be one of Given Sought Make, as A C, portions of the. Sides may lelves, unless one of them be first known. The The AB: BC::S,C:S, A, which two Angles therefore may be found. When Sides A and A B, the Side opposite to C, the AB, B. given Angle is longer than C B, BC, che Side opposite to the foughe and C, Angle, the fought Angle is less 3 the An than a Right one. But when it gle op is shorter, because the Sine of an posite to Angle, and that of its Comple ment to two Right Angles, is the them. fame, the Species of the Angle A must be firit known, or the Solu tion will be ambiguous. The The Vid.Fig, to Prop. 1 3. BL=AB: two Angles BC-AB::T,A+C:T, A A-C Sides A and AB, C. Whence is known the Difference BC, of the Angles A and C, whose Sum is given, and io (by Prop. inter 15.) the Angles themselves will jacent be given. Angle B. Sides Angles. cular be drawn from the Vertex to the Bale, and find the SegBC, ments of the Base by Prop. 14. A C. viz, make as BC:AC+AB:: 5 AC-AB: DC-DB. And o BD, DC, are given from the Analogy, and thence the Angles ABD, A CD, will be given by the Resolution of Right-angled Triangles. THE 2 4 and the U 3 THE E LE M E N T S OF Spherical Trigonometry . DEFINITIONS. 1.THE Poles of a Sphere are two Points in the Superficies of the Sphere, that are the Extremes of the Axis. II. The Pole of a Circle in a Sphere is a Point in the Superficies of the Sphere, from which all Right Lines that are drawn to the Circumferenee of the Circle are equal to one another. III. A great Circle in a Sphere is that whose Plane passes through the Centre of the Sphere, and whose Centre is the same with that of the Sphere. IV. A spherical Triangle is a Figure compre bended under the Arcs of three great Circles in a Sphere. V. A spherical Angle is that which, in the Su perficies of the Sphere, is contained under two Arcs of great Circles; and this Angle is equal to the inclination of the Planes of ibe said Circles, PRO PROPOSITION I. otber. each Circle, and so will cut them into two equal Paris, Coroll. Hence the Arcs of two great Circles in the Superficies of the Sphere, being less than Semicircles, do not comprehend a Space; for they cannot, unless they meet each other in two opposite Points; that is, unless they are Semicircles. F 1 PROPOSITION II. a Right Line C D to the Centre thereof, the the Circle A F B ; then, because in the Triangles Interval of a Quadrant ; for, since the Angles them, viz. the Arcs CĞ, CF, will be Quadrants. 2. Great Circles, that pass thro’ the Pole of some other Circle, make Right Angles with it; and, contrariwise, if great Circles make Right Angles with fome other Circle, they shall pass thro' the Poles of that other Circle ; for they must neceffarily pass through the Right Line DC. |