SHADOW. 



Now let W Y cut the plane A X in Z ; and in this ex- 

 ample the vai ilhinjr lino. V X, is parallel to the van>(hing 

 line of the plane E I : therefore the planes, roprefenced by 

 E I and X U W Y, interfedt each other in a line parallel 

 to the pifture ; and, therefore, the reprelentatioii of fuch 

 an interfeftion is parallel to X Y, or to R S, the vaniihing 

 line of the plane E I. 



Given the vanilhing lines, A B, CD, E F, {Jig. 3.) 

 of three planes, G H I K, L M N O, and M N I Q R, 

 the common interfeftion, NO, of the planes GHIK 

 and L M N O ; alfo the interfeaions, N I and M N, 

 of the planes GHIK and L M N O, with the plane 

 M N I Q R ; the reprefentation, ab, of a line in the plane 

 L M N O ; the point of light, c ; c ci, z line parallel to 

 the pifture ; and </, the point where it interfefts the plane 

 M N I Q R : to find the fhadow of the line on the plane 

 G H I K. 



Firft, find the reprefentation of a ray of light parallel to 

 the pifture, thus : draw dc parallel to A B, cutting M N 

 at e ; draw ef parallel to E F : then if ab be not parallel 

 to ef, produce ba to /, and join fc, which is the ray 

 required. Secondly, find the vanifhing line of a plane of 

 ihade palling through the line ab, and the ray /c, thus: 

 produce a b to meet C D in D, which is the vanifhing 

 point oi ab; through D draw DF parallel to / c, and 

 D F will be the vanifhing line of the plane required. And, 

 laftly, find the (hadow of A B upon the plane GHIK, 

 thus : produce O N and ab to meet in g ; from F, through 

 g, draw the line Y hi; and from the point of light, c, draw 

 cbh and cat; then h i will be the (hadow of the line, as 

 required. 



For de being parallel to A B, the vanilhing line of the 

 plane M N I Q R, de will be parallel to the pifture ; and 

 Cnce ef is drawn parallel to E F, the vanifhing line of the 

 plane L M N O, ef will be parallel to the pidure ; and 

 becaufe ba meets ef in/, fc is a ray of light parallel to 

 the pifture, meeting the line a b ; and becaufe C D is the 

 vanifhing line of the plane L M N O, and ab is in the 

 plane LMNO, therefore the vanifhing point of a'b is in 

 C D, and confequently at D, where a b produced meets 

 C D : and becaufe D is the vanifhing point of a b, the 

 vanifhing line of the plane of fhade will pafs through D 

 parallel to fc : but F is the interfeftion of the vanifhing 

 line of the plane of fhade, with the vanifhing line E F of 

 the plane GHIK, on which the fhadow is projefted, 

 therefore F is the vanifhing point of the fhadow on the 

 plane GHIK; and becaufe g is the interfedlion of ab 

 with the plane GHIK, the fhadow wiU commence at g, 

 and confequently drawing ¥ghi gives the direflion of the 

 fhadow ; and lailly, becaufe e is the luminous point, the 

 rays cat znd cbh will terminate the fhadow. 



As D would be the vanifhing point of all lines parallel 

 to the original of a i in the plane reprefented by L M N O ; 

 and as different reprefentations could not meet the hne ef 

 in the fame point, the ray cf will have different pofi- 

 tions, and confequently D F, which is drawn parallel 

 thereto ; and as the point D is ilationary, the point F will 

 be variable. 



Given the reprefentation of three reftangular planes, 

 forming a folid angle, the reprefentation of a point of 

 light or candle, and the feat of the light on one of the 

 planes ; to find the feat of the light on the other two 

 planes. 



Let the three planes be A B C D, A B G F, A F E D, 

 (Jig. 4.) it is evident that every two adjoining planes have three 

 edges parallel to each otl)er, one common to both, which is 

 their line of concourfe j thefe edges will therefore vanifh in 



a point or be parallel to each other, according as the origi- 

 nal planes are oolique or parallel to the pitture ; let the ori- 

 ginal planes be obliquely fituated ; therefore produce the 

 fides CD, BA, GF, of the two adjoininjr planes A BCD, 

 A B G F, and they will all meet in V, their vanifhing point ; 

 alio produce the fides D E, A F, B G, of the two ad- 

 joining planes D A F E, F A B G, and they will meet in 

 W, thi'ir vanifhing point ; likewite produce the fides C B, 

 DA, E F, and they will meet in X, their vanifhing point. 



Ltt L be a luminous point, and S its leat in the plane 

 A B C D : draw S X, cutting A B in a ; draw a W, and 

 draw L X, cutting aW in S', then S' is the feat of the 

 luminous point in the plane A B G F : draw S V, cutting 

 A D in ^ ; draw b W, and L V, cutting each other in S -, 

 then S* is the feat of the luminous point in the plane 

 ADEF. 



Becaufe the plane A B C D reprefents a reftangle, and V 

 is the vanifhing point of the one fide, and X that of the other ; 

 all the lines drawn to X will reprefent right angles with the 

 lines which vanifh in V ; therefore Sa and A B reprefent a 

 right angle in the plane A B C D. For the fame realon, a S 

 reprefents a right angle in tlie plase A B G F, and fince the 

 planes A B G F and A B C D are at right angles, the angle 

 S a S' will reprefent a right angle ; and becaufe a S reore- 

 fents a perpendicular to A B, a S ' and S L will reprefent 

 parallel lines ; and fince L S ' and S a have the fame vanifh- 

 ing point X, the original cf L S' is parallel to the original 

 of S a ; but S a reprefents a perpendicular to the plane 

 A B G F, therefore L S ' alfo reprefents a perpendicular to 

 the plane A B G F ; and becaufe the point S ' is in the plane 

 A B G F, S" is the feat of the luminous point L, in the 

 plane A B G F. In the fame manner it may be fhewn that 

 S ' is the feat of the luminous point in the plane ADEF. 



Given the reprefentation c d o( 3. line perpendicular to the 

 original of the plane A B C D, and the vanifhing point W 

 of the line, and the point d, where the line meets the plane 

 A B C D, a luminous point L, with its feat S, alfo upon 

 the plane A B C D ; to find the fhadow of the line CD upon 

 the faid plane. 



Draw S<^ and L<r to meet each other in e, then de will be 

 the fhadow of the line cd, as required. In the fame manner, 

 if /^ reprefent a line perpendicular to the plane A B G F, 

 and g the point where it meets the plane A B G F, g /> will 

 be the fhadow of the line, by drawing Ly and S'g to meet 

 in /;. 



This method is general for any pofition of the original 

 planes, with refpeft to the pifture ; and this pofition of the 

 planes, in refpeCl of each other, is that which moft fre- 

 quently occurs in praftice. 



Let A B C D {Jg. 5.) be the infide of a room, fhewing 

 five fides, one, E F G H, being parallel to the pifture, and 

 the other four perpendicular to it ; C' is the centre of the 

 pifture. 



Let L be the light of a candle, S its feat upon the floor ; 

 then to find the feat of the light on all the other four fides. 

 Through S draw ab parallel to V L', the vanifhing line of 

 the horizon, cutting B F at a, and C G at b ; draw a S ' and 

 ^S' parallel to YZ, the vanifhing line of the two vertical 

 planes ; through L, the point of hght, draw S ' S ', then S ' 

 is the feat of the light in the plane A B F E, and S' the 

 feat of the light in the plane C D H G. Produce C S to 

 meet BC in c ; draw c ^ parallel to Z Y, and join dC } 

 draw S S^ parallel to Y Z ; then S^ is the feat of the light 

 in the plane A E H D ; let C S cut the line F G in ^ ; draw 

 fS* parallel to Z Y, cutting LC' in S-», then S'* will be the 

 feat of the light on the plane E F G H. Then to projedt a 

 prifm flandiag perpendicular to any of thefe planes, fuppofe 



that 



