CORNELL 
 
 UNIVERSITY 
 
 LIBRARY 
 
 GIFT OF 
 
 Ifrs. K. 1, Turner 
 
 R^s Airrs impm 
 
* MMMMMiK^^^ and p 
 
 089 
 
Cornell University 
 Library 
 
 The original of tiiis book is in 
 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924020549089 
 

A TEXT-BOOK 
 
 SHADES AND SHADOWS, 
 
 PERSPECTIVE. 
 
 PREPARED FOR THE USE OF STUDENTS IN 
 TECHNICAL SCHOOLS 
 
 JOHN E. HILL, M.S., M.C.E., 
 
 Professor of Civil Engineeritig, Brown University. 
 
 SECOND EDITION, REVISED AND ENLARGED. 
 FIRST THOUSAND. 
 
 l^ 
 
 a 
 
 NEW YORK : t ^ '< 
 
 JOHN WILEY & SONS. _<^ 
 
 London : CHAPMAN & HALL, LiMiTEa ^ ^ 
 1900. , ^^^ 
 
 \ c ■ 
 
 .- y ^r 
 
Copyright, 1S96, 
 
 BY 
 
 JOHN E. HILL, 
 
 ^:[>.l N. K.^ 
 
 .t 
 
 i>f/^lr 
 
 
 ROBFET DEUMMOND, ELECTROTYPER AND PRINTER, NEW YORK. 
 
PREFACE TO THE FIRST EDITION. 
 
 This text-book is designed to furnish a short course in 
 Shades and Shadows, and One-Plane Perspective, for students 
 who have studied the elements of Descriptive Geometry. 
 Prominence has been given to the fact that the subjects 
 treated are branches or applications of Descriptive Geometry 
 and follow that subject in logical sequence. 
 
 The author is greatly indebted to many friends who have 
 kindly assisted in the preparation of the work, and especially to 
 Messrs. C. W. Comstock and C. W. Sherman, Instructors in 
 Civil Engineering, Cornell University. 
 
 The method of perspective given is that elaborated in 
 " Modern Perspective " by Professor W. R. Ware, and the 
 method of shading used is to be found in the works of M. 
 Jules Pillet. To many works, but especially to those men- 
 tioned, the author desires to express his obligation. 
 
 J. E. H. 
 
 Ithaca, N. Y., March i, 1894. 
 
PREFACE TO THE SECOND EDITION. 
 
 For this edition the book has been throughly revised and 
 in the main rewritten. The most notable changes are in the 
 notation of Shades and Shadows ; the extension and addition 
 of certain problems ; and the addition of examples of shaded 
 objects showing the application of the principles of shading. 
 
 My thanks are due Professor H. S. Jacoby, of Cornell Uni- 
 versity, for many suggestions. 
 
 J. E. H. 
 Providence, R. I., July i, 1896. 
 
TABLE OF CONTENTS. 
 
 SHADES AND SHADOWS. 
 CHAPTER I. 
 
 PAGE 
 
 g§ 1-22, General Principles, 'Definitions AND Conventions., i-ii 
 
 CHAPTER II. 
 §§ 23-33. Shadows OF Right Lines ■ 12-21 
 
 CHAPTER III. 
 §§ 34-47. Shadows OF Curves 22-4>2 
 
 PERSPECTIVE. 
 
 CHAPTER IV. 
 §§ 48-77. General Principles, Definitions and Conventions. . 43-54 
 
 CHAPTER V. 
 §§ 78-94. Measurement of Lines in Perspective 55-6i 
 
 CHAPTER VI. 
 §§ 95-ioo«. Parallel, Angular AND Oblique Perspective 62-79 
 
 CHAPTER VII. 
 
 ^§ loi-iog. Perspective of Circles 80-88 
 
 CHAPTER VIII. 
 §§110-116. Perspective of Shadows 89-96 
 
 APPENDIX. 
 §§ 117-121. Shading 97-101 
 
SHADES AND SHADOWS. 
 
 CHAPTER I, 
 GENERAL PRINCIPLES, DEFINITIONS AND CONVENTIONS. 
 
 Shadows. 
 
 1. Shades and Shadows is primarily an application of 
 Descriptive Geometry. (See §§ i6 and 19.) 
 
 2. The subject defined. The subject treats of one of the 
 means employed by which the representation of any object 
 may be given the appearance of reality. This result is here 
 accomplished by giving certain tints or colors to one or both 
 of the orthographic projections of the object upon the co- 
 ordinate planes, and by the determination of its shadow either 
 upon those planes, upon the object itself, or upon both. 
 
 Shading or tinting is treated in the Appendix. 
 
 3. Shades and Shadows in Nature. Non-luminous ob- 
 jects are seen by the light which they reflect and refract.* It 
 is evident that without light an object is invisible. It is also 
 evident that if an object be illuminated by but one source of 
 light, e. g., the sun, that the portion of the object towards the 
 sun will appear light or bright, and the portion on the oppo- 
 
 * The detailed reasons for some of the statements in this chapter may be 
 found in any of the larger text-books on Physics. 
 
2 SHADES AND SHADOWS. 
 
 site side, or out of the light, will appear dark; also that 
 various portions will appear to have different tints depending 
 upon the relative positions which they bear to the sun. On 
 a sphere the various tints blend and there are no sharp con- 
 trasts ; on an hexagonal prism the tints on the several sides 
 are sharply contrasted. 
 
 Because of this variation in the gradation of tints or colors 
 in the appearance of objects, we are able to distinguish one 
 from another, and were it not for this phenomenon all objects 
 would appear flat, i. e., like a plane. 
 
 In many cases we may determine by analogy the form of 
 an object from its shadow; e.g., one would never mistake the 
 shadow of a tree for that of a building. If then the proper 
 shading be given to the representation, and the shadow be 
 determined, we have a picture of the object. 
 
 4. Sunlight The light from the sun is transmitted to the 
 earth along slightly curved and slightly divergent lines, because 
 the atmosphere is not homogeneous and the sun is not in- 
 finitely distant. We may consider, however, for drafting pur- 
 poses that the lines of light are parallel right lines, as the error 
 caused by such assumption is not appreciable for any drafting 
 that would be made. 
 
 5. Conventions. The majority of draftsmen desiring uni- 
 form results have adopted the following conventional method 
 of representation. It is unnecessary to discuss the scientific 
 accuracy of these assumptions ; let it be taken for granted 
 that they supply the want and give the desired results. 
 
 Conventions. 
 
 (yd) The source of light is considered to be a fixed point 
 situated at an infinite distance from the object to be repre- 
 sented. 
 
PRINCIPLES, DEFINITIONS AND CONVENTIONS. 3 
 
 (^) All lines of light emanating from the source are par- 
 allel right lines. 
 
 (c) The source of light is further limited in position by- 
 being so placed with reference to the coordinate planes that 
 a line of light forms one of the diagonals of a cube, two of 
 whose faces are the coordinate planes. If the spectator were 
 in the first dihedral angle looking at V, the source of light 
 would be at an infinite distance above and behind his left 
 shoulder ; the direction of a line of light would be the diago- 
 nal of the cube above mentioned running from the near upper 
 left vertex to the far lower right vertex. Fig. i gives a per- 
 
 Fig, t. 
 
 spective view of a cube, two of whose faces, gfe i and b c e i , 
 are in H and V respectively, and the diagonal R through a 
 and e gives the direction of a line of light to which all lines 
 of light are parallel. 
 
 [d) All lines of sight for either coordinate plane are parallel 
 right lines perpendicular to that plane. 
 
 {e) All objects situated in the second, third, or fourth di- 
 hedral angles are considered invisible. 
 
 (y) All objects are to be considered opaque. 
 
 6. Rays. Lines of light are called Rays of light, or simply 
 Rays. (See § 5, sec. (b)) 
 
4 SHADES AND SHADOWS. 
 
 7. Plane of Rays. Any plane containing a ray of light 
 is a Plane of rays, for such a plane contains an infinite number 
 of lines parallel to the ray, and consequently an infinite num- 
 ber of rays. 
 
 8. Pencil of Rays. Any collection of rays other than a 
 plane is called a Pencil of rays. (By some authors " cylinder 
 of rays.") 
 
 9. Line of Shade. The Line of Shade of a surface or solid is 
 the locus of the points of tangency of all rays which can be 
 drawn tangent to that surface or solid. NOTE: The word 
 tangent as used in this connection means tangent in its geo- 
 metric sense, and also touching without immediately intersect- 
 ing. The general idea of tangent rays is, however, always 
 maintained, because the rays determining the line of shade do 
 not intersect the object before touching, and do not im- 
 mediately intersect after touching. (See Fig. i.) The line 
 of shade of the cube is composed of the six edges fd, 
 do, cb, bi, ig, and gf, determined by six planes of rays 
 passed through them, and is an example of " touching without 
 immediately intersecting." If a plane of rays, were passed 
 through the edge ag\\. would immediately intersect the cube, 
 consequently «^ is not a part of the line of shade. The line 
 of shade of a sphere is a great circle determined by a tangent 
 pencil (cylinder) of rays. 
 
 The line of shade divides the object into two parts, one 
 illuminated, or in the light, and the other unilluminated, or in 
 the dark ; the light being excluded from the unilluminated parb 
 by the object. 
 
 The line of shade of a plane surface is the perimeter of the 
 surface ; of a plane figure, the lines of that figure. 
 
 10. Umbra. The f/wi^ra or indefinite shadow of an object 
 is that portion of space from which light is excluded by the 
 
PRINCIPL-ES, DEFINITIONS AND CONVENTIONS. 5 
 
 object. The umbra of a point is a line ; of a right Hne is a 
 plane, etc. The umbra of a surface or solid is bounded by the 
 pencil of rays tangent to the object. 
 
 11. Shadow. The Shadow of an object is the intersection 
 of the umbra with the surface of any object. If no object is 
 situated in the umbra of an object there is no shadow. A 
 shadow is always a point, line, or surface. (See Plate II, Fig. 
 II.) The shadow of the lower edge, e b, of the parallelopiped 
 is partly on the three front faces and one side face of the 
 tinder prism and partly on H. Care must be taken to obtain 
 the complete shadow in every problem. 
 
 12. Primary and Secondary Shadows. A distinction is 
 made between the primary shadow, or, briefly, the shadow, 
 and the secondary shadow. The former is the intersection of 
 the umbra with the first object which it meets, and the latter 
 is a subsequent intersection. In many problems the secondary 
 shadow is found in order to more easily determine the primary 
 shadow. 
 
 13. Line of Shadow. The Line of Shadow is the perimeter 
 of the shadow, or the intersection of the tangent pencil of rays 
 Tvith the object on which the shadow is cast. 
 
 14. Shade. The Shade is that portion of the object from 
 which light is excluded by the object itself, and which would 
 not be illuminated were any part of the object removed. The 
 shade is bounded by the line of shade. (See § 9.) 
 
 15. It follows from §§ 9 and 13 that the line of shadow is 
 the shadow of the line of shade. 
 
 16. Oblique projection. Since shadows are determined by 
 lines (rays) which are oblique to the coordinate planes, the 
 method is one of oblique projection, and the problems pre- 
 sented are similar to those of Descriptive Geometry involving 
 
6 SHADES AND SHADOWS. 
 
 intersections. This fact should be constantly in mind whea 
 studying this subject. (See § i.) 
 
 17. Conventional direction of rays. Since a ray forms 
 the diagonal of a cube as shown on Fig. i (see § 5, sec. (c)), the 
 projections of a ray make angles of 45° with G. L. The ray itself 
 makes angles of 35" 15' 52" with H, V, or a profile plane ; this, 
 angle is called for convenience. 
 
 Trigonometric functions of 0. 
 
 (See Fig. i. Each edge of the cube is unity.) 
 
 I V 2 
 
 sin ^ = -^ = 0.57735. cos ^= —= = 0.81649.. 
 
 ^3 ^3 
 
 tan = —7= = 0.7071 1 = cos 45° = sin 45°. 
 
 V2 
 
 V^ I 
 
 sec ^ = — £ = 1.22475. cos 2 6* = - = 0,33333.. 
 
 V2 3 
 
 18. Advantages of conventional direction of rays. By 
 
 using the conventional direction of rays the construction of 
 many problems may be shortened ; e. g., to find the shadow of 
 the point a (Plate I, Fig. 3) : As a^u is greater than a^ u, from 
 a^ lay off a^ p = cC u, draw p a^y parallel to G. L., and «°«si 
 making 45" with G. L. The intersection of the two lines, a^i, 
 is the shadow of a required, because a' u is the distance of the 
 point above H, and if a right-angled triangle be constructed 
 with aa^ = a7 u for the altitude and a ray for the hypotenuse, 
 the base in H is a^ agy. 
 
 In certain problems it may be advantageous to disregard 
 the conventional position of the source of light and assume 
 some other position in order to bring out more clearly, either 
 
PRINCIPLES, DEFINITIONS AND CONVENTIONS. 7 
 
 the character of the object or a particular detail. Because 
 such cases are rare and their treatment requires mature judg- 
 ment no departure is made in this work from the ordinary- 
 method of procedure. Usually a change in the position of the 
 object rather than in the position of the light will give the de- 
 sired result. 
 
 19. The general problem. In general the shadow of an 
 object is found by passing a pencil of rays tangent to the ob- 
 ject and determining the intersection of this pencil with the 
 object which receives the shadow ; this method involves, first, 
 the determination of the line of shade, and second, its shadow. 
 When the line of shade is a right line and the surface receiving 
 the shadow is a plane, the problem is simple and may be stated 
 thus : To find the intersection of a plane passed through a 
 given line with a given plane. When the line of shade is a 
 curve and the surface receiving the shadow is curved the 
 problem is more difificult ; stated in Descriptive Geometry 
 language it is : To find the intersection of any surface with a 
 cylinder. 
 
 In problems relating to curves or surfaces of the second 
 degree, the axes or diameters of parts of the shadows may 
 often be found and the solution greatly simplified. As a last 
 resort a number of rays may always be passed through the line 
 of shade and their intersections with the object receiving the 
 shadow found. This method should not be employed how- 
 ever, unless necessary, as the great number of construction 
 lines required increases the chances for error and has a ten- 
 dency to mar the appearance of the drawing. 
 
 The line of shade in many cases is readily determined by 
 inspection. When this cannot be done, or uncertainty exists 
 in regard to lines probably composing the line of shade, the 
 shadow of the entire contour line of the object may be found 
 
an 
 
 SHADES AND SHADOWS. 
 
 d the parts which cast shadows within the perimeter of the 
 shadow rejected. The general test is, that no part of the line 
 of shade casts a shadow within the line of shadow. 
 
 20. Notation. The following notation will be employed 
 throughout Shades and Shadows : 
 
 (a) The projections of a point on the coordinate planes will 
 be designated by small letters with the exponents « and ^ rep- 
 resenting the horizontal and vertical projections respectively ; 
 e. g. , the projections of the point a are a^ and a^. 
 
 (b) The projections of the revolved position of a point will 
 be designated by the letter of the point with the subscript „ ^ 
 or 3 and the exponents ^ and ' ; e. g., a^ and «/. 
 
 (c) The projection of a point upon a new coordinate plane 
 will be designated by the letter of the point with the exponent 
 ^1 or ■^i, °' or ^« ; e. g., «"! or d^^. ' 
 
 {d') The shadow of a point on one of the coordinate planes 
 will be designated by the letter of the point with the subscript 
 g ; e. g., «s. When the shadows* of a point on both coordinate 
 planes are required they will be designated by the letter of the 
 point with the subscripts si and si\ e. g., «si and a^z. 
 
 {e) The projections of the shadow of a point on a surface 
 other than the coordinate planes will be designated by the let- 
 ter of the point with the exponents " and ^ and the subscripts 
 si and s2, etc., e.g., a\, ajj. When only one projection of the 
 shadow is required it will be designated by the letter of the 
 point with the subscript s ; e. g., a^. 
 
 {/) A line will be designated by a capital letter, and its 
 projections, revolved position and shadows by the exponents 
 or subscripts used for points. See {a), {p], (c), {d), (e). 
 
 {g) The trace of a plane will be designated by a capital let- 
 ter preceded by a capital H or V; e. g., HP or VP. 
 
 * Primary and Secondary shadows (see § 12) usually. 
 
PRINCIPLES, DEFINITIONS AND CONVENTIONS. 9 
 
 (Ji) The trace of a plane in a revolved •^os\\\ox\ will be desig- 
 nated by the letter of the plane with the subscript j, , or 3, and 
 preceded by a capital H or F; e. g., HP^ or FP,. 
 
 (?) The trace of a plane on a w^ze/ coordinate plane will be 
 ■designated by the letter of the plane, preceded by a capital H 
 ■or V with the subscript 1, j or , ; e. g., H^P or V^P. 
 
 21. Drafting Conventions. 
 
 (i) Given and required lines, if visible, are represented by 
 heavy full lines ; if invisible, by heavy long dashes. 
 
 (2) Auxiliary lines, visible and invisible, are represented by 
 light dashes. Projecting lines and lines indicating the paths of 
 rotated points are short dashes and other auxiliary lines long 
 dashes. 
 
 (3) Traces of given and required planes, if visible, are rep- 
 resented by heavy full lines ; if invisible, by one long dash and 
 two short dashes, also heavy. 
 
 (4) Traces of auxiliary planes — planes used as a means for 
 solving the problem — are represented by light broken lines 
 consisting of one long dash and two short dashes alternately. 
 
 (5) Intersections of given and required planes, if visible, are 
 represented by heavy full lines ; if invisible, by one long dash 
 and one short dash, also heavy. 
 
 (6) Intersections of auxiliary planes, or of auxiliary with 
 given or required planes, are represented by light broken lines 
 consisting of one long dash and one short dash alternately. ■ 
 
lO SHADES AND SHADOWS. 
 
 (7) The long dashes should measure about one-eighth of aii 
 inch and the short dashes about one twenty-fourth of an inch, 
 t. e., about one-third as long as the long dashes. The spaces 
 between the dashes should be very short. 
 
 (8) New ground lines are represented by heavy full lines. 
 
 (9) All objects of which the shadows are to be found are 
 considered opaque. 
 
 Recapitulation. Given and required lines, traces, and inter- 
 sections, if visible, are distinguished from the invisible in form 
 only, the weight remaining the same. When visible all of- them 
 are represented by full lines. 
 
 When given or required, but invisible, they differ from the 
 auxiliary in weight only, the form remaining the same. Traces 
 have one long dash and two short dashes, intersections one 
 long dash and one short dash, and other lines are composed of 
 either long or short dashes. 
 
 All given and required lines, etc., are heavy, and all auxil- 
 iary lines are light. 
 
 22. Problems in Drafting Room. The work in the draft- 
 ing room may often be greatly facilitated and improved by 
 assigning problems in which the positions of points, etc., are 
 designated by means of coordinates, as in analytic geometry. 
 
 The position of a point in space is known when its distance 
 from three planes (no two of which are parallel) is known, 
 hence to designate the position of a point in a problem in 
 shades and shadows (descriptive geometry), we may refer it to 
 H, V, and a profile plane, P. Following the notation of ana- 
 lytic geometry, G. L. is the axis of x, the horizontal trace of 
 P\s the axis of ^, and the vertical trace of Pis the axis of z. 
 The point a with the following reference, x = 2", y = 3", z = 
 6", means that the point is in the first dihedral angle 2" to the 
 right ol P, 3" in front of Tand 6" above H. The point b 
 
PRINCIPLES, DEFINITIONS AND CONVENTIONS. IT 
 
 with X = 3", J = — 3", z = — 4", means that the point is in. 
 the third dihedral angle 3" to the right of P, 3" behind V, and 
 4" below U. The designation of lines and planes readily fol- 
 lows. It is preferable to assume that the reference profile 
 plane {P) passes through the left border line of the sheet upon 
 which the drawing is to be made. 
 
CHAPTER II. 
 Shadows of Right Lines. 
 
 23. Problem I. To find the shadow of a line on the coordi- 
 nate planes. Plate I, Fig. 3. 
 
 Analysis. The shadow of a line is determined by the in- 
 tersection of a plane of rays through the line with the object 
 receiving the shadow ; or, in other words, the shadow is the 
 locus of the intersections of all rays passed through the line 
 with the object receiving the shadow. The object in this case 
 is the coordinate planes, hence if the shadows of any two 
 points of the line on H and on V be found and joined by a 
 line, the required shadow is determined. Or the shadow of 
 one point of the line may be found on H and on V, and 
 joined with the H and V traces of the line respectively, for 
 since the traces of a line are points in the coordinate planes, 
 each is its own shadow on the plane in which it is situated, 
 and the shadow of the line is determined as before. 
 
 Construction. Let M be the given line. Take any two 
 points of the line as a and b and through each pass a ray. 
 The projections of the rays pass through the projections of 
 the points and make angles of 45° with G. L. The intersection 
 (trace) of the ray through a with H is a^l and with V is a^^. 
 The intersection (trace) of the ray through b with H is b^ and 
 •with V is b^Y Joining a^^ b^^ we have the shadow of M on H, 
 and joining a^ bg^, the shadow of M on V. The parts of the 
 shadow on H behind V and on V below H are secondary (see 
 
SHADOWS OF RIGHT LINES. 13 
 
 § 12), hence the shadow of M on the coordinate planes is the 
 line agifisbsi or M^x- 
 
 In this problem, as in many others, by finding the second- 
 ary shadow the solution is simplified. The method of § 18 
 might have been employed with advantage. 
 
 Another Solution. Since «gi and b^i are the H traces of 
 rays passed through two points of the line M and a^^ and b^i 
 are the V traces of the same rays, a^^ bg^ and a^^ b^i are the H 
 and V traces respectively of a plane of rays passed through 
 M, and hence are the shadow of M on H and V. The visible 
 portion of these traces and hence the primary shadow is the 
 broken line Mg^. 
 
 24. Problem II. To find the shadow of a plane surface on 
 the coordinate planes. Let the given surface be a triangle. 
 Plate I, Fig. 4. 
 
 Analysis. The line of shade of the surface is the perim- 
 eter of the surface (§ 9) and the line of shadow is the 
 shadow of the perimeter (§ 15), hence the shadow of a triangle 
 is determined by the shadows of its three sides. If we find 
 the shadows of the vertices of the triangle we will have the 
 shadows of two points in each side, hence the shadows of the 
 vertices determine the shadow of the triangle. 
 
 Construction. Let abc h^ the given triangle. Find the 
 shadows of the vertices. The shadow of a is a^ on H, of b is 
 bs\ on V, and of c is c^ on H, all found as in Prob. i, § 23. 
 Join «s Cg for the shadow of ac. As a^ and c^ are on .Awhile 
 ^si is on V, it is necessary to find the shadow (secondary) of b 
 on H, or of a and c on V, in order to determine the direction 
 of the shadow of ab and of cb on H. The shadow o{ b on H 
 is b^, hence a^ m^ and c^ n^ are the primary or visible shadows 
 of ab and cb respectively on H. To complete the shadow of 
 the triangle draw ;«s b^^ and Wg bsu the primary shadows of ab 
 
14 SHADES AND SHADOWS. 
 
 and cb respectively on V. The required line of shadow is 
 
 25. Problem III. 7^^ find the shadow of a solid on the 
 coordinate planes. Let the solid be a cube. (Simplest case.) 
 Plate I, Fig. 5. 
 
 Analysis. The line of shadow of a solid is the shadow of 
 the line of shade (§ 15), hence first determine, if possible, the 
 line of shade. In complicated problems in solids it is some- 
 times difficult if not impossible to determine the line of shade 
 without virtually finding the shadows of various lines of the 
 object. The criterion to be employed is : A line of the soli(^ 
 is a part of the line of shade when its shadow is a part of the 
 perimeter of the shadow of the solid. A line of the solid is 
 not a part of the line of shade when its shadow falls within 
 the perimeter of the shadow of the solid. For rectilinear 
 solids we may usually determine the position of the line of 
 shade by inspection if we remember the definition of § 9. 
 
 The line of shade of the cube is found by inspection to be 
 the edges dj,jg,gf,fb, and the edges in H, ab and «d? which 
 coincide with their shadows. The proof of this statement will 
 be given below. Having determined the line of shade, pro- 
 ceed as in Probs. I and II. 
 
 Construction. Given the cube as shown with one face in 
 H and the other faces perpendicular or parallel to V. The 
 shadow of the edge ^' is d^j^, oijgisj^g^, oig/isg^f, oi fb 
 is fs 5", found as in Prob. I by joining the shadows of the two 
 points of each edge. The shadows of the edges ab and ad 
 coincide with the lines. The line of shadow is, therefore, 
 d^JsKsfs l>^ ^^ d^ on H. The visible shadow is bounded by the 
 line d'^jsgsfs b^ ^ d^. 
 
 Proof of statement in analysis: — Suppose the edge ej to 
 be part of the line of shade, its shadow is c^j\ which falls 
 
SHADOWS OF RIGHT LINES. 1$ 
 
 within the line of shadow, hence ej is not a part of the line of 
 shade. (See § 19.) 
 
 A plane of rays through ej immediately intersects the 
 cube, hence ej is not a part of the line of shade. (See § 9.) 
 In the same manner it may be shown that the edges ae, ef, 
 «tc., are not parts of the line of shade. 
 
 Note : From the solution of this problem we see several 
 facts which are principles of geometry, {a) If a line is par- 
 allel to a plane its shadow on that plane is parallel to the line. 
 {V) Parallel lines cast parallel shadows on the same plane, etc. 
 
 26. Problem IV. To Jind the shadow of a vertical prism 
 en the coordinate planes. Plate I, Fig. 6. 
 
 Analysis. The analysis of this problem is similar to that 
 of Prob. III. 
 
 Construction. The line of shade is found by inspection to 
 be the edges cj and af, three sides of the upper base jk, kl, 
 and If, and the two sides of the lower base ab and be ; which 
 statement may be proved as in Prob. III. The shadow of cj 
 is c^y'si, oijk \sjsi n^ k^i (see Prob. I), of kl is k^i 4i ,oilfls l^ifi, 
 of af is fi ps a^, of ab is a" b", and of be is b^ c^. The complete 
 line of shadow is now determined. The visible portion of the 
 shadow is bounded by the line c^jsi n^ k^ m^ d^ p^ «" e^ d^ c^ 
 
 27. The shadow of the upper base of the prism on //"and 
 on Fis shown and may be used as the solution of the prob- 
 lem : To find the shadow of a horizontal polygon on one or 
 both of the coordinate planes. The shadow (visible) of the 
 polygon on the coordinate planes is the lineygi n^ k^i ^sifsi ?s^si7si- 
 (See Prob. II.) 
 
 28. Problem V. To find the shadow of a plane surf ace on 
 Ml oblique plane. Let the surface be a triangle. Plate I, 
 Fig. 7. 
 
 Analysis. See analysis of Prob. II, § 24. The shadow of 
 
i6 SHADES AND SHADOWS. 
 
 a point on a plane is the intersection of a ray through the 
 point with the plane, and of a line on a plane is the intersection 
 of a plane of rays through the line with the plane. Hence two 
 methods of solution are available : either find the shadows of 
 the vertices of the triangle by means of rays, or the shadows of 
 the sides of the triangle by means of planes of rays. In general,, 
 the first method is the simpler. 
 
 Construction. Let a b c\t& the given triangle, and Q the 
 plane on which the shadow is cast. To find the shadow of c 
 on Q pass a vertical plane of rays Z through the point. This 
 plane intersects the plane Q in the line M{M'', HZ). Hence 
 ^5, the intersection of a ray through c with M, is the shadow of 
 c required. In a like manner pass the vertical planes of rays 
 Fand X through a and b respectively. As the vertical planes 
 of rays Z, Y, and X are parallel, their intersections with the 
 plane Q are parallel, hence U" and T" may be drawn through 
 the intersection of the F traces parallel to Jkf. A ray through 
 a intersects [/ at a^, and a ray through b intersects T at b^.. 
 The shadow of the triangle is, therefore, a^b^c^. 
 
 29. The student should constantly have in mind the fact 
 that the solutions of the following problems are exactly alike l 
 " To find the shadow of a point on a plane " and " To find the 
 intersection with a given plane of a line passed through a given 
 point and parallel to a given hne." " To find the shadow of 
 a right Hne on a plane " and " To find the intersection with a. 
 given plane of a plane passed through a given line and parallel 
 to a given line." 
 
 30. Problem VI. To find the shadow of one line upon 
 another line. Plate I, Fig. 8. 
 
 Analysis. If one line casts a shadow upon another line a 
 plane of rays through the first line intersects the second line,, 
 otherwise the two lines are in parallel planes of rays and the 
 
SHADOWS OF RIGHT LINES. IJ 
 
 second cannot receive a shadow from the first. If a plane of 
 rays be passed through each hne these planes intersect and the 
 line of intersection is a ray, because the line of intersection of 
 any two planes of rays is a ray. Hence, if the shadows of the 
 two lines on the same object be found and a ray passed through 
 their intersection, the required shadow and the point which 
 casts the shadow is determined by the intersections of the ray 
 with the two lines. A limiting case of the above is when the 
 two lines are in the same plane of rays, in which case the sec- 
 ond line is the shadow of the first line. 
 
 Construction. Let M and N be the given lines. The 
 shadows of tliese lines on H are M^, and N^ respectively, which 
 intersect at the point o^^. A ray through o^ is the line of in- 
 , tersection of the planes of rays which determined the shadows 
 of the lines. This ray intersects i\^ at i^si and M sX o ; hence 
 the shadow of M on N is o^^. 
 
 31. Problem VII- To find the shadow of a hexagonal' 
 abacus on a vertical hexagonal prism. Plate I, Fig. 9. 
 
 Analysis. See analysis of Prob. V, § 28. Only the shadow 
 of the abacus on the prism is considered. It is evident from 
 an inspection of the figure that the lower edges ef,fa, and ab 
 of the abacus compose the line of shade for the purpose named. 
 Having found the line of shade the problem becomes : To 
 find the shadow of a line on a plane (§ 28). 
 
 Construction. Let M, N, O, etc., be a vertical prism, and 
 a be, etc., be the abacus. As the faces of the prism are vertical 
 the intersection of the H projection of a ray through any 
 point of the line of shade, as f, with the H projection (base) of 
 the prism gives the H projection of the shadow of that point 
 on the prism. Through / pass a ray : it intersects the prism at 
 /s; hence /s is the shadow of /on the prism. Through k,j\ g, 
 a, and i pass rays : they intersect the prism at k^,j\, gs, a^, and 
 
15 SHADES AND SHADOWS. 
 
 4, respectively; hence Kfsjsgs ^s 4 's the line of shadow required 
 of which the portion g^ a^ 4 is invisible on V projection and all 
 is invisible on H projection. The points k,j,g, and? were 
 determined as follows: Take the line of shade /«: it is evident 
 tliat its shadow falls on three faces of the prism, PQ, QL, and 
 LM; hence one point oi fa must cast a shadow on Q and one 
 point on L. Since Q is perpendicular to H through Q^ draw 
 the //"projection of a ray: it intersects /" a" aty"; hence /is 
 that point of y<a: which casts a shadow on Q. Similarly for the 
 points g, k, and i. In general, the shadow of a line changes 
 direction whenever the object receiving the shadow changes 
 direction. As the line of shade of a rectilinear solid is a 
 broken line, its shadow on an object usually changes direction 
 whenever the line changes direction. 
 
 32. Problem VIII. To find the shadow of a vertical prism 
 surmounted by a parallelopiped. Plate II, Fig. 11. 
 
 Analysis. This problem is a combination of Probs. IV and 
 VII. The parallelopiped overhangs the prism which contains 
 a recess. 
 
 The line of shade of the parallelopiped may be determined 
 by inspection, but that of the prism can only be approximated 
 in this manner, for a shadow is cast by the parallelopiped upon 
 that which would otherwise be a part of the line of shade. 
 Hence the shadow on the prism is determined before the line 
 of shade of the prism. As the visible shadow is the only part 
 which is usually required for a representation of an object, in 
 this problem the invisible shadow is not determined, except as 
 much as is necessary for the solution. 
 
 Construction. The line of shade of the parallelopiped is 
 composed of the edges ac, ex, x d, de, eb, and b a. The 
 shadow of « c is a^p^ c^, of ex is c^ 4 (the remainder of the shadow 
 is invisible), of ;r^is invisible, oi ba is b^a^. A ray through e 
 
SHADOWS OF RIGHT- LINES. 19 
 
 pierces the prism at e^, which is the shadow of e required. Since 
 the edge ^ ^ is perpendicular to V and the left front face of the 
 prism is parallel to V, the shadow oi e d or^ this face must pass 
 through ^s at an angle of 45° with G. L.\ hence egf^, is the 
 shadow. Another method for finding j^ is that given in Prob. 
 VII, § 31. The shadow of fd is invisible. The edge ^^ is 
 parallel to G. L., and three of the front faces of the prism are 
 parallel to V\ hence the shadow oi e b on these faces is parallel 
 to G. L. Through e^ draw ^s^si SiWdJ^kgy parallel to G. L., 
 since the two front faces are in the same plane. The point i 
 of the edge eb casts the shadow 4 on the edge D of the prism 
 (see Prob. VII, § 31); hence i^gs^ parallel to G. L., and limited by 
 a ray through the edge B, is the shadow on the inner face of 
 the recess and i^j^ is the shadow on the right face of the recess. 
 The remainder of the edge e b,or k b, casts a shadow k^i b^ on H. 
 This completes the line of shadow of the parallelepiped. The 
 line of shade of the prism is composed of parts of the edges A, 
 B and C and parts of the lower base. As the shadows of m 
 and n, two points of the lower base, on H are within the H 
 projection of the parallelepiped, it is evident from an inspec- 
 tion of the figure that the entire shadow of this base is within 
 the projection ; hence there is no visible shadow of this part of 
 the line of shade, and only the three edges need be considered. 
 The shadow of A is m^ks^, limited by a ray through k, and the 
 visible portion is u^ k^^ ; of 5 is g^^ ^s on the inner face of the re- 
 cess, and limited by a ray through g and the base of the prism ; 
 of Cis «s?s<'s. limited by a ray through o and the visible portion 
 is Ts q^ ts- The shadow on the prism is seen only in V projection, 
 and is bounded by the line /^ e^g^^, part of R' , part of 71" nf , 
 ^'s g&kj^K\, part of A", part of h^d", thence down C^ to/^. 
 On the coordinate planes the visible shadow is bounded by the 
 line b^ a^ps c^ 4 nt' is ?s ''s c^ i's Ki K 
 
20 SHADES AND SHADOWS. 
 
 33. Problem IX. To find the shadow of a chimney and a 
 dormer on a roof. Plate II, Fig. 12. 
 
 Analysis. The solution of this problem shows the use of a 
 profile plane as the roof is assumed parallel to G. L. 
 
 If a plane of rays be passed through a point its intersection 
 with any plane is a locus ot the shadow of the point on that 
 plane, also a ray through the point is a locus of the shadow of 
 the point ; hence the intersection of these loci is the shadow of 
 the point on the plane. Similar reasoning applies to the 
 shadow of a line on a plane. 
 
 Construction. Let the plane W be the plane of the roof 
 parallel to G. L., E, F, etc., be the chimney perpendicular to 
 H, and bf e t, etc., be the dormer, which is supposed to be faced 
 with stone. The relative positions are shown clearly by the 
 three projections. 
 
 The line of shade of the chimney is Eadc F. Through a 
 pass a plane of rays parallel to G. L., it intersects the roof Win 
 G, found as follows : Project the point a on a profile plane P, 
 revolve into Fand obtain a^^, through «/ draw a^ x^, making 
 an angle of 45° with G. L.: this is the revolved position of the 
 intersection of the plane of rays through a with the plane P, 
 since a plane of rays parallel to G. L. bisects the first dihedral 
 angle. Revolve back to original position and thus determine 
 X, which is one point in the line of intersection G, which may 
 now be drawn through x parallel to G. L. A ray through a 
 intersects G at ag, which is the shadow of a on the plane W. 
 Since E is perpendicular toH, a plane of rays through E is per- 
 pendicular to H; hence the //'projection of its shadow or a^a^ is 
 incHned at an angle of 45° with G. L. The //"projection of the 
 visible shadow is a"/" and /?«", as the part f"f^ is on the 
 shadow of the dormer. The F" projection of the shadow of E 
 runs from m along the roof to the face of the dormer, thence 
 
SHADOWS OF KIGHT LIISES. 21 
 
 oip the face of the dormer along El, thence becomes invisible 
 but again appears in the line fsol. A similar construction 
 applies to the remainder of the line of shadow. The line of 
 shade of the dormer is o ibf and the rear part (see figure) of et, 
 the shadow of which is found by the method used before. 
 
 Another method of solution is that given in Prob. V, § 28, 
 in which a profile plane is not used. 
 
CHAPTER III. 
 Shadows of Curves. 
 
 34. Problem X. To find the shadow of a sphere on the 
 coordinate planes. Plate II, Fig. 13. 
 
 Analysis. The pencil of rays tangent to a sphere is evidently 
 a cylinder having for its axis 'a ray through the center of the 
 sphere and for its right section 2l great circle oi the sphere. The 
 plane of the line of shade is therefore a meridian plane of the 
 sphere perpendicular to a ray. Having determined the line of 
 shade the problem then becomes: to find the shadow of a 
 circle on the coordinate planes. 
 
 Construction. Given the sphere A as shown. Project the 
 sphere on a new V plane coinciding with a vertical plane of 
 rays through the center. The new V projection is A7'^. 
 Through the center pass a ray : it intersects H at o^y ; hence 
 c^iOgi is the projection of the ray on V^. (This projection 
 might also have been obtained by passing a line through d'^ 
 making an angle of 35° 15' 52" with G,. Z,.) The line c^v^^d^^ 
 perpendicular to o^'^o^^^ is the intersection of the plane of the 
 line of shade with V^. 
 
 Passing rays through c and d, and using the new F plane, 
 the shadow c^d^, of a diameter of the line of shade is deter- 
 mined, and hence a diameter (axis in this case) of the shadow. 
 As the diameter dc is in a plane perpendicular to H, the 
 diameter ab parallel to H is perpendicular to it; hence the 
 shadow of a b, or a^b^, is the diameter (axis in this case) which. 
 
SHADOWS OF CURVES. 23 
 
 Is conjugate to c^d^ of the shadow. The points a^ and b^ are 
 the intersections of ^^ £7si ^g drawn perpendicular to c^d^ with 
 the H projections of rays through a and b. We know that 
 c^dg^ and b^a^, are not only diameters, but axes of the shadow, 
 because « ^ is parallel to //and t a? is in a plane perpendicular 
 to a b and to H; hence the shadows of these lines are perpen- 
 dicular to each other (see § 35). Constructing an ellipse with 
 c^d^ and b^a^ as axes the shadow of the sphere on H is 
 determined. 
 
 As rays are equally inclined to the coordinate planes and 
 the plane of the line of shade is perpendicular to the rays 
 and the line of shade is a circle, the shadow of the sphere on V 
 must be an ellipse equal in all respects to that on H. Hence, 
 since the shadow of the center on V\s o^^, lay off on the V projec- 
 tion of the ray through the center, Osij\ = Os,\C^ —■ Os\d^ = o^f^, 
 and on a line through 0^^ perpendicular toy^ o^^f^ lay off o^^g^^^ 
 0^2 ^s = ''si ^s =^ ''si '^s- -^^ ellipse constructed on the axes 
 Js/s 'ind gg e^ is the shadow of the sphere on V. The shadow 
 of the sphere is bounded by the line of shadow Cg b^psg^fg m^ Cg, 
 which is the solution sought. 
 
 35. Proposition I. T/ie oblique projections {or shadows) 
 of any pair of rectangular diameters of a circle or conjugate 
 diameters of an ellipse are conjugate diameters of the oblique 
 projection (or shadow) of that circle or ellipse. 
 
 36. Proposition II. If two surf aces intersect each other in 
 two curves one of zvhich is a curve of the second degree, both 
 curves are of the second degree, and the surfaces are of the second 
 degree. 
 
 This proposition for use in the subsequent problems may 
 be stated thus:* When we cut off by a plane and remove 
 a portion of a surface of the second degree, such as a cylinder, 
 
 *See "A Course in Shades and Shadows," by William Watson. 
 
24 SHADES AND SHADOWS. 
 
 cone, etc., the shadow of the section cast upon the interior 
 surface so exposed is a plane curve, and consequently one of 
 the second degree. 
 
 37. To construct an ellipse having given any pair of its con- 
 jugate diameters. Plate II, Fig. 14. 
 
 Construction. Let a b and c d\>Q the conjugate diameters of 
 an ellipse intersecting at o, the center. With ^ as a center and 
 radius equal to the semi-longer diameter describe a circle. 
 Draw n in perpendicular \.o ab at o, and join inc and nd. 
 Draw any line as wr q parallel to n m and intersecting a b 
 and the circle, then draw t rp parallel to do c and qp and w t 
 parallel \.o mc and n d. The points / and t are points of the 
 ellipse. 
 
 38. To construct the axes of an ellipse having given any pair 
 of its conjugate diameters. Plate I, Fig. lO. 
 
 Analysis and Construction. Let a^^b^ and c^d^ be conjugate 
 diameters of an ellipse in H. Draw G. L. through b^ parallel 
 to Cs,d^; then G. L. is tangent to the ellipse. 
 
 As only one ellipse can be constructed on a given pair of 
 conjugate diameters, as a^b^, and c^d^, and as these lines may 
 be considered the shadows of a pair of rectangular diameters 
 in an infinite number of ellipses, we may assume for the 
 purpose of our demonstration that these lines are the shadows 
 of a pair of rectangular diameters of a circle situated in a plane 
 which passes through G. L. On this assumption the diameter 
 of the circle which casts the shadow is equal to c^d^. NOTE: 
 It is not necessary to determine in any case the actual ellipse 
 or circle which casts the shadow, as we wish to find only the 
 magnitude and direction of the axes of the shadow ; also, the 
 rays of light are not assumed in the conventional direction for 
 the same reason. 
 
 If the radii of the circle be produced they will intersect 
 
SHADOWS OF CURVES. 25 
 
 G. L. and the shadows of the produced radii will run from the 
 intersections on G. L. to o^, the shadow of the center. A certain 
 pair of rectangular radii cast rectangular shadows, which are 
 the axe:; of the elliptical shadow. As each radius and its shadow 
 intersects G. L. at the same point, the required pair of radii 
 and their shadows may each be inscribed in a semi-circle of the 
 same radius and having a common diameter in G. L. 
 
 To find the semi-circles revolve the plane of the circle cast- 
 ing the shadow about G. L. into H, the circle takes the position 
 as shown with center at o^, join 0^ o^, bisect 0^ o^ and draw the 
 perpendicular g i intersecting G. L. at i, with i as a center and 
 radius equal to io^ = io^ describe a circle which gives the semi- 
 circles required. Draw (3j/and 0^ e, fo^p^ and e 0^ m^, also q^ q^ 
 and n^ n^ parallel to 0^0^, then q^pa and n^ m^ are the required 
 axes. 
 
 39. To find the position [not magnitude) of the axes of an 
 Mipse, having given any pair of its cotijugate diameters.* Plate 
 II, Fig. 15. 
 
 Construction. Let oa and obh^ the semi-conjugate diame- 
 ters of an ellipse. Produce aVo c making oaXac^=ob'Xob. 
 
 Draw man parallel to ob and through o and c describe a 
 circle having its center in m a n. Draw m and o n, which are 
 the indefinite axes of the ellipse. 
 
 40. Problem XI. To find the shadow of the edge of a 
 hemispherical shell on the interior surface. Plate III, Fig. 16. 
 
 Analysis. The line of shade or part of the edge which casts 
 a shadow on the interior surface is determined by the diameter 
 of the line of shade of the sphere which is parallel to H (in 
 this case) and is the semi-circumference of the edge. 
 
 The shadow is a conic section (§ 36), in this case the arc of 
 
 *See Salmon's Conic Sections. 
 
26 SHADES AND SHADOWS. 
 
 a circle, because the only plane curve which can be cut from a 
 sphere is a circle. 
 
 Construction. Given the hemispherical shell ab c, with the 
 plane of the edge parallel to H. The diameter b c perpendicu- 
 lar to the rays and parallel to ZT is a diameter of the line of 
 shade and also of the shadow, as it joins two points b and c on 
 the line of shadow. (See Prob. X, § 34.) The radius o« is 
 perpendicular to be, hence its shadow oa^, on the plane of the 
 shadow of the edge, is a semi-diameter of the shadow conju- 
 gate to b c; but b c and a^ are perpendicular, hence the axes 
 of the projection of the shadow on H are determined. The V 
 projections of the axes be and o a^ are conjugate diameters of 
 the F projection of the shadow. To find the shadow of « on 
 the interior, either pass a meridian plane of rays through a and 
 revolve parallel to F about an axis perpendicular to /f through 
 the center of the shell, or project on a F" plane of rays 
 through G^. Z,. The V^ projection of a is «^i. Through «^i 
 draw R'\ making an angle of 35° 15' 52" with G^. Z,.: the in- 
 tersection tfji with the contour is the F, projection of the 
 shadow required ; hence a^ is the shadow of a on the interior. 
 
 Short Construction. Draw «s' o^^ and project flgi on a^i m at 
 
 n. The angle M(7^i«Ji ^ 2 angle o^'a^iaji. Therefore o^in=: 
 
 r r 
 
 o'^ flji cos 2 p = -. Hence we may at once lay off c^a^ = _ 
 
 3 3 
 
 and construct an ellipse. The shadow of the edge on the in- 
 terior is b a^c found by constructing ellipses on axes in H pro- 
 jection and conjugate diameters in V projection. Only the 
 shadow in //'projection is visible. 
 
 41. Problem XII. To find the shadow of an oblique cylin- 
 drical surface on the coordinate planes and the shadow of the 
 upper base on the interior af the surface. Plate III, Fig. 17. 
 
 Analysis. The line of shade is determined by two tangent 
 
SHADOWS OF CURVES. 2/ 
 
 planes of rays and two pencils (cylinders) of rays through the. 
 bases. The cylinder, being hollow, part of the shadow of the 
 upper base is on the interior surface and part on the coordinate 
 planes. To determine the elements of shade, pass a plane 
 through the axis (or element) of the cylinder and a ray ; this 
 plane is parallel to the tangent planes, which are now readily 
 determined. 
 
 The problem "to pass a. plane of raj/s tangent to a cylinder 
 or cone " is the same problem as " to pass a plane tangent to 
 a cylinder or cone and parallel to a given line." From § 36 
 we know that the shadow on the interior is a conic section, in 
 this case an ellipse, and hence ap/ane curve. 
 
 Construction. Given the oblique cylindrical surface with 
 one base in and the other parallel to H, as shown. 
 
 Shadow on the coordinate planes. The shadow of a the cen- 
 ter of the upper base on H is a^^ ; hence b^a^^ is the H trace of 
 a plane of rays through the axis, because it contains the H 
 trace of the axis and the H trace of a ray through a. The H 
 traces of the two planes parallel to this plane and tangent to 
 the cylinder are (^r^ and e^ x^, and the elements of shade are 
 therefore cdan^ef, which cast shadows c^r^d^ and e^x^f^ 
 respectively. The line of shade (edge) of the upper base is 
 divided in two parts by the elements of shade, the shadow of 
 fgd being on the coordinate planes and the shadow oifj'd 
 on the interior. 
 
 Draw the rectangular diameters df and J g: their shadows 
 on V are conjugate diameters of the shadow of the upper 
 base; hence, constructing an arc of an ellipse on d^f^ and j\g^ 
 limited by the points ^^ and d^, we have the complete shadow 
 of the surface on the coordinate planes. 
 
 Shadow on the interior surface. Since df is a diameter of 
 the line of shade and contains two points of the shadow it is a 
 
28 SHADES AND SHADOWS. 
 
 diameter of the shadow. The shadow oij, the extremity of t-he 
 rectangular diameter gj, \sj\, found as follows: Through the 
 element tj pass a plane of rays ; it intersects the cylinder in the 
 element wg\ a ray through/ intersects w^ at /s, hence /s is 
 the shadow of/ on the interior. A plane through fd and j^ 
 is the plane of the line of shadow (see § 36), because the curve 
 of shadow is a plane curve, and this plane contains three points 
 of that curve. 
 
 Produce /s a to k on the element //, then/g a = ak and df 
 and/s/fe are conjugate diameters of the elliptical shadow. A 
 semi-ellipse constructed on d^ f^ and cFj\ is the //projection 
 of the shadow, and a semi-ellipse on/^ d?"' and a^'/s is the V 
 projection of the shadow. The visible shadow on the interior 
 is bounded by the line/^'/g u d^j'^f^, and on the coordinate, 
 planes by the line c^ r^ y i" x^ z c^- 
 
 42. Another solution for finding shadow on interior of the 
 surface of preceding problem. Same figure. 
 
 Analysis. The intersection of the umbra of any curve with 
 another curve is the shadow of the first curve on the second 
 (see § 30). 
 
 Hence the intersection of a pencil of rays through the upper 
 base of the cylinder with any other section is the shadow of 
 the base on that section. 
 
 Construction. Pass any auxiliary plane through the cylinder 
 as the horizontal plane X\ it cuts out the section B. The 
 shadow of the upper base on the plane X is A^, the circum- 
 ference of a circle of radius equal to that of the base, since the 
 plane of the base and Xare parallel. A^ and B intersect at/^ 
 and q^, hence these points are two points of the required 
 shadow. A sufficient number of planes to accurately determine 
 the line of shadow must be employed. 
 
 43. Problem XIII. To find the shadow of a conical surface 
 
SHADOWS OF CURVES. 29 
 
 on the coordinate planes and tJie shadow of the base on tJie interior 
 of the surface. Plate III, Fig. 18. 
 
 Analysis. This problem presents a fact concerning tangent 
 planes to cones which is often overlooked. The elements of 
 shade are determined by two tangent planes of rays. As these 
 tangent planes are planes of rays each must contain the ray 
 the which passes through the vertex of the cone, hence this ray is 
 line of intersection of the two planes. Using the conventional 
 direction of rays (§ 5) this line of intersection or ray cannot be 
 parallel to the coordinate planes, but must intersect them. 
 The ZT and F traces of the tangent planes must pass through 
 the corresponding traces of this line, hence the //^ traces of 
 these planes cannot be parallel, and the F traces cannot be 
 parallel. As in the preceding problems, the shadow on the in- 
 terior is a conic section, in this case an ellipse. 
 
 Notice the fact that in this problem the line joining a and 
 3 is a chord and not diajtieter of the line of shadow as the line 
 fd\n the preceding problem (§ 41). 
 
 Construction. Let the conical surface be given as shown 
 with axis perpendicular to H, and base the circumference 
 of a circle parallel to H. To find the elements of shade pass 
 a ray through the vertex o : it intersects the plane of the base 
 at 7'; henceya andjb drawn tangent to the base are the inter- 
 sections of the tangent planes of rays with the plane of the base, 
 and a and o b are the elements of shade. The shadows oi o a 
 and b are o^ 4 a^ and o^ k^ b^ respectively. The shadows of any 
 two rectangular diameters of the base as y"^ and c ^ give con- 
 jugate diameters of the shadow of the base. Hence an ellipse 
 constructed on fg dg and c^ e^ as conjugate diameters is the 
 shadow of the base on V. But as the line of shade A is di- 
 vided in two parts by the elements of shade and the shadow of 
 
so SHADES AND SHADOWS. 
 
 the ^xc alb is on the interior, the shadow of the base on Fis 
 limited by the shadows of a and b. Hence Os K b^ d^ c^ «s 4 ^'s Js 
 the shadow on the coordinate planes required. 
 
 Shadow on the interior surface. The chord a b oi the line 
 of shade is a chord of the interior line of shadow, since it joins 
 two points a and b of that shadow. To find diameters of the 
 line of shadow revolve the meridian plane containing a ray 
 about the axis of the conical surface parallel to V. The point 
 /, one extremity of the diameter of the line of shade perpen- 
 dicular to a b, takes the position /j, and a ray through / in re- 
 volved position is /^ /gj, hence the revolved position of the 
 shadow of / is 4, and the true position is 4. The plane of the 
 line of shadow, therefore, passes through the points a, b, 4 (see 
 § 36). In revolved position (perpendicular to Fin this case) 
 this plane contains the line a^ b^ (vertically projected at gX) and 
 the point 4ii and intersects the revolved position of the element 
 <? / at w,, hence 4i '^1 is the revolved position of a diameter 
 (axis) of the elliptical shadow. Bisect 4i «i at w?, ; a chord of 
 the cone /, r^ through m^ perpendicular to 4, ^i and in the 
 plane a^b^,g^ is the revolved position of another diameter 
 (axis) of the shadow. Revolve back to the original position 
 and we have the lines l\ri^, i' r^ for conjugate diameters 
 of the V projection, and If n^, i^ r^ for the axes of H pro- 
 jection of the shadow. Constructing arcs of ellipses thus 
 determined and limited by the points a and b, we have the 
 shadow on the interior, only the H projection of which is 
 visible. 
 
 The method of § 42 might also have been employed for 
 finding the interior shadow. 
 
 The small figure above Fig. 18 shows the shadow of a right 
 circular cone on the coordinate planes. 
 
SHADOWS OF CURVES. 31 
 
 44. Problem XIV. To find the shadow of a horizontal 
 circle on a diagonal plane.* Plate III, Fig. 19. 
 
 Analysis and Construction. A diagonal plane is a vertical 
 
 plane inclined 45° to V. Let m n be a diameter, perpendicular 
 
 to a ray, of the horizontal circle abed: its shadow on the 
 
 diagonal plane P is parallel to //'and equal in length to in n, since ' 
 
 mn is parallel to P. The shadow of the rectangular diameter 
 
 p q ox\ P \i perpendicular to H and in length = pq tan 0. The 
 
 V projection of the shadow of in n is in\ n\ parallel to H 
 
 and in length ^-mn sin 45° = vin tan 9, and the F projection 
 
 of the shadow oipq is perpendicular to //'and of length — p^q^ 
 
 — pqtan 6. But m n =/^, being diameters of the same circle; 
 
 iience inn tan = pq tan 0, and the V projection of the 
 
 shadow of the circle a b c d on the diagonal plane is a circle of 
 
 r 
 radius = r tan = —-=. This method may be used in Probs. 
 
 V2 
 
 XII, XIII, and XVI. 
 
 45. Problem XV. To find the shadow of a surface of revo- 
 lution on the coordinate planes. First case : The surface concave 
 towards the axis. Plate IV, Fig. 20. 
 
 Analysis. The typical surface of revolution taken is the 
 torus or annulus. 
 
 Meridian plane method for determination of the line of 
 shade. If a ray be projected (orthographically) on any merid- 
 ian plane of the surface the projecting plane is a plane of rays 
 (§ 7), and if planes be passed parallel to this plane and tangent 
 to the surface the points of tangency are points of the line of 
 shade (§ 9). The line of shadow on the coordinate planes is 
 not a conic section, hence it is most readily determined by the 
 
 * This method for solving certain problems is due M. Jules Pillet, ficole 
 Nationale des Fonts et Chaussees. 
 
32 SHADES AND SHADOWS. 
 
 intersections of individual raj's through the line of shade. As 
 this part of the problem presents no difficulty, the line of 
 shadow is not shown for the outer curve. 
 
 Construction. Given the torus as shown with axis perpen- 
 dicular to H. Let Y be any meridian plane of the torus and / o 
 a ray intersecting this plane at o. The projection oipoon 
 the plane Y\swo, found by dropping a perpendicular from/ 
 to the plane. Revolve the plane F about the axis of the torus, 
 X, until parallel to V. The line wo takes the position w^o\ 
 hence the lines Q^,M^, P,, A'', parallel to this line and tangent 
 in F projection to the contour of the torus are the revolved 
 positions of the intersections of tangent planes of rays with the 
 meridian plane Y, hence the revolved positions of four points 
 of the line of shade are determined. Revolving back we have 
 the points q, r, a and t of the line of shade. The points in the 
 meridian plane Z found by the same method are /, k, m and n. 
 In this manner all points of the line of shade, which is com- 
 posed of the two curves Iq nt I on the concave and krmakon 
 the convex side of the torus were determined. 
 
 Analysis of inner line of shade. It does not follow that a 
 line of shade which fulfils only certain geometrical conditions 
 casts a shadow ; this is paiticularly true of surfaces convex to- 
 wards the axis, like the inner part of a torus or the figure of 
 Prob. XVI (Fig. 21). The complete shadow of krmak on 
 H is k^y i^iins^bgizk^. This line crosses itself at i^ and b^ 
 which shows that the parts of the line of shade irdi^i and 
 bfa bsi do not cast shadows on H. From i to r and from b 
 to f the shadow is on the surface of the torus, while from d 
 to 4i and from a to bgi the line is in shadow, and hence can- 
 not cast a shadow. 
 
 From r to d and from f to a the H projection is convexed 
 towards the axis, and if a plane of rays be passed parallel to Z 
 
SHADOWS OF CURVES. 33 
 
 between these points and the inner circle of the torus it will cut 
 the line of shade in'four points. 
 
 Such a plane cuts a curve from the torus similar to that 
 shown in Fig. 20a (the right and left ends of the curve are not 
 shown). To this curve four rays may be drawn tangent as 
 shown : one at a which is a point on the line of shade and casts 
 a shadow on the curve near b, and the other three at b, c and d 
 which are not points on a true line of shade, although geometri- 
 cally so determined, because the rays intersect the torus before 
 touching the curve. The true line of shade on the convex 
 surface of the torus is the line fbkir. The shadow on the 
 torus may be found by the methods of previous problems or 
 by the secant plane method as indicated. 
 
 46. Problem XVI. To find the shadow of a surface of 
 revolution on the coordinate planes. Second case: The surface 
 convex towards the axis. Plate IV, Fig. 21. 
 
 Analysis. The typical surface taken is a scotia or base of 
 a column, generated (in this case) by the revolution of two tan- 
 gent arcs of circles joined to two lines about an axis situated 
 in the plane of all the lines. (See figure). 
 
 Method of inscribed tangent surfaces. If two surfaces of 
 revolution are tangent along the circumference of a circle their 
 lines of shade intersect, for the pencils of rays tangent to the 
 surfaces intersect at some point (or points) common to the two 
 surfaces. As the line of shade of a sphere or cone is readily 
 determined, inscribed tangent spheres or cones maybe used to 
 find the line of shade of the given surface. The problem is. 
 naturalU' divided into four parts : — First : To find the complete- 
 line of shade. Second : To find the shadow of the upper cylin- 
 der on the surface and on the coordinate planes. Third : To 
 find the shadow of the double curved portion. Fourth : To find 
 the shadow of the lower cylinder. 
 
34 SHADES AND SHADOWS. 
 
 Construction. First : The line of shade of the upper cylin- 
 der is composed of the two elements y z and Y and half of each 
 of the bases B and C. The line of shade of the lower cylin- 
 der is composed of the two elements W and Z, half of the lower 
 base and parts of the upper base A limited by the elements 
 W and Zand the shadow of the concave portion. Line of shade 
 of the double curved portion. Let acd he. any horizontal sec- 
 tion of the surface cut by the horizontal plane T, also the line 
 (circle) of tangency of an inscribed tangent sphere F. The 
 radius of this sphere, ao, is determined by drawing iao in the 
 principal meridian plane of the surface (J/) pei-pendicular to a 
 tangent (not shown) to the surface at a. The plane of the 
 line of shade of the sphere is perpendicular to a ray (§ 34) ; 
 hence its traces are perpendicular to the respective projections 
 of a ray, hence 0^ V perpendicular to R" is the F projection 
 of the intersection of the plane of the line of shade of the 
 sphere with the principal meridian plane of the surface M. The 
 line b pierces the plane oi a cd (circle of tangency) at b ; then 
 d^ b^ c^ perpendicular to i?" is the ZT projection of the inter- 
 section of the plane of the line of shade of the sphere with the 
 plane of the circle of tangency, since the plane of the circle of 
 tangency is parallel to H\ hence d and ^, the points of intersec- 
 tion oi d b c and this circle, are points of the line of shade re- 
 quired, for they are points of the line of shade of the sphere and 
 of the line of tangency of the sphere and the given surface. In 
 a like manner the points /, 2,^,4, etc., were determined. The 
 line, of shade of the double-curved portion is C2l6^d4^c. 
 Alternate method by inscribed tangent cones. Let the circle 
 acd (see before) be the line (circle) of tangency and also the 
 base of an inscribed tangent cone. The vertex of the cone is 
 e^, <5", found by producing the elements seen in V projection 
 until they intersect the axis. (Note : If the base of the cone 
 
SHADOWS OF CURVES. 35 
 
 were assumed above a horizontal plane through k the vertex 
 would be below, instead of above, the base.) The shadow 
 of the cone on the plane 7" of the base is found by drawing e^ c 
 and es,d tangent to the base (see §43), hence ec and ed are 
 the elements of shade. These elements intersect the line (circle) 
 of tangency of the cone and given surface, or in other words 
 intersects the base of the cone, at c and d ; hence these points 
 are points in the required line of shade and coincide with the 
 points found by the sphere method. 
 
 Second : Shadow of the upper cylinder. The shadows of 
 the elements J)/ ^^ and Y are ys,Zs and Fg respectively. The 
 shadow of the upper base B on F is part of an ellipse ^gj lim- 
 ited by the point z^ and G. L., and constructed on the shadows 
 '2s?s> ^sfs of the rectangular diameters n q and rf. On H 
 the shadow is B^^i, an arc of a circle, since B is horizontal. 
 The shadow of the lower base C on V is C^^t ^ part of an ellipse 
 constructed on the shadows w^x^, t^ti^ of the rectangular di- 
 ameters w X and t u ; on H the shadow is composed of two 
 arcs of the same circle whose center is p^^ and radius = p^ 
 /g=/ if, since C is horizontal. The shadow on the double- 
 curved surface is determined by the method of § 42. Let Z 
 be any horizontal secant plane ; it cuts from the surface the 
 circumference G. The shadow of C on this plane is found by 
 constructing an arc with/gi, the shadow of / on the plane, as 
 a center and radius = radius of C. This arc intersects G at 
 ■m^ and 4 ; hence these points are two points of the required 
 shadow. Other points were determined by the same method. 
 The shadow of C on the surface is Qj. The line of shade of 
 the double-curved portion intersects C^ at the points 2 and 
 5, at which points rays are tangent to the curve C^. Only 
 that part of C^i which is on the left of or above the line 2, 5 is 
 on the outside of the scotia ; the remainder of the line (on the 
 
36 SHADES AND SHADOWS. 
 
 right of or below 2, 5) can only exist as a shadow on the sup- 
 position that the scotia is hollow and that the upper cylinder 
 is removed ; under these conditions this part of the shadow _ 
 of Cis on the interior. 
 
 Third : Through the points /, _?, 4, and 6 of the line of 
 shade of the double-curved portion draw tangent rays ; these 
 rays divide the line into four parts, as follows: The part above 
 /, 6 is on the interior or convex side of the surface, which is 
 readily seen by drawing a ray tangent to the surface at some 
 point above these points, e.g. in the meridian plane of rays 
 through /, /s2 (see § 45). Such a ray is within the scotia and 
 intersects the curve cut from the surface by the plane in a 
 point on the side opposite the point of tangency. At the 
 point / (and 6) the point of tangency and the point of inter- 
 section of the ray coincide, hence at this point the line of 
 shade changes from the convex to the concave side of the sur- 
 face. The parts J 2 ) and 6^4 are on the concave side while 
 the remainder (below ^, 4) is on the interior or convex side. 
 We have shown that the shadow of C on the surface intersects 
 the line of shade at the points 2 and 5; hence the part of the 
 line of shade 2 i 6 ^ (i. e. all above 2,5) is in shadow and 
 cannot cast a shadow even though the parts 2 / and 5 6 are on 
 the concave side of the surface, hence the only portions of the 
 entire line which cast a shadow are the parts 2 5 and 5 4. The 
 shadows of 2 _J and 5 4 are partly on the coordinate planes and 
 partly on the surface. The shadow on the surface is deter- 
 mined by the method of §§ 30 and 42, thus : Find the shad- 
 ows on H by individual rays ; they are 2s ^^ and 5^ 4^. Pass 
 any horizontal secant plane as U below the points _J, 4, it cuts 
 the circumference L from the surface. The shadow oiLon H 
 is Zs, which intersects the shadow of the line of shade at ^s2 
 andysji a ray through each of these points intersects Z at g^i 
 
SHADOWS OF CURVES. 37 
 
 and/si, hence these points are points of the required shadow. 
 The shadow thus determined is J^<^, only part of which is vis- 
 ible as shown. The shadow on HisJ^i limited by Qi and A^ 
 which is the shadow of the base A (see below.) 
 
 Fourth : The shadows of the element Z and W a.ve Z^ and 
 W^ respectively, and of the upper base A is A^, as shown. 
 As the lower base D is in H, it is its own shadow. 
 
 The complete shadow both on the surface and the coordi- 
 nate planes is now determined. The visible shadow on the 
 coordinate planes is bounded by the lines Z^As/siC^i Y^B^^ 
 B^ , thence along the V contour of the surface to G. L., thence 
 along G. L. to 4> thence CsiJs\As W^ A^. The visible 
 shadcAV on the surface is, in V projection, bounded by the 
 lines Z^, part of A"', Jl^, 4", ^ , Cjj, thence along the V con- 
 tour of the surface to f, C^ Y", thence along B' to r", r^ u^ , 
 thence along the ^contour ol the sJrrace Lo G. L., thence 
 along G. L. to t^ In H projection the visible shadow of the 
 surface is bounded by the lines y"2 C^J^^^- 
 
 47. Problem XVII. To find the shadotv on the siirface of 
 a triangular-threaded screw. Axis vertical. Plate IV, Fig. 22. 
 
 Analysis. The surfaces of a triangular-threaded screw are 
 warped helicoids, and the outside and the root of the thread 
 are helices. 
 
 Planes tangent to a warped helicoid at points on the same 
 helix make equal angles with the axis and equal angles with 
 H. (See Des. Geom.) If a series of meridian planes of the 
 helicoid be passed perpendicular to these tangent planes, the 
 lines cut from the tangent planes make a constant angle with 
 H equal to the angle which the tangent planes make with H; 
 and with the elements passing through the points of tangency 
 a constant angle, since the relative positions are the same in 
 all cases. Hence the angle between the H projection of any 
 
38 SHADES AND SHADOWS. 
 
 line of intersection of the meridian and tangent planes and 
 the ^projection of the corresponding element is constant. If 
 now a plane of rays tangent at some point of the given helix 
 be determined, the element containing the point of tangency 
 and hence the point of tangency may be determined. Having 
 determined the line of shade, construct the line of shadow by 
 individual rays. 
 
 Construction. Let A be an element of one of the helicoidal 
 surfaces of the screw and C the helix on which a point of the 
 line of shade is to be determined. 
 
 The element A pierces H at a, and a tangent line to the 
 helix C aX b (the intersection of A and C) pierces /f at (^, 
 hence H T \s the horizontal trace of a tangent plane. H U is. 
 the U trace of a meridian plane perpendicular to the tangent 
 plane T, hence the angle x oj is the constant angle made by 
 the H projection of the intersection of the two planes with the 
 H projection of the element passing through the point of tan- 
 gency. The angle which D, the line of intersection of the 
 planes T and U, makes with H is equal to the angle which any 
 tangent plane to the helicoid at a point of the helix C makes, 
 with H. Revolve D about the axis X; a cone is generated 
 with vertex at {d', o) and base Q in H. A ray M through the 
 vertex of the cone pierces ZT at ^; hence ^Z*, through ^ and 
 tangent to the base, is the H trace of a plane of rays tangent 
 to the cone, and hence is parallel to the //"trace of a plane of 
 rays tangent to the helicoid at some point of C. 
 
 The H trace of a meridian plane perpendicular to the plane 
 P is o/; hence making the angle yoin^ = angle x oy we have 
 the element which passes through the required point of tan- 
 gency. This element intersects the helix C at m; hence m is 
 the point of tangency and therefore a point of the line of 
 shade. 
 
SHADOWS OF CURVES. 39 
 
 A point of the line of shade on the helix B is determined in 
 the same manner, thus : H Q'ls the /^ trace of a tangent plane 
 through the element F on which is the point ti of B, and o i is 
 the //"trace of a perpendicular meridian plane; hence ioii^ is 
 the constant angle. The line G is the intersection of the me- 
 ridian and tangent planes and pierces H aX i; hence C^ is the 
 base* andy, o the vertex of a cone generated by rotating G 
 about the axis, and H R is the H trace of a plane of rays 
 tangent to the cone. 
 
 The line o k is the H trace of a meridian plane perpendicu- 
 lar to the plane R ; hence laying off the angle k o g^ = i ou^ we 
 have the element which intersects the helix B 3Xg, thus deter- 
 mining another point of the line of shade. 
 
 To determine an intermediate point of the line of shade : 
 Let K^ be the H projection of an intermediate helix. For 
 convenience pass an auxiliary horizontal plane Z intersecting 
 the helix at w. The intersection with the plane Z oi a. tangent 
 plane to the helicoid, through the element A at some point of 
 the hehx K, is L and of a perpendicular meridian plane is oz; 
 hence zoy is the constant angle. The intersection of the me- 
 ridian and tangent planes is H, which revolved about the axis 
 generates a cone with base T on the auxiliary plane Z. U is 
 the intersection with Z oi z. plane of rays tangent to the cone, 
 and ^7- of the perpendicular meridian plane; hence laying off 
 the angle r o t" = angle z o y the element containing the re- 
 quired point t is determined. The Fprojection of t is t" on the 
 V projection of the element (not shown). The line of shade on 
 the lower thread \s gtm and part of the helix B on the right, 
 of^. 
 
 As the several threads of the screw are alike in all respects, 
 
 * Almost coincides with C^. 
 
40 SHADES AND SHADOWS. 
 
 g^ t^ m^ is the H projection of all the lines of shade on the 
 front heHcoids, hence npq is the line of shade of the second 
 thread. The visible line of shade is composed of the lines 
 in tg, np q and part of the helices on the right of ^ and n. 
 
 The Shadow. The shadows of npq and N on H are q^a, 
 ps2 , etc. (this line is on the right of the border, and hence is not 
 shown). The shadow of any element P of the helicoid .Zis P^. 
 This shadow intdrsects the shadow of the line of shade at/sj; 
 hence passing a ray through ps2 the shadow p^i on the ele- 
 ment P is determined. (See § 30.) In this manner the shadow 
 on V projection q^ p^si «s 4 was determined. The shadow on 
 the lower thread is of exactly the same form. The JI projec- 
 tions of the shadows and the shadows on H are not shown. 
 
 Problems. 
 
 1. Find the shadow of an oblique line on an oblique plane. 
 
 2. Find the shadow of a line parallel to G. L. on an oblique 
 plane. 
 
 3. Find the shadow of a line situated in a profile plane on 
 the coordinate planes. 
 
 4. Find the shadow of a pyramid, base on H, on the coor- 
 dinate planes. 
 
 5. Find the shadow of an inverted pyramid, base parallel to 
 H, on the coordinate planes. 
 
 6. Find the shadow of a pyramid, base parallel to H, on the 
 coordinate planes. 
 
 7. Given a square pyramid on a square pedestal, find the 
 complete visible shadow. 
 
 8. Given a square pyramid on a cylindrical pedestal, find 
 the complete visible shadow. 
 
 9. Find the shadow of any parallelopipeji on the coordinate 
 planes. 
 
SHADOWS OF CURVES. 4 1 
 
 10. Find the shadow of a hollow hexagonal prism on the 
 interior surface and on the coordinate planes. 
 
 11. Find the shadow of a vertical cross. 
 
 12. Given a vertical cross on a cylindrical pedestal, find the 
 complete visible shadow. 
 
 13. Given a square column surmounted by a square abacus, 
 find the complete visible shadow. 
 
 14. Given a hexagonal column surmounted by a square 
 abacus, find the complete visible shadow. 
 
 15. Find the complete shadow of a rectangular box with 
 its lid partly raised. 
 
 16. Given a book-case containing three shelves, find the 
 complete shadow. 
 
 17. Given a square-topped table with four square legs, find 
 the complete shadow. 
 
 18. Find the complete shadow of any building having the 
 plan and elevation. 
 
 19. Given a hollow cylinder, axis parallel to G. L., find the 
 complete visible shadow. 
 
 20. Given a hollow cylinder, axis perpendicular to H or V, 
 find the complete visible shadow. 
 
 21. Find the shadow of a cone, axis perpendicular to V, on 
 the coordinate planes. 
 
 22. Find the shadow of any oblique cone on the coordinate 
 planes. 
 
 23. Given a hollow cone, axis parallel to G. L., base on left, 
 find the complete shadow. 
 
 24. Given a vertical cylinder surmounted by a square 
 abacus, find the complete visible shadow. 
 
 25. Given a cylindrical column surmounted by a cylindrical 
 abacus, find the complete visible shadow. 
 
42 SHADES AND SHADOWS. 
 
 26. Given a cylindrical column surmounted by a hexagonal 
 abacus, find the complete visible shadow. 
 
 27. Find the shadow of an ellipsoid of revolution, axis per- 
 pendicular to H or V. 
 
 28. Find the shadow of the frustum of a cone. 
 
 29. Find the complete shadow of the frustum of a hollow 
 cone. 
 
 30. Given a small vertical cylinder intersecting a large hori- 
 zontal cylinder, find the complete visible shadow. 
 
 31. Find the complete visible shadow of a groined arch. 
 
 32. Find the complete visible shadow of a cylindrical niche 
 capped by a quarter sphere. 
 
 33. Find the shadow on the interior of a recess of any 
 shape. 
 
PERSPECTIVE. 
 
 CHAPTER IV. 
 
 GENERAL PRINCIPLES, DEFINITIONS AND CONVENTIONS. 
 
 48. Perspective in Nature. The phenomena of perspec- 
 tive in nature are so familiar that they scarcely require descrip- 
 tion. St.and at one end of a long straight street and look 
 towards the other end ; the rows of trees on the sides, the 
 fronts of the buildings, the sidewalks and curb-stones, seem tO' 
 approach each other and grow gradually smaller as they recede 
 from the spectator. Straight railroad tracks make an excel- 
 lent example of perspective, especially if they be several miles 
 long. If you stand on one rail and look along the tracks all 
 the rails will apparently meet it at a distant point. At the sea 
 shore or on shipboard the sky and water seem to meet at the 
 horizon — the effect is the same as if the sky and water were 
 
 'two vast horizontal planes. It is this phenomenon or effect of 
 distance which the draftsman tries to reproduce in a perspec- 
 tive drawing. 
 
 49. Deductions. Examples of natural perspective might 
 be multiplied indefinitely, but all teach the same facts, z. e., 
 that any series of parallel lines if sufficiently produced appar- 
 ently approach each other and meet in a point, and that any 
 series of parallel planes if sufficiently extended apparently ap- 
 proach each other and meet in a line ; also, that if a line be 
 
 43 
 
44 PERSPECTIVE. 
 
 passed through the eye parallel to any series of parallel lines, 
 it will pass through their meeting point, and that if a plane be 
 passed through the eye parallel to any series of parallel planes, 
 it will pass through their' meeting line. 
 
 As lines or planes may be extended in two directions an 
 infinite distance from any given point, there are two meeting 
 points or lines 1 80° removed from each other for each series, 
 but as the spectator can only look in one direction at a time it 
 is not necessary to consider the second case. In the case of 
 parallel planes the meeting line would become the circumfer- 
 ence of a circle because planes may be infinitely extended in 
 all directions. 
 
 50. Methods. The method of perspective in general use 
 by draftsmen is called One-plane Perspective, and difiersin many 
 essentials from that used by artists, which is called Cylinder or 
 Multiplane Perspective. 
 
 The drawing produced by the One-plane method is the 
 same as that which would be made by placing a sheet of trans- 
 parent paper between the object to be represented and the 
 spectator, and covering or concealing every line which is seen 
 on the object by a line on the paper. NOTE : For purposes 
 of representation the paper might just as well be placed be- 
 hind the object as in front of it, although not usually so con- 
 sidered. In the first case the drawing is smaller than the ob- 
 ject, in the latter case larger. 
 
 In Multi-plane Perspective a transparent cylindrical surface 
 is supposed to be placed between the spectator, whose eye is 
 in the axis of the cylinder, and the object ; lines are then 
 drawn covering or concealing those on the object, and the cyl- 
 inder is developed. This method is essentially that used by 
 artists, but the difficulty of considering it geometrically is 
 great and the practical application tiresome. 
 
GENERAL PRINCIPLES, DEFINITIONS, ETC. 45 
 
 51. Relation to Geometry. Suppose a plane placed be- 
 tween a given point and an object, and the object projected 
 on the plane by means of lines drawn from the point to the 
 object ; the drawing obtained will be a duplicate of that made 
 by the process described in § 50. In this case the natural 
 method has been considered geometrically. The lines from 
 the point to the object, i. e., the projecting lines, form a cone 
 because they radiate from a point, hence the object is conically 
 projected. If the object to be projected be a point, the pro- 
 jecting cone becomes a line ; if the object be a line, the pro- 
 jecting cone becomes a plane, etc. 
 
 52. Definition. One-plane * Perspective is Conical projection. 
 Note : If the student will remember that One-plane Perspective 
 mtSins projection from a fixed center upon a plane, much of the 
 difificulty sometimes found with the subject may be avoided. 
 
 In Descriptive Geometry the projecting lines are parallel, 
 in Perspective the projecting lines radiate from a point. 
 
 53. Picture-Plane. The Picture Plane (also called Per- 
 spective Plane and Plane of Projectioii) is the plane upon which 
 the object is projected or the drawing is made. It is usually 
 assumed vertical. 
 
 54. Perspective. The conical projection of an object on 
 the picture plane is called the Perspective of the object. 
 
 Any point or line in the picture plane may be considered 
 to be the Perspective of some point or line in space. 
 
 55. Station Point. The Station Point (also called the 
 Point of Sight) is the point from which the object is projected 
 on the picture plane. It is the vertex of the projecting cone 
 of the object. This point may, in general, be placed anywhere 
 within a finite distance of the picture plane, but the position 
 
 * Also called Linear Perspective. 
 
46 PERSPECTIVE. 
 
 once selected must remain fixed for any one drawing. In 
 order that a perspective drawing may appear natural or give 
 one the correct impression of the object represented, the eye 
 of the spectator must be placed at the station point, otherwise 
 the drawing will appear distorted. 
 
 56. Center of the Picture. The Center of the Picture is 
 the orthographic projection of the station point on the picture 
 plane. 
 
 57. Axis. The Axis of the picture is the perpendicular 
 projecting the station point on the picture plane. 
 
 58. System. Any series of parallel lines is called a System of 
 lines ; any series of parallel planes is called a System of planes. 
 
 59. Axis of a System. The Axis of a System of planes 
 is any line perpendicular to the system. 
 
 60. Vanishing Point. Given a System of lines and a 
 point : If a plane be passed through each line and the point 
 the common intersection of the planes is parallel to the System 
 of lines. Given a System of lines, the Station Point and the 
 Picture Plane : If a plane be passed through each line of the 
 system and the station point, the intersection of each plane 
 with the picture plane is the Perspective of the line through 
 which the plane is passed (§ 54). But these planes have a 
 common intersection which pierces the picture plane in a point, 
 and the intersection of each plane with the picture plane passes 
 through this point ; hence the Perspective of each line passes 
 through this point. The point of intersection of the perspec- 
 tives of a system of lines is called the Vanishing Point of that 
 system or of any line of that system. 
 
 The Vanishing Point of a line may also be defined as the 
 perspective (§ 54) of that point on the line which is at an in- 
 finite distance from the station point. Since perspective is 
 conical projection and parallel lines intersect only at infinity. 
 
GENERAL PRINCIPLES, DEFINITIONS, ETC. 47 
 
 the projection (perspective) of this point of intersection on tlie 
 picture plane must be determined hyaline through the station 
 point parallel to the given line. 
 
 6i. Determination of Vanishing Points. Since the 
 common intersection of the projecting (conical projection) 
 •planes of a system of lines is parallel to the system and passes 
 through its vanishing point (§ 60), for any system of lines the 
 "vanishing point is determined by the intersection with the 
 picture plane of a line of the system through the station point. 
 
 62. Vanishing Points of Systems of Lines Parallel 
 to the Picture Plane. The perspectives of a system of lines 
 parallel to the picture plane are parallel to each other and to 
 the system, for the common intersection of their projecting 
 planes is parallel to the picture plane and therefore intersects 
 it only at infinity. This fact limits but does not violate the 
 preceding rule (§61), 
 
 63. Vanishing Trace. The Vanishing Trace of a plane 
 is the locus of the vanishing points of all lines in the plane. 
 
 64. Horizon. The Horizon is the vanishing trace of hori- 
 zontal planes or the locus of the vanishing points of horizontal 
 lines. It is the horizontal line in the picture plane which 
 passes through the center of the picture (§ 56). 
 
 65. Determination of Vanishing Traces. For any sys- 
 tem of planes the vanishing trace is determined by the inter- 
 section with the picture plane of a plane of the system through 
 the station point. (See § 61.) 
 
 66. Relative Position of Vanishing Points and Van- 
 ishing Traces. From §§ 6i and 63 it follows that the vanish- 
 ing point of a line is in the vanishing trace of any plane which 
 contains that line. Also, that if a line be the intersection of 
 two planes its vanishing point is the intersection of the vanish- 
 ing traces of the two planes. 
 
48 PERSPECTIVE. 
 
 67. Initial Point. The Initial Point of a line is the point 
 in which the line pierces the picture plane. As the picture 
 plane is usually assumed vertical, initial point is only another 
 name for vertical trace. Throughout this book the initial point 
 of a line is the vertical trace of the line, and the initial line (§ 68) 
 of a plane is the vertical trace of the plane. As the initial point 
 of a line is in the picture plane it is one point in the perspective 
 of the line. 
 
 68. Initial Line. The Initial Line of a plane is the line 
 in which the plane intersects the picture plane. The initial 
 line of a plane is parallel to the vanishing trace of the plane 
 (§§ 63 and 65). 
 
 69. The Initial Point of a line determines the position of 
 the perspective of the line. The Vanishing Point of a line 
 determines the direction of the perspective of the line. 
 
 70. Principal Lines of an Object. The Principal Lines 
 of an object are the systems of lines which most frequently 
 appear in the representation of the object. 
 
 Take, for example, a building such as a factory: the princi- 
 pal lines shown in the drawing belong at most to but three 
 systems. First : On the front of the building, the horizontal 
 lines, such as the courses of brick, water-tables, window-sills, 
 etc. Second : On the side of the building, the horizontal lines 
 corresponding to those on the front. Third : On the front and 
 side, the vertical lines, such as the edges of the building, window- 
 frames, etc. 
 
 Hence in order to make a perspective drawing of an object 
 which consists essentially of systems of right lines, it is neces- 
 sary in general to determine but three vanishing points. 
 
 71. Position of Object to be Represented. Objects 
 which have three (or less) systems of principal lines may be 
 placed, with reference to the picture plane, in three general 
 
GENERAL PRINCIPLES, DEFINITIONS, ETC. 49 
 
 positions. First : With the systems parallel to, and perpendic- 
 ular to, the picture plane. Second : With one system parallel 
 to, and two systems oblique to, the picture plane. Third : 
 With the three systems oblique to the picture plane. 
 
 72. Parallel or One-Point Perspective. The repre- 
 sentation of an object situated as in Case I, § 71, is called 
 Parallel or One-Point Perspective, because but one vanishing 
 point is required to make the drawing. 
 
 73. Angular or Two-Point Perspective. The repre- 
 sentation of an object situated as in Case II, § 71, is called 
 Angular or Two-Point Perspective, because two vanishing points 
 are required to make the drawing. 
 
 74. Oblique or Three-Point Perspective. The repre- 
 sentation of an object situated as in Case III, § 71, is called 
 Oblique or Three-Point Perspective, because three vanishing 
 points are required to make the drawing. 
 
 75. Notation. 
 
 (i) The position of the station poiitt is denoted by e. 
 
 (2) The center of the picture is denoted by e^ (see § 56). 
 
 (3) I" general, when the projections of an object are attached 
 to the drawing, the perspective of appoint is denoted by the letter 
 of the point with the exponent ' ; e. g., a', b', etc. 
 
 (4) The perspectives of a system of lines are denoted by the 
 letter of the system with the exponents ', '', ', etc. ; e. g., R", 
 R^ etc. 
 
 (5) The vanishing point of a system of lines is denoted by a 
 capital V with an exponent which is the letter of the system, 
 e. g., VK, yP etc. 
 
 (6) The vanishing trace oi a system of planes is denoted by 
 a capital T succeeded by the letters of any two systems of lines 
 in the planes, e. g., T R P., T M N, etc. 
 
5° PERSPECTIVE. 
 
 (7) The initial point of a line is denoted by a capital I with 
 an exponent which is the letter of the line; e.g., P, P, etc. 
 
 (8) The initial line of a plane is denoted by a capital I suc- 
 ceeded by the letters of any two lines in the plane, or of any 
 line in the plane, or by the letter of the plane ; e.g., I R B, I M, 
 I E, etc. 
 
 (9) K point of distance for a system of lines is denoted by a 
 capital D with an exponent, which is the letter of the system ; 
 e. g, D^ D^ etc. 
 
 (10) A point of proportional distance for any line is denoted 
 by a capital P with an exponent, which is the letter of the line; 
 e.g., P^ P^ etc. 
 
 76. Perspective is Descriptive Geometry. Plate V, 
 Fig. 23. This figure is the Descriptive Geometry representa- 
 tion of three lines, Q, W, and U, and of a plane P, passed 
 through the intersecting lines Q and W. 
 
 To find the vanishing points of the lines Q, W, U, and the 
 vanishing trace of the plane P. Plate V, Fig. 231^. Let the 
 picture plane be the V coordinate plane of Fig. 23, and also use 
 the H coordinate plane of the same figure. Let e^, e^ be the or- 
 thographic projections of the station point (§ 55), which is about 
 one and one-half inches in front of the picture plane, e' is the 
 center of the picture (§ 56). Through the station point pass 
 a line parallel to Q: it pierces the picture plane at V^ ; hence 
 V*^ is the vanishing point (§§ 60 and 61) of Q and of all Hues 
 parallel to Q, i. e., of the system Q. In the same manner the 
 vanishing points V^ and V^ of the lines f/and W^ respectively, 
 were determined. 
 
 The line e^W^ is the horizon, since U a.r\d Q are horizontal 
 (§ 64), and T W Q) passing through V^ and V^, is the vanishing 
 trace of the plane P and of all planes parallel to P, i. e., of the 
 system P, since it contains the vanishing points of two lines of 
 
GENERAL PRINCIPLES, DEFINITIONS, ETC. $1 
 
 that plane (§§ 65 and 66). TWQ is parallel to VP 
 
 (Fig- 23)- 
 
 It is thus evident that the determination of a vanishing 
 point is the determination of the V trace of a line passed 
 through a given point, and that the determination of the vanish- 
 ing trace of a plane is the determination of the J^ trace of a 
 plane which passes through two given lines intersecting at a 
 given point. 
 
 77. To find the perspective of a rectangular prism by 
 direct projection. Plate V, Fig. 26. 
 
 Analysis. Since perspective is conical projection or projec- 
 tion from a given point upon a plane (§§ 51, 52, 54), it follows 
 that if a line be drawn from the given point (station point, § 5S) 
 to each point of the figure, that the intersections of these lines 
 with the plane (picture plane, § 53) give the perspective of the 
 figure. This is the same as finding the V traces of a number 
 of lines which radiate from a point. In this method both pro- 
 jections (//' and F) of the object are required, and are usually 
 placed so that the object is in either the second or third 
 dihedral angle. 
 
 A much simpler solution than the above, which may be 
 cahed direct projection, will be explained in the next chapter. 
 
 Construction. Let ab dcik gfa be a rectangular prism 
 situated in the second dihedral angle. The top and bottom 
 faces are parallel to H and the other four faces are vertical 
 planes inclined at angles of 60° and 30° to V as shown. The 
 station point (§ 5S) is ^ and the picture plane (§ 53) is V. From 
 e draw a line to a : it intersects the picture plane ( V") at d\ hence 
 d is th.& perspective of the point a. From e draw a line to /fe: it 
 intersects the picture plane (F) at ^'; hence k" is the prospective 
 of k. In like manner draw lines to all the corners of the prism 
 and find their intersections with the picture plane (F). 
 
52 PERSPECTIVE. 
 
 These intersections or perspectives joined in the proper 
 manner give the perspective d ¥ d^ c" i' k' g^ f^ d of the prism. 
 
 Note the following facts : The perspective lines «"/', c' i\ 
 d' U,3.x\A F g' converge and meet at V^', which is the vanishing 
 point of these lines (§ 60). This is as it should be, because the 
 corresponding Hnes on the prism are parallel, and hence must 
 have a common vanishing point. Through e (station point) 
 draw a line parallel to one of these lines on the prism, e. g., af: 
 it intersects the picture plane at V^, which is the same vanish- 
 ing point as before. Hence we reach the same result whether 
 we begin by determining the vanishing point of a system of lines 
 (§ 61), or whether we find the vanishing point from the per- 
 spectives of the lines. Similarly the vanishing point of the hnes 
 inclined to the right is V^ (not shown, about one inch to right 
 of border line on V^^'*'). The perspective lines d i>\ c' d\ t' k\ 
 a.ndf'g' are parallel, which is correct, because the correspond- 
 ing lines on the prism are parallel to the picture plane (V); 
 hence their vanishing point is at infinity, for if a line be drawn 
 through the station point (e) parallel to these lines it will inter- 
 sect the picture plane only at infinity. The systems af, c i, etc., 
 and ac,bd, etc., are horizontal, hence the line V*^ e^ is the hori- 
 zon (§ 64) or the vanishing trace (§ 63) of the planes afica.nd 
 bgk d, because it contains the vanishing points (V^ and V^) of 
 two or more lines in each plane. This is also true from the 
 fact that the two planes afic and bgkd are horizontal ; hence 
 if a plane be passed through the station point parallel to 
 these planes it will intersect the picture plane in the line V'^ e', 
 or the horizon. 
 
 As the planes abgf a.nA cdki are vertical their vanishing 
 trace must be perpendicular to G. L. And as V is the vanish- 
 ing point of a line in each plane, Y^ z is the vanishing trace of 
 
GENERAL PRINCIPLES, DEFINITIONS, ETC. 53 
 
 these planes. This fact might also have been established by 
 finding the vanishing point of any other line in either plane, 
 £. g., one of the diagonals of either of the two faces. 
 
 The V trace of the line af, or the point in which the line 
 pierces the picture plane, is q^; hence cf is the initial point 
 .(§ 6"]) of the line af. The perspective of this line, «'/"', passes 
 through q^, because q"' is a point in the line and also a point in 
 the picture plane, in other words, if and the perspective of (f 
 •coincide. 
 
 Similarly m' is the initial point of /? and the perspective 
 /' «' passes through it, also li' is the initial point of gk and 
 the perspective g^ ¥ passes through it. Since iif and tC are 
 the initial points (F traces) of two lines in the plane y"_^/^2, the 
 line in^ vf iC must be the initial line (§ 68) or f^ trace of the 
 plane fgki. Also m^ q' is the initial line of the plane a fie. 
 The line in^ (f is horizontal, which is correct, since the vanish- 
 ing trace V'^^'^ is horizontal, and hence parallel to in'' q'^ (§ 69). 
 We thus establish the fact that the vanishing trace and 
 initial line of a plane are parallel, which must be true 
 because the plane which determines the vanishing trace is 
 parallel to the plane which has the initial line and the 
 intersections of parallel planes with the same plane are 
 parallel. The line in^ ri' which is the initial line of the plane 
 f gk i is also parallel to the vanishing trace (not shown) of the 
 plane. The line if cf is the initial line of the plane afgb be- 
 cause it contains the initial point f of a line in the plane and 
 is parallel to N^ z which is the vanishing trace of the plane. 
 We might also have determined this initial line by drawing a 
 line through if parallel to a b, since the initial point o{ ab is at 
 infinity. Note also the fact that the initial point of a line is at 
 the intersection of the initial lines of any two planes which con 
 
54 PERSPECTIVE. 
 
 tain the line, e.g., nf , and that the vanishing point of a line is 
 at the intersection of the vanishing traces of any two planes 
 which contain the line ; e.g., V"^ is the vanishing point of af 
 and the intersection of if^V'^ and ^•V'^. 
 
 For the duplication of this problem by the method of 
 measurement see Prob. I, § 95. 
 
CHAPTER V. 
 
 THE MEASUREMENT OF LINES IN PERSPECTIVE. 
 
 78. The perspectives of lines which lie in the picture plane 
 coincide with the lines, and hence are equal to them in length. 
 In general, however, a line and its perspective are unequal in 
 length. 
 
 If by any means we may transfer distances from the per- 
 spectives of lines which are actually in the picture plane to the 
 perspectives of other lines which are not in the picture plane, 
 a method of measuring or laying off distances in perspective is 
 effected. 
 
 79. Proposition I. If parallels to the base of a triangle 
 be drawn, the corresponding segments on the two sides are in 
 a constant ratio. (See Plate VII, Fig. 34.) 
 
 If the triangle be isosceles the ratio is unity, hence the cor- 
 responding segments on the two sides are equal. (See Plate 
 VII, Fig. 35)- 
 
 80. Proposition II. A line drawn through the intersection 
 of the diagonals of a parallelogram parallel to two of its sides 
 bisects the other two sides and the parallelogram. (See Plate 
 VII, Fig. 36.) This method of subdivision may be continued 
 as shown in the figure. 
 
 81. Proposition III. If one side of a parallelogram be di- 
 vided in any way at one end, equal divisions may be laid off at 
 the other end by means of two diagonals. (See Plate VII, 
 
 Fig- 37-) 
 
 82. To duplicate Proposition I in perspective. Plate V, 
 Fig. 24. Isosceles triangle. 
 
 55 
 
56 PERSPECTIVE. 
 
 Analysis and Construction. Let a z x \)& an isosceles tri, 
 angle situated in the third dihedral angle and having one of its 
 equal sides, a x,in V (or picture plane.) Assume G. L. to be 
 the horizon, e the station point, and e'' the center of the picture 
 
 (§ 56). 
 
 Find the perspective of the triangle. The vanishing point 
 oi a z {/) is V'' found by passing a line through the station 
 point e parallel to / (see § "jj). Since a' is in the picture 
 plane it is the initial point of y, and hence one point in the per- 
 spective ofy; hence d^'V'' is the indefinite perspective of J. 
 The vanishing point o{ x z {F) is D'', found by passing a line' 
 through the station point parallel to F, and x^ is the initial 
 point of F; hence, ;i:''D'' is the indefinite perspective of the line. 
 The hnes a'V'' and x^ T)^ intersect at z\ which is therefore the 
 perspective of the point z. 
 
 Since a^f is in the picture plane, a^ x^ is the perspective, or 
 in other words the line is its own perspective. Hence a^ z^ x^ is 
 the perspective of the triangle az x. But the triangle a z x '\% 
 isosceles, hence d^ x^ = d" s' in perspective, that is, the per- 
 spective length d' z^ = the true length d' x''. Parallels to the 
 base (i^) of the triangle vanish at D'^; hence if we lay off any 
 distance on d" x^, as for example a^ c =: 1/2", and draw c m D'', 
 we will again have an isosceles triangle in which a^ c = 1/2" 
 = a^ m in perspective. Also lay off any other distance as 
 a^ d= i^', draw ^D^ intersecting J' at/; then a' d— ij" = 
 d^ p in perspective. 
 
 Notice that T J F is the vanishing trace of the plane of the 
 triangle because it contains the vanishing points of two sides 
 of the triangle, and that d' x^ is the initial line of the plane of 
 the triangle, and also that T J F and cC x"^ are parallel (see §§ 
 68 and Tj). 
 
 As the chief requirement is to find a means to lay off dis- 
 
THE MEASUREMENT OF LINES IN PERSPECTIVE. 57 
 
 tances on the perspective /', any line in the picture plane 
 other than a x which would complete an isosceles triangle 
 having a z for one of its equal sides may be used. Let 
 ay = ax = as he such a line; then we have the isosceles tri- 
 angle a 2 J. The vanishing point of the base /^ is D'''; hence 
 n' z'y" is the perspective of the triangle azy, and we may use 
 the side a"'y"', which is in the picture plane, to lay off distances 
 to be transferred (by parallels to the base) to the perspective 
 side/'. 
 
 82a. Usually in a problem where measurements are nec- 
 essary we have the following data (Fig. 24) : The vanishing 
 point of the line as V-', the indefinite perspective of the hne as 
 y ', the vanishing trace of a plane containing the line as T J F, 
 the horizon as G. L. {G. L. is the horizon only for convenience), 
 and the station point e. It is required to transfer measure- 
 ments to the line y in perspective by means of an isosceles 
 triangle. We know that the line T J F is parallel to the in- 
 itial line of the plane of the triangle, and hence is parallel to one 
 of the equal sides ; we also know that the line joining the station 
 point with the vanishing point V''^ of the line is parallel to the 
 line and hence parallel to the other equal side ; hence all we 
 are required to find is the base or vanishing point of the base 
 of the triangle having for its equal sides e N^ , and part of T J F 
 equal in length to e V'', to form a triangle whose sides are re- 
 spectively parallel to the triangle in space which has the side 
 
 /. 
 
 Revolve the plane containing T J F and e about T J F 
 (which is the F trace of such plane) into F (picture plane): the 
 station point e revolves to e^, and the line eN^ to ^, V''. 
 
 Now lay off V'^ D'' = V'' ^,, and join D-^ e^, and we have an 
 isosceles triangle V''D''^£j. But D''^ is a point in the picture 
 plane, and hence the vanishing point of the base e D'' of the 
 
58 PERSPECTIVE. 
 
 isosceles triangle e V'' D'^. We have thus determined the same 
 point, D'', for the vanishing point of the base of the isosceles 
 triangle that we had in § 82. 
 
 The point D-'' or D'' is called a Point of Distance i^ 83), and 
 the line (f x" or a"" y is called a Line of Measures (§ 87). 
 
 83. Point of Distance. A Point of Distance for a given 
 oblique line is the vanishing point of the base of an isosceles 
 triangle whose equal sides are the given line and the initial line 
 of any plane which contains the given line. From § 'is2a we 
 may derive the following definition : A Point of Distance for a 
 given oblique line is the vanishing point of the base of an isos- 
 celes triangle whose equal sides are, the line from the station 
 point to the vanishing point of the given line, and the vanish- 
 ing trace of any plane containing the given line. 
 
 84. Since a line may be contained in an infinite number of 
 planes having an infinite number of vanishing traces, it follows 
 that a Point of Distance for a line may be any point in the 
 picture plane situated as far from the vanishing point of the 
 line as the vanishing point of the line is from the station point. 
 
 85. When- the given line \% parallel to the picture plane its 
 vanishing point is at infinity; hence the equal sides of the 
 isosceles triangle are parallel, and the point of distance may 
 be any point on the vanishing trace of a plane containing the 
 line (see §§ 95 and 97). 
 
 86. Point of Half Distance. Plate V, Fig. 24. Suppose 
 the lines revolved, etc., as in § 82«. Bisect V D"^, and draw 
 e^ D^ ; a triangle e, D^'' V-^ is formed, of which the side V'^D^'^ 
 is one-half e^ V-'. Di-^ is the vanishing point of the side e Di'' 
 of the triangle. Hence if we lay off on a^ x" distances to half 
 scale and draw lines to Di"^ intersecting /', the same subdi- 
 visions are effected as in § 82 ; e.g., lay off a' f= \ inch, draw 
 fiW intersecting /' at m, then a' m in perspective = 2^"'/, 
 
THE MEASUSEMENT OF LINES IN PERSPECTIVE. 59 
 
 because the sides of the triangle a7 fm are parallel respectively 
 to the sides of the triangle e D^'^ V'^. The point D^'^ is called 
 a Point of Half Distance. If one-quarter of V'' e^ were laid off 
 on V'' D'' from V'', a Point of Quarter Distance would be ob- 
 tained, and the distances ox\ ax would be laid off to quarter 
 scale. This process may, of course, be continued indefinitely, 
 and Points of Three-Quarter Distance, One Tenth Distance, etc.,, 
 obtained. 
 
 87. Line of Measures. A Line of Measures for a given 
 line is the initial line of the plane containing the given line 
 and its point of distance. Hence a line of measures for an 
 oblique line is a line in the picture plane which passes through, 
 the initial point of the line and xs parallel to the line joining 
 the vanishing point of the given line with its point of distance 
 (see § 8212). 
 
 88. Plane of Measures. A Plane of Measures is a plane in 
 which the measurements which are to be transferred to the- 
 perspective are made. 
 
 Throughout this book the Plane of Measures is supposed to- 
 coincide with the picture plane. 
 
 89. Scale. The scale to which the drawing is made may be 
 anything convenient, such as i inch to 10 feet, i inch to i foot, 
 etc., or even i inch to i inch, and depends entirely upon the 
 purposes for which the drawing is required. A decimal or 
 chain scale is, however, preferable to any other although not 
 used in practice. 
 
 90. Proportional Measures. It is often necessary to di- 
 vide a given line into a certain number of equal or propor- 
 tional parts irrespective of the actual length of any part. For 
 this purpose a method somewhat shorter than the method of 
 equal measures (§ 82 et seg.) is given. 
 
6o PERSPECTIVE. 
 
 To duplicate Proposition i in Perspective. Plate V, Fig. 25. 
 Any triangle. See § %2a. 
 
 Analysis and Construction. Let V"^ be the vanishing point of 
 a line, and T J the vanishing trace of any plane passed through 
 the line/. Revolve the plane of T J and e about TJ as an axis 
 into the picture plane : the point e revolves to e^, and the Hne 
 e V-' to e^ V-'. Draw any line as e^ P^ thus cojnpleting a triangle 
 e^ V' P-^. Revolve the triangle back to its true position, and 
 P''^ is the vanishing point of the base e '?^ . 
 
 Let V b be the perspective of a line of the system /which 
 is to be divided into three (or any number) of equal (or un- 
 equal) parts. 
 
 The point V is the initial point of/', but as far as the solu- 
 tion of the problem is concerned may be anywhere on the 
 perspective line /'. Through P draw P^ parallel to T J, draw 
 Y^ b d: then the perspective triangle V b d'vs, similar to the tri- 
 angle ^ V'' P''. Divide P(3?into three equal parts (the method 
 of geometric division is shown), and from the points of division 
 draw / P''^ and ^ P'' intersecting/' at m and n respectively: 
 then V p : p q : etc., : : V nt : m n : etc., in perspective. The 
 point P-"^ is called a Point of Proportional Distance, and the line 
 F d is called a Line of Proportional Measures. Any other point 
 as P''' might have been taken on T J, and the same subdivisions 
 affected, as shown in the figure. Instead of using the line 
 F d through the initial point of /, we may use any other line 
 parallel to T J, as in g, and obtain the same result. 
 
 91. Point of Proportional Distance. A Point of Propor- 
 tional Distance for an oblique line is the vanishing point of the 
 base of a triangle, the sides of which are the given line, and 
 any line parallel to the vanishing trace of a plane which con- 
 tains the given line. 
 
THE MEASUREMENT OF LINES IN PERSPECTIVE. 6r 
 
 92. Line of Proportional Measures, A Line of Propor- 
 tional Measures for an oblique line is a line drawn through any 
 point of the given line parallel to the vanishing trace of any- 
 plane containing the given line. 
 
 93. Of any two perspective lines having the same vanishing 
 point, one may be taken as the trace of a plane passing through 
 the other ; and if a third line be drawn parallel to the first 
 through any point of the second, any parts taken upon this 
 third line may be transferred to the second in their true propor- 
 tions by means of a point of proportional distance taken upon 
 the first.* 
 
 94. It is evident from the preceding paragraphs that lines 
 which are parallel to the picture plane may be sub-divided 
 directly. 
 
 * Modern Perspective, by W. R. Ware. 
 
CHAPTER VI. 
 
 PARALLEL, ANGULAR, AND OBLIQUE PERSPECTIVE. 
 
 95. Problem I. To find the perspective of a rectangular 
 prism. Angular perspective. Plate V, Fig. 27. 
 
 Analysis. This problem is a duplication of § "jy (Fig. 26) 
 by the method of measurements described in the last chapter. 
 The prism is supposed to rest on a horizontal plane through 
 G. L. which corresponds to the line vi" (f of Fig. 26. All 
 dimensions are given in Fig. 26. The principal (and only) 
 lines of the prism are the edges which belong to three systems 
 (§ 70). Four of the edges are horizontal and inclined to the 
 right [K lines), four are horizontal and inclined to the left 
 (iV lines), and four are parallel to the picture plane and perpen- 
 dicular to H {A lines). Hence the drawing is to be made in 
 angular perspective (§ 73), and only the two vanishing points 
 of the horizontal edges are required. 
 
 Construction. Let V V^ be the horizon, assume the station 
 point 2y (= e^ X, Fig. 26) in front of the picture plane, and let 
 e^ be the center of the picture (§ 56). Consider a horizontal 
 plane passed through V^ e^ N^ ; then e^ is the projection of the 
 station point on this plane, and also its true position, and 
 e^ e^ = 2|-". This horizontal plane through the horizon is 
 shown in a number of the following problems and is useful 
 in determining vanishing points, etc. 
 
 Since K and N are horizontal, their vanishing points are 
 on the horizon (§ 64) ; hence by drawing through e^ a line 
 
 K making an angle of 60° with V V^ the vanishing point V^ of 
 
 62 
 
PARALLEL, ANGULAR, AND OBLIQUE PERSPECTIVE. 63 
 
 the edges inclined to the right is determined, and through i\ a 
 line TV perpendicular to ^the vanishing point V^ of the edges 
 inclined to the left is determined. A point of distance for ^is 
 D'^, found by laying off V^ D^ = V^ e^ on the horizon (§§ 82-84), 
 and a point of distance for N is D'', found by laying off V^ D'^ 
 
 In practical work it is usually much easier to compute the 
 positions of vanishing points and points of distance than to 
 determine them by geometric methods. In the present problem 
 the angles which the edges make with the picture plane are 
 given ; hence ^'^ V^ = 2.5 tan 30" = 1.44", e^Y^ = 2.5 tan 60° 
 
 = 4.33", V D^' = .^'^ ,=: 5.00", and V^ D^ =^^^=2.89". 
 ^ ^^ sm 30 sm 60 ^ 
 
 These distances may be laid off at once. 
 
 Since the prism rests on a plane through G. L., this line 
 
 must be the initial line (§ 68) of the lower face of the prism ; 
 
 and as it is parallel to the horizon which contains the vanishing 
 
 points and points of distance of the horizontal lines of this 
 
 face, it is also a line of measures (§ 87) for this face. The 
 
 point a (Fig. 26) is i|-" to the left of the station point and 3/4" 
 
 behind the picture plane. Hence lay off ji/ P (Fig. 27) = i-|". 
 
 The perspective of the point « is on a perpendicular to the 
 
 picture plane through P, and as all perpendiculars vanish 
 
 at e^ (V^), 1^ e^ must contain the perspective of the point. 
 
 A point of distance for 1^ e"' is V''^"^, found by laying off 
 
 ^vY45''L_ g^g'^^ since the perpendicular is in the plane of the 
 
 (lower) face. Lay off l^J — 3/4" and draw/ V*'°-^ intersecting 
 
 I^ e^ at rt' ; we thus have an isosceles triangle formed in which 
 
 I^y= P«' in perspective; hence «' is the perspective of the 
 
 point required (§ 82). Draw the indefinite lines K\ N\ and 
 
 A^ for the indefinite perspectives of the edges of the prism 
 
 ■which intersect at a (Fig. 26). 
 
64 PERSPECTIVE. 
 
 We know, that the initial points of the lower edges are on 
 C. Z., because G.L. is the initial line of the lower face ; also 
 that this line is a line of measures and we have determined the 
 points of distances for TV' and K\ The initial point of N' is F', 
 found by producing N' to G. L. Draw D*' a^ p intersecting N^ 
 at d ; then P' dp is an isosceles triangle in which P'/ = P' a} 
 in perspective. Lay off / m = i^" (= a/, Fig. 26) and draw 
 m D^ intersecting yV' at/'; then/ m — «'/' in perspective, being 
 corresponding segments on the equal sides of an isosceles tri- 
 angle (§§ 79 and 82), and the perspective of the lower left edge 
 is determined. It was of course unnecessary to find the initial 
 point P' except to show how the isosceles triangle was formed 
 as we wished to lay off a distance (rom a\ To lay off the 
 length of K' draw D^ a' n, lay ofi ng— 1/2" (=: ac, Fig. 26),. 
 draw ^D"^ intersecting K' at <:', then «^= «'c' in perspective. 
 We may now draw N' K", which intersect at i\ and thus com- 
 plete the perspective of the lower face. Draw the indefinite 
 lines/' ^', t ' k\ and c' d^ perpendicular to G. L. {i. e., parallel to 
 the corresponding lines of Fig. 26) for the perspectives of the 
 edges which are parallel to the picture plane. To lay off /'^' : 
 The point in is the initial point of the line /«/' D^ (see before), 
 and this line passes through the point /' of /'_^'. The initial 
 poifit of /'^' is at infinity; hence mr parallel to/'^' is the 
 initial line of a plane containing/'^', and also a line of meas- 
 ures for/ g^. Lay off m r = 3/4" {—fg, Fig. 26), draw D" r 
 intersecting/'^' at ^', then ;« r = /' _§-' in perspective. This 
 is true, because mr and/'^' are actually parallel, and mD^ 
 and r D^ are parallel in perspective. 
 
 Note : The measurement of a line parallel to the picture is 
 effected by laying off distances on the initial line of any plane 
 containing the line (§ 85), and transferring them by means of 
 
PARALLEL, ANGULAR, AND OBLIQUE PERSPECTIVE. 65 
 
 parallels to the perspective of the line. In the case of lines 
 parallel to the picture plane the initial line {?, parallel to the 
 perspective line. 
 
 Similarly we may lay off' c';^' as follows : P'^ is the initial 
 point of N", which passes through the point c' of c^ (P; hence 
 JN2 iN4jg ^ ]if,g Qf measures fortV, to which it is parallel. Lay 
 off F^F^ = 3/4" (= cd, Fig. 26), draw piV intersecting 
 c^ d^ at d\ then c' ^"^ is the perspective required. Draw N^ 
 {e-'d') intersecting A' at 5', also K' {d' d'), N' and /^ thus 
 completing the perspective of the prism, which is a duplicate 
 of Fig. 26. 
 
 Instead of erecting the perpendiculars to G.L., a' b\ f g^, 
 i' k\ and c' d', and laying off the height of the prism upon one of 
 them, we may find the perspective of the upper face directly as 
 follows : Since the prism is 3/4" high and the plane of the up- 
 per face is horizontal, lay off m r = 3/4" and draw r F^ par- 
 allel to G. L.; then r 1™ is the initial line of the upper face, and 
 also its line of measures, and we may duplicate in this plane 
 the work of finding the perspective of the lower face. 
 
 After the perspective of the upper face is determined by 
 this method join the corresponding points in the two faces to 
 complete the perspective. 
 
 96. Object at 45°. When an object is so placed that a 
 number of its principal lines (§ 70) are horizontal and inclined 
 at an angle of 45° to the picture plane, the determination of the 
 vanishing points and points of distance is much simplified, as 
 these points are symmetrically placed with reference to the 
 center of the picture. 
 
 The vanishing points of horizontal lines making angles of 
 45" with the picture plane are on the horizon at a distance 
 from the center of the picture equal to the length of the axis 
 (§ 57) ! hence, if the center of the picture and one of these 
 
^ PERSPECTIVE. 
 
 vanishing points be given or assumed, the position of the sta- 
 tion point is known. 
 
 Plate VI, Fig. 28. The horizon is the horizontal hne 
 passing through e", the center of the picture, and V^°^ and 
 V*""^ are the vanishing points of horizontal lines making 
 angles of 45° to the right and left. These points are usu- 
 ally denoted V''^°» and V*=°^ or simply V^=°. V' '^ e" — V«°i-^^ 
 = length of axis = 3 inches (in this case) ; hence the station 
 point is 3 inches in front of the picture plane. T W J and 
 T F O are the vanishing traces of diagonal planes (§ 44), and 
 T Y Z is the vanishing trace of profile planes. Lines in pro- 
 file planes and making angles of 45° with the picture plane 
 vanish at V^ and V^; hence V«°« e' = e" N^ = e' V^ = V^=°^^\ 
 and V*=° V^ = V"° V^ = etc. = the distance that the station 
 point is from V*°°^, or V*^°^ = the distance that a point of 
 distance (§§ 83 and 84) for R, L, Z, or Y lines is from V*=°^, 
 V*^°^, V^, or V^ respectively. With V*'°^ as a center and 
 radius = V^^°^ V^ describe a circle (only an arc is shown); this 
 circle is the locus of the points of distance for the system of 
 lines vanishing at V*^°^. 
 
 Similarly with V'''°^ as a center and radius = V^'° V"*^ de- 
 scribe a circle which is the locus of the points of distance for 
 the system vanishing at V*'""^- The symmetry of the 45° posi- 
 tion is thus shown. 
 
 The vanishing point of any line in a diagonal plane may be 
 readily determined if the angle which it makes with a horizon- 
 tal plane be known, as follows : The line if inclined to the 
 right vanishes in T W J, if inclined to the left it vanishes in 
 T F O (§ 63). 
 
 If the station point be revolved about T W J into the pic- 
 ture plane, D^^°^ is its revolved position, since V*^°^ qw-r _ 
 V^'^ e ; hence through D*^°'* draw any Hne, as D*^°^ V'', making 
 
PARALLEL, ANGULAR, AND OBLIQUE PERSPECTIVE. 6/ 
 
 the given angle (30° in this case) with the horizon. V'^, 
 the intersection of D*'°^ V'' and T W J, is the vanishing point 
 required. Similarly V^, V°, and V'^ are the vanishing points of 
 lines in diagonal planes inclined down to the right and up and 
 down to the left respectively, and making angles of 30° with a 
 horizontal plane. We might at once have laid off the distances 
 Y45°K v"^ = V*^°^ V-' = V^"^ V° = V*'°'^ V^ = 3 V2 tan 30" = 
 2.45". V'' D*^°^ = V'^f; hence, if with V'^ as a center and ra- 
 dius V'' D^°^ we describe a circle, the locus of the points of 
 distance for the system vanishing at V''^ is obtained. Notice 
 that this circle passes through V^. Why ? 
 
 T F R, T L W, TOR, and T L J are the vanishing traces 
 ■of planes containing horizontal lines inclined at angles of 45° to 
 the picture plane, and lines in diagonal planes inclined at an- 
 gles of 30° to the horizontal plane. 
 
 Another advantage in placing an object at 45° is that in 
 this position the station point is at a maximum distance from 
 the picture plane, and the drawing is concentrated within nar- 
 rower limits; e.g., Plate V, Fig. 27, the object is placed at 30° 
 and 60°, and the station point is at the distance e^ e^ from the 
 picture plane ; if now the station point be moved to ^,, its dis- 
 tance from the picture plane is ^^ D^ > e^ e^, and the vanishing 
 points, V^ and V*^, used for the 30°, 60° position, are now 
 used for the 45° position, since they are equally distant 
 from ^j, and V^ e^ V^ is a right angle, being inscribed in a 
 semicircle. 
 
 97. Different Ways of Measuring the Same Line. 
 Plate VI, Fig. 28. 
 
 Analysis and Construction. Let J be the indefinite perspec- 
 tive of a line which vanishes at V''^, and let R be the perspec- 
 tive of its horizontal projection on a plane through I^; then P 
 is the initial point of R, and I W J perpendicular to the hori- 
 
68 PERSPECTIVE. 
 
 zon (i. e., parallel to T W J) is the initial line of a plane contain- 
 ing R and J. 
 
 Note the fact that the vanishing point "of a line and the 
 vanishing point of its horizontal projection are in the same 
 perpendicular to the horizon. 
 
 Produce J until it intersects I W J at F, which is the in- 
 itial point oij. It is required to lay off 2.5" on y from the 
 point/. Since V'' is the vanishing point of the line, TWJ, 
 T L J, and T Q J, all passing through V, may be considered 
 the vanishing traces of planes containing the line, and V^, D''^, 
 and V on the " locus of D''" are the corresponding points of 
 distance (see § 96). {a) We have already found the line of 
 measures I W J parallel to T W J; hence draw N^ p i inter- 
 secting I W J at i, from i lay off ik ^ 2.5", draw k V^ inter- 
 secting y at q; then/^ in perspective = 2.5". [B) Through F 
 draw I L J parallel to T L J for a line of measures, draw 
 D''//, from / lay off /^= 2.5" and draw^^^D""; then /) ^ in 
 perspective = 2.5" as before, (c) Through F draw I Q J par- 
 allel to T Q J, draw V/ r, from r lay off r ^ = 2.5" and draw 
 tqY'^; then/ ^ in perspective = 2.5". Three ways of obtaining 
 the same result have thus been shown. 
 
 The measurement of the line L, which is in a horizontal 
 plane through F, is also shown. V^ is a point of distance, be- 
 ing on " locus of D^ " (see § 96), and F 3, parallel to T L U and 
 passing through F, the initial point of the line, is a line of 
 measures ; then w x in perspective = j/ ^. 
 
 Let A be the perspective of a hne parallel to the picture 
 plane and B the perspective of its projection on a horizontal 
 plane, of which G. L. is the initial line. It is required to lay 
 off a certain distance on A from n to the left. 
 
 Suppose we measure the hne by parallels (§ 85) situated in 
 profile planes (compare with § 95). [NOTE: We make this as- 
 
PARALLEL, ANGULAR, AND OBLIQUE PERSPECTIVE. 69 
 
 sumption only for convenience in drawing; any other assumed 
 position of the parallels would have answered the purpose.] 
 As the vanishing trace of profile planes is T Y Z (a line 
 through e^ perpendicular to the horizon), the point of distance 
 or vanishing point of the parallels to be used must be on 
 T Y Z. Through n, I pass a profile plane, its initial line is 
 I N M; hence a line of measures for A must have one extrem- 
 ity in I N M, because this line is the initial line of a profile 
 plane through one extremity of A, and hence must contain the 
 parallel which passes through n. Assume V**^, or any point on 
 T Y Z, as the vanishing point of the parallels, draw V**^ n d in- 
 tersecting I N M at of; then ^ is the initial point of this line. 
 Through d draw dj parallel to A for a line of measures, lay off 
 the required distance dj and drawy'^^V"^; then </y= n vi in 
 perspective. The same measurement is found by using V-"-. 
 
 98. Problem II. To find the perspective of a cube. Parallel 
 and angular perspective. Plate VII, Figs. 30-33 inc. 
 
 (rt) Analysis. The four equal cubes shown are supposed to 
 be placed in a room ; two are on or near the floor and two are 
 fastened to the ceiling. The length of the edge of each cube 
 is 4 feet and the room is 14 feet high. The scale of the draw- 
 ing is I inch = 2 feet. The station point is 7 feet in front of 
 the center of one wall which is taken as the picture, plane and 
 behind which, in the room, the cubes are placed. The lower 
 left cube (Fig. 30) has one face in the floor, and one face par- 
 allel to, and I foot behind, the wall. The lower right cube 
 (Fig. 31) has one edge in the floor and one edge in the wall, 
 and the face passing through these two edges is inclined 45° 
 to the floor. The upper cubes (Figs. 32 and 33) are similarly 
 situated in regard to the ceiling and wall; each has one face in 
 the ceiling, one edge parallel to, and 1.5 feet behind, the wall, and 
 the faces passing through this edge are inclined 45° to the wall. 
 
"JO PERSPECTIVE. 
 
 Hence Fig. 30 is in parallel perspective, and Figs. 31, 32,. 
 and 33 are in angular perspective (§§ 72 and 73). 
 
 Construction. The center of the picture is e', G. L. is the 
 floor line, and T Q Z is the ceiling line. 
 
 {b) Fig. 30. This cube is in parallel perspective (§ 72); 
 hence four edges {P) vanish at e\ and eight edges {A and B) 
 are parallel to the picture plane. A point of distance for P 
 lines is V*=°^ (or V*5°e)_ since e'Sf''"^ = €> e ^ 3.5" (= 7' to 
 scale). Draw P' e^ , the indefinite perspective of one of the 
 edges of the base ; on G. L., the line of measures corresponding 
 to V*'°^ lay off F' I" = 0.5" (= i' to scale), draw I" Y'^"^ in- 
 tersecting P^^^ at b; then b is i' behind the wall or picture- 
 plane. Lay off F' F* = 2" (= 4' to scale), draw p-* e', draw 
 A^ parallel to G.L. intersecting P* e^ at a; then A^ is the per- 
 spective of the near edge in the floor. The diagonal of the 
 base inclined to the left vanishes at V^^"-"^, because it is a hori- 
 zontal line inclined at an angle of 45° to the picture plane ; 
 hence draw b V^'""^ intersecting P^ e^ at k, and the perspective 
 of the edge P^ is determined. Draw A* parallel to A^ and in- 
 tersecting /" at/", thus completing the perspective of the base. 
 We might have laid oft piy= 2" and drawn/ V^^"'^ intersecting- 
 P' aty, etc. The line \^^ m perpendicular to G.L. is the in- 
 itial line of the face /", B\ and hence a line of measures for £^. 
 -Lay off Pi l^ = 2" (= 4' to scale), draw P^ e' and £\. 
 which intersect each other at c; then 3 c is the perspective of 
 the edge £\ Draw B', B', B* perpendicular to G. L. and par- 
 allel to B^ ; also draw P\ A\ P\ and A^ to complete the figure 
 (see § 95). 
 
 The diagonals- of the upper and lower faces vanish at V-^' 
 and V^ (see before). The diagonals Z of the side faces vanish 
 at V^, since e^ V^ = e^ Y^ = 3.5", and the other two diagonals 
 (not shown) vanish at V^. Since Z^ and Z' vanish at V^, and 
 
PARALLEL, ANGULAR, AND OBLIQUE PERSPECTIVE. ",'1 
 
 A' and A' are parallel to the picture plane and horizontal, the 
 vanishing trace of the plane ^4' Z'' ^^ is TQZ; and since D 
 vanishes at N^, and B^ and B'' are vertical, the vanishing trace 
 of the plane D B^ is T L Q; hence V^, the intersection of these 
 two vanishingtraces, is the vanishing point of the diagonal Q of 
 the cube (§ 66). Similarly the diagonal W^ vanishes at V^, 
 etc. 
 
 (c) Measurement of the Diagonals. The vanishing trace of 
 a plane containing Z'' is T Y Z ; hence D^, found by laying off 
 V^ D^ =. V^ Vi- = 4.9s" (§ 84), is a point of distance for Z\ 
 Since I"»? is the initial line of the plane of the face /-" B^ (see 
 before), it is the line of measures (§ 87) corresponding to D^ ; 
 hence draw D^ bn and D^ I in intersecting I^' in at n and in re- 
 spectively ; then n m = 2.83" is the true length of the diagonal 
 Z ^. The vanishing trace of a plane containing R} is the horizon, 
 since R} is horizontal; hence the required point of distance is 
 D«°^ found by laying off V^ D« = V^ D^ = 4.95". G. L. is the 
 line of measures; hence draw 'D^fp and Ti^aq intersecting 
 G.L. at p and q respectively; then /^ = »;;/ = 2.83" is the 
 length of R required. Since the diagonals W^ and Q^ inter- 
 sect, T W Q is the vanishing trace of a plane containincr W^, 
 and D^, found by laying off V^ D^ = the hypotenuse of a 
 triangle of which V"^ ^"^ and e e" are the sides = 6.06", is the 
 point of distance. Since P^ is the initial point of P", it is one 
 point of the initial line of the plane W^ , Q\ which contains P' ; 
 hence I W" C parallel to T W Q is the line of measures. 
 Draw D^/r and D^/z z intersecting I W C at r and 2 respect- 
 ively ; then rt — 3.46" is the length of W\ T L Q is the vanish- 
 ing trace of a plane containing Q\ and D<5 is a point of distance 
 since V<i DQ = V^ D^. The hne I L' Q' through I" (the initial 
 point of Z') is the line of measures; hence draw D*^ (5 z^ and 
 B'^sw; then uiv — ri= the length of Q. The diagonal C^ 
 
7^ PERSPECTIVE. 
 
 is parallel to the picture plane, and also to T W Q by construc- 
 tion, and is contained in the plane /" C /". Hence I W C 
 is the line of measures, and the initial points of P and P' deter- 
 mine the length of C ; hence P' P' = 2.83" = 3/^ in perspec- 
 tive. (See § 97.) 
 
 In order to either lay off distances on a line in perspective 
 or to measure the perspective of a line, it is essential to find 
 first the initial point of the line, or the initial line of a plane 
 containing the given line. When by the nature of the draw- 
 ing the initial point of the line is given, we may at once pro- 
 ceed to lay off the distance on the initial line (line of measures) 
 of any plane which contains the line, using a point of distance 
 on the vanishing trace of this plane. But in the majority of 
 problems the portion of the line already drawn or to be meas- 
 ured does not intersect the picture plane {i. e., initial point un- 
 determined) ; and as this initial point is unnecessary (except 
 when we wish to measure the distance along the line from the 
 picture plane), we may determine the line of measures by the 
 initial point of any line in the plane which is most readily 
 found. The plane to be chosen is usually the one which con- 
 tains a number of lines of the figure, for its initial line may be 
 used as a line of measures for all lines in the plane. 
 
 Take, for example, the measurement of the diagonal W^ : 
 From previous construction we have the initial point (F^) of 
 P^, and we know that P' and W^ intersect at/"; hence we may 
 at once draw the line of measures I W C ' parallel to T W Q, 
 which contains the vanishing points of /" and W^. 
 
 If we use the plane h kfc, the construction is more difficult. 
 This plane vanishes at G. L., because V^ is on G. L. and A'' 
 and A^ are parallel to the picture plane. A point of distance 
 is D^ (on G. L.). From previous construction we do not know 
 the initial point of any line in this plane ; hence we may as well 
 
PARALLEL, ANGULAR, AND OBLIQUE PERSPECTIVE. 73 
 
 find the initial point of W^. R} is the perspective of the pro- 
 jection of W^ on the floor (through G. Z.), and I^' is the initial 
 point of R} ; hence P* I^^ perpendicular to G. L. is the initial 
 line of a plane containing R^ and W^; hence I^' is the initial 
 point of W^, being the intersection of the line with the initial 
 line of a plane containing the line. Through I^' draw I W A' 
 parallel to G.L. for the line of measures, draw D^/z(^and 
 from Playoff (^^=3.46", draw ^D^ intersecting W^ ^^f', 
 then dg ^= 3.46" = hf'v!\ perspective. The same result would 
 have been obtained by drawing I W A* through the initial 
 point of the diagonal k k, which is at the intersection of k h, 
 I P' Z^ and I W A*. 
 
 Note : The two problems " to measure a line in perspec- 
 tive" and " to lay off distances on the perspective of a line '' 
 are identical as far as the drawing is concerned. In the first 
 we draw lines through the extremities of the perspective and 
 find the intercept on the line of measures ; in the second we 
 draw lines from the extremities of the distance measured on 
 the line of measures and find the intercept on the perspective. 
 In regard to the line of reasoning to be employed in each 
 problem one may be called the inverse of the other. 
 
 {d) Fig. 31. (See analysis.) As one edge (^') is in the pic- 
 ture plane (wall) and one edge is in the floor, and the face 
 containing them is inclined 45° to the floor, A" is above G. L. 
 a distance = 1.41", = 4/2 (to scale) ; hence lay off 1.41" above 
 G.L. and draw ^4' parallel to G.L. and 2" in length. The Z 
 edges, being inclined upwards at an angle of 45" to the picture 
 plane, and being in profile planes, vanish at V^ (see before), and 
 the Y edges, being inclined downwards, vanish at V"''^. 
 
 Draw the indefinite perspectives Y\ V, Z^, and Z'' of the 
 four edges intersecting A '. Through b draw I B' Z' perpen- 
 dicular to G. L.; I B' Z' is the initial line of the plane contain- 
 
74 PERSPECTIVE. 
 
 ing V and Z'', and hence a line of measures for these lines, 
 D^ is the point of distance (see before) corresponding to this, 
 line of measures. On I B'Z^ lay off bu^ 2", draw 71 D^ in- 
 tersecting Z^ aty"; then bf is the perspective of the upper right 
 edge. Since the diagonal of the right face which runs from f 
 is vertical, draw B^ perpendicular to G. L.\ it intersects Y^ at 
 c, hence b c is the perspective of the edge F^ The remainder 
 of the construction follows directly, as it is unnecessary to 
 make other measurements. 
 
 {/) Measurement of the Diagonals. Aline of measures for 
 B^ is I B'Z" (see before). Since B^ is parallel to the picture 
 plane, take any point on T Y Z, as D^, for a point of distance ;, 
 draw Tt^ftc and Ti"^ cm intersecting I B' Z'' at u and tn respect- 
 ively; then m u = p g (Fig. 30) = 2.83" = the length of the 
 diagonal B'. The diagonal C is parallel to the picture plane. 
 The initial point of a perpendicular to the picture plane 
 through f is n, because I B' Z' is the initial line of the profile 
 plane Z" V; hence I C parallel to C is the initial line of a 
 plane containing C\ and therefore a line of measures. Draw 
 e^ a/> intersecting I C at/, then np ^= r i (Fig. 30) = 3-46". 
 The diagonal C is horizontal and vanishes at V ; hence the 
 vanishing trace of a plane containing G^ is the horizon, and D"*, 
 
 found by laying off V« D» = V* V^ = -^^ = 4.20", is the 
 
 cos d "f ^ ' 
 
 point of distance. A^ is the line of measures, hence draw 
 T}^ gq intersecting I G' M' (or A-) at q ; then b q— pn = 3.46" 
 is the length required. The Jf diagonals of the two faces in- 
 clined down from A"- and A^ vanish at V'", found by laying off 
 yY yM _ ^v yw _ 4.^5"^ since these diagonals are in the faces 
 of the cube inclined 45° to the picture plane. Similarly the 
 iV diagonals vanish at V^. The diagonal J/' is in the plane 
 A" A''; hence I G' M' {A) is the initial line and line of meas- 
 
PARALLEL, ANGULAR, AND OBLIQUE PERSPECTIVE- 75 
 
 ures. A point of distance corresponding to this line is on 
 G. L. at a distance from V™ = 7", hence draw D** a r ; then b r 
 — 2.83" = 3 « in perspective. The diagonal M'' is in the plane 
 A^ A'. The initial point of a line {¥') in this plane is P', 
 hence I M'' Y* parallel to A' is a line of measures; then by us- 
 ing D'" we obtain xj = 2.83" for the length of M\ 
 
 A point of distance for iV' is D'* (on G. L.), found by laying 
 off VN D" = 7" ; then dt =. 2.83" is the length of N\ 
 
 {/) Figs. 32 and 33. The two upper cubes are symmetric- 
 ally placed with reference to T Y Z, and similiarly placed with 
 reference to the picture plane and ceiling; hence the explana- 
 tion for one suffices for the other. 
 
 One edge of the cube is vertical and 1.5' behind the picture 
 plane. To find this edge draw the perpendicular i c^, lay off 
 k z -^ 0.75", and draw ^r V*°^ intersecting ke' -aX a; then a is- 
 0.75' behind the picture plane, and the vertical line E" is the 
 indefinite perspective of the edge. The L and R edges are 
 horizontal and inclined at an angle of 45° to the picture plane; 
 hence they vanish at V*^°^ and V*^°^ respectively. Draw U 
 and R\ A point of distance for Z' is D*^"^ and T Q Z is the 
 line of measures, since the face L' R' is in the ceiling. Draw 
 D'^ a m, from m lay oS m n = 2", draw n D^ intersecting L' at 
 d; then ad is the perspective sought. A line of measures for 
 i?' is I U' B', hence lay off zj> = 1" and draw/ V^ intersecting 
 B^ at b; then ab m perspective = 2". A line of measures for 
 72' is T Q Z and the point of distance is D^, hence draw D^ a r 
 and from r lay off rj = 2", draw j D^ intersecting i?' at c\ 
 then a c\s the perspective of the edge required. The cube is 
 completed by drawing i?^ D ; then E', R', etc. 
 
 {g) Measurement of the Diagonals. The C/ diagonals van- 
 ish on T W R at V^ (not shown), 4.95" below V^- A point 
 of distance, D^, is found by laying off V^ D^ = 7". The line 
 
•j6 PERSPECTIVE. 
 
 of measures for Z7, \s,zq, hence draw VP aw and W fq inter- 
 secting ^■^ at w and q respectively; then wq =/^ (Fig. 30) = 
 2.83" = the length of the diagonal U ^. The diagonal /' van- 
 ' ishes at V'^ (not shown, 4.9S" above V^), a point of distance is 
 D-f on T W R, and the line of measures is ^r ^ (I U' B'). 
 
 It is evident that D-' c intersects zq beyond the limits of the 
 drawing, hence we employ a point of half distance Vi^l^ , found 
 by laying off D^D^/^^ = 1/2 D^ V = 3.5". Draw Yy'ftUs and 
 D1/2J h X (the point x is just above D) intersecting z q ^\. s and 
 
 TV Q 
 
 X respectively; then .y;f = — - =1.42" = one half the length of 
 
 y. The diagonal (9' vanishes at V^, and a point of distance is 
 found by laying off V^ DO = V° V«°. Since C, is in a profile 
 plane which contains the edges B^ and E", a perpendicular to 
 the picture plane through a determines the line of measures 
 I O' B' parallel to T Y Z. Draw D° ay and D° Ih intersecting 
 I O' B' at y and h respectively; then y h = ri (Fig. 30) = 3.46". 
 The length of the diagonal K^ isgi (on T W R) = 2.83". 
 
 (See § 105 for construction of the circles inscribed on the 
 faces of the cube in Fig. 33.) 
 
 99. Successive Perspective Plans. It is often necessary 
 in order to save confusion in the drawing and render the 
 construction more simple to make a perspective plan of the 
 object at some distance either below or above the desired posi- 
 tion. 
 
 The plan of a large building, for example, is frequently 
 quite complicated, and as at least half is invisible from a given 
 position, it is unnecessary to show that portion in the finished 
 drawing. The complete plan, however, is needed in order to 
 obtain the details of the roof, the chimneys, etc. 
 
 Plate VI, Fig. 29. Angular perspective. Given the vanish- 
 ing points, etc., as shown, also given a plan. The perspective 
 
PARALLEL, ANGULAR, AND OBLIQUE PERSPECTIVE. 77 
 
 drawing is made to the same scale as the plan in this particular 
 case. The plane of the plan is horizontal and the point a is in the 
 picture plane. The points a, b, c, etc., of the plan correspond to 
 the points a', (5\r', etc., of the perspective on the horizontal 
 plane through G. L., and the method of measurement is readily 
 traced. 
 
 Take any new ground line, as G^. Z,., parallel to G. L.\ from 
 d erect a perpendicular to G. L. (and H) intersecting G^.L^. at 
 «^ From a' lay out the perspective of the plan by measure- 
 ments as before. Since a' was determined by a perpendicular 
 to G. L. through a^, all the other points, as U', c\ d'^, etc., might 
 have been similarly determined; e.g., draw i?', erect the per- 
 pendicular b'' b^ intersecting 72' at b'' ; draw D and a perpen- 
 dicular through c' intersecting U at /, etc. Hence it is evident 
 that measurements were unnecessary to determiine the second 
 perspective plan. The geometric statement of the preceding 
 is : If a cylinder (general sense) be perpendicular to H, all hori- 
 zontal sections are equal, and the horizontal projections of 
 these sections coincide with the base of the cylinder on H. 
 
 100. Problem III. To find the perspective of a cube. Ob- 
 lique perspective. Plate VIII, Fig. 38. 
 
 Analysis and Construction. All the faces and edges are ob- 
 lique to the picture plane (§ 74). Each edge is 2" long. Let 
 F, R, Q be three edges of the cube intersecting at the point a 
 which is in the picture plane. Assume the vanishing points 
 V"^ and V^ of the edges F^ and Q' respectively; then T F Q is 
 the vanishing trace of the plane of those edges. Assume V'^, 
 the center of V^ V^, to be the vanishing point of the diagonal 
 N'' which bisects the angle F^ Q. From these assumptions it 
 follows that the center of the picture is on V^ V-'' drawn 
 through V perpendicular to T F Q. Assume e^ as the center 
 of the picture; hence the axis (§ 57) is one side of a right- 
 
78 PERSPECTIVE. 
 
 angled triangle of which the other side and hypotenuse are 
 e"" V^ and V^ Y^ respectively. With V^ as a center and radius 
 V'^ V^ describe an arc of a circle V^ e^ V^, draw e' e^ perpen- 
 dicular to V^V^ intersecting the arc at f,, then ^"^^, is the 
 length of the axis. The edgei?' is perpendicular to the face F^ Q 
 and to the diagonal iV'; hence i?' vanishes on V*^V^ De- 
 scribe an arc of a circle through V*^ e^ with center in V^ V^; 
 then V^ is the vanishing point of R', since V'^ e^ V^ is a right 
 angle, being inscribed in a semicircle. 
 
 e^ V^ is the distance of the station point from V^; hence 
 with V^ as a center and V^ e^ as radius describe a circle which 
 is the locus of the points of distance, D^, for the system R. 
 With V^ as a center and radius V^ e^ describe a circle, thus ob- 
 taining the locus of D'5; and with V^ as a center and radius 
 V^ e^ describe a circle, thus determining the locus of D^. We 
 may now proceed to draw the cube. 
 
 Through a, the vertex in the picture plane, draw Q, Ji\ F\ 
 the indefinite perspectives of the edges which intersect at a. 
 Draw the line of measures I Q' R' through a parallel to T Q R, 
 lay off a n = 2" and draw n D^ intersecting i?' at 3 ; also lay 
 off « / = 2" and draw t D*^ intersecting Q' at d; the perspec- 
 tives of two edges are now determined. Through a draw the 
 line of measures I F' R' parallel to T F R, lay off a m = 2" and 
 draw ;« D^ intersecting F^ aty, thus determining the perspective 
 of the third edge. The drawing is finished as in the preceding 
 problems, as all the necessary measurements have been made. 
 
 Instead of assuming e'' we might have assumed V^ ; then 
 ^^ may be found as follows: Describe semicircles through V^, 
 V, V« ; y% V^, V« and V«, e„ V^. 
 
 From the points of intersection o and/ of these semicircles 
 draw oV^ and/ V*^ intersecting at e'', the center of picture 
 required. Let the student prove this statement. 
 
PARALLEL, ANGULAR, AND OBLIQUE PERSPECTIVE. 79 
 
 lOOa. Second Solution. Fig. 3812:. The essential points in 
 either solution are the determination of the position of the 
 station point (having assumed the vanishing points of two 
 edges which are not parallel), and the position of the third 
 vanishing point. 
 
 Suppose the vanishing point of the diagonal iV' does not 
 bisect V V^ (Fig. 38), but has the position shown in Fig. 38a. 
 
 To find the position of the station point: On V^ V^ de- 
 scribe a semicircle which must contain the revolved position 
 (i?j) of the station point, since F'^ and Q are perpendicular. A 
 line from e^ to V must bisect the right angle between F^ 
 and Q ; hence the angle V^ e^ V^ = the angle V^ «■, V = 45°. 
 Hence in the triangles b e^r and r e^x we have the following pro- 
 portion : zr:r X =^ sin § : sin a = cos a : sin a = cot a. Through 
 r draw re = rx and perpendicular to zx, draw^rc; then cot a = 
 
 — . The line eic intersects the semicircle at e,, which is there- 
 r c 
 
 fore the revolved position of the station point in the picture 
 
 plane ; hence the center of the picture may be anywhere on the 
 
 line fj (f drawn perpendicular to V^V^. The position of the 
 
 third vanishing point readily follows. 
 
CHAPTER VII. 
 
 PERSPECTIVE OF CIRCLES. 
 
 loi. The problem, to find the perspective of a circle, is the 
 Descriptive Geometry problem, to find the intersection of a 
 cone with a plane (§ 51). In perspective the intersecting plane 
 is the picture plane, and the cone is given by its vertex (the 
 station point) and circular base in a plane which is usually 
 oblique to the picture plane. 
 
 102. In general the perspective of the center of a circle is not 
 the center of the perspective of the circle, {qx the same reason 
 that in general the axis of a cone does not pass through the 
 center of an oblique section. 
 
 103. Proposition IV. If any variable tangent to a central 
 conic section meet two fixed parallel tangents, it will intercept 
 portions on them, whose rectangle is constant and equal ' to the 
 square of the semi-diameter parallel to them.* Plate VIII, Fig. 
 
 39- 
 
 Construction. Let <« t be a diameter of the ellipse abed, 
 M and Q tangents at the extremities of the diameter a c, and 
 iV any other tangent intersecting Jf and Q at _/ and j' respect- 
 ively; then afy. cg=^ {oby. 
 
 Hence, if we have the tangents to any diameter of an ellipse 
 and any third tangent, the length of the semi-conjugate diam- 
 eter may be readily found (Fig. 39a:). af— af (Fig. 39), 
 fc=gc (Fig. 39). On ac asa diameter describe a circle, draw 
 
 * Salmon's Conic Sections. 
 
 80 
 
PERSPECTIVE OF CIRCLES. 81 
 
 fm perpendicular to ac, then {fmy = {ob)" (Fig. 39) = afx. 
 fc. 
 
 104. If we have six or more tangents to a small ellipse and 
 the corresponding points of tangency, the ellipse maybe drawn 
 accurately without finding diameters. 
 
 105. Problem IV. Given a cube in perspective to describe 
 circles on the visible faces. Plate VII, Figs. 33 and 33(3;. 
 Method of tangents (§ 104). 
 
 Analysis and Construction. For the construction of the 
 cube see § 98/. It is required to inscribe circles (in perspec- 
 tive) in R'R\ R^ R\ and E> B\ the visible faces of the cube. 
 
 Fig. 33«. To find the tangents and points of tangency 
 (§ 104) inscribe a circle in a square (2" sides) of the same size 
 as that of the face of the cube. Draw the diagonals n a and 
 m h and the four tangents to the circle which are perpendicular 
 to these diagonals, and we have eight tangents and eight points 
 of tangency, y, c, d, i, 0, k,g, and /, which are to be transferred 
 to the perspective. 
 
 Fig. 33- The lengths of D and R^ were laid off by means 
 of the line of measures T Q Z ; in nla the length of L\ and r g- 
 is the length of i?'. Hence lay off np — np (Fig. S3a),po = 
 po, q =^ q and qm =^ q m. From the points p, 0, q draw 
 lines to D'^ intersecting L\ and thus transfer the distances laid 
 off on m n to U. Similarly for R^. 
 
 The same subdivision may be effected by proportional 
 measures (§ 90) as follow^ : Through a draw at, a. line of pro- 
 portional measures, parallel to the horizon, on a t lay off «;r„ 
 xt, etc., = ax,xt, etc. (Fig. 33a) respectively, draw t c F^ ; 
 then P^ is a point of proportion distance. From the points of 
 subdivision on a t draw lines to P^ intersecting i?' a.tj,j, and s,. 
 thus determining the same points as before. 
 
 To divide B', which is parallel to the picture plane (§ 94), 
 
82 PERSPECTIVE. 
 
 proceed as follows : Through a (Fig. no) draw ab = ab '(Fig. 
 33), join bh, from the points of division draw parallels to b h 
 intersecting ab zs shown. Transfer the distances thus obtained 
 directly to 5' (Fig. 33), and the required subdivision is effected. 
 
 Complete the octagon on each face and draw the diagonals 
 to determine the points of tangency not on the edges of the 
 cube. Note the fact that the tangents (Fig. 33«)/ /^ and/t are 
 parallel to nih, and hence have the same vanishing point, and 
 that q i and sf are parallel to n a, and hence have the same 
 vanishing point. On each face draw a smooth curve (ellipse) 
 tangent to the octagon at the determined points of tangency, 
 and thus obtain the required solution. 
 
 106. Problem V. To find the perspective of a circle. Plate 
 VIII, Figs. 40-43« inclusive. 
 
 General Analysis. The same horizon, vanishing points, etc., 
 are used for all the figures. The center of the picture is e^, and 
 the station point is 4" in front of the picture plane i. e., ^'^ V^° 
 = 4". The diameter of each circle is 1.5". 
 
 («) First Case. When the circle is in a vertical plane, and 
 the perspective of the center is on the principal horizon. Fig. 
 40. 
 
 Analysis. Circumscribe a square about the circle with its 
 sides vertical and horizontal. Assume that the horizontal sides 
 vanish at V*^°^, thus placing the circle in a vertical plane inclined 
 45° to the picture plane, and assume that one of the vertical 
 sides is in the picture plane. The circumscribed square will 
 give two parallel tangents to the perspective of the circle, and 
 two tangents which are not parallel ; hence diameters of the 
 elliptical perspective are readily found by § 103. 
 
 Construction. Let ab= 1.5" be the side of the square in 
 the picture plane, and R' and R^ the indefinite perspectives of 
 the horizontal sides. Since the perspective of the center is to 
 
PERSPECTIVE OF CIRCLES. 83 
 
 be on the horizon, ap = bp. Through b draw a Hne of meas- 
 ures, b n, parallel to the horizon, lay o'S. bin ■= m n = 0.75", and 
 draw m Yi^ and n D^ intersecting i?' at f and c respectively. 
 Draw //^ and c(/ perpendicular to the horizon; then abcd\s 
 the perspective of the circumscribed square. Since /^ is per- 
 pendicular to the tangents a b and c d, it is an axis of the ellipse. 
 Through 0, the center oi p q, draw go r parallel to a b, and lay 
 
 ofi or = (7^ = Vdg X <^p for the major axis. An ellipse con- 
 structed oh p g aiid^r as axes is the perspective of the circle 
 required, e^ in this case coincides with the perspective of the 
 center of the circle; hence the axis of the picture (§ 57) is the 
 axis of the projecting cone of the circle. 
 
 (a) Second Case. Fig. 40a. Analysis. The position of the 
 circle is so chosen that the picture plane cuts a sub-contrary 
 section from the projecting cone. One point of the circle is 
 in the picture plane. A circumscribed square is used as before. 
 
 Construction. Let the plane of the circle be inclined at an 
 angle of 60° to the picture plane towards the right. The per- 
 spective of the center is 2.31" (4 tan 30°) to the left of e^ at 
 D^, since the axis of the projecting cone must be inclined at 
 an angle of 60° (in this case) to the picture plane in order that 
 the picture plane cut a sub-contrary section. The triangle 
 lying in a horizontal plane through the horizon and formed by 
 the axis of the projecting cone, the horizon, and a diameter of 
 the given circle, is equilateral (in this case) ; hence to find the 
 point of the circle in the picture plane lay off D*^ r = 0.75", and 
 r is the point desired. Through r draw arb=^ 1.5" for the 
 side of the circumscribed square which is in the picture plane. 
 Draw iV' and iV". Through b draw b m parallel to the horizon, 
 draw m D*' intersecting N at c, and draw c d parallel to « (5 to 
 complete the square. Bisect r/ at 0; with (7 as a center and 
 radius o r describe a circle, which is the perspective required. 
 
84 PERSPECTIVE. 
 
 (a) Third Case. When the circle is in a vertical plane 
 and the perspective of the center is not on the horizon. 
 Fig. 41. 
 
 Analysis and Construction. Let the perspective be a du- 
 plicate of Fig. 40, except that the center is below the 
 horizon. Let a b , on a b (Fig. 40) produced, be the side of the 
 circumscribed square in the picture plane, draw R^ and R' lirB- 
 ited by dc (Fig. 40) produced. Then a b c d is the perspective 
 of the circumscribed square. Through/, the center of ab, draw 
 R". Produce ab to t, making b t =^ c g, on p t describe a semi- 
 circle, draw b r perpendicular to p t\ then b r \s the length of 
 the semi-diameter conjugate to p g (§ 103). Through i?, the 
 center of / ^, drawee / parallel to a (5, and lay off <?/ = ij^ = 
 b r for the diameter conjugate to p g. An ellipse constructed 
 on these diameters is the perspctive of the given circle. 
 
 (V) First Case. When the circle is in a horizontal plane. 
 Fig. 42. Analysis. In order to obtain diameters of the per- 
 spective easily the sides of the circumscribed square should be 
 parallel to, and perpendicular to, the picture plane. 
 
 Construction. The perspective of the circumscribed square 
 is abed, found by employing a point of one-half distance, 
 D'/^^, instead of V^, for the measurement of the sides perpen- 
 dicular to the picture plane. The parallel tangents (see be- 
 fore) are be and ad, and c d is 3l third tangent. A mean pro- 
 portional between cf and dj \seg; hence through z, the 
 center oijf, draw t ir parallel to a d, and lay off «V = i t ^= eg. 
 On fj and t r as conjugate diameters construct an ellipse 
 which is the required perspective. 
 
 {b) Second Case. When the circle is in a horizontal plane. 
 Figs. 43 and 43a. 
 
 Fig. 43 is a duplicate of Fig. 43a in perspective. The cir- 
 cle is circumscribed by an octagon (see § 105), and an ellipse 
 
PERSPECTIVE OF CIRCLES. 85 
 
 is drawn through the points of tangency in perspective. As 
 the construction is like that of § 105, no description is given. 
 
 107. Apparent Distortions. Fig. 44 shows the perspec- 
 tives of four horizontal circles constructed as in § 105. The 
 circles are equal and their centers are on the same straight 
 line. The perspectives present a twisted or distorted appearance, 
 although the drawing is accurate, due to the fact that in look- 
 ing at the drawing the eye is not usually placed exactly at the 
 station point (§ 55). This distortion is much more apparent in 
 curves than in straight lines, although it exists equally in both. 
 From an artistic standpoint this figure shows the limitations 
 of One-plane Perspective. In order to remedy this apparent 
 distortion and make the drawing " look right " from other 
 positions than the station point, the perspectives should be 
 turned so that their axes are parallel. The amount of change 
 necessary to effect the correction depends greatly upon the ar- 
 tistic sense of the draftsman, and no rules will be given for 
 guidance, except that the drawing should first be made accu- 
 rately and the changes made afterwards. 
 
 When the perspectives of the centers of circles are on the 
 horizon (see Figs. 40 and 40«), no correction is needed, fof 
 the axes of the ellipses are parallel. 
 
 108. Problem VI. To find tlie perspective of a sphere. Plate 
 IX, Fig. 45. 
 
 Analysis. This problem in Descriptive Geometry would be 
 stated thus : To find the intersection with F of a cone tan- 
 gent to a given sphere, the vertex of the cone being a given 
 point. To solve the problem in perspective we employ partly 
 the direct method of Descriptive Geometry and partly the 
 method of measurements heretofore given. 
 
 Consider the picture plane to be the Fcoordinate plane, and 
 the H coordinate plane to be a horizontal plane through G. L. 
 
86 PEKSPECTJVE. 
 
 The sphere, 1.5" diameter, is in the second dihedral angle, and 
 the projections coincide. The vanishing points, etc., are as 
 shown, and the station point is 3" in front of the picture plane. 
 
 Construction. Pass a cone tangent to the sphere having its 
 vertex at e. B^, D^ are the H projections of the contour ele- 
 ments, and the broken line to the right of, and parallel to, 
 y-H <;H^H jg ^jjg jj- projection of a chord of the circle of tan- 
 gency which is parallel to ^ in a horizontal meridian , plane of 
 the sphere. Project the sphere and tangent cone on a new V 
 plane through G^. L^. parallel to the axis of the cone ; then 
 h^^e^^d^^ is the new F projection of the apparent contour of 
 the cone, and V^c'-^d^^ is the projection and diameter of the 
 circle of tangency. The H projection of this diameter is 
 h^ c^ d^, which is an axis of the H projection of the circle. 
 Draw f^ c^ g^ perpendicular to d^ b^, and lay off c^ f^ = c^ g"' 
 = c"^ V'^\ the ellipse constructed on these lines for axes is the 
 ZTprojection of circle of tangency. The problem now becomes : 
 To find the perspective of a circle which is in an oblique 
 plane. The plane Q perpendicular to the axis of the tangent 
 cone is a plane parallel to the plane of the circle of tangency. 
 T N Q parallel to V Q is the vanishing trace of the system Q, 
 and hence the vanishing trace of the plane of the circle of tan- 
 gency. 
 
 Consider a square circumscribed about this circle with two 
 of its sides horizontal ; then the other two sides are lines of 
 greatest declivity of the plane of the circle. The horizontal 
 sides vanish at V*^, and the inclined sides at V^ (not shown), 
 the intersection of T N Q and T O Q; since the II projections 
 of the inclined sides being perpendicular to H Q, vanish at V°. 
 D*^, found by laying off V D^ =/ e^, is a point of distance for 
 iVlines, and D'^ (not shown), found by laying off e'' D** = V e^ 
 is a point of distance for Q lines. 
 
PERSPECTIVE OF CIRCLES. 87 
 
 Fig. 45(2 shows an octagon circumscribed about a circle 
 which is equal to the circle of tangency and gives the measure- 
 ments necessary to draw the perspective (§ 105). The per- 
 spective of the H projection of the center of the sphere is 
 0'"^, the intersection of (e^o^, e^ n) with the picture plane {V), 
 Horizontal lines perpendicular to the horizontal sides of the 
 circumscribed square, i. e., lines parallel to the H projections 
 of the inclined sides, vanish at V° (see before), and their point 
 of distance is D°. 
 
 Through 0^^ draw D° o^^j, lay offyV = h i ,j q = i k, and j'n 
 ■^ hk — ky, draw /D°, q D°, and n D° intersecting (9", the 
 perspective of the H projection of a line through the center of 
 the circumscribed square parallel to the inclined side, at c/"', 
 <:'", and V^ respectively. These points are the perspectives of 
 the H projections of d, c, and b respectively. To find the per- 
 spectives of the points b, c,d, take any point on the horizon, as 
 D^, for a point of distance, and draw D^ b^^ v, D-^ c'" w, and 
 D^ d^^ X intersecting G. L. at v, w, and x respectively. From 
 these points erect perpendiculars to G. L., lay off distances 
 equal to b'^''y, c^^k, and d^-^h, and from the points thus obtained 
 draw lines to D'^ intersecting (9' at i5', t\ and (^'respectively, 
 which are the perspectives of b, c, and d respectively of the cir- 
 cle of tangency, or the points b, c, and d of Fig 45^. Draw iV', 
 TV", and N^, the indefinite perspectives of the horizontal sides of 
 the circumscribed square, and of a line parallel to them 
 through the center of the circle. Since N'^ is horizontal, its 
 line of measures is s s parallel to G. L., and at a distance above 
 G. L. equal to hd''^. Draw D^ d^ u, lay off u s = ti z = df 
 (Fig. 45a), and draw s D*^ and z D^ intersecting N'' at m and r; 
 then m d' r is one side of the circumscribed square in perspec- 
 tive. Draw Q' and Q'' to complete the perspective of the 
 square. The perspective of the octagon (not shown) is now 
 
88 PERSPECTIVE. 
 
 readily found. The ellipse as shown is the perspective of the 
 sphere. 
 
 Alternate Method. After finding the two projections (only 
 H projection shown) of the circle of tangency of the projecting 
 cone we may use the direct method of §77 to find the per- 
 spective of the circle; i. e., find the intersection (base)of the 
 cone with the picture plane (F). Let a be any point on the 
 circle of tangency, through e (the station point) and a draw 
 the line F, which intersects the picture plane at a' ; hence «' is 
 the perspective of a. (See § 107.) 
 
 109. Method of Coordinates. If the position of a point 
 relative to three rectangular axes be known, its perspective 
 may be readily determined. This method for the determina- 
 tion of perspectives is known as the Method of Coordinates. It 
 is used to find the perspective of any irregular object which 
 cannot be readily determined by other methods, and for 
 sketching in contours. 
 
 Construction. Plate IX, Fig. 46. Let X, Y, Z be the rect- 
 angular axes corresponding to the axes of Analytic Geometry. 
 (See § 22.) 
 
 Two of the axes are in the picture plane, and the third 
 passes through their intersection perpendicular to the picture 
 plane. The axes are divided according to any convenient 
 scale, and parallels to the axes are drawn through the points of 
 division. 
 
 To determine the perspective of a point the coordinates of 
 which are + 3, —3,-1-2 proceed thus : Plus 3 means that the 
 point is in a vertical plane through 3/, — 3 that the point is 
 in a vertical plane through 3 q, and -\- 2 that the point is in a 
 horizontal plane through 2 r ; hence the perspective of the point 
 is a, the intersection of the three planes. 
 
CHAPTER VIII. 
 
 PERSPECTIVE OF SHADOWS. 
 
 110. For definitions, etc., concerning shadows see §§ 5-17 
 inclusive. 
 
 111. Vanishing Point of Rays. The vanishing point of 
 rays of light iS in the vanishing trace of vertical planes inclined 
 to the right at an angle of 45° to the picture plane, and at a 
 distance below the horizon equal to the length of the axis of 
 the picture (see Fig. i, page 3), since rays are equally in- 
 clined to H and V (or picture plane) and their projections 
 make angles of 45° with G. L. 
 
 The H projections of rays vanish at V*^°-^- 
 
 112 Determination of the Perspectives of Shadows. 
 The determination of the perspective of the shadow of any 
 right-line object involves usually the determination of the 
 vanishing point of the line of intersection (§§ 11 and 13) of a 
 plane of rays passed through each division of the line of shade 
 with the plane which receives the shadow ; e. g., if the shadow 
 is on H and the line of shade is vertical, the vanishing point 
 of the shadow is V*'"^, because this point is the intersection of 
 the vanishing trace of /f with the vanishing trace of a plane of 
 rays through the vertical line of shade (§ 66). 
 
 113. Perspective of the Shadow of a Point. The fol- 
 lowing is a general method for finding the perspective of the 
 shadow of a point on a plane : Through the perspective of the 
 point draw the perspective of a ray, and through the perspec- 
 tive of the projection of the point on the plane draw the per- 
 
90 PERSPECTIVE. 
 
 spective of the projection of a ray on the plane ; the intersec- 
 tion of these two lines is the perspective of the shadow of the 
 point on the plane.* This method follows directly from the 
 fact that the intersection of a line with a plane is the intersec- 
 tion of the line with its projection on the plane. 
 
 114. Problem VII. To find the perspective and the per- 
 spective of the shadow of a vertical prism mounted on a paral- 
 lelopiped. Angular perspective. Plate X, Fig. 47. 
 
 Analysis. The parallelepiped measures 2" X 2" X 1.06", 
 and prism 0.5" X 0.5" X I.39"- The horizon is the lower of 
 the two horizons shown in the figure. The vertical faces of the 
 two bodies are inclined 30° and 60° to the picture plane, and 
 the vanishing points of the principal lines are as shown, except 
 V'*', which is not within the limits of the drawing. The station 
 point is 4" in front of the picture plane. As the method for 
 finding the perspective is similar to that given in previous 
 problems, no explanation is necessary, and we may immedi- 
 ately proceed to find the perspective of the shadow. 
 
 The line of shade is determined by inspection, as in Shades 
 and Shadows. Part of the shadow is on H and part is on the 
 parallelepiped. 
 
 Construction. The line of shade of the parallelopiiped is 
 composed of the edges A", U', M', A", and the two left edges 
 in H. Rays vanish at V^, and the H projection of rays vainish 
 at V*5°«. 
 
 The vanishing trace of the plane containing A^ and a ray is 
 T R X, and the vanishing trace of the plane on which the 
 shadow of A^ is cast is the horizon, since the shadow of A^ is 
 on H ; hence V^, the intersection of T R X and the horizon, 
 is the vanishing point of the shadow. Through a draw R\ the 
 
 * Descriptive Geometry, A. E. Church. 
 
PERSPECTIVE OF SHADOWS. QI 
 
 indefinite perspective of the shadow of A^ on H, and through 
 /draw the perspective of a ray intersecting R at f^ ; then a/^ 
 is the perspective of the shadow required. V vanishes at V°\ 
 hence the vanishing trace of a plane of rays through W is 
 TUX. The shadow of U' is on H and vanishes at V^', since W 
 is horizontal ; hence U\ is the indefinite perspective of the 
 shadow. The perspective of a ray through o intersects U\ at 
 0^ ; hence /s o^ is the perspective of the shadow of V on H. 
 Since M^ is horizontal and vanishes at V"" (not shown), the 
 perspective of the shadow of Af' on H vanishes at V'"; hence 
 draw the indefinite perspective o^ V'". It is now evident that 
 the perspective of the shadow of the rear edge, A^, is invisible, 
 and therefore it is not drawn. 
 
 The line of shade of the prism is composed of the edges 
 A\ Al,j k, k I, dc, and c b. Since the upper base, W M\ of 
 the parallelepiped is horizontal, the shadow of A'' on this plane 
 vanishes at V (see before); hence through b draw the indefi- 
 nite perspective, i?^ of the shadow. K' intersects M^ at u^ 
 the perspective of a ray through u intersects o^ V at u^ ; then 
 K' limited by the perspective of a ray through/ is the per- 
 spective of the shadow of ^^ on H. 
 
 Or, through b^ draw b"^ V^ {K°) intersecting o^Y^ at u^, etc. 
 The shadow of j k on H vanishes at V^, hence draw j\ V^ and 
 kSr^ intersecting at k^ , then/g k^ is the perspective of the shadow 
 of J k. 
 
 The line k I vanishes at V"*^, hence draw k^ V" and / V^ inter- 
 secting at 4, then k^ 4 is the perspective of the shadow of k I. 
 Since A" is vertical, draw i?' {d^ 4 V^) to complete the shadow 
 on H in perspective. The perspective of the shadow of A^ on 
 the upper base of the parallelepiped is i?', found by drawing 
 
 The perspective of the visible shadow on H is i?', U^, o^ u^. 
 
92 PERSPECTIVE. 
 
 R\js Ki '^s4> ^, and M"-, which is its own shadow. The per- 
 spective of the visible shadow on the upper base of the paral- 
 lelepiped is bu and a small part of K!'. 
 
 115. Problem VIII. Given a vertical rectangular column 
 near a small building having a projecting roof : To find the per- 
 spective and the perspective of the shadow. Angular perspective. 
 Plate X, Fig. 48. 
 
 Analysis. The vanishing points, etc., are as shown, and the 
 station point is 4" in front of the picture plane. One edge of 
 the column is in the picture plane, and from the center of the 
 rear edge of the upper base a guy-line {F) extends to the 
 ground. The column is in 30°, 60° perspective, and the build- 
 is in 45"^ perspective as shown. The special problem is to de- 
 termine the perspectives of the shadows of the column and 
 guy-line on the building. 
 
 Construction. Let abed, f B\ B' be the perspective 
 of the column with the guy F^ running from /to the ground, 
 which is a horizontal plane through G. L., at g. Also, let the 
 perspective of the building be as shown, with m^ j^ P k^ m^ 
 for the perspective plan (§ 99) of the roof. The vanishing trace 
 of the plane of the left side of the roof is T L J, and of the 
 right side is T L W. V'^ is on T W R above the horizon. 
 
 The shadow of the building. The line of shade is composed 
 of the lines /', /^ L\ L\ A", A\ and g v^ 2 l\ The indefinite 
 perspective of the shadow of A' is v^ V^- The perspective of 
 the shadow of m (the corner of the roof) is Wg, found by § 113, 
 i. e., the intersection of the perspective of a ray through m 
 with the perspective of the ff projection of a ray through m^. 
 Since L^ is horizontal and vanishes at V", its shadow on H 
 vanishes at V", hence through m^ draw m^ V"^ intersecting v^ V 
 at Ps, then v^ ps is the perspective of the shadow of A^. The 
 vanishing trace of a plane of rays through y is T W R, hence 
 
PERSPECTIVE OF SHADOWS. 93 
 
 the shadow of /" on H vanishes at V^. Draw R\ through n 
 draw the perspective of a ray intersecting E^ at n^, then m^ n^ is 
 the perspective of the shadow of /^ V vanishes at V^ hence 
 fig V^ is the indefinite perspective of its shadow. The per- 
 spective of a ray through o intersects «s V^ (Z's) at Og. Through 
 Os draw R* for the perspective of the shadow of y. The re- 
 mainder of the shadow on H is invisible. The vanishing point 
 of the shadow of V on the right side of the building is V^, the 
 intersection of T W R and T L X, hence through p, the inter- 
 section of the plane A^ A'' with IJ, draw X' for the perspective 
 of the required shadow on the right side of the building. 
 Through v draw v V^ for the perspective of the shadow of U 
 on the left side. 
 
 Shadow of the Column. Since E" and B' are vertical their 
 shadows on //vanish at V^, hence through a and b draw /?' 
 and K respectively. The perspectives of rays through x and 
 c intersect i2' and E^ at x^ and c^ respectively, thus determining 
 the perspective's of the shadows of E" and B' on H. Since x d 
 vanishes at V^, its shadow on H vanishes at V^, hence draw 
 x^ V^ ; and since d c vanishes at V*, its shadow on H vanishes 
 at V*, hence draw V* ^s d^ intersecting x^ V^ at d^. We now 
 have the complete shadow of the column on H in perspective. 
 The intersections of planes of rays through B" and B'' with the 
 side A^ A^ of the building are parallel to B^ and B^ and hence 
 are parallel in perspective, therefore draw B\ and B\ for the 
 perspectives of the shadows of B^ and B^ on the side of the 
 building. Since itf- k^ is the perspective plan of L\ the plane 
 m^ m k k^ is parallel to A" A', hence erecting perpendiculars 
 from 5° and r", the shadows s and r of B^ and B' respectively 
 on D are determined. The vanishing trace of a plane of rays 
 through B^ or 5' is T W R and the vanishing trace of the left 
 side of the roof is T L J, hence the vanishing point of the 
 
94 PERSPECTIVE. 
 
 shadows of E" and B" on this side of the roof is V-' (not shown), 
 the intersection of T W R and TL J. Through s and r draw 
 y andy* respectively for the required perspectives. Similarly 
 the perspectives of the shadows of B\ and B' on the right side 
 of the roof vanish at V^, the intersection of T W R with 
 T L W, hence through i and h draw iN^ and h V^ respect- 
 ively for the required perspectives. These two lines are limited 
 by;7. 
 
 Shadow of the Guy. /^vanishes at V^, hence the vanishing 
 trace of a plane of rays through F \% T F N. The vanishing 
 point of the shadow of ^ on H is V^ (not shown), the 
 intersection of T F N and the horizon, hence draw N^ for 
 the perspective of the shadow on H. Or, find the perspective 
 of the shadow of /on H and join with g. The shadow of F 
 on the H projecting plane of D, or the plane in^in k/^, van- 
 ishes at V^, hence t u is the perspective of the shadow of F on 
 that plane. The vanishing point of the shadow of F on the 
 left side of the roof is the intersection of T F N and T L J (not 
 shown, beyond the left border), hence through u draw E\ The 
 vanishing point of the shadow of F on the right side of the 
 roof is V^ the intersection of TFN and TLW, hence from 
 the point where £■' intersects Z" draw a hne to V^ and limit it 
 by the intersection with //. Or, to find the shadow of the guy 
 on the roof, produce W to y and draw yw\^, which is the 
 perspective of the intersection of the plane of the right side of 
 the roof with H. N^ intersects yY^ at w, hence through w 
 draw V^w intersecting D, etc. 
 
 Il6. Illustration. Plate XI shows the perspective of a C. 
 B. & Q. R. R. standard furniture car drawn by the methods out- 
 lined in the preceding paragraphs. The scale is f" = i'. The 
 principal lines are in 30°, 60° angular perspective ; the 30° van- 
 ishing point is about 23" to the right of e^, and the 60° vanishing 
 
PERSPECTIVE OF SHADOWS. 95 
 
 point is about 7" to the left of e'. e' is 2.5" above O (see 
 drawing), and the station point is 13" (nearly) in front of the 
 picture plane. No part of the car or track is in the picture 
 plane. 
 
 Problems. 
 
 Each problem may be stated in six different ways, depending 
 upon the position of the object relative to the picture plane. 
 {a) Parallel perspective, {b) Angular perspective, (c) Oblique 
 perspective; (i) Part of the object is in the picture plane, (2) 
 No part of the object is in the picture plane. When the state- 
 ment of the problem is succeeded by the letters {a 1), etc., it 
 means that the drawing is to be made in parallel perspective 
 {a), and that a point or line of the object is in the picture plane 
 (i), etc. Each problem is to be solved in the positions (« i), 
 {a 2), (b i), and [pi), unless otherwise stated. 
 
 All dimensions are to be given and the perspectives are to 
 be drawn to scale. 
 
 1. Find the perspective of a pyramid, base on H. 
 
 2. Find the perspective of an inverted pyramid, base par- 
 allel to H. 
 
 3. Find the perspective of a vertical hexagonal prism. 
 
 4. Find the perspective of a hexagonal prism resting on 
 one face in H. 
 
 5. Find the perspective of a vertical cross. 
 
 6. Find the perspective of a square column surmounted 
 by a square abacus. 
 
 7. Find the perspective of a rectangular box with its lid 
 partly raised, {c i), {c 2), etc. 
 
 8. Find the perspective of a bookcase containing three 
 shelves. 
 
 9. Find the perspective of a square-topped table with four 
 square legs. 
 
96 PERSPECTIVE. 
 
 10. Find the perspective of a flight of steps. 
 
 11. Find the perspective of a triangle situated in an oblique 
 plane, {c i), {c 2), etc. 
 
 12. Find the perspective of a vertical circular cylinder. 
 
 13. Find the perspective of a circular cone, axis vertical. 
 
 14. Find the perspective of an ellipsoid of revolution, axis 
 vertical. 
 
 15. Find the perspective of any building. 
 
 16. Find the perspective of a groined arch, (a i), (« 2). 
 
 17. Find the perspective and the perspective of the shadow 
 in any one of the preceding problems. 
 
APPENDIX. 
 
 Shading. 
 
 117. Hypothesis.* («) All objects to be shaded are to be 
 considered unpolished ; the character of the surface being simi- 
 lar to that of a plaster cast. 
 
 {b) The sun is the only source of direct light and is fixed in 
 position (§ 5). 
 
 (c) Objects are also illuminated by reflected light from the 
 minute particles which are everywhere present in the atmos- 
 phere. 
 
 Were it not for reflected light the shade of an object 
 would be black ; hence these particles illuminate the shade or 
 portions in shadow and cause them to be visible. Reflected 
 light, being much less intense than direct light {U), does not 
 cause shadows and does not affect the line of shade. 
 
 (d) For the illumination of the shhde the minute reflecting 
 particles (c) may be considered, with but slight error, to be con. 
 centrated in a point diametrically opposite the sun or source of 
 light. (See f and g^ 
 
 (e) The only kinds of light which are supposed to reach an 
 object are from the source {b) and from atmospheric particles 
 {c). All surrounding objects which might reflect light and thus 
 
 * This method of shading is due M. Jules Fillet, Ecole Nationale des 
 Fonts et Chauss6es, to whose works the student is referred for a more extended 
 treatment embracing the effect of distance, color, etc. 
 
 ^7 
 
98 APPENDIX. 
 
 alter the appearance of the given object are considered to be 
 removedi 
 
 (/) The apparent illumination of an object depends only on 
 the intensity of the source of light and the angle which the rays 
 make with a normal to the surface. 
 
 ig) The illumination of any element (any very small area) 
 of a surface is proportional to the cosine of the angle which the 
 incident ray makes with a normal to the surface at that 
 element. 
 
 Ii8. Deductions. Using the sphere as a typical surface, 
 we make the following deductions from § 117. (See Plate XII, 
 Fig. 49.) 
 
 {a) The apparent darkest part of the sphere is the line of 
 shade o, because the cosine of the angle between an incident 
 ray and normal is a minimum, i. e., = O, at any point of the 
 line ; the apparent lightest part is the point 8, because the 
 cosine of the angle between the incident ray and a normal at 
 this point is a maximum, i. e., — i. From 8 to o the shading 
 gradually increases in depth, and from o to — 5 it gradually 
 decreases in depth ; the part O to — 5, however, is much darker 
 than the part o to 5, since it is unilluminated by direct light. 
 
 {b) If any object be placed so as to exclude light from the 
 sphere, the shadow is as much darker as that portion would 
 be lighter without the interposed object, for the shadow is 
 illuminated only by reflected light from atmospheric particles. 
 
 119. Method of Shading. Plate XII, Figs. 49 and 49a. 
 Let the diameter of the sphere which coincides with a ray be 
 divided into sixteen equal parts, and let planes be passed per- 
 pendicular to the diameter (ray) through the points of division. 
 Fig. 49« shows the lengths of the axes of the projections 
 of the circles cut by the secant planes. 
 
 Call the line of shade o, the ellipses (in projection) which 
 
SHADING. 99 
 
 are illuminated by direct light i, 2, 3, 4, etc., and the ellipses 
 (in projection) which are illuminated only by reflected light 
 — I, — 2, — 3, — 4, etc. We are now ready to apply the tints 
 of india ink to produce the proper shading. 
 
 {a) To obtain the tint to be applied to the shade rule a 
 small sheet of white paper with equidistant lines, thus : 
 
 so that the width of each line is one half the width of the 
 white space between any two lines. In a small beaker place 
 six portions of water (a brushful may serve as a measure) and 
 enough india ink to obtain a shade equivalent in depth to that 
 produced by the equidistant lines on the ruled sheet. Call 
 this tint No. o and apply to the shade of the object, i. e., be- 
 tween the o line and the right contour. Add one portion of 
 water to tint No. o, call this tint No. i, and apply between the 
 lines I and — i after the o tint is dry. Add two portions of 
 water to No. i, call this tint No. 2, and apply between the 
 lines 2 and — 2 after No. i is dry. Add three portions of 
 water to No. 2, call this tint No. 3, and apply between the 
 lines 3 and - 3 after No. 2 is dry, etc. This method gives a 
 banded appearance to the drawing. 
 
 In order to give a uniformly decreasing depth of tint proceed 
 thus ; First place the No. o tint over the portion in the shade 
 as before and allow this tint to dry. Then between the lines 
 I and —1 apply tint No. i, but instead of allowing it to dry 
 "draw out" the edges with a brush moistened with water, and 
 thus eliminate the sharp lines which would otherwise appear 
 at I and — i; after the "drawing out " is effected and the 
 
lOO APPENDIX. 
 
 tint is allowed to dry lay on tint No .2, "draw out," etc. 
 Either method is readily acquired after a few preliminary 
 trials. 
 
 (^) Suppose an object placed between the sphere and the 
 sun so as to cut off direct light from a small portion of the 
 area included between the lines 5 and 7. This portion is now 
 in shadow and must be shaded according to the statement of 
 § \\U. 
 
 First, over the area inclosed by the line of shadow (or sim- 
 ply shadow) lay on tint No. o. Second, lay on tints Nos. I, 2, 
 3, 4, 5, 6, 7, as if the lines 5, 6, 7 were in the shade; /. e., over 
 7 to 6 lay on tints Nos. i, 2, 3, 4, 5, 6, 7, arid over 6 to 5 lay 
 on tints Nos. i, 2, 3, 4, 5, 6. Third, over the shadow lay on a 
 tint stronger than No. o, call this tint No. S. Fourth, to tint 
 No. S add one portion of water, call this tint No. 6', and Jay on 
 between 7 and 6 ; to tint No. 6' add two portions of water, call 
 this tint No. 5', and lay on between 7 and 5, etc. 
 
 120. Shading of Any Surface. The proper shading for 
 
 any surface may be determined from the sphere type (Plate 
 
 XII, Fig. 49) by inscribing a tangent sphere in the surface 
 
 and finding the intersections of the secant planes passed through 
 
 '"the sphere with the surface. 
 
 Plate XII, Fig. 50, gives the shading lines on a vertical cir- 
 cular cylinder. A tangent sphere is inscribed in the cylinder, 
 with center at the center of the base (on i7) of the cylinder. 
 Fifteen equidistant planes are passed perpendicular to a ray and 
 their intersections with the base determined, thus giving the 
 line 5, 6, 6, 5, 4, 3, 2, i, o, — i,— 2, — 3, and — 4. It is at once 
 evident that the lightest part of the cylinder is between 6 and 
 6 corresponding to the zone 6 on the sphere (§ 119). 
 
 PIat,e XII, Fig. 50a, gives the proper shading lines on a 
 circular cylinder, axis parallel to <9. Z. C . . . . D, with corre- 
 
SHADING. lOI 
 
 spending shading lines, is for H projection and E . . . . F, wifh 
 corresponding shading lines, is for V projection. 
 
 121. Plate XII, Fig. 51, shows the shading on a sphere in 
 F" projection by the "drawing out" method; and Fig. 52 
 shows the shading on a vertical cylinder by the " banded " 
 method. 
 
PLATE I 
 
 G— 
 
 G^ 
 
 L 
 
 G- 
 
 
 J— L 
 
 G- 
 -L 
 
 V-L 
 
 
 
 \ 
 \\ 
 
 -0-' / ^|- L 
 
 
 \ 
 \ 
 
 o;t 
 
 / 
 
 \ 
 
 / 
 
 Fiff. 10. 
 
PLATE II. 
 
 Ft (J. i:$. 
 
 HW 
 
PLATE in 
 
 •L 
 
IMATE IV. 
 
PLATE V. 
 
 / 
 
 / \\ 
 
 I I \ \ 
 
 //I 
 
 / 
 
 ~1^ 
 
 F'uf. 24. 
 
 \ 
 
 \ 
 \ 
 
 
 
 
 Fuj. 25. 
 
 » \ 
 
 ,\ 
 
 \X-^:Ae>- - 
 
 
 
 ./ W /^ \/ 
 
 / 
 
 \ \ 
 
 \i 
 
 w 
 
 \ 
 
 \p^' 
 
 v>; 
 
 
 / i An V\ .>^ \ \ i 
 
 \ \ 
 
 
 I 
 j 
 
 i 
 
 Fig, 26. 
 
 / • 
 
 / 
 
 n^,^ 
 
 / I 
 
 / I 
 
 / j 
 
 / 
 
 \ 
 
 \, 
 
 •i T- 
 
 \ 
 
 / 
 
 \ 
 
 ./ 
 
 
 M./ 
 
 Fig. 23 
 
 Fig. 23a. 
 
 e" 
 
 9c. 
 
 1 1* 
 
 Horizon \J 
 
 \ 
 
 
 
 \ 
 
 ■^nr9- 
 
 .xr-; 
 
 1^ 
 
 
 ! A i \/N\ 
 
 \ 
 
 \, 
 
 \ ! 
 
 "\ jq" x; 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 c. 
 
 / 
 
 / 
 
 ' — z^ 
 
 !/ 
 
 / 
 
 •L 
 
 \ 
 \ 
 A 
 \ 
 
 \ 
 \ 
 
 .-i|^-: 
 
 / 
 
 IN'K' 
 
 e' D" 
 
 / 1 -^y \ \ 
 
 y 
 
 I 
 
 \ / 
 
 
 1 1 
 
 .^f\\\ 
 
 \ 
 
 p ,p 
 
 \ 
 
 \ 
 
 Fig.2r. 
 
 \ 
 
 j q |N' 
 
 r 
 
 \ 
 
 \. 
 
 '^\\ 
 
 ^^^ 
 
 I 
 I 
 I 
 
 J, I / 
 
 'y 
 
PLATE VI. 
 
 y->50L 
 
 Fi(j. 29. 
 
 j 
 
 
 
 
 i 
 
 k 
 
 1 
 
 
 f 
 d 
 
 9 
 
 
 m 
 
 
 
 n 
 
 • c 
 
 
 
 
 
 -J 
 
 a 
 
 
 R 
 
 
 b 
 
PLATE VII. 
 
PLATE VIII. 
 
 Fi(j. 43a. 
 
 Fifj. 40a. 
 
 "i — — _ 
 
 Fif/.40. 
 
 G — L ^ ^^ y ^ V- ' — ^ ^ ^ \ \, \ \ . 
 

 ■^ 
 
 V 
 
 Fifj. 45 
 
 \ 
 
 / 
 
 / 
 
 / 
 
 Fig. 45(f. 
 
 ^'.-"T'^^"/^ 
 
 X 
 
 \ 
 
 \ 
 
 \ 
 
 X 
 
 / 
 
 / \N 
 
 \ 
 
 \. 
 
 \ 
 
 /\ 
 
 
 X 
 
 "% 
 
 / 
 
 ^J/ 
 
 
 \, 
 
 \ v" 
 
 -e — 
 
 \ 
 
 V* 
 
 ■f 
 
 \ 
 
 / 
 
 / 
 
 / 
 
 / 
 
 / 
 / 
 
 
 .r ,r// 
 
 N 
 
 PLATE IX. 
 
 Fig. 46. 
 
 :\\ 
 
 
 .^ + -V 
 
 ^ 
 
 i- 
 
 \ 
 
 4- 
 
 + 
 
 -v 
 \ 
 
 \ 
 
 ^ \^ 
 
 -V 
 
 Xi xX\ 
 
 \ 
 
 \ 
 1^ 
 
 e" 
 
PLATE X. 
 
 
 -1^ 
 
 -. O 
 
 ^ 
 
 <^ 
 
 
 y 
 
 V- ^/ \ 
 
 
 / 
 
 
 \ 
 
 — r 
 
 y45»R I 
 
 
 e' 
 ■9- 
 
 
 D450L 
 -9 
 
 w 
 
 / 
 
 \ 
 
 
 ./ 
 
 /V//../r. 
 
 is-- 
 
 ./ 
 
 / 
 
 
 \l 
 
 ! / 
 
 I /^ 
 
 \ 
 
 \. 
 
 I 
 
 N 
 
 Fig. 48. 
 
 /> 
 
 ^ 
 
 v^ 
 
 ^ 
 
 // 
 
 <^.^ 
 
 -/ 
 
 y45<>R 
 
 c! 
 
 •\ 
 
 TF.y,- 
 
 :^ 
 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 --^^. 
 
 \ 
 
 N. 
 
 N. 
 
 ■■-^^ 
 
PLATE XI. 
 
PLATE ATI. 
 
 G- 
 
 M<j. 49. 
 
 Fifj. 49a. 
 
 
 
 / 
 
 / 
 
 / 
 
 / 
 
 --r?V 
 
 Fig. 51. 
 
 Fi(j. ok 
 
 t> 
 
 
 Fig. 50. 
 
 >c>\ //K A 
 
 / /;<>(//K x"" \ 
 
 i\\ \ \ \X -t \\ 
 
 Fig. 50a. 
 
 / 
 
 // 
 
 / 
 
 / //^>^ 
 
 A 
 
 ^y >^ 
 
 V 
 
 / 
 
 t" 
 
 \ 
 
 X, 
 
 \ 
 
 X / 
 
 / 
 
 X 
 
 \/ / 
 
 K / X / _/ 
 
 1/ / / \ /' 
 
 Y 
 
 \v\-^ 
 
 ^x \ ^- \ v^'^ 
 
 / 
 / 
 
 Y / 
 
 V / 
 
 / 
 
 / 
 
 / 
 
 / 
 
 X/ - . 
 
 \ 
 
 / / 
 
 X X >\" 
 
 / / / A- 
 
 ' / / / \ 
 
 'A' 
 
 
 \ 
 
 V/ 
 
 / 
 
 X 
 
 ,x 
 
 
 'V 
 
 / 
 
 _ .- -B E _ _ 
 
 2 34 4 3 
 
 o 
 
SHORT-TITLE CATALOGUE 
 
 OF THE 
 
 PUBLICATIONS 
 
 OF 
 
 JOHN WILEY & SONS, 
 
 New York. 
 London: CHAPMAJST & HALL, Limited. 
 
 ARKANGED UNDER SUBJECTS. 
 
 Descriptive circulars sent on application,^ 
 
 Books marked with an asterisk are sold at net prices only. 
 
 All books are bound in cloth unless otherwise stated. 
 
 AGRICULTURE. 
 
 Cattle Feeding — Daikt Practice — Diseases of Animals — 
 Gardening, Hokticulthbb, Etc. 
 
 Armsby's Manual of Cattle Feeding 12mo, H 75 
 
 Downing's Fruit and Fruit Trees. \. ; 8vo, 5 00 
 
 Grotenfelt's The Principles of Modern Dairy Practice. CWoU.) 
 
 13mo, 2 00 
 
 Kenip's Landscape Gardening ..... 13mo, 2 50 
 
 Maynard's Landscape Gardening 13mo, I 50 
 
 Steel's Treatise on the Diseases of the Dog 8vo, 3 50 
 
 " Treatise on the Diseases of the Ox 8yo, 6 00 
 
 Stockbridge's Roclis and Soils 8vo, 2 50 
 
 WoU's Handbook for Farmers and Dairymen 12mo, 1 50 
 
 ARCHITECTURE. 
 
 Building — Carpentry— Stairs— Ventilation — Law, Etc. 
 
 Birkmire's American Theatres— Planning and Construction. 8vo, 3 00 
 
 " Architectural Iron and Steel ...8vo, 3 50 
 
 " Compound Riveted Girders 8vo, 2 00 
 
 " Skeleton Construction iij Buildings .8vo, 3 00 
 
 1 
 
Birkmirc'sPlannln^ and.Copstruction^f High Oflce B^dlpgs, 
 ■ -= ~ ' - ^- - , ^ ' . / . . .'] „, ; ■ 8to, 
 
 Briggs' Modern Am. School Building 8vo, 
 
 Carpenter's Heating and Ventilating of Bi|ildings 8vo, 
 
 Freitag's Architectural Engineering 8vo, 
 
 " Tlie Fireprooflng of Steel Buildings .8vo, 
 
 Gerhard's Sanitary, House Inspeptioa. , il6mo, 
 
 " Theatre Fires and Panics 12mo, 
 
 Hatfield's American House Carpenter. :. Svo, 
 
 Holly's Carpenter and Joiner. . .1. ......!..'.', , ■ ,18flio, 
 
 Kidder's Architect and Builder's Pocket-book. ,.16mo, morocco, 
 
 Merrill's Stones for Building and Decoration Svo, 
 
 Monckton's Stair Building — Wood, Iron, and Stone 4to, 
 
 "Wait's Engineering and Architectural Jurisprudence Svo, 
 
 Sheep, 
 Worcester's Small Hospitals — Establishment and Maintenance, 
 including Atkinson's Suggestions for Hospital Archi- 
 tecture ■.-..'.'..'.!..'.'.'..■. 12mo, 
 
 World's Columbian. Exposition of 1893 Large 4to, 
 
 ARMY, NAVY, Etc. 
 
 MiLiTAKT Engineeeing— Okdnance — Law, Etc. 
 
 *BrufE's Ordnance and Gunnery .Svo, 
 
 Chase's Screw Propellers Svo, 
 
 Cronkhite's Gunneryfor Non-com. Officers 32mo, morocco, 
 
 ■* Davis's Treatise on Military Law Svo, 
 
 Sheep, 
 
 * " Elements of Law 8vo, 
 
 De Brack's Cavalry Outpost Duties. (Can.). . . .33mo, morocco, 
 Dietz's Soldier's First Aid .16mo, morocco, 
 
 * Dredge's Modern French Artillery. . . .Large 4to, half morocco, 
 
 " Eecord of the Transportation JSxhibits Building, 
 World)s Columbian Exposition of 1893.. 4to, h^lf morocco, 
 
 Dumnd's Resistance and Propulsion of Ships Svo, 
 
 Dyer's Light Artillery 12mo, 
 
 Hoff's Naval Tactics Svo, 
 
 *Ingalls's Ballistic Tables 8vo, 
 
 n so 
 
 4 00 
 
 3 00 
 
 2 50 
 
 3 50 
 
 ; 1 00 
 
 1 50 
 
 5 00 
 
 75 
 
 4 00 
 
 5 00 
 
 4 00 
 
 6 00 
 
 6 50 
 
 1 25 
 
 2 50 
 
 6 00 
 
 3 00 
 
 3 00 
 
 7 00 
 
 7 50 
 
 2 50 
 
 2 00 
 
 1 25 
 
 15 00 
 
 10 00 
 
 5 00 
 
 3 00 
 
 1 50 
 
 1 50 
 
14 00 
 
 7 50 
 
 3 00 
 
 4 00 
 
 6 00 
 
 1 50 
 
 10 
 
 2 50 
 
 4 00 
 
 1 50 
 
 2 CO 
 
 2 50 
 
 1 50 
 
 2 00 
 
 1 00 
 
 Ingalls's Handbook of Problem^ In fiirect Fire 8to, 
 
 Miiliau's Permanent Fortifications. (Merour.).8vo, half morocco, 
 
 "* Mercur'a Attack of Fortified Places 12mo, 
 
 J. • 
 
 Elements of the Art of War 8vo, 
 
 Metcalfe's Ordnance and Gunnery. . .-. 12mo, with Atlas, 
 
 Murray's A Miinual for Courts-Martial ICmo, morocco, 
 
 " Infantry Drill Regulations adapted to the Springfield 
 
 Rifle, Caliber .45 . . . .33mo; paper, 
 
 * Phelps's Practical Marine Surveying - 8vo, 
 
 Powell's Array Officer's Examiner ' .12mo, 
 
 Sllarpe's Subsistliig Armies ...■.....; .33mo, morocco, 
 
 Wheeler's Siege Operations 8vo, 
 
 Winthrop's Abridgment of Military Law..; . ..... . . . 13iiio, 
 
 Woodhull's Notes on Military Hygiene ; 16mo, 
 
 Young's Simple' Elements of Navigation. . . . i; .16mo, moroCco, 
 
 " " " " " first e'dition 
 
 ASSAYING. 
 
 Smeltiko — Ore Dressing— AiiLOTS, Etc. 
 
 Fletcher's Qiianti^ Assaying with the Blowpipe. .16mo, morocco, 1 50 
 
 Furman's Practical Assaying ....,.; .Bvo, 3 00 
 
 Kunhardt's Ore Dressing Svo, 1 50 
 
 .O'DriscoU's Treatment of Gold Ores. ...J. v. Svo, 3 00 
 
 Eicketts and Miller's Notes on Assaying, ;.l8vo, 3 00 
 
 Thurston's Alloys, Brasses,, and iBronies.i w....; ..8vo, 2 50 
 
 Wilson's Cyanide Processes 12mo, 1 50 
 
 " The Chlorination. Process. .;,.. 12mo, 1 SQ 
 
 ASTRONOMY. 
 
 Practical, Theoretical, and Descriptive. 
 
 {yraig's Aziii>uth> ....:.:.;. vs .4to, 8 50 
 
 Doolittle's, Practical Astronomy Svo, 4 00 
 
 Gore's Elements of Geodesy.,. .......' .Svo, 2 50 
 
 Hayford's Text-book of Geodetic Astrononiy Svo. 3 00 
 
 * Michie and Harlow's Practical Astronomy Svo, 3 00 
 
 ** White's Theoretical and Descriptive Astronomy 12mo, 3 00 
 
 3 
 
' BOTANY. ,;,;,. 
 
 ;>,.ii GARDENlNa FOB LaDIBS; EtC'. . 
 
 Baldwin's Otchids of.New Englaiid.;)ji! J. :•. li.; ..: .vi Small 8to, $!■ 50 
 
 Thome's Structural.Botany. ....i'.'J-'.ViXUi.i:'.^ 16mo, 2 25 
 
 ■Westermaier's iQeneral B6tany. (SchneWier,.i)>.bi . a..'u iSTo,.' 2 00 
 
 BRIDGES, ROOFS, j^tc. , 
 Cantilevee — Draw.— Highway — Suspension, 
 , , J , (Sec oZso Engineeiuijg, p., 7.) ,, >,;,>'! 
 
 Boiler's Highway Bridges... . .,, ., ii<... . . .Svo, ,2,.00 
 
 *, " The T.l<^mie.s,Riyer Bridge. .... , ,, . . .4t0(.ipaperf 500 
 
 Burr's Stresses in Bridges. , 8vo, 3 50 
 
 Crehore's,,}jI,echamcs,of .the, Girder, .v^,.,.r. .,.;. ..,...,.„,. . .,.,.,. .8vo,, 5,00 
 
 Dredge's Thames Bridges .i. . 7;pa,i:ts,,perpart,, 1 35 
 
 DuBois'grSitriesses iu Framed Structures, i..;.SmaU 4to,. 10 00 
 
 Eosf er's /V^^opden Trestle Bridges ; 4to, 5 00 
 
 Greene's Arches in Wood, etc : Svo, 2 50 
 
 " Bridge Trusses (•)!••!•■•, r ^'°' ^ ^^ 
 
 " Boof Trusses Svo, 125 
 
 Howe's Treatise oil Arches !„ ..........Svo, 4 00 
 
 Johfason's Modern Framed iSttuctures.'i .-.-!/. ShialH4to, 10 '00 
 
 Merriman'Tfi; -Jacoby's Text-book of Roofs -aind Bridges." '=i' 
 
 Part I., Stresses....... , , '.i.-^; .'i. ;.'.iSvo, iaiSO 
 
 Meri-iman'ifc Jacoby's Text-book df 'Roofs ■ and' Bridges. 
 
 ■- Part'IL, Graphic Staticai-iii »■....).'...':.' '..Svo, 2 50 
 
 Merriman '^& Jacoby's Text-bo©k' of Roofs- and Bridges; ■ 
 
 ■ Part III., .Bridge Design ;'.....;....... .Svo, 250 
 
 Merrimani' & Jacoby's . Text-book of "R6of s ' I 'and ■ Bridges. 
 Part IV., Continuous, Draw, Cantilever, Su.spension, and 
 
 Arched Bridges y^^,|,.,.^.(.j.^,- j^_^^ Svo, 3 50 
 
 * Moriaon's The Memphis Bridge Oblong 4to, 10 00 
 
 Waddell's Iron Highway Bridges .'. '. .'.'*. .Svo, 4 00 
 
 " De Pontibus (a Pocket-book for Bridge Engineere). • 
 
 ''■i'16mo, morocco, 3 00 
 
 " Specifioations for Steel Bridges '•. ..:'. .(In press.) 
 
 Wood's Construction of Bridges and Roofs. :8vo, 3 00 
 
 Wright's Designing of Draw Spans. • Parts I. and II.. Svo, each 2 50 
 
 ■' " ■ " -.!'.'" ' •' Complete...... ......Svo, 3 50 
 
 4 
 
CHEMISTRY— BIOLOGY— PHARMACfYT-SANITARY. SCIENCE^ 
 
 Qualitative — QuANTrrATivE'-'-OiRaX.Nic— Inokganic, Etc. : . i 
 
 Adriance's Laboratory €aldulatioris.'. . '.' !'. ISmo, $1 35 
 
 ^lien's Tables for Iron Analysis.". '. Svo, 3 00 
 
 Austen's, IStotes for Chemicj^LStfi^ent^ ..; 12mo, ' .% 50 
 
 Bolton's Student's. Guide in Quantitative Analysis. .8vo, 1 50 
 
 Boltwood's Elementary Electro ChenlTstry'.'; ... . . .(Id the press.) 
 
 Classen's Analysis by Electrolysis. (Herrick'arid l3olt\vood.).8vo, 3 00 
 
 Colin's Indicatbrs and' Test- papers , 13rG0 2 00 
 
 Cratts'sQtj^litative Analysis^!/ (Schaeffer.),..,. ..,.,,,,..'.... ,12mo, 1 50 
 
 Davenport's , Statistical Methods yn'xik Special iReference to Bio- i ,.■ 
 
 ■' ■ logicalrVariations 13mo, morocco, 135 
 
 Direehsel's Chemical Reactions. iCMerrill'.).'. . ':..'. 13mo, 1 25 
 
 Erdmanu's Introduction to Chemical Preparations. (Dtiiilap.) 
 
 • ■r •■ ■ ■' .^ ..... ..ii ,,.1 12rao,, 1 25 
 
 Breseniusls .Quantitative Chemical Analy^P, . (j^llen. ) Svo, 6. 00 
 
 ■ ^" Qualitative " " -'(Johnson.) Svo, 3 00 
 
 -" ■ " ' .< ' •< ' ■(Wells.)' ' Trans. 
 
 16th German Edition Svo, 5 00 
 
 Puertes's "Water and Public Hqalth,.,.. ,.,.... 12mo, 1 50 
 
 Gill's Gas and Fuel Analysis 13mo, 1 35 
 
 Hammarsten's Physiological Oheniistryj (Maudel.). . . .. i . .Svoj 4 00 
 
 Helm's Principles of Mathematical ^Iiemistry. (Morgan). 13mo, 1 50 
 
 Hopkins' Oil-Chemist's Hand-book ,. , .Svo, 3 00 
 
 Jjadd's Quantitative Chemical Analysis. , 12mo, 1 00 
 
 Iianidauer's "Spectrum Analysis. (Tingle. ) '. . . . .Svo, 3 00 
 
 LiSb's Electrolysis and Electrosynthesis of Organic Compounds. 
 
 (Loreuz.) ,.......,.,..,,,.,.„.,..... .l2mo, 1 00 
 
 Mf.ndel's Bio-chemical Laboratory : 13mo, 1 50 
 
 Mason's Water-supply ' .8vo, 5 09 
 
 ' " Examination of Water. .. !' 13mo, 135 
 
 Meyer's Radicles in"Carbon Compounds. (Tingle.) ..,12ino, 1 00 
 
 Mixter's Elementary Text-book of Chemistry , ISmo, 1 50 
 
 Morgan's The Theory of Solutions and its Results. . .12mo, 1 00 
 
 " Elements of Physical Chemistry .ISiho, 2 00 
 
 IN-Ichols'S Water-supply (Chemical and Sanitary) : . . . .Svo, 2 50 
 
 O'Brine's lialioratory Guide to Chemical Analysis ,. .Svo, 2 OO 
 
 Pinner's Organic Chemistry. (Austen.) 12mo, 1 50 
 
 5 
 
f 3 OO 
 
 1 Uft 
 
 75 
 
 2 (TO 
 
 2 50 
 
 2 00 
 
 3 00 
 
 2 50 
 
 3 OO 
 
 1 50 
 
 1 50 
 
 i 50 
 
 3 50 
 
 3 00 
 
 2 50 
 
 2 00 
 
 FSole's Calorific Power of Fuels ...../..........,.. ."6V6; 
 
 Richards's CostorLiring as Modified by Sanitary Science;. 12mo. 
 
 " and Woodman'8 Air, Water, and Food {Inpivss.) 
 
 lUcketts and Russell's Notes on Inorganic Chemistry (Noa- 
 
 metallic) OUong 8vo, morocco, 
 
 Ruddimail's Incompatibilities iu Prescriptions. .. 1. ...... . .8vo, 
 
 Scliimpf's Volumetric Analysis. .....'. ; . .■ 12mo, 
 
 Spencer's Sugar Manufacturer's Handbook;.. .,.16nio, morocco, 
 " Handbook , for Cliemists of Beet . Srugar Houses. 
 
 ^6mo, morocco, 
 
 Stockbridge's Rocks and Soils , .8vo, 
 
 •Tillman's Descriptive GeneralCbcmistrj' .8vo, 
 
 Van Deventer's Physical Chemistry for Beginners. (Bftltwood.) > 
 
 13mo, 
 
 Wells's Inorganic Qualitative Analysis........ ....;; ]r2mo,. 
 
 " Laboratory Guide in (Qualitative Cbeinipal Analysis. 
 
 ' Svo, ' 
 
 Whipple's Microscopy of DrinldDg-waler Svo, 
 
 Wiechmanu's Chemieal Lecture Notes 12mo, 
 
 " Sugar Analysis Small.fivo, 
 
 WuUing's Inqvganic Phar. and Med. Chemistry il2nio. 
 
 DRAWINd. 
 
 Elemehtaky — Geometeical — Mechanical — Topographical, i ; ; 
 
 Hill s Shades and iShadows arid Perspective 8vo, 2 m 
 
 MaeCord's Descriptive Geometry. ..'J.-..'-.'; 8vo, 3' 00 
 
 Kinematics -. ......li. ...8v0/ 5 00 
 
 ' ' Mechanical Drawing .,;.......... .,,,.,, .8vo, ,., 4 , 00 
 
 Mahan's Industrial Drawing. (Thompson.'),.. ._... 2 vols., 8yo^.. 3,50 
 
 Reed's Topographical Drawing. (H. A). 4to, 5 OS 
 
 Reid's A Course in Mechanical Drawing J .8vo. 2 OO 
 
 ■ "' Mechanical Drawing and Elementary Machine Design. ' ■ ' 
 .■%^o. {In, the press.) 
 
 Smifli's Topographical Drawing. (Macmillan.) , .,. ... .8vo, 2 50 
 
 "'Sy^aij-ren's Descriptive Geometry ...2 vols., 8vo, 3 50 
 
 " Drafting Instriiineuts ...!.. ..i2mo, 125 
 
 Free-hand Drawing .;..'.'.'.:.'. '. '. 13mb, 1 00 
 
 '• !" Linear Perspective. .....'-.■:'.!■.... .■. 12mo, 100 
 
 " Machine Construction. . .iji. J .1 .2-vols., 8to, 7 50 
 
 Plane Problenjs,,, ;. ., ! .,. 12mo, 1 25 
 
 Primary Geometry,... l^mo^ 75 
 
 '" Problems ancl "Theorems ...... . . ...fe'vo, 2 50 
 
 " PrAjectiou Drawing ..';': '...'!'.';..... .13m6,' 1 56 
 
■Warren's Shades and Shadows 8vo, $3 00 
 
 ' ' Stereotomy— Stone-cutting 8vo, 2 50 
 
 Whelpliey's Letter Engraving . . . ; 12nio, 2 00 
 
 Wilson's Free-hand Perfective {In press.) 
 
 ELECTRICITY AND MAGNETISM. 
 
 Illumination — Batteries — Phssics^ — Railways. 
 
 Anthony and Brackett's Text-hook of Physics. (Magie.) Small 
 
 8vo, 3 00 
 
 Anthony's Theory of Electrical Measurements 12mo, 1 00 
 
 Barker's Deefp- sea Soundings .8vo, 2 00 
 
 Benjamin's Voltaic Cell.. .8vo, 3 00 
 
 " History of Electricity ...Svo, 3 00 
 
 Ciassen'sAnalysisby Electi'olysis. (.Herrick audBoltwood.)8vo, 3 00 
 Crehore and Squier's Experiments with a New Polarizing Photo- 
 Chronograph 8vo, 3 00 
 
 Dawson's' Electric Railways and Tramways. Small, 4to, half 
 
 morocco, 
 * Dredge's Electric Illuminations. . . .2 vols.,'4to, half morocco, 
 
 " " " Tol. II..; ; 4to, 
 
 Gilbert's De magnete; (Mottelayi) 8voj 
 
 Holman's Precision of Measurements. .8vo, 
 
 " Telescope-inirror-scale Method Large Svo, 
 
 LiJb's Electi'olysis and Electrosynthesis of Organic Compounds. 
 
 (Lorenz. ) 12mo, 
 
 *Michie's Wave Motion Relating to Sound and Light. Svo, 
 
 Morgan's The Theory of Solutions and its Results 12mo, 
 
 Niaudet's Electric Batteries. (Fishback.) 12mo, 
 
 Pr.itt and' Alden's Street-railway Road-beds Svo, 
 
 Reagan's Steam and Electric Locomotives .12mo, 
 
 Thurston's Stationary Steam Engines for Electric Jjighting Pur- 
 
 _' poses Svo, 
 
 *Tillman's Heat. . ;.. Svo, 
 
 ENGINEERING. 
 
 Civil — MBCHANiCAii^SANiTAKT, Etc. 
 
 {See 'also' Bkidges, p. 4; Hydradlics, p. 9; MATBitiALa of En- 
 gineering, p. 10 ; Mech-anics and Machinery, p. 12 ; Steam 
 Engines and Boilers, p. .14.) 
 
 Baker's Masonry Construction ; Svo, $5 00 
 
 " Surveying Instruments 12mo, 3 00 
 
 7 
 
 13 50 
 
 35 00 
 
 7 50 
 
 3 50 
 
 2 00 
 
 75 
 
 1 00' 
 
 4 00 
 
 1 00 
 
 2 50 
 
 2 00 
 
 3 00 
 
 2 50 
 
 1 50 
 
Black's U. S. Public Works Oblong 4to, $ 500 
 
 Brooks's Street-railway Locatiou. 16iiio, morocco, 1 50 
 
 Butts's Civil Engineers' Field Book. 16mo, morocco, 3 50 
 
 Byrne's Highway Construction .'.''...,...,... .8vb, 5 00 
 
 " Inspection of Materials and Workmanship 16mo, 3 00 
 
 Carpenter's Experimental Engineering 8vo, 6 00 
 
 Church's Mechanics of Engineeriijg-^Solids>and Fluids. ...8vo, 6 00 
 
 " Notes and Examples in Mechanics .8vo, 2 CO 
 
 Crandall's Earthwork Tables . . . .'. : 8vo, 1 50 
 
 " 'The Transition Curve , 16mo, morocco, , 1 50 
 
 * Dredge's Penn. Railroad Construction, etc. Large 4to, 
 
 half morocco, 30 00 
 
 * Drinker's Tunnelling 4to, half morocco, - 35 00 
 
 Eissler's Explosives — Nitroglycerine and Dynamite. ..... . .8vo, , 4 00 
 
 Frizell's Water Power {In press.) ■ 
 
 Folwell's Sewerage ,. ., , 8vo, 3 00 
 
 " Water-supply Engineering... , .,8vo, 4 00 
 
 Fowler's Coffer-dam Process for Piers ., . .8vo. 3 50 
 
 Gerh ard's Sanitary House Inspeqtion , . ■. . 13mo, 1 00 
 
 Godwin's Railroad Engineer's Field-book 16mo, morocco, 3 50 
 
 Gore's Elements of Geodesy 8vo, 2 50 
 
 Howard's 'transition Curve Field-book 16mo, morocco, 1 50 
 
 Howe's Retaining Walls (New Edition.) ,13m;0,. 1 35 
 
 Hudson's Excavation Tables. Vol. II. ■./. 8vo, 1 00 
 
 Hutton's Mechanical Engineeiing of Power Plant^ .8vo, 5 00 
 
 " Heat and Heat Engines......... ,< 8vo, 5 00 
 
 Johnson's Materials of Construction '. Large 8vo, 6 00 
 
 " 'Theory and Practice of Surveying Spiall 8vo, 4 Oft 
 
 Kent's Mechanical Engineer's Pocket-book 16mo, morocco, 5 OO 
 
 Kiersted's Sewage Disposal ISmo, 1 25 
 
 Mahan's Civil Engineering. (Wood.) , 8vo, 5 00 
 
 Merriman and Brook's Handbook for Surveyors. . . .16mo, mor., 3 00 
 
 Merriman's Precise Surveying aud Geodesy 8vo, 3 50 
 
 " Sanitary Engineering 8vo, 3 00 
 
 Nagle's Miinual for Railroad Engineers 16mo, morocco, 3 00 
 
 Ogden's Sewer Design 13mo, 3 00 
 
 Patton's Civil Engineering 8vo, half morocco, 7 50 
 
 Patton's Foundations .;. .i. .'...'.. 8vo, 5 00 
 
 Philbrick's Field Manual for Engineers (In press.) 
 
 Pratt and Aldeu's Street-railway Road-beds .8vo, 2 00 
 
 Rockwell's Roads and Pavements in France 12mo, ,1 35 
 
 Searles's Field Engineering 16mo, morocco, 3 00 
 
 Railroad Spiral. 16mo, morocco, 1 50 
 
 Slebert and Biggin's Modern Stone Cutting and Masonry. . .8vo, 1 50 
 
 Smart's Engineering Laboratory Practice ISmo, 2 50 
 
 Smith's Wire Manufacture and Uses Small 4to, 3 00 
 
 8 
 
Spalding's Roads and Pavements 12m.o, |2 00 
 
 . " Hydraulic Cement 13mo, 3 00 
 
 Taylor's Prismoidal Formulas and Earthwork. . , ... ..!'.. .'.8vo, 1 50 
 
 Tliurston's, Materials of Construction , 8vo, 5 Op 
 
 * Trautwine's Civil Engineer's Pocket-book. . . .idmo, morocco, 5 00 
 
 * " Cross-section ,■•/•,•■' Sheet, 25 
 
 * " Excavations and Embankments 8vo, 2 00 
 
 * " Laying Out Curves. . : . ;. 12mo, iho'rocco, 2 50 
 
 Waddell's De Pontibus (A Pocket-book for Bridge Engineers); 
 
 16mo, morocco, 3 00 
 
 Wait's Engineering and Architectural Jurisprudence Svo, 6 00 
 
 ' ■ ' Sheep, 6 50 
 
 " Law of Field Operation in Engineering, etc. . .(In press.) 
 
 Warren's Stereotomy — Stone-cutting Svo, 2 50 
 
 Webb's Engineering Instruments. New Edition. 16mo, riiorooco, 1 25 
 
 " Railroad Construction .'. .'Svo, 4 00 
 
 Wegmanu's Construction of Masonry Dams 4to, 5 00 
 
 Wellington's Location of Railways. '..'. .'/. f. .' Small Svo, 5 00 
 
 Wheelier's Civil, Engineering. ..■.,.., ,. ,.| Svo, 4 00 
 
 Wilson's Topographical Surveying .Svo, 3 50 
 
 Wolff's Windmill as a Prime Mover .Svo, 8 OO 
 
 HYDRAULICS. 
 
 WaTEB- WHEELS — WINDMILLS — SERVICE PiPE — DRAINAGE, EtC. 
 
 (See also Engineering, p. ".) 
 Bazin's Experiments upon the Contraction of the Liquid Vein. 
 
 (Tratilwlne.). . ; , Svo/ 2 00 
 
 Bdvey's Treatise on Hydraulics .. .' Svo, 4 00 
 
 Cofflu's Graphical Solution of Hydraulic Problems 12mo, 3 50 
 
 Ferrel's Treatise on the Winds, Cyclones, and Tornidoesi . .Svo, 4 00 
 
 Folwell's Water Supply Engineering..; ' Svo, 4 00 
 
 Fuertes's Water and Poblic Health 12mo, 1 50 
 
 Ganguillet & Kutter's Flow of Water. (Hering & Trautwine.) 
 
 Svo, 4 00 
 
 Hazen's Filtration of Public Water Supply Svo, 3 00 
 
 Herschel's 115 Experiments '. Svo, 2 00 
 
 Kiersted's Sewage Disposal 12mo, 1 35 
 
 Mason's Water Supply Svo, 5 00 
 
 " Exattiination of Water 12mo, 1 25 
 
 Merriman's Treatise on Hydraulics Svo, 4 00 
 
 Nichols's Water Supply (Chemical and Sanitary) . i Svo, 2 50 
 
 Turueaure and Russell's Water-supply .{In press.) 
 
 Wegmann's Water Supply of the City of New Tork 4to, 10 00 
 
 Weisbach's Hydraulics. (Du Bois. ) Svo, 5 00 
 
 Whipple's Microscopy of Drinking Water ....'....; .Svo, 3 50' 
 
 9 
 
Wilson's Iirigatlon Engineeiiug .gvo, ^4 OO 
 
 " Hydraulic and Placer Miuing ..lymo, 3 OO 
 
 Wolff's 'Windmill as a Prime Mover. 8vo, 3 00 
 
 Wood's Tlieory of Turbines 8vo, 3 50 
 
 LAW. 
 
 Abchitectuke — Engineering — Mjlitabt. 
 
 Davis's Elements of Law '. ..8vo, 3 50' 
 
 " Ti-eatlse on Military Law 8yo, 7 00 
 
 Sheep, 7 50 
 
 Murray's A Manual for Courts-martial .16mo, morocco, 1 50 
 
 Wait's Engineering and Architectural Jurisprudence. ..... Svq, 6 OO 
 
 Sheep, 6 50 
 
 " Laws of Field Operation ill Engineeiiug (In press.) 
 
 Winthrop's Abridgment of Military Law ISmo, 2 50- 
 
 MANUFACTURES. 
 
 B6ILEKS — ExPLOSrVBS^-lKON — StKEL— SUGAR — WoOLI^EKS, EtC. 
 
 Allen's Tables for Iron Analysis , 8vo, 
 
 Beaumont's Woollen and Worsted Manufacture 12mo, 
 
 Holland's Encyclopsedia of Founding TpiTji^. ... 12mo, 
 
 " The Iron Founder ]2mo, 
 
 " " " " '■ Supplement V.'.lSmo, 
 
 Bouvier's Handbook on Oil Painting; 12mo, 
 
 Eissler's Explosives, Nitroglycerine and Dynamite .Svo, 
 
 Ford's Boiler Makiug for Boiler Makers .18mo, 
 
 Metcalfe's Cost of Manufactures .8vo, 
 
 Metcalf's Steel — A Manual for Steel Users.. 12mo, 
 
 *Eeisig's Guide to Piece Dyeiug. . ; Svo, 
 
 Spencer's Sugar Manufacturer's Handbook . . .;.16mo, morocco, 
 
 " Handbook for Chemists of Beet Sugar Houses. 
 
 - ! 16mo, morocco, 
 
 Thurston's Manual of Steam Boilers Svo, 
 
 Walke's Lectures on Explosives. ;, 8yo, 
 
 West's American Foundry Practice l?mD, 
 
 " Moulder's Text-book 12mo, 
 
 Wiechmann's Sugar Analysis ,.. Small Svo, 
 
 Woodbury's Fire Protection of Mills ,.,. .8vo, 
 
 MATERIALS OF ENQINEERINQ 
 
 STliftNGTH^ — Elasticity — Resistance, Etc. 
 (See also Engineering, p. 7.) 
 
 Baker's Masonry Construction. . . ., , Svo, 3 00 
 
 Beardslee and Kent's Strength of Wrought Iron. Svo, 1 50 
 
 10 
 
 3 op- 
 
 1 50 
 
 3 OO 
 
 2 50 
 
 2 50 
 
 3 00 
 
 4 OO 
 
 1 00 
 
 5 00 
 
 2 OO 
 
 25 OO 
 
 3 OO 
 
 aoo 
 
 5 OO 
 
 4 00 
 
 3 50 
 
 3 50 
 
 3 50 
 
 2 50 
 
7 50 
 
 |5 00 
 
 5 00 
 
 6 OO 
 
 10 00 
 
 6 OO 
 
 7 50 
 
 7 50 
 
 5 00' 
 
 4 00- 
 
 1 OO 
 
 5 OO 
 
 1 25 
 
 2 00 
 
 5 00' 
 
 8 00 
 
 2 00 
 
 3 50 
 
 2 50 
 
 2 00 
 
 Bbvey's Strength of Matevials. 8vo, 
 
 Burr's Elasticity and Resistance of Materials '. 8vo, 
 
 Byrae's Highway Construction 8to, 
 
 Church's' Mechanics of Engineering — Solids and Fluids 8vo, 
 
 Du Bois'a Stresses in Framed Structures. . L. Small 4to, 
 
 Johnson's Materials of Construction 8vo, 
 
 Lanza's Applied Mechanics '. 8vo, 
 
 Marlens's Testing Materials. (Henning.). . . . ...;;. .2 vols. , 8vo, 
 
 Merrill's Stones for Building and Decoration. ..';.. 8vo, 
 
 MeiTiman's Mechanics of Materials ■; 8vo, 
 
 " Strength of Materials ...;.."; 12nio, 
 
 Pattou's Treatise on Foundations :.'. 8vo, 
 
 Rockwell's Roads and Pavements in France 12mo, 
 
 Spalding's Roads and Pavements .'. . .12mo, 
 
 Thurston's Materials of Construction ...... .8vo, 
 
 Materials of Engineering .,.,,3 vols., 8vo, 
 
 Vol. I., Non-irietallic '. 8vo, 
 
 Vol. II., Iron and Steel. ;....... .8vo, 
 
 Vol. III., Alloys, Brasses, and Bronzes 8vo, 
 
 Wood's Resistance of Materials, 8vo, 
 
 MATHEMATICS. 
 
 'Calculus — Geometky — Tkigokombthy, Etc. 
 
 Bs^ker's Elliptic Functions 8vo, 1 50 
 
 *Bass's Differential Calculus 12mo, 4 00 
 
 Briggs's Plane Analytical Geometrj- 12mo, 1 00 
 
 Chapman's Theory of Equations. . . , 12mo, 1 50 
 
 Compton's Logarithmic Computations, , . . .12mo, 1 50 
 
 Davis's Introduction to the Logic of Algebra .8vo, 1 50 
 
 Halsted'S ^Elements of Geometry ,...8yo, 1 75 
 
 " Synthetic Geometry 8vo, 150 
 
 Johnson's CJurve Tracing 12mo, 1 00' 
 
 " Differential Equations — Ordinary' and Partial. 
 
 Small 8vo, 3 50 
 
 "[' ." Integral Calculus ,. , 12mo, 1 50 
 
 ' " " " Unabridged. Small 8vo. {In press.) 
 
 Least Squares ■ ■ • l^mo, 1 50 
 
 *Ludlow''s Logarithmic and Other Tables. (Bass.) 8vo, 2 OO 
 
 * " 'Trigonometry with Tables. (Bass.) .,., 8vo, 3 00 
 
 *Mahan's Descriptive Geometry (Stone Cutting) , 8vo, 1 50, 
 
 Merriman and Woodward's Higher Mathematics 8vo, 5 OO 
 
 Mcrriman's Method of Least Squares 8vo, 2 00 
 
 Rice.and Johnson's Differential and Integral Calqulus, 
 
 , 2 vols, in 1, small 8vo, 2 50 
 11 
 
Bice and Johnson's Differential Calculus , Small 8yo, |3i 00 
 
 , ", Abridgaient of Differential Calculu^: , 
 
 BmalliSvQ, ,1 5iQ 
 
 Totten's Metrology, ..'^.|. ,...„. ii,., 8to, 2. 50 
 
 Warren's Descriptive Geometry^,,,.,........., 3 vols., 8vo,^ .:?;50 
 
 " Drafting Instruments.. .,,13rnOn,,,-,l, 35 
 
 " Free-hand Drawing... 1.3»io, 1 OQ 
 
 " Iiineaj:, Perspective .12mo, 1 QO 
 
 i ," Primary Geometry, . . .,. , ,«l 13mo, ,75 
 
 " Plane Problems , l^mp,.. .^l 25 
 
 " Problems and Theorems , ^Svo, 3 50 
 
 , " Projection Drawing. 12mo, 150 
 
 Wood's Cojprdinate Geometry. • • >^^0i ,?.Oft 
 
 " 'Trigonometry , ..., , . .l^naiD,,. , 1 00 
 
 Wpplf 's Descriptive Geometry .Large; 8vo,, , 8 00 
 
 MECHANICS-MACHINERY. ; 
 
 Text-books and PRACTiCAji Works, ; , 
 (5ee also Engineering, p. 7.) 
 
 Baldwin's Steam Heating for Buildings.,. ;. 12mo, 8 50 
 
 Barr's Kinematics of Machinery 8vo, 2 50 
 
 Benjamin's Wrinliles and Recipes ..^ .,.,, 13mo, 3 00 
 
 Chordal's Letters to Mechanics '. 13mo, 3 00 
 
 Church's Mechanics of Engineering. .8vo, 6 00 
 
 " Notes and Examples in Mechanics. , 8vo„ 2 00 
 
 Crehore's Mechanics of the Girder ...].!. .8vo, 5 00 
 
 Cromwell's Belts and Pulleys 13mo, 1 50 
 
 Toothed Gearing 13mo, 150 
 
 Cbnipton's First Lessons in Metal Working... .13m'o, 1 50 
 
 Compton and De Grobdt's Speed Lathe.. .' . i'. 13mo, 1 50 
 
 Dana's Elementary Mechanics 12mo, 1 50 
 
 Dingey's Machineiy Pattern Making '.'."... 12mo, • 2 00 
 
 Dredge's Trans. Exhibits Building, World Exposition. 
 
 ' ' Large 4tO, half morocco, 10 00 
 
 Du Bois's Mechanics. Vol. I., Kinematics 8vo, 3 50 
 
 " " Vol. IL, Statics 8vo, 4 00 
 
 Vol. Ill, Kinetics.... ..8vo, 3 50 
 
 Fitzgerald's Boston Machinist. 18mo, 1 00 
 
 Plather's Dynamometers.,. , 12mo, 3 00 
 
 •' liope Driving..: 1. . . .12ipo, 3 00 
 
 flail's Car Lubrication... 12mo, 1 00 
 
 Hblly's Saw Filing 18mo, ' 75 
 
 Johnson's Theoretical Mechanics. An lllementary Treatise. 
 "^ {In iJie press.) 
 
 Jones's Machine Design. Part I. , Kihematica .8vo,' ' ' 1 SO 
 
 ■' ' 12 
 
$3 00 
 
 7 50 
 
 5 00 
 
 4 00 
 
 5 OO 
 
 4 00 
 
 1 50 
 
 3 00 
 
 3 00 
 
 3 00 
 
 1 00 
 
 7 50 
 
 5 00 
 
 Jones's Machine Design. Part II., Strength and Proportion of 
 
 Miichine Parts ;. 8vo, 
 
 Lanza's Applied Mechanics ..'..'....' '. . .' 8vo, 
 
 MacCord's Kinematics .' 8vo, 
 
 Merriman's Mechanics of Materials Bvo, 
 
 Metcalfe's Cost of Manufactures 8vo, 
 
 •Michie's Analytical Mechanics.. .;.........'. 8vo, 
 
 Richards's Compressed Air ....:!...' ; .13mo, 
 
 Robinson's Principles of Mechanisin 8vo, 
 
 Smith's Press-working of Metals.. .-„..:... ;8vo, 
 
 'I'hUrston's'ri-i'ction and Lost Work 8vo, 
 
 "' ' " ■ The Animal as a Machine ...J;' ,13nio, 
 
 Warren's Machine Construction '.'..- 3 vols. , 8vo, 
 
 Weisbach's Hydraulics and Hjwiraulic Motors. (Du Bois.)..8vo, 
 " 'Mechanics of Engineering. Vol. III., Part I., 
 
 Sec.L (Klein.) ...;...;. 8?o, 5 00 
 
 Weisbach's Mechanics of Engineering. Vol. III., Part I., 
 
 Seb.IL (Klein.) ' ......Svo, 5 00 
 
 Weisbach's Steam Etgines. (Du Bois.) Svo, 5 00 
 
 Wdod's Analytical Mechanics .'. . Svo, 3 00 
 
 " Elementary Mechanics ;...13mo, 125 
 
 " " " Supplement and Key 12m6, 135 
 
 METALLURGY. 
 
 ■ Iron— Gold— Silvbe^-Alloys, Etc. 
 
 Allen's Tables for Iron Analysis Svo, 
 
 Eglesto^i's ,091^ j^ii4j Mercury. . . .j. .,...,.. .... > ..; , ...Large Svo, 
 
 " Metallurgy of Silver Large Svo, 
 
 * Kerl'si Metallurgy — Copper and lion , Svo, 
 
 * " '" . Steel, Fuel, etc.... .,.^..^.„.., Svo, 
 
 Kunhardt's Ore Dressing in Europe Svo, 
 
 Metcalf's Steel— A Manual for Steel Users. . .... .1'. ..12mo, 
 
 O'Driscoll's Treatment of Gold Ores Svo, 
 
 Thurston's Iron and Steel. .......'.' Svo, 
 
 ' ■• Alloys .............Svo, 
 
 Wilson's Cyanide Processes J . v! 12mo, 
 
 MINERALOGY AND MINING. 
 
 Mine Accidents — Ventilation — Orb Dressing, Etc. 
 
 Barringer's Minerals of Commercial Value Oblong morocco, 2 50 
 
 Beard's Ventilation of Mines.. 13mo, 3 50 
 
 Boyd's Resources of South Western Virginia. i Svo, 3 00 
 
 " Map of South Western Virginia Pocket-book form, 3 00 
 
 Brush and Penfield's Determinative Mineralogy. New Ed. Svo, 4 00 
 
 13 
 
 3 00 
 
 .7 50 
 
 7 50 
 
 15 00 
 
 15 00 
 
 1 50 
 
 2 00 
 
 2 00 
 
 3 50 
 
 2 50 
 
 1 50 
 
Chester's Catalogue pf Minerals .§y^, 
 
 Pap?r, 
 
 " Dictionary of the Names of Minerals §vp, 
 
 Dana's American Localities of Minerals , • -Large Hvq, 
 
 " Descriptive Mineralogy. (E.S,) LargeSvo, hal.fjiiofwp^o, 
 " First Appendix to System of Mine^'a,lqgy. .,. Largp 8vo, 
 
 " Mineralogy and Petrography. (J. D.) , 13mo, 
 
 " Minerals and How to Study Them. (E. 8.)... . ...l^no, 
 
 " Text-book of Mineralogy. (E- S.)...ljfew Edition, 8vo,. 
 * Drinker'&Tunnelling, Explosives, Conjipoujnds, an,d Roplc Drills. 
 
 4tp, half morocco, 
 
 Egleston's Cfjitalogue of Minerals a;id, Synonyms ,,. (.8vo, 
 
 Eissler's Explosives — Nitroglycerine an4 Dynajnite ,,. , .-Svo, 
 
 H'lssak's Rock-fpfjniing Minei,-a,lfi. , (Smi_th.) Sjfiall Svo, 
 
 Ihlseng's Maiiufil of Mining ,.-..... .... i./.8vo, 
 
 Eunbardt^s Ore Dressing in Europe .8vo, 
 
 O'DriscoU's Treatment of Gold Ores , ,...,. 8 vo, 
 
 "*,Penfield's Record of Mineral Tests Pa,per, Svo^- 
 
 Boseubuscb's Microscopical Physiogrj^phy ot Min(;ia\s aijd,. 
 
 Rocks. (Iddings.) — -.^yp,,- 
 
 Savyyer's Accidents in Mines , Ljirgo 8vo, 
 
 Stockbridge's Rocks and Soils. ....,.-..- Svo, 
 
 "Walke's Lectures ou Explosives ■.8vo, 
 
 "Williams's Lithology j ...;./. Svo, 
 
 "Wilson's Mine Ventilation... .......... .13mo, 
 
 " Hydraulic and'Placer Mining '.......... .12mo, 
 
 STEAM AND ELECTRICAL ENGINES, BOILERS, Etc. 
 
 Stationaky — Mabinb^^Locomotivb — Gas Engines; ■ Etc. 
 (See aZ«!> Engineekino, p. 7.) 
 
 Baldwin's, Steam Heating for Buildings. . , j,-. . . . . . 12ijio, . i . 8-5.9 
 
 •Clerk's Gas Engine , , ,. ijmalj Svo, 4 QQ 
 
 Eord's Boiler Making for Boiler Makers , 18nio, 1 00 
 
 Hemenway's Indicator Practice ,5,3mo, 2 00 
 
 Kneass's Practice and Theory of the Injector , -.Syji , 1 50 
 
 MacCord's Slide Valve 8vo, 2 00 
 
 Meyer's Modern Locomotive Cp^i^truction. . ..., , 4to, 10 00 
 
 Peabody and Miller's Steam-boilers 8vo„ 4 00 
 
 Peabody*s Tables of Saturated Steam. .8voj 1 00 
 
 " Thermodynamics of the Sileam Engine Svo, &,00 
 
 " Valve Gears for the Steam Engine, , ...Svo, 8 50 
 
 " Steam-engine ludioatov i3mo, 150 
 
 Pray's Twenty Years with the Indicator Large Svo, 3 50 
 
 P\ipin and Osterbevg's Thermodynamics 13mo, 1 25 
 
 14 
 
 $1 25 
 
 50 
 
 S ,0& 
 
 l.,# 
 
 12 50 
 
 1 op 
 
 2 00 
 
 M9 
 
 4 CO 
 
 25 00 
 
 3 50 
 
 4 00 
 
 .2,00 
 
 4 00 
 
 1 50 
 
 3 00 
 
 50 
 
 ',5'OP 
 
 7 00 
 
 2 50 
 
 4 00 
 
 3 00 
 
 1 25 
 
 2 50 
 
Heagan's Steam and Blectrip Locomotives , ISmo, $2 00 
 
 BOutgen's Thermodynamics. (DuBois.) 8to, 5 00 
 
 Slaclair's Locomotive Runniilg , ISmo, 2 00 
 
 Snow's Steam-boiler Practice 8vb. 3 00 
 
 Thurston's Boiler Explosions 12mo, 1 50 
 
 " Engine and Boiler Trials 8vo, 5 00 
 
 " Manualof the Steam Engine. Piirt I., Structure 
 
 and Theory 8vo, 6 00 
 
 " ' Manual of the Steam Engine. Part II., Design, 
 
 Construction, and Operation 8vo, 6 00 
 
 3 parts, 10 00 
 
 Thurston's Philosophy of the Steam Engine 12mo, 75 
 
 " Reflection ou' the Motive Power of Heat. (Caruot.) 
 
 13mo, 1 50 
 
 " Stationary Steam Engines 8vo, 2 50 
 
 " <■ Steam-boiler Construction and Operation 8vo, 5 00 
 
 Spangler's Valve Gears 8vo, 3 50 
 
 Weisbach's, Steam Engine. .(Du Bois.) ; 8vo, 5 00 
 
 "Whitham's Steam-engine Design 8vo, 5 00 
 
 Wilson's Steam Boilers. (Flather.) 12mo, 2 50 
 
 "Wood's Thernjodynfimics, Heat Motors, e)t(^. , ■8vq, , 4 00 
 
 TABLES, WEIGHTS, AND MEASURES. 
 
 '■'• For Actuaries, Chemists, Engineers, Mechanics — Metric 
 Tables, Etc. 
 
 ^driance's Laboratory Calculations 12mo, 1 35 
 
 Allen's Tables for Iron Analysis 8vo, 3 00 
 
 Bixby's Graphical Computing Tables Sheet, 25 
 
 Compton's Logarithms '. 13mo, 1 50 
 
 Crandall's Railway and Earthwork T<ibles 8vo, . 1 50 
 
 Egieston's Weights and Meas.ures. '18mo, 75 
 
 Fisher's Tafele qf Cubic Yards Cardboard, 35 
 
 Hudson's Excavation Tableg. Vol.11 8vo, 100 
 
 ,Jolinson'§ Stadia and Earthwork Tables , 8vo, 1 25 
 
 .Ludlow's XiOgarithmic and Other Tables. (Bass.) 13nio, 3 00 
 
 .Totteu's Metrology 8vQ, , 3 50 
 
 VENTILATION. 
 
 Steam Heating — House Inspection — Mine Ventilation. 
 
 Baldwin's Steam Heating 13mo, 3 50 
 
 Beard's Ventilation of Mines 13mo, 3 50 
 
 Carpenter's Heating and Ventilating of Buildings .^vo, 3 00 
 
 •Gerhard's Sanitary House Inspection 13mo, 1 00 
 
 Wilson's Mine Ventilation 13mo, 1 35 
 
 15 
 
|5 00 
 
 3 00 
 
 1,50 
 
 4 00 
 
 2 50 
 
 1 00 
 
 1 00 
 
 3 00 
 
 MISCELLANEOUS PUBLICATIONS. 
 
 Alcott's Gems, Sentiment, Language.../ i.. . .GHt edges, 
 
 .'Davis's Elements of Law .;.•., 8vo, 
 
 Emmon's Geological Guide-book of the Rocky Mountains. :8vo, 
 
 Ferrel's Treatise on the Winds .8vo, 
 
 Haines's Addresses Delivered before the Am. Ry. Assn. ..12mo, 
 Mott's The Fallacy of the Present Theory of Sound. .Sq. 16mo, 
 
 Ricliards's Cost of .Living l-2mo, 
 
 Ricketts's Histoiy of Rensselaer Polytechnic Institute. i. . . .8vo, 
 Rotherham's The New Testament Critically Emphasized. 
 
 13mo, 1 50 
 " The Erpphasized BTew, Test. A new translation. 
 
 Large 8vo, 3 00 
 
 Totteu's Au Important 'Question in Metrology. 8 vo, 3 50 
 
 * Wiley's Yosemite, Alaska, and Yellowstone 4to, 3 00 
 
 HEBREW AND CHALDEE TEXT-BOOKS. 
 
 For Schools and Theological Seminaries. 
 
 Gesenius's Hebrew and Chaldee Lexicon to Old Testament. 
 
 (Tregelles.) , Small^ 4to, half morocco, 5 00 
 
 Green's Elementary Hebrew Grammar. . . ..'.'....'. ...'.'.. .13mo, 1 35 
 
 " , Grammar of the Hebrew Language (New Edition). 8vo, 3 00 
 
 " Hebrew Chrestomathy 8vo, 3 00 
 
 Letteris's Hebrew Bible (Massoretic Notes in English)., 
 
 8vQ, arabesque, 3 35 
 
 MEDICAL. 
 
 Hamnlarsten's Physiological Chemistry. (Mandcl.) 8vo, 4 00 
 
 Mott's Composition, Digestibility, and Nutritive Value of Food. ' 
 
 Large mounted chart, 1 25 
 
 Ruddiman's Incompatibilities in Prescriptions 8vo, 3 00 
 
 Steel's Treatise on the Diseases of the Ox. 8vo, 6 00 
 
 " ' Treatise on the Diseases of the Dog 8vo, 3 50 
 
 Woodhull's Military Hygiene .16mo, 1 50 
 
 Worcester's Small Hospitals — Establishment and Maintenance, 
 including Atkinson's Suggestions for Hospital Archi- 
 tecture .13mo> 1 25 
 
 16