f

PRO

336

PRO

through K to l, touching the straight line A x and d i at thepoints i and l ; and aikid will represent the shadow, asrequired.

Example 5. — Figure 5, exhibits the shadow of a dormerwindow with a kirb-roof top : find the shadows, M, k, i, ofthe points e, f, g, as in the last example; and aikmefgawill represent the shadow required.

It is hardly possible to conceive a method more expeditious,than the foregoing, of finding the indefinite representation ofthe shadow, by producing the line which throws it, and theintersection of the plane, till they meet each other in theplane in which the shadow is thrown.

Example 6.— Plate XII. Figure 1, No. 1, is the elevationof a prismatic object, as the walls of a building, which formtwo prisms attached to each other; the lower prism is termi-nated with an inclined plane, in the manner of a roof. Theplan, No. 2, shows the end of the higher prism, to which thelower prism is attached, to be at an obtuse angle with theplane of the front.

Let the ray, b f, No. 2, be drawn on the plan from thecorner b, to meet the side CD at f ; then f is the projectedpoint of b on the plan ; draw f g, the elevation of the point f,cutting the top of the lower wall at g ; draw l m, the eleva-tion of the sun’s ray, as in the preceding examples ; producethe intersection, h i, of the inclined plane to meet the line, b l,which projects the shadow in k ; through the points, k and g,draw kg m, meeting the elevation of the sun’s rays at m.Now, as the line, l o, of the top of the end of the prism pro-jects the next shadow, produce lo, and the intersection, i h,to meet each other in n; draw m n, cutting the upper termi-nation, h q, of the inclined plane at r ; and fgmrhief willrepresent the shadow required.

Example 7, Figure 2, No. 1, is a plane figure, representingthe gable of a house, of which b a is the base; f g, g h, h u,are the intersections of several planes; f g, parallel to b c,is the intersection of a vertical plane, as the face of a wall;G h and h u are the intersections of two inclined planes, form-ing two sides of a kirb-roof. Now, to find the shadow, bythe intersecting points of the lines which project the shadowwith the plane on which the shadow is thrown, it is evidentthat the line, B C, will project a shadow on the plane, whoseintersection is f g, parallel to fg; and as b c, No. 2, repre-sents b c, No. 1, draw p q parallel to b c, cutting the lowerinclined plane at q. Then, supposing the side, b c, also tothrow a shadow upon the lower inclined plane, produce itsintersection, h g, No. 1, and the line, b c, which throws theshadow, to meet in K ; produce C b, No. 2, to k, and draw k kparallel to b a ; then through the points, k and q, draw kq r ;draw C C from C, No. 1, to c, No. 2, parallel to b a, No. 1 ;from the point, C, No. 2, draw the elevation, c r, of the sun’srays. C d, No. 1, is the next line, which throws the shadowstill upon the lower inclined plane; therefore, produce c dand g h to meet each other in M; draw m m parallel to b a,cutting bc at m; draw also the indefinite shadow, r m, cuttingthe intersection of the two inclined planes at s, No. 2 ; thenthe line, c d, will also project a shadow upon the upperinclined plane; therefore, produce h d and c d, No. 1, tomeet each other in n ; draw n n parallel to b a, cutting b cat n ; draw the indefinite shadow, s n; then d, No. 2, beingthe corresponding point of d, No. 1, draw the elevation of theray, d t, No. 2; and the point, t, No. 2, will be the shadowof d, No. 1. de, No. 1, is the next line that projectsa shadow, and u is the intersection of d e and h u, and u thecorresponding point to u, No. 1 ; therefore join t u andb p q r s t u will represent the shadow required.

Example 8, Figure 3, represents an example, similar toFigure 1, but with two inclined planes; the plan is here

! first

rtf’

omitted, as being only used in the projection of the j*‘ n y di 3 'the ray, G l, parallel to a b, is drawn at pleasure a ^ 0 t"'°tance from f h, to meet the intersection, H P> 0 j- ne aadjoining planes in l ; produce J h to meet the p twhich throws the shadow in d ; draw d l, andto s ; draw the elevation of the ray b s ; then B 0 l9 ^ t>, e !line which throws the shadow ; produce b c and cU t tb e

each other in e ; draw the indefinite shadow, S E, ^ g Cintersection, i Q, of the two inclined planes at ’ t j ]er ef° re ’will also project a shadow upon the upper plane , , #oe , to

produce b c, and the intersection, J k, of the upp e P cu 1 'meet each other in t ; draw the indefinite shadow,^^

the

ting the top, kb, at n ; then fgismbk will re P^g plaa e ’projection of the shadow by the edges a b, b C, oiA b c, as required. P A ® C ’

This figure may either represent the vertical pla® ^ cCO rd'or perpendicular to the first vertical plane, fob > 0 o>

ing as the sides are parallel or oblique to the Prepresentation. r.auve

Example 9. Eigure 4.—Let A b c d e, be the ho ^ ill®gable ; K L m n the upright of a wall ; A B the ba t {begable ; N m the base of the wall; and supp° se g0 tl» #tedge, b c, is joined to N k at any given angle, ^ jt bethe point b may coincide with the point N. ’ u pc>n ^required to find the shadow of the gable, A b C d placewall, k l m n, the side, B c, of the gable “ e ‘ ° j gide 3 >parallel to the edge, N k, of the wall: as the ver t ^ r0 rfOa e and b c, of the gable are parallel, the sba N U j

upon the plane, klmb, will also be parallel to t > ° Q jj E >therefore, taking n g at pleasure, draw g h P a ^ a , v e fa

eleva’tion^ ^ c ° rres P°nding' point of E, dra ” i. e theshadow of t) 3 ray t!le sun > and the point h : eC tS■' ■ , e P°J n t E. ed being the next line thatp ^ e

1 - - intersection

the shadow, produce e d and b c, the lntei^--^ ^ j, a

plane, to meet in f ; find f, the corresponding P° re sp on

draw the indefinite shadow, h f ; find d, the c cU tting ^point of d, and draw the elevation, d i, of thro"' 3 theat i; c being the concourse of the line w j anej find ^ jjishadow with the intersection, b C, of the P ll ’ c i g h ^corresponding point of c, and join '’.pj. f

represent the shadow of the lines A e, e d, an gabl e tExample 10. Figure 5. — A bcde represen ^ ^ u P r '| v ea building, as in the preceding example ; ® T t j, e resp eC ( jjewall and inclined side of a roof ; F G and G supP° 3 ’ ng 'redintersections of the wall and the roof ; “ ier) ’ j t j g refi" 1 , 0edge, e d, of the gable to be attached to s > p G) ot ,jjto find the form of the shadow; the intersee gjjado'^ g yplane being parallel to a b, the breadth o op ofl

also be parallel; therefore, taking the distan tl‘ e ;

at pleasure, draw n o parallel to s T; then, 0 { the eresponding point of b, draw the e ^ evatl ? n ’ g ], a do"'» P 1 ° oJ id'bc being the next line which projects tfie^ thethe intersection, f g, to meet b c in g j ia( j 0 w, 0 ’' aC liD el

point of l, and draw the indefinite 8 * n the

the

UlUYY IUC umw**--- nf>Il A

b w at p; bc now projects the shadow up ^ ^ ai> , ^plane; therefore, produce the intersecting K ; hedge, B c, that throws the shadow, to *^ e an( j p ^ r . a , 0 f C,the corresponding point of K ; through n( jj n g P 0lI) n dingindefinite shadow kp q; c being the corresp^^^ *

draw the elevation, e q, of a ray; ’ t i ie sha d

point of I ; and join i q, which cotnp‘ e e u at e

both planes. ne( { toi^ 1 % ,A th e

Problem XXX.—A cylinder being c . a P F sideS , to J 1 ’ Jo*abacus, jutting alike over each of l ie ^ va ({ 0 n an g f th erepresentation of the shadow; the e e i e vat t ° n Jbeing given, as also the plan ansun’s rays.