to the tangent of the angle ABC opposite to that fide; and as the radius is to the co-tangent of the angle ABC, fo is the tangent of the fame angle ABC to the radius, (Cor. 2. def. Pl. Tr.) by equality, the fine of the fide AB is to the co-tangent of the angle ABC adjacent to it, as the tangent of the other fide AC to the radius. PROP. XVIII. FIG. 13. IN right-angled spherical triangles the fine of the hypothenufe is to the radius, as the fine of either fide is to the fine of the angle opposite to that fide. Let the triangle ABC be right angled at A, and let AC be ei. ther of the fides; the fine of the hypothenuse BC will be to the radius as the fine of the arch AC is to the fine of the angle ABC. Let D be the centre of the sphere, and let CG be drawn perpendicular to DB, which will therefore be the fine of the hypothenuse BC; and from the point G let there be drawn in the plane ABD the straight line GH perpendicular to DB, and let CH be joined: CH will be at right angles to the plane ABD, as was shewn in the preceding propofition of the straight line FA: Wherefore CHD, CHG are right angles, and CH is the fine of the arch AC; and in the triangle CHG, having the right angle CHG, CG is to the radius as CH to the fine of the angle CGH: (1. Pl. Tr.) But since CG, HG are at right angles to DGB, which is the common section of the planes CBD, ABD, the angle CGH will be equal to the inclination of these planes; (6. def. 11.) that is, to the spherical angle ABC. The fine, therefore, of the hypothenuse CB is to the radius as the fine of the fide AC is to the fine of the opposite angle ABC. QE. D. COR: Of these three, viz. the hypothenuse, a side, and the angle oppofite to that fide, any two being given, the third is alfo given by prop. 2. PROP. XIX. FIG. 14. IN right-angled spherical triangles, the co-fine of the hypothenuse is to the radius as the co-tangent of either of the angles is to the tangent of the remaining angle. Let ABC be a spherical triangle, having a right angle at A, the co-fine of the hypothenuse BC will be to the radius as the co-tangent of the angle ABC to the tangent of the angle ACB. Describe the circle DE, of which B is the pole, and let it meet AC in. F, and the circle BC in E; and fince the circle BD paffes through the pole B of the circle DF, DF will also pafs through the pole of BD. (13.18. 1. Theod. sph.) And fince AC is perpendicular to BD, AC will also pass through the pole of BD; wherefore the pole of the circle BD will be found in the point where the circles AC, DE meet, that is, in the point F: The arches FA, FD are therefore quadrants, and likewise the arches BD, BE: In the triangle CEF, right-angled at the point E, CE is the complement of the hypothenuse BC of the triangle ABC, EF is the complement of the arch ED, which is the measure of the angle ABC, and FC the hypothenuse of the triangle CEF, is the complement of AC, and the arch AD, which is the measure of the angle CFE, is the complement of AB. But (17. of this) in the triangle CEF, the fine of the fide CE is to the radius, as the tangent of the other fide is to the tangent of the angle ECF opposite to it, that is, in the triangle ABC, the co-fine of the hypothenuse BC is to the radius, as the co-tangent of the angle ABC is to the tangent of the angle ACB. Q. E. D. COR. 1. Of these three, viz. the hypothenuse and the two angles, any two being given, the third will also be given. COR. 2. And since by this propofition the co-fine of the hypothenuse BC is to the radius as the co-tangent of the angle ABC to the tangent of the angle ACB. But as the radius is to the co-tangent of the angle ACB, so is the tangent of the same to the radius; (Cor. 2. def. Pl. Tr.) and, ex æquo, the co-fine of the hypothenuse BC is to the co-tangent of ( ( of the angle ACB, as the co tangent of the angle ABC to the radius. PROP. XX. FIG. 14. IN right-angled spherical triangles, the co-fine of an angle is to the radius, as the tangent of the fide adjacent to that angle is to the tangent of the hypothenuse. The same construction remaining ; in the triangle CEF. (17. of this) the fine of the fide EF is to the radius, as the tangent of the other fide CE is to the tangent of the angle CFE opposite to it; that is, in the triangle ABC, the co-fine of the angle ABC is to the radius as (the co-tangent of the hypothenuse BC to the co-tangent of the fide AB, adjacent to ABC, or as) the tangent of the side AB to the tangent of the hypothenuse, since the tangents of two arches are reciprocally proportional to their co-tangents. (Cor 1. def. Pl. Гг.) Cor. And since by this proposition the co fine of the angle ABC is to the radius, as the tangent of the fide AB is to the tangent of the hypothenuse BC; and as the radius is to the cotangent of BC, so is the tangent of BC to the radius; by equality, the co-fine of the angle ABC will be to the co-tangent of the hypothenuse BC, as the tangent of the fide AB, adjacent to the angle ABC to the radius. PROP. XXI. FIG. 14. IN right-angled spherical triangles, the co-fine of either of the fides is to the radius, as the co-fine of the hypothenufe is to the co fine of the other fide. The fame construction remaining; in the triangle CEF, the fine of the hypothenuse CF is to the radius, as the fine of the fide CE to the fine of the oppofite angle CFE; (18. of this) that is, in the triangle ABC the co-fine of the side CA is to the radius as the co-fine of the hypothenuse BC to the co-fine of the other fide BA. E. D. PROP. PROP. XXII. FIG. 14. IN right angled spherical triangles, the co-fine of efther of the fides is to the radius, as the co-fine of the angle opposite to that fide is to the fine of the other angle. The fame construction remaining; in the triangle CEF, the fine of the hypothenuse CF is to the radius as the fine of the fide EF is to the fine of the angle ECF opposite to it; that is, in the triangle ABC, the co-fine of the fide CA is to the ra dius, as the co-fine of the angle ABC opposite to it, is to the sine of the other angle. E. D. OF Of the CIRCULAR PARTS. IN any right angled spherical triangle ABC, the complement Fig. 15. of the hypothenuse, the complements of the angles, and the two fides, are called The circular parts of the triangle, as if it were following each other in a circular order, from whatever part we begin: Thus, if we begin at the complement of the hypothenuse, and proceed towards the fide BA, the parts following in order will be the complement of the hypothenuse, the complement of the angle B, the fide BA the fide AC, (for the right angle at A is not reckoned among the parts), and, lastly, the complement of the angle C. And thus at whatever part we begin, if any three of these five be taken, they either will be all contiguous or adjacent, or one of them will not be contiguous to either of the other two: In the first cafe, the part which is between the other two is called the Middle part, and the other two are called Adjacent extremes. In the second cafe, the part which is not contiguous to either of the other two is called the Middle part, and the other two Opposite extremes. For example, if the three parts be the complement of the hypothenuse BC, the complement of the angle B, and the side BA; fince these three are contiguous to each other, the complement of the angle B will be the middle part, and the complement of the hypothenuse BC and the fide BA will be adjacent extremes: But if the complement of the hypothenuse BC, and the fides BA, AC be taken; since the complement of the hypothenuse is not adjacent to either of the fides, viz. on account of the complements of the two angles B and C intervening between it and the fides, the complement of the hypothenuse BC will be the middle part, and the fides, BA, AC opposite extremes. The most acute and ingenious Baron Napier, the inventor of Logarithms, contrived the two following rules concerning these parts, by means of which all the cafes of rightangled spherical triangles are refolved with the greatest cafe. The rectangle contained by the radius and the fine of the middle |