PROPOSITION III. Pole 4; then the Arc CF, intercepted between or CDF. Vid. Fig. to Prop. 1. TH HE Arcs A C, A F (by Cor. 1. Prop. 2.) are Qua drants; and, confequently, the Angles A DC, ADF, are Right Angles : Wherefore (by Def. 6. El. 11.) the Angle CDF (whose Measure is the Arc CF)is equal to the Inclination of the Planes A CB, A FB, and also equal to the Spherical Angle CAF, or CBF. W. W. D. Coroll. 1. If the Arcs A C, A F, are Quadrants, then shall A be the Pole of the Circle passing thro' the Points C and F; for AD is at Right Angles to the Plane FDC (by 4 El. 11.) 2. The vertical Angles are equal : for each of them is equal to the Inclination of the Circles : also the adjoining Angles are equal to two Right Angles. PROPOSITION IV. Triangles shall be equal and congruous, if they have two Sides equal to two Sides, and the Angles comprekended by the two Sides also equal. P R O POSITION V. Side, together with the adjacent Angles in one PROPOSITION VI. Triangles mutually equilateral, are also mutually equiingular. PROPOSITION VII. In Tosceles Triangles, the Angles at the Base oil equal. PRO PROPOSITION VIII. And if the Angles at the Base be equal, then the Triangle shall be Isosceles. THES E five laft Propofitions are demonstrated in the same manner, as in plane Triangles. PROPOSITION IX. Any Two Sides of a Triangle are greater than the ibird. FO OR the Arc of a great Circle is the shorteft Way, between any two Points in the Superficies of the Sphere. PROPOSITION X. Semicircle. be produced till they meet in D: Then shall the Arc ACD, which is greater than the Arc AC, be a Semicircle. PROPOSITION XI. Tbe three Sides of a Spherical Triangle are less than a whole Circle. FOR BD+DC is greater than BC (by Prop: 9:); and, adding on each Side B A +AC; then DBA +DCA, that is a whole Circle, will be greater than BA+BC+AC, which are the three sides of the Spherical Triangle A B C. PROPOSITION XII. In a Spherical Triangle A B C, the greater Angle A is subtended by the greater Side. M AKE the Angle B A D=Angle B ; then shall AD=BD (by 8 of this); and to BDC=DA+ DC, and these Arcs are greater than A C. Wherefore the Side B C, that fubtends the Angle B AC, is greater than the Side A C, that subtends the Angle B. PRO PROPOSITION XIII. In any Spherical Triangle ABC, if the sum of the Legs AB and BC be greater, equal, or less, than a Semicircle, then the internal Angle at ibe Bale BAC shall be greater, equal, or less, than tbe external and opposite Angle BCD; and to the Sum of the Angles A and ACB fall also be greater, equal, or less, than two Right Angles. F. IRST, let A B+AC=Semicircle=AD; then BC=B D, and the Angles B C D and D equal (by 8 of this); and therefore the Angle BCD chall be=Angle A. Secondly, let A B+B C be greater than A BD; then shall B C be greater than BD; and so the Angle D (that is, the Angle A, by 12 of this) shall be greater than the Angle BCD. In like manner we demonftiate, if A B+B C be together less than a Semicircle, that the Angle A will be less than the Angle BCD: And because the Angles B C D and BCA are two Right Angles, if the Angle A be greater than the Angle BCD, then shall A and B C A be greater than two Right Angles; if the Angle A=BCD, then shall A and BCA be equal to two Right Angles, and if A be less than BCD, then wijl A and B C A be less than two Right Angles. W. W.D. PROPOSITION XIV. In any Spherical Triangle G HD, the Poles of the Sides, being joined by great Circles, do constitute another Triangle X M N, which is the Supplement of the Triangle G HD, viz. the Sides NX, XMN, and NM, shall be Supplements of the Arcs that are the Measures of the Angles D, G, H, to the Semicircles, and the Arcs ibat are the Measures of the Angles M, X, N, will be the Supplements of the Sides G H, GD, and HD, to Semicircles. ROM the Poles G, H, D, let the great Circles X CAM, TMNO, X KBN, be described ; then, FR -3 then, because G is the Pole of the C rcle X CAM, we shall have GM=Quadrant (Cor. 1. Prop. 2.); and fince H is the Pole of the Circle T MO, then will HM be also a Quadrant, and fo (by Cor. 1. Prop. 3.) M shall be the Pole of the Circle GH. In like manner, because D is the Pole of the Circle X BN, and H the Pole of the Circle TMN, the Arcs DN, HN, will be Quadrants; and so (by Cor. 2. Prop. 3.) N thall be the Pole of the Circle HD. And because GX, D X, are Quadrants, X will be the Pole of the Circle GD. These Things premised, Because NK=Quadrant, and X B=Quadrant (by Cor. 1. Prop. 2.); then will N K+X B, that is, N X +K B=two Quadrants, or a Semicircie; and so N X is the Supplement of the Arc K B, or of the Measure of the Angle HDG, to a Semicircle. in like manner, becaule M C=Quadrant, and XA=Quadrant, then will MC+XA, that is, X M+AC=iwo Quadrants, or a Semicircle ; and, consequently, X M is the Supplement of the Arc A C, which is the Measure of the Angle H GD. Likewise, fince MO, NT,=are Quadrants, we shall have M OUN T= OT+NM= Semicircle : And therefore N M is the Supplement of the Arc O T, or of the Measure of the Angle GHD to a Semicircle. W.WD. Moreover, because DK, HT, are Quadrants, DK +HT, or KT+HD, are equal to two Quadrants, or a Semicircle ; therefore K T, or the Measure of the Angle X NM, is the Supplement of the Side HD to a Semicircle. After the same manner, it is demonstrated, that 0 C, the Measure of the Angle XMN, is the Supplement of the Side GH; and B A, the Measure of the Angle X, is the Supplement of the Side GD. W. W.D. PROPOSITION XV. Equiangular Spherical Triangles are also equi lateral. FOI OR their Supplementals (by 14 of this) are equila. teral, and therefore equiangular allo; and so them. felves are likewise equilateral (by Part 2. krop. 14). PRO PROPOSITION XVI. Tbe three Angles of a Spherical Triangle are greater than two. Righi Angles, and less than £1x. FO OR the three Measures of the Angles G, H, D, together with the three sides of the Triangle XNM, make iniee Sernicircles (by !4 of this); but the three sides of the Triangle XN M are lets than *two Semicircles (by II of this). Where!ore the three Meatures of the Ang'es G H, D, are greater than a Semicircle ; ad fo the Angles G, H, D, are greater than two Righ: Angl. s. The sec nid Puit of the Proposition is manifeft; for, in every Spherical Traile, the external and internal Angles, together, inly make six Right Angles: V'herefore the internal Aagl s are less than fix Right Angles. PROPOSITION XVII. If from the Point R, not being the P le of the Circle AFBE, there all ike dres RA, RB, RG, RV, of great circ'es 10 tkr Clicum irence of that Circle ; tben the gre.110/ 0 1 0;e resis R A, which palesikrough the Pub Cibeke.fi and the Remainder of 11 is the bali and bise that are more distant from the grea'ifi are less than those which are nearer lo ib. and they make an obtuse Angle with the former Circle AFB, on the Side next to the greatest sirc. Vid. Fig. to Prop. 1. BEcause 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 7 El. 3.) be greater than SG, and SG greater than SV. Whence, in the Rightangled plane Triangles RSA, RSG, RSV, we thall have R 59+S Aq; or R A q, greater than RSq+ or R G.q and so R A will be greater than RG, and the Arc R A greater than the Arc RG. In like manner, R Sq+S G q, or R G q, shall be greater than SGą, |