e t is 
 the more direct of the two ; but there is no other 
 term which would not cause confusion. 
 
 Lastly, in order to complete our knowledge of the 
 position of an object, the vertical height of some 
 
INTEODUCTION. 15 
 
 point in it, above or below the eye, must be given ; 
 that is to say, either d p or d q in Fig. 2. * : this I 
 shall call the '^ vertical dista>nce " of the point given. 
 In all perspective problems these three distances, and 
 the dimensions of the object, must be stated, other- 
 wise the problem is imperfectly given. It ought not 
 to be required of us merely to draw a room or a 
 church in perspective ; but to draw this room from 
 ^Jiis corner, and that church on that spot, in per- 
 spective. For want of knowing how to base their 
 drawings on the measurement and place of the object, 
 I have known practised students represent a parish 
 church, certainly in true perspective, but with a nave 
 about two miles and a half long. 
 
 It is true that in drawing landscapes from natiure 
 the sizes and distances of the objects cannot be accu- 
 rately known. When, however, we know how to 
 draw them rightly, if their size were given, we have 
 only to assume a rational approximation to their 
 
 * p and Q being points indicative of the place of the tower's base 
 and top. In this figure both are above the sight-line ; if the tower 
 were below the spectator both would be below it, and therefore 
 measured below d. 
 
16 THE ELEMENTS OF PERSPECTIVE. 
 
 size, and the resulting drawing will be true enough 
 for all intents and purposes. It does not in the least 
 matter that we represent a distant cottage as eighteen 
 feet long, when it is in reality only seventeen ; but 
 it matters much that we do not represent it as 
 eighty feet long, as we easil}?^ might if we had not 
 been accustomed to draw from measurement. There- 
 fore, in all the following problems the measurement 
 of the object is given. 
 
 The student must observe, however, that in order 
 to bring the diagrams into convenient compass, the 
 measurements assumed are generally very different 
 from any likely to occur in practice. Thus, in 
 Fig. 3., the distance d s would be probably in practice 
 half a mile or a mile, and the distance T s, from the 
 eye of the observer to the paper, only two or three 
 feet. The mathematical law is however precisely the 
 same, whatever the proportions ; and I use such pro- 
 portions as are best calculated to make the diagi-am 
 clear. 
 
 Now, therefore, the conditions of a perspective 
 problem are the following : 
 
INTEODUCTION. 17 J 
 
 3 
 
 1 
 
 The Sight-line an given, Fig. 1. ; j 
 The Sight-point s given ; 
 
 The Station-point T given ; and ^ 
 The three distances of the ohject*, direct, late- 
 ral, and vertical, with its dimensions, given. ^ 
 
 The size of the picture, conjecturally limited by \ 
 
 the dotted circle, is to be determined afterwards at i 
 
 our pleasure. On these conditions I proceed at ; 
 
 once to construction. I 
 
 * More accurately, " the three distances of any point, either in the ' 
 
 object itself, or indicative of its distance." i 
 
18 
 
 THE ELEMENTS OF PERSPECTIVE. 
 
 PROBLEM L 
 
 TO FIX THE POSITION OF A GIYEN POINT. 
 
 Fig. 4. 
 
 Let p, Fig. 4., be the given point. 
 
 Let its direct distance be d T ; its lateral distance 
 to the left, D c ; and vertical distance beneath the 
 eye of the observer, c p. 
 
 * More accurately, " To fix on the plane of the picture the appa- 
 rent position of a point given in actual position." In the headings 
 of all the following problems the words " on the plane of the picture " 
 
TO FIX THE POSITION OF A GIVEN POINT. 19 
 
 [Let G H be the Sight-line, s the Sight-point, and 
 T the Station-point.] * 
 
 It is required to fix on the plane of the picture the 
 position of the point P. 
 
 Arrange the three distances of the object on 
 your paper, as in Fig. 4. f 
 Join c T, cutting g h in q. 
 From Q let fall the vertical line q p'. 
 Join p T, cutting Q p in p^ 
 p' is the point required. 
 
 are to be understood after the words " to draw." The plane of the 
 picture means a surface extended indefinitely in the direction of 
 the picture. 
 
 * The sentence within brackets will not be repeated in succeed- 
 ing statements of problems. It is always to be understood. 
 
 f In order to be able to do this, you must assume the distances 
 to be small ; as in the case of some object on the table : how 
 large distances are to be treated you will see presently ; the mathe- 
 matical principle, being the same for all, is best illustrated first on a 
 small scale. Suppose, for instance, p to be the corner of a book on 
 the table, seven inches below the eye, five inches to the left of it, and 
 a foot and a half in advance of it, and that you mean to hold your 
 finished drawing at six inches from the eye; then t s will be six 
 inches, t d a foot and a half, d c five inches, and c p seven. 
 c 2 
 
20 
 
 THE ELEMENTS OF PEKSPECTITE. 
 
 If the point p is above the eye of the observer 
 instead of below it, c P is to be measured upwards 
 
 Fig. 5. 
 
 from c, and Q p' drawn upwards from q. The con- 
 struction will be as in Fig. 5. 
 
 And if the point P is to the right instead of the 
 left of the observer, d c is to be measured to the right 
 instead of the left. 
 
TO FIX THE POSITION OF A GIVEN POINT. 21 
 
 The figures 4. and 5., looked at in a mirror, will 
 show the construction of each, on that supposition. 
 
 Now read very carefully the examples and notes 
 to this problem in Appendix I. (page 97.). I have 
 put them in the Appendix in order to keep the 
 sequence of following problems more clearly trace- 
 able here in the text; but you must read the fii'st 
 Appendix before going on. 
 
 C 3 
 
22 
 
 THE ELEMENTS OF EEESPECTIVE. 
 
 PROBLEM II. 
 
 TO DRAW A EIGHT LINE BETWEEN TWO GIYEN POINTS. 
 
 a D' 
 
 Fig. 6. 
 
 Let a b, Fig. 6., be the given right line, joining the 
 given points A and b. 
 
 Let the direct, lateral, and vertical distances of 
 the point A be t d, d c, and c A. 
 
RIGHT LINE BETWEEN GIVEN POINTS. 23 
 
 Let the direct, lateral, and vertical distances of 
 the point b be t d', d c', and c' b. 
 
 Then, by Problem I., the position of the point A 
 on the plane of the picture is a. 
 
 And similarly, the position of the point B on the 
 plane of the picture is 6. 
 
 Join a h. 
 
 Then a 6 is the line required. 
 
 COROLLARY L 
 
 If the line a b is in a plane parallel to that of the 
 picture, one end of the line A b must be at the same 
 direct distance from the eye of the observer as the 
 other. 
 
 Therefore, in that case, d t is equal to d' t. 
 
 Then the construction will be as in Fig. 7.; and 
 the student will find experimentally that a 6 is 
 now parallel to A B.* 
 
 * For by the construction at : ailiBT: 6t; and there- 
 fore the two triangles ab t, abi, (having a common angle ai^b,) 
 are similar. 
 
 C 4 
 
24 THE ELEMENTS OF PERSPECTIVE. 
 
 And that a 6 is to A b as t s is to t d. 
 C C D 
 
 Fig. 7. 
 
 Therefore, to draw any line in a plane parallel to 
 that of the picture, we have only to fix the position 
 of one of its extremities, a or 6, and then to draw from 
 a or 6 a line parallel to the given line, bearing the 
 proportion to it that T s bears to t d. 
 
RIGHT LINE IN HORIZONTAL PLANE. 
 
 25 
 
 COEOLLARY II. 
 
 If the line A b is in a horizontal plane, the vertical 
 distance of one of its extremities must be the same 
 as that of the other. 
 
 Therefore, in that case, A c equals B c' (Fig. 6.). 
 
 And the construction is as in Fig. 8. 
 
 Eig. 8. 
 
 In Fig. 8. produce a 6 to the sight-line, cutting the 
 sight-line in v; the point v, thus determined, is 
 called the Vanishing-Point of the line A b. 
 
26 THE ELEMENTS OF PERSPECTIVE. 
 
 Join T V. Then the student will find experi- 
 mentally that T V is parallel to A b.* 
 
 COEOLLAEY III. 
 
 If the line a b produced would pass through some 
 point beneath or above the station-point, c d is to 
 D T as c' d' is to d' T ; in which case the point c 
 coincides with the point cf, and the line a 6 is 
 vertical. 
 
 Therefore every vertical line in a picture is, or 
 may be, the perspective representation of a ho- 
 rizontal one which, produced, would pass beneath 
 the feet or above the head of the spectator.! 
 
 * The demonstration is in Appendix 11. Article I. 
 
 t The reflection in water of any luminous point or isolated object 
 (such as the sun or moon) is therefore, in perspective, a vertical line; 
 since such reflection, if produced, would pass under the feet of the 
 spectator. Many artists (Claude among the rest) knowing some- 
 thing of optics, but nothing of perspective, have been led occasion- 
 ally to draw such reflections towards a point at the centre of the 
 base of the picture. 
 
VANISHING-POINT OF HORIZONTAL LINES. 
 
 27 
 
 PROBLEM III. 
 
 TO FIND THE YANISHING-POINT OF A GIYEN HORIZONTAL 
 LINE. 
 
 Fig. 9. 
 
 Let a b. Fig. 9., be the given line. 
 
 From T, the station-point, draw TV parallel to 
 A B, cutting the sight-line in v. 
 
 V is the Vanishing-point required.* 
 
 * The student will observe, in practice, that, his paper lying flat 
 on the table, he has only to draw the line t v on its horizontal sur- 
 face, parallel to the given horizontal line A b. In theory, the paper 
 
28 THE ELEMENTS OF PERSPECTIVE. 
 
 COEOLLAKY I. 
 
 As, if the point b is first found, v may be deter- 
 mined by it, so, if the point v is first found, b may 
 
 should be vertical, but the station-line s t horizontal (see its defini- 
 tion above, page 10.) ; in which case t v, being drawn parallel to 
 A B, will be horizontal also, and still cut the sight-line in v. 
 
 The construction will be seen to be founded on the second Corol- 
 lary of the preceding problem. 
 
 It is evident that if any other line, as m n in Fig. 9., parallel 
 to A B, occurs in the picture, the line t v, drawn from t, parallel to 
 M N, to find the vanishing-point of m n, will coincide with the line 
 drawn from t, parallel to a b, to find the vanishing-point of ab. 
 
 Therefore a b and m n will have the same vanishing-point. 
 
 Therefore all parallel horizontal lines have the same vanishing- 
 point. 
 
 It will be shown hereafter that all parallel inclined lines also 
 have the same vanishing-point; the student may here accept the 
 general conclusion — ^^ All parallel lines have the same vanishing- 
 point.'' 
 
 It is also evident that if a b is parallel to the plane of the picture , 
 TV must be drawn parallel to gh, and will therefore never cut gh. 
 The line a b has in that case no vanishing-point : it is to be drawn 
 by the construction given in Fig. 7. 
 
 It is also evident that if a b is at right angles with the plane of 
 the picture, t v will coincide with t s, and the vanishing-point of 
 A B will be the sight-point. 
 
RIGHT LINE IN HORIZONTAL PLANE. 
 
 29 
 
 be determined by it. For let ab. Fig. 10., be the 
 given line, constructed upon the paper as in Fig. 8. ; 
 
 and let it be required to draw the line a h without 
 using the point c'. 
 
 Find the position of the point A in a. (Pro- 
 blem I.) 
 
 Find the vanishing-point of A b in v. (Problem 
 
 III.) 
 
30 THE ELEMENTS OF PERSPECTIVE. 
 
 Join a V. 
 
 Join B T, cutting a y in b. 
 
 Then a 6 is the line required.* 
 
 COKOLLAEY II. 
 
 We have hitherto proceeded on the supposition that 
 the given line was small enough, and near enough, to 
 be actually drawn on our paper of its real size ; as in 
 the example given in Appendix I. We may, however, 
 now deduce a construction available under all cir- 
 cumstances, whatever may be the distance and length 
 of the line given. 
 
 From Fig. 8. remove, for the sake of clearness, the 
 lines c' d', b v, and t v ; and, taking the figure as 
 here in Fig. 11., draw from a, the line a R parallel to 
 A B, cutting B T in R. 
 
 * I spare the student the formality of the reductio ad absurdum, 
 which would be necessary to prove this. 
 
RIGHT LINE IN HOEIZONTAL PLANE. 
 
 31 
 
 Fig. 11. 
 
 Then a r is to A b as a t is to a t. 
 
 — — as c T is to c T. 
 
 — — as T s is to T D. 
 
 That is to say, a R is the sight-magnitude of 
 
 AB. 
 
 * For definition of Sight- Magnitude, see Appendix I. It ought 
 to have been read before the student comes to this problem ; but I 
 refer to it in case it has not. 
 
32 THE ELEMENTS OF PERSPECTIVE. 
 
 Therefore, when the position of the point A is 
 fixed in a, as in Fig. 12., and a v is drawn to the 
 
 B 
 
 vanishing-point; if we draw a line a r from a, 
 parallel to A b, and make a R equal to the sight- 
 magnitude of A B, and then join r t, the line R T 
 will cut a V in 6. 
 
 So that, in order to determine the length of a b, 
 we need not draw the long and distant line A b, but 
 
RIGHT LINE IN HORIZONTAL PLANE. 
 
 33 
 
 only a E parallel to it, and of its sight-magnitude ; 
 which is a great gain, for the line A b may be two 
 miles long, and the line a r perhaps only two 
 inches. 
 
 COKOLLARY III. 
 
 In Fig. 12., altering its proportions a little for the 
 sake of clearness, and putting it as here in Fig. 13., 
 
 Fig. 13. 
 draw a horizontal line a r' and make a r' equal to 
 
 OtR. 
 
 Through the points r and h draw r' m, cutting the 
 
34 THE ELEMENTS OF PERSPECTIVE. 
 
 sight-line in M. Join t v. Now the reader will 
 find experimentally that v m is equal to v T.* 
 
 Hence it follows that, if from the vanishing-point 
 V we lay off on the sight-line a distance, v M, 
 equal to v t; then draw through a a horizontal 
 line a R^ make a r' equal to the sight-magnitude of 
 A B, and join r' m; the line r' m will cut a v in 6. 
 And this is in practice generally the most convenient 
 way of obtaining the length of a 6. 
 
 COEOLLAEY IV. 
 
 Eemoving from the preceding figure the unneces- 
 sary lines, and retaining only r'm and a\, as in 
 Fig. 14., produce the line a r' to the other side of 
 a, and make a x equal to a r'. 
 
 Join X h, and produce x 6 to cut the line of sight 
 in N. 
 
 Then as x r' is parallel to M N, and a r' is equal 
 to a X, V N must, by similar triangles, be equal to v M 
 (equal to v T in Fig. 13.). 
 
 * The demonstration is in Appendix II. Article II. p. 137. 
 
DIVIDING POINTS. 
 
 35 
 
 Therefore, on whichever side of v we measure the 
 distance v T, so as to obtain either the point M, or the 
 point N, if we measure the sight-magnitude a r' or 
 a X on the opposite side of the line a v, the line 
 joining r' m or x n will equally cut a v in 6. 
 
 Fig. 14. 
 
 The points m and n are called the "Dividing- 
 Points" of the original line AB (Fig. 12.), and we 
 resume the results of these corollaries in the fol- 
 lowing three problems. 
 
 i> 2 
 
36 
 
 THE ELEMENTS OF PEESPECTIVE. 
 
 PROBLEM IV. 
 
 TO FIND THE DIVIDING-POINTS OF A GIVEN HORIZONTAL 
 LINE. 
 
 Fi^. 15. 
 
 Let the horizontal line ab (Fig. 15.) be given in 
 position and magnitude. It is required to find its 
 dividing-points. 
 
 Find the vanishing-point v of the line A b. 
 
 With centre v and distance v t, describe circle 
 cutting the sight-line in m and N. 
 
 Then M and N are the dividing-points required. 
 
 In general, only one dividing-point is needed for 
 use with any vanishing-point, namely, the one 
 nearest s (in this case the point m). But its op- 
 posite N, or both, may be needed under certain cir- 
 cumstances. 
 
GITBN LINE IN HOKIZONTAL PLANE. 
 
 37 
 
 PROBLEM y. 
 
 TO DRAW A HORIZONTAL LINE, GIVEN IN POSITION AND 
 MAGNITUDE, BY MEANS OF ITS SIGHT-MAGNITUDE 
 AND DIVIDING- POINTS. 
 
 B 
 
 iig. 16. 
 
 Let ab (Fig. 16.) be the given line. 
 
 Find the position of the point A in a. 
 
 Find the vanishing-point v, and most convenient 
 dividing-point M, of the line A b. 
 
 Join a V. 
 
 Through a draw a horizontal line a h' and make 
 a h' equal to the sight-magnitude of A ^. Join h' M, 
 cutting a V in 6. 
 
 Then a 6 is the line required. 
 
 » 3 
 
38 
 
 THE ELEMENTS OF PERSPECTIVE. 
 
 Fig V7 
 
 Supposing it were now required to draw a line A c 
 (Fig. 17.) twice as long as A b, it is evident that the 
 sight-magnitude a d must be twice as long as the 
 sight-magnitude a h' ; we have, therefore, merely to 
 continue the horizontal line a b% make h' d equal to 
 a h'y join c m', cutting a v in c, and a c will be the line 
 required. Similarly, if we have to draw a line A D, 
 
GIYEN LINE IN HORIZONTAL PLANE. 39 
 
 three times the length of ab, adf must be three 
 times the length of a h\ and, joining dfu, ad will be 
 the line required. 
 
 The student will observe that the neai'er the por- 
 tions cut off, he, cdf &c., approach the point v, the 
 smaller they become ; and, whatever lengths may be 
 added to the line A d, and successively cut off from 
 a V, the line a v will never be cut off entirely, but 
 the portions cut off will become infinitely small, and 
 apparently "vanish" as they approach the point v; 
 hence this point is called the " vanishing " point 
 
 COEOLLARY II. 
 
 It is evident that if the line A d had been given ori- 
 ginally, and we had been required to draw it, and 
 divide it into three equal parts, we should have had 
 only to divide its sight-magnitude, a d\ into the three 
 equal parts, a If, h' d, and c d', and then, drawing to m 
 from y and </, the line a d would have been divided 
 as required in h and c. And supposing the original 
 
 D 4 
 
40 THE ELEMENTS OF PERSPECTIVE. 
 
 line A D be divided irregularly into any number of 
 parts, if the line ad' he divided into a similar num- 
 ber in the same proportions (by the construction given 
 in Appendix I.), and, from these points of divi- 
 sion, lines are drawn to m, they will divide the line 
 ad in. true perspective into a similar number of pro- 
 portionate parts. 
 
 The horizontal line drawn through a, on which the 
 sight-magnitudes are measured, is called the " Mea- 
 
 SURING-LINB." 
 
 And the line a d, when properly divided in b aAd 
 e, or any other required points, is said to be divided 
 "IN PERSPECTIVE RATIO " to the divisious of the 
 original line A d. 
 
 If the line a v is above the sight-line instead of 
 beneath it, the measuring-Hne is to be drawn above 
 also : and the lines 6' M, cf M, &c., drawn down to the 
 dividing-point Turn Fig. 17. upside down, and it 
 will show the construction. 
 
TRIANGLE IN HORIZONTAL PLANE. 
 
 41 
 
 PROBLEM VI. 
 
 TO DRAW ANT TRIANGLE, GIVEN IN POSITION AND 
 MAGNITUDE, IN A HORIZONTAL PLANE. 
 
 Fig. 18. 
 
 Let ABC (Fig. 18.) be the triangle. 
 
 As it is given in position and magnitude, one of its 
 sides, at least, must be given in position and magni- 
 tude, and the directions of the two other sides. 
 
42 THE ELEMENTS OF PERSPECTIVE. 
 
 Let A B be the side given in position and mag- 
 nitude. 
 
 Then A B is a horizontal line, in a given position, 
 and of a given length. 
 
 Draw the line A b. (Problem V.) 
 
 Let a 6 be the line so drawn. 
 
 Find V and V, the vanishing-points respectively 
 of the lines A c and b c. (Problem III.) 
 
 From a draw a v, and from h, draw h v', cutting 
 each other in c. 
 
 Then a 6 c is the triangle required. 
 
 If A c is the line originally given, a c is the line 
 which must be first drawn, and the line V h must be 
 drawn from V' to c and produced to cut a 6 in h, 
 Similai-ly, if b c is given, v c must be drawn to c and 
 produced, and a h from its vanishing-point to 6, and 
 produced to cut acm a. 
 
RECTILINEAR QUADRILATERAL FIGURE. 43 
 
 PROBLEM VII. 
 
 TO DRAW ANT RECTILINEAR QUADRILATERAL FIGURE, 
 GIVEN IN POSITION AND MAGNITUDE, IN A HORIZONTAL 
 PLANE. 
 
 Let ab c d (Fig. 19.) be the given figure. 
 
 Join any two of its opposite angles by the line B c. 
 
 Draw first the triangle ABC. (Problem VI.) 
 
 And then, from the base bc, the two lines bd, 
 
 c D, to their vanishing-points, which will complete the 
 
 figure. It is unnecessary to give a diagram of the 
 
 construction, which is merely that of Fig. 18. dupli- 
 
44 THE ELEMENTS OF PERSPECTIVE. 
 
 cated ; another triangle being drawn on the line A c 
 or BC. 
 
 COKOLLAKY. 
 
 It is evident that by this application of Problem 
 VI. any given rectilinear figure whatever in a 
 horizontal plane may be drawn, since any such figure 
 may be divided into a number of triangles, and the 
 triangles then drawn in succession. 
 
 More convenient methods may, however, be gene- 
 rally found, according to the form of the figure 
 required, by the use of succeeding problems ; and for 
 the quadrilateral figure which occurs most frequently 
 in practice, namely, the square, the following con- 
 struction is more convenient than that used in the 
 present problem. 
 
SQUARE IN HORIZONTAL PLANE. 
 
 45 
 
 PROBLEM YIIL 
 
 TO DRAW A SQUARE, GIVEN IN POSITION AND MAGNITUDE, 
 IN A HORIZONTAL PLANE. 
 
 Fig. 20. 
 
 Let a b c d. Fig. 20., be the square. 
 
 As it is given in position and magnitude, the 
 position and magnitude of all its sides are given* 
 
46 
 
 THE ELEMENTS OF PERSPECTIVE. 
 
 Fix the position of the point A in a. 
 
 Find V, the vanishing-point of ab; and M, the 
 dividing-point of A B, nearest s. 
 
 Find V , the vanishing-point of A c; and N, the 
 dividing-point of A c, nearest s. 
 
 Draw the measuring-line through a, and make 
 a U, a c', each equal to the sight-magnitude of A b. 
 
 (For since A b c d is a square, A c is equal to A b.) 
 
 Draw a Y and c' n, cutting each other in c. 
 
 Draw a v, and 6' m, cutting each other in h. 
 
 Then ac,ab, are the two nearest sides of the square. 
 
 Now, clearing the figure of superfluous lines, we 
 have ab, ac, drawn in position, as in Fig. 21. 
 
 V 
 
 - T 
 
 Fi^. 21. 
 
RECTANGLE IN HORIZONTAL PLANE. 47 
 
 And because A b c D is a square, c d (Fig. 20.) is 
 parallel to A b. 
 
 And all parallel lines have the same vanishing- 
 point. (Note to Problem III.) 
 
 Therefore, v is the vanishing-point of c d. 
 
 Similarly, V is the vanishing-point of B D. 
 
 Therefore, from b and c (Fig 22.) draw b v', c v, 
 cutting each other in d. 
 
 Then abed is the square required. 
 
 COEOLLAKY I. 
 
 It is obvious that any rectangle in a horizontal 
 plane may be drawn by this problem, merely making 
 a b', on the measuring-line. Fig. 20., equal to the 
 sight-magnitude of one of its sides, and a d the 
 sight-magnitude of the other. 
 
48 THE ELEMENTS OF PERSPECTIVE. 
 
 COEOLLARY 11. 
 
 Let abed, Fig. 22., be any square drawn in 
 perspective.* Draw the diagonals a d and b c, cut- 
 ting each other in c. Then c is the centre of the 
 square. Through c, draw e / to the vanishing- 
 
 point of a b, and g hto the vanishing-point of a c, 
 and these lines will bisect the sides of the square, so 
 that a ^ is the perspective representation of half the 
 side ab; a e is half a c ; c ^ is half c d ; and 6 / is 
 half b d. 
 
RECTILINEAR QUADRILATERAL FIGURE. 49 
 
 COKOLLAEY III. 
 
 Since A b c d, Fig. 20., is a square, b A c is a right 
 angle ; and as t y is parallel to A b, and t y' to a c, 
 y' T Y must be a right angle also. 
 
 As the ground plan of most buildings is rectan- 
 gular, it constantly happens in practice that their 
 angles (as the corners of ordinary houses) throw the 
 lines to the vanishing-points thus at right angles ; 
 and so that this law is observed, and v T v' is kept a 
 right angle, it does not matter in general practice 
 whether the vanishing-points are thrown a little 
 more or a little less to the right or left of s : but it 
 matters much that the relation of the vanishing- 
 points should be accurate. Their position with re- 
 spect to s merely causes the spectator to see a little 
 more or less on one side or other of the house, 
 which may be a matter of chance or choice ; but their 
 rectangular relation determines the rectangular shape 
 of the building, which is an essential point. 
 
50 
 
 THE ELEMENTS OF PERSPECTIVE. 
 
 PROBLEM IX. 
 
 TO BRAW A SQUARE PILLAR, GIVEN IN POSITION AND 
 MAGNITUDE, ITS BASE AND TOP BEING IN HORIZONTAL 
 PLANES. 
 
 h 
 
 Let a h, Fig. 23., be tlie square pillar. 
 
 Then, as it is given in position and magnitude, the 
 position and magnitude of the square it stands upon 
 
 must be given (that 
 is, the line A b or A c 
 in position), and the 
 height of its side A e. 
 Find the sight- 
 magnitudes of A B 
 Fig. 24. and A e. Draw the 
 two sides ah, a c, of the square of the base, by 
 Problem VIII., as in Fig. 24. From the points 
 a, h, and c, raise vertical lines ae, cf, b g. 
 Make a e equal to the sight-magnitude of A e. 
 
SQUARE PILLAR IN HORIZONTAL PLANE. 51 
 
 Now because the top and base of the pillar are in 
 horizontal planes, the square of its top, F G, is parallel 
 to the square of its base, b c. 
 
 Therefore the line E P is parallel to A c, and E G to 
 
 AB. 
 
 Therefore e p has the same vanishing-point as A c, 
 and E G the same vanishing-point as a b. 
 
 From e di'aw ef to the vanishing-point of a c, 
 cutting cf in /. 
 
 Similarly draw e ^ to the vanishing-point of a 6, 
 cutting b g in g. 
 
 Complete the square gf in h, by drawing gh to 
 the vanishing-point of ef, and fh to the vanishing- 
 point of e g, cutting each other in h. Then a ghf is 
 the square pillar required. 
 
 COKOLLAEY. 
 
 It is obvious that if a E is equal to A c, the whole 
 figure will be a cube, and each side, aefc and aegb, 
 will be a square in a given vertical plane. And by 
 
 B 2 
 
52 THE ELEMENTS OF PERSPECTIYE. 
 
 making A b or a c longer or shorter in any given pro- 
 portion, any form of rectangle may be given to either 
 of the sides of the pillar. No other rule is therefore 
 needed for drawing squares or rectangles in vertical 
 planes. 
 
 Also any triangle may be thus drawn in a vertical 
 plane, by enclosing it in a rectangle and determining, 
 in perspective ratio, on the sides of the rectangle, 
 the points of their contact with the angles of the 
 triangle. 
 
 And if any triangle, then any polygon. 
 
 A less complicated construction will, however, be 
 given hereafter.* 
 
 * See page 131. (note), after you have read Problem XVI.. 
 
PYEAMID ON SQUARE BASE. 
 
 53 
 
 PROBLEM X. 
 
 TO DRAW A PYRAMID, GIYEN IN POSITION AND MAG- 
 NITUDE, ON A SQUARE BASE IN A HORIZONTAL PLANE. 
 
 Let a b, Fig. 25., be the four-sided pyramid. As it is 
 given in position and magnitude, the square base 
 on which it stands must be given in position and 
 magnitude, and its vertical height, c d.* 
 
 Draw a square pillar, a b G e. Fig. 26, on the 
 square base of the pyramid, and make the height 
 of the pillar af equal to the vertical height of 
 the pyramid c d (Problem IX.). Draw the diagonals 
 
 * If, instead of the vertical height, the length of a d is given, the 
 vertical must be deduced from it. See the Exercises on this Problem 
 in the Appendix, p. 106. 
 
 E 3 
 
54 
 
 THE ELEMENTS OF PEKSPECTIVE. 
 
 Fiff. 26. 
 
 G F, H I, on the top of the square pillar, cutting each 
 other in c. Therefore c is the centre of the square 
 F G H I. (Prob. VIII. CoR 11.) 
 • Join C E, C A, C B. 
 
 Then A b c E is the pyramid required. If the base 
 of the pyramid is above the eye, as when a square 
 spire is seen on the top of a church-tower, the con- 
 struction will be as in Fig. 27. 
 
 Fig. 27. 
 
CURVE IN HORIZONTAL OR VERTICAL PLANE. 55 
 
 PROBLEM XL 
 
 TO DRAW ANY CURVE IN A HORIZONTAL OR VERTICAL 
 PLANE. 
 
 r e a 
 
 Fig. 28. 
 
 Let a b, Fig. 28., be the curve. 
 
 Enclose it in a rectangle, c d e F. 
 
 Fix the position of the point c or d, and draw the 
 rectangle. (Problem VIII. Coroll. I.)* 
 
 * Or if the curve is in a vertical plane, Coroll. to Problem IX. 
 As a rectangle may be drawn in any position round any given curve, 
 its position with respect to the curve will in either case be regulated 
 by convenience. See the Exercises on this Problem, in the Appendix, 
 p. 115. 
 
 B 4 
 
56 THE ELEMENTS OF PERSPECTIVE, 
 
 Let c D E F, Fig. 29., be the rectangle so drawn. 
 If an extremity of the curve, as A, is in a side 
 E p. p of the rectangle, divide the 
 side c E, Fig. 29., so that A c 
 shall be (in perspective ratio) 
 to A E as A c is to A E in Fig. 
 28. (Prob. V. Cor. II.) 
 Similarly determine the 
 points of contact of the curve and rectangle e, /, g. 
 
 If an extremity of the curve, as b, is not in a 
 side of the rectangle, let fall the perpendiculars B a, 
 B 6 on the rectangle sides. Determine the corre- 
 spondent points a and b in Fig 29., as you have 
 already determined A, B, e, and /. 
 
 From 6, Fig. 29., draw b b parallel to CD*, and 
 from a draw a B to the vanishing-point of D F, 
 cutting each other in b. Then b is the extremity of 
 the curve. 
 
 Determine any other important point in the curve, 
 as p, in the same way, by letting fall p q and p r on 
 the rectangle's sides. 
 
 * Or to its vanishing-point, if c d has one. 
 
CIRCLE IN HOEIZONTAL PLANE. 
 
 57 
 
 Any number of points in the curve may be thus 
 determined, and the curve drawn through the series ; 
 in most cases, three or four will be enough. Prac- 
 tically, complicated curves may be better di*awn in 
 perspective by an experienced eye than by rule, as 
 the j&xing of the various points in haste involves 
 too many chances of error ; but it is well to draw a 
 good many by rule first, in order to give the eye its 
 experience.* 
 
 COKOLLAKY. 
 
 If the curve required be a circle. Fig. 30., the rect- 
 angle which encloses it will 
 become a square, and the 
 curve will have four points 
 of contact, A b c d, in the 
 middle of the sides of the 
 square. 
 
 Draw the square, and as a 
 square may be drawn about 
 
 * Of course, by dividing the original rectangle into any number 
 of equal rectangles, and dividing the perspective rectangle similarly, 
 
58 
 
 THE ELEMENTS OF PEESPECTIYE. 
 
 C 
 
 HP' B 
 
 Q' F 
 
 D 
 
 K 
 
 ^ 
 
 /-^A^^-^ 9/ 
 
 Fig. 31. 
 
 a circle in any position, draw it with its nearest side, 
 E G, parallel to the sight-line. 
 
 Let E F, Fig. 31., be the square so drawn. 
 
 Draw its diagonals E F, G h; and through the 
 centre of the square (determined by their intersec- 
 tion) draw A B to the vanishing-point of G F, and 
 c D parallel to E G. Then the points A B c D are 
 the four points of the circle's contact. 
 
 On E G describe a half square, e l ; draw the semi- 
 
 the curve may be approximately drawn without any trouble ; but, 
 when accuracy is required, the points should be fixed, as in the 
 problem. 
 
CIBCLE IN VERTICAL PLANE. 5& 
 
 circle K A L ; and from its centre, R, the diagonals r e, 
 R Q, cutting the circle in x, y. 
 
 From the points x y, where the circle cuts the 
 diagonals, raise perpendiculars, p a?, Q 2/5 to E G. 
 
 From p and Q draw P p', Q q', to the vanishing-point 
 of G F, cutting the diagonals in m, n, and 0, jp. 
 
 Then m, n^ 0, p are four other points in the 
 circle. 
 
 Through these eight points the circle may be 
 drawn by the hand accurately enough for general 
 purposes ; but any number of points required may, 
 of course, be determined, as in Problem XI. 
 
 The distance e P is approximately one seventh of 
 E G, and may be assumed to be so in quick practice, 
 as the error involved is not greater than would 
 be incurred in the hasty operation of drawing the 
 circle and diagonals. 
 
 It may frequently happen that, in consequence of 
 associated constructions, it may be inconvenient to 
 draw E G parallel to the sight-line, the square being 
 perhaps first constructed in some oblique direction. 
 In such cases, q g and e p must be determined in 
 
60 THE ELEMENTS OP PEKSPECTIYE. 
 
 perspective ratio by the dividing-point, the line E G 
 being used as a measuring-line. 
 
 \_Obs. In drawing Eig. 31. the station-point has been taken much 
 nearer the paper than is usually advisable, in order to show the 
 character of the curve in a very distinct form. 
 
 If the student turns the book so that e g may be vertical, Fig. 
 31. will represent the construction for drawing a circle in a vertical 
 plane, the sight-line being then of course parallel to g l ; and the 
 semicircles a d b, a c b, on each side of the diameter a b, will 
 represent ordinary semicircular arches seen in perspective. In that 
 case, if the book be held so that the line e h is the top of the square, 
 the upper semicircle will represent a semicircular arch, above the 
 eye, drawn in perspective. But if the book be held so that the line 
 G F is the top of the square, the upper semicircle will represent a 
 semicircular arch, below the eye, drawn in perspective. 
 
 If the book be turned upside down, the figure will represent a 
 circle drawn on the ceiling, or any other horizontal plane above the 
 eye; and the construction is, of course, accurate in every case.] 
 
DIYISION OF CIRCLE INTO EQUAL PARTS. 61 
 
 PROBLEM XII. 
 
 TO DIVIDE A CIRCLE DRAWN IN PERSPECTIYE INTO ANY 
 GIVEN NUMBER OF EQUAL PARTS. 
 
 Let a b. Fig. 32., be the circle drawn in perspective. 
 It is required to divide it into a given number of 
 equal parts ; in this case, 20. 
 
 Let K A L be the semicircle used in the construc- 
 tion. Divide the semicircle K A L into half the 
 number of parts required ; in this case, 10. 
 
 Produce the line e g laterally, as far as may be 
 necessary. 
 
 From 0, the centre of the semicircle K A L, draw 
 radii through the points of division of the semicircle, 
 p, q, r, &c., and produce them to cut the line e G in 
 p, Q, R, &c. 
 
 From the points p Q R draw the lines p p', Q q', r r', 
 &c., through the centre of the circle A B, each 
 cutting the circle in two points of its circumference. 
 
 Then these points divide the perspective circle as 
 required. 
 
DIVISION OF CIRCLE INTO UNEQUAL PARTS. 
 
 63 
 
 If from each of the points p, q, r, a vertical were 
 raised to the line e G, as in Fig. 31., and from the 
 point where it cut e G a line were drawn to the 
 vanishing-point, as Q q' in Fig. 31., this line would 
 also determine two of the points of division. 
 
 If it is required to divide a circle into any number 
 of given unequal parts (as in the points A, b, and c. 
 Fig. 33.), the shortest way is thus to raise vertical 
 
 Fig. 33. 
 
 lines from A and b to the side of the perspective 
 square x Y, and then draw to the vanishing-point, 
 cutting the perspective circle in a and b, the points 
 required. Only notice that if any point, as a, is on 
 the nearer side of the circle A b c, its representative 
 
64 THE ELEMENTS OF PERSPECTIVE. 
 
 point, a, must be on the nearer side of tlie circle ahc\ 
 and if the point b is on the farther side of the circle 
 A B c, 6 must be on the farther side oi ah c. If any, 
 point, as c, is so much in the lateral arc of the 
 circle as not to be easily determinable by the vertical 
 line, draw the horizontal c P, find the correspondent 
 p in the side of the perspective square, and draw jp c 
 parallel to x y, cutting the perspective circle in c. 
 
 COEOLLARY. 
 
 It is obvious that if the points p', q', k, &c., by which 
 the circle is divided in Fig. 32., be joined by right 
 lines, the resulting figure will be a regular equilateral 
 figure of twenty sides inscribed in the circle. And if 
 the circle be divided into given unequal parts, and 
 the points of division joined by right lines, the re- 
 sulting figure will be an irregular polygon inscribed 
 in the circle with sides of given length. 
 
 Thus any polygon, regular or irregular, inscribed 
 in a circle, may be inscribed in position in a per- 
 spective circle. 
 
SQUARE WITHIN SQUARE. 
 
 65 
 
 PROBLEM XIIT. 
 
 TO DRAW A SQUARE, GIVEN IN MAGNITUDE, WITHIN A 
 LARGER SQUARE GIVEN IN POSITION AND MAGNITUDE ; 
 THE SIDES OF THE TWO SQUARES BEING PARALLEL. 
 
 Let a b, Fig. 34., be the sight-magnitude of the 
 side of the smaller square, and a c that of the side 
 of the larger square. 
 
 Draw the larger square. Let d e F G be the square 
 so drawn. 
 
 Join E G and d f. 
 
 On either d e or d G set off, in perspective ratio, 
 D H equal to one half of b c. Through h draw h k to 
 
66 
 
 THE ELEMElNTS OP PEESPECTIVE. 
 
 the vanishing-point of d e, cutting d f in i and e G in 
 K. Through i and K draw i M, K L, to vanishing-point 
 of D G, cutting D p in L and e G in M. Join l m. 
 
 Then i K L M is the smaller square, inscribed as 
 :required.* 
 
 COEOLLAEY. 
 
 If, instead of one square within another, it be re- 
 quired to draw one circle within another, the dimen- 
 sions of both being 
 given, enclose each 
 circle in a square. 
 ■^'S- '^^' Draw the squares 
 
 first, and then the circles within, as in Fig. 36. 
 
 * If either of the sides of the greater square is parallel to the 
 
 plane of the picture, as d g 
 xL r 
 
 in Fig .35., d G of course 
 
 must be equal to a c, and 
 D H equal to — , and the 
 construction is as in Fig. 
 Fig. 35. 35. 
 
 D H 
 
TRUNCATED CONE. 
 
 67 
 
 PROBLEM XIV. 
 
 TO DEAW A TRUNCATED CIRCULAR CONE, GIVEN IN 
 POSITION AND MAGNITUDE, THE TRUNCATIONS BEING 
 IN HORIZONTAL PLANES, AND THE AXIS OF THE CONE 
 VERTICAL. 
 
 JiET A B c D, Fig. 37., be the portion of the cone 
 required. 
 
 Fig. 37. 
 
 As it is given in magnitude, its diameters must be 
 
 given at the base and summit, A b and c d ; and 
 
 its vertical height, CE.* 
 
 * Or if the length of its side, A c, is given instead, take a e, Fig. 
 37., equal to half the excess of a b over c d ; from the point e raise 
 the perpendicular ce. With centre a, and distance a c, describe a 
 circle cutting c e in c. Then c e is the vertical height of the portion 
 of cone required, or c e. 
 
 F2 
 
68 
 
 THE ELEMENTS OE PERSPECTIVE. 
 
 And as it is given in position, the centre of its 
 base must be given. 
 
 Draw in position, about this centre *, the square 
 
 pillar afd, Fig. 38., 
 making its height, h g, 
 equal to c E ; and its 
 side, a 6, equal to a b. 
 
 In the square of its 
 base, abed, inscribe a 
 circle, which therefore 
 is of the diameter of 
 the base of the cone, A b. 
 In the square of its top, efgh, inscribe concen- 
 trically a circle whose diameter shall equal c d. 
 (CoroU. Prob. XIII.) 
 
 Join the extremities of the circles by the right 
 lines kl,nm. 
 
 Then klnm is the portion of cone required. 
 
 * The direction of the side of the square will of course be regu- 
 lated by convenience. 
 
CYLINDERS AND POLYGONAL PILLARS. 69 
 
 COEOLLARY I. 
 
 If similar polygons be inscribed in similar posi- 
 tions in the circles hn and I'm (CoroU. Prob. XIL), 
 and the corresponding angles of the polygons joined 
 by right lines, the resulting figure will be a por- 
 tion of a polygonal pyramid. (The dotted lines in 
 Fig. 38., connecting the extremities of two diame- 
 ters and one diagonal in the respective circles, oc- 
 cupy the position of the three nearest angles of a 
 regular octagonal pyramid, having its angles set on 
 the diagonals and diameters of the square a d, 
 enclosing its base.) 
 
 If the cone or polygonal pyramid is not truncated, 
 its apex will be the centre of the upper square, as in 
 Fig. 26. 
 
 COROLLARY IL 
 
 If equal circles, or equal and similar polygons, be 
 inscribed in the upper and lower squares in Fig. 38., 
 the resulting figure will be a vertical cylinder, or a 
 
 F 3 
 
70 THE ELEMENTS OF PERSPECTIVE. 
 
 vertical polygonal pillar, of given height and dia- 
 meter, drawn in position. 
 
 COKOLLARY III. 
 
 If the circles in Fig. 38., instead of being inscribed 
 in the squares h c and fg, be inscribed in the sides of 
 the solid figure h e and df, those sides being made 
 square, and the line hd of any given length, the 
 resulting figure will be, according to the construc- 
 tions employed, a cone, polygonal pyramid, cylin- 
 der, or polygonal pillar, drawn in position about a 
 horizontal axis parallel to b d. 
 
 Similarly, if the circles are drawn in the sides g d 
 and e c, the resulting figures will be described about 
 a horizontal axis parallel to a 6. 
 
INCLINED LINES. 71 
 
 PROBLEM XV. 
 
 TO DEAW AN INCLINED LINE, GIYEN IN POSITION AND 
 MAGNITUDE. 
 
 We have hitherto been examining the conditions 
 of horizontal and vertical lines only, or of curves en- 
 closed in rectangles. 
 
 We must, in conclusion, investigate the perspective 
 of inclined lines, beginning with a single one given in 
 position. For the sake of completeness of system, I 
 give in Appendix II. Article III. the development 
 of this problem from the second. But, in practice, 
 the position of an inclined line may be most conve- 
 niently defined by considering it as the diagonal of 
 a rectangle, as A b in Fig. 39., and I shall there- 
 fore, though at some sacrifice of system, examine it 
 here under that condition. 
 
 If the sides of the rectangle A c and A d are given, 
 the slope of the line A b is determined ; and then its 
 position will depend on that of the rectangle. If, as 
 
 F 4 
 
72 
 
 THE ELEMENTS OF PERSPECTIVE. 
 
 in Fig. 39., the rectangle is parallel to the picture 
 plane, the line A b must be so also. If, as in Fig. 
 40., the rectangle is inclined to the picture plane, the 
 line A B will be so also. So that, to fix the position 
 of A B, the line A c must be given in position and 
 magnitude, and the height A d. 
 
 If these are given, and it is only required to draw 
 the single line A b in perspective, the construction is 
 entirely simple ; thus : — 
 
 Draw the line A c by Problem I. 
 
 Let A c. Fig. 41., be the line so drawn. 
 
 From a and c raise the vertical lines a d, 
 
 c h. Make a d equal to the sight-mag- 
 
 /7 nitude of A d. From d draw d b to the 
 
 'J?- 41. vanishing-point of a c, cutting b c inb. 
 
 Join a b. Then a b is the inclined line required. 
 
INCLINED LINES. 
 
 73 
 
 If the line is inclined in the op- D 
 posite direction, as d c in Fig. 42., we 
 have only to join d c instead of a 6 
 in Fig. 41., and d c will be the line 
 required. 
 
 I shall hereafter call the line A c, when used to 
 define the position of an inclined line A b (Fig. 40.), 
 the " relative horizontal " of the line A b. 
 
 Observation. 
 
 In general, inclined lines are most needed for gable 
 roofs, in which, when the conditions are properly 
 stated, the vertical height of the gable, x T, Fig. 43., 
 is given, and the base y/ b' Y 
 
 line, A c, in position. When 
 these are given, draw A c ; 
 raise vertical A d; make A' 
 A D equal to sight-magni- 
 tude of X T ; complete the 
 perspective-rectangle A d b c ; join A b and d c (as by 
 dotted lines in figure) ; and through the intersection 
 
 Fi"r. 43. 
 
74 
 
 THE ELEMENTS OF PERSPECTIVE. 
 
 of the dotted lines draw vertical x y, cutting d b in y. 
 Join AY, c Y ; and these lines are the sides of the 
 gable. If the length of the roof A a' is also given, draw 
 in perspective the complete parallelepiped a' d' b c, 
 and from y draw Y y' to the vanishing-point of A a', 
 cutting d' b' in Y^ Join a! y, and you have the slope 
 of the farther side of the roof. 
 
 The construction above the eye is as in Fig. 44. ; 
 the roof is reversed in direction merely to familiarise 
 the student with the different aspects of its lines. 
 
 Fig. 44. 
 
VANISHING-POINT OF INCLINED LINES. 75 
 
 PROBLEM XVI. 
 
 TO FIND THE VANISHING-POINT OF A GIVEN INCLINED 
 LINE. 
 
 If, in Fig. 43. or Fig. 44., the lines A T and a' y' 
 be produced, the student will find that they meet. 
 
 Let p. Fig. 45., be the point at which they meet. 
 
 From p let fall the vertical p v on the sight-line, 
 cutting the sight-line in v. 
 
 Then the student will find experimentally that v 
 is the vanishing-point of the line A c* 
 
 Complete the rectangle of the base A c', by draw- 
 ing a' c' to V, and c c' to the vanishing-point of A A'. 
 
 Join t' c'. 
 
 Now if Y c and y' c' be produced downwards, the 
 student will find that they meet. 
 
 Let them be produced, and meet in p'. 
 
 Produce p v, and it will be found to pass through 
 the point p'. 
 
 * The demonstration is in Appendix II. Article IIL 
 
76 
 
 THE ELEMENTS OF PEESPECTIVE. 
 
 Therefore if a y (or c t). Fig. 45., be any in- 
 clined line drawn in perspective by Problem XV. 
 
 Fie. 45 
 
VANISHING-POINT OF INCLINED LINES. 77 
 
 and A c the relative horizontal (a c in Figs. 39, 40.), 
 also drawn in perspective. 
 
 Through v, the vanishing-point of A c, draw the 
 vertical p p' upwards and downwards. 
 
 Produce A y (or c t), cutting p p' in p (or p'). 
 
 Then p is the vanishing-point of A Y (or p' of c y). 
 
 The student will observe that, in order to find the 
 point p by this method, it is necessary first to draw 
 a portion of the given inclined line by Problem XY. 
 Practically, it is always necessary to do so, and, 
 therefore, I give the problem in this form. 
 
 Theoretically, as will be shown in the analysis of 
 the problem, the point p should be found by draw- 
 ing a line from the station-point parallel to the given 
 inclined line : but there is no practical means of 
 drawing such a line ; so that in whatever terms the 
 problem may be given, a portion of the inclined line 
 (ay or c y) must always be drawn in perspective 
 before p can be found. 
 
78 
 
 THE ELEMENTS OF PEESPECTIYE. 
 
 PROBLEM XVII. 
 
 TO FIND THE DIVIDINa-POINTS OF A GIVEN INCLINED 
 LINE. 
 
 X 
 
 s 
 
 / 
 
 D 
 
 V B' 
 
 Fig. 46. 
 
 Let p, Fig. 46., be the vanishing-point of the in- 
 clined line, and v the vanishing-point of the rela- 
 tive horizontal. 
 
 Find the dividing-points of the relative horizontal, 
 D and d'. 
 
 Through p draw the horizontal line x T. 
 
 With centre p and distance D p describe the two 
 arcs D X and d' y, cutting the line x Y in x and Y. 
 
DIYIDING-POINTS OF INCLINED LINES. 
 
 79 
 
 Then x and t are the dividing-points of the in- 
 clined line.* 
 
 Obs. The dividing-points found by the above 
 rule, used with the ordinary measuring-line, will lay 
 off distances on the retiring inclined line, as the 
 ordinary dividing-points lay them off on the retiring 
 horizontal line. 
 
 Another dividing-point, peculiar in its application, 
 is sometimes useful, and is to be found as follows : — 
 
 Fig. 47. 
 
 Let A B, Fig. 47., be the given inclined line drawn 
 in perspective, and A c the relative horizontal. 
 * The demonstration is in Appendix II., p. 138. 
 
80 THE ELEMENTS OF PEKSPECTIVE. 
 
 Find the vanishing-points, v and e, of a c and 
 A B ; D, the dividing-point of Ac; and the sight- 
 magnitude of A c on the measuring-line, or a c. 
 
 From D erect the perpendicular d f. 
 
 Join c B, and produce it to cut d e in f. Join e f. 
 
 Then, by similar triangles, d f is equal to e v, and 
 E F is parallel to d v. 
 
 Hence it follows that if from d, the dividing-point 
 of A c, we raise a perpendicular and make d f equal 
 to E V, a line c f, drawn from any point c on the 
 measuring-line to F, will mark the distance ab on 
 the inclined line, A b being the portion of the given 
 inclined line which forms the diagonal of the vertical 
 rectanofle of which A c is the base. 
 
SIGHT-LINE OF INCLINED PLANES. 81 
 
 PROBLEM XVIIL 
 
 TO FIND THE SIGHT-LINE OF AN INCLINED PLANE IN 
 WHICH TWO LINES ARE GIVEN IN POSITION.* 
 
 As in order to fix the position of a line two points in 
 it must be given, so in order to fix the position of 
 a plane, two lines in it must be given. 
 
 Let the two lines be A b and c d, Fig. 48. 
 
 As they are given in position, the relative hori- 
 zontals A E and F must be given. 
 
 Then by Problem XVI. the vanishing-point of A b 
 is v, and of c d, v'. 
 
 Join V V and produce it to cut the sight-line 
 in X. 
 
 Then v x is the sight-line of the inclined plane. 
 
 Like the horizontal sight-line, it is of indefinite 
 length ; and may be produced in either direction as 
 
 * Read the Article on this problem in the Appendix, p. 132, 
 before investigating the problem itself. 
 
 a 
 
82 
 
 THE ELEMENTS OF PERSPECTIVE. 
 
 Fig. 48. 
 
 occasion requires, crossing the horizontal line of sight, 
 if the plane continues downward in that direction. 
 
 X is the vanishing-point of all horizontal lines in 
 the inclined plane. 
 
VANISHING-POINT OF STEEPEST LINES. 
 
 83 
 
 PROBLEM XIX. 
 
 TO FIND THE VANISHING-POINT OF STEEPEST LINES IN AN 
 INCLINED PLANE WHOSE SIGHT-LINE IS GIVEN. 
 
 Let V X, Fig. 49., be the given sight-line. 
 
 Produce it to cut the horizontal sight-line in x. 
 
 Therefore x is the vanishing point of horizontal 
 lines in the given inclined plane. (Problem XVIII.) 
 
 G 2 
 
84 THE ELEMENTS OF PERSPECTIYE. 
 
 Join T X, and draw T Y at right angles to t x. 
 
 Therefore T is the rectangular vanishing-point 
 corresponding to x.* 
 
 From T erect the vertical t p, cutting the sight- 
 line of the inclined plane in p. 
 
 Then p is the vanishing-point of steepest lines in 
 the plane. 
 
 All lines drawn to it, as Q p, k p, n p, &c.j are the 
 steepest possible in the plane ; and all lines drawn to 
 X, as Q X, X, &c., are horizontal, and at right angles 
 to the lines p Q, P k, &c. 
 
 * That is to say, the vanishing-point of horizontal lines drawn at 
 I'ight angles to the lines whose vanishing-point is x. 
 
VANISHING-POINT OF STEEPEST LINES. 85 
 
 PROBLEM XX. 
 
 TO FIND THE VANISHING-POINT OF LINES PERPENDICU- 
 LAR TO THE SURFACE OF A GIVEN INCLINED PLANE. 
 
 As the inclined plane is given, one of its steepest 
 lines must be given, or may be ascertained. 
 
 Let A B, Fig. 50., be a portion of a steepest line 
 in the given plane, and v the vanishing-point of its 
 relative horizontal. 
 
 Through v draw the vertical G F upwards and 
 downwards. 
 
 From A set off any portion of the relative hori- 
 zontal A c, and on A c describe a semicircle in a 
 vertical plane, A d c, cutting A b in e. 
 
 Join E c, and produce it to cut G F in F. 
 
 Then f is the vanishing-point required. 
 
 For, because a e c is an angle in a semicircle, it is 
 a right angle ; and therefore the line e f is at right 
 
 G 3 
 
86 
 
 THE ELEMENTS OF PEKSPECTIYE. 
 
 Fig. 50. 
 
 angles to the line A b ; and similarly all lines drawn 
 to F, and therefore parallel to E F, are at right angles 
 with any line which cuts them, drawn to the vanish- 
 ing-point of A B. 
 
VANISHING-POINT OF STEEPEST LINES. 87 
 
 And because the semicircle adc is in a vertical 
 plane, and its diameter A c is at right angles to the 
 horizontal lines traversing the surface of the inclined 
 plane, the line e c, being in this semicircle, is also 
 at right angles to such traversing lines. And there- 
 fore the line e c, being at right angles to the steepest 
 lines in the plane, and to the horizontal lines in it, 
 is perpendicular to its surface. 
 
 O 4 
 
88 THE ELEMENTS OF PERSPECTIVE. 
 
 The preceding series of constructions, with the 
 examples in the first Article of the Appendix, put it 
 in the power of the student to draw any form, how- 
 ever complicated *, which does not involve intersection 
 of curved surfaces. I shall not proceed to the ana- 
 lysis of any of these more complex problems, as they 
 are entirely useless in the ordinary practice of artists. 
 For a few words only I must ask the reader's further 
 patience, respecting the general placing and scale of 
 the picture. 
 
 As the horizontal sight-line is drawn through the 
 sight-point, and the sight-point is opposite the eye, 
 the sight-line is always on a level with the eye. 
 
 * Asin algebraic science, mucli depends, in complicated perspec- 
 tive, on the student's ready invention of expedients, and on his quick 
 sight of the shortest way in which the solution maybe accomplished, 
 when there are several ways. 
 
PLACING AND SCALE OF PICTURE. 89 
 
 Above and below the sight-line, the eye compre- 
 hends, as it is raised or depressed while the head 
 is held upright, about an equal space; and, on 
 each side of the sight-point, about the same space is 
 easily seen without turning the head ; so that if a 
 picture represented the true field of easy vision, it 
 ought to be circular, and have the sight-point in its 
 centre. But because some parts of any given view 
 are usually more interesting than others, either the 
 uninteresting parts are left out, or somewhat more 
 than would generally be seen of the interesting parts 
 is included, by moving the field of the picture a 
 little upwards or downwards, so as to throw the 
 sight-point low or high. The operation will be 
 imderstood in a moment by cutting an aperture in 
 a piece of pasteboard, and moving it up and down 
 in front of the eye, without moving the eye. It 
 will be seen to embrace sometimes the low, some- 
 times the high objects, without altering their per- 
 spective, only the eye will be opposite the lower 
 part of the aperture when it sees the higher objects, 
 and vice versa. 
 
90 THE ELEMENTS OF PERSPECTIVE. 
 
 There is no reason, in the laws of perspective, 
 why the picture should not be moved to the right 
 or left of the sight-point, as well as up or down; 
 But there is this practical reason. The mo- 
 ment the spectator sees the horizon in a picture 
 high, he tries to hold his head high, that is, in its 
 right place. When he sees the horizon in a picture 
 low, he similarly tries to put his head low. But, if 
 the sight-point is thrown to the left hand or right 
 hand, he does not understand that he is to step a 
 little to the right or left ; and if he places himself, as 
 usual, in the middle, all the perspective is distorted. 
 Hence it is generally unadvisable to remove the 
 sight-point laterally, from the centre of the picture. 
 The Dutch painters, however, fearlessly take the 
 license of placing it to the right or left ; and often 
 with good effect. 
 
 The rectilinear limitation of the sides, top, and 
 base of the [picture is of course quite arbitrary, as 
 the space of a landscape would be which was 
 seen through a window; less or more being seen 
 at the spectator's pleasure, as he retires or advances. 
 
PLACING AND SCALE OF PICTURE. 91 
 
 The distance of the station-point is not so arbi- 
 trary. In ordinary cases it should not be less than 
 the intended greatest dimension (height or breadth) 
 of the picture. In most works by the great masters 
 it is more ; they not only calculate on their pictures 
 being seen at considerable distances, but they like 
 breadth of mass in buildings, and dislike the sharp 
 angles which always result from station-points at 
 short distances.* 
 
 Whenever perspective, done by true rule, looks 
 wrong, it is always because the station-point is too 
 near. Determine, in the outset, at what distance the 
 spectator is likely to examine the work, and never 
 use a station-point within a less distance. 
 
 There is yet another and a very important reason, 
 not only for care in placing the station-point, but for 
 that accurate calculation of distance and observance 
 
 * The greatest masters are also fond of parallel perspective, that 
 is to say, of having one side of their buildings fronting them full, and 
 therefore parallel to the picture plane, while the other side vanishes 
 to the sight-point. This is almost always done in figure back- 
 grounds, securing simple and balanced lines. 
 
92 THE ELEMENTS OF PERSPECTIVE. 
 
 of measurement whicli have been insisted on through- 
 out this work. All drawings of objects on a reduced 
 scale are, if rightly executed, drawings of the ap- 
 pearance of the object at the distance which in true 
 perspective reduces it to that scale. They are not 
 small drawings of the object seen near, but drawings 
 the real size of the object seen far off. Thus if you 
 draw a mountain in a landscape, three inches high, 
 you do not reduce all the features of the near 
 mountain so as to come into three inches of paper. 
 You could not do that. All that you can do is to 
 give the appearance of the mountain, when it is so 
 far off that three inches of paper would really hide 
 it from you. It is precisely the same in drawing 
 any other object. A face can no more be reduced 
 in scale than a mountain can. It is infinitely 
 delicate already ; it can only be quite rightly ren- 
 dered on its own scale, or at least on the slightly 
 diminished scale which would be fixed by placing 
 the plate of glass, supposed to represent the field 
 of the picture, close to the figures. Correggio and 
 Kaphael were both fond of this slightly subdued 
 
PLACING AND SCALE OF PICTURE. 93 
 
 magnitude of figure. Colossal painting, in which 
 Correggio excelled all others, is usually the en- 
 largement of a small picture (as a colossal sculpture 
 is of a small statue), in order to permit the subject 
 of it to be discerned at a distance. The treatment 
 of colossal (as distinguished from ordinary) paintings 
 will depend therefore, in general, on the principles 
 of optics more than on those of perspective, though, 
 occasionally, portions may be represented as if they 
 were the projection of near objects on a plane 
 behind them. In all points the subject is one of 
 great difficulty and subtlety; and its examination 
 does not fall within the compass of this essay. 
 
 Lastly, it will follow from these considerations, 
 and the conclusion is one of great practical import- 
 ance, that, though pictures may be enlarged, they 
 cannot be reduced, in copying them. All attempts 
 to engrave pictures completely on a reduced scale 
 are, for this reason, nugatory. The best that can 
 be done is to give the aspect of the picture at the 
 distance which reduces it in perspective to the 
 size required ; or, in other words, to make a drawing 
 
94 THE ELEMENTS OF PERSPECTIYE. 
 
 of the distant effect of the picture. Grood paint- 
 ing, like nature's own work, is infinite, and unre- 
 duceable. 
 
 I wish this book had less tendency towards the 
 infinite and unreduceable. It has so far exceeded 
 the limits I hoped to give it, that I doubt not the 
 reader will pardon an abruptness of conclusion, 
 and be thankful, as I am myself, to get to an end 
 on any terms. 
 
APPENDIX, 
 
 I. 
 
 PRACTICE AND OBSEEVATIONS. 
 
 II. 
 
 DEMONSTEATIONS. 
 
PRACTICE AND OBSERVATIONS ON THE 
 PRECEDING PROBLEMS. 
 
 Problem L 
 
 An example will be necessary to make this problem clear 
 to the general student. 
 
 The nearest corner of a piece of pattern on the carpet 
 is 41 feet beneath the eye, 2 feet to our right and 3^ 
 feet in direct distance from us. We intend to make a 
 drawing of the pattern which shall be seen properly 
 when held 1^ foot from the eye. It is required to fix 
 the position of the corner of the piece of pattern. 
 
 Let AB, Fig. 51., be our sheet 
 of paper, some 3 feet wide. 
 Make st equal to 1^ foot. 
 Draw the line of sight through 
 s. Produce t s, and make d s 
 equal to 2 feet, therefore t d 
 equal to 3J feet. Draw d c, 
 equal to 2 feet; cp, equal to 
 4 feet. Join t c (cutting the 
 sight-line in q) and t p. 
 
 Let fall the vertical q p', 
 then p' is the point required. 
 
 Fig. 51. 
 
98 
 
 PRACTICE AND OBSERVATIONS 
 
 [App. I. 
 
 If the lines, as in the figure, fall outside of your sheet 
 of paper, in order to draw them, it is necessary to at- 
 tach other sheets of paper to its edges. This is incon- 
 venient, but must be done at first that you may see your 
 way clearly ; and sometimes afterwards, though there 
 are expedients for doing without such extension in fast 
 sketching. 
 
 It is evident, however, that no extension of surface 
 could be of any use to us, if the distance t d, instead 
 of being SJ feet, were 100 feet, or a mile, as it might 
 easily be in a landscape. 
 
 It is necessary, therefore, to obtain some other means 
 of construction ; to do which we must examine the prin- 
 ciple of the problem. 
 
 In the analysis of Fig. 2., in the introductory remarks, I 
 used the word " height " only of the tower, qp, because it 
 was only to its vertical height that the law deduced from 
 
 Fig. 52. 
 
 the figure could be applied. For suppose it had been a 
 pyramid, as o qp, Fig. 52., then the image of its side, qp. 
 
Prob. I.] ON THE PRECEDING PROBLEMS 99 
 
 being, like every other magnitude, limited on the glass a b 
 by the lines coming from its extremities, would appear 
 only of the length q' s ; and it is not true that q' s is to Q p 
 as T s is to TP. But if we let fall a vertical qd from q, so as 
 to get the vertical height of the pyramid, then it is true 
 that q' s is to Q D as T s is to T D. 
 
 Supposing this figure represented, not a pyramid, but 
 a triangle on the ground, and that qd and qp are horizontal 
 lines, expressing lateral distance from the line t d, still the 
 rule would be false for q p and true for q d. And, simi- 
 larly, it is true for all lines which are parallel, like Q d, 
 to the plane of the picture a b, and false for all lines which 
 are inclined to it at an angle. 
 
 Hence generally. Let p q (Fig. 2. in Introduction, p. 11) 
 be any magnitude parallel to the plane of the picture ; 
 and p' q' its image on the picture. 
 
 Then always the formula is true which you learned in 
 the Introduction : p' q' is to p Q as s T is to d t. 
 
 Now the magnitude P dash q dash in this formula I call the 
 " SIGHT-MAGNITUDE " of the line P Q. The student must fix 
 this term, and the meaning of it, well in his mind. The 
 " sight-magnitude " of a line is the magnitude which bears 
 to the real line the same proportion that the distance of 
 the picture bears to the distance of the object. Thus, if a 
 tower be a hundred feet high, and a hundred yards off; and 
 the picture, or piece of glass, is one yard from the spectator, 
 between him and the tower ; the distance of picture being 
 then to distance of tower as 1 to 100, the sight-magnitude 
 of the tower's height will be as 1 to 100 ; that is to say, 
 
 H 2 
 
100 PRACTICE AND OBSERVATIONS [App. I. 
 
 one foot. If the tower is two hundred yards distant, the 
 sight-magnitude of its height will be half a foot, and 
 so on. 
 
 But farther. It is constantly necessary, in perspec- 
 tive operations, to measure the other dimensions of 
 objects by the sight-magnitude of their vertical lines. 
 Thus, if the tower, which is a himdred feet high, is 
 square, and twenty-five feet broad on each side ; if the 
 sight-magnitude of the height is one foot, the measure- 
 ment of the side, reduced to the same scale, will be the 
 hundredth part of twenty-five feet, or three inches : and, 
 accordingly, I use in this treatise the term " sight-mag- 
 nitude " indiscriminately for all lines reduced in the same 
 proportion as the vertical lines of the object. If I tell you 
 to find the " sight-magnitude " of any line, I mean, always, 
 find the magnitude which bears to that line the proportion 
 of s T to D T ; or, in simpler terms, reduce the line to the 
 scale which you have fixed by the first determination of 
 the length s t. 
 
 Therefore, you must learn to draw quickly to scale 
 before you do anything else ; for all the measurements of 
 your object must be reduced to the scale fixed by st 
 before you can use them in your diagram. If the object 
 is fifty feet from you, and your paper one foot, all the lines 
 of the object must be reduced to a scale of one fiftieth 
 before you can use them ; if the object is two thousand 
 feet from you, and your paper one foot, all your lines 
 must be reduced to the scale of one two-thousandth before 
 you can use them, and so on. Only in ultimate prac- 
 
Prob. 1.] ON THE PRECEDING PROBLEMS. 101 
 
 tice, the reduction never need be tiresome, for, in the case 
 of large distances, accuracy is never required. If a build- 
 ing is three or four miles distant, a hairbreadth of accidental 
 variation in a touch makes a difference of ten or twenty- 
 feet in height or breadth, if estimated by accurate perspective 
 law. Hence it is never attempted to apply measurements 
 with precision at such distances. Measurements are only 
 required within distances of, at the most, two or three 
 hundred feet. Thus it may be necessary to represent a 
 cathedral nave precisely as seen from a spot seventy feet in 
 front of a given pillar ; but we shall hardly be required 
 to draw a cathedral three miles distant precisely as seen from 
 seventy feet in advance of a given milestone. Of course, if 
 such a thing be required, it can be done ; only the reductions 
 are somewhat long and complicated : in ordinary cases it is 
 easy to assume the distance s t so as to get at the reduced 
 dimensions in a moment. Thus, let the piUar of the nave, 
 in the case supposed, be 42 feet high, and we are required 
 to stand 70 feet from it : assume s t to be equal to 5 feet. 
 Then, as 5 is to 70 so will the sight-magnitude re- 
 quired be to 42; that is to say, the sight-magnitude 
 of the pillar's height will be 3 feet. If we make s t 
 equal to 2^ feet, the pillar's height wiU be 1^ foot, and 
 so on. 
 
 And for fine divisions into irregular parts which cannot 
 be measured, the ninth and tenth problems of the sixth 
 book of Euclid wiU serve you: the following construction 
 is, however, I think, more practically convenient : 
 
 The line ab (Fig. 53.) is divided by given points, a, h, c, 
 
 H 3 
 
102 
 
 PRACTICE AND OBSERVATIONS 
 
 [App. I. 
 
 into a given number of irregularly unequal parts ; it 
 is required to divide any other line, CD, into an equal 
 number of parts, bearing to each other the same propor- 
 tions as the parts of ab, and arranged in the same 
 order. 
 
 Fijr. 53. 
 
 Draw the two lines parallel to each other, as in the 
 figure. 
 
 Join A c and b d, and produce the lines AC, b d, till they 
 meet in p. 
 
 Join a p, J P, c P, cutting cd in /, g, h. 
 
 Then the line CD is divided as required, in/, g, h. 
 
 In the figure the lines a b and c d are accidentally per- 
 pendicular to A P. There is no need for their being so. 
 
 Now, to return to our first problem. 
 
 The construction given in the figure is only the quickest 
 mathematical way of obtaining, on the picture, the sight- 
 
Prob. 1.] ON THE PRECEDING PROBLEMS. 103 
 
 magnitudes of dc and PC, which are both magnitudes 
 parallel with the picture plane. But if these magnitudes 
 are too great to be thus put on the paper, you liave only 
 to obtain the reduction by scale. Thus, if t s be one foot, 
 T D eighty feet, d c forty feet, and c p ninety feet, the dis- 
 tance Q s must be made equal to one eightieth of d c, or 
 half a foot ; and the distance Q p', one eightieth of c P, or 
 one eightieth of ninety feet ; that is to say, nine eighths of 
 a foot, or thirteen and a half inches. The lines c t and P t 
 are thus practically useless, it being only necessary to 
 measure q s and q p, on your paper, of the due sight- 
 magnitudes. But the mathematical construction, given 
 in Problem L, is the basis of all succeeding problems, and, 
 if it is once thoroughly understood and practised (it can 
 only be thoroughly understood by practice), all the other 
 problems will follow easily. 
 
 Lastly. Observe that any perspective operation what- 
 ever may be performed with reduced dimensions of every 
 line employed, so as to bring it conveniently ^vithin the 
 limits of your paper. When the required figure is thus 
 constructed on a small scale, you have only to enlarge it 
 accurately in the same proportion in which you reduced 
 the lines of construction, and you will have the figure con- 
 structed in perspective on the scale required for use. 
 
 H 4 
 
104 PRACTICE AND OBSEEYATIONS [App. I. 
 
 PEOBLEM IX. 
 
 The drawing of most buildings occurring in ordinary- 
 practice will resolve itself into applications of this problem. 
 In general, any house, or block of houses, presents itself 
 under the main conditions assimied here in Fig. 54. There 
 will be an angle or corner somewhere near the spectator, as 
 A B ; and the level of the eye will usually be above the base 
 of the building, of which, therefore, the horizontal upper 
 lines will slope down to the vanishing-points, and the base 
 lines rise to them. The following practical directions wiU, 
 however, meet nearly all cases : — 
 
 Let A B, Fig. 54., be any important vertical line in the 
 block of buildings ; if it is the side of a street, you may 
 fix upon such a line at the division between two houses. 
 If its real height, distance, &c., are given, you will proceed 
 with the accurate construction of the problem ; but usually 
 you will neither know, nor care, exactly how high the 
 building is, or how far off. In such case draw the line A b, 
 as nearly as you can guess, about the part of the picture it 
 ought to occupy, and on such a scale as you choose. Divide 
 it into any convenient number of equal parts, according to 
 the height you presume it to be. If you suppose it to be 
 twenty feet high, you may divide it into twenty parts, and 
 let each part stand for a foot ; if thirty feet high, you may 
 divide it into ten parts, and let each part stand for three 
 
Pbob. IX.] ON THE PRECEDING PROBLEMS. 
 
 105 
 
 feet ; if seventy feet high, into fourteen parts, and let each 
 part stand for five feet; and so on, avoiding thus very 
 minute divisions till you come to details. Then observe 
 
 B 1 23 4567 
 
 Fiff. 54. 
 
 how high your eye reaches upon this vertical line ; sup- 
 pose, for instance, that it is thirty feet high and divided 
 into ten parts, and you are standing so as to raise your 
 head to about six feet above its base, then the sight-line 
 may be drawn, as in the figure, through the second division 
 from the groimd. If you are standing above the house, 
 draw the sight-Hue above b ; if below the house, below a ; 
 at such height or depth as you suppose may be accurate (a 
 yard or two more or less matters little at ordinary distances, 
 while at great distances perspective rules become nearly 
 useless, the eye serving you better than the necessarily 
 
106 PEACTICE AND OBSERVATIONS [App. I. 
 
 imperfect calculation). Then fix your sight-point and 
 station-point, the latter with proper reference to the scale 
 of the line A B. As you cannot, in all probability, ascer- 
 tain the exact direction of the line a v or b v, draw the 
 slope B V as it appears to you, cutting the sight-line in v. 
 Thus having fixed one vanishing-point, the other, and the 
 dividing-points, must be accurately found by rule ; for, as 
 before stated, whether your entire group of points (vanish- 
 ing and dividing) falls a little more or less to the right or 
 left of s does riot signify, but the relation of the points to 
 each other does signify. Then draw the measuring-line b g, 
 either through a or B, choosing always the steeper slope of 
 the two ; divide the measuring-line into parts of the same 
 length as those used on a b, and let them stand for the 
 same magnitudes. Thus, suppose there are two rows of 
 windows in the house fi-ont, each window six feet high by 
 three wide, and separated by intervals of three feet, both 
 between window and window and between tier and tier ; 
 each of the divisions here standing for three feet, the lines 
 drawn from b G to the dividing-point D fix the lateral di- 
 mensions, and the divisions on A b the vertical ones. For 
 other magnitudes it would be necessary to subdivide the 
 parts on the measuring-line, or on a b, as required. The 
 lines which regulate the inner sides or returns of the win- 
 dows (a, h, c, &c.) of course are drawn to the vanishing- 
 point of B F (the other side of the house), if r b v represents 
 a right angle ; if not, their own vanishing-point must be 
 found separately for these returns. But see Practice on 
 Problem XL 
 
Prob. IX.] ON THE PRECEDING PROBLEMS. 
 
 107 
 
 Interior angles, such as ebc, Fig. 55. (suppose the 
 comer of a room), are to be treated in the same way, 
 each side of the room having its measurements separately 
 ^ b c C 
 
 D 
 
 carried to it from the measuring-line. It may sometimes 
 happen in such cases that we have to carry the measure- 
 ment up. from the corner b, and that the sight-magnitudes 
 are given us from the length of the line a b. For in- 
 stance, suppose the room is eighteen feet high, and there- 
 fore A B is eighteen feet ; and we have to lay off lengths 
 of six feet on the top of the room wall, b c. Find d, the 
 dividing-point of b c. Draw a measuring-line, b f, from 
 b ; and another, g c, anywhere above. On b f lay off b g 
 equal to one third of a b, or six feet ; and draw from d, 
 through G and b, the lines g ^, b 5, to the upper measuring- 
 line. Then ^ 6 is six feet on that measuring-line. Make 
 i c, c A, &c., equal to h g\ and draw c e, A/, &c., to d, 
 cutting B c in e and f^ which mark the required lengths 
 of six feet each at the top of the wall. 
 
108 
 
 PKACTICE AND OBSERVATIONS 
 
 [Apf. I. 
 
 PROBLEM X. 
 
 This is one of the most important foundational problems 
 in perspective, and it is necessary that the student should 
 entirely familiarise himself with its conditions. 
 
 In order to do so, he must first observe these general 
 relations of magnitude in any pyramid on a square base. 
 
 Let agh', Fig. 56., be any pyramid on a square 
 
 The best terms in which its mag- 
 nitude can be given, are the length 
 of one side of its base, a h, and 
 its vertical altitude (cd in Fig. 25.) ; 
 for, knowing these, we know all 
 the other magnitudes. But these 
 F^o- ^6- are not the terms in which its size 
 
 will be usually ascertainable. Generally, we shall have 
 given us, and be able to ascertain by measurement, 
 one side of its base a h, and either a g the length of 
 one of the lines of its angles, or B G (or b' g) the 
 length of a line dra'svn from its vertex, G, to the middle 
 of the side of its base. In measuring a real pyramid, A G 
 will usually be the line most easily foimd ; but in many 
 architectural problems B G is given, or is most easily 
 ascertainable. 
 
Prcb. X.] ON THE PRECEDINa PROBLEMS. 
 
 109 
 
 Y'lcr. 57. 
 
 Observe therefore this general construction. 
 
 Let A B D E, Fig. 57., be the square base of any 
 pyramid. 
 
 Draw its diagonals, a e, b d, p 
 
 cutting each other in its centre, 
 c. 
 
 Bisect any side, a b, in F. 
 
 From F erect vertical f g. ' 
 
 Produce f b to h, and make f h 
 equal to a c. 
 
 Now if the vertical altitude 
 of the pyramid (c d in Fig. 25.) 
 be given, make f g equal to this 
 vertical altitude. 
 
 Join G B and G h. 
 
 Then g b and g h are the true magnitifdes of G b and 
 GH in Figure 56. 
 
 If GB is given, and not the vertical altitude, with 
 centre b, and distance g b, describe circle cutting f g in 
 G, and F G is the vertical altitude. 
 
 If G H is given, describe the circle from h, with distance 
 G H, and it will similarly cut f g in G. 
 
 It is especially necessary for the student to examine 
 this construction thoroughly, because in many complicated 
 forms of ornaments, capitals of columns, &c., the lines b g 
 and G H become the limits or bases of curves, which are 
 elongated on the longer (or angle) profile G h, and short- 
 ened on the shorter (or lateral) profile b g. We will 
 
no 
 
 PRACTICE AND OBSERVATIONS 
 
 [App. I. 
 
 take a simple instance, but must previously note another 
 construction. 
 
 It is often necessary, when pyramids are the roots of 
 some ornamental form, to divide them horizontally at a 
 given vertical height. The shortest way of doing so is 
 in general the following. 
 
 Let A E c, Fig. 58., be any pyramid on a square base 
 ABC, and ADC the square pillar used in its construction. 
 
 Fiff. 58. 
 
 Then by construction (Problem X.) bd and af are 
 both of the vertical height of the pyramid. 
 
 Of the diagonals, f e, d e, choose the shortest (in this 
 case D e), and produce it to cut the sight-line in v. 
 
Prob. X.] ON THE PEECEDING PROBLEMS. Ill 
 
 Therefore v is the vanishing-point of d e. 
 
 Divide d b, as may be required, into the sight-magni- 
 tudes of the given vertical heights at which the pyramid 
 is to be divided. 
 
 From the points of division, 1, 2, 3, &q., draw 
 to the vanishing-point v. The lines so drawni cut the 
 angle line of the pyramid, b e, at the required elevations. 
 Thus, in the figure, it is required to draw a horizontal 
 black band on the pyramid at three fifths of its height, 
 and in breadth one twentieth of its height. The line b d 
 is divided into five parts, of which three are counted from 
 B upwards. Then the line drawn to v marks the base of 
 the black band. Then one fourth of one of the five parts 
 is measured, which similarly gives the breadth of the 
 band. The terminal lines of the band are then drawn 
 on the sides of the pyramid parallel to ab (or to its 
 vanishing-point if it has one), and to the vanishing-point 
 of BC. 
 
 If it happens that the vanishing-points of the diagonals 
 are awkwardly placed for use, bisect the nearest base line 
 of the pyramid in b, as in Fig. 59. 
 
 Erect the vertical d b and join g b and d g (g being the 
 apex of pyramid). 
 
 Find the vanishing-point of d g, and use d b for division, 
 carrying the measurements to the line g b. 
 
 In Fig. 59., if we join ad and d c, adc is the vertical 
 profile of the whole pyramid, and b d c of the half pyra- 
 mid, corresponding to f g b in Fig. 57. 
 
112 
 
 PEACTICE AND OBSERYATIONS [App, I. 
 
 Fiff. 59. 
 
 Fig. 60. 
 
 We may now proceed to an architectural example. 
 
 Let A H, Fig. 60., be the vertical profile of the capital 
 of a pillar, a b the semi-diameter of its head or abacus, 
 and F D the semi-diameter of its shaft. 
 
 Let the shaft be circular, and the abacus square, do"WTi 
 to the level e. 
 
 Join B D, E F, and produce them to meet in G. 
 
 Therefore e c G is the semi-profile of a reversed pyra- 
 mid containing the capital. 
 
 Construct this pyramid, with the square of the abacus, 
 in the required perspective, as in Fig. 61.; making ae 
 equal to a e in Fig. 60., and ak, the side of the square, 
 equal to twice a b in Fig. 60. Make e g equal to c G, 
 and e d equal to c d. Draw d f to the vanishing-point of 
 
Prob. X.] ON THE PRECEDING PEOBLEMS. 
 
 113 
 
 the diagonal dv (the figure is too small to include this 
 vanishing-point), and f is the level of the point F in 
 Fig. 60., on the side of the pyramid. 
 
 Fi^. 61, 
 
 Draw Fm, fw, to the vanishing-points of a h and ak. 
 Then f n and f m are horizontal lines across the pyramid 
 at the level f, forming at that level two sides of a square. 
 
 Complete the square, and within it inscribe a circle, as 
 in Fig. 62., which is left unlettered that its construction 
 may be clear. At the extremities of this draw vertical 
 lines, which will be the sides of the shaft in its right 
 place. It will be found to be somewhat smaller in dia- 
 meter than the entire shaft in Fig. 60., because at the 
 
114 
 
 PRACTICE AND OBSEEVATIONS 
 
 [A PP. I. 
 
 centre of the square it is more distant than the nearest 
 edge of the square abacus. The curves of the capital may- 
 then be drawn approximately by the eye. They are not 
 quite accurate in Fig. 62., there being a subtlety in their 
 
 Eiff. 62. 
 
 junction with the shaft which could not be shown on so 
 small a scale without confusing the student ; the curve on 
 the left springing from a point a little way round the circle 
 behind the shaft, and that on the right from a point on 
 this side of the circle a little way within the edge of the 
 shaft. But for their more accurate construction see Notes 
 on Problem XIV. 
 
Prob. XL] ON THE PRECEDING PROBLEMS. 
 
 115 
 
 PROBLEM XI. 
 
 It is seldom that any complicated curve, except occa- 
 sionally a spiral, needs to be drawn in perspective ; but 
 the student will do well to practise for some time any 
 fantastic shapes which he can find drawn on flat surfaces, 
 as on wall-papers, carpets, &c., in order to accustom him- 
 self to the strange and great changes which perspective 
 causes in them. 
 
 The curves most required in architectural drawing, 
 after the circle, are those of pointed arches; in which, 
 however, all that wiU be generally needed is to fix the 
 
 Fior. 63. 
 
 apex, and two points in the sides. Thus if we have to 
 
 draw a range of pointed arches, such as a p b, Fig. 63., 
 
 I 2 
 
116 
 
 PRACTICE AND OBSERVATIONS 
 
 [App. I. 
 
 draw the measured arch to its sight-magnitude first neatly 
 in a rectangle, a b c d ; then draw the diagonals a d and 
 B c ; where they cut the curve draw a horizontal line (as 
 at the level e in the figure), and carry it along the range 
 to the vanishing-point, fixing the points where the arches 
 cut their diagonals all along. If the arch is cusped, a 
 line should be drawn at r to mark the height of the cusps, 
 and verticals raised at g and h, to determine the interval 
 between them. Any other points may be similarly deter- 
 mined, but these will usually be enough. Figure 63. 
 shows the perspective construction of a square niche of 
 good Veronese Gothic, with an uncusped arch of similar 
 size and curve beyond. 
 
 Fisj. 64. 
 
 In Fig. 64. the more distant arch only is lettered, as the 
 construction of the nearest explains itself more clearly to 
 
Prob. XL] ON THE PRECEDING PROBLEMS. 117 
 
 the eye without letters. The more distant arch shows 
 the general construction for all arches seen underneath, 
 as of bridges, cathedi-al aisles, &c. The rectangle a b c d 
 is first drawn to contain the outside arch ; then the depth 
 of the arch, a a, is determined by the measuring-line, and 
 the rectangle, abed, drawn for the inner arch. 
 
 A a, B J, &c., go to one vanishing-point ; A b, ab, &c., 
 to the opposite one. 
 
 In the nearer arch another narrow rectangle is drawn 
 to determine the cusp. The parts which would actually 
 come into sight are slightly shaded. 
 
 1 3 
 
118 
 
 PEACTICE AND OBSERYATIONS 
 
 [App. r. 
 
 PROBLEM Xiy. 
 
 Several exercises will be required on this important 
 problem. 
 
 I. It is required to draw a circular flat-bottomed dish 
 narrower at the bottom than the top ; the vertical depth 
 being given, and the diameter at the top and bottom. 
 
 d 
 
 Fig. 65. 
 
 Let a 5, Fig. 65., be the diameter of the bottom, a c the 
 diameter of the top, and ad its vertical depth. 
 
Prob. XIV.] ON THE PRECEDING PROBLEMS. 119 
 
 Take a d in position equal to a c. 
 
 On A D draw the square a b c d, and inscribe in it a 
 circle. 
 
 Therefore, the circle so inscribed has the diameter of 
 the top of the dish. 
 
 From A and d let fall verticals, a e, d h, each equal 
 to a d. 
 
 Join E H, and describe square e f G h, which accordingly 
 will be equal to the square a BCD, and be at the depth 
 a d beneath it. 
 
 Within the square e f G h describe a square i k, whose 
 diameter shall be equal to a h. 
 
 Describe a circle within the square i k. Therefore the 
 circle so inscribed has its diameter equal to a & ; and it is 
 in the centre of the square e f g h, which is vertically 
 beneath the square a b c d. 
 
 Therefore the circle in the square i k represents the 
 bottom of the dish. 
 
 Now the two circles thus drawn will either intersect 
 one another, or they will not. 
 
 If they intersect one another, as in the figure, and they 
 are below the eye, part of the bottom of the dish is seen 
 within it. 
 
 To avoid confusion, let us take then two intersecting 
 circles with(fut the enclosing squares, as in Fig. QQ. 
 
 Draw right lines, ah, c d, touching both circles ex- 
 ternally. Then the parts of these lines which connect 
 the circles are the sides of the dish. They are drawn in 
 
 1 4 
 
120 PKACTICE AND OBSERYATIONS [App. 1. 
 
 Fig. 65. without any prolongations, but the best way to 
 construct them is as in Fig. 66. 
 
 Fig. 66. 
 
 If the circles do not intersect each other, the smaller 
 must either be within the larger or not within it. 
 
 If within the larger, the whole of the bottom of the 
 dish is seen from above, Fig. 67. a. 
 
 If the smaller circle is not 
 within the larger, none of the 
 bottom is seen inside the dish, 
 b. 
 
 If the circles are above in- 
 stead of beneath the eye, the 
 bottom of the dish is seen be- 
 neath it, c. 
 
 If one circle is above and 
 
 another beneath the eye, ^^ ' 
 
 neither the bottom nor top -,. __ 
 
 ^ Fig. 67. 
 
 of the dish is seen, d. Un- 
 less the object be very large, the circles m this case 
 will have little apparent curvature. 
 
 II. The preceding problem is simple, because the lines 
 of the profile of the object (a b and c d, Fig. 66.) are 
 
Prob. XIV.] ON THE PRECEDING PROBLEMS. 
 
 121 
 
 straight. But if these lines of profile are curv^ed, the 
 problem becomes much more complex : once mastered, 
 however, it leaves no farther difficulty in perspective. 
 
 Let it be required to draw a flattish circular cup or 
 vase, with a given curve of profile. 
 
 The basis of construction is given in Fig. 68., half 
 of it only being drawn, in order that the eye may seize 
 its lines easily. 
 
 Two squares (of the required size) are first drawn, one 
 above the other, with a given vertical inter\^al, a c, be- 
 tween them, and each is divided into eight parts by 
 its diameters and diagonals. In these squares two circles 
 are drawn; which are, therefore, of equal size, and one 
 above the other. Two smaller circles, also of equal size, 
 are drawn within these larger circles in the construction of 
 the present problem ; more may be necessary in some, 
 none at all in others. 
 
 It will be seen that the portions of the diagonals and 
 
122 
 
 PKACTICE AND OBSERVATIONS 
 
 [A pp. I. 
 
 diameters of squares which are cut off between the circles 
 represent radiating planes, occupying the position of the 
 spokes of a wheel. 
 
 Now let the line a e b, Fig. 69., be the profile of the 
 vase or cup to be drawn. 
 
 D 
 
 E 
 Fig. 69. 
 
 Enclose it in the rectangle c d, and if any portion of it 
 is not curved, as a e, cut off the curved portion by the 
 vertical line e f, so as to include it in the smaller rect- 
 angle FD. 
 
 Draw the rectangle a c b d in position, and upon it 
 construct two squares, as they are constructed on the 
 rectangle a c d in Fig. 68. ; and complete the construction 
 of Fig. 68., making the radius of its large outer circles 
 equal to a d, and of its small inner circles equal to a e. 
 
 The planes which occupy the position of the wheel- 
 spokes will then each represent a rectangle of the size of 
 
Prob. XIV.] ON THE PRECEDING PEOBLEMS. 123 
 
 F D. The construction is shown by the dotted lines in 
 Fig. 69. ; c being the centre of the uppermost circle. 
 
 Within each of the smaller rectangles between the circles, 
 draw the curve e b in perspective, as in Fig. 69. 
 
 Draw the curve x y, touching and enclosing the curves 
 in the rectangles, and meeting the upper circle at y* 
 
 Then x y i^ the contour of the surface of the cup, and 
 the upper circle is its lip. 
 
 If the line x y \% long, it may be necessary to draw 
 other rectangles between the eight principal ones ; and, if 
 the curve of profile ab is complex or retorted, there 
 may be several lines corresponding to x ?/, enclosing the 
 successive waves of the profile ; and the outer curve will 
 then be an undulating or broken one. 
 
 III. All branched ornamentation, forms of flowers, ca- 
 pitals of columns, machicolations of round towers, and 
 other such arrangements of radiating curve, are resolva- 
 ble by this problem, using more or fewer interior circles 
 according to the conditions of the curves. Fig. 70. is an 
 example of the construction of a circular group of eight 
 trefoils with curved stems. One outer or limiting circle 
 is drawn within the square e D c f, and the extremities of 
 the trefoils touch it at the extremities of its diagonals and 
 diameters. A smaller circle is at the vertical distance b c 
 below the larger, and a is the angle of the square within 
 which the smaller circle is drawn ; but the square is 
 
 * This point coincides in the figure with the extremity of the 
 horizontal diameter, but only accidentally. 
 
124 PRACTICE AND OBSERVATIONS [App. I. 
 
 to 
 
Prob. XIV.] ON THE PEECEDING PROBLEMS. 125 
 
 not given, to avoid confusion. The stems of the tre- 
 foils form di'ooping cur\^es, arranged on the diagonals 
 and diameters of the smaller circle, which are dotted. 
 But no perspective laws will do work of this intricate 
 kind so well as the hand and eye of a painter. 
 
 rV. There is one common construction, however, in 
 which, singularly, the hand and eye of the painter almost 
 always fail, and that is the fillet of any ordinary capital or 
 base of a circular pillar (or any similar form). It is rarely 
 necessary in practice to draw such minor details in per- 
 spective; yet the perspective laws which regulate them 
 should be understood, else the eye does not see their con- 
 tours rightly until it is very highly cultivated. 
 
 Fig. 71. will show the law with sufficient clearness; it 
 represents the perspective construction of a fillet whose 
 profile is a semicircle, such as f h in Fig. 60., seen above 
 the eye. Only half the pillar with half the fiUet is dra^vn, 
 to avoid confusion. 
 
 Q is the centre of the shaft. 
 
 p Q the thickness of the fillet, sight-magnitude at the 
 shaft's centre. 
 
 Eound p a horizontal semicircle is drawn on the dia- 
 meter of the shaft a b. 
 
 Eound Q another horizontal semicircle is drawn on dia- 
 meter cd. 
 
 These two semicircles are the upper and lower edges of 
 the fillet. 
 
 Then diagonals and diameters are drawn as in Fig. 68., 
 
126 
 
 PRACTICE AND OBSERVATIONS [App. i. 
 
 Fig. 71. 
 
 and, at their extremities, semicircles in perspective, as in 
 Fig. 69. 
 
 The letters a, b, c, d, and e, indicate the upper and 
 exterior angles of the rectangles in which these semi- 
 circles are to be drawn ; but the inner vertical line is not 
 dotted in the rectangle at c, as it would have confused 
 itself with other lines. 
 
 Then the visible contour of the fillet is the line which 
 encloses and touches* aU the semicircles. It disappears 
 
 * The engraving is a little inaccurate ; the enclosing line should 
 touch the dotted semicircles at a and b. The student should draw 
 it on a large scale. 
 
Prob. XIV.] ON THE PRECEDING PROBLEMS. 127 
 
 behind the shaft at the point h, but I have dra^vn 
 it through to the opposite extremity of the diameter 
 at d. 
 
 Turned upside down the figure shows the construction 
 of a basic fillet. 
 
 The capital of a Greek Doric pillar should be drawn fi-e- 
 quently for exercise on this fourteenth problem, the curv^e 
 of its echinus being exquisitely subtle, while the general 
 contour is simple. 
 
128 PRACTICE AND OBSERYATIONS [App. I. 
 
 PROBLEM XVI. 
 
 It is often possible to shorten other perspective operations 
 considerably, by finding the vanishingrpoints of the in- 
 clined lines of the object. Thus, in drawing the gabled 
 roof in Fig. 43., if the gable a y c be drawn in per- 
 spective, and the vanishing-point of a y determined, it is 
 not necessary to draw the two sides of the rectangle, 
 a' d' and d' b', in order to determine the point y' ; but 
 merely to draw yy' to the vanishing-point of a a' and 
 a' y' to the vanishing-point of a y, meeting in y', the point 
 required. 
 
 Again, if there be a series of gables, or other figures 
 produced by parallel inclined lines, and retiring to the 
 point V, as in Fig. 72. *, it is not necessary to draw 
 each separately, but merely to determine their breadths 
 on the line A V, and draw the slopes of each to their 
 vanishing-points, as shown in Fig. 72. Or if the gables 
 are equal in height, and a line be draAvn from y to v, the 
 construction resolves itself into a zigzag drawn alternately 
 to p and Q, between the lines y v and a v. 
 
 The student must be very cautious, in finding the 
 vanishing-points of inclined lines, to notice their relations 
 to the horizontals beneath them, else he may easily mis- 
 take the horizontal to which they belong. 
 
 ♦ The diagram is inaccurately cut. y v should be a right line. 
 
Prob. XVI.] ON THE PRECEDING PROBLEMS. 
 
 129 
 
 ,' I /I 
 
 / / /HI' 
 
 / / /////« 
 
 / / ////// 
 
 / / / ///'/' 
 
 / / / //''•'// 
 
 / / / / /// / 
 
 / / / /^/ 
 
 / / ' I'lH 
 
 /y / / //// 
 
 V:-^ / /// / 
 
 \\\m 
 
 Fis. 7>. 
 
 \ A \ » 
 
 V>» 
 
 
 \ \ » \ 
 
 \\\\ 
 
 
 » \ ^ » 
 
 
 
 V *» \ 
 
 .v«\\ 
 
 
 \ a\ 
 
 < >\«1 
 
 
 \\\ 
 
 \'\'« 
 
 
 V A 
 
 /'''W 
 
 
 \\ 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 v\\ 
 
 
 \\\ 
 
 ^^ 
 
 
 
 
 Q 
 
130 
 
 PKACTICE AND OBSEKVATIONS [App. I. 
 
 Thus, let A BCD, Fig. 73., be a rectangular inclined 
 plane, and let it be required to find the vanishing-point 
 of its diagonal b d. 
 
 B 
 
 Fig. 73. 
 
 Find V, the vanishing-point of a d and b c. 
 
 Draw A E to the opposite vanishing-point, so that D a E 
 may represent a right angle. 
 
 Let fall from b the vertical b e, cutting a e in E. 
 
 Join e d, and produce it to cut the sight-line in v'. 
 
 Then, since the point e is vertically imder the point b, 
 the horizontal line e d is vertically under the inclined 
 line b d. So that if we now let fall the vertical v' p 
 
Prob. XVL] ON THE PRECEDING PROBLEMS. 131 
 
 from v', and produce b d to cut v' P in p, the point p 
 will be the vanishing-point of b d, and of all lines parallel 
 to it.* 
 
 * The student may perhaps understand this construction better 
 by completing the rectangle a d f e, drawing d f to the vanishing- 
 point of A E, and E F to V. The whole figure, b f, may then be 
 conceived as representing half the gable roof of a house, a f the 
 rectangle of its base, and a c the rectangle of its sloping side. 
 
 In nearly all picturesque buildings, especially on the Continent, 
 the slopes of gables are much varied (frequently unequal on the two 
 sides), and the vanishing-points of their inclined lines become very 
 important, if accuracy is required in the intersections of tiling, 
 sides of dormer windows, &c. 
 
 Obviously, also, irregular triangles and polygons in vertical planes 
 may be more easily constructed by finding the vanishing-points of 
 their sides, than by the construction given in the corollary to Pro- 
 blem IX. ; and if such triangles or polygons have others concen- 
 trically inscribed within them, as often in Byzantine mosaics, &c., 
 the use of the vanishing-points will become essential. 
 
 K 2 
 
132 
 
 PRACTICE AND OBSERVATIONS [App. I. 
 
 PROBLEM XVIII. 
 
 Before examining the last three problems it is necessary 
 that you should understand accurately what is meant by the 
 position of an inclined plane. 
 
 Cut a piece of strong white pasteboard into any irregular 
 shape, and dip it in a sloped position into water. However 
 you hold it, the edge of the water, of course, will always 
 draw a horizontal line across its surface. The direction of 
 this horizontal line is the direction of the inclined plane. 
 (In beds of rock geologists call it their " strike.") 
 
 Next, draw a semicircle on the piece of pasteboard ; draw 
 its diameter, ab, Fig. 74., and a vertical line from its 
 
 centre, c d ; and draw some other lines, c E, c f, &c., from 
 the centre to any points in the circumference. 
 
 Now dip the piece of pasteboard again into water, and, 
 
Prob. XV III-] ON THE PRECEDINa PROBLEMS. 133 
 
 holding it at any inclination and in any direction you 
 choose, bring the surface of the water to the line a b. 
 Then the line c d will be the most steeply inclined of all 
 the lines dra^vn to the circumference of the circle ; G c 
 and H c will be less steep ; and e c and f c less steep still. 
 The nearer the lines to c d, the steeper they will be ; and 
 the nearer to a b, the more nearly horizontal. 
 
 When, therefore, the line a b is horizontal (or marks 
 the water surface), its direction is the direction of the 
 inclined plane, and the inclination of the line d c is the 
 inclination of the inclined plane. In beds of rock geolo- 
 gists call the inclination of the line dc their " dip." 
 
 To fix the position of an inclined plane, therefore, is to 
 determine the direction of any two lines in the plane, a b 
 and c D, of which one shall be horizontal and the other at 
 right angles to it. Then any lines drawn in the inclined 
 plane, parallel to a b, will be horizontal ; and lines drawn 
 parallel to c d will be as steep as c d, and are spoken of in 
 the text as the " steepest lines " in the plane. 
 
 But farther, whatever the direction of a plane may be, 
 if it be extended indefinitely, it will be terminated, to the 
 eye of the observer, by a boundary line, which, in a hori- 
 zontal plane, is horizontal (coinciding nearly with the 
 visible horizon) ; — in a vertical plane, is vertical ; — and, 
 in an inclined plane, is inclined. 
 
 This line is properly, in each case, called the " sight-line" 
 of such plane ; but it is only properly called the " horizon " 
 in the case of a horizontal plane : and I have preferred 
 K 3 
 
134 PRACTICE AND OBSERVATIONS. [App. I. 
 
 using always the term " sight-line," not only because more 
 comprehensive, but more accurate ; for though the curva- 
 ture of the earth's surface is so slight that practically its 
 visible limit always coincides with the sight-Hne of a 
 horizontal plane, it does not mathematically coincide with 
 it, and the two lines ought not to be considered as theo- 
 retically identical, though they are so in practice. 
 
 It is evident that all vanishing-points of lines in any 
 plane must be found on its sight-line, and, therefore, that 
 the sight-line of any plane may be found by joining any 
 two of such vanishing-points. Hence the construction of 
 Problem XVIII. 
 
11. 
 
 DEMONSTRATIONS WHICH COULD NOT CONVE- 
 NIENTLY BE INCLUDED IN THE TEXT, 
 
 THE SECOND COROLLARY, PROBLEM IL 
 
 In Fig. 8. omit the lines c d, c' d', and d s ; and, as here in 
 Fig. 75., from a draw a d parallel to a b, cutting b t m d ; 
 and from d draw d e parallel to b c. 
 
 Fiff. 75. 
 
 k4 
 
136 ADDITIONAL DEMONSTRATIONS. [App. II. 
 
 Now as 0? ^ is parallel to a b — 
 
 AC : ac 11 BC : de; 
 but A c is equal to b c' — 
 
 /, ac = de, 
 
 Now because the triangles acv, bc'v, are similar — 
 a c I be' II ay : by; 
 and because the triangles der, be' t are similar — 
 de I be' 11 dr : bT. 
 
 But « c is equal to 6? e — 
 
 /. ay I by :: dT I bT; 
 ,•« the two triangles a b d, 5 t v, are similar, and their 
 angles are alternate ; 
 
 /, T V is parallel to a d. 
 
 But a c? is parallel to a b — 
 
 ,% T V is parallel to a b. 
 
Prob. III.] ADDITIONAL DEMONSTRATIONS. 137 
 
 11. 
 
 THE THIRD COROLLARY, PROBLEM III. 
 
 In Fig. 13., since an is by construction parallel to a b 
 in Fig. 12., and t v is by construction in Problem III. 
 also parallel to a b — 
 
 /, a R is parallel to t v, 
 
 .'. abR and t J v are alternate triangles, 
 
 ,\ an I TV :: ab : by. 
 
 Again, by the construction of Fig. 13., an' is parallel 
 to MV — 
 
 .*, abR' and m 5 v are alternate triangles, 
 /, an' : MY 11 ab I by. 
 
 And it has just been shown that also 
 
 aR : Ty y, ab : by — 
 /. a r' ; M V : : a R : TV. 
 
 But by construction, a r' = aR — 
 
 .*. M V = T V. 
 
138 
 
 ADDITIONAL DEMONSTRATIONS. 
 
 [App. II. 
 
 III. 
 
 ANALYSIS OF PROBLEM XV. 
 
 We proceed to take up the general condition of the second 
 problem, before left unexamined, namely, that in which 
 the vertical distances bc' and AC (Fig. 6. page 22.), as 
 well as the direct distances t d and t d' are unequal. 
 
 In Fig. 6., here repeated (Fig. 76.), produce c' b down- 
 wards, and make c' e equal to c A. 
 
 C D' 
 
Prob. XV.] ADDITIONAL DEMONSTRATIONS. 139 
 
 Join A E. 
 
 Then, by the second Corollary of Problem II., ae is a 
 horizontal line. 
 
 Draw T V parallel to a e, cutting the sight-line in v. 
 /, V is the vanishing-point of a e. 
 
 Complete the constructions of Problem II. and its second 
 CoroUary. 
 
 Then by Problem II. ab is the line a b drawn in 
 perspective ; and by its Corollary a e is the line a e drawn 
 in perspective. 
 
 From V erect perpendicular v p, and produce ab to cut 
 it in p. 
 
 Join TP, and from e draw e/ parallel to a e, and cutting 
 A Tin/. 
 
 Now in triangles e b t and a e t, as eb is parallel to 
 e b and e/ to a e ; — eb ', efH eb I ae. 
 
 But T V is also parallel to a e and ty to eb. 
 
 Therefore also in the triangles a p v and « v t, 
 eb l efllFV I YT. 
 
 Therefore pv;vt::eb;ae. 
 
 And, by construction, angle tpv=:Z.aeb. 
 
 Therefore the triangles t v p, a e b, are similar; and T p 
 is parallel to a b. 
 
 Now the construction in this problem is entirely general 
 for any inclined line a b, and a horizontal line a e in the 
 same vertical plane with it. 
 
140 
 
 ADDITIONAL DEMONSTRATIONS. 
 
 [App. II. 
 
 So that if we find the vanishing-point of a e in v, and 
 from V erect a vertical v P, and from t draw t p parallel 
 to A B, cutting V p in p, p will be the vanishing-point of 
 AB, and (by the same proof as that given at page 27.) 
 of all lines parallel to it. 
 
 Next, to find the dividing-point of the inclined line. 
 I remove some unnecessary lines from the last figure 
 and repeat it here, Fig. 77., adding the measuring-line 
 
 Fig. 77. 
 
 a M, that the student may observe its position -with 
 respect to the other lines before I remove any more of 
 them. 
 
Pkob. XV.] ADDITIONAL DEMONSTEATIONS. 
 
 141 
 
 Now if the line a b in this diagram represented the 
 length of the line A b in reality (as a b does in Figs. 10. 
 and 11.), we should only have to proceed to modify 
 Corollary III. of Problem II. to this new construction. 
 We shall see presently that a b does not represent the 
 actual length of the inclined line a b in nature, neverthe- 
 less we shall first proceed as if it did, and modify our 
 result afterwards. 
 
 In Fig. 77. draw a d parallel to a b, cutting b t in J. 
 
 Therefore a c? is the sight-magnitude of a b, as a r is 
 of A B in Fig. 11. 
 
 Remove again from the figure all lines except P v, v T, 
 p T, ah, ad, and the measuring-line. 
 
 Fig. 78. 
 * Set off on the measuring-Hne a m equal to a d. 
 
142 ADDITIONAL DEMONSTRATIONS. [App. II. 
 
 Draw p Q parallel to a m, and through h draw m q, 
 cutting p Q in Q. 
 
 Then, by the proof already given in page 32., p q=p t. 
 
 Therefore if p is the vanishing-point of an inclined line 
 A B, and Q p is a horizontal line drawn through it, make 
 p Q equal to p t, and a m on the measuring-line equal to 
 the sight-magnitude of the line a b in the diagram^ and 
 the line joining m q will cut « p in ^. 
 
 We have now, therefore, to consider what relation the 
 
 length of the line a b in this diagram, Fig. 77., has to the 
 
 length of the line a b in reality. 
 
 Now the line a e in Fig. 77. represents the length of 
 
 a E in reality. 
 
 But the angle a e b, Fig. 77., and the corresponding 
 
 angle in all the constructions of the earlier problems, is 
 
 in reality a right angle, though in the diagram necessarily 
 
 represented as obtuse. 
 
 Therefore, if from e we draw e c, as in Fig. 79., at 
 
 B right angles to a e, make e c=e b, 
 
 and join a c, A c will be the real length 
 
 of the line a b. 
 
 E Now, therefore, if instead of a m in 
 
 Fig. 78., we take the real length of 
 
 T,. ^„ A B, that real length will be to a m as 
 
 Fig. 79. ' ^ 
 
 A c to A B in Fig. 79. 
 And then, if the line drawn to the measuring-line p Q is 
 still to cut a p in h, it is evident that the line p q must 
 
Prob. XV.] ADDITIONAL DEMONSTBATIONS. 143 
 
 be shortened in the same ratio that a m was shortened ; 
 and the true dividing-point will be q' in Fig. 80., fixed 
 so that q' p' shall be to Q p as « m' is to a m ; am' repre- 
 senting the real length of a b. 
 
 But a m' is therefore to a w as a c is to a b in Fig. 79. 
 
 Therefore p q' must be to p q as a c is to a b. 
 
 But p Q equals p t (Fig. 78.) ; and p v is to v T (in Fig. 
 78.) as BE is to A E (Fig. 79.). 
 
 Hence we have only to substitute p v for e c, and v t 
 for A E, in Fig. 79., and the resulting diagonal a c will 
 be the required length of p q'. 
 
 It will be seen that the construction given in the text 
 (Fig. 46.) is the simplest means of obtaining this mag- 
 nitude, for V D in Fig. 46. (or v m in Fig. 15.) = v t by 
 construction in Problem FV. It should, however, be 
 observed, that the distance p q' or p x, in Fig. 46., may 
 
144 ADDITIONAL DEMONSTRATIONS. [App. II. 
 
 be laid on the sight-line of the inclined plane itself, if 
 the measuring-line be drawn parallel to that sight-line. 
 And thus any form may be drawn on an inclined plane 
 as conveniently as on a horizontal one, with the single 
 exception of the radiation of the verticals, which have a 
 vanishing-point, as shown in Problem XX. 
 
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 The Education of the Human Race. Now 
 
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 fervid eloquence which has so materially contri- 
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 " Tlie ■ Two Paths ' contains much eloquent de- 
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 " The meaning of the title of this book is, that 
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 Beauty, 
 
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 work treats chiefly of mountain scenery, and | artists and writers."— «j)cc^a^or. 
 discusses at; length the principles involved in the | " Th-) fourth volume brings fresh stores of 
 
 pleasure we derive from mountains and their 
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 " Mr. Buskin's work will send the painter more 
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 will learn to admire, and mere admirers will learn 
 how to criticise : thus a public will be educated. — 
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"VTOEKS I>TJB3LISIIED BY 
 
 WORKS OF MR. RJJSKl^— continued. 
 
 The Stones of Venice. 
 
 Complete in Three Volumes, Imperial 8vo, with Fifty-three Plates and 
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 EACH VOLUME MAY BE HAD SEPABATELY, 
 
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 " The ' Stones of Venice ' is the production of an 
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 awe of God, and delight in nature ; a knowledge, 
 love, and just estimate of art; a holding fast to 
 fact and repudiation of hearsay; an historic 
 breadth, and a fearless challenge of existing social 
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 paralleled."— (Sjjccia^or. 
 
 " This book is one which, perhaps, no other man 
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 are convinced, elevate taste and intellect, raise 
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 "By 'The Seven Lamps of Architecture.* we 
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 of instructive matter, as well as the artist. The 
 author of this work belongs to a class of thinkers 
 of whom we have too few amongst us."— 
 JExaminer, 
 
 " Mr. Ruskin's book bears so unmistakeably the 
 marks of keen and accurate observation.of a true 
 and subtle judgment and refined sense of beauty, 
 joined with so mucli earnestness, so noble a sense 
 of the purposes and business of art, and such a 
 commanit of rich and glowing language, that it 
 cannot but tell powerfully in producing a mora 
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 deeper insight into its artistic principles.*'— 
 Guardian. 
 
 Notes on the Picture Exhibitions of 1859, 
 
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 Xnb\x\,e<X"—Athe7i(Bum. 
 
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 "This book, daring, as it is, glances keenly at 
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 LIFE AND CORRESPONDENCE OF 
 LORD METCALFE. By J. W. 
 Kate. New and Cheap Edition, 
 in 2 vols., small post 8vo, with 
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 size and price the new edition is a great in.prove- 
 ment on the original vrork."—Econo)nist. 
 
 "This edition is revised with care and judgment. 
 Mr. Kaye has judiciously condensed that portion 
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 empire."— G/o6e. 
 
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 PAPERS OF THE LATE LORD 
 METCALFE. By J. W. Kate. 
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 LIFE AND CORRESPONDENCE OF 
 SIR JOHN MALCOLM, G.C.B. 
 
 By J. W. Kate, 2 vols., 8vo, with 
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 " Mr. Kaye's biography is at once a contribution 
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 BRITISH RULE IN INDIA. Sixth 
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 *,* A reliable class-hook for examination in the 
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 SUGGESTIONS TOWARDS THE 
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 INDIA. By Harriet Martineau. 
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 "As the work of an honest able writer, these 
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 " Genuine honest utterancRS of a clear, sound 
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 Daily News. 
 
 EIGHT MONTHS' CAMPAIGN 
 AGAINST THE BENGAL SE- 
 POYS, DURING THE MUTINY, 
 1857. By Colonel GeorcxE Bour- 
 CHiER, C.B., Bengal Horse Ar- 
 tillery. With plans. Post Svo. 
 Price 7s. 6d. cloth. 
 
 " Col. Bourchier has given a right manly, fair, 
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 "Col. Bourchier describes the various opera- 
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 " None who really desire to be more than very 
 superficially acquainted with the rise and pro- 
 gress of the rebellion may consider their studies 
 complete until they have read Col. Bourchier. The 
 nicely engraved plans from the Colonel's own 
 sketches confer additional value upon his contri- 
 bution to the literature of the Indian war,"— 
 Leader, 
 
 9 
 
■VTOKKIS I>XJBIL.ISIIEr) BY 
 
 NEW WORKS ON INDIA AND THE EAST- 
 
 Continued. 
 
 PERSONAL ADVENTURES DURING 
 THE INDIAN REBELLION, IN 
 ROHILCUND,FUTTEGHUR,AND 
 OUDE. By W. Edwards, Esq., 
 B.C.S. Fourth Edition, post 8vo. 
 Price 65. cloth. 
 
 " For touching incidents, hair-breadth 'scapes, 
 and the pathos of suffering almost incrediole, 
 there has appeared nothing hke this little hook of 
 personal adventures. For the first time we seem 
 torealize the magnitude of the afflictions which 
 }.ave belallen our unhappy counti-ymen in the 
 East. The terrible drama comes before us, and we 
 are by turns bewildered with horror, stung to 
 
 fierce indignation, and melted to tears 
 
 We have here a tale of suffering such as may have 
 been equalled, but never surpassed. These real 
 adventures, which no effort of the imagination 
 can surpass, will find a sympathising public."— 
 AtheneBum. 
 
 "Mr, Edwards's narrative is one of the most 
 deeply interesting episodes of a story of which 
 the least striking portions cannot be read without 
 emotion. He tells his story with simplicity and 
 manliness, and it bears the impress of that 
 earnest and unaffected reverence to the will and 
 hand of God, which was the stay and comfort 
 of many otlier brave hearts."— Gwaj-drnw. 
 
 " The narrative of Mr. Edwards's sufl'ering and 
 escapes is full of interest; it tells many a painful 
 tale, but it also exhibits a man patient under ad- 
 versity, and looking to the God and Father of us 
 all for guidance and support."— i^c/ct-^ic Review. 
 
 "Among the stories of hair-breadtli escapes in 
 India this is one of the most interesting and 
 touchin s"— Examiner. 
 
 "A fascinating little hoo\i."— National Review, 
 
 "A very touching narrative."— Z,«i. Gazette. 
 
 "No account of it can do it justice."— G^o6e. 
 
 A LADY'S ESCAPE FROM GWA- 
 LIOR DURING THE MUTINIES 
 OF 1857. By Mks. Cooplakd. 
 Post 8vo. Price 10s. 6d. 
 
 " A plain, unvarnished tale, told in the simplest 
 xaBXiuer."— Press. 
 
 " This book is valuable as a contribution to the 
 history of the great Indian re bellion."—^i/(eji<E«<>». 
 
 " The merit of this book is its truth. . . . Ic 
 contains some passages that never will be read 
 by Englishmen without emotion."— Examiner. 
 
 THE CHAPLAIN'S NARRATIVE OF 
 THE SIEGE OF DELHI. By the 
 Rev. J. E. W. RoTTON, Chaplain 
 to the Delhi Field Force. Post 
 8vo, with a plan of the City and 
 Siege Works. Price 10s. 6<i. cloth. 
 
 "A simple and touching statement, which bears 
 the impress of truth in every word. It has this 
 advantage over the accounts m hich have yet been 
 published, that it supplies some of those personal 
 anecdotes and minute details which bring the 
 events home to the understanding."— .(4 ^Ae^zfjewm. 
 
 "'The Chaplain's Narrative' is remarkable for 
 its pictures of men in amoral and religious aspect 
 during the progress of a harassing siege and 
 when suddenly stricken down by the enemy or 
 disease."— spectator. 
 
 "A plain imvarniahed record of what came 
 under a Field Chaplain's daily observation. Our 
 author is a sincere, hardworking, and generous 
 minded man, and his work will be most acceptable 
 to the friends and relations of the many Christian 
 1 eroes whose fate it tells, and to whose later 
 hours it alludes."— Leader. 
 
 " A book which has value as a careful narrative 
 by an eye witness of one of the most stirring 
 episodes of the Indian campaign, and interest as 
 an earnest record by a Christian minister of 
 some of the most touching scenes which can come 
 under observation."— itferary Gazette. 
 10 
 
 THE AUTOBIOGRAPHY OF LUT- 
 FULLAH, A MOHAMEDAN GEN- 
 TLEMAN, WITH AN Account op 
 HIS Visit to England. Edited 
 by E. B. Eastwick, Esq. Third 
 Edition, small post 8vo. Price 5s. 
 cloth. 
 
 "Thank you, Munshi Lutfullah Khan ! We 
 have read your book with wonder and delight. 
 Your adventures are more curious than you are 
 aware. , . . But your book is chiefly striking 
 for its genuineness. . . . Th3 story will aid, in 
 its degree, to some sort of understanding of the 
 Indian insurrection. Professor Eastwick has done 
 a grateful service in making known this valuable 
 volume."— Athenceum. 
 
 "Read fifty volumes of travel, and a thousand 
 imitations of the Oriental novel, and you will not 
 get the flavour of Eastern life and thought, or the 
 zest of its romance, so perfectly as in Lutfullah's 
 book. "— Ze« c/er . 
 
 "This is a remarkable book. We have auto- 
 biographies in abundance of Englishmen, French- 
 men, and Germans ; but of Asiatics and Mahome- 
 tans, few or none. ... As the autobiography 
 of a Mahometan mulla, it is in itself singularly 
 interesting. As the observations of an eye- 
 witness of our Indian possessions and our policy 
 and proceedings in the peninsula, it possesses a 
 valueofits own, quite distinct from any European 
 memovinls on the same subjects."— -S^aMf?ara. 
 
 "This is the fresliest and most original work 
 that it has been our good foi-tune to meet with lor 
 long. It Dears every trace of being a most genuine 
 account of the feehngs and doings of the author. 
 The whole tone of the book, the turn of every 
 thought, the association of ideas, the allusions, 
 are all fresh to the English reader; it opens up a 
 new vein, and many will be astonished to find 
 how rich a vein it is. Lutfullah is by no means an 
 ordinary specimen of his Tace."—Econon'i8t. 
 
 "This veritable autobiography, reads like a mix- 
 ture of the Life and Adventure of Gil Bias, with, 
 those of the Three Calendars."— GZofte. 
 
 "As an autobiography, the hook is very curious. 
 It bears the strongest resemblance to Gil Bias of 
 anything we have ever read."— Spectator. 
 
 THE CRISIS IN THE PUNJAB. 
 
 By Frederick H. Cooper, Esq., 
 C. S., Umritsir. Post 8vo, with 
 Map. Price 7s. 6c?. cloth. 
 
 " The book is full of tei-rible interest. Tlie nar- 
 rative is written with vigour and earnestness, 
 and is full of the most tragic interest."— 
 Economist. 
 
 " One of the most interesting and spirited books 
 which have sprung out of the sepoy mutiny."— 
 Globe. 
 
 THE DEFENCE OF LUC KNOW: 
 
 A Staff -Officer's Diary. By 
 Captain Thomas F, Wilson, 13th 
 Bengal N.I., Assistant Adjutant- 
 Gtneral. Sixth Thousand. With 
 plan of the Residency. Small post 
 8vo. Price 2s, 6d. 
 " Unadorned and simple, the story is, neverthe- 
 less, an eloquent one. This is a narrative not to 
 be laid down until the last line has been read." — 
 Leader. 
 
 " The Staff-Officer's Diary is simple and brief, 
 and has a special interest, inasmuch as it gives a 
 fuller account than we nave elsewhere seen of 
 those operations which were the chief human 
 means of salvation to our friends in Lucknow. 
 The Staff- Officer brings home to us, by his details, 
 the nature of that un-ierground contest, upon the 
 result of which the fate of the beleaguered garrison 
 especially depended."— i?a;owj«tcr. 
 
SMITH, EILDEK .^VI>rr> CO. 
 
 NEW WORKS ON INDIA AND THE EAST— 
 
 Continued. 
 
 THE LIFE OF MAHOMET AND 
 HISTORY OF ISLAM TO THE 
 ERA OF THE HEGIRA. By 
 
 William Muir, Esq., Bengal Civil 
 Service. 2 vols., 8vo. Price 32s. 
 cloth. 
 
 "The most perfect life of Mahomet in the 
 "English language, or perhaps in any other. . . . 
 The work is at once learned and interesting, and 
 it cannot fail to be eagerly perused by all persons 
 having any pretensions to historical knowledge." 
 — Observer. 
 
 VIEWS AND OPINIONS OF BRIGA- 
 DIER-GENERAL JACOB, C.B. 
 
 E'lited by Capt. Lewis Pellt. 
 Demy 8vo. Price 12s. cloth. 
 
 "The statesmanlike \'iew8 and broad opinions 
 enunciated in this work would command attention 
 tinder any circumstances, but coming from one of 
 such experience and authority they are doubly 
 valuable, and merit the consideration of legis- 
 lators and politicians."— iSmw. 
 
 " The facts in this book are worth looking at. 
 If the reader desires to take a peep into the inte- 
 rior of the mind of a great man, let him make 
 acquaintance with the * Views and Opinions ol 
 General Jacob.' "—Globe. 
 
 " This is truly a gallant and soldierly book ; very 
 Napierish in its self-confidence, in its capital 
 sense, and in its devotedness to professional 
 honour and the public good. The book should be 
 studied by all who are interested in the choice of 
 a new government for India."— Da i^y News. 
 
 THE PARSEES : their Histoky, 
 Religion, Manneks and Customs. 
 By DosABHOT Framjee. Post 
 8vo. Price 10s. cloth. 
 
 "Otir author's account of the inner life of the 
 Parsees will be read with interest."— Z)ai^y Newn. 
 
 " A very curious and well written book, by a 
 young Parsee, on the manners and customs of 
 his own race."— National Review. 
 
 "An acceptable addition to our literature. It 
 gives information which many will be glad to 
 have carefully gathered together, and formed into 
 ft shapely vrhole."— Economist. 
 
 THE VITAL STATISTICS OF THE 
 EUROPEAN AND NATIVE AR- 
 MIES IN INDIA. By Joseph 
 Ewart, M. D., Bengal Medical 
 Service. Demy 8vo. Price 9s. 
 cloth. 
 
 " A valuable work, in which Dr. Ewart, with 
 equal industry and skill, has compressed the 
 essence and import of an immense mass of de- 
 tails ."—Sp cc^ tor. 
 
 " One main object of this most valuable volume 
 is to point out the causes which render the Indian 
 climate so fatal to European troops."— Critic. 
 
 INDIAN SCENES AND CHARAC- 
 TERS, Sketched from Life. 
 By Prince Alexis Soltykoff. 
 Sixteen Plates in Tinted Litho- 
 graphy, with Descriptions. Edited 
 by E. B. Eastwick, Esq., F.R.S. 
 Colombier folio, half-bound in 
 morocco, prints, 3/. 35. ; proofs 
 (only 50 copies printed), 4/. 4«. 
 
 NARRATIVE OF THE MISSION 
 FROM THE GOVERNOR-GENE- 
 RAL OF INDIA TO THE COURT 
 OF AVA IN 1855. With Notices 
 of the Country, Government, 
 AND People. By Capt. Henry 
 Yule, Bengal Engineers. Imperial 
 8vo, with 24 plates (12 coloured), 
 50 woodcuts, and 4 maps. Ele- 
 gantly bound in cloth, with gilt 
 edges, price 2/. 12s. 6d. 
 
 " A stately volume in ijorgeous golden covers. 
 Such a book is in our times a rarity. Large, 
 massive, and beautiful in itself, it is illustrated 
 by a sprinkling of elegant woodcuts, and by a 
 
 series of admirable tinted lithographs 
 
 We have read it with curiosity and gratification, 
 as a fresh, full, and luminous report upon the 
 condition of one of the most interesting divisions 
 ol Asia beyond the Ganges."— .<liAe/<OBM/». 
 
 " Captain Yule has l)ronght to his narrative a 
 knowledge of many things, which is the main 
 help to observation. He has a taste in archi- 
 tecture, art, and the cognate sciences, as well as 
 much information on the history and religion of 
 the Bui-mese. . . . His description of these 
 things, especially of the antiquities, are not only 
 curious in themselves, but lor the speculations 
 they open up as to origin of the Burmese style, 
 and the splendour of the empire, centuries ago."— 
 Spectator. 
 
 " Captain Yule, in the preparation of the splendid 
 volume before us, has availed himself of the labours 
 of those.who preceded him. To all who are desirous 
 of possessing the best and fullest account that 
 has ever been given to the public, of a great, and 
 hitherto little known rej;ion of the globe, the 
 interesting, conscientious, and well-written work 
 of Captain Yule will have a deep interest, while 
 to the political economist, geographer, and mer- 
 chant it willbe indispensable."— i!<'a;ami«er. 
 
 TIGER SHOOTING IN INDIA. By 
 
 Lieutenant William Rice, 25th 
 Bombay N. I. Super royal 8vo. 
 With 12 plates in chromo-litho- 
 graphy. Price 21*. cloth. 
 
 " These adventures, told in handsome large 
 print, with spirited chromo-lithographs to illus- 
 trate them, make the volume before us as pleasant 
 reading as any record of sporting achievements 
 we have ever taken in hand."— AthencBum. 
 
 "A remarkably pleasant book of adventures 
 during several seasons of ' large game ' hunting 
 in Rajpootana. The twelve chromo-lithographs 
 are very valuable accessories to the narrative; 
 they have wonderful spirit and freshness."— 
 Globe. 
 
 "A good volume of wild sport, abounding in. 
 adventure, and handsomely illustrated with, 
 coloured plates from spirited designs by the 
 author."— Examiner, 
 
 THE COMMERCE OF INDIA WITH 
 EUROPE, AND ITS POLITICAL 
 EFFECTS. By B. A. Irving, 
 Esq. Post 8vo. Price 7*. 6c?. cloth. 
 
 " Mr. Irving's work is that of a man thoroughly- 
 versed in his subject. It is a historical hana- 
 book of the progress and vicissitudes of European 
 trade with India.."— Economist. 
 
 11 
 
"WOKKS rUBLISHEr) BY 
 
 WORKS ON INDIA AND THE EAST. 
 
 THE ENGLISH IN WESTERN INDIA: 
 
 BEING THE EaRLY HiSTORY OF THE 
 
 Factory at Surat, of Bombay. 
 By Philip Anderson, A.M. 2ad 
 edition, 8vo, price 145. cloth. 
 
 ' Quaint, curious, and amusing, this volume 
 f'eseribes, fi-om old manuscripts and obscure 
 books, the life of English mercliants in an Indian 
 factory. It contains fresli and amusing gossip, 
 all bearing on events and characters of historical 
 ixavortiinoG."—AtAe7iat(,m. 
 
 ''a book of permanent vaAne."— Guardian. 
 
 LIFE IN ANCIENT INDIA. By Mrs. 
 Speir. With Sixty Illustrations 
 by G. Scharf. 8vo, price I5s., 
 elegantly bound in cloth, gilt edges. 
 
 "AVTioever desires to have the best, the com- 
 pletest, and the most popular view of what 
 Oriental scholars have made known to ug respect- 
 ing Ancient India must peruse the work of Mrs. 
 Sjieir; in which he wil' find the story told in 
 clear, correct, and unaffected English. The book 
 is admirably got np."—Hxaminer. 
 
 THE CAUVERY, KISTNAH, AND 
 GODAVERY: being a Keport 
 ON THE Works constructed on 
 THOSE Rivers, for the Irrigation 
 of Provinces in the Presidency 
 of Madras. By U. Baird Smith, 
 F.G.S., Lt.-Col. Bengal Engineers, 
 &c., &c. In demy 8vo, with 19 
 Plans, price 285. cloth. 
 
 "A most curious and interesting work."— 
 Economist. 
 
 THE BHILSA TOPES ; or, Buddhist 
 Monuments of Central India. 
 By Major Cunningham. One vol., 
 Svo, with Thirty-three Plates, 
 price 305. cloth. 
 
 "Of the Topes opened in various parts of India 
 none have yielded so rich a harvest olT important 
 infonnation as those of Bhilsa, opened by Major 
 Cunningham and Lieut. Maisey; and wliicli are 
 described, with an abundance of highly curious 
 
 fraphic illustrations, in this most interesting 
 ook."— Examiner. 
 
 THE CHINESE AND THEIR REBEL- 
 LIONS. By Thomas Taylor 
 Meadows. One thick volume, Svo, 
 with Maps, price IBs. cloth. 
 "Mr. Meadows' book is the work of a learned, 
 conscientious, and observant person, and reaUy 
 important in many respects."— Tiwea. 
 
 "^Mr. Meadows has produced a work which 
 Reserves to be studied by all who would gain a true 
 appreciation of Cliinese character. Information 
 IS sown broad-cast through every page."— 
 AthencEum. 
 
 ADDISON'S TRAITS AND STORIES 
 OF ANGLO-INDIAN LIFE. With 
 Eight Illustrations, price 55. cloth. 
 "An entertaining and instructive volume of 
 Indian anecdotes."— 3f if// ar^ Spectator. 
 
 "Anecdotes and stories well calculated to 
 illustrate Anglo Indian life and the domestic 
 manners and nabitsof Hindostan "— Observer, 
 
 " A pleasant collection of amusing anecdotes." 
 - Critic. 
 
 12 
 
 TRACTS ON THE NATIVE ARMY 
 OF INDIA. By Brigadier- General 
 Jacob, C.B. Svo, price 25. 6</. 
 
 ROYLE ON THE CULTURE AND 
 COMMERCE OF COTTON IN 
 INDIA. Svo, price I85. cloth. 
 
 ROYLE'S FIBROUS PLANTS OF 
 
 INDIA FITTED FOR CORDAGE, 
 
 Clothing, and Paper. Svo, price 
 125. cloth. 
 
 ROYLE'S PRODUCTIVE RE- 
 SOURCES OF INDIA. Super 
 royal Svo, price 145. cloth. 
 
 ROYLE'S REVIEW OF THE MEA- 
 SURES ADOPTED IN INDIA FOR 
 THE IMPROVED CULTURE OF 
 COTTON. Svo, 25. 6d cloth. 
 
 A SKETCH OF ASSAM: 
 
 with some Account of the Hill 
 Tribes. Coloured Plates, Svo, 
 price 145. cloth. 
 
 BUTLER'S TRAVELS AND ADVEN- 
 TURES IN ASSAM. One voL Svo, 
 with Piates, price 125. cloth. 
 
 DR. WILSON ON INFANTICIDE IN 
 WESTERN INDIA. Demy Svo, 
 price 125. 
 
 WARING ON ABSCESS IN THE 
 LIVER. Svo, price 35. 6cf. 
 
 LAURIE'S SECOND BURMESE 
 WAR — RANGOON. Post Svo. 
 with Plates, price 25. 6c?. cloth. 
 
 LAURIE'S PEGU. Post Svo, price 
 145. cloth. 
 
 IRVING'S THEORY AND PRACTICE 
 OF CASTE. Svo, price 55. cloth, 
 
 THE BOMBAY QUARTERLY 
 REVIEW. Nos. 1 to 9 at 55., 10 to 
 1 4, price 65. each. 
 
 BAILLIE'S LAND TAX OF INDIA. 
 
 According to the Moohummudan 
 Law. Svo, price 65. cloth. 
 
 BAILLIE'S MOOHUMMUDAN LAW 
 OF SALE. Svo, price 145. cloth. 
 
 BAILLIE'S MOOHUMMUDAN LAW 
 OF INHERITANCE. Svo, price 
 95. cloth. 
 
SIMITJEC, ELDEJR J^H^TJO^ CO. 
 
 MISCELLANEOUS, 
 
 ANNALS OF BRITISH LEGIS- 
 LATION, A Classified Summary 
 OF Parliamentary Papers. El. 
 hy Professor Leone Levi. The 
 yearly issue consists of 1,000 pages, 
 super royal 8vo, and the Subscrip- 
 tion is Two Guineas, payable in 
 advance. The Thirty-fourth Part 
 is just issued, commencing the 
 Third Year's Issue. Volumes L to 
 IV. may be had, price 4/. As. cloth. 
 
 "A series that will, if it be always managed as 
 it now is hy Profes-<or Levi, last as lon^ as tiero 
 remains a Legislature in Great Britain. These 
 Annals arc to give the essence of work done and 
 information garnered for the State during each 
 legislative year, a summary description ot every 
 Act passed, a digest of the vital fsicts contained 
 in every Blue Book issued, and of all documents 
 relating to the public business of the country. 
 Tlie series will live, while generations of men die, 
 if it be maintained in its old age as ably and as 
 conscientiously as it is now In its youth." — 
 Examiner. 
 
 "The idea was admirable, nor does Ihe execu- 
 tion fall short of the plan. To accomplish this 
 effectively, and at the same time briefly, was not 
 an easy task ; but Professor Levi has undertaken 
 it with great success. The work is essentially a 
 jfuide. It will satisfy those persons who refer to 
 it merely for general purposes, w hile it will direct 
 the research of others wnoso investigations take 
 a wider iti,nse."—Athen(eum. 
 
 CAPTIVITY OF RUSSIAN 
 PRINCESSES IN SHAMIL'S 
 SERAGLIO. Translated from the 
 Russian, by H S. Edwards. With 
 an authentic Portrait of Shamil, a 
 Plan of his House, and a Map. Post 
 8vo, price 10s. 6c/. cloth. 
 
 " a book than which there are few novels more 
 interesting. It is a romance of the Caucasus. 
 The account of life in the hou«e of Shamil is full 
 and very entertaining; and of Shamil himself we 
 see raueh."— Examiner. 
 
 "The story is certainly one of the most curious 
 we have read; it contains the best popular notice 
 of the social polity of Shamil and the manners of 
 his people."— Leader. 
 
 "The narrative is well worth reading."— 
 Atbenceum, 
 
 SHARPE'S HISTORIC NOTES ON 
 THE OLD AND NEW TESTA- 
 MENT. Third and Eevised Edition. 
 Post 8vo, price 7s. cloth. 
 
 •• An inestimable aid to the clergyman, reader, 
 city-rai?5sionary, and Su"da.v-school teacher." 
 —Illustrated News of tfie World. 
 
 " A learned and sensible hoo):.."— National Be- 
 view. 
 
 ELLIS'S (WILLIAM; RELIGION IN 
 COMMON LIFE. Post 8vo, price 
 7s. 6d. cloth. 
 
 " A. book addressed to young people of the 
 tipper ten thousand upon social duties."— 
 J:,xaminer. 
 
 "Lessons in Political Economy for young people 
 by a skilful hund."— Economist. 
 
 THE OXFORD MUSEUM. By 
 
 Henry W. Aclaxd, M.D., and 
 John Ruskin, A.M. Post Svo, 
 with three Illustrations. Price 
 2s. 6d. cloth. 
 
 I " Everyone who cares for the advance of true 
 ; learning, and desires to note an onward step, 
 I should buy and read this little \olume."—Jfoj'M- 
 I ina Herald. 
 
 " There is as much significance in the occasion 
 
 of this little vol ume as interest in the book itselt.'* 
 
 —Spectator. 
 
 THE ENDOWED SCHOOLS OF 
 
 IRELAND. By Harriet Mak- 
 
 TixEAU. Svo. Price 3s. 6t/., cloth 
 
 boards. 
 
 "The friends of education will do well to pos- 
 sess themselves of this \)OQ]i.."— Spectator. 
 
 PARISH'S (CAPT. A.) SEA 
 OFFICER'S MANUAL. Second 
 Edition, Small Post Svo, price 5s. 
 cloth. 
 
 "A very lucid and compendious manual. "We 
 Duld recommend youths mt 
 life to study iW-Athenceum. 
 
 would recommend youths intent upon a seaifarins 
 
 "A little book that ought to be in great request 
 among young seamen."— JiTxami/i^'. 
 
 ANTIQUITIES OF KERTCH, 
 
 AND Researches in the Cim- 
 jfERiAN BospnoRus. By Duncan 
 McPherson, M.D., of the Madras 
 Army, F.RG.S., M.A.L Imp. 4to, 
 with Fourteen Plates and numerous 
 Illustrations, including Eight 
 Coloured Fac-Similes of Relics of 
 Antique Art, price Two Guineas. 
 
 "It is a volume which deserves the careful 
 attention of every student of classical antitiuity. 
 No one can fail to be pleased with a work whiCii 
 has so much to attract the eye and to gratify the 
 
 love of beauty and elegance in design 
 
 The book is got up wiih great care and taste, 
 and forms one of the handsomest works that have 
 recently issued from the English press."— 
 Saturday Review. 
 
 WESTGARTH'S VICTORIA, 
 
 AND THE Australian Gold Mines 
 IN 1S57. Post Svo, with Maps, price 
 10s. &d. cloth. 
 
 "Mr. Westgarth has produced a reliable and 
 readable book well stocked with iulormation, and 
 pleasantly interspersed with incidents of travel 
 and views of colonial life. It is clear, sensible, 
 and suggestive."— j4<A€«««ot. 
 
 " A lively account of the most wonderful bit of 
 colonial experience that the world's history has 
 furnished."— Examiner. 
 
 "We think Mr. Westgarth's book much the 
 best which has appeared on Australia since the 
 great crisis in its hi^Xory."— Saturday Review. 
 
 " A rational, vigorous, illustrative report upon 
 the progress of the greatest colony iu Australia." 
 —Leader. 
 
 "The volume contains a large amount of 
 statistical and practical information relating to 
 \ictoria.."— Spectator. 
 
 n 
 
AX-OItKlS I>TJBI.ISIIEX) BY 
 
 MISCELLANEOUS— con^mwed'. 
 
 TAULER'S LIFE AND SERMONS. 
 
 Translated by Miss Susanna Wink- 
 woRrH. With a Preface by the 
 Rev. Charles Kingsley, Small 
 4to, printed on Tinted Paper, and 
 bound in Antique Style, with red 
 edges, suitable for a Present. 
 Price 75. 6c?. 
 " Miss Winkworth has done a service, not only 
 to church histoi-y and to literature, hut to those 
 who seek simple and true-hearted devotional 
 reading, or who desire to kindle their own piety 
 through the example of saintly men, hy producing 
 a very instructive, complete, and deeply interest- 
 ing life of Tauler,and by giving to us also a sample 
 of Tauler's sermons tastefully and vigorously 
 translated."— Gwardiaw. 
 
 " No difference of opinion can be felt as to the 
 intrinsic value of these sermons, or the general 
 interest attaching to this book. The Sermons 
 are weU selected, and the translation excellent." 
 —AthenoBum, 
 
 CHANDLESS'S VISIT TO SALT 
 LAKE : BEING A Journey across 
 THE Plains to the Mormon 
 Settlements at Utah. PostSvo, 
 with a Map, price 2s. 6rf. cloth. 
 
 " Mr. Chandless is an impartial observer of the 
 Mormons. He gives a full account of the nature 
 of the country, the religion of the Mormons, their 
 government, institutions, morality, and the singu- 
 lar relationship of the sexes, with its conse- 
 quences."— Crirtc. 
 
 "Those who would understand what Mor- 
 monism is can do no better than read this 
 authentic, though light and lively volume."— 
 Leader. 
 
 " It impresses the reader as t&HYdMl.'"— National 
 Review. 
 
 DOUBLEDAY'S LIFE OF SIR 
 ROBERT PEEL. Two volumes, 
 8vo, price 18s. cloth. 
 
 " It is a good book of its kind. . . . It is well 
 worth reading, and very pleasantly and sensibly 
 written."— Saturday Review. 
 
 "This biography is a workof great merit, con- 
 scientiously prepared, plain, clear, and practically 
 interesting ."—Xeader. 
 
 " It is a production of great merit, and we hail 
 it as a most valuable contribution to economical 
 and statistical science."— £WtisA Quarterly. 
 
 CAYLEY'S EUROPEAN REVOLU- 
 TIONS OF 1848, Crown Svo, 
 price 6s. cloth. 
 
 " Mr. Cay ley has evidently studied his subject 
 thoroughly, he has consequently produced an 
 interesting and philoso_phical, though unpretend- 
 ing history of an important epoch." — JVcjc 
 Q,uarterly. 
 
 " Two instructive volumes."— 06aen;er. 
 
 BUNSEN'S (CHEVALIER) SIGNS 
 OF THE TIMES ; or. The Dan- 
 gers TO Religious Liberty in 
 THE Present Day. Translated by 
 Miss Susanna Winkvtorth. One 
 volume, Svo, price 5s. cloth. 
 
 " Dr. Bunsen is doing good service, not only to 
 his country but to Christendom, by sounding an 
 alarm touching the dangers to religious liberty in 
 the present state of the world."— £ri<i8A (Quar- 
 terly, 
 
 14 
 
 THE COURT OF HENRY VIIL : 
 
 BEING A Selection of the 
 Despatches op Sebastian Gius- 
 tinian, Venetian Ambassador, 
 1515-1519. Translated by Raw- 
 don Brown. Two vols., crown Svo, 
 price 21s. cloth. 
 
 " It is seldom that a page of genuine old history 
 is reproduced for us with as much evidence of 
 painstaking and real love of the subject as in the 
 selection of despatches made and edited by Mr. 
 Rawdon Brown."— 2'me«. 
 
 "Very interesting and suggestive volumes."— 
 British Quarterly Review. 
 
 "Most ably edited."— J'raser's Magazine. 
 
 PAYN'S STORIES AND SKETCHES. 
 
 Post Svo, price 2s. 6rf. cloth. 
 
 "A volume of pleasant reading. Some of the 
 papers have true Attic salt in them."— Literary 
 Gazette. 
 
 "Mr. Payn is gay, spirited, observant, and shows 
 no little knowledge of men and books."— deader. 
 
 "A most amusing volume, full of humorous 
 adventure and pleasant satire,"— Press. 
 
 STONEY'S RESIDENCE IN TAS- 
 MANIA. Demy Svo, with Plates, 
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 18 
 
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 handles."— -E"c/^citc Review. 
 
 " It matters little as to the machinery with 
 which a writer works out his purpose, provided 
 that purpose be laudable and the execution of the 
 work good. Both conditions are perfectly fulfilled 
 in the work before us ; the sentiment is pure and 
 true, the moral excellent, and the style incompa- 
 rably hee^ntiTvLl."— Illustrated News of the World. 
 
 " We cannot recommend to our readers a plea- 
 santer book for an evenings instruction and 
 amusemeut."— £ady'« Newspaper. 
 
 AFTER DARK. By Wilkie Collins. 
 Price 2s. 6rf. cloth. 
 
 " Mr. Wilkie Collins stands in the foremost rank 
 of our younger writers of fiction. He tells a 
 story well and forcibly, his style is eloquent and 
 picturesque; he has considerable powers of pa- 
 thos; understands the art of construction: la 
 never wearisome or wor ^y. and has a keen insight 
 into character."— Z»ai/y iVctc«. 
 
 "Stories of adventure, we'l varied, and often 
 striking in the incidents, or with thrilling situa- 
 tions. They are about as pleasant reading as a 
 novel reader could AeisxTe."— Spectator. 
 
 " Mr. Wilkie Collins has been happy in the choice 
 of a thread whereon to string the pearls ; we read 
 it almost as eagerly as the stories themselves. 
 Mr. Collins possesses a rare faculty I'art de 
 conter. No man living better tells a story."— 
 Lea er. 
 
 "Mr. Wilkie Collins takes high rank among 
 the who can invent a thrilling story and tell it 
 with brief simplicity. The power of commanding 
 the faculties of the reader is exercised in nearly 
 all these stories."— Gtofte. 
 
 "Their great merit consists either in the effec- 
 tive presentation of a mystery, or the effective 
 working up of striking situaXions."— Westminster 
 Review. 
 
 " ' After Dark ' abounds with genuine touches 
 of naXuvG."— British Quarterly. 
 
 "These stories possess all the author's well- 
 known beauty of style and dramatic power."— 
 New Quarterly. 
 
 PAUL FERROLL. Fourth edition, 
 price 2s. cloth. 
 
 " We have seldom read so wonderful a romance. 
 We can find no fault in it as a work of art. It 
 leaves us in admiration, almost in awe, of the 
 power* of its author."— iV^ew? Quarterly. 
 
 " The art displayed in presenting Paul Ferroll 
 throughout the story is beyond all praise."— 
 Exrjminer. 
 
 "The Incidents of the book are extremely well 
 mn.ntLged."—AtheniEum. 
 
 " The fruit of much thoughtful investigation is 
 represented to us in the character of Paul 
 
 Ferroll We do not need to he told how 
 
 he felt and why he acted thus and thus ; it wiU 
 be obvious to most minds from the very opening 
 pages. But the power of the story is not weak- 
 ened by this early knowledge : rather is it 
 heightened, since the artistic force of contrast is 
 grand and fearful in the two figures who cling so 
 closely together in their fond human love."— 
 Morning Chronicle. 
 
 SCHOOL FOR FATHERS. 
 
 By Talbot Gwynne. Price 2s. el. 
 
 " 'The School for Fathers ' is one of the cleverest, 
 most brilliant, genial, and instructive stories that 
 we have read since the publication of 'Jane 
 Eyre.' "—Eclectic Review. 
 
 " The pleasantest tale we have read for many a 
 day. It is a story of the Tatler and Spectator 
 days, and is very fitly associated with that time 
 of good English literature by its manly feeling, 
 direct, unaffected manner of writing, and nicely- 
 managed, well-turned narrative. The des. riptions 
 are excellent; some of the country painting is as 
 fresh as a landscape by Alfred Constable, or an 
 idyl by l&rmYson."— Examiner. 
 
 ''A capital picture of town and country a 
 century ago; and is emphatically the ft-eshest, 
 raciest, and most artistic piece of fiction that has 
 lately come in our yray ."—Nonconformist. 
 
 PREPARING FOR PUBLICATION. 
 
 THE TENANT OF WILDFELL HALL. By Acton Bell. {Now ready.) 
 
 KATHIE BRANDE : the Fireside History of a Quiet Life. By Holme 
 Lee, Author of " Sylvan Holt's Daughter." 
 
 BELOW THE SURFACE. By Sir Arthur Hallam Elton, Bart., M.P. 
 
 19 
 
"W OEKIS I»XJBI.ISIIED BY 
 
 NEW NOVELS. 
 
 (to be had at 
 
 AGAINST WIND AND TIDE. By 
 Holme Lee, Author of "Sylvan 
 Holt's Daughter." (Now ready.) 
 
 EXTREMES. By Miss E. W. Atkin- 
 son, Author of '* Memoirs of the 
 Queens of Prussia." 2 vols. 
 
 "A nervous and vigorous style, an elahorate 
 deliuealion of character under many varieties, 
 spirited and well-sustained dialogue, and a care- 
 luUy-consiructed plot; if these have any charms 
 for our readers, they will not forget the swiftly 
 gliding hours passed in perusing ' Extremts.' "— 
 Morning Fost. 
 
 " We have no hesitation in placing this hook 
 high ahove the ephemeral sto-ies with which 
 from time to time the circulating libraries are 
 inur.dateil. The story is not so intense as that of 
 'Jane Eyre,' nor are the characters so pro- 
 nounced as those in ' Adam Bede,' and yet we 
 think 'Extremes' will bear comparison with 
 either of the two. There is throughout the whole 
 story the trace of great power and delicate 
 l)erception of minute shades of character, which 
 
 filace Miss Atkinson high in the ranks of con- 
 emporary novelists."— iadies' Newspaper. 
 "'Extremes 'is a novel written with a soher 
 purpose, and wound up with a moral. The 
 purpose is to exemplify some of the errors arising 
 from mistaken zeal in religious matters, and the 
 evil consequences that flow from those errors."— 
 Spectator. 
 
 " The machinery of the plot is well imagined 
 and well worked out, and, we need scarcely add, 
 well calcul ted to afford gratiti cation to the 
 reader."— Press. 
 
 THE TWO HOMES. By the Author 
 of " The Heir of Vallis." 3 vols. 
 
 " There is a great deal that is very good in this 
 book— a great deal of good feeling and excellent 
 design. . . . There are some good pictures of 
 Madeira, and of life and society tliere: and there 
 ai e evidences of much painstaking and talent."— 
 Athenceum. 
 
 " ' Ihe Two Homes ' is a very clever novel. . . 
 Madeira furnishes Mr. Mathews with a fertile 
 theme for his dc-criptive powers. The dialogue 
 is good: the characters all speak and act con- 
 sistently with their natures."— iearfcr. 
 
 "'The Two Homes' is a novel of more than 
 ordinary merit, and is written throughout in a 
 careful and elegant style."— Morning Post. 
 
 THE DENNES OF DAUNDELYONN. 
 
 By Mrs. Charles J. Peoby. 3 vols. 
 
 "This is a novel of more than average merit. 
 There is considerahle knowledge of character, 
 power of description, and quiet social satire, ex- 
 hibited in its pages." — Press. 
 
 " ' The Dennes of Daundelyonn ' is a very read- 
 able book, and will be immensely popular. . . . 
 It has many beauties which deservedly recom- 
 mend ii to the novel reader."— C»-jiic. 
 
 '• ' The Dennes of Daundelyonn ' is a hook writ- 
 ten with great vigour and freshness."— ieader. 
 
 " There is mr re cleverness and variety in these 
 volumes than in twenty average novels.'^'— G^ he. 
 
 COUSIN STELLA J or. Conflict. 
 By the Author of " Violet Bank." 
 3 vols. 
 
 "An excellent novel, written with great care; 
 the interest is well sustained to the end, and the 
 characiers are all life-like. It is an extremely 
 well-written and well-conceived story, with quiet 
 power and precision of touch, with freshness of 
 interest and great laerit."— Athenceum. 
 
 "* Cousin Stella* has the merit, now becoming 
 rarer and rarer, of a comparative novelty in its 
 subject; the interest of which will secure for this 
 novel a fair share of V^v^laxxty ."— Saturday 
 Review. 
 
 20 
 
 ALL LIBRARIES.) 
 
 CONFIDENCES. By the Author of 
 " Rita." 
 
 "Decidedly both good and interesting. The 
 hook has a fresh and p'easant air about it : it is 
 written in an excellent tone, and there are touches 
 of pathos here and there which we must rank 
 M'ith a higher style of composition than that 
 usually attainei in works of this class."— JY^ew 
 Quarterly Revieic. 
 
 "This new novel, by the author of 'Rita,' dis- 
 plays the same combination of ease and power in 
 the delineation of character, the same life-like 
 dialogue, and the same faculty of constructing an 
 interesting story."— Spectator. 
 
 "•Confidences' is written in the most pleasing 
 manner of any novel we have read for years 
 past."— Leader. 
 
 "A clever book, and not too long."— Examiner. 
 
 TRUST FOR TRUST. By 
 
 A. J. Barrowcliffe, Author of 
 " Amberhill." 3 vols. 
 
 " The story is adrrirably developed. Theinterest 
 never flags, the incidents are natural without 
 being commonplace, and the men and woman talk 
 and act like human beings."— Press. 
 
 " It is seldom we find, even in this great age of 
 novel writing, so much that is pie isant and so 
 little to object to ns in ' Trust for Trust.' It con- 
 tains much original thought and fresh humour." 
 —Leader. 
 
 " The story evinces vigour of description and 
 power of writing."— ii^erary Churchman. 
 
 ELLEN RAYMOND; or, Ups and 
 Downs. By Mrs. Vidal, Author 
 of ''Tales for the Bush," &c. 
 3 vols. 
 
 " The plot is wrought out wi'h wonderful inge- 
 nuity, and the dillerent characters are sustained 
 in perfect keeping to the cnA."— Illustrated News 
 of the World. 
 
 " The characters are good, the style pure, cor- 
 rect, brisk, ana easy."— Press. 
 
 "Mrs. Vidal displays resource, imagination, 
 and power in no common degree. * * • There is 
 more power and strength put forth in ' KUen 
 Raymond' than perhaps in any lady's book of 
 this generation."— ^'atwrdaj- Review. 
 
 "This novel will find a great many admirers." 
 —Leader. 
 
 LOST AND WON. By Georgiana 
 M. Craik, Author of "Riverston." 
 1 vol. 2nd Edition, 
 
 "Nothing superior to this nrvel has appeared 
 during the prestnt se son."— Leader. 
 
 " Miss Craik's new story is a good one and in 
 point 01 ability above the average of ladies' novels." 
 —Daily News. 
 
 " The language is good, the narrative spirited, 
 the characters are lairly delineated, and the 
 dialogue has considerable dramatic force."— 
 Saturday Review. 
 
 " This is an improvement on Miss Craik's first 
 work. The story is more compact and more 
 interesting."— ui thenceum. 
 
 THE MOORS AND THE FENS. 
 
 By F. G. Tr AFFORD. 3 vols. 
 
 " This novel stands out much in the same way 
 that 'Jane Eyre' did. . . . The characters are 
 drawn by a mind which can realize fictitious 
 characters with minute intensity."— iatwrduy 
 Review. 
 
 "It is seldom that a first fiction is entitled to 
 such applause as is 'The Moors and the Fens,' 
 and we shall look anxiously for the writer's next 
 essay."- Critic. 
 
 " The author has the gift of telling a story, and 
 'The Moors and the Fens* will be read." — 
 Athenaum. 
 
felktlTH, EI.X>ER JSJNTf CO. 
 
 NEW '^OYELS— continued. 
 
 AN OLD DEBT. By Floeence 
 Dawson. 2 vols. 
 
 " A powerfully written novel ; one of the best 
 which has recently proceeded from a female 
 hand. . , . The dialogue is vigorous and 
 spirited."— J/orwJHC Post. 
 
 "There is an energy and vitality about this 
 work which distinguish it from the common 
 head of novels. Its terse vigour sometimes recals 
 Miss Bronte, hut in some respects Miss Florence 
 Dawson is decidedly superior to the author oi 
 ■ Jane Eyre.' "Saturday Review. 
 
 "Ttiis novel is written wiih great care and 
 painstaking; it evinces considerable powers of 
 reflection. The style is good, and the author 
 passesses the power of depicting emotion."— 
 AtbencBum. 
 
 "A very good seasonable noyaV— Leader. 
 
 SYLVAN HOLT'S DAUGHTER. 
 
 By Holme Lee, Author of " Kathie 
 Brande," &c. 2nd edition. 3 vols. 
 
 "The well-established reputation of Holme 
 Lee, as a novel writer, will receive an additional 
 elory from the publication of 'Sylvan Holt's 
 Daughter.' It is a charming tale of country life 
 and character."— G/o/jc. 
 
 " There is much tliat is attractive in ' Sylvan 
 Holt's Daughter,' much that is graceful and re- 
 tined, much that is fresh, healthy, and natural." 
 —Press. 
 
 "The conception of the story Ins a good deal of 
 originality, and the characters avoid common- 
 place types, without being unnatural or improba- 
 ble. The heroine herself is charming. It is a 
 novel in which there is n uch to interest and 
 please."— A'ew QuurteHy Review. 
 
 "A novel that is well worth reading, and which 
 possesses the cardinal virtue of being extremely 
 iLteresting."— A^AencEwm. 
 
 "A really sound, good book, highly finished, 
 true to nature, vigorous, passionate, honest, and 
 sincere."— DmSWh University Magazine. 
 
 MY LADY : a Tale of Modern 
 Life. 2 vols. 
 
 "'My Lady' is a fine specimen of an English 
 matron, exhibiting that union of strength and 
 gentleness, of common sense and romance, of 
 eiiergy and grace, which nearly approaches our 
 ideal of womanhood."— Prfsa. 
 
 " ' .My Lady' evinces charming feeling and deli- 
 cacy of touch. It is a novel that will be read with 
 interest."— A<AeM<Eur». 
 
 "The story is told throughout with great 
 strength of feeling, is well written, and has a 
 plot which is by no meacs common-place."— 
 Examiver. 
 
 " There is some force and a eood deal of fresh- 
 ness in ' My Lady.* The characters are distinctly 
 drawn, and often wear an appearance of indi- 
 viduality, or almost personality. The execution 
 is fresh and powerful."— >j)eef"<'>r. 
 
 "A tale of some xio^vev."— National Review. 
 
 "It is not in every novel we can light upon a 
 style so vigorously graceful— upon an intelligence 
 80 refined without littleness, so tenderly truthful, 
 which has sensibility rather thm poetry; but 
 which is also most subtly and searchingly power- 
 ful."— /^ttft/iw University Magazine. 
 
 "Care has been bestowed on the writing, which 
 is pleasant and flowing. The descriptions of nature 
 ai'e tru hful ar.d delicately 6xdkV,'a."— Economist. 
 
 GASTON BLIGH. By L. S. Lavenu, 
 Author of " Erlesmere." 2 vols. 
 
 " ' Gaston Bligh ' is a good story, admirably 
 told, full of stirring incident, sustaining to the 
 Close the interest c.f a very ingenious plot, and 
 abounding in clever sketches of character. It 
 sparkles with wit, and will reward perusaL"— 
 Critic. 
 
 "Thestoryis told with great power; the whole 
 book sparkles with esprit ; and the characters 
 talk like gentlemen and ladies. It is very enjoy- 
 able reading."— Press. 
 
 THE PROFESSOR. By Cukeer 
 Bell. 2 vols. 
 
 "We think the author's friends have shown 
 sound judgment in publishing the ' Professor,' 
 now that she is gone. ... It shows the first 
 germs of conception, which afterwards expanded 
 and ripened into the great creations of her imagi- 
 nation. At the same time her advisers were 
 equally right when they counselled her not to 
 publish it in her lifetime. . . . But it abounds 
 in merits."— Su^wrday Revietv. 
 
 " The idea is original, and we every here and 
 there detect germs of that power which took the 
 world by storm in ' Jane Eyre.' The rejection of 
 the 'Professor' was, in our opinion, no less ad- 
 vantageous totheyoungauthoress than crcditabld 
 to the discernment of the booksellers."— Prea«. 
 
 "Anything which throws light upon the growth 
 and composition of such a mind cannot bo other- 
 wise than interesting. In the ' Professor ' we may 
 discover the germs of many ti"ains of thinking, 
 which afterwards came to be enlarged and 
 illustrated in subsequent and more perfect 
 works."— Criiio. 
 
 "There is much new insight in it, mr.ch ex- 
 tremely characteristic genius, and one character, 
 moreover, of fresher, lighter, and more airy 
 grace."— Economist. 
 
 " We have read it vrith the deepest interest ; 
 and confidently predict that this legacy of Char- 
 lotte Brontg's genius will renew and confirm the 
 general admii-ation of her extraordinary powers." 
 —Eclectic. 
 
 BELOW THE SURFACE. 3 vols. 
 
 "The book is unquestionably clever and enter- 
 taining. The writer develops from first to last 
 his double view of human life, as coloured by the 
 manners of our age. ... It is a tale superior 
 to ordinary novels, in its practical application to 
 the phases of actual Mte."— Athenaeum. 
 
 " There is a great deal of cleverness in this story ; 
 a much greater knowledge of country life and 
 character in its various aspects and conditions 
 than is possessed by nine-tenths of the novelists 
 who undertake to describe it."—Si,ectator. 
 
 "The novel is one that keeps the attention fixed, 
 and it is written in a genial, often playful tone. 
 The temper is throughout excellent."— £xajni«e»*. 
 
 "This IS a book which possesses the rare merit 
 of being exactly what it claims to be, a story of 
 English country life ; and, moreover, a very well 
 told story."— Daily News. 
 
 " 'Below the Surface' merits liigh praise. It is 
 full of good things; good taste — good feeling- 
 good writing— good notions, and high morality." 
 — Globe. 
 
 " Temperate, sensible, kindly, and pleasant."— 
 Saturday Review. 
 
 "A more pleasant story we have not read for 
 many a day."— British Quarterly, 
 
 CHANCES. 
 
 of "The Fair 
 
 THE THREE 
 
 By the Author 
 Carew." 3 vols. 
 
 " This novel is of a more solid texture than 
 most of its contemporaries. It is full of good 
 sense, good thought, and good writing."— /S^afe*- 
 man. 
 
 " Some of the characters and romantic situa- 
 tions are strongly marked and peculiarly original. 
 . . . It is the great meiit of the authoress that 
 the personages of her tale aie human and real."— 
 Leader, 
 
 THE CRUELEST WRONG OF ALL- 
 
 By the Author of " Margaret ; or, 
 Prejudice at Home." 1 vol. 
 
 " The author has a pathetic vein, and there is a 
 tender sweetness in the tone of her narration."— 
 Leader. 
 
 " It has the first requisite of a work meant to 
 amuse : it is amusing,"— G/oOe. 
 
 21 
 
"WOi&KlS I^XJJBILISPIED BY 
 
 NEW l^OYELS^continued. 
 
 KATHIE BRANDE : a Fireside His- 
 tory OP A Quiet Life. By 
 Holme Lee. 2 vols. 
 
 " • Kathie Brande ' is not merely a very interest- 
 ing novel— it is a very wholesome one, for it 
 teach 98 virtue by example."— Critic. 
 
 "Throughout 'Kathie Brande' there is much 
 sweetness, and considerable power of description." 
 —Saturday Review. 
 
 " 'Kathie Brande' is intended to illustrate the 
 J aramount excellence of duty as a movin.^ prin- 
 ciple. It is full of beauties."— Dai^y News. 
 
 '' Certainly one of the best novels that we have 
 lately tg&A." —Guardian. 
 
 EVA DESMOND ; or, Mutation. 
 3 vols. 
 
 "A more beautiful creation than Eva it would 
 be difficult to imagine. The novel is undoubtedly 
 full of interest."— jWorniwe' Post. 
 ' " There is power, pathos, and originality in con- 
 ception and catastrophe."- ieader. 
 
 THE NOBLE TRAYTOUR. 
 
 A Chronicle. 3 vols. 
 
 " An Elizabethan masquerade. Shakespeare, 
 the Queen, Essex, Raleigh, and a hundred nobles, 
 ladies, and knights of the land, appear ou the 
 stage. The author has imbued himself with the 
 spirit of the times."— Leader. 
 
 " The story is told with a graphic and graceful 
 pen, and the chronicler has produced a romance 
 not only of great value in a historical point of 
 view, but possessing many claims upon the atten- 
 tion of the scholar, the antiquary, and the general 
 reader,"— Posi. 
 
 PERVERSION ; or, The Causes and 
 Consequences op Infidelity. By 
 the late Rev. W. J. Conybeare. 
 3 vols. 
 
 'This story has a touching interest, which 
 
 booh 
 
 -A thencEum. 
 
 lingers with the reader after ne has closed the 
 
 OK."—. 
 
 The 1 
 ling f 
 
 ardia 
 
 '■ It is long, very long, since we have road a 
 
 3f ~ ■ ■ ' 
 
 ' The tone is good and healthy ; the religious 
 feeling sound and true, and well sustained."— 
 Quardian. 
 
 narrative of more power than this."— British 
 Quarterly Review. 
 
 "This is a good and a noble book."— iVew 
 Quarterly. 
 
 THE WHITE HOUSE BY THE SEA : 
 
 A Love Story. By M. Betham- 
 Edwards. 2 vols. 
 
 "A tale of English domestic life. The writing is 
 very good, graceful, and unaffected ; it pleases 
 without startling. In the dialogue, people do not 
 harangue, but talk, and talk naturally.''- Cririe. 
 
 " Tlie narrative and scenes exhibit feminine 
 spirit and quiet truth of delineatiou."—5'i)eetator. 
 
 MAUD SKILLICORNE'S PENANCE. 
 
 By Mary C. Jackson, Author of 
 "The Story of My Wardship." 
 2 vols. 
 
 " The style is natural, and displays considerable 
 dramatic power."— Critic. 
 
 "It is a well concocted tale, and will be very 
 palatable to novel rea,6iei's."—Jldorning Post. 
 
 THE ROUA PASS. By Erick 
 Mackenzie, 3 vols, 
 
 " It is seldom that we have to notice so good a 
 novel as the ' Roua Pass,' The story is well con- 
 trived and well told ; the incidents are natural and 
 varied; several of the characters are skilfully 
 drawn, and that of the heroine is fresh, powerful, 
 and original. The Highland scenery, in which 
 the plot IS laid, is described with truth and feeling 
 —with a command of language which leaves a 
 vivid ivo.i>ression."—Saturday Review. 
 
 " The peculiar charm of the novel is its skilful 
 painting of the Highlands, and of life among the 
 Highlanders, Quick observation and a true sense 
 of the poetiy in nature and human life, the 
 author has."— Examiner. 
 
 " The attractions of the story are so numerous 
 and varied, that it would be difficult to single out 
 anj^ one point of it for attention. It is a brilliant 
 social picture of sterling scenes and striking 
 adventures."— .Sm«. 
 
 RIVERSTON. By Georgiana M. 
 Craik. 3 vols. 
 
 "A decidedly good novel. The book is a very 
 clever one, containing much good writing, well 
 discriminated sketches of character, and a story 
 told so as to bind the reader pretty closely to the 
 text."— Examiner. 
 
 " Miss Craik is a very lively writer : she has wit, 
 and she has sense, and she has made in the 
 beautiful young governess, with her strong will, 
 saucy independence, and promptness of repartee, 
 an interesting picture,"— Press. 
 
 " Miss Craik writes well ; she can paint cha- 
 racter, passions, manners, with considerable 
 effect : her dialogue flows easily and expressively." 
 —Baity News, 
 
 " The author shows great command of language, 
 a force and clearness of expression not often met 
 with. . . , We offer a welcome to Miss Craik, 
 and we shall look with interest for her next 
 work."— AthencBum. 
 
 FARINA. By George Meredith. 
 1 vol. 
 
 "A masque of ravishers in steel, of robber 
 knights ; of water-women, more ravishing than 
 lovely. It has also a brave and tender deliverer, 
 and a heroine proper for a romance of Cologne, 
 Those who love a real, lively, audacious piece of 
 extravagance, by way of a change, will enjoy 
 ' Farina,.'"— AthencEum. 
 
 FRIENDS OF BOHEMIA; 
 
 OR, Phases of London Life. By 
 E. M. Whitty, Author of " The 
 Governing Classes." 2 vols. 
 
 " Mr, Whitty is a genuine satirist, employing 
 satire for a genuine purpose. You laugh with him 
 very much ; out the laughter is ft-uity and ripe in 
 thought. His style is serious, and his cast of 
 mind severe. The author has a merriment akin 
 to that of Jaques and that of Timon."—Athen(Bum. 
 
 THE EVE OF ST. MARK. A 
 
 Romance of Venice. By Thomas 
 
 DOUBLEDAY, 2 Vols. 
 " ' The Eve of St, Mark ' is not only well written, 
 but adroitly constructed, and interesting. Its 
 tone is perhaps too gorgeous; its movement is too 
 much that of a masquerade; but a mystery is 
 created, and a very loveable heroine is pour- 
 trayed."— ,4i(Aen<E«)n. 
 
 NOVELS FORTHCOMING. 
 
 A NEW NOVEL. By Nathaniel Hawthorne, Author of "The Scarlet 
 
 Letter," &c. 3 vols. 
 
 A NEW NOVEL. By the Author of " My Lady," 3 vols. 
 
 And other Works of Fiction hj Popular Authors. 
 
 22 
 
SMIITH, ELDER ^^IISTD CO. 
 
 NEW BOOKS FOR YOUNG READERS. 
 
 THE PARENTS' CABINET of Amusement and Instruction for Young 
 Persons. New edition, carefully revised, in 12 Shilling Volumes, each, 
 complete in itself, and containing a full page Illustration in oil colours, 
 fnth wood engravings, in ornamented boards. 
 
 AMUSING STORIES, all tending to the development of good qualities, and the avoidance of faults. 
 BIOGRAP1IICA.L ACCOUNTS OF REMARKABLE CHARACTERS, interesting to Young People. 
 SIMPLE NARRATIVES OF HISTORICAL EVENTS, suited to the canacity of children. 
 ELUCIDATIONS OF NATURAL Hf STORY, adapted to encourage habits of observation. 
 FAMILIAR EXPLANATIONS OF NOTABLE SCIENTIFIC DISCOVERIES AND MECHANICAL 
 
 INVENTIONS. 
 LIVELY ACCOUNTS OF THE GEOGRAPHY. INHABITANTS, AND PRODUCTIONS O? 
 
 DIFFERENT COUNTRIES. 
 
 Miss Edgewobth's Opinion of the Pabbwts' Cabhtet :— 
 
 "I almost feel afraid of praising it as much as I think it deserves. . , . There is so much 
 variety in the book that it cannot tire. It alternately excites and relieves attention, and does not lead 
 to the bad habit or frittering away the mind by reciuiring no exertion from the reader. . , . Whoever 
 your scientific associate is, he understands his business and children's capabilities right well. . . . 
 Without lecturing, or prosing, you keep the right and the wrong clearly marked, and hence all 
 the sympathy of the young people is always enlisted on the right side." 
 
 *^* The work is now complete in 4 vols., extra cloth, gilt edges, at 3*. 6c?. 
 each; or in 6 volumes, extra cloth, gilt edges, price 2*. 6d. each. 
 
 By the Author of " Round the Fire," &c. 
 I. 
 
 UNICA : A Story for a Sunday 
 Afternoon. With Four Illus- 
 trations. Price 35. cloth. 
 
 " The character of Unica is charmingly con- 
 ceived, and the story pleasan'ly to\d."— Spectator. 
 
 " An excellent and exceedingly pretty stoi-y for 
 clnldreu."— Statesman. 
 
 "This tale, like its author's former ones, will 
 find favour in the nursery."— .4 tAeKoswnt. 
 
 II. 
 
 OLD CINCERBREAD AND THE 
 SCHOOL- BOYS. With Four 
 Coloured Plates. Price 3s. cloth. 
 
 "'Old Gingerbread and the School-boys' is 
 delightful, antt the drawing and colouring of the 
 pictorial part done with a spirit and correctness." 
 —Prets. 
 
 "This tale is very good, the df'scriptions being 
 natural, with a feeling or country freshness."— 
 Spectatar. 
 
 " The book is wellgotup, and the coloured plates 
 are very pretty."— G7o6e. 
 
 "An excellent boys' book ; excellent in its moral, 
 chaste and simple in its language, and luxuriously 
 inu&tra.ted."-Illu«trated A etc* of the World. 
 
 "A very lively and excellent tale, illustrated 
 with very delicately coloured pictures." — 
 Economist. 
 
 " A delightful stoiy for little boys, inculcating 
 bouevolent feelings to the poor."— £ciect?ci?erieic. 
 
 III. 
 
 WILLIE'S BIRTHDAY? showing now 
 A Little Boy did what he Liked, 
 
 AND HOW HE EnJOYED IT. With 
 
 Four Illustrations. Price 2s. 6<f. cl. 
 
 UNCLE JACK, THE FAULT KILLER. 
 
 With Four Illustrations. Price 3s. cl. 
 
 "An excellent little book of moral improvement 
 made pleasant to children: it is far beyond the 
 common-place moral tale in design and execution." 
 —Globe. 
 
 ROUND THE FIRE: Six Stories 
 for Young Readers. Square 
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 24