middle part, is equal to the rectangle contained by the tangents of the adjacent parts. 1 RULE II. 1 The rectangle contained by the radius, and the fine of the middle part is equal to the rectangle contained by the cofines of the oppotite parts. Fig. 16. These rules are demonstrated in the following manner. First, Let either of the fides, as BA, be the middle part, and therefore the complement of the angle B, and the fide AC will be adjacent extremes. And by Cor. 2. prop. 17. of this, S, BA is to the Co-T, B, as T, AC is to the radius, and therefore RX S, BACO-T, BXT, AC. The fame fide BA being the middle part, the complement of the hypothenuse, and the complement of the angle C, are oppofite extremes; and by prop. 18. S. BC is to the radius, as S, BA to S, C; therefore RXS, BAS, BCXS, C. Secondly, Let the complement of one of the angles, as B, be the middle part, and the complement of the hypothenuse, and the fide BA will be adjacent extremes : And by Cor. prop. 20. CoS, B is to Co-T, BC, as T, BA is to the radius, and therefore RX Co S, B = Co-T, BCXT, ВА. Again, Let the complement of the angle B be the middle part, and the complement of the angle C, and the fide AC will be opposite extremes: And by prop. 22. CoS, AC is to the radius, as Co S, B is to S, C: And therefore RX Co S, B=Co S, ACXS, C. Thirdly, Let the complement of the hypothenuse be the middle part, and the complements of the angles B, C, will be adjacent extremes: But by cor. 2. prop. 19. CoS, BC is to CoT, Cas to Co-T, B to the radius: Therefore RX Co S BCCOT, В X Со-Т, С Again, Let the complement of the hypothenuse be the middle part, and the fides AB, AC will be opposite extremes : But by prop. 21. Co S, AC is to the radius, Co S, BC to Co S, BA; therefore R X Co S, BC = COS, BAX COS, AC. Q. E. D. Σ SOLU. 509 SOLUTION of the Sixteen CASES of right Fig. 16. angled Spherical triangles. GENERAL PROPOSITION. IN a right angled spherical triangle, of the three fides, and three angles, any two being given besides the right angle, the other three may be found. In the following table the solutions are derived from the preceding propofitions. It is obvious that the fame folutions may be derived from Baron Napier's two rules above demonstrated, which, as they are easily remembered, are commonly used in practice. 3 B, C AC S, C: Co S, B:: R: COS, AC: By 22. and R:COS, BA:: COS, AC: COS, BC 21. and if both BA, AC be greater or less than a 4 BA, AC BC quadrant, BC will be less than a quadrant. But if they be of different affection, BC will be greater than a quadrant. 14. 5 BA, BC AC 6 BA, AC B COS, BA: R:: Co S, BC: COS, AC 21. S, BA: R:: T, CA: T, B. 17. and B is Cafe 16 BC, C в CoS, C: R:: T, AC: T, BC. 20. And BC is less or greater than a quadrant, according as C and AC or Cand B are of the fame or different affections. 14. I. T,BC: R:: T, CA: COS, C. 20. If BC be less or greater than a quadrant, CA and AB, and therefore CA and C, are of the fame or different affection. 15. R: S, BC:: S, B: S, AC. 18. And AC is of the fame affection with B. S, B: S, AC::R: S, BC: 18. S, BC: R:: S, AC: S, B:18. And Bis of the fame affection with AC. T, C: R:: Co T, B: Co S, BC. 19. And according as the angles B and C are of different or the fame affection, BC will be greater or less than a quadrant. 14. R: Co S, BC:: T, C: CoT, B. 19. If BC be less or greater than a quadrant, C and B will be of the fame or different affection. 154 The second, eighth, and thirteenth cafes, which are commonly called ambiguous, admit of two solutions: For in these it is not determined whether the fide or measure of the angle fought be greater or less than a quadrant. PROP. XXIII. FIG. 16. IN spherical triangles, whether right angled or oblique angled, the fines of the fides are proportional to the fines of the angles opposite to them. First, Let ABC be a right-angled triangle, having a right angle at A; therefore by prop. 18. the fine of the hypothenufe BC is to the radius (or the fine of the right angle at A) as the fine of the fide AC to the fine of the angle B. And, in like manner, the fine of BC is to the fine of the angle A, as the fine of AB to the fine of the angle C; wherefore (11.5.) the fine of the fide AC is to the fine of the angle B, as the sine of AB to the fine of the angle C. Secondly, Let BCD be an oblique-angled triangle, the fine of either of the fides BC, will be to the sine of either of the other two CD, as the fine of the angle D opposite to BC is to the fine of the angle B oppofite to the fide CD. Through the point C, let there be drawn an arch of a great circle CA perpendicular upon BD; and in the right-angled triangle ABC (18. of this) the fine of BC is to the radius, as the fine of AC to the fine of the angle B; and in the triangle ADC (by 18. of this:) And, by inverfion, the radius is to the fine of DC as the fine of the angle D to the sine of AC: Therefore, ex æquo perturbate, the fine of BC is to the fine of DC, as the fine of the angle D to the fine of the angle B. Q. E. D. PROP. XXIV. FIG. 17. 18. IN oblique-angled spherical triangles, having drawn a perpendicular arch from any of the angles upon the oppofite fide, the co-fines of the angles at the base are proportional to the fines of the verticle angles. Let Fig. 17.18. Let BCD be a triangle, and the arch CA perpendicular to the bafe BD; the co-fine of the angle B will be to the co-fine of the angle D, as the fine of the angle BCA to the fine of the anole DCA. For by 22. the co-fine of the angle B is to the fine of the angle BCA as (the co-fine of the side AC is to the radius; that is, by Prop. 22. as) the co-fine of the angle D to the fine of the angle DCA; and, by permutation, the co fine of the angle B is to the co fine of the angle D, as the fine of the an gle BCA to the fine of the angle DCA. Q. E. D. T PROP. XXV. FIG. 17. 18. HE fame things remaining, the co-fines of the fides BC, CD, are proportional to the co-fines of the bafes BA, AD. For by 21. the co-fine of BC is to the co-fine of BA, as (the co-fine of AC to the radius; that is, by 21.as) the co-fine of CD is to the co-fine of AD: Wherefore, by permutation, the co-fines of the fides BC, CD are proportional to the co-fines of the bases BA, AD. Q.E. D. T PROP. XXVI. FIG. 17. 18 HE same conftruction remaining, the fines of the bafes BA, AD are reciprocally proportional to the tangents of the angles Band D at the base. For by 17. the fine of BA is to the radius, as the tangent of AC to the tangent of the angle B; and by 17. and in verfion, the radius is to the fine of AD, as the tangent of D t the tangent of AC: Therefore, ex aequo perturbate, the fint of BA is to the fine of AD, as the tangent of D to the tan gent of B. i PROP |