SHADCn\^. 



iifiially added on the face of the quadrat. Its defcription 

 and life, fee under Quadrat, and Altitude. 



Shadows, The DoBrlne of, in PerfptR'ive, is the theory 

 and praftice of reprefeiiting fhadows, as projefted from a 

 given point at a finite diilance, fuch as a candle, or as pro- 

 jected from the fun, where the diftance, thougli not infinite, 

 is, for the fake of fimplicity, coiifidered as luch, in order 

 that the rays may be all parallel ; or othcrwife, for this pur- 

 pofe, the rays may be fuppofed as proceeding from all points 

 of fpace in parallel lines. 



A line of fhade is the line deprived of light by an opaque 

 point oppofed to the luminary. 



A plane of fhade is an opaque or dark plane, occafioned 

 by the privation of light from the intcrpofition of a ttraight 

 line oppofed to the luminary ; and hence it is evident, that 

 every plane of fhade will pafs through the luminary. 



To find the fhadows upon the furfaces of bodies occa- 

 fioned by the privation of the fun's rays. 



Given the vanifliiiig line of a plane, the vanilhing point 

 of the fun's rays, the vanifliing point of the feat of a ray 

 on the plane, the reprefentation of a point in fpace, and the 

 repreientation of the feat of the point in the plane whofe 

 vanifliing line is given ; to find the reprefentation of the flia- 

 dow upon the plane of the piftiire. 



Join the vanifliing point of the line to the vanifhing 

 point of lines perpendicular to the plane, whofe vanifliing 

 line is given, and you will thus obtain the vanifliing line of 

 another plane, in which is the original of the feat of tlie 

 point, and the original of the line in projeftion ; and there- 

 fore the interfeftion of the vanifhing line given of the plane 

 on which the feat of the line required to be drawn and the 

 vanifhing line found is the vanifhing point of the feat of the 

 line. Therefore, draw a (traight line through the feat of 

 the point given in projedlion to the vanifliing point found, 

 and the line thus drawn will be the whole reprefentation of 

 the feat. 



This propofition is evident, fince the vanifhing line of 

 every plane perpendicular to the plane whofe vanifliing line 

 is given, will pafs through the vanifhing point of lines per- 

 pendicular to that plane ; and fince the feat of the original 

 line, on the original of the plane given, is formed by a plane 

 paffing through the original line perpendicular to the given 

 plane interfefting therewith ; therefore the vanifliing line of 

 this perpendicukr plane will pafs through tlie va!iifliing 

 point of lines perpendicular to the original of the plane 

 given ; but when two points in a vanifliing line are given, 

 the whole of the vanifhing line is given, being the ftraight 

 line pafling through thefe points. 



A general knowledge of the fhadows of lines upon 

 planes in any pofition ought firft to be acquired ; but as 

 the relation of lines and planes to the horizon is generally 

 given, it will be neceflary to find the relation of thefe lines 

 and plane? to one another; and here it will be proper to ob- 

 ferve, that whatever be the number of planes, the vanifliing 

 point of the fun's rays will remain unchangeable, or in the 

 fame pofition in refpeft of the firll vanifiiing line, and will 

 be common to all the different planes ; but every difierent 

 plane will have its own vanifhing point for the feat of the 

 fun's rays in that plane, and that vanifliing point will 

 be in the vanifhing line of that plane. As vertical and ho- 

 rizontal planes occur moil frequently in praAice, thtl'e will 

 require particular attention. 



Given the inclination of a plane to the plane of the 

 pifture, both being perpendicular to the ori;/iiial plane, and 

 the feat and inclination of a ftraight Inn- in the plane of the 

 horizon ; to determine the vanifhing point of the feat of the 

 line on the vertical plane, and the vanifhing point of the line. 

 Voi..XXXn. 



Let the fcheme, N" i. {Plate I. Shadows, Jig. i.) repre- 

 fent the vanifiing plane, and N° 2. the pl-iie of the piAure. 

 In the vanifhing plane, N° I, let v 1 be the vanifhing 

 line, e the point of fight or place of the eye, yl B the in- 

 terfeftion of the original vertical plane, inclined to the plane 

 of the pifture in the angle A g\. Let yl D he the feat of 

 the line, as given in pofition, to the horizon : make the angle 

 D AF equal to the inclination of the line to the plane of the 

 horizon; draw Z) 7^ perpendicular to AD, and D B per- 

 pendicular to A B ; produce I) B to K ; make B K equal 

 to D F, and join A K, which is the feat of the line on the 

 vertical plane. • Draw el parallel to A B, and draw Ih per- 

 pendicular to v 1 : in V 1, make 1 m equal to 1 e, and make 

 the angle 1 m h equal to B A K, and h will be the vanifh- 

 ing point of the feat of the line. Draw e v parallel to 

 D A, and V i perpendicular to v 1 : make v n, in the vanifh- 

 ing line, equal to v e ; make the angle v n i equal to the 

 angle D A F, which the original line makes with the plane 

 of the horizon. Draw e © perpendicular to v I, meeting 

 vl in 0. 



In the plane of the pifture N" 2, let V L be the vanifh- 

 ing line anfwering to v 1, N° i : in V L make choice of any 

 convenient point, 0> for the centre of the pifture : make 

 L equal to o I, N° i, and V equal to o v, N'' i : draw 

 L H and V I perpendicular to V L, then H is the vanifh- 

 ing point of the feat of the line, and I the vanifhing point 

 of the line itfelf. 



The points H and I will be both on the fame fide of the 

 vanifhing line of the horizontal planes. 



This problem is the fame when the feat and altitude of a 

 ray of the fun arc given, and the inclination of a vertical 

 plane to the plane of the pifture ; to find the vanifhing point 

 of a ray of light, and the vanifliing point of the feat of the 

 fun's rays. 



When the fun is on the fame fide of the piftnre with the 

 fpedlalor, the vanifliing point of the feat of the rays, and 

 the vanifliing point of the rays, will be below the vanifliing 

 line V L ; but when on the other fide of the pifture, the 

 vanifhing point of the rays and the vanilhing point of their 

 feat will be above V L. 



The following problem unites that of finding the vanilh- 

 ing points of the feat of a line, and the vanifliing point of 

 the line itfelf, with the vanifhing point of the feat of the 

 fun's rays and the vanifliing point of the rays, as relatinr 

 to the plane given. 



Given the inclination of a plane to the plane of the pic- 

 ture, both being perpendicular to the original plane, the 

 feat and inclination of a llraight line, and the feat and in- 

 clination of the fun's rays, both to the plane of the horizon ; 

 to determine the vanifliing point of the feat of the fun'j 

 rays, the vanifliing point of the feat of the line on the ver- 

 tical plane, as alfo the vanifhuig point of the fun's rays and 

 vanifiiing point of the line itfelf. 



It is evident, that the vanilhing point of the feat of the 

 fun's rays, and the vanifiiing point of the feat of the line, 

 are both in the vanifhing line of the plane, which is a ftraight 

 line perpendicular to the vanifliing line of the horizon ; fince 

 the original of the feat of a ray, and the original of the 

 feat of the line, arc both in the original plane : and if the 

 line be parallel to the original ])lane, the vanilhing point of 

 the feat of the line will be in the interfeftion of the vanilh- 

 ing line of the vertical plane with thai of the horizon. 



Join V S, {fi^- 2.) and let it meet A B in / ; draw b s and 

 a S, cutting each other in c, and h c \i the fiiadow of the liue 

 required. 



For the vanifiiing point of the line that projcfls the 



fhadow and the vanilhing point of the fun'« rays, are in the 



3 A vanifhing 



