Co-f. CB : rad. (: : 20-f. BD : 20-f. CD) :: Co-fa 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 XT. CD AB x radius = + rad. co-f. AC x 20-. BC. But (by Theor. 1.) radius : co-f. C x T.AC Co-L.C:: T. AC : T. CD = rad. Co-. C XS. AC Co-f. AC (by Corol. 1. Prop. 1.) which last being substituted for its equal, we shall have, S. CA X S. CB x co-f. C CO-1. AB x rad. = rad. + Co-f. AC x co-f. BC; from whence, if each term be multiplied by radius, the truth of the proposition will appear manifeft. There is another way of demonstrating this proposition, 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 fum, and AF the difference, of the two sides of a Spherical triangle ABC, and V be put -to denote the versed fine of the vertical angle, and R the radius ; then will V = RX Ço-. AF-C0-4: AB S. AC XS. BC Co-f. AF-60-). AE S. AC X S., BC It It appears from the last R F B : but the and consequently V = Which is the first 2R x co-f. AF-CO-s. AB CO-L. AF - Co-f. AE Moreover, since RX AB + AF x S. 2 AB-AF (by the fame) it follows that V is likewise S. AC X S, BC Hence, because IR x V is = the square of the fine of įC (by Prop. 1.) it follows that iq. S. ¿C= R? x S. AB + AF * S. AB - JAF S. AC x S. BC From whence we have the following theorem, for solving the uth case of oblique triangles, where the three sides are given, to find an angle. As the rectangle of the lines of the two sides, including the proposed angle, is to the rectangle under the fines of half the base plus half the difference of the sides, and half the base minus balf the difference of the sides; fo is the Square of radius, to the Square of the fine of half the required angle. COROLLARY 2. R*XC0-f.AF-CO-LAB Moreover, because Vis= S. AC X S. BC we shall have R’: S. AC x S. BC::V: Co-f. AF - Co-f. AB; which gives the following theorem, for finding a lide, when the opposite angle, and the other two sides, are given. As the square of radius, is to the rectangle of the fines of the two sides including the given angle; so is the versed fine of that angle, to the difference of the co-fines for versed fines) of the difference of those sides, and the fide required. COROL COROLLARY 3. 2R x Co-f. AF - CO-CAB, co-f. AF – co-f. AE and the 2R X Co-f. AB –V x co-f. AE W Tbe END. 1 LONDON: Printed by Luxe HANSAR D, 1799. MATHEMATICAL Books Printed for F. WINGRAWE; Successor to Mi. Nourse, in the Strand, у T* revised, 8vo. 6 s. 3. The Doctrine and Application of Fluxions; 2 vols. 8vo. the fourth Edition. 4. The Nature and Laws of Chance. Svo, new Edition, 3 s. sewed. 5. The Doctrine of Annuities and Reversions deduced from general and evident Principles, with useful Tables, 8vo: the tecond Edition, 3 s. fewed. 6. The Supplement to ditro, 8vo. 2 s. 6 d. sewed. 2. Select Exercises for young Proficients in the Mathema. tics, a new Edition, 40 which is prefixed, an Account of the Life and Writings of the Author, by CHARLES HUTTON, F.R. S. 8vo. 6 s. (N. B. The Life may be kad separate, price 6d.) 8. Miscellaneous Tracts on some curious and very interest. ing Subjects, 4to. 75. sewed. The Eight preceding Books are written by Mr. THOMAS SIMPSON, F. R. S. 9. The Mathematical Works of the late Mr. WILLIAM EMERSON. 10. The Elements of Euclid; also the Book of Euclid's Data, in like Manner corrected. By ROBERT Simson, M. D. The tenth Edition 8vo.. 11. The Mathematical Repository, by James DODSON, 3. Volonies 12 mo. 12 s. 12. The Elements of Navigation; containing the Theory and Practice, with all the necessary Tables, by John ROBERT The fixth Edition, carefully revised and corrected by Mr. WILLIAM WALES; 2 vols.. voy al 8vo. £.1. . 1. 45. 13. A General Treatise of Menfuration, by J. ROBERTSON. The third Edition, 12mo, 3 s. 6 d. 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 JORN LAWSON, collected in one Volume 4to. 12 s. 6 d. boards. SON. |