TRIGONOMETRY.
765
and the spherical triangle ABC is right-angled at B, where CA isthe hypothenuse, and BA, BC, are the legs. To the arches GE,CB, draw the tangents HF, OB, and the sines GM, Cl, on theradii DF, DB; also draw BL the sine of the arch AB, and CKthe sine of AC : and then join IK and OL. Now [IF, OB, GM,Cl, are all perpendicular to the plane ADFB. And HD, GK,PL, lie all in the same plane ADGC. Also FD, IK, BL, lie allhi the same plane ADGC. Therefore the right-angled trianglesHFD, CIK, ODL, having the equal angles 1 IDF, CKf, OLB,are similar. And CK : DG :: Cl : GM ; that is, as the sine ofthe hypothenuse : rad. : : sine of a leg : sine of its opposite angle.For GM is the sine of the arc GF, which measures the angleCab. Also, LB : DF : : BO : FI 1 ; that is, as the sine of a leg:radius: : tangent of the other leg : tangent of its opposite angle.Q. F.. d. Hence it follows, that the sines of the angles of any ob-lique spherical triangle ACD (fig. 17 ,) are to one another, direct-ly, as the sines of the opposite sides. Hence it also follows, thatin right-angled spherical triangles, having the same perpendicular,the sines of the bases will be to each other, inversely, as the tan-gents of the angles at the bases.
Theorem Ii. In any right-angled spherical triangle ABC,(hg. 18 ,) it will be,^as radius is to the co-sine of one leg, so is the'o-sine of the other leg to the co-sine of the hypothenuse. Hence,*1 two right-angled spherical triangles ABC, CBD, (fig. 17 ,)have the same perpendicular BC, the co sines of their hypo-tenuses will be to each other, directly as the co-sines of theirbases.
Theorem HI. In any spherical triangle it will be, as radius's to the sine of either angle, so is the co-sine of the adjacent leg!o the co-sine of the opposite angle. Hence, in right-angledspherical triangles, having the same perpendicular, the co-sines ofthe angles at tile base will be to each other, directly, as the sines°f the vertical angles.
Theorem IV. In any right-angled spherical triangle it willbe, as radius is to the co-sine of the hypothenuse, so is the tangentof either angle to the co-tangent of the other angle. As the sumof the sines of two unequal arches is to their difference, so is thetangent of half the sum of those arches to the tangent of half theirdifference : and as the stun of the co-sines is to their difference, sois the co-tangent of half the sum of the arches to the tangent ofhalf the difference of the same arches.
Theorem V. In any spherical triangle ABC, (figs. 19,20,) as theco-taiment of half the sum of the sides to tangent of half their differ-ence so is the co-tangent of half the base to the tangent of the dis-tance (D E) of the perpendicular irom the middle ot the base. Since
• 1 .Ac “I- lie
the last proportion, by permutation, becomes co-lang.-T__
: co-tang. AE : : tang.- - : frng. DE, and as the tangents
of any two arches are, inversely, as their co-tangents; .it fol-
AC - BC
lows, therefore, that tangent AE : tangent —-:: tangent
At
---: tangent I)E ; or, which is the same thing expressed
without the algebraic formula, that the tangent of half the base is tothe tangent of half the sum of the sides, as the tangent of half thedifference of the sides to the tangent of the distance of the perpen-dicular from the middle of the base.
Theorem VI. I11 any spherical triangle ABC, (fig. 19,) itwill be, as the co-tangent of half the sum of the angles at the baseis to the tangent Qf half their difference, so is the tangent of halfthe vertical angle to the tangent of the angle which the perpendi-cular CD makes with the line CF bisecting the vertical angle.In respect to the construction of spherical triangles, we refer thereader to the article Projection of the Sphere, where propo-sitions, shewing the method of delineating and measuring the sidesand angles of spherical triangles, will be found. We shall not,therefore, in this place, give the graphical solution of spherical tri-angles, but shall add the following observations and definitions,and then give the tables of analogies by which the arithmetical so-lution is obtained. When three great circles of the sphere mu-tually intersect, there will be formed a great variety of sphericaltriangles, of which the areas, and the properties, could not be as-certained by plane trigonometry ; but come under considerationas parts of spherical surfaces, terminated by circles, of which toform a correct idea, the following definitions should he clearlv un-derstood. 1 st, The poles of a sphere are two points in the super-fices of the sphere, that are the extreme of the axis. 2d, Thepole of a circle in a sphere is a point in the superficies of thesphere, from which all right lines that are drawn to the circum-ference of the circle are equal to one another. 3 d, A great circlein a sphere, is that whose plane passes through the centre of thesphere; and whose centre is the same as that of the sphere. 4th,A spherical triangle is a figure comprehended under the arcs ofthree great circles in a sphere. 3 th, A spherical angle is thatwhich, in the superficies of the sphere, is contained under twoarcs of great circles; and this angle is equal to tire inclinations ofthe planes of the said circles. 6th, The measure of a spherioalangle is I he arc of a great circle, contained between the circlesforming the angle, and of which the angular point is denominatedthe pole.
Case
Given
Sought
"“1
Solution
1
The hyp. AC and oneangle A
The opposite leg
BC
As radius : sine hyp. AC :: sine A : sine BC (by theformer part of theorem 1.)
2
The hyp. AC and outangle A
The adjacent leg
AB
As radius : co-sine of A :: tang. AC : tang. AB (bythe latter part of theorem 1.)
3
The hyp. AC and oneangle A
The other angle
C/
As radius : co-sinc of AC :: tang. A : co-tang. C (bytheorem 4 .)
4
The hyp. AC and oneleg AB
The other leg
BC
As co sine AI 5 : radius:: co-sine AC : co-sine BC (bytheorem 2.)
5
The hyp. AC and oneleg AB
The opposite angle
As sine AC : radius sine AB : sine C (by the for-mer part of theorem 1.)
G
The hyp. AC and oneleg AB
The adjacent angle
A
As tang. AC : tang. AB :: radius ; co sine A (bytheorem 1.)
The