1 is the Diameter, and O V the Verfed Sine of P Z S the Angle Fig. Sought And the Right Lines NC, KL, and RG, being Paral- XXIX. lel, by the Work, their Inter-fegments M B and R C, and alfo MH and S C, are proportional. THEOREM IV. In all Spherical Triangles; As the Rectangle of the Sines of the Sides containing the Angle enquired, Is to the Square of the Radius: So is the Difference between the Verfed Sine of the Bafe, and the DEMONSTRATION. Let the Sides of Triangle A E K be given, and let the Angle at Fig. A be enquired; and from O, let fall the Perpendicular O B. Now then, OE being the Difference of the Sides AE and A K, equal to AO; the Right Line OQ is the Right Sine thereof, and E Q the Verfed Sine. In like manner, SM is the Right Sine, and EM the Verfed Sine of E S; that is, of the Eafe EK; and M Qor OB, is the Difference of thofe Verfed Sines. OK, is the Verfed Sine of the Angle OA K, in the Measure of the Parallel O F; and D X is the Verfed Sine of the fame Angle in the measure of a Great Circle, whofe Diameter is H D. Now then, because of their like Arches, it fhall be As DC DX:: OL: 0 K: And because NE and OK are Parallel, as alfo E C and OB, the Angles BOK and C E N are equal: And the Triangles E CN and OK B like; and therefore the Sides N E, E C, O B, MQ and OK, are Proportional: And it will be M As XXX. Fig. XXX. Fig. As LO DC: OK: D X, As LONE: DC EC::OK*OB: DX *OK: As LONE: DC* EC::OB: DX, The Verfed Sine of the Angle fought. Which was to be Demonftrated. THEOREM V. In all Spherical Triangles, As the Rectangle of the Sines of the Sides containing the enquired Is to the Square of the Radius; So is the Rectangle of the Sines of the half Sum, and half Difference of the Bafe, and Difference of the Legs, To the Rectangle made of the Radius, and half the Verfed Size of the Angle enquired. DEMONSTRATION.. It is already proved by the laft Theorem, XXXI. That, As LO NE to the Square of the Radius: And therefore alfo, LONE: Sq. of Rad.:: half O B: half D X.. And now in the following Diagram, OEGH, the Sum of ES and OE, is the double Measure of the Angle BSO; and the Arch OS is the Difference between the Bafe ES and E O, the Difference of the Sides A K and A E; And, As R half OS:: OH: half O B. And, Therefore, LONE: Sq. Rad.:: half OB Rad.: half D X * Rad. And And also, LONE: Sq. R.:: half OS half O H*: half D X * R. THEOREM VI In all Spherical Triangles, As the Rectangle of the Sines of the Sides containing the enquired Is to the Square of Radius: So is the Rectangle of the Sines of the half Sum, and half Difference of the Bafe, and Difference of the Legs, To the Square of the Sine of half the Angle enquired. DEMONSTRATION. It is already proved by the laft Theorem, That Fig. XXXI. Fig. LO*NE: Sp. R.:. half OS half OH : half D X*R. XXXI. But the Rectangle made of Radius, and half the Verfed Sine of an Arch, is equal to the Square of the Sine of half the Arch: As in the foregoing Diagram; let the Arch given be DT, then is DX the Verfed Sine of that Arch; and D F the Right Sine of half the Arch; and the Triangles D F R and DTX are like. DR: DF: Therefore, half DT (=D F): half D X: And, DR half D X, is equal to the Square of D F. Therefore, LONA: Rq. :: half OS halfOH: the Square of DF. Which was to be Demonftrated. M 2 CHAP. CHAP. VII. The Solution of the Twelve Cafes of Oblique-angled Spherical Triangles, without any Regard had to a Perpendicular let fall, whereby to reduce it into Two Right-angled Triangles. Fig. ND the Spherical Triangle which I fhall make use of is II. A XXXII. that noted with Z S P, whofe Sides and Angles, both in Sexagenary Degrees and Minutes, and Decimal Parts alfo, are as in this Table is expreffed. Fig. CASE I. Two Sides, ZS and Z P, with the Angle P, oppoXXXII. fite to one of them, being given; To find the Angle S, oppofite Cafe I. to the other. Fig. XXXII. CASE II. Two Angles, S and P, with the Side Z P, oppofite to one of them, being given; To find the Side Z S, oppofite to the other Analogy. As the Sine of the Angle S, 36 Deg. 8 Min. Co-Ar. So is the Sine of the Angle P, 46 Deg. 18 Min. 0.2293936 9.6104465 98591180 9.6989581 CASE |