rce X C A M, or. I. Prop. 2.); ile TM O, then (by Cor. 1. Prop. rcle GH. In like the Circle X BN, IN, the Arcs DN, (by Cor. 2. Prop. 3.) HD. And because will be the Pole of the uremised, wad X B=Quadrant (by - K+XB, that is, NX Semicircle; and fo NX cKB, or of the Measure Semicircle. In like manant, and X A=Quadrant, at is, X M+AC=iwo Quaand, consequently, X M is Arc A C, which is the Mea. GD. Likewise, since MO, we shall have MOUNTcle: And therefore N M is the COT, or of the Measure of the micircle, W. W D. : DK, H T, are Quadrants, DK ID, are equal to two Quadrants, erefore K T, or the Measure of is the Supplement of the Side HD ifter the same manner, it is demon the Measure of the Angle XMN, it of the Side GH; and B A, the Angle X, is the Supplement of the .W.D. POSITION XV. . Spherical Triangles are also equi lateral. ir Supplementals (by 14 of this) are equilaid therefore equiangular also; and so them. kewise equilateral (by Part 2. Prop. 14). PRO PROPOSITION III. If a great Circle ECF be described about the Pole 4 ; tben the Arc CF, intercepted between A C, CF, is ibe Measure of the Angle C AF, or CDF. Vid. Fig. to Prop. 1. TH E Arcs A C, A F (by Cor. 1. Prop. 2.) are Qua drants; and, confequently, the Angles ADC, A DF, 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 ACB, AFB, 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 comprehended by the two Sides also equal. PROPOSITION V. Also Triangles shall be equal and congruous, if one Side, together with the adjacent Angles in one Triangle, be equal to one side, and the adjacent Angles of the other Triangle. PROPOSITION VI. Triangles mulually equilateral, are also mutually equiingular. PROPOSITION VII. In Tosceles Triangles, the Angles at the Base one equal. PRO PROPOSITION VIII. And if the Angles at the Base be equal, then the Triangle shall be lfosceles. TH HES E five last Propositions are demonstrated in the same manner, as in plane Triangles. PROPOSITION IX. Any Two Sides of a Triangle are greater than the tbird. FOR OR the Arc of a great Circle is the shortest Way, : between any two Points in the Superficies of the Sphere. PROPOSITION X. Semicircle. ABC, be produced till they meet in D: Then shall the Arc A CD, which is greater than the Arc AC, be a Semicircle. PROPOSITION XI. The three Sides of a Spherical Triangle are less tban a wbole Circle. FOR BD:+DC is greater than B (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. MAKE the Angle B A D=Angle B; then shall AD=BD (by 8 of this); and to B DC=D A to DC, and these Arcs are greater than A C. Wherefore the Side B C, that subtends the Angle B AC, is greater than the Side A C, that subtends the Angie 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 tbe internal Angle at ibe Bale BAC shall be greater, equal, or less, than the external and opposite Angle BCD; and to the Sum of the Angles A and A C B fall also be greater, equal, or less, than two Right Angles. FIRST, let A B+AC=Semicircle=AD; then BC=BD, and the Angles BCD and D equal (by 8 of this); and therefore the Angle B C D thall be=Angle A. Secondly, let A B+B C be greater than ABD; 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 demonftrate, 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 B C A are two Righe Angles, if the Angle A be greater than the Angle BCD, then shall A and B C A be greater than iwo Right Angles; if the Angle A-BCD, then fhall A and BCA be equal to two Right Angles, and if Abe less than BCD, then will A and BCA be less than two Right Angles. W. W.D. PROPOSITION XIV. In any Spherical Triangle G HD, tbe 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, 10 the Semicircles, and the Arcs that are the Measures of ibe Angles M, X, N, will be the Supplements of the Sides GH, GD, and H D, to Semicircles. FROM the Poles G, H, Di let the great Circles then, 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 TM O, then will H M be also a Quadrant, and so (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 shall be the Pole of the Circle HD. And because GX, D X, are Quadrants, X will be the Pole of the Circle G D. These Things premised, Because N K=Quadrant, and X B=Quadrant (by Cor. 1. Prop. 2.); then will N K+X B, that is, NX +K B=two Quadrants, or a Semicircie ; and so NX is the Supplement of the Arc K B, or of the Measure of the Angle HDG, to a Semicircle. In like manner, because 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 HGD. Likewise, since MO, NT,=are Quadrants, we thall have M ON T= OT+N M=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. W D. Moreover, because DK, H T, are Quadrants, D K +HT, or KT+HD, are equal to two Quadrants, or a Semicircle ; therefore KT, or the Measure of the Angle XNM, is the Supplement of the Side HD to a Semicircle. After the same manner, it is demonftrated, 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. FO OR their Supplementals (by 14 of this) are equila teral, and therefore equiangular also; and so them. felves are likewise equilateral (by Part 2. Prop. 14). PRO |