PRO

335

PRO

*Kl tO A

t ay ) | c > and join g b ; and G b will represent the sun’sT| lia plane of which the intersection is d b.

(] P r °Mem is necessary in the three preceding examples,^ 0 ‘i 3 u„ le . seat; and inclination of the sunboi

on the

5u . _ _ — __is given

Jotl, h| ’ * n . or< Jer to find the orthography of the ray upon1 V,u Vati ° n3 ’ v ’ z -> u P on the front and end.

°n a , Rl1 XX YI .—To find the representation of the shadow'Kl ur niatic solid, by a plane cutting the prism perpen-Pla.n e to * (s arrises: given the elevation of the prism on aQs a bto *'** arrises, and parallel to one of its sides :

A e *ec« le fis ure of the plane which throws the shadow, with^ e *»£se °? surfaces of the prism on that plane, and

Jl. f e s un's rays, both on the plane and on the elevation.

O^ti.e- ■

tW f . t! ‘eir

figure of the plane and the elevation so that anycorresponding points, or parts, may be in a lineDraw the representation

°fth ft n ^' cu ^ nr to the intersection.

SZ from all the angular points on the figure of the

ip 0ll( jj^ Inee t the sectional line of the prism ; draw the cor-

Ir 0 Q, ti, e ° f e P r esentations of the rays on the elevation, and^ U, fc , Points where the representations from the angles^tati!; n U,le meet the section, draw lines parallel to the repre-ss 0J1 ,? the arrises of the prism to cut the corresponding8l, a j le elevation ; then, by joining the successive points,fequired will be determined.

^G, t] 1( . Pff 1- Figure 2.— Let a bcd be the plane, and e f,

* 6 Cn .“{‘erseetions of two otlier planes at right angles tou et K l n o be the elevation of the prism ; the lines

‘bsu^hand :

h . a to KE > representing the arrises corresponding to"'at o j s a . r P°ints, g, f, e, on the section, — that is to say, theA a , 1( j 111 the same straight line with on, the point f withtl Qe Par if P°' nt E w 'th k L; also, let klo represent theo'® lin e \ e ! to the plane of elevation, and let k p representshadg hitersection ; make K U equal to the breadth of* Hie p] a ' V a t the bottom ; draw the ray, d i, on the figure

Will :

1 an d d t perpendicular to k p, cutting kp at t ;

: e l«vnt; re P resen t the point which throws the shadow onvw t Q, the ray of the sun, on the elevation ;and u S parallel to K P, cutting k m at s ;

JOi,,' -uei t0DT

gl 0 Q • 4.1

■No* ’ len K u s Q v o will be the representation of the

s £*o% eq " ired -

' a ft Upon f '—Figure 3 shows the shadow of a cliimney-^allo] a , e r °of of a house, the sides of the shaft beingo >er P enc *‘ cu ' ar t0 [ h e front.

(pit o —Figure 4 shows the shadow of a chimney-

0 f S 8 h a f t , a .roof, which has two inclinations; the sides ofPerf 0rn) . e,n o situate as in Example 2. The principles^dim, these two examples are the same as in thep e is Lc f q6m > Example 2.

i>n dlcu ) -XX VII.— To find the indefinite shadow of a lines,^! n aei 0n a ^J 0 , ^ le horizon on an inclined plane: given the

hff ruy s the plane to the horizon, and the seat of thebe{ n t le intersection of the inclined plane with theftion V Parallel to the intersection of the plane of repre-«<e l ^ e horizon.

• Figure 2, No. 3.—Draw any straight line,

Xl.

^ake

A B to a c, as the sine is to the cosine of the

i n( e 0 f 'Ration ; make the angle, cad, equal to that ofpotion 6 sun’s rays, with a line perpendicular to the°^the horizon and the plane of representation ;> l ’ ri< iefii, 4 < ! r P >ent ^r c ula r to A C, and join B d ; and b d will be^^Presentation required,

f^^XXVIII.— To find the shadow of a plane recti-hjfvh u,h- e , intersecting another plane figure: a pointl\? es th e h f ^ te s i ia< i° w i s to pass on the plane whichfinadow, and the orthography of the rays of the^ ffiven.

c ' w hich throws the shadow meets the intersection

of the planes, draw a straight line from the point of concoursethrough the given point, and the line thus drawn will be theindefinite representation of the shadow. But if the linewhich throws the shadow does not meet the intersection ofthe planes, continue the line or the intersection of the plane,or both of them, as the case may require, to meet each other,and draw a straight line from the point of concourse throughthe given point, as before; and the line thus drawn will bethe shadow required.

Problem XXIX. — Given the orthography of a deter-minate line, and the indefinite representation of its shadow,also, the orthography of the suits rays, to find the limits ofthe shadow.

From one, or either of the terminations of the line, as thecase may require, draw the orthography of a ray to cut ormeet the indefinite shadow of the line, and the length com-prehended between the point, or points, thus found in theshadow, will be the extremity, or will terminate the length,as required.

Example 1.— To find the shadow of a chimney-shaft onthe roof of a house.

Plate XI. Figure 1.—Find the indefinite shadow of thevertical line, f L, No. 2, thus : let A c, No. 3, be parallelto f l : find b d, as in Problem XXYII : draw f o, No. 2,parallel to b d, No. 3 : now let the parallel of the orthographyof the sun’s rays be given : from l draw l O, the orthographyof the sun’s rays, and the point o is the termination of theshadow, f O ; produce F G and l m to meet each other in N ;draw o n, and it will give an indefinite representation of theshadow of the edge, l m, of the top ; draw the orthography,mq, of the ray, parallel to l o, cutting ok at Q; continuethe intersection, g h, and the edge, m l, of the top, to meeteach other in p; drawQP, and it will give the indefiniterepresentation of the edge of the top, and of the other verticalplane not seen ; draw h r parallel to f o, and foqkh willrepresent the shadow required.

Example 2.—Figure 2, is the shadow of a dormer windowwith a rectangular front, and the top inclined towards thefront: find the indefinite shadow, ag, as in the last example,and the orthography, e G, of the sun’s rays; join b g, and A G bwill represent the shadow, as required.

Example 3 .—Figure 3, is the shadow of a dormer window,with a pediment top : find a g, the shadow of A e, as before;then d e being the next line that projects a shadow, A b isthe intersection of the plane, abode; therefore, produce d eand b A to meet in F ; join F G, which produce to h ; find thetermination, h, by the ray, d h ; and A G h d E will repre-sent the shadow, as required.

Note .—If tiie lines d e, e g, had been in the same straightline, or had formed a salient angle, instead of a re-entrantone, the upper side of the shadow would have been the sameas in the last example.

Example 4.— Figure 4, exhibits the shadow of a dormerwindow with a semicircular head: find the shadow, a i, ofthe perpendicular line a f ; let O be the centre of the semi-circular head; draw di to touch the head of the window,at d, parallel to the orthography of the sun’s rays; draw o dperpendicular to d l, and D will be the point of contact.—Now, let it be required to find the shadow of any intermediatepoint, e, of the arc of the circle: a b is the intersection ofthe plane a b c d e f ; therefore draw the chord, e f, andproduce A b and f, f to meet each other in h ; produce h ito K ; and draw the orthography, e k ; and the point k willrepresent the shadow of the point e. In like manner, drawthe chord de, and produce de and BA to meet each otherin g ; join g k ; produce GKtoi; and L will represent theshadow of the point d : draw a curve from the point i,