For, the rectangle contained by the sines of any two sides is to the square of the radius, as the rectangle contained by the sines of half the perimeter and its excess above the base, to the square of the cosine of half the angle opposite to the base, by Prop. 30; and as the rectangle contained by the sines of the two excesses of half the perimeter above each of the sides to the square of the sine of half the angle opposite to the base, by Prop. 29: therefore, the rectangle contained by the sines of half the perimeter and its excess above the base, is to the rectangle contained by the sines of the two excesses of half the perimeter above each of the sides, (as the square of the cosine to the square of the sine of half the angle opposite to the base, that is,) as the square of the radius to the square of the tangent of half the angle opposite to the base. Q. E. D. * Solution of the several Cases of Right Angled and Oblique Angled Spherical Triangles. GENERAL PROPOSITION. In a right angled spherical triangle, of the three sides and three angles, any two being given, besides the right angle, the other three may be found. 1. Rules for the sixteen Cases of Right Angled Spherical Trigonometry. See Fig. 16. The two quantities, in the same analogy, marked with asteriscs, are both of the same affection; that is, both at the same time less, or both at the same time greater, than 90°. Cases 7, 8, 9, are doubtful, for two triangles may have the given things, but have the things sought in one of them the supplements of the things sought in the other. In the remaining cases, if the two given quantities which occur in the previous terms of the same analogy are both of the same affection, the last term will be less than 90°; but if they are of different affection, the last term will be greater than 90°. These limitations are founded on the 13th, 14th and 15th Propositions. 2. Rules for the sixteen Cases of Right Angled Spherical 3. Rules for the Twelve Cases of Oblique Angled Spherical Triangles. Rule. Draw the perpendicular CD from C the unknown angle, not required, on AB. R: cos A tan AC: tan AD, less or greater than 90o, as A and AC are of the same or of different affection. BD ABAD. sin BD sin AD: : tan A: tan B, of the same or of different affection, as AD is less or greater than AB. cos AD: cos BD:: cos AC: cos BC, less or greater than 90°, as A and BD are of the same or of different affection. Draw the perpendicular CD, from C the end of the given side next to BC, the side sought. R: cos AC: tan A: cot ACD, less or greater than 90o, as AC and A are of the same or of different affection. BCD ACB ACD. cos BCD: cos ACD:: tan AC: tan BC, less or greater than 90o, as A and BCD are of the same or of different affection. sin ACD: sin BCD cos A: cos B, of the same or of different affection with A, as ACD is less or greater than ACB. The affection of B sin BC: sin AC:: sin A: sin B. R: cos AC:: tan A: cot ACD, less or greater than 90)o, : : R: cos A tan AC: tan AD, less or greater than 90°, as A and AC are of the same or of different affection. cos AC cos BC: cos AD: cos BD, greater than 90°, if one or all of the 3 terms AD, AC, BC, are greater than 90°; otherwise less than 90°. AD+BD AB, doubtful. If AD+BD exceed 180°, take their difference; if BD is greater than AD, take their sum for A B. Sought. 9. 10. 11. opposite to the Rule. sin B sin A:: sin AC: sin BC, doubtful; unless it can be determined by this rule, that according as A+B is greater or less than 180°, AC+BC is also greater or less than 180°. : The side ABR cos A:: tan AC: tan AD, less or greater than 90o, The third angle ACB. An angle 12. A side AC. R: cos AC: tan A: cot ACD, less or greater than 90°, as AC and A are of the same or of different affection. cos A cos B: sin ACD: sin BCD, doubtful. ACB =ACD+BCD. If ACD+BCD exceed 180o, take their difference; if BCD be greater than ACD, take their sum for ACB. Let S denote the sum of the three sides; then will −sin ( ¦ S—AC). sin ( ¦ S—AB). rad2 =√ [sin (} 1. sin A, 2. cos A4 3. tan A sin AC. sin AB sin S. sin (1S—BC). rad2 sin AC. sin AB 4. Rules for the first Ten Cases of Oblique Angled Spherical Triangles, expressed in general terms. |