S H A

428

SH A

because g is the intersection of a b with the plane ghxk, theshadow will commence at g, and consequently drawing f ghigives the direction of the shadow ; and, lastly, because c isthe luminous point, the rays cat and cbh will terminate theshadow.

As d would be the vanishing-point of all lines parallel tothe original of a b in the plane represented by lmso;and as different representations could not meet the lineef in the same point, the ray cf will have differentpositions, as will consequently D F, which is drawn parallelthereto; and as the point d is stationary, the point f willbe variable.

Proposition VIII.— Given the representation of three rect-angular planes, forming a solid angle, the representation ofa point of light, or candle, and the seat of the light on oneof the planes; to find the seat of the light on the other twoplanes.

Figure 4.—Let the three planes be A b c d, a b g f, a f e d.It is evident that every two adjoining planes have three edgesparallel to each other, one common to both, which is theirline of concourse; these edges will therefore vanish in a point,or be parallel to each other, according as the original planesare oblique or parallel to the picture; let the original planesbe obliquely situated; therefore produce the sides CD,ba,gf,of the two adjoining planes abcd,abgf, and they will allmeet in v, their vanishing-point; also produce the sides d e,a f, b g, of the two adjoining planes date, fabg, and theywill meet in w, their vanishing-point; likewise produce thesides c b, d a, e f, and they will meet in x, their vanishing-point.

Let i be a luminous point, and s its seat in the planeabcd : draw sx, cutting ab in a; draw a W, and drawt x, cutting a tv, in 8 1 , then S 1 is the seat of the luminous pointin the plane abgf; draw s v, cutting ad in b ; draw b w,and i v, cutting each other in s 2 , then s 2 is the seat of theluminous point in the plane adsf.

Because the plane abcd represents a rectangle, and v isthe vanishing-point of the one side, and x that of the other;all the lines drawn to x will represent right angles with thelines which vanish in v; therefore s a and a b represent aright angle in the plane abcd. For the same reason, a s 1represents a right angle in the plane abgf; and since theplanes abgf and abcd are at right angles, the angle! s a s 1 will represent a right angle; and because a s

j represents a perpendicular to a b, a s 1 and s i will represent

parallel lines ; and since i S 1 and s a have the same vanishing-point, x, the original of I s 1 is parallel to the original of sa;but s a represents a perpendicular to the plane abGe, there-fore i s 1 also represents a perpendicular to the plane abgf;and because the point s' is in the plane a b g f, s' is the seatof the luminous point l, in the plane abgf. In the samemanner it may be shown that s 2 is the seat of the luminouspoint in the plane adef.

Proposition IX.— Given the representation, c d, of a lineperpendicular to the original of the plane abcd, and thevanishing-point, w, of the line, and the point, d, where theline meets the plane abcd, a luminous point i, with its seats, also upon the plane abcd; to find the shadow of the lineC d upon the said plane.

Draw s d and i c to meet each other in e, then d e will bethe shadow of the line c d, as required. In the same manner,if fg represent a line perpendicular to the plane abgf, andg the point where it meets the plane abgf, gh willbe the shadow of the line, by drawing if and s'g tomeet in h.

This method is general for any position of the originalplanes with respect to the picture; and this position of the

freflUen^

planes in respect of each other, is that which most rr c Hoccurs in practice. .

Figure 5 .—Let A b c d be the inside of a room, s ^ t ),efive sides, one, efgh, being parallel to the picture, . tlir eother four perpendicular to it; c‘ being the centreof th e ^ o0 r;

Let l be the light of a candle, s its seat upon the ^:then, to find the seat of the light on all the other fp u ^ t (iethrough s draw a b parallel to v l 1 , the vanishing j D( j b s’horizon, cutting bf at a, and CG at b; draw <* s 0 ] a ne 9 ’parallel to Y z, the vanishing line of the two vertica P j 0 fthrough L, the point of light, draw s' S 2 , then S 1 is t 16 . t jnthe light in the plane abfi, and s 2 the seat of tl8 ( j ra w c ^the plane cbhg. Produce c s to meet b C in c > thenparallel to z y, and join d c 1 ; draw s s 3 parallel to Y "’ ut thes 3 is the seat of the light in the plane a e h d ; let ® tbe”line f G in e ; draw e s 4 parallel to z y, cutting L c 111 q+en>s 4 will be the seat of the light on the plane e f 6 &' 0 f th e3eto project a prism standing perpendicular to any ^ th eplanes, suppose that which stands on the floor: ^ g gii

seat, s, draw s i, meeting c G in o; draw or para ^ th e

and draw the ray l mr; then r will be the shado_ ^point m ; draw S h, cutting C G in p; draw ps P ara 0 f the

nnr? Hrnw T. n. f»nltir»or ns nt. . 9 ? then & 18 til© Sll^ .

and draw L n, cutting p s at s; then s is thepoint n ; also draw s g, meeting c G in q; draw Q

-

to z y, and l k, meeting qt at t; then t is the shadow^ g j |8 -point k; join rs and s t, which will complete thedow of the prism upon the floor, and on the wall* _^ g 0 „ the

The principle of finding the shadows of the P rl ® eC tiother sides, is the same, and will be obvious toThe truth of the method has already been shown.

SHAFT, (from the Saxon sceaft,) that part o ^frufwhich, in the classic examples, may be denomma e . jt 1 3turn of a conoid, situate between the base and capalso called the fust, trunk, or body of the c0 ^ uin I 1 ' ^ one*^’’ r eBy some architects, columns are diminished ti° S° nl _upwards; this occasions a very gouty npP ea ? B t j ie co"architects and builders, however, have fallen lD gtra ;gIit h atrary error, by making the sides of columns m a j, re f a ce jfrom the base to the capital. Mr. Revely, in 13 g tha 1 a .the third volume of Stewart’s Athens, expressly j g jj e dthe columns he had seen in Greece were “iin ^ str a 'S^a gentle curve. The curve is so gentle, tha ^ is aline, which is a tangent at the bottom of the ^ 0 f

parallel to the axis, but falls nearer to it at jjmini^ 11column than at the bottom. For the method 0 ^

the shaft of a column, see Column. u pon V a fr ,.

The method of drawing the shafts of colum uge f u i adin the most expeditious manner, will be a ve ytion to this article. , height V mO

Figure 1.— To represent a fluted column, ^ 0 of th ecolumn, its diameter at the bottom, and the radiameters, being given. _ . c0 lumn> P , )e

Let A b represent the height and axis ot rat jo ot aduceAB toe, make ab + bc to B C jn t ie j ; d>' a .diameter at the base, to its diameter at the c P^] eS to Jline through A, and another through B, at rig ^ gaC h siset half the diameter of the column from A . 0 parts , e i >it, and divide the whole length of this line in . pj-qi eC ij a esenting the ratio of the flutes orthographic ^ get; , e ptfrom the points of division draw lines to C, rep rer

passing through b, and the lines thus drawn e

the shaft of a column as fluted. . g, the ^

In this example ab + bc is to BC as tioie sthe point c will be found by repeating A B ( | |r e e

tion-

A to c.

Figures 1,2, 3, 4, represent a rangeof which are here supposed to be drawn j

of coin*" V fo 1 '