SHADOW. 



L U, the vanifhing line of the vertical planes B F G I and 

 D E M N of the walls, in U ; V R, the vanifhing line of the 

 gables, in R ; S L, the vanifhing line of the main roof, in Z ; 

 and T L', the vanifhing line of the pent-houfe,' in Z'. We 

 are now prepared for drawing the (hadow of the pole W X 

 upon the horizontal plane and upon the building. 



Produce A B to meet W X in X, then X will be the point 

 where the pole rells upon the ground or horizontal plane : 

 draw X Y, cutting D E in a ; draw U a, cutting D M in 

 b ; draw b R, cutting M Q in r ; draw c Z', cutting P Q in 

 d; draw U d, cutting G I at I ; and draw I Z, cutting the 

 ridge I H aty"; then y^abc def will be the whole fhadow 

 of the pole. 



For, fince the fhadow firft begins at the foot of the pole 

 or line in the plane of the horizon, and fince the interfedtion 

 of the vanifhing line of a plane on which the fhadow is to 

 be thrown, and the interfeftion of the vanifhing line of the 

 plane of fhadc, gives the vanifhing line of the fhadow upon 

 that plane ; now Y is the interfeftion of the vanifhing line 

 of the plane of fhadc with the vanifhing line of the horizon ; 

 therefore Y is the vanifhing point of the fhadow of the line 

 W X upon the plane of the horizon. The next plane on 

 which the fliadovv is thrown is D E N M ; now L U is the 

 vanifhing hne of the plane D E N M, and U is the point 

 where the vanifliing line of the plane of fhade cuts L' U ; 

 therefore U is the vanifhing point of the fhadow upon the 

 plane DEN M. The next plane on which the fhadow is 

 projefted is the plane C D M Q : now V R is the vanifhing 

 line of the plane C D M Q, and it interfefts the vanifhing 

 line of the plane of fhade in R ; therefore R is the vanifhing 

 point of the fhadow upon the plane C D M Q. The next 

 furface on which the fhadow is projefted is the plane, 

 M N P Q, of the roof of the pent-houfe : now Z ' is the 

 interfedtion of the vanifhing line of the plane of fhade 

 with the vanifhing line of the plane M N P Q ; therefore 

 Z ' is the vanifhing point of the fhadow on the plane 

 MNPQ. Tlie next furface on which the fhadow is pro- 

 jedled is the plane B F G I of the wall ; but U has already 

 been fhewn to be the vanilliing point of the fliadow. The 

 plane of the roof is the lafl furface on which the fhadow 

 I is projedled : now S L is its vanifhing line, and it meets the 

 vanifhing line of the plane of fliadc in Z, therefore Z is the 

 vanifhing point of the fhadow upon the roof. 



In carrying the fhadow of a line acrofs feveral planes, it 

 will not be furprifing if fome little inaccuracy takes place 

 from the obliquity of interfedtions : it might be a great 

 chance, that when the part of the fliadow d I, which falls 

 upon the plane B F G L, is drawn from the vanifhing point 

 U, through the point <•/, that it will meet the pole at I, as 

 it ought to do. To remedy this, begin with the fhadow I d, 

 and proceed in the reverfe order, until it meets the line W X 

 at X, which it mijfl in principle, and will not be liable to 

 vary much in pradtice. 



The points which diredl the fhadows upon the feveral 

 planes might alfo be found by the methods fhewn in the 

 article Projection'. 



The following obfervations will be ufeful in the pradlice 

 of fliadows. 



When a ftraight line that throws a fhadow is parallel to 

 the pidture, it is then reprcfented parallel to the original. 

 In this cafe it has no vaniflung point ; or, in other words, 

 the vanifhing point of the line may be faid to be at an in- 

 finite diilance : and, therefore, inflead of the vanifhing point 

 of the line being joined to the vanifhing point of the fnn's 

 rays, draw a ilraight line from the vanifhing point of the 

 fun's rays parallel to the projedtion of the line winch throws 

 fhe fhadow, which will be the vanifhing line of tlic plane of 



fhade ; and therefore the interfedtion of the vanifhing line 

 of the plane of fhade with the vanifhing line of tlie plane on 

 which the fhadow is to be thrown, will give the vanifhing 

 point of the fhadow on that plane, after the fame analogy 

 as lines which are inchned to the pidlure. This cafe is 

 fimilar to that of the fun's rays being parallel to the 

 pidlure : for here alfo the vanifhing point of the rays is at 

 an infinite diflance ; but as the plane of fhade will ftill have 

 a vanifhing line, this line will be found by drawing a ftraight 

 line through the vanifhing point of the line that throws the 

 fhadow parallel to the fun's rays, as already fhewn in a 

 former example. 



ShaJoius projeSed from a given Poirt ; as by the Light of a 

 Candle or Lamp. — It is evident, if the reprefentation of the 

 luminous point be given, and its feat upon any plane, alfo 

 the reprefentation of any point in fpace, and its reprefenta- 

 tion upon that plane, the fhadow of the point will be found 

 by drawing a ftraight line from the luminous point through 

 the point in fpace, and by drawing another ftraight line 

 from the feat of the luminous point through the feat of 

 the point in fpace ; and the interfedtion of the two lines thus 

 drawn will reprefent the fhadow of the point upon the plane. 

 But when the relation of feveral planes reprefented in a 

 pidlure, the reprefentation of the light with its feat, and 

 the reprefentation of a point in fpace with its feat, are 

 given, to projedl the fliadow of the point on the other planes, 

 other confiderations become neceflary. 



For this purpofe, let A B C D (Jig. 2.) be the infide of 

 a room, confiiting internally of the vertical planes, A H, E I, 

 F K, G C, and of the horizontal planes A E F G B and 

 D H I K C : alfo, let L be the luminous point, and M its 

 feat in the plane A E F G B. In order to form an idea of 

 the point L, in refpedl of the other planes, it is necefTary 

 to have the interfedtion of a line drawn through L, in a 

 given pofition with one of the planes. Thus, if it is known 

 that the ftraight line L a, parallel to the pidlure, cuts the 

 plane of the wall B K, in the point a ; the pofition of the 

 point L to any of the other planes may be eafily deter- 

 mined, as follows. 



Through a draw ab parallel to the vanifhing line NO, 

 of the plane B K, cutting B G, the interfeftion of the 

 planes B K and A G, in i ; through b draw b M parallel to 

 P Q, the vanifliing line of the floor, cutting A E, tlie inter- 

 fedtion of the planes A G and A H, in c ; alfo F E, the in- 

 terfedtion of the planes A G and E I, in d. Draw ce pa- 

 rallel to N O, the vanifhing line of the plane A H ; and df 

 parallel to R S, the vanifhing line of the plane E I. Then, 

 becaufe the interfedling and vanifliing lines of any plane are 

 parallel to each other, and becaufe a line drawn parallel to 

 the interfedling line is parallel to the pidlure ; therefore the 

 reprefentations of all the hncs, ab, be, or bd, ce, and ejy 

 are all parallel to the pidlure, and in a plane pafling through 

 the luminous point L. 



Given the reprefentation of any ftraight line T H, and 

 the points V and W, where the lines T V and U W, 

 drawn parallel to the pidlure and to each other, meet the 

 plane AG, whofe vanifliing line, P Q, is given, to find 

 the vanifhing point, X, of the line T U. 



Draw W V, cutting P Q, the vanifhing line of the plane 

 A G, in Y ; and draw Y X parallel to T V or U W, meet- 

 ing the line U T, produced in X, the vanifhing point re- 

 quired. 



To make this appear, it is evident thst the vanifliing line 

 of a plane pafTing through W and V, mufl alfo pafs 

 through Y ; and likewife the vanifliing line of a plane pafling 

 through UW, mud be parallel to it : wherefore Y X is 

 the vanifliing line of the plane, which paffcg through V< W. 

 ^ A 2 Now 



