SHADOW. 



dow is the deeper. Hence, the intenfity of the (hadow is 

 meafured by the degrees of light that fpace is deprived of. 

 In reahty, the {hadow itfelf is not deeper, but it appears 

 fo, becaufe the furrounding bodies are more intenfely illu- 

 minated. 



4. If a luminous fphcre be equal to an opaque one, which 

 it illumines, the (hadow this latter projefts will be a cylin- 

 der ; and, of confequence, will be propagated ftill equal to 

 itfelf, at whatever diftance it extends ; fo that, if it be cut 

 in any place, the plane of the feftion will be a circle equal 

 to a great circle of the opaque fphere. 



J. If the luminous fphere be greater than the opaque one, 

 the Ihadow will be conical. If, therefore, the {hadow be 

 cut by a plane parallel to the bafe, the plane of the feftion 

 will be a circle ; and tliat fo much the lefs as it is at a greater 

 diftance from the bafe. 



6. If the luminous fphere be lefs than the opaque one, the 

 fliadow will be a truncated cone : confequently it grows ftill 

 wider and wider ; and therefore, if cut by a plane parallel 

 to its bafe, that plane will be a circle fo much the greater 

 as it is farther from the bafe. 



7. To find the length of the (hadow, or the axis of the 

 fhady cone, projefted by a lefs opaque fphere, illumined by 

 a larger ; the femidiameters of the two, as C G and I M, 

 [Plate XX. Optics, Jig. 1.) and the diftances between their 

 centres G M, being given : 



Draw F M parallel to C H ; then will I M = C F ; and 

 therefore F G will be the difference of the femidiameters 

 G C and I M. Confequently, as F G, the difference of the 

 femidiameters, is to G M, the diftance of the centres ; fo 

 is C F, or I M, the diameter of the opaque fphere, to M H, 

 the diftance of the vertex of the (hady cone, from the cen- 

 tre of the opaque fphere. If then, the ratio of P M to M H 

 be very fmall, fo that M H and P H do not differ very con- 

 fiderably, H M may be taken for the axis of the fhadowy 

 eone : otherwife the part P M muft be fubtrafted from it, 

 to find which, feek the arc L K, which is the meafure of 

 the angle L M K, or M H I, and this angle is one of the 

 angles of the right-angled triangle M H I, the fides of 

 ■which, M I and M H, are known ; for this, fubtrafted from 

 a quadrant, leaves the arc I Q, which is the meafure of 

 the angle IMP. Since then, in the triangle M I P, which 

 is reftangnlar at P, befides the angle I M Q, we have the 

 fide I M ; the fide M P is eafily found by plain trigono- 

 metry. 



E. g. If the femidiameter of the earth be M I =:; I ; the 

 femidiameter of the fun will be r= 117; and therefore G F 

 = 111; and of confequence M H = 217; fince then M P 

 is found by calculation to bear a very fmall ratio to M H ; 

 for the angle M I P = K M L, may be taken equal to the 

 apparent femidiameter of the fun, becaufe of the fun's 

 great diftance, and its confiderable magnitude, in propor- 

 tion to the globe M ; and therefore, MP: MI :: fine of 

 16' : radius, i.e. :: 217 : l, nearly; and as M H is about 

 217 times M I, P M may be neglefted, and P H may be 

 taken to be 217 femidiameters of the earth. See Eclipse 

 of the Moon. 



Hence, as the ratio of the diftance of the opaque body, 

 from the luminous body G M, to the length of the fhaduw 

 M H, is conltant ; if the diftance be diminilhed, the length 

 of the fhadow muft be diminifhed hkewife. ConCcquently, 

 the fhadow continually decreafes as the opaque body ap- 

 proaches the luminary. 



8. To find the length of the fhadow projefted by an 

 opaque body T S {Jig. 2.) ; the altitude of the luminary, 

 t. gr. of the fun above the horizon, viz.. the angle S V T, 

 and that of the body, being given. Since, in the reftangled 



triangle S T V, which iS reftangular at T, we have given 

 the angle V, and the fide T S ; the length of the fhadow 

 T V is had by trigonometry. 



Thus, fuppofc the altitude of the fun 37° 45', and the 

 altitude of a tower 178 feet ; T V will be found 230 feet 

 nearly. 



9. The length of the (hadow T V, and the height of the 

 opaque body T S, being given ; to find the altitude of the 

 fun above the horizon. 



Since, in the reftangled triangle STV, reftangular at 

 T, the fides T V and T S are given ; the angle V is found 

 thus : as the length of the (hadow T V, is to the altitude 

 of the opaque body T S, fo is the whole fine to the tangent 

 of the fun's altitude above the horizon. Thus, if T S be 

 30 feet, and T V 45, TVS will be found 41^ 49'. 



10. If the altitude of the luminary, e. gr. the fun above 

 the horizon TVS, be 45 ', the length of the (hadow T V 

 is equal to the height of the opaque body, the triangle in 

 this cafe being ifofceles. 



1 1. The length of the (hadows T Z and T V of the fame 

 opaque body T S, in different altitudes of the luminary, are 

 as the co-tangents of thefe altitudes. 



Hence, as the co-tangent of a greater angle is lefs than 

 that of a lefs angle ; as the luminary rifes higher, the (hadovr 

 decreafes ; whence it is, that the meridian (hadows axe longer 

 in winter than in fummer. 



12. To meafure the altitude of any objeft, e. gr. a tower 

 A B {Jig. 3.) by means of its (hadow projefted on an hori- 

 zontal plane. 



At the extremity of the (hadow of the tower C, fix a 

 ftick, and meafure the length of the (hadow A C ; fix an- 

 other ttick in the ground of a known altitude D E, and 

 meafure the length of the fhadow thereof E F. Then as 

 E F is to A C, fo is D E to A B. If, therefore, A C be 

 45 yards, E D 5 yards, and E F 7 yards ; A B will be 

 3 2 J- yards. 



13. The right (hadow is to the height of the opaque 

 body, as the cofine of the height of the luminary to the 

 fine. 



14. The altitude of the luminary being the fame in both 

 cafes, the opaque body A C {Jig. 4. ) will be to the verfed 

 fhadow A D, as the right fhadow E B to its opaque body 

 D B Hence, 1. The opaque body is to its verfed (hadow, 

 as the cofine of the altitude of the luminary to its fine ; 

 confequently the verfed (hadow A D is to its opaque body 

 A C, as the fine of the altitude of the luminary to its cofine. 



2. If D B = A C ; then will D B be a mean proportional 

 between E B and A D ; that is, the length of the opaque 

 body is a mean proportional between its right fhadow and 

 verfed fliadow, under the fame altitude of the luminary. 



3. When the angle C is 45', the fine and cofine are equal ; 

 and, therefore, the verfed (hadow is equal to the length ot 

 the opaque body. 



15. A right fine is to a verfed fine of the fame opaque 

 body, under the fame altitude of the luminary, in a dupli- 

 cate ratio of the cofine to the fine of the altitude of the 

 luminary. 



Right and verfed (hadows are of confiderable ufe in mea- 

 furing : as by their means we can coinmodioufly enough 

 meafure altitudes, both acceffible and inacceffible, and that 

 too when the body does not projeft any fhadow. The 

 right (hadows we ufe, when the fhadow does not exceed 

 the altitude ; and the verfed fhadows, when the fhadow is 

 greater than the altitude. On this footing is made an in- 

 ftrument called the quadrat, or line of Jhadows ; by mean* 

 of which the ratios of the right and verfed fhadow of any 

 object, at any altitude, are determined. This inftrument ii 



ufually 



I 



