a. of 1. 
 
 DEPARTrOEOT of AR-CUITCCTORE 
 
 SWAbtS And SWAb0ai5 
 
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Digitized by the Internet Archive 
 
 in 2011 with funding from 
 
 University of Illinois Urbana-Champaign 
 
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SHALES ..FD SH<^DOrs 
 Notes Arr^.n^cd For 
 Tlic Dep-.rt-vor.t of Arcliitcctr.re 
 
 UNIl^RSXTr OF ILLINOIS — 
 
 Mimoegr?cphecl by 
 
 Student Supply Store 
 Ghr.mpc^.ign, ill. 
 
1 
 

 1. 
 
 •, rIOTvf' M.d:Mi(\^ m :^aLPK PAHNING 
 
 1. IlITRODUCTION ; 
 
 Objects are QYi-it»le to the ' eye owing to the re- 
 flection from their surface 'ro.ys of light, These reflected rays 
 strike upon the retina of the eye and give the sensation of sight. 
 The greatest source of light is the sun, and rays from it strike 
 all bodies on the earth at various angles. If the angles of impact 
 of all the rays efriking on the. surface be ^:- the .'same, the surface 
 appears flat, technically ' "plane" . If the angles of impact are 
 different then the surface appears "curved". 
 
 Since therefore, the pleasant or unpleasant effect 
 of any object v.'ill depend upon the producing of an effect of 
 pleasing or unpleasing lighting upon the eye, it becomes of 
 primary importance that the students of those branches of art 
 that deal xjxVa objects in the round, vjith objects subject for 
 their effect upon the play of light and shade upon thail' surface, 
 namely, Architecture and Sculpture, be thoroughly familiar vjith 
 the phenomena of light and shade. 
 
 In addition to this it is essential that he v;ho; - 
 conceives an idea m.ust be able to express that idea forcibly and 
 convincingly to others, else his idea becomes of no importance. 
 The painter expresses an idea in one vjay, the -vTritcr in another, 
 the sculptor in another, the architect in still another. The 
 Sculptor v7orks. out his idea in clay, then piaster, then in stone 
 or in bronze; the architect has certain conditions set before 
 him; he v/orks ' out the idea in the drafting room and working under 
 his direction, the builder oxcc^^tes that idea in permanent 
 materials of wood, brick, and stone. 
 
 A building docs not consist merely of the lines by 
 which it is represented in geometrical drawing, but of masses 
 and these are better and more quickl" represented by tints than 
 by mere line drawing and in order that the final results may be 
 that pleasing creation that distinguishes the thing of art from 
 that of pure utility-that make of it architcctmre-the architect 
 must be thoroughly familiar with those feb:if:gal phenomena which 
 will make of his creation a thing of beauty and must be able to 
 represent those phenomena on drav/ings . 
 
 In order that a representation of light and shade 
 may be made upon geometrical drawings, the architect brings into 
 use an application of certain principles of descriptive geometry, 
 using that science, however, not as a means alone, but as a means 
 toward an end, and baaing the general underlying: principles of 
 the application upon a careful observ:^nceof natural phenomena- 
 adopting convenient conventions--never violating the fundamental 
 
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2. 
 
 §?;'^^ AIIT 'v_^U)OVS, 
 principles of "n.-^.turr.l -;- ?r-Oi.iena. 
 
 2. .LIGHT_RAYS: 
 
 The ray® of light ivhich .-^.re deo.lt v/ith in reference 
 to anyr-- one body'rxre first-direct or incident rays, second- 
 tangential t?ays, -third-lateral rays. Direct rays impinge directly 
 upon the body. Tangential raye are those tangent to its surface. 
 Lateral rays do. .not strike the surface, "but go on to illuminate 
 bodies beyond, "^'aturally, in the study of Sh^ides " and Shadovjs , 
 direct and tangential rays are those of im.portance. Fig. 1 
 
 3. SHADi: ArlD SHADOWS ; 
 
 If rays of light are excluded from certain portions 
 of a surface by the shape of that surface itself, then the dark- 
 ened portion is said to be in shade, if the surface, by projection 
 beyond or position betv^een the source of light and another surface, 
 exclude light from any portion of the second surface, then the 
 first on? is said to"cast shadovr over the second". Shadovrs reveal 
 by their' extent the relative position of as '.'jell as the shape of 
 surfaces. Shade reveals merely the shape of separate surfaces. 
 
 4. KE/iSOIIS FOR AS_SU?^?TI_Oj£ OF 
 
 c:^KTA III "cokv-':jttt ous . ' 
 
 Any object placed in a fixed position on the earth, 
 as is of course every building, vTill be subject to a continual 
 change of lighting, to a continually changing play of light, shade 
 and shadow. If on the geometrical or line drav;ing of an element 
 architectural, each separate artist ;7ere to assume a direction 
 for the rays of light that are presuj-tied to illui'iinate an object-- 
 for naturally an assumption of directions of light is prerequisite 
 to the Tv'orking out of shades and shado"'fs geometrically-then to 
 each'drav/ing y:ould have to be added a statement of the assumptions 
 made. The assumption of certain directions of the rays v^rould be 
 cause of great difficulties in" the geometiical solution and a 
 geheral confusion v/ouia result, though each distinct method or 
 means 'ivould itself be correct. Since the finished result of the 
 artist's efforts must be easily and clearly readable by those 
 who do not understand the methods used in acheiving the result 
 set before them; since the v/orking out of shadou's is in any event 
 a long and complicated process, and since, therefore, the adoption 
 of any conventions tending to make for the dra.ftsman greater else 
 of v/orking, are of prime importance, and for these reasons solely 
 
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5. 
 
 SHADES Mi'D SHADDWS 
 
 cert''.in conventions as to direction of light have been universally 
 adopted, Tiie student, hovfever, must not come to believe that the 
 adoption for sake of convehience of particular conventions affect 
 in any ivay the general principles underlying the Study of Shades 
 and Shadows. 
 
 5. THE RAY A^ 45fo 
 
 In the study, therefore, of Shades and Shadovrs, the sun is 
 assumed to be the source of light, and the rays assumed to take 
 a dov;nv;ard direction, and to the right parr-^allel to the diagonal 
 of a cube (Fig. 2 ), The angle which the ray makes vjith its owa 
 projection .is, therefore, 35% 15' 52'', and is Icnovm visually as 
 the angle ;|:. or the"true angle of the ray". 
 
 The projections of this one ray are three, and each 
 projection naturally raakes an angle of ^5% v/ith the horizontal, 
 and since they originate at inf inity--the sun--the pi'ojections 
 of all rays In any particular plane are parallel. (Fig, 3) 
 Expressed architecturally, there is one ray vith a front ' elevatiQn, 
 a side elevation and a plan as can be readily understood. The use 
 of the ray iii this particular dilr-ection, gives not only ease of 
 construction, but also ease of interpretation of projections of 
 surfaces on beyond the other, for naturally the v/idth of the 
 shador; cast by one plane surface over another v/ill be exactly 
 equal to the projection of the first surface in front of the 
 second. Having established a definite basis for study there can 
 no7' be discussed the various procedures necessar^' for the actual 
 finding of Shades and Shadov;s under any given condition. To the 
 study the architectural student must bring more than a mere 
 laaoTJledge of the processes of descriptive geometry. He must bring 
 a sense of analysis and thoughtfulness that will enable him to 
 discover in each problem those particular elements that are 
 necessary for the rapid solution of the problem, but he must 
 above all else cultivate a knovvledge of the general shapes of 
 actual Shades and Shadov;s and a common sense way of looking at ■ 
 any problem presented. 
 
 By a m.ere application of principles of descriptive geometry 
 the finding of the precise piercing point of a line on a surface- 
 any profile in shades and shadows can be solved, but since 
 rapidity of thought ahd action as well as accuracy of result are 
 the prime requisites of the architectural draftsman every effort 
 must be made to learn the shortest and the easiest methods of 
 acheiving a given result. Analysis and acquired experience, 
 thoughfulness , Imagination, observation of natural phenomena, 
 these all will give rapidity and accuracy. 
 
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4. 
 
 THE FINDING OK GEQIiIETRICAL DKAVflNGS 
 OP SH/.DES AND S'rl/DOWS 
 
 BY GEO'ITITRICAL MEANS 
 
 6. DEFINITIONS, 
 
 "A plane of Rays" is a plane -A'hich raay be considered as 
 made up of the rays passing through adjacent points of a straight 
 line, (Fig, 4) 
 
 "Point of loss"-The point of intersection of a shadov/ v/ith 
 a shade line, or of a shadovf or shade line virith the line of 
 division betv/een a lighted and unlighted sr.rface. (Fig. 5) 
 
 "Invisible shddorrs" or shadoivs in space" that portion 
 
 of space from iThixih light is excluded by a body in direct light. 
 
 (Pig. 5) 
 
 7. I'iET'riODS OP CONSTRUCTION. 
 
 The finding of sbadovjs comprises tv/o distinnt operations. 
 They are in order of consti-^uction: ^ 
 
 1. Ti.ie finding on the object itself of the line 
 v;hich separates the lighted part from that 
 shaded part, known in consequence as the 
 separation or separatrix or "shade line", 
 
 2. The finding of the outlines of the s^adov;s 
 cast by the object on a foreign surface, that 
 surface being usually in the hoticontal or 
 vertical plane, knovm as the "shadow'' , 
 
 Since it has been assumed that the light rays fall in 
 a definite position oblique to the vertical and horizontal plane 
 of the projeotion and since architectural details are made up 
 for the most part of regular surfaces, either planes at right angler 
 to each other and parallel to or at right angles to the general ■ 
 surfaces of revolution v/hose elements are parallel or perpendicular 
 to the same plane, when it is evident that in general the shadow 
 of any object over another object in an 6blique projection of 
 the first obgdot,iand that, as can be readily seen, tHe outline 
 2f t he first shadov; is t he sha dow of the s h ade line of the objec t 
 cashing the shadow."' 
 
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5. 
 
 SHADES AilD SH/>DOWS 
 8. THE SHADOWS £>E POINTS. 
 
 The shadow of a point on any surface is found by 
 passing a ray (straight line) through that point and finding v/here 
 that ray pierces the surface. The point of piercing must 
 necessarily be at the shado?/ of the point.. (Fig. 7) 
 
 9. THE SH/iDOW OF LIIIES. 
 
 A straight line is by definition made up of a number of 
 points, St) that the pas#ing of separate rays through each of 
 the points v;ould make a plane of rays, the intersection of 
 which plane v/ith the surfaces in question would give the line 
 of shadow of the line considered. The length of the shadov; would 
 be determined by the shadows of the two points at the ends of 
 the line or by the limits of the surfaces on which the shadow 
 falls (Fig. 8). 
 
 10. THE sh;.dows of surp;.ces. 
 
 Since ■^. surface is limited by lines, the shadow of a 
 surface can evidently be found by finding the shadows of 
 boundary lines of that surface. 
 
 11. THE SK.;D0WS of SOLIDS. 
 
 a) Polyhedra. 
 
 ^^re solids bounded by portions of intersecting 
 planes. The lines of demarcation between light 
 and shade viii natur^.lly be along lines of 
 intersection. The outlines of the shado'ws of 
 a polyhedron viill be determined by finding 
 the shadows of those lines of intersection that 
 divide the lighted from the unlighted pofction 
 of the polyhedra (Fig, 9). 
 
 b) Surfaces of Revolution . 
 
 The shade lines on surfaces of revolution are 
 formed by rays tangent to the surfaces. The 
 ovitline of the shadov? v;ill evidently depend 
 upon the finding of the shadow of that line 
 of tangency. (Fig. 10). 
 
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6. 
 
 SHADES AND SHADOWS 
 
 12. THE GEIIERAL PKOBLEli 
 
 Prom the preceding discussions it should be evident that 
 the general problem involved is that of ''representing the rays 
 which pass through points in the shade line of an object r-.nd 
 finding the points at which these rays strike another object. 
 Generally spealcing, this is not a difficult problem in 
 descriptive geometry, and is one quite v/ithin the power of an 
 architectural draftsman of a little experience, if he will keep 
 cler.rly in mind the nr.ture of the problem he is to solve. "He is 
 c.pt to entangle himself in trying to remember rules and methods 
 by which to reach a solution"", (T'cGoodv/in) . S tatement of certain 
 more or less evident corollaries of p receding discussion. 
 
 A thorough understanding of the general corollaries 
 stated below will be of inestimable benefit in quick and ready 
 analysis of problems. 
 
 1. All straight lines and planes may be considered as 
 being of indefinite extent. Parts not of such lines 
 and plahes lying beyond parts having actual 
 existence in cases considered will be termed " 
 "imaginary". .{Pig. 11). 
 
 2. (a) A point which is not in light cannot cast a 
 
 real shadow. 
 
 (b). Every real shadov/ line must cast a real shadow 
 and this real shadow cannot lie within another 
 real shadow, Host of the blunders in casting 
 shadows are due to a neglect ownunderstanding 
 of these two statements. For instance, in Fig. 
 12 it is evident that point "a" cannot cast a 
 shadov/ on the wall. 
 
 3. (a) The shadow of a straight line on a plane may 
 
 be determined by the shadovj of any two of its 
 points ofc the plane, ^'^aturally the shadows of 
 the points at the ends of the line are the 
 ones most advantageous to find.- 
 
 (b) The shadows of any line on any surface may be 
 determined by finding the shadows of adjacent 
 points of the lines. As could be expected, this 
 method is a very cumbersome one, and such 
 shadows will be, wherever possible, found by 
 less lengthy processes. 
 
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7. 
 
 SHADES AND SHADOVfS 
 
 13. THE GENERAL METHOD . 
 
 Sought for results may be acheived by cumbersome or by 
 easy methods. So in the finding of shades and shadows on architect- 
 ural drav/ings, certain methods have been found to give results easil' 
 and accurately. Those general methods found to be the most applicable 
 are: 
 
 1) The method of oblique projections, 
 
 2) "^e method of circujascribing surfaces. 
 
 3) The method of auxiliary shadov;s, 
 
 4) The slicing method. 
 
 Each separate method ivill be discussed in detail in its 
 proper sequence, and with its proper applications. All of the 
 methods or but one method may be conveniently used in the casting 
 of shadows on any one' object. The use of a little common sense and 
 visualizing faculty are essential ifi the student is to do any 
 particular probl"!u problem accurately and quickly. 
 
 14. THE METHOD OP OBLIQUE PROJI^CTION. 
 
 The method of oblique projection consists simply in 
 drawing on the projections of the object the forty-five degree line 
 representing the rays tanrent to an object or passing through itd 
 shade line and then in finding the points ;There the rays strike any 
 uther object involved in the problem, these points of interesction 
 giving the outline of the shadov;. 
 
 Thffis method is simple and direct, but n.'/hrr'-.j ly can be 
 used only -when the plan or side elevation can be represented by a 
 line. Otherwise it is impossible to find cirectly points at wHicE rays 
 s tr'ike the given surface. For example, in plan the surface of a 
 cylinder with vertical elements can be represented by a circle but 
 the surface of a torus, scotia, or cone cannot. Hence in the latter 
 case.s some method other than that of 'direct projection must be used 
 if shades or shadovTS are to be found on these surfaces. Theoretically 
 this method requires the finding of the shadows of all points in a 
 line, but practically, under the assumption made in this study-- 
 namely; rays of fixed direction parallel v;ith each other--the 
 shadov/s of certain points and lines on certain surfaces in certain 
 portions will be always the sr^.me. They maj be stated as follows: 
 
 1) The elev'^tion of the shadov/ of a point on a vertical 
 plane v-iill '\lways lie on a forty-five degree line to the right 
 of the elevation of the point in front of the plane. (Fig. 13). 
 
8. 
 
 SHADES MID SHADOWS 
 
 2) The shadow on a given plane of any line which is 
 parallel to that plane is a line equal and parallel to the given 
 line and lies to the right of the line a distance equal to the 
 distance of the line in front of the plane. Fig. 14). 
 
 parallel. 
 
 3) The shadows of parallel lines in any plane will be 
 
 4) The shadow of a line perpendicular to an elevation 
 plane will in front elevation be always a forty-five degree line no 
 matte'r v/hat be the fonii or posi tion of the o b jects r e ceiving t he 
 shadow. The shadow of a line being formed by the intersection of a 
 plane of rays, through the line with the surface considered, in this 
 particular case, will coincide, of course, with the elevation of the 
 plane of rays this plane will, therefore, be itself perpendicular 
 
 to the surface receiving the shadow and will appear in elevation as a 
 forty-five degree line. Eig.lS). 
 
 5) The shadow in plane of line perpendicular to the plan 
 plane should be a forty-five d e gree line . The reasoning given in 
 (4) should be sufficient to malte the point clear. Fig. 16). 
 
 6) The shadow of a vertical |bine on an inclined plane 
 v/hose horizontal lines are parallel to the elevation plane is an 
 
 V inclined plane whose slope is equal to that of the given plane. 
 The most frequent application to architectural problems is found 
 in shadows of dormers and chimneys on roofs. (Fig. 17), 
 
 7) The shadow of a vertical line on a series of horizont' I. 
 
 mouldings is equal in front elevation to the profile of the right 
 
 section of the m.ouldings. (Fig. 18). The shadow line of course moves 
 
 to the right as the contour recedes. This shadow of such -frequent <> 
 
 occurence in architectHral problems is too often . drawn. 
 
 incorrctly 
 
 8) The shadows of horizontal :>lines either parallel or 
 perpendicular to the elevation plane, on a vertical plane receding 
 diagonally to the left at an angle of forty-five degrees--are of 
 the parallel lines forty-five degree lines, sloping dovmward and 
 
 to the left, and of tjie perpendicular lines forty-five degree lines 
 sloping down-ward and to the right, (Fig. 19). These shadows are 
 often used in finding of auxiliary shadows. 
 
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9.. 
 
 SHADES AND SHADOWS 
 
 9) ^he shdde line on a curved surface whose elements are 
 horizontal or vertical straiccht lines, is found by drawing the 
 elevation of a ray tangent to the profile of the surface. (Fig, 20), 
 Evidently the shadoiv cast over such a surface "by a straight line 
 •vvhich is parallel to the elements of the surface can be found as 
 shown. (Fig. 21), 
 
 With the knowledge, therefore, of the definite positions 
 that the shadows of these lines vjhich form boundary lines are to 
 practically all surfaces appearing in architectural vrork take, the 
 finding of shadows of even the most complicated surfaces becomes / 
 comparatively easy. Often the deter.nination of the shadow of a \ 
 single point xvill suffice for the determinc?.tion of an entire gr^up / \ 
 of shadov/s. An attempt must always be made to use as little as 
 possible either plan or side elevation in determining shado-^firs-- 
 for they are often in architectural vjork at different scale from 
 the front elevations, and if used to a large extent would have to 
 be redra.wn--a lengthy and entirely uncalled for proceeding. 
 
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10. 
 
 SHADES Am SHADOVJS 
 Nomenclature 
 
 For the purpose of clearness in reading of diagrams 
 given, the followinj;; nomenclature has been adopted: 
 
 R - Ray of lic'ht in space at Conventional Angle. 
 
 R^.- Front Elevation of Ray. 
 
 R2- Side elevation of Ray. 
 
 R3- Plan of Ray. 
 
 (|) - True angle of Ray^' 
 -:■ V - Vertical plane of projection or Front Elevation Plan^ 
 
 P - Profile or Side Elevation Plane. 
 
 H - Horizontal or plan plane. 
 
 X - Any other plane. 
 Let A Any point in space. 
 Then Aj - Any point' in front elevation. 
 
 A2 - Same point in plan. 
 
 A3 - Same point in side eleVation. 
 Let. Als- Front elevation of shadov; of point A.' 
 
 A2S Plane of shadovf of point A. 
 
 Ajg Side elevation of shadow of point A. 
 
 GL- Ground Line-Line of intersection of V Plane and 
 
 H plane. 
 
 NOTE: In lettering of all problem', plates this nomenclature is to 
 be follo'vved. 
 
 The Architectural terms, front elevation (or usually 
 Elevation), Side Elevation , and Plan" ^ are to be used in preference 
 to the terms V Projection, P projection, and H projection. 
 
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11. 
 
 THE SHADOVfS OF CIRCLES . 
 
 1^. SHADO'^S OF CIRCLES IN PLANES PARALLEL 
 TO PLANE TiECEIVING SHADDW7 
 
 It is quite evident that the shadov/ line of, for instance, 
 a circular flat disl-: on a plane v/ill be formed "bj the intersection 
 of a cylinder of rays v/ith the plane in question, and that the 
 intersection of the cylinder u-ill be a circle or an ellipse depend- 
 ing upon v/hether the disk vfere in a plane parallel to the given 
 plane or in a plane at; an anjV.le vvith the ^iveh plane, (Fig, 22). 
 It is also evident that in the first instance, the shadow line will 
 be a circle of exactly the same radius as the disk casting the 
 shadovRji. and that, therefore, the finding of the shadov; of the center 
 of the circle will be sufficient to determine the complete shadow. 
 The arch is the common architectural form in -which circles occur in 
 such a position. 
 
 16. SKADOITS OK VERTICAL AND HORIZONTAL PLALIES . 
 
 TiTien the shadow line is an ellipse, by methods of direct 
 projection from plan or dide elevation as auxiliaries can be found 
 a nuinber of points of shadov-'s of the circumference of the circle. 
 Usually circular forms occur in architectural work in planes 
 perpendicular to the vertical plane of projection and parallel to 
 the horizontal, or in planes perpendicular to both planes of pro- 
 jection, and the shadovj of such circles are usually cast on a 
 vertical or elevation plane, though sometimes on a horizontal or 
 plan plane. Since the center of the circle is a point, it is quite 
 easy to determine its shadow, which determines naturally the center 
 of the ellipse ?shadow. The major and minor axes of the ellipse of 
 shado'w are then determined. 
 
 The architect must, of cours, determine if possible bjf 
 methods of reasoning, those shado-w points that "will be of greatest 
 importance, and through them must construct the ellipse of shadow. 
 The simplest and most accurate method of determination is as follows. 
 (Fig. 23). 
 
 The shadows of the circumscribed and the inscribed squares 
 are first found, using the already found shadow of the center of 
 the circleas a point for syrmnetrical construction. The shadov/s of 
 the median and diagonal lines are easily found. The points at \"jhich 
 the ellipse of sha.doi7 crosses the diagonals is found as shown in the 
 figures. The tangents, v/hich of course are parraia^el to the diagonals. 
 are usually drawn' to serve as a guide in freehand construction of the 
 ellipse of shadow. Since the circle is a contintkous curve, if through 
 any inaccuracy of construction points of shadow found do not give 
 continuoTis curve, then these points must be disregarded and the curve 
 dra;vn through the greatest n\imber that lie on a continuous curve. 
 
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 Tr; 
 
12* 
 
 SHADES AND SHADOWS. 
 
 17. SHADOWS ON INCLINED PLANES . 
 
 The same methods of reasoninrr, as used for shadows of 
 circles on vertical and horizciital planes give the construction 
 of the shadovf on an inclined plane, as shown in Fig. 24, 
 
 18. SHADOWS ON 45-^ AU XI LIARY PLANE . 
 
 '^he shadcvr or. a vortical plane at 45 degrees passing 
 through its center,, of a circle in a horizontal plane perpendicular 
 to the elevation plane is a circle (Fig, 25), 
 
 19. CONC LUSI ON, 
 
 It T^Uf^t above all -"Ise be remembered that the shadow of 
 any circle must, of course, be completely within the shadow of the 
 circumscribed square and will be tangent to that shadow at points 
 where the original t:ircle is tangent to the circumscribed square, 
 Time honored blunder 3 i"u the casting of the eaadov/s of circles may 
 be almost entirely avoided by the accurate finding of the shadow 
 of the circur!iEcriT;ed square even though the inscribed square is not 
 found. 
 
13, 
 
 20. THE SHADES ON MID SHADQVfS OF SURFACES OF REVOLUTION. 
 
 Surfaces of revolution are created "by revolving lines 
 straight or curved or both about a fixed axis or series of axes, 
 . In the forms comraonly met v;ith in architectural objects, the 
 surfaces of revolution are generally either vertical or horizontal, 
 so that the shapes created are more simple to deal xvith than those 
 created v;hen the axes are inclined, '^e study, therefore, vill be 
 confined to right cylinders, cones, spheres, tori and sco$;ias. The 
 shades on and shadoivs of certain of these surfaces can easily be 
 found by an application of some one or all of the methods mentioned 
 at the beginning of the discussion. 
 
 21. THE SHADES ON AN UPRIGHT CYLINDER. (Fig. 26). 
 
 It is quite evident that the surface of an upright 
 cylinder can be represented in plan by a circle, and the surface 
 of a horizontal cylinder in side elevation by a circle. Hence the 
 method of oblique projection can be applied in the finding of the 
 shades and the shadovrs. The lines of on the cylinder -aill evidently 
 be determined by tv.'o planes of rays tangent to the surface, They v/ill 
 come tangent along a vertical or horizontal element of the cylinder 
 and can be represented in plan or side elevation as the case may be 
 by lines tangent to the plan or side elevation of the line represent- 
 ing the surface of the cylinder. From the points of tangency thus 
 determined are secured the lines of shades. One line of shade is of 
 course invisible in the front elevation. The other is a little less 
 than 1/6 of the v;idtheof the elevation of the cylinder to the left 
 of the right profile of the cylinder. This proportion is a con- 
 venient one to remember, 
 
 22. THE SHADOWS OF CYLIND~RS. (Fig. 26). 
 
 The outline of the shadow of a cylindB±cal surface on a 
 plane can evidently be determined by finding the shadows of the tvo 
 shade lines-~vjhich are straight lines, and the shadovjs of those 
 portions of the circular outline of the top and base that lie betv.'een 
 the points at which the lines of shade cross the bases. These will 
 be elliptical, and it is best to determine by methods already given 
 for the shadows of entire circles. As can readily be seen, the ^■'. ■-. ' 
 width of the elvation of the shadovf of a cylinder on a vertical 
 planerwill be equal to the diagonal of a square having the diameter 
 of the cylinder for a side and -Till also be symmetrical about the 
 shadow of the axis of the cylinder. This fact can be conveniently 
 put into use in many cases. 
 
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 J. i 
 
 •'■.".■' c^ ■".■■ " 'f (''■'r'J 
 
 r , - 
 
 
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14. 
 
 23. THE SHADOWS ON CYLII'DRIC.'.L SURFACES. 
 
 1. The Sh?.do-as of a Straight Line on an Upright Cylinder, 
 
 The shadov: of a straight line parallel to both the V and H 
 planes nill be a circle v/hose radius is equal to the radius of the 
 cylinder and the elevation of vrhose center lies on the elevation 
 of the axis of the cylinder, belo:'.' the elevation of the line a 
 distance equal to the distance of the line in front of the axis of 
 the cylinder. This should be clear from Fig, 28. 
 
 The shadov; on the surface of a line perpendicular to the 
 elevation plane and parallel to the plan plane is a line at 45/'/! 
 (Art. 14, paragraph 4). The shadow of ' the end of the line would be 
 found by direct projection. (Fig, 28). The shado'.v of any other 
 straight line would be formed by the direct projection onto the 
 surface of the cylinder of enough points of shadov; to determine 
 the curve of shadov;. 
 
 2. The shadov; on an Upright Cylinder of a Larger Cylinder 
 whose axis coincides v;ith the Axis of the Smaller 
 Cylinder. 
 
 Evidentl-y the shadow line vrill be a curve rf no easily constnur 
 ed geometrical form. Hence it becomes necessary to find of that curve 
 by means of direct projection enough points to determine the directici 
 of the curve. Since the outline of the shadov; v;ill be determined 
 by the shadov; ofi a certain portion of the circle bounding the lovjer 
 surface of the c^^linder, by means of direct projection from points 
 in this circle on the sv\rface can be found any nuraber of points 
 desired. However, certain points are of more importance in determin-t 
 ing shape of the shadov; than others. Those points are naturally 
 enough, the points v;here the shadow crosses the lighted profile of the 
 cylinder, the point v;here it crosses the shade line, and the point 
 vjhere it crosses the elevation of the axis, and v;here it is closest 
 to the elevation of the line casting the shadov;, 3y inspection it 
 ca.n be seen that the highest point of shadov; lies on the diagobai 
 axis en the left of the center, for on that line the rays strike the 
 
 :F 
 
 surface at the true angle, --(The angle "p ). 
 
 Therefore , it is determined first of all vrhat points on the 
 circle cast shadov; on these lines mentioned and thenas many 
 additional points are determined as are deemed necessary to the 
 correct drawing of the shadow. If the object be small, then n?.tviu:-lly 
 the determination of the four points mentioned is sufficient: if the 
 object be large, then more points must be determined. The method 
 of determination is shown in Fig» 29, The p©int on the profile line 
 
f 'i;l • 
 
 J v" .' ' i 
 
 )j..' 
 
 / 
 
 ^ «rr< r ■•• 
 
 
 ■/■ 
 
 •» ; 
 
 
15. 
 
 and that on the axis of the cylinder are at the same distance beloiv 
 the line casting the shad or;. The determination, therefore, of the 
 position of either one v/ill be sufficient to determine the position, 
 of the other. 
 
 24. THE SHADES AND SHADOWS OF HOLLQH CYLINDERS . 
 
 In the d ra;7ing of the sections of buildings it is often neces- 
 sary to determine the shades and shadovrs of hollov/ cylinders, as for 
 instance in the section of the cupola of a dome, or that of a hori- 
 zontal barrel vault or that of an arch. The methods of determining 
 the points necessary to give the correct general shapes of the shades 
 and shadov;s on such cylindrical surfaces are shown in Fig. 30, 31, 3 2, 
 
 25. TIffi lETHOD OF AUXILIARY SHADOWS 
 
 The principles upon -which the application of this method is 
 based are :- 
 
 1. If upon any surface of revolution a series of auxiliary 
 curves be drawn, the shadov; of the surface will include 
 the shadovjs of all the au::iliaries, and v;ill be tangent 
 to those that cross the shade line of the surface at 
 points vjhich are the shadows of the points of crossing, 
 
 2. The point of intersection of tvjo shadov/ lines is the shad - 
 ow of the point of intersection of these lines if they are 
 intersecting lines; if the shadow of the point where the 
 shadow of one of the lines crosses the other line, if they 
 are not interecting lines. If, therefore, the shadows of 
 tvJO lines intersect or come tangent in a point, the position 
 of the point of tangency or the point of intersection of 
 the tvjo lines casting the shadow may be determined by trac- 
 ing back along the ray to the line in question. Evidently 
 ease of construction and the necessity of accurate drawing 
 malce it necessary to choose auxiliary lines -whose shadows 
 may be easily and accurately determined. 
 
 26.' THE SHADES ON AND SHADOVIS OF CONES. 
 
 The Conical forms ordinarily met with in architectural work 
 are those -with vertical axis--as for example, the roof of a cylind- 
 rical tower, the lower part of a wall lamp, etc. The d iscussion to 
 follow -will be limited, therefore, to upright cones. 
 
;'" " ' t -L. 
 
 ^-^n •:/.-; ,_f.-,-. .-, 
 
 
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 >'''. 
 
 I 'J 
 
 
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 ■i'ff 
 
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 ■ / ^ 
 
 -' :^0 J, ;./. 
 
 ■..( + 
 
 -' '• 
 
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 -*' ' ■■ v; 
 
 - - »'..■■: 
 
 i 
 
 V . -^ >- 
 
 ■■:■' 'ilO ^v.,;' 
 
 '- :. . 
 
 - 'vVti 
 
 ■'■■-■ r I r'i ~ 
 
 ■^■^ V*r:, 
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 >^:J 
 
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 T'.,; ■ V -ifr, I 
 
 ■f.t 
 
 "'.f*'- - ;};^',; . 
 
 ' I. 
 
 * it . ; 
 
 '•i'JOj 
 
16. 
 
 The surface or c. cone cannot be represented in any one of the 
 three planes of projection "b;' lines. Therefore, the shade lines 
 cannot bo found by the processes applied to the cylinder. Prom 
 inspection it can be seen that the sha,de lines v/ill be forned by 
 planes of rays tangent to the surface of the cone. Tliey will be 
 strai^jht lines passing ;^hrou£h the apex and crossing the line of 
 the base. Since in projection, the shade of the apex vrill coincide 
 v/ith the projections of the apex, then, in order to find the pro- 
 jection of the shade lines it" becomes necessary to determine only 
 poj,nts at ;7hich the shade lines 
 
 cross the projection of the base. To secure those points, the 
 method -of auxiliary shado•■^rs is used. 
 
 The outline shado" of the cone on any plane v;ill be determined 
 by shc.doi^s of the base, the apex ?\nd the shade line. If, therefore, 
 as in Pia. 33, the shadovr of the apex and the shaov; of the base be 
 cast independently upon any given plane, 'then the shadov/s of the 
 shade lines riust he Als,Bls, -.nd Als,Cls. t'oints 3Is, and Cls, 
 com?aon to the shadovrs of the shade line and the shadow of the ba,se, 
 nwxstl^9 the shado'.vs of points coTL--on to the shade line and base, 
 namely, the shadows of points common to the shade where the shade 
 lines cross the base line. 3y passing back from 3ig^ and Cig along 
 rays of light to the base ftin^be found points D and C> and thas 
 shade lines A h and A C constructed. 
 
 In finding the points 3 and C geometrically some method that 
 ■'/ill give absolute accuracy must be used. Therefore, to find the 
 points 3 and C in elevation the shado'vs of the cone on either a 
 horizontal plane thron'^h Its base, or on a 45 degree vertical plane 
 through its a;:is arc used as auxiliaries. Figs. 35 a.nd 36 illustrate; 
 the application of the method vfherein is used as an auxiliary, the 
 shadov,' on the horizontal plane. (Pig. 35) In the geometrical con- 
 struction shovna in Fig, 36, the shadow of the cone Bg^* -^29' ^2,, i^ 
 simply, for the sake of convenience superposed 6n the elevation of 
 the cone and points Bl and Cx thus deter:;iined. 
 
 In Fig. 38 J the points Ei and Pi are first determined by cast- 
 ing the shadovr of the cone oh the 45 degree auxiliary plane, then 
 since the shadovf A1,E1 III PI is that of the base of the cone on the 
 45/- auxiliary plane, then from El and PI in elevation, rays are 
 passed at 45',.. to cut the elevation of the base, p.nCi thereby determine 
 the position of the points 3i and Ci. Tlie shade lines AiBl and AiC]_ 
 can then be drav/n. 
 
 27. COKES ^"ITHOUT V-ISI3LE SIlADE LIUES . 
 
 If, as in Pig, 39, the profile lines of the cone make v/iththe 
 hoiPizontal an angle of 45%, then there is in front elevation no 
 visible shade line. Tiie shade lines in plan are, however, as sho".vn. 
 If, as ■ x *J'''. in Fig, 40, the profile lines make an angle of <t> 
 or less tha A (J) -vith the horizontal, then the cone has no shade 
 line in plaiior elevation. 
 
 I f 
 
.,.,. p 
 
17. 
 
 28. THE SHADOVfS OF CONES. 
 
 The outline of the shadow on any surface can be secured by cas 
 onto that surface the shadows of the apex, of the shade lines am 
 the profile of the base. If the surface on which the shadow fallf^ 
 a plane, the shadow of the axis of the cone should be first detei - 
 mined I then on the shadow line can be determined the shadow of the 
 apex then the shadow of the base. Through the shadow of the apex 
 straight lines drawn tangent to the shadow of the base will complete 
 the shadow of the cone. 
 
 xn£ 
 
 of 
 
 be 
 
 29. THE SHADES AND SHADOVfS OF S PHERES. 
 
 The shade lines of a s^ihere is evidently a great circle of the 
 sphere and is symmetrical about the two forty-five degree axes in 
 the plane and in elevation. It is readily understood that ' the point 
 vvhere the shade lineof any double curved surface of revolution toucho? 
 the contour lines of the surface in plan or elevation is of course t|, 
 be found at the point of the contour at which the plan or elevation 
 of a ray is tangent to it. 
 
 Therefore, Eig.41b, to find on the plan the points where the 
 shade lines come tangent to the plan of the contour draw the rays R2, 
 tangent to the plan of the sphere. The points thus found evidently 
 give, (Fig, 41a), on the elevation of the equator the points of shade 
 Ci&Di, and by symmetry the points Ei and Fi- Points Ai and B]_ are 
 determined by drawing rays HI tangent to the elevation of the sphere. 
 Thus are determ.ined six points on the shade line. To determine the 
 points xhich will give the length of the minor axis of the ellipse, 
 two equilateral triangles with their apexes at Al and 31 are construct 
 ed, and points Gl and HI thus determined. Through these eight points 
 the ellipse of shade can then be drawn. The short construction for 
 the points requil;'ed is shov/n in Pig, 41c. The outline of the shadow 
 of the ellipse on any surface will 'be the shadow of the shade line 
 of the sphere, and sufficient points could be determined by direct 
 projection from plan and elevation to fix the shape of the curve, 
 The shadow on a vertical or horizontal plane, hov/ever, is generally 
 found by means of the construction shown in Fig. 41d, and is 
 sufficiently accurate provided the shape of the ellipse of shadow 
 is drawn approximately as shown on the figure. 
 
 Many mistakes are made by beginners in determining for instance 
 the line of shade on a dome, v/hich is of course only part of a sphere. 
 The visible shade line can be determined accurately only by completing- 
 the snhere. 
 
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 T:/. 
 
 ' X ' ■ 
 
 ^o'lxr 
 
 i:ss- 
 
 J ■'" 
 
 -.roi- 
 
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 ■: .•) 
 
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 3^' 
 
 yt 
 
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18, 
 
 30. raE SLICING HETHOD. 
 
 In this method the object casting shadovr and the object receiv- 
 ing the shadov/ are first cut through b^' a vertical planes pari^allel 
 to the rays of the light. The elevations of the lines of intersection 
 of the planes v;ith the' surfaces in question are then determined and 
 then points of shade and shadoxv are determined on these slices. 
 
 The process is explained by Pig, 42, the plan of the obj'^ct 
 is shovrn at (a), Tlie plan of a vertical plane at 45;/!' passed through 
 the object nould be represented by the line (1). The elevation of 
 the line of intersection of that plane viith the surface given would 
 be the dotted line (l!*, Tlie shadow on the scotia ?70uld be cast by thp 
 circular edge x y. The shadow of point Bl vfould evidently lie at the 
 point Bis '-fbere the elevation of ray through (3i) strikes the line 
 of intersection (1). Plane (2) vfould give intersection (2), and the 
 elevation of a ray through CI v;ould give the shadovf C]_g, By thB 
 use of a sufficiently large number of accurately constructed slices 
 enough points of shadow ■m,?.y be determined to fix the curve of the r. 
 shadov/ line. 
 
 Because it is a quite easily understood, this method of determn - 
 ing shade and shadov.r is apt to be abused by the beginner. Firstly, i .. 
 must be distinctly remembered that the accurate construction of each 
 slice is ", slow, tedious process and that the line of. slicing when 
 determined v/ill be sufficient to give but one point of shade or 
 shadow and, secondly, that a point of shade determined by drawing a 
 ray tangent to such a slice line v;ill not be accurate in its position. 
 Vihile the position of the slicing planes may be chosen so as to give 
 the most easy construction and valuable results, yet the slicing 
 method is one to be used only v/hen noj. other xzan be supplied. It 
 cannot be used '^.t .small scale. The shapes of shadow and shade lines 
 determined by this method must be so determined at a scale that will 
 allov; of accurate construction and -nay then be copied at small scale 
 by proportion. 
 
 3y means of this method c"n be determined the shades and shadows 
 of scotias, and those on the more co"nplicated vase forms. However, 
 those shadov/s en often be m.ore ' accurately determined by the use 
 of the 45degree auxiliary plane. 
 
 31. THE SHADOF OP A CIRCULAR IIICIIE 
 
 v:iT:ii:^ & spherical head 
 
 The outline of the shadow will be formed by the shadow of the 
 circular outline of th.e head of the niche -^.nc. by' the shadov; of the 
 strai. ht li:..:..' •■'■<:.-.h rsp/'csents the left hand side of the niche. Tlie 
 lines casting the shadow can be represented in plan by the line 
 
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 C r 
 
 
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 ^ -rx 
 
 ■ • f : 
 
 -r I I ! 
 
 
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 =:!:•.(: ^ 
 
 
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 ^. .-f ... , ,-. r^ ■ 
 
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 M-r,- vi^. 
 
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19. . 
 
 A2 C2 B2, Fig. 4 6. The Portion of the niche in elevation up to 
 line Al 01 Bl can be represented in plan by a semicircle, i-'oints of 
 sbadov; Als Els Dls can be found by direct projection from the plan 
 and elevation. The curve of shadov; v;ill evidently start at Als and 
 end at Gls vrhere the ray R]_ comes tangent to the elevation of the 
 profile. To find points of shadov/ on the spherical head of the niche 
 auxiliary vertical planes such as x and y are passed through the 
 surface of the niche and the elevation of the lines of their inter- 
 section viith the surface of the niche found. On these planes are 
 then cast the shadow of the semicircular outline of the head of the 
 niche and the points Fls and Gls, thus Pis and Gls are points of 
 shadov/ of the semicircular head <£ the niche because they are points 
 common to the surface in question and to the shadov; of the semicir- 
 cular head. 
 
 At small scale the points of shadov/ to be determ.ined are /Is, 
 Dls, and Cls. Als and Cls are easily determined, and Dls may be 
 placed on the elevation of a ray through On a distance from 01 
 equal to l/3 of the radius of the niche. (Pig,47). Tne shadov/ line 
 must not cross the outline at Cls but must be tangent to the outline 
 at that point. 
 
 32. THE SHADES ON AW THE SHADOV'S OP HORIZONTAL TORI . 
 
 As in the case v/ith all double curved surfaces of revolution, 
 the shade line v/ill be symmetrical in plan about the two 45 degree 
 axes. If there can be d etermined accurately several points on that 
 ±Lade line, then same can be drav/n v/ith accuracy through the points 
 thus determined. 
 
 33. THE 5ADE LINE ON i TORUS . 
 
 From the theorem that the point v/here the §i ade line of any 
 d ouble curved surface of revolution touches the contour line of the 
 surface in plan or elevation, is found at the point of the contour 
 at v/hich the plan or elevation of a ray is tangent to the contour, 
 there can be determined the position in elevation cf the 'ooints El 
 Al Pig. 4 3, By symmetrical construction can be found the ooints 
 CI, HI. Prom the plan bj; drav/ing rays tangent to the plan of the 
 contour, vhich is the equatorial circle of the torus, can be located 
 points Gl and Dl. If nov/, there c^n be c etermined the hir'.ir.st znd 
 lov/est points of ±i ade then the shade line can be constructed -/ith 
 sufficient accuracy , It is evident that the lovest and the hi^hect 
 points v/ ill lie in plan somev/here on the axis of the torus that is 
 parallel to the rays of light and that point v/ill be v/here the ray 
 cf light stri>:es the surface of the torus at the angle 0. If the 
 torus be then revolved about its vertical axis unt?.l the point 
 comes into the vertical plane of projection then can be determined 
 the horizontal element on "vhich the point lie s. It lies on the 
 45 degree axis in plan. 3y finding the plan or 
 
+ ? ' 
 
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 ■'•■ 3 
 
 
 f*V 
 
 V ■■ 
 
 Ai 
 
20. 
 
 horizontal elements it deternines the position of the points 
 desired, namely, Bl and Fl. The construction is shown in Fig. 43 
 
 34. ^tie SHADprr OF A TORUS . 
 
 The outline of the shadov; of a torus is the shadow of its 
 shade line, and can be cast on any surface by the direct projection 
 of points in plan and elevation. Hov;ever, as shov/n in Fig. 44, the 
 shador line so nearly corresponds to shadov/ of the equatorial circle 
 of the torus that at small scale the shadovj of the equatorial circle 
 can be used as the shadov/ of the torus. 
 
 55. THT OVAL CURVE _0F THE TORUS . 
 
 The Shadovr of a torus on the 45 degree auxiliary plane is 
 found by casting on that plane the shadovjs of various horizontal 
 elements of the torus and dravjing the envelope of those shadows. The 
 construction is shovm in Fig. 45. This ±i ad ovx is used frequently as 
 an auxiliary, particularly in the find ing of the shadov/s of straight 
 lines and circles on the surface of tori. 
 
.'],T 'Vr, 
 
 r ■ rs' 
 
 rx 
 
 'So-i., 
 
 ■' '<■ y 'r 
 
 
 r:. r~ 
 
 ::r^r.,-..- 
 
 '•* '•'^- '- ?i' 
 
THE: "Rays 
 Op Light. 
 
 YlGUi;>£: Z 
 ThEtCdmvention 
 PlKECTION 
 
 It If: Ray ^ 
 
 LlCMT 
 
 6HADKand5HAD0W5 
 PLATt I. 
 
 L= Lateral Rav^^. 
 
 T=TANCEMy f^AY& 
 
 T = Incident Ray5 
 
 LE-TABCDH-IGHBEACUBL 
 AG--PlAGDNALOfCuBIL- 
 
 R= Ray 0f Light Paralul 
 
 To Direction c7Piagonal 
 I^rPROJECTioN Of Ray On V 
 
 Rj\NL Or Elevation Or \^av. 
 R/P^ojictionOfRavOu H 
 
 Flake Or Plan Of ]^AY 
 1^= PROJECTION Of Pay On P 
 
 FUkie0r5ideIllvatio\^0f!^,y 
 / AGC True AncleOpRa^ 
 
 Oi^ Angle 'I'. 
 
 Geometric CoN5TeucTioN Fo^ Angl£<^ 
 
 F\GUR.& 5 
 ThErTHREtPROJECTIONS OfRaY R. 
 
 Side 
 
 fpONT 
 
 Plan 
 
If 
 
 FiGURI: 4. 
 PLANE 0PRAY5- 
 . DEinNiyioN! 
 
 5HADI:5 8 SHADOWS 
 
 FiGURfc 5- 
 PO\NT<5 OMOSS- 
 R)iNTp a b.C.d ore- POINTO Of L055. 
 
 NVI5IBLE Shadow 
 
 ^'*^ V15IBLE Shadow 
 
 TlGURt 6 . 
 
 lNVI5lBLt AND VISIBLE SHADOWS. 
 DtFiNlJION" 
 
 FIGURE: 7- 
 
 flGUR.E: 8 
 INX SHADOW Of ABFALL5 ON PLAMt 
 IN "5 SHADOW 0f-AB?ALL5 PARTLY OM 
 GROUND .PARTLY ON CURVEP 5URFAC& « 
 

Figure 9 
 
 6HADE5 ^ SHADOW. 
 Plate IE 
 
 
 DXYC 15 AN Imaginary T 
 
 Of THtTOPO?THt box- y , i .^ -^i. 
 
 Imagina^^Y IinE:-Thl5hadgw 0? 
 
 THE Cube May BE BEi'Ond LINE LK 
 
 AND THE: PORTION mOND WOULD 
 
 BEIMACINARV. 
 
 \ \ 
 
 flCUR.E 12 
 
 • •^ 
 
 FlGUl^tlO 
 
<5hADL§?< -Shadows 
 
 Plate E 
 
 THE.6hAD0v/5 Of 
 Bl) Jtf Toyi/the 
 
 PhieSPECTIVEr SKETCH 
 APPOINT lN5i^ACL 
 
 IhE Shadow Of Anv Point 
 
 f IGUCt 14 
 Shadows Of Lines P.aralell TctL£vATioNR.ANt 
 
 F)GUR.t 15 
 
 Shadow Of A Imt PECPtNOicuLAR TotuEVATioN Plane. 
 
i 
 
6HADE5 & 5HAP0W5 
 PlateV 
 
 p3iDt Elevation 
 
 F"TJ15P 16 
 
 ihE-OHAD0Vv5 Uf-M LiNL I^El^PtNDICULAli'IbPLAN 
 
 Plane On PLAN?LANt, 
 
 A^ 
 
 
 
 
 
 As' 
 
 \ 
 
 -> 
 
 
 / 
 
 
 / 
 
 K 
 
 
 i^ 
 
 ^35 
 
 Figure 17 
 
 Shadow Of VEBTicALLiNt On iNCUNLDR-ANfc 
 
 1 I M 
 
 S-;^---^-Ki^^ 
 
 % -^- 
 
 \ 
 
 
 te- 
 
 :^ 
 
 PT 
 
 ^ 
 
 az". 
 
 
 ?IGUl^El8 
 
 Shadow Of Vleticai- Line On Cecils 
 
 Of HORlZOtHyAl- M0UL-DINC5. 
 
i 
 
X 
 
 CHADE5?)f5HAD0W5 
 
 Plate ?I 
 
 5ij J ft- Fo3<fythe 
 
 X. 
 
 / ^r-P^A 
 
 Figure I9fd) 
 
 The Shadow Of ALinl Pacallll To Elevation 
 And Plan R ake5 On APla^E At 45'' 
 
 
 B„ 
 
 \^' 
 
 -r 
 
 H 
 
 X 
 
 \. 
 
 .B.. 
 
 A,., 
 
 T h 
 
 FlGWl^E: 19(b) 
 
 The Shadow OrALiNfcPiiePE.NDicuLAi3.lb Elevation 
 
 PLANt On A Plane At 45° 
 
 ^\ 
 
 y 
 
 /L 
 
 flGUREZO 
 
 \ 
 
 — jX— 
 
 tLtVATiOS 
 
 ^^ 
 
 FiGUStZl 1 
 
 Persplctwe 
 
 The: OH APES AND Shadows Of 
 AHoRizoNjAiIiRO'jpOf Mouldings . 
 
IF 
 
 Distance of O in. 
 front- of 'V . 
 
 V 
 
 Plate W ' 
 
 V 
 
 jf^M^M' 
 
 
 \ \ 
 
 CrUCLE In fLANE-pARALLELTb^^LANE 
 
 On WTtrctt 3ftADc^x^ f^ cast 
 
 A,fi.B. 
 
 f.^^kf'^. 
 
 C,fl.D>. 
 
 flGUU£.22 
 
 ClliCLE ;Nl>LANEpE£P£NDrCl7LAQ. lb PLANE 
 
 On WffTat,5fiADow b Cast. 
 
 \ 
 
 Di5T Of OIw Promt O PLAKfi. 
 
 «.r s^ rn 
 
 -f-^ 
 
 1 ^\ 1 
 
 ^ 
 
 
 ll>. 
 
 
 
 PGUI^E 25(a) 
 vShadowOpDiagowl 
 
 AD ISAfta^UONTAL UNC 
 
 Distance of Ca 
 frrnr of 
 
 CJiRcLE Jv Plane PoiPENDia/LAH. To Both t, 
 Plan s^ ^-""--^ Elevation 
 
 ^TH^E>7>ivt iVj<L^ ,5radowOeDiaoojja:. , 
 1 \ "^ I £;qUALTo PiAMETER. 
 
 PlGf,niE.23Cc) 
 To Find 5tf?=^ ■.'.- ;f Point r; 
 Ust Center 0.. AndWi l'5 
 
 0,sC,s TrtTiU"N,,lHU5roLij: 
 J)TiAv;- J-tADOw jFCniCLTM:- 
 
 3a2;BEi).5gtJA^E. 
 
 >kp rlGUCE2.5it: Al^ .6jat?TtFf.c5 
 
 J OTCLEIN PLANE P£1^PENDIC0LA^. "fcLETTtRSiN »=;& 
 
 ro flEWriON ^PAliALLEi. To PLAN 2.3(c^ . 
 
 A' 
 
 fLwt-X. i 
 
 Plan 
 
 \ r b; 
 
 FrGTn^25. 
 
 /Ci» pj ^ 
 
 F[GUUE24 
 SHADOW Of Circle On Incuned Plan& 
 
 NO 
 
 ^J 
 
f.» 
 
4L4e 
 
 -H 
 
 1 
 
 
 }rz~7 — 
 
 1~y Di^tande from 
 
 c.k- 
 
 I 
 
 &1. 
 
 ^ax,iy 0|T cylinder 
 |j^ "^^ fo VI p. jrze . 
 
 kSraDE^vS ScvStMDOWvS 
 
 
 I 7t 
 
 Cylinder 1 vV \ i ^i/ 
 
 j£»Y Jtt fbr>/-Y^^^ 
 
 FIGURE: Z/. 
 
 P.IGURB 25 
 
 fr^HP 
 
 Kail 
 
 A:f^%. ^--' 
 
 PieuRB 29 
 
 W&ll Line? 
 
 1 — [ h 
 
 yi 
 
 L 
 
 ^fiAi:)EK3 &r ^fW)ows In Hollow 
 Cylindesc^s. 
 
 FiGUUEt 30 
 
 FlGUREv5 
 
 •> 
 
 / 
 
 \ 
 
 \ 
 
 ^. 
 
 \ 
 
 \ 
 
 ^ 
 
 
 f^rcuREi 32 
 
ftiGURa. 55 
 
 vSrTADP.^ ^K^UAhOWS 
 
 Plat^K 
 
 Bv JtiPbr^vtiie. 
 
 PlGURB 34. 
 
 \^Prauua 55 
 
 Shade i-ine*-^ 
 
 // »SttADELrNtONA 
 
 Con&ByUselOp 
 
 >5llADOW* On H PLAMEi. 
 
 Figure 56 
 
 GaOMtTftlCAL 
 
 CON»STUUCTI0N 
 
 b Figure 59 
 
 FlGt/BE 5b i 
 
 Geombti^jcal 
 CorisTUucTroN 
 
 3ftAD£LlN£ Q\iK 
 Cone By Dse Of- 
 SffAPOv^^ On 4-5° 
 AuxiLiAiiY Plane 
 
 riGURP 57 
 
 FlGUI^ 40 
 
I 
 
flGLT^Ei 41(a) 
 
 DiVtiJtnce of Center 
 \o>- sSphers from Plane 
 
 K5!mm.$ bc^Stm^ows 
 
 
 Of- »5phere,. 
 
 PiGUREi tb) 
 
 The Points 
 iN Plan 
 
 f IGURE- 41 (c) 
 
 Th£ ii5HADE.»5 And 
 JiiADOw^ At 
 •Small Scalel 
 
 Plan Op vSphereT]''^ 
 
 riQL>B& C4l (d) 
 v5ttAD ow On V Plane:, 
 
 a 4. 
 
 9 
 
 h 
 
 f IGURfi 42 (b) 
 
 1 1 rt'H ' ^ f ^ :" - v. '~ - '^ - ~S 
 
 A. .. 1/ / / / 
 
 X 
 
 x^ 
 
 y ^^ , yp, ^. 4^ 
 
 X. 
 
 Tne vSujiNG M&Tttop 
 
 ©^"^ 
 
"SfE iShadeLine On ATo^u.5 _ 
 
 The Tctru^ = n^Shadcw 
 Op Tovivs On\43° 
 
 ^on^frvjc iota for ^rue 
 OQcilffi cfT rt 
 
 Center linf ofi toiuw «P,® © 
 
 T^ic equatond circle of +h.c forvkT 
 
 XI liar Y Qrcley* 
 
 oY Auxiliury 
 
 Circles 1,2,5,4, etc 
 la^S" plane. 
 
 filevotioa of y"b<adow 
 ineon. 45° plane. 
 
 v5mDow Of Tti 
 Eqimtorial 
 
 *3ftADOW Of ToRUvS 
 
 kSecussd I)y 
 Auxiliary c 
 
 ;:^,_,S^.^^ J>L^ 
 
Of THE 
 —r/ Of iLLiWOIS 
 
}r 
 
 
 \ \ Dh^dow or- 
 
 j|f Shac oa o-f O, or plane Y 
 
 hau=|ow o-^ O, on. p [oifze X 
 
 AT C)N]ALL DC^Ltr 
 
 PL>qA|E-X>[ 
 
 Veiltic/^L- 
 
 Fl^m of- Nicnt 
 
 Fjgur^e- 47 
 
 The 5h/)DOw:5 
 
 OM THE-CoRi^- 
 THMAI COLUM/^. 
 
C. .... 
 l";VERSiTY Of iLLINOI? 
 
Y^\j^jijji»''" 
 
■■■^Sh ^^. 
 
 'i/^ 
 
 .« ;^ 
 
;j!i*' 1 
 
 •^•ji. 
 
 '^^ 
 
 University of illinois-ubbana 
 
 Q516 61L6S COOl 
 
 Slides ar>iish.<loW5 nol.s .rr.ngeil lor 
 
 3 0112 089506593