First, it will be, Rad.: Co-f. A :: T. AC: T. AB (by Theor. 1.) and therefore Rad. + Co-f. A: Rad. Co-f. A: T. AC+ T. AB: T. AC-T. AB: 1 2 C B Whence, by arguing as in the laft Prop. it will appear, that, Co-tang A: Tang. ¦ A :: Rad. + Co-f. A: Rad.Co-f. A (:: T. AC + T. AB: T. AC-T. AB): :S. AC+ AB: S. AC-AB (by Prop. 4.). Hence it appears, that, As the Co-tangent of half the given Angle, is to its Tangent, fo is the Sine of the Sum of the Hypothenuse and adjacent Leg, to the Sine of their Difference. PROP. XXIII. The Hypothenufe AC and the Sum, or Difference, of the two adjacent Angles being given, to find the Angles. Let EC be perpendicular to BC; and, then it will be, Rad. Co-f. AC:: T. A: T. ACE (by Theor. 5.) From whence, by reasoning as above, we fhall, alfo, have, Co-tang.AC: Tang. E B AC:: S. A+ACE: S. A-ACE; whereof the two last Terms, by fubftituting 90° — A C B for ACE, will become S. 90°+A-ACB (Co-f. ACB -A) and S. A+ACB-90° refpectively. Whence it appears, that, As the Co-tangent of half the Hypothenufe, is to its Tangent, fo is the Co-fine of the Difference of the Angles at the Hypothenufe, to the Sine of the Excess of their Sum above a Right-angle. COROLLAR Y. Hence, if the Angles be fuppofed equal, then it will be, as Radius: Tang. AC:: Tang. AC: Sin. 2A-90°. PROP. XXIV. 2 In two right-angled spherical Triangles ABC, ADE, having one Angle A common, let there be given the two Perpendiculars BC, DE and the Sum, or Difference, of the Hypothenufes AC, AE, to determine the Triangles. gent of half the Sum of the two Perpendiculars, is to the Tangent of half their Difference, fo is the Tangent of half the Sum of the two Hypothenufes, to the Tangent of half their Difference. PROP. XXV. In two right-angled spherical Triangles ABC, ADE, having the fame Angle A at the Bafe, let there be given the two Perpendiculars BC, DE and the -Sum or Difference of the Bases AB, AD, to determine the Bafes (fee the preceding Figure). Since, ご Since, T. DE: T. BC:: S. AD: S. AB (by Theor. 4. and Equality); therefore is T. DET. BC: T. DET. BC:: S. AD+ S. AB: S. AD -S. AB; whence (by Prop. 4. and the Lemma in p 28.) it will be, S. DE+ BC: S. DE-BC:: T. AD+AB AD-AB 2 T. 2 ; that is, As the Sine of the Sum of the two Perpendiculars, is to the Sine of their Difference, fo is the Tangent of balf the Sum of the two Bafes to the Tangent of half their Difference. PROP. XXVI. The Product of the Square of Radius and the Co-fine of the Bafe of any Spherical Triangle ABC, is equal to the Product of the Sines of the two Sides and the Cofine of the vertical Angle, together with the Product of Radius and the Co-fines of the fame Sides. Co-f. CB Rad. (:: Co-f. BD: Co-s. CD) :: Co-f. Rad. + But (by Theor. 1.) Radius: A B x Radius = Co-f. AC x Co-f. BC. Co-f. C:: T. AC: T. CD = Co-f. Cx S. AC Co.f. AC Co-f. A Bx Rad. = Rad. (by Corol. 1. Prop. 1.) which laft being fubftituted for its Equal, we. fhall have, S. CAX S. CBx Co-f. C Rad. + Co. f. ACx Co-f. BC; from whence, if each Term be multiplied by Radius, the Truth of the Propofition will appear manifeft. There is another Way of demonftrating this Propofition, from the Orthographic Projection of the Sphere; but that is a Subject which neither Room, nor Inclination, will permit me to treat of here. PROP. XXVII. If AE be the Sum, and AF the Difference, of the two Sides of a spherical Triangle ABC, and V be put to denote the verfed Sine of the vertical Angle, and R R2x Co-f. AFCo-f.AB the Radius; then will V = S. ACX S. BC 2 R × S. AB + ÷ A F × S. ÷ AB — ¦ AF. S. ACX S. BC. 3 It 2 Sum of the two former of the three laft Terms is and confequently V = R2x Co-f. AF - Co-f. AB S. AC x S. BC Which is the first Cafe. Again, because S. AC x S. BC is = Rx Co-f. AF-Co-f. AE (by Corol. 3. to Prop. 2.) we shall, also, have V2 Rx Co-L. AF-Co-f. AB Co-f. AF - Co-f. AE Which is the fecond Cafe. Moreover, fince Rx Co-f. AF-Co-f. AB is S. A B+ AF AB-AF 2 2 x S. (by the fame) it follows that V is likewise I 2 2 Rx S. AB+ AF ×S. AB — AF' S. AC x S. BC 2. E. D. Co |