FOI PROPOSITION XVI, The three Angles of a Spherical Triangle.are greater than two. Right Angles, and less than fix. OʻR the three Measures of the Angles G, H, D, together with the three sides of the Triangle XNM, make iniee Semicircles (by !4 of this); but the three sides of the triangle XN M are lets than *two Semicircles (by it of this). Wherelore the three Mealues if the Ang'es G H, D, are greater than a Semicircle ; a: di fo the Angits G, H, D, are greater than two Righ: Ang! s. The fec od part of the Propofition is manifeft; for, in every Spherical Tracle, the external and internal Angles, together, cinly make six Right Angles: Wi'herefure the internal Angls are less than six Right Angles. PROPOSITION XVII. If from the Point R, nat being the P le of the Circle AFBE, there full ike dres RA, RB, RG, RV, of great circ'es totb-Cielm ovence of that Circle ; then the grenie GI thoje trcsis R A, which pales through the Pu! Cikke.fi and ibe Remainder of 11:s the lat; ond : bise that are more distant from the greatesi are less than those which are nearer to it and they make an obiuse Angle with the former Circle AFB, on the Side next to the greatest irc. Vid. Fig. co Prop. 1. BEcause Ecause C is the Pole of the Circle AFB, then shall CD and RS. which is parallel thereto, be perpendicular to the Plane A FB. And if S A, SG, SV, be drawn, then shall SA (bv ; El. 3.) be greater than SG, and SG greater than SV. Whence, in the Rightangled plane Triangles RSA, RSG, RSV, we thall have R S9+S A q; or R A q, greater than RSq+ SG qi and so R A will be greater than RG, and the Arc R A greater than the Arc R G. In like manner, RS9+SG 9, or RG 9) shan 9, or RG than RS9+SV q, or R Vq; and fo RG shall be greater than RV, and the Arc RG greater than the Arc R V. 2dly, The Angle RG A is greater than the Angle CGA, which is a Right Angle (by Cor. Prop. 3.); and the Angle RVA is greater than the Angle CVA, which aito is a Right Angle. Therefore the Angles RGA, RV A, are obruse Angles. PROPOSITION XVIII. . In Spherical Triangles. A GR, A GX, Right. angled ar C, the Legs containing ihe Right Angle are of the same Affection with the opposite Angles ; that is, if the Legs be greater or less than Quadrants, iben, accordingly, will the Angles opposite to them be greater or less than Right Angies. Vid. Fig. to Prop. 1. F: OR if A C be a Quadrant, then will be the Pole of the Circle AFB; and the Angles AGC, A V C, will be Right Angles. If the Leg A R be greater than a Quadrant, then shall the Angle AGR be greater than a Right Angle (by 17 of this); and if the Leg A X be less than a Quadrant, the Angle A G X shall be less than a Right Angle. PROPOSITION XIX. be of the same Affection (and consequently the Pole of the Leg AR: then will RF be a Quadrant, which is greater than R V (by 17 of this). PROPOSITION XX. If they be of a different Affection, then shall the H puthenuje be greater than a Quadrant. Vid. o Prop. 1. the Tr wigle ARG, the Hypothenuse RG sier than R F, which is a Quadrant. PRO. Given (Sought the drant, then shall B A and CA be 21. of the fame Affection; if greater, for a different: But B A is given, also given. 18. BAB AC. R:S, BA::T, B:T, A C, of By Piop. 7 Ithe same Kind with B. 18. AC,B. BA T, B:T, CA:: R:S, A B, By Prop. ambiguous. 127. BC be less than a Quadrant, the 28, and Bi 27, and A C be given. AC, C. BC. Cor. C:R::T, CA: 1, BC. By Prop' And so, if the Angle C and C A, 28, 205 10 be of the same Affection, then B C 21. lif of a different, greater. ihen CA and B A, and confe-21. 12 Given Sought besides the Right Angle. B,C,B. AC. R:S, BC::S, B:S, A C, of By Prop. the fame Species with B. 29, and 18. AC, B BC. S, B:S, AC:: R:S, BC, am- By Prop. 13 biguous. 29. BC, B. S, BC:R::S, AC :S, B, of By Prop. 141 AC. Ithe same Species with CA. 129. BC. T, B : Cot. B :: R: Col. B C. By Prop. And so, if the Angles B and C are 30, 19, of the same Affection, then thall and 20. BC be less than a Quadrant, if of a different, greater. And so, if B C be less than thall be of the fame Affection ; 16 if greater, of a different. Bu B, C. 151 a 30, and Of the Solution of Right-angled Spherical Triangles, by the five circular Parts. T HE Lord Neper (the noble Inventor of Loga rithms) by a due Consideration of the Analogics by which Right-angled Spherical Triangles are solved, found out two Rules, easy to be remembered, by means of which, all the sixteen Cases may be solved : For since in these Triangles, besides the Right Angles, there are three Sides, and two Angles; the two Sides comprehending the Right Angle, and the Complements of the Hypothenuse, and the two other Angles, were called by Neper, Circular Parts; and when there are given any two of the said 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 calle ed Adjacent Extremes ; or is separated from them, and then are called Opposite 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 opposite Extremes. Also, if the Complement of the Hypothenuse B C be supposed the middle Part, then the Complements of the Angles B and C are adjacent Extremes, and the Legs A B, A C, are opposite Extremes. In like manner, lupposing 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 Hypothenuse B C, are opposite Extremes to the faid middle Part. These Things premised, R U L E I. In any Right-angled Spherical Triangle, the Rest angle under the Radius, and the Sine of the middle Part, is equal to the ReEtangle under the Tangents of the adjacent Parts. RUL E |