Since T. DE: T. BC :: S. AD: S. AB (by Theor. 4. and equality); therefore is T. DE + T. BC: T. DET. BC:: S. AD+ S. AB: S. AD - S. AB; whence; (by Prop. 4. and the lemma in p. 30.) it will be, S. DE+ BC: S. DE AD + AB 2 : T. AD-AB ; that is, BC:: T. As the fine of the fum of the two perpendiculars, is to the fine of their difference; fo is the tangent of half the fum of the two bafes, to the tangent of half their difference. PROP. XXVI. The product of the fquare of radius and the co-fine of the base of any spherical triangle ABC, is equal to the product of the fines of the two fides and the co-fine of the vertical angle, together with the product of radius and the co-fines of the fame fides. co-f. CB rad. (:: co-f. BD: co-f. CD):: co-f. AB: co-f. AC (by Corol. to Theor. 2.) whence, by multiplying means and extremes, we have co-f. S. CB x co-f. AC x T. CD AB X radius= co-f. AC x co-f. BC. rad. But (by Theor. 1.) radius co-f Cx T. AC + co-f. CT. AC: T. CD rad. co-f. AC X 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 fphere; but that is a fubject which neither room, nor inclination, will permit me to treat of here. PROP. XXVII. If AE be the fum, and AF the difference, of the two fides of a spherical triangle ABC, and V be put to denote the verfed fine of the vertical angle, and R Rx co-f. AF-co-f. AB S. AC X S. BC the radius; then will V = fum of the two former of the three laft terms is =co-f. AF x R (by Cor. 1. to Prop. 2.); therefore it will be co-f. AB x R co-f. AF XR S. AC x S. BC × V and confequently V = R R2 x co-f. AF co-f. AB S. AC X S. BC Which is the first Rx cafe. Again, because S. AC x S. BC is co-f. AF — co-f. AE (by Corol. 3. to Prop. 2.) we 2R X co-f. AF co-f. AB fhall, also, have V = Which is the fecond cafe. co-f. AF-co-f. AE Moreover, fince R x co-f. AF co-f. AB is = S. 2 (by the fame) it follows that V is likewife 2R × S.1⁄2AB + ÷AF × S. AB — AF S. AC X S. BC 2 2. E. D. COROL Hence, because R x V is the fquare of the fine of C (by Prop. 1.) it follows that fq. S. C= R2 × S. AB + ÷AF × S. ÷AB - AF S. AC X S. BC From whence we have the following theorem, for folving the 11th cafe of oblique triangles, where the three fides are given, to find an angle. As the rectangle of the fines of the two fides, including the propofed angle, is to the rectangle under the fines, of half the bafe plus half the difference of the fides, and half the bafe minus half the difference of the fides; fo is the fquare of radius, to the fquare of the fine of half the required angle. COROLLAR Y 2. Moreover, because V is= Raxco-f.AF-co-f.AB S. AC X S. BC we shall have R2: S. AC x S. BC :: V: co-f. AF -co-f. AB; which gives the following theorem, for finding a fide, when the opposite angle, and the other two fides, are given. As the fquare of radius, is to the rectangle of the fines of the two fides including the given angle; fo is the verfed fine of that angle, to the difference of the co-fines (or verfed fines) of the difference of thofe fides, and the fide required. COROL COROLLARY 3. Laftly, because V= 2R x co-L. AF-co-f. AB co-f. AF- co-s. AE we fhall, by transforming the equation, and putting W for (2R V) the verfed fine of BCE (the fupplement of the vertical angle) have co-f. 2R x co-f. AB-W x co-L. AF AE= co-fine AF = and the 2R X co-f. AB - V x co-f. AE W From whence the fides themselves may be determined, when their fum, or difference, is given, with the base and vertical angle. The EN D. ERRATU M. |