The three Angles of a Spherical Triangle are greater than two Right Angles, and less than fix. FOR OR the three Measures of the Angles G, H, D, together with the three Sides of the Triangle XN M, make three Semicircles (by 14 of this); but the three Sides of the Triangle X N M are lefs than two Semicircles (by 1 of this). Wherefore the three Measures of the Angles G H, D, are greater than a Semicircle; and fo the Angies G, H, D, are greater than two Right Angls. The fee and Part of the Propofition is manifeft; for, in every Spherical Trangle, the external and internal Angles, together, only make fix Right Angles: Wherefore the interual Angles are lefs than fix Right Angles. XVII. PROPOSITION If from the Point R, not being the Ple of the Circle AFBE, there full the Arcs RA, R B, RG, RV, of great circles to the Circumference of that Circle; then the greatest of thoje Arcs is RA, which paffes through the Poly C there if; and the Remainder of it is the lat; and ibise that are more diftant from the greatest are less than thofe which are nearer to it, and they make an obtufe Angle with the former Circle AF B, on the Side next to the greatest Arc. Vid. Fig. to Prop. 1. BEcaufe C is the Pole of the Circle AFB, then fhall CD and RS, which is parallel thereto, be perpendicular to the Plane A FB. And if S A, S G, SV, be drawn, then fhall SA (by 7 El. 3.) be greater than SG, and S G greater than S V. Whence, in the Rightangled plane Triangles R.S A, RSG, RSV, we thall have R Sq+S Aq; or R Aq, greater than RS q+ SGq, or RG. q; and fo RA will be greater than R G. and the Arc R A greater than the Arc R G. In like manner, R Sq+S Gq, or RG q, fhall be greater than than RS q+S V q, or RV q; and fo R G fhall be greater than RV, and the Arc RG greater than the Arc R V. 2dly, The Angle R GA is greater than the Angle CGA, which is a Right Angle (by Cor. Prop. 3.); and the Angle RVA is greater than the Angle CV A, which alfo is a Right Angle. Therefore the Angles RG A, RV A, are obtufe Angles. PROPOSITION XVIII. In Spherical Triangles A G R, A G X, Rightangled at C, the Legs containing the Right Angle are of the fame Affection with the oppofite Angles; that is, if the Legs be greater or less than Quadrants, then, accordingly, will the Angles oppofite to them be greater, or less than Right Angles. Vid. Fig. to Prop. 1. FOR if AC be a Quadrant, then will C be the Pole of the Circle A F B; and the Angles A G C, AV C, will be Right Angles. If the Leg AR be greater than a Quadrant, then fhall the Angle A GR greater than a Right Angle (by 17 of this); and if the Leg A X be lefs than a Quadrant, the Angle AG X fhall be less than a Right Angle. be PROPOSITION XIX. If two Legs of a Right-angled Spherical Triangle be of the fame Affection (and confequently the Angles), that is, if they are both less, or both greater, than a Quadrant, then will the Hypothenuje be lejs than a Quadrant. Vid. Fig. to Prop. I. IN N the Triangle AR V, or BR V, let F be the Pole of the Leg AR: then will R F be a Quadrant, which is greater than R V (by 17 of this). PROPOSITION XX. If they be of a different Affection, then fhall the Hypothenuje be greater than a Quadrant. Vid. o Prop. I. in the Triangle ARG, the Hypothenufe RG ter than R F, which is a Quadrant. PRO. Given Sought befides the 5 Right BA, A C. Cof. BAR:: Col, BC: Cof. By Prop. B C. BA, 6 CA. B. CA. If BC be lefs than a Qua-26, and S, BA:R::T, CA: T, B, of By Prop. 18. BA,B AC. R: S, BA :: T, B: T, A C, of By Prop. the fame Kind with B. 27, and 18. AC,B. BA. T, B: T, CA::R:S, A B, By Prop. Jambiguous. 27. BC,C. A C. R : Co. C :: T, BC: T, CA. It By Prop. BC be less than a Quadrant, the 28, and Angles C and B are of the fame 21. Affection; if greater, of a different. Therefore, if the Species of the Angle B be given, then will AC be given. AC, C. B C. Cof. C: R:: T, CA: T, BC. By Prop And fo, if the Angle C and CA, 28, be of the fame Affection, then B C21. if of a different, greater. 201 T, BC: T, CA::R: Cof. C. By Prop. 12 B,C,B. A C. RS, BC::S, BS, AC, of By Prop. the fame Species with B. 29, and 18. AC, B B C. S, B: S, AC:: R: S, BC, am-By Prop. 13 14 B C, A C. B, C. B. 15 1 16 biguous. 29. S, BC: R:: S, AC: S, B, of By Prop. the fame Species with C A. 29. BC. T, B: Cot. B: R: Cot. B C.By Prop. R: Col. B C :: T, &: Cot. B.By Prop. Of the Solution of Right-angled Spherical ΤΗ HE Lord Neper (the noble Inventor of Logarithms) by a due Confideration of the Analogies by which Right-angled Spherical Triangles are folved, found out two Rules, eafy to be remembered, by means of which, all the fixteen Cafes may be folved: For fince in these Triangles, befides the Right Angles, there are three Sides, and two Angles; the two Sides comprehending the Right Angle, and the Complements of the Hypothenufe, and the two other Angles, were called by Neper, Circular Parts; and when there are given any two of the faid Parts, and a third is fought; one of these three, which is called the Middle Part, either lies between the two other Parts, which are called Adjacent Extremes; or is feparated from them, and then are called Oppofite Extremes: So if the Complement of the Angle B (Fig. to Prop. 25.) be supposed the middle Part, then the Leg A B, and the Complement of the Hypothenuse B C, are adjacent extreme Parts; but the Complement of the Angle C, and the Sides A C, are oppofite Extremes. Alfo, if the Complement of the Hypothenufe B C be fuppofed the middle Part, then the Complements of the Angles B and C are adjacent Extremes, and the Legs A B, A C, are oppofite Extremes. In like manner, fuppofing the Leg AB the middle Part, the Complement of the Angle B and A C are adjacent Extremes; for the Right Angle A doth not interrupt the Adjacence, because it is not a circular Part. But the Complement of the Angle C, and the Complement of the Hypothenufe BC, are oppofite Extremes to the faid middle Part. Thefe Things premifed, RULE I. In any Right-angled Spherical Triangle, the Redangle under the Radius, and the Sine of the middle Part, is equal to the Rectangle under the Tangents of the adjacent Parts. RULE |