Fig. XXX. Fig. As LO:DC::OK:D X, It will alfo be As LONE: DC * EC :: OK*OB:DX *OK: Or rejecting the common Altitude O K, it will be The Versed Sine of the Angle fought. Which was to be Demon- 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 Base, and Difference of the Legs, To the Rectangle made of the Radius, and half the Verfed Sine of the Angle enquired. DEMONSTRATION.. It is already proved by the last Theorem, So is OB: DX. And therefore also, LONE: Sq. of Rad. :: half OB: 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 Base ES and E O, the Difference of the Sides A K and A E; And, As R: half OS:: OH: half OB. And, LONE: Sq. Rad. :: half OB Rad.: half D X * Rad. LONE: Sq. R.:: half OS * half O H *: half D X * R. XXXI. Which was to be Demonstrated. THEOREM VI. In all Spherical Triangles, As the Rectangle of the Sines of the Sides containing the enquired Angle, 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. Fig. It is already proved by the last Theorem, That But the Rectangle made of Radius, and half the Versed 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 D T, then is DX the Versed Sine of that Arch; and D F the Right Sine of half the Arch; and the Triangles DFR and DTX are like. Therefore, DR:DF:: half DT (=DF): half D X: Therefore, LO*NA: Rq.:: half O S * halfO H: the Square of D F. Fig. XXXII. CHAP. VII. The Solution of the Twelve Cases of Oblique-angled Spherical Triangles, without any Regard had to a Perpendicular let fall, whereby to reduce it into Two Right-angled Triangles. A ND the Spherical Triangle which I shall make use of is that noted with ZSP, whose Sides and Angles, both in Sexagenary Degrees and Minutes, and Decimal Parts also, are as in this Table is expressed. Fig. CASE I. Two Sides, ZS and ZP, with the Angle P, oppo XXXII. fite to one of them, being given; To find the Angle S, opposite Cafe I. to the other. Fig. ΧΧΧΙΙ. Cafe II. CASE II. Two Angles, S and P, with the Side ZP, oppofite 10 one of them, being given; To find the Side Z S, opposite to the other Analogy. As the Sine of the Angle S, 36 Deg. 8 Min. Co-Ar. Is to the Sine of ZP, 24 Deg. 4 Min. So is the Sine of the Angle P, 46 Deg. 18 Min. To the Sine of Z S, 30 Deg. 0.2293936 9.6104465 98591180 19.6989581 CASE |