by geometrical construction, and subsequently derive all others from it. Let a, b, c, be the sides, and A, B, C, the angles of a spherical triangle, as usual. From the vortex of the angle C let tangents be drawn to the arcs a and b; and from the centre of the sphere let the radii through the vertices of the angles A and B be drawn and produced to meet the tangents; and let the points where they meet the tangents be connected by a right line D. The produced radii are evidently the secants of the sides a, b, the radius of the sphere being unity, and the parts of the tangents intercepted between the secants and the vertex of C are the tangents of the sides. The angle under the tangents is equal to C, and that under cos. A = = sin.c sin. (s—b)sin. (s—c) sin.b sin.c sin.s sin.(s sin.b sin.c a) sin.(sa) sin. (sc), sin.a sin.c sin.8 sin. (8 -- b) sin.a sin.c [4]; sin.(A+B) sin. A cos. B sin. B cos. A, sin. a sin.b cos. C: = sin.s sin.(s—c), sin.(sb)+ sin. (s—a) = 2sin. c cos.(a - b), sin.(s-b)-sin.(sa) sec. a sec.b tan. a tan.b. cos.a cos.b' sin. a sin.b give = cos.a cos.b' cos.a cos.b+sin.a sin.b cos. C—cos.co. This principle being successively applied to each of the three angles A, B, C, gives a system of three formule, which may be expressed thus: Cos.a cos.A sin.b sin.c - cos.b cos. c = 0, cos. (b+c) + sin.b sin.c— cos.b cos.c= cos. (bc)cos. a+ sin.b sin. c(1+ cos. A) = 0,2 cos.a+sin.b cos. (bc). cos.a-sin.b sin. c(1— cos. A) cos. (b+c)-cos.a= 2sin. (a+b+c) sin. (b+c-a), cos. (b-c-cos. a 2sin. (a+bc) sin.(a+c—b), 2sin. 1(a+b—c) sin.(a+c-b), 1 + cos.A = 2cos.'A, 1-cos. A 2sin.A. Hence, if cos. (A-B) = sin. C 2 cos.c cos. (a+b) 2cos.c sin.(a - b), } [5]. } sin C sin.(a+b) = sin. c [6] respectively, we obtain (30.) By dividing the formulæ of [5] by those of tan.(A+B) = tan.(A-B) = cos. (a—b) cos.(a+b) sin.(a-b) sin.(a+b) cot. C cot. C (31.) The polar triangles furnish a rule by which every group of formulæ expressing relations between the sides and angles of a spherical triangle can be converted into another group giving other relations between the same quantities. Any formula may be applied to the polar triangle by changing the sides into the supplements of the angles, and vice versa. But since the sine and cosecant of the supplement of an angle are the same as those of the angle itself, and the cosine, tangent, cotangent, and secant of the supplement only differ from those sin. b sin.9c sin. "A=4sin.8 sin. (s—a)sin.(s—b) [2] of the angle itself in sign, it follows that it is al [3] sin. B. sin. C. cos.2 a=cos. (S—B) cos. (S—C) sin. B. sin. C. sin. a=—cos. S. cos. (S-A) sin Bsin.C.sin'a--4cos.S cos. (S-A)cos.(S-B)cos.(S-C) the surrounding circle are placed a. b, and — cos.(SB) cos. (SC) cos. S. cos. (S — A) 2 [ sin. A sin. B sin.C cos. (A-B) tan.c sin.(A+B) -tan.c tan. (a - b) SECTION V. 2 T ---- 2 2 that is, the sides and the complement of the hypo- 2 We shall now prove that the two following forOn the Solution of Right Angled Spherical Trian- mulæ are true, and include all the ten cases before gles. Napier's rules. These are called Napier's rules, and are generally announced thus: 1. "The rectangle under the radius and the sine. of the middle part is equal to the rectangle under the tangents of the adjacent extremes." 2. 66 The rectangle under the radius and the sine of the middle part is equal to the rectangle under the cosines of the opposite extremes." The radius being unity does not appear in the formulæ. Taking each of the five circular parts as middle successively, and making the proper substitutions in the above formulæ, we obtain the ten following equations; which solve the ten cases of right-angled triangles, and are adapted to logarithmic computation. 1. cos.c cot. A cot. B. cos. B small, or if more than ordinary accuracy be requi- Let the computed quantity be expressed as a function of the given quantity, and let the differential co The angle C being determined, the side c may efficient be found with respect to the given quantity as be found by the second case. SECTION VII. On the Relations between the small Variations in the (34.) We have already shown, that in all determinate problems respecting the solution of triangles, it is indispensably necessary that three of the six parts of the triangle should be known, and in plane triangles, one at least of three must be a side. În practice, these data, always obtained originally by observation and measurement, are liable to error from obvious and inevitable causes. It is true that, from the great excellence of instruments, and the almost inconceivable accuracy of modern observation, these errors are extremely minute, yet, in cases where great precision is requisite, it becomes necessary to determine the effects which small er rors in the data will produce upon the computed quantities, and to select the data and quæsita in such a manner, that given errors in the one shall entail upon the other the smallest possible errors. The principles of the differential calculus present easy means for attaining this end. Let us suppose that of the three data, two have been obtained with sufficient accuracy, but the third, x, is liable to an error of a given amount, which we shall call h. Let u be the sought quantity. Two of the three data being considered constant, the sought quantity u may be considered as a function of the third, x, so that u = F(x). The quantity becoming x + h, let the quantity u become u', we have u' = F(x + h) + A... + A3 where A,, A2, A3, 1.2 h3 .... 1.2.3 a variable, the error in the computed quantity will be determined by multiplying the error in the given quantity by this differential co-efficient. If two of the data be liable to given errors, the effect upon the sought quantity may be computed on similar principles, by considering the sought quantity as a function of the two data so liable to error, and differentiating it with respect to these as two independent variables; the differential of the sought quantity thus found will represent the error to which it is liable, the differentials of the data representing their respective errors which are supposed to be very small and given. It is evident that the same method extends to th case where all the data are liable to given small errors. In this case the sought quantity is to be regarded as a function of three variables, and its dif ferential found as before. The principles which have just been established furnish a method by which, when a triangle plane or spherical is subject to minute variations in its To determine the relation between the minute variations of the side of a plane right-angled triangle and the opposite angle, the remaining side being considered constant. = Let a and A be the side and angle which are subject to variation, and b the constant side. a= btan.A, da bsec. 'Ada, which is the variation sought. To determine this for any given value of a, let b be eliminated by the two equations, and the result is are the successive differ daa (cot.A+tan.A)ɗA. = If x be supposed to represent the true value of that part of the triangle which is liable to the error h, then a+h will be the quantity given by observation, and u will be the true value of the sought quantity, and u' its computed value. Hence u'-u is the error sought, which is therefore represented by the above series. Since h in practice is always a very small quantity, this series converges rapidly, and therefore a small number of its initial terms may be assumed as equivalent to the whole, without sensible error. The number of terms to be taken for the whole depends entirely on the magnitude of the error h. In most cases it is sufficient to take the first term only, but in case h be not extremely which is the relation required. Two sides of a plane triangle being given to investigate the relation between the small variations of the included angle and the opposite side. Let a and b be the given sides, and C the included angle, SECTION VII. On the Computation of Trigonometrical Tables. Trigonometrical computation is conducted by the aid of computed tables from which the numerical values of the sines, tangents, &c. of angles referred to some given radius may be found. Of these tables there are two kinds; those in which the immediate values of the sines, &c. are registered, and which are called tables of natural sines, &c. and those in which the logarithms of the sines, &c. are registered, and which are called tables of logarithmic sines, &c. It is not proposed here to enter minutely into the details of the methods of constructing trigonometrical tables, but only to point out in a general way the application of the formula by which the successive terms of a table may be computed, and the methods of checking the errors, whether of the computist or the printer, in those tables which have been already computed: It very rarely happens that the values of the sines or cosines of angles can be exactly expressed by integers or finite decimals. An approximation in decimals can, however, be always obtained to any degree of accuracy which may be required. The ordinary tables give the values continued to seven places of decimals; but tables have been computed extending to ten, and even to fifteen decimal places. If the sines of angles, which are nearly equal to 90°, or the cosines of very small angles be required to that degree of approximation which would determine the results to seconds and tenths, it will be necessary to extend the computation to twelve decimal places, the radius being unity. If we suppose that the series of angles of which the sines and cosines are to be tabulated, are in arithmetical progression the common difference, and the number of places in the approximate values will depend each upon the other. If A be an angle of the table, and x the common difference, then several successive angles of the table will be A A+x, A + 2x, A + 3x. Now if upon calculation it be found that the sines of several of these successive angles agree in the first seven places, it is plain that seven places do not give a sufficient approximation to distinguish angles differing by so small a quantity as x. Let us suppose that the computed value of the sines of A, A+ and A+ 2x, were the same as far as seven places, but that the seventh place in sines (A + 3x) were different; it is obvious that in this case it would be useless to tabulate sin. (A+), sin. (A +2x), and that if the approximation be limited to seven places, sin. (A + 3x) should succeed sin. A, and therefore that the common difference should be 3x; or if the common difference x be retained, the approximation must be continued until the last digit of sin. (A + x) differ from that of sin. A. It should also be observed, that the degree of approximation necessary to distinguish angles, having a given difference x, also depends on the values of the angles themselves, since the variation of the sine is very slow with respect to the variation of the arc, if it be nearly 90°, and that of the cosine if it be very small. In calculations requiring an extreme degree of accuracy, therefore, it is frequently necessary to compute the values of the sines or cosines to a greater number of places than are given in the tables. If a be the least angle in the proposed table, and the successive terms be the multiples of x, 2x, 3x, 4x, &c. any three successive tabulated angles will be (n−1)2, nữ, (n+1), and we have the relation between their sines and cosines; = sin.(n+1)x sin.(n-1)x + 2sin.x cos. nx, cos.(n + 1)x = cos.(n − 1)x · 2sin.x sin. nx. By these formulæ, if the first term sin.æ be known, and also the sines of (n-1)x and nx, the sine of (n + 1)x may be computed. That is, if the first term of the tables, and any two successive terms be known, all the succeeding ones may be computed. By substituting successively 1, 2, 3, . . . . for n, we obsin. 2x = 2sin.x cos.x cos.2x = 1—2sin. x Š tain * The notation (2), (3), (4), &c. is here used to express 1.2, 1.2.3, 1.2.3.4, &c. VOL. XVIII. PART I. L* |