SOLUTION of the CASES of OBLIQUE-ANGLED I GENERAL PROPOSITION. N an oblique.angled triangle, of the three fides and three angles, any three being given, the other three may be found, except when the three angles are given; in which cafe the ratios of the fides are only given, being the fame with the ratios of the fines of the angles oppofite to them. 4 GIVEN. SOUGHT. 2 ACXCB: ACq+CBq -ABqR: Co S, C. If ABq+CBq be greater than ABq. FIG. 16. 2 ACXCB ABg-ACq -CBq+R: Co S, C. I AB, BC, CA, A, B, C, the ABq be greater than ACq× the three fides. three angles. CBq. FIG. (17. 4.) Otherwife. Let AB+BC+AC=2P. P+P-AB : P—AC + P-BC Rq: Tg, C, and hence C is known. (5.) Otherwife. Let AD be perpendicular to BC. 1. If ABq be lefs than ACq+CBq. FIG. 16. BC: BA + AC :: BAAC BD-DC, and BC the fum of BD, DC is given; therefore each of them is given (7.) 2. If ABq be greater than ACq+CBq. FIG. 17. BC: BA+AC:: BA-AC: BD +DC; and BC the difference of BD, DC is given, therefore each of them is given. (7.) And CA: CD : R: Co S, C. (1.) and C being found, A and B are found by cafe 2. or 3. SPHERICAL SPHERICAL TRIGONOMETRY. TH DEFINITIONS. I. HE pole of a circle of the fphere is a point in the fuperficies of the fphere, from which all straight lines drawn to the circumference of the circle are equal. II. A great circle of the fphere is any whofe plane, paffes through the centre of the sphere, and whofe centre therefore is the fame with that of the sphere. III. A spherical triangle is a figure upon the fuperficies of a sphere comprehended by three arches of three great circles, each of which is lefs than a femicircle. IV. A fpherical angle is that which on the superficies of a sphere is contained by two arches of great circles, and is the fame with the inclination of the planes of these great circles. As they have a common centre their common section will be a diameter of each which will bifect them. TH PRO P. II. FIG. 1. HE arch of a great circle betwixt the pole and Let ABC be a great circle, and D its pole; if a great circle DC pass through D, and meet ABC in C, the arch DC will be a quadrant. Let the great circle CD meet ABC again in A, and let AC be the common fection of the great circles, which will pafs 493 |