S TE

446

S T E

upon c, as a centre, with the distance c e, describe an arc, f;and upon a, as a centre, with the distance a b, describeanother arc, cutting the former at f ; join f a and f c i thenthe angle a f c will be the measure of the third side. Nowif the triangle abd be turned round the line a b, thetriangle c b e round c b, and the triangle acf round a c,until the points d, e, f, coincide, each of the two planes, abdand cbe, will be perpendicular to the plane A b c ; there-fore, two of the sides of a solid angle will be given, oneperpendicular to the other, to find the inclination of the ver-tical plane with that of the liypothenusal. Proceed, there-fore, as in the last problem, and find the angle g h k, whichwill be the inclination of the two planes cbe and cap. Inthe same manner may the inclination of the planes abdand A c f be found.

Note. —The triangle abc represents the spherical triangle,i of which A b and b c are the tangents of two arcs; and theangle abc is the spherical angle contained by the arcs, ofwhich A b and b c are tangents.

Problem V.— The three sides of a spherical triangle beinggiven ; to find the angles.

Figure 4. No. 1 and 2.—Make the three angles, a b c, c b e,i and e b F, equal to the three sides of the spherical triangle,that is, to contain the same number of degrees. On b, as acentre, with any radius, A b, describe an arc, a f ; draw a c| and f e tangents at A and f ; join c e ; draw the straight

| line G H equal to C e ; on the centre G, with the tangent a c,

I describe an arc at i; and on the centre h, with the tangent f e,

! describe another arc, cutting the former at i. Join g i and h i ;

j draw I k and i L perpendicular to I G and i h, making them

equal to A b or b f ; join k g and l h. Now if the triangles,

| G i K and II i l, be raised on the lines G I and n I, until the

| points k and l coincide; then each of the triangles, gik

I and hil, will be perpendicular to the triangle gih. Pro-

i ceed, therefore, as in the first Proposition , to find the angles,

i which, in the representation of the spherical triangle igh,

I are represented by G and h.

j Scholium . — Since each of the extreme angles may be made

■ the middle angle in No. 1, the triangle ghi, No. 2, may be

i laid down in three different figures, each of which will have

I as many angles included by each pair of tangents. These

| three angles, made in each separate triangle, are the measures

of the three spherical triangles; but this mode requires morelines than what are described in the above Proposition.

There is another method of finding the angles of a sphericaltriangle, when the three sides are given, pointed out byBishop Horsley, at page 215, of his Elementary Treatises.The substance of it is as follows : Draw a right angle; makeone of the legs equal to the difference of the cosines of thesides containing the required angle ; the hypothenuse beingequal to the chord of the third side. Upon the remainingperpendicular side, as a base, construct a triangle, whose twoother sides are equal to the sines of the sides containing theI ; required angle ; then the angle contained by the sines willj ! be the measure of the spherical angle. This may be veryj ! easily accomplished by means of a scale of sines and chordsi from Gunter’s scale.

i | Problem YI.— Two angles, and a side opposite to one| | of them, being given; to find the other two sides and thej i remaining angle.

I Figure 5.—Make the angle abc equal to the sphericalangle next to the given side; draw b d and b e perpendicularto b a and b c ; make b d of any length, and b e equal to it;

J and make the angle bd a equal to the measure of the givenj side ; draw af perpendicular to B C, cutting it in f ; make

i the angle fag equal to the complement of the other given

angle; on the centre f, with the distance F G, describe an

arc, ghi; draw c h e a tangent to the arc at H, the ^ w illin the second Proposition ; join a c ; and the angle c ^jtb

be the measure of the included side. On the centre ■*’the distance c e, describe an arc at k ; and on the ® t j, ewith the distance A D, describe another arc, cU tb eformer at k ; join a k and k c ; then will A Kmeasure of the third side of the spherical triangle. , s jje;

Problem VII.— Given two angles, and the contatto find the other three parts. _ dra w ® ^

Make abc, Figure 3, one of the given angles ; u et of

perpendicular to b c; make bec equal to the n - n t, <?>

degrees contained in the given side ; in bc take pet'

draw Gl perpendicular to C E, cutting it at I, an j t0 g J>pendicular on the other side of it ; make G H eq ,jra' vand the angle giik equal to the other given ang ^ ^ n eC K A, as in Proposition III., and ABC will btriangle representing the spherical one. tang entS ’

Now, because abc is the angle included by * q : 0 jn P A!draw bd perpendicular to b a, and equal to b e, ij n ^ ; 3 th ethen b d a is the measure of the side, of which j eg crib 6tangent. On the centre a, with the distance ’ nCe C*an arc at f ; and on the centre C, with the d‘ j 0 j I1 f Adescribe another arc, cutting the former at F.and f c ; then afc will be the measure of the t u ^ gfitProblem VIII.— Two sides, and an angle fPP (g.of them, being given ; to find the three remaining P - n i D g tb eFigure 6.—Draw AC, representing the side a a 8 c

given angle, and A b perpendicular to it; make t e . E vequal to the given side ; in A c take any point, ’ n( jicul aperpendicular to bc, cutting it in d, and EG P e P e qu»l t0to A C ; make e f equal to e d ; and the angle E ^ ce oi {e [the given angle; draw the line cgh ; on A, an ar e,with the tangent of the other given side, descri e ^ an dand if it cut the straight line C h in two P oin H ’ an d e( l U jjoin A K and A H ; draw A I perpendicular to A jjgtanc® 11 ’to ab; join i ii ; on the centre h, with the j. gtan ce c ’describe an arc at m ; and on the centre c, with t ie ii Hi 0 fdescribe another arc, cutting the former n ’the me asur ®. r e

C m ; then the angle c a k, or C a h, will be g ^eas”

the spherical angle included by the tangents. j ca l arl ? ,

of the angle A h c, or a k c, representing the S P ^ ; 3opposite the given side, shown by the tangeuby Proposition I. The angle C m h, or c rt j e jnd uof the remaining side, viz., that opposite * 6 ° . nb®

by the tangents. . . f or if A ® c

Note. —This case is not always ambiguous , cut ,

equal to, or greater than A c, the arc k ii w 1 Qne tria* 1 ®),in one point ; and, therefore, there can only^ only t0

ii c; and in this case also there can l,c F“* J . r) f ie rict''

or, if the angle A H c be a right angle, A .ii c; and in this case also there can be only ,- ca l tf*Problem IX. — The three angles of a S Pbeing given; to find the three sides. . an d & eS .gcb

Take the supplements of each of the ang ’ e qual t (i 0a triangle, by Proposition V., whose sides a^ W |H be ; 9supplements ; and the measure of its fln ° oU< rlit- *supplements of the sides of the triangle - rII ii)l

demonstrated by writers on spherical trig 0 ” given ,[,«Though the author of this Dictionary h _ re lat> n Sdemonstrations of the preceding Prop 0 ® 1110 ' as ’Lb K

geometrical construction of spherical tr o en o a . ofhave swelled the article too far, he ^ 10 P e ^ ear con ce P s gnt!>'been said to enable any one, who has a rep’’ ^o s®

the parts of a spherical triangle, to ^ e ® c . rl pa rts f t joa 0tion of it, and to find the measure ot 1 P T °l eC n s a 1

oa?’. tio a

tllO P

easy manner, without having recourse . gect lu '', gC trthe sphere, which frequently runs into P 1

from their difficulty of description, ren

ot I