THEOREM III. As radius is to the cosine of cne of the legs, so is the cosine of the other leg to that of the hypotenuse. That is, R: cos. AB :: cos. AC : cos. BC; Or, R : cos. AC :: cos. AB : cos. BC. COR. If two light-angled spherical triangles have one common leg, the cosines of their hypotenuses are as the cosines of their other legs. THEOREM IV. As radius is to the sine of either angle, so is the cosine of the adjacent leg to the cosine of the other angle. That is, R: sin. B :: cos. AB : cos. C ; Or, R : sin. C :: cos. AC : cos. B. COR. If two right-angled spherical triangles have one common leg, the cosines of the angles, opposite to this leg, are to each other as the sincs of the adjacent angles. THEOREM V. As radius is to the sine of one of the legs, so is the tangent of its adjacent angle to the tangent of the other leg. That is, R sin. AB : :: tang. B : tang. AC; Or, R R: sin. AC :: tang. C : tang. AB. COR. I. If two right-angled spherical triangles have one common leg, the sines of their other legs are reciprocally as the tangents of the angles at these legs. COR. COR. 2. If they have one common angle, the tangents of their legs, opposite to this angle, are as the sines of the legs adjacent to it. THEOREM VI. As radius is to the cotangent of one of the angles, so is the cotangent of the other angle to the cosine of the hypotenuse; or, which is the same, radius is to the cosine of the hypotenuse, as the tangent of one angle is to the cotangent of the other. NOTE. These six Theorems are sufficient for the solu tions of all the cases of right-angled spherical triangles. THEOREM VII. The product of radius and the sine of the middle part is equal to the product of the tangents of the conjunct extremes, or to that of the cosines of the disjunct extremes.* NOTE. * DEMONSTRATION. This Theorem may be demonstrated by substituting for each particular term the value, specified in the small table belonging to the demonstration of the first Theorem. Thus, if we assume AB, in the figure belonging to the definitions, for the middle part, AC and B are the conjunct extremes, and BC and C the disjunct extremes; then, according to this Theorem, Rx sin. AB tang. AC x cot. B≈sin. BC x sin. C. The expressions of the values of these quantitics being taken from BP the aforesaid table, we have RX GP QPX BP GPXQP NOTE. This Theorem is general, and equally applica ble to every case of Rectangular Spherical Trigonometry. It is NAPIER's Method of the five circular parts, and is sometimes called the Catholic Proposition. THEOREM VIII. The angles at the hypotenuse are always of the same affection with their opposite sides; and the hypotenuse is less or greater than a quadrant, according as the legs are of the same or different affection.* NOTE. BQX BPXGQ ; each of which expressions is evidently the GQ X BQX GP same, when reduced to its lowest terms. In the same manner may any other case be proved. Therefore the Theorem is true in all cases. Q. E. D. which are supposed to be 90°; that is, they will be of the same affection with their opposite sides. To prove, that, when the legs AB and AC are both acute, their opposite angles will be so also; let the side AC be produced to F, or till it be 90°. Then, as the point F will be the pole of the arc AB, the angle B will also be 90°; and consequently the angle ABC, which is less than the angle ABF, will be acute. We may prove in the same manner, that the angle at C is acute, when the opposite side AB is acute. NOTE. The converse of this Theorem is true in all its parts. And when the hypotenuse is exactly a quadrant, one or each of the legs is 90°. PROBLEM. In a right-angled spherical triangle, any tivo of the six parts being given, beside the right angle, to find the other three. This Problem has six cases. 1. When the hypotenuse and a leg are given. 2. When the hypotenuse and an angle are given. 3. When It is equally evident, that the angles B and C, in the triangle BaC, right-angled at a, are obtuse, when the opposite sides aB, aC, are obtuse. If one of the sides Ab bé obtuse, and the other side AC acute, as in the right-angled triangle bAC, the angle at C will also be obtuse, and that at b acute. For, having taken the arc AG=90° on ABb, and drawn the arc GC from the point G to the point C, the angle ACG will be right, since G is the pole of the arc AC; whence it follows, that the angle ACb will be obtuse. For the same reason the angle abC, in the right-angled triangle baC, will be obtuse, it being opposite to the obtuse side aC; and consequently the supplement of it AbC acute, that is, of the same affection with its opposite side. Therefore the angles at the hypotenuse are of the same affection with their opposite sides. Q. E. 1o. D. 2. It is evident, that EC, considered either as the hypotenuse of the triangle BAC, or that of the triangle BaC, is less than BF; and that the hypotenuse bC, in the triangle baC, is greater than bF; whence it follows, that the hypotenuse of any rightangled spherical triangle is always less than 90°, when the two legs are of the same affection; and greater, when they are of different affection. Q. E. 2°. D. 3. When a leg and its opposite angle are given. 5. When the two legs are given. 6. When the two angles are given. These six cases have sixteen analogies; and the solutions of them, by the general Theorem of NAPIER, immediately follow. CASE I. EXAMPLE. Given the hypotenuse AC=55° 8', and the leg BC=32° 12′; to find the rest. Projection of the triangle.* 1. At the circumference of the primitive. Set BC from x to y; reduce y to z, and draw the quadrant zm. Set the hypotenuse from A to o and p, and draw the parallel oCp; through C, the point of its intersection with zm, draw the right circle BCI, and the oblique circle ACd. B 2. At *In each of the following examples of the six Cases, the triangle is projected at the circumference, at the centre, and in the plane, or between the centre and circumference, of the primitive. |