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 × T. CD AB × radius rad. co-f. AC x co-f. BC. But (by Theor. 1.) radius : co-f. CT. AC: T. CD = co-f. Cx T. AC rad. (by Corol. 1. Prop. 1.) which laft co-f. C x S. AC co-f. AC co-f. AB x rad. — being fubftituted for its equal, we shall have, 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 R* x J. AF - AB the radius; then will V = S. AC × S. BC fum of the two former of the three last terms is co-f. AF x R (by Cor. 1. fore it will be co-f. AB x R S. AC x S. BC x V R to Prop. 2.); thereco-f. AF x R and confequently V = Which is the firft Rx co-f. AF-co-f. AB S. ACS. BC cafe. Again, because S. AC x S. BC is R x co-f. AF-co-f. AE (by Corol. 3. to Prop. 2.) we 2Rx cof. AF-co-f. AB fhall, alfo, have V = Which is the fecond cafe. co-f. AF co-f. AE co-f. AF co-f. AB is = S. AB-AF 2 2 -(by the fame) it follows that V is likewife 2R x S. AB+ AF x S. AB S. AC x S. BC AF. 2. E. D. COROL COROLLARY I. Hence, because Rx V is the fquare of the fine of C (by Prop. 1.) it follows that fq. S. C= Rx S. AB+ AF x 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 base minus half the difference of the fides; fo is the fquare of radius, to the fquare of the fine of half the required angle. COROLLARY 2. Moreover,because Vis= R'xco-f.AF-co-LAB S. ACX S. BC we fhall have R: S. AC x S. BC:: V: co-f. AF co-f. AB; which gives the following theorem, for finding a fide, when the oppofite 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 those fides, and the fide required. COROL COROLLARY 3. Laftly, because V= 2R x co-f. AF-co-f AB, co-f. AF-co-f. 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 cof. 2R x co-f. ABW x co-f. AF AE= co-fine AF = V , 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 bafe and vertical angle. The END. LONDON: Printed by LUKE HANSARD, N° 6, Great Turnstile, Lincoln's-Inn-Fields. 1799. MATHEMATICAL Books Printed for F. WINGRAWZ; T Succeffor to Mr. NOURSE, in the Strand, HR Elements of Geometry; the fifth Edition, carefall revifed, 8vo. 6 s. 2. A Treatise of Algebra, 7th Edition, 8vo. 7 s. y 3. The Doctrine and Application of Fluxions, 2 vols. 8vo. the fourth Edition. 4. The Nature and Laws of Chance. 8vo. new Edition, 35. lewed. 5. The Doctrine of Annuities and Reverfions deduced from general and evident Principles, with ufeful Tables, 8vo. the fecond Edition, s. fewed. 6. The Supplement to ditto, 8vo. 2s. 6d. fewed. 7. Select Exercifes for young Proficients in the Mathematics, a new Edition, to which is prefixed, an Account of the Life and Writings of the Author, by CHARLES HUTTON, F. R. S. 8vo. 6s. (N. B. The Life may be had separate, price 6d.) 8. Mifcellaneous Tracts on fome curious and very interesting Subjects, 4to. 7 s. fewed. The Eight preceding Books are written by Mr. THOMAS, 9. The Mathematical Works of the late Mr. WiLLÍAM EMERSON. 10. The Elements of Euclid; alfo the Book of Euclid's Data, in like Manner corrected. By ROBERT SIMSON, M. D. The tenth Edition 8vo. 11. The Mathematical Repofitory, by JAMES DODSON, 3 Volumes 12mo. i2 s. 12. The Elements of Navigation; containing the Theory and Practice, with all the neceflary Tables, by JoHN ROBERTThe fixth Edition, carefully revised and corrected by Mr. WILLIAM WALES, 2 vols. royal 8vo. £. 1. 45. SON. 13. A General Treatife of Menfuration, by J. ROBERTSON. The third Edition, 12mo, 3 s. 6d. " 14. A Treatise of Algebra, by COLIN MACLAURIN. The fixth Edition, 8vo. 8 s. 15. The Works of the late Mr. JoHN LANDEN, in 2 vols. 4to. £.2. 15 s. bound. 16. The Mathematical Tracts of the late Reverend JoHN LAWSON, collected in one Volume 4to. 12 s. 6 d. boards. BRITISH |