PRO

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PRO

Problem XXIV.— Given the inclination of a line to twoplanes at right angles to each other ; to find the seat of theline on each of the planes, and the intersection of the planes.

Figure 4.—Let abc, No. 2, be the inclination, or angle,made by the line with one of the planes, and def. No. 3,the angle it makes with the other.

From the angular point, b, No. 2, take b c, of any length,and from the angular point, e, No. 3, take e f equal to bc ;in No. 2, draw c a perpendicular to a b ; and, in No. 3,draw f d perpendicular to E d; in No. 1, draw any line, G it;and through g, any point in g it, draw h i perpendicularto gk; make gh equal to the sine, df, of one of the anglesof No. 3, and G i equal to the sine, a c, of the other, No. 2 ;from h, with the radius, a b, No. 2, describe an arc, cuttingG k at k ; and join h k and k i ; then g k will be the inter-section of the planes, and it h and it i will be the seats of theline on each of them.

In the following problems, respecting shadows, two planesof projection are always supposed to be given, viz., the seat ofthe rays of light on each of the planes, and the intersectionof the planes, unless otherwise announced.

Practical examples of shadows.

Example 1.— To find the shadow of a rectangular prismattached to a wall, one of the sides of the prism coincidingwith the surface of the wall.

Figure 5. —Let h i k l m N be the plan, the straightline, H I M N, that of the wall, and I k l m that of the prism;let a b c d be the elevation of the prism, and v w the inter-section of the plane of elevation with that of the ichnography,parallel to h n, and k l in the plan.

To find the shadow of any point, b, on the elevation, cor-responding to l on the plan. —Draw the ichnography, l n, ofa ray of light; also, from b, draw B f, the elevation of theray, at any given angle with the intersection v W; join bl;draw f n parallel to b l ; and the point f will be the shadowof the point b on the elevation.

In the same manner, find e and G, the shadows of thepoints A and C; join e f and f g; and A B C G F e A will bethe shadow required.

Or, find the shadow, f, of the point b, as before; draw theindefinite representations, A e and c G, of the extreme rays ;then, because the arrises, or edges, of the prism are parallelto the wall, or to the surface on which the shadows arethrown, the representations of the edges of the shadows willbe parallel to the representation of the arrises on the planeof elevation ; therefore draw f e parallel to A b, and f gparallel to b C ; and aefgc will be the edge of the shadow,as before.

Example 2. Plate IX.—The shadow of Figure 1 is foundin the same manner as in the last example ; the body whichthrows the shadow being a prism, with its planes paralleland perpendicular to the surface on which the shadow isthrown ; AsiLO is therefore the outline of the shadowthrown by a prism, represented by f g h i on the plan, andby abcd on the elevation. The figure here introducedrepresents a cantiliver projecting from a wall.

Example 3. — To find the shadow of a cantiliver, formedas in the last example, the plane of elevation being placedfbliquely to the surface on which the shadow is thrown.

Figure 2.—Let e f i k be the representation of the surface®n which the shadow is projected, placed obliquely to theline v w, the intersection of the orthographic planes ; letf g HI be the plan of the cantiliver, and a b c d its elevation,the points A and D on the elevation corresponding with thepoint H on the plan, and the points b and c correspondingwith the point f. The end of the cantiliver is represented

the P 1 ?”:

by d ef a, and the root by c g h b ; the point I, oB poi 11being represented by g and h on the elevation, and ^ tl' ef, on the plan, by b and c on the elevation. b dra"' 6 ^shadow of any corresponding lines, F G andtiie parallel of the seat of the sun s rays on the P a .j’cul arthe shadowed surface, ek in L ; draw L m perpen '^ ^ {bev w ; from A, the corresponding point of G> draw j 0elevation of the ray ; join b m; draw g o parade tb efldraw m n parallel to a b, and n o parallel to A se ntat‘ o11hmnog will be the shadow required ; for the repr re p r es6°*of the two edges of the shadows are parallel to thetation of the two edges of the end of the can ^^ ve ^,' geC tion 0

Example 4. Figure 3 — Let a b d e be the m

any vertical surface, on which a shadow is to b e : e eti 11 i>also let b c d be the plan of a triangular prism Juftfrom that surface ; and ghif its elevation ; nn q ,shadow of the perpendicular line, as before ; aIlCl ■ . e( \. .

then will pn G be the shadow of the prism, as

Example 5. Figure 4.— G H 1 k l is the elevatioi ^ 0 [ flrectangular prisms joined to each other in t * ie - nn erpediment; ghiil is the outline, and Imn tj> e * ul .f a ce ° 9forming the tympanum ; a b e f is the plan of the eC -which the shadow is to be thrown, parallel toline, v w ; bcde is the plan of the pediment, ol p fflvV <?line C D, representing the face, is parallel to A F.hp, 1 g, and n r, the elevation of the rays ; a s °’ pe ndi c ,ray d v on the plan, meeting a f at v ; v p g P e ^ 0 f 6 jto the intersecting line, cutting the rays, HP an< jT en g 0 ?

vation, at p and G : draw p o parallel to G H 5 \

will Vio +lio rtf tlm ahnHnw. 9.S in

ke ,l :

1*

a 4

will be the extremity of the shadow, as infind the shadows from the inner edges, l m and n ^ j, r,equal to go, or h p, or I G, as each of the po' n 3> \\ e \ to ^equally distant from the surface ; draw r s P a ^ sb aand r t parallel to nm : then will srt represenfrom the under arrises of the inclined parts. ^ re® 49 ? fyeExample 6.— Figure 5, is the shadow of seV ® _ on e oflar prisms crossing each other at equal ang e ’ g t, 9 do'faces of each is parallel to the plane on whic 0 pP oS * te ( jo'' r

thrown, and also to the plane of elevation; t j, e s4) . a

parallel faces, are attached to the plane on w n r pend> c j a3 tis thrown, and, therefore, the other faces are F^ the ^to such plane. The shadows are to be foun ^ t j 10 a f >8 0 {example, viz., by drawing indefinite rays * rot ?of 0 ,^ g 0points on the elevation, and finding the ® jjuesthem; then the rest will be completed by P althe termination. „Mnna^ ar ,Je, h‘<

Example 7.— To find the shadow of a re ^ ie cen jjuSFigure 6. — Find the point, l, the shadow 0 ^j t h tb e Mthrough l draw op perpendicular to k l; them ^ iB eetl ?m 9 0of the interior circle, describe the j t j,’the ra ^ 0 ^

said inner circle in w and u ; also, from h " Q q p ; ^t c \ethe exterior circle, describe the semicircle, ^ t l> e cl i, a da^and p N parallel to l k, and these lines will eX tei'>° r 3m and n ; then m o Q p n will represent t e 0 fi

required. ttl nd^ cli la^ s .

Problem XXY.— Given the suns sea . tv) °the plane of the horizon, and the intersect 10 the j thright angles with each other, and perpendwu ^ ea cto find the representation of the sun s ray . xs eC gvertical planes. . the i nt fl > g f 9 !'e

Plate X. Figure 1.— Let AD and D B the s u ’ gleof the two vertical planes, and a b the sea to A

on the horizon ; make the angle, A B 0, e< l t0 A®’, j 0the sun’s altitude ; draw a C perpendicu^ ^perpendicular to a d : make A e equal

to AC*

W -rjo,nth e rP

which will represent the ray of the

which the intersection is a d : produce