by geometrical construction, and subsequently de sin. A = rive all others from it. s— sin.b sin.c. Let a, b, c, be the sides, and A, B, C, the angles of a spherical triangle, as usual. From the vortex sin. A 2[sin.s sin.(s—a)sin.(8—6) sin(s—c)] of the angle C let tangents be drawn to the arcs a [3] sin.a sin. a sin.b sin.c and b; and from the centre of the sphere let the radii through the vertices of the angles A and B This formula being a symmetrical function of be drawn and produced to meet the tangents; and the sides of the triangle will remain unchanged, if let the points where they meet the tangents be con b be changed into a or C, and vice versa. Hence nected by a right line D. The produced radii are we infer in general, that sin. A sin.B sin. C evidently the secants of the sides a, b, the radius of [4]; the sphere being unity, and the parts of the tan sin.a sin.b sin.c gents intercepted between the secants and the ver (28.) By [2] tex of C are the tangents of the sides. The angle sin. LA sin.(s — b)sin.(3-0) under the tangents is equal to C, and that under sin.b sin.c the secants to C. sin.8 sin.( sa). By applying the principle (75.) to the two plane cos. 4A sin.b sin.c triangles of which D is a common side, and of which sin.(8-a) sin.( sc), the remaining sides are the tangents and secants of sin." B a and b, we obtain sin.a sin.c Do = tan. Sa+tan.”—2 tan. a tan. b cos. C. sin.8 sin.(8--b) cos."IB D' – sec. *a+sec. ab- 2 sec. a sec. b cos. C. sin.a sin.c Subtracting the former from the latter, and ob- which, with serving the conditions. sin.}(A + B) = sin. J A cos.B + sin. B cos. 1A, sec. a — tan.'d = 1, sin.a sin.b cos."{C = sin.8 sin.(3-0), sec.ob-tan. 1, sin.(8-5)+ sin.( sa) = 2sin.dc cos. (a - b), 1 sec. a sec.b = sin. (8--6) --- sin.(-a) = 2cos. c sin./(a - b), cos. a cos.6' give sin. a sin.b cos. C tan. a tan.b. sin. (A+B) cos. (a--) cos.acos.b' cos.c [5]. we obtain cos. C sin. (A-B) = sin./(a−b) cos. a cos.b + sin.a sin.b cos.C-cos.c =0. sin.c This principle being successively applied to each (29.) By a process precisely similar, we obtain of the three angles A, B, C, gives a system of three cos.}(A+B) sin. formulæ, which may be expressed thus: cos. (a+b) cos. C sinc cos. [6]. 16 cos. B sin.lc sin.la cos.c cos. a=0 (1,] įsin. (a+b) cos.}(A-B) sin.c which formulæ are the foundation of 'spherical [6] respectively, we obtain (30.) By dividing the formulæ of [5] by those of trigonometry. (26.) By [1] cos. (a-6) tan. (A+B) = cos.a - cos. A sin.b sin.c-cos.b cos. C=0, cos. (a+b) [7] sina-6) cos. (b + c) + sin.b sin.c- cos.b cos.c= = 0, tan.(A-B) = cot. C sin. (a+b) cos.(b-c) - cos.a-sin.b sin.clmcos. Aj = 0. (31.) The polar triangles furnish a rule by which cos. (b+c)—cos.a=2sin. f(a+b+c) sin. $(6+—a), every group of formulæ expressing relations be (bnc)—cos.d = 2sin. 1(a+bc) sin.dia+b), tween the sides and angles of a spherical triangle 1 + cos.A = 2cos.'JA, 1 - cos. A = 2sin." A. can be converted into another group giving other Hence, if relations between the same quantities. Any fors= }(a + b + c), .:(8-a)= }(b + (-a), mula may be applied to the polar triangle by chang(8—b) = ja +Cb, (8-0)= (a + b—c), ing the sides into the supplements of the angles, and we obtain vice versa. But since the sine and cosecant of the sin.b sin.c cos.”LA £ sin. 8 sin.(sma) supplement of an angle are the same as those of sin.b sin.c sin.) A = sin.(s—b)sin.( sc) the angle itself, and the cosine, tangent, cotangent, sin. b sin.'c sin."A=4sin.: sin.( sa)sin.(5%) and secant of the supplement only differ from those sin.(s—c) [2] of the angle itself in sign, it follows that it is altan.JA sin.(8-b)sin.( sc) lowed in any formulæ to change a, b, c, into A, B, sin.8 sin.(sma) C, and vice versa, provided that the signs of all coIt is evident that four formulæ analogous to these sines, tangents, cotangents, and secants, be changed. are applicable to each of the three angles, and may all the formulæ which have been established in this The changes thus indicated being effected upon be derived from these by merely changing the leto section, the result will be a series of analogous forters. mulæ. (27.) By the third of the group [2] we obtain It may be observed in general, that if a formula VOL. XVIII. PART I. K* } } } C cor.£C ? +} COS. 37 2 2 2 la cos.B. be a symmetrical function of the sides and angles, Let ABC be a spherical triangle right angled at this change in the parts produces no change in the C, and let the sides and angles be expressed as bithwhole. erto. Let the triangle be imagined to be placed This observation applies to [4.] within a circle, which is merely used to mark the orTo make this transformation on [2,] [3,] [7,] let der of certain quantities, to which, and to their order S= A + B + C); thus determined, we shall have occasion to refer. and let the sides and angles of the supplemental Opposite to the arrows which point from the sides to polar triangle be a', b', c', A', B', C', с sa'= (S—A, ) B -B -A B IC B A B sin. B. sin. C. sin.'ļa= cos. S. cos. (SMA) sinoBsin”.C.sin’a=-4cos.S cos. (S-A)cos.(S—B)cos.(S—C) the surrounding circle are placed a. b, and --, cot.";a= cos.(S – B) cos. (S – C) that is, the sides and the complement of the hypocos. S. cos. (S - A) thenuse. Opposite to the arrows which point from sin.a 2 [ cos. S. cos. (S-Acos.)(S – B) cos. (S-C)] the angles are *-A, and B, that is, the com 2 sin. A sin. A sin.B sin.C. plements of the angles. These quantities which -) tan.}(a + b) thus surround the circle are called circular parts. cos(A+B) Any one of these being taken as middle pari (M,) sin. (A-B) [11]. those which are next to it on each side going round tan. (a - b) ) the circle are called adjacent extremes (A, A'); and The formulæ [7] and [11] are called “Napier's the remaining iwo are called opposite extremes (0, Analogies,” that mathematician having been the O'). Thus, if A be the middle part, the adfirst to establish them, and to apply them to the solution of spherical triangles, jacent extremes will be cand b; and the opposite B and a. 2 We shall now prove that the two following for. On the Solution of Right Angled Spherical Trian mulæ are true, and include all the ten cases before gles. Napier's rules. mentioned; sin.M= tan. A tan.A', (32.) In a right angled spherical triangle there sin.M = cos. O cos. O'. are five quantities which may become the objects These are called Napier's rules, and are generally of computation, scil. two angles and three sides. announced thus; Any two of these five quantities being known or 1. “The rectangle under the radius and the sine discoverable, the other three may, in general, be of the middle part is equal to the rectangle under computed. The solution of right angled triangles, the tangents of the adjacent extremes." therefore, is resolved into as many cases as there 2. “ The rectangle under the radius and the sine are different combinations of two to be made from of the middle part is equal to the rectangle under 5.4 the cosines of the opposite extremes.” The radius being unity does not appear in the We triangles, and are adapted to logarithmic compushall first explain these two celebrated rules, and tation. then show that they comprise all the cases. 1. cos.c = cot. A cot. B. 2 7 =10. 1 cos kia—b)cot.C tan.}(A—B)=sin.:(a+b) 2. cos.c = cos.a cos.b. 40.cot.ja= Ico cos.(S—B) cos.(S-C) cos. S cos.(S-A) Ill. Given two sides and the included angle. tan. (A+B= cot cos. (a+b) 8. cos. B = tan.a cot.c. sin. Ia-6) 9. sin. a tan.b cot. B. cot.ic Having determined the sum and difference of the remaining angles by any of these systems, the an gles may be immediately found by addition and To determine the remaining side c, having pre- sin.a sin. (= sin, C. triangles. There would then be as many distinct sin. A systems of data in the solution of oblique spherical IV. Given two angles and the included side. triangles as there could be combinations of three 6.5.4 made from six cos.d(A-B) tan.i(a+b) = = 20. They may be, how. tan.ic. 1.2.3 cos. (A+B) ever, reduced to a smaller number of more compre tan.}(ab) sin.}(A-B) hensive classes. It is obvious that the three data tan.jc. sin. (A+B) The sum and difference of the sides being found, the sides themselves can be determined by addition and subtraction. To determine the remaining angle, having pre- viously determined the side à or l as above, we have V. Two sides and the angle opposite one of them. sin. A sin. B sin.ca sin.c. sin.a sin.b V. Given two sides and the angle opposed to one of them. sin.b sin. A msin.A. sin.a 19. sin. A=N sin.(s—- b)sin.(s—c), Since an angle is to be determined in this case sin.b sin.c from its sine, the result is equivocal; the sine being 2o.cos. A= sin.8 sin.(s--a) common to an angle and its supplement. sin.b sin.c To determine the side c, we have cos. A = cos.b cos.c + sin.b sin.c cos.A, 30. sin.Az V'sin.3 sin.(3-asin.(s—b)sin.(s—c). sin. 6 sin. o = cos.c + tan.b sin.c cos.A. cos.b Let 4o. tan. A= Jsin.(8—6) sin.(s— —c tan.b cos. A = tano, sin.s sin.(sa) cos.c cos.8 + sin.c sin.d These formulæ are all suited to logarithmic cal cos.b cos. Leto be the difference between c and e. Hence COS.Q = COS.A cos.bo The side c being found, the angle C may be de- termined by the first case. VI. Given two angles and the side opposed to one of them. Let the angles be A, B, and the side a. To determine 6, we have 2 sin.B sin.b sin.a sin. B sin.CV-cos. Scos.(S-A)cos.(S—B)cos(s-C.) sin. A 2 COS. COS.a cos. A cos. B The result is also equivocal in this case, and for small, or if more than ordinary accuracy be requithe same reasons as in the last. site, the first two terms are taken; this, however, is To determine the angle C, we obtala by the sup- seldom necessary, so that we may in general assume plemental triangle u U= A, h, cos.o = tan.B cos.a, which gives the following rule for determining the cos. A sin.C cos.8 sin.A cos.C error in a computed quantity, produced by a small error in one of the given quantities from whence it cos.B sin. is derived: sin. (C-1) = sin. Let the computed quantity be expressed as a func tion of the given quantity, and let the differential coThe angle C being determined, the side c may efficient be found with respect to the given quantity as be found by the second case. a variable, the error in the computed quantity will be determined by multiplying the error in the given quanSECTION VII. tity by this differential co-efficient. If two of the data be liable to given errors, the On the Relations between the small Variations in the effect upon the sought quantity may be computed Sides and Angles of Triangles. on similar principles, by considering the sought (34.) We have already shown, that in all deter- quantity as a function of the two data so liable to error, and differentiating it with respect to these minate problems respecting the solution of trian as two independent variables; the differential of the gles, it is indispensably necessary that three of the six parts of the triangle should be known, and in sought quantity thus found will represent the error to which it is liable, the differentials of the data plane triangles, one at least of three must be a side. In practice, these data, always obtained originally representing their respective errors which are sup posed to be very small and given. by observation and measurement, are liable to error It is evident that the same method extends to th from obvious and inevitable causes. It is true that, case where all the data are liable to given small er. from the great excellence of instruments, and the rors. In this case the sought quantity is to be realmost inconceivable accuracy of modern observation, these errors are extremely minute, yet, in garded as a function of three variables, and its dif ferential found as before. cases where great precision is requisite, it becomes The principles which have just been established necessary to determine the effects which small er. furnish a meihod by which, when a triangle plane rors in the data will produce upon the computed quantities, and to select the data and quæsita in sides or angles, the relation between these varia or spherical is subject to minute variations in its such a manner, that given errors in the one shall tions and the sides and angles themselves may alentail upon the other the smallest possible errors. ways be investigated and expressed by equations, The principles of the differential calculus present so that when there are sufficient data, any one of easy means for attaining this end. Let us suppose the variations may be derived from the others. that of the three data, two have been obtained with The following examples will illustrate the appli. sufficient accuracy, but the third, x, is liable to an cation of this principle. error of a given amount, which we shall call h. Let u be the sought quantity. Two of the three data To determine the relation between the minute varibeing considered constant, the sought quantity u ations of the side of a plane right-angled triangle and may be considered as a function of the third, x, so the opposite angle, the remaining side being considthat ered constant. u (2). The quantity « becoming x + h, let the quantity u Let a and a be the side and angle which are subbecome u', we have ject to variation, and b the constant side. u' = F(x + h) btan.A, :;: da = bsec. 'Ada, h h3 which is the variation sought. To determine this .:u-- U=A, for any given value of a, let b be eliminated by the 1.2 where A,, A,, A3, two equations, and the result is are the successive differential co-efficients of F (x). da = a (cot. A+tan.A)dA. dA. sin. 2A If x be supposed to represent the true value of that part of the triangle which is liable to the error Two sides of a plane triangle being given to invesh, then 2th will be the quantity given by observation, and u will be the true value of the sought tigate the relation between the small variations of the included angle and the opposite side. quantity, and u' its computed value. Hence u' - u is the error sought, which is therefore represented Let a and b be the given sides, and C the included by the above series. Since h in practice is always angle, '; a very small quantity, this series converges rapidly, C = a +6° 2abcos.C and therefore a small number of its initial terms ..: cdc= ab sin. CdC may be assumed as equivalent to the whole, with. But šab sin. C being the area of the triangle if p be out sensible error. The number of terms to be taken the perpendicular from C upon c, we have for the whole depends entirely on the magnitude of pc absin.C the error h. In most cases it is sufficient to take •: dc = pdc, the first term only, but in case h be not extremely which is the relation required. + A... + As 1.2.3 + 2a tain SECTION VII. if it be nearly 90°, and that of the cosine if it be On the Computation of Trigonometrical Tables. very small. In calculations requiring an extreme degree of accuracy, therefore, it is frequently necesTrigonometrical computation is conducted by the sary to compute the values of the sines or cosines aid of computed tables from which the numerical to a greater number of places than are given in the values of the sines, tangents, &c. of angles referred tables. to some given radius may be found. Of these tables If x be the least angle in the proposed table, agd there are two kinds; those in which the immediate the successive terms be the multiples of X, 2x, 3x, values of the sines, &c. are registered, and which 4x, &c. any three successive tabulated angles will be arc called tables of natural sines, &c. and those in which the logarithms of the sines, &c. are regis (n − 1)x, nx, (n + 1)x, tered, and which are called tables of logarithmic and we have the relation between their sines and sines, &c. It is not proposed here to enter minutely cosines; into the details of the methods of constructing tri sin.(n + 1). sin.(n2 1). + 2sin.x cos. Nx, gonometrical tables, but only to point out in a gene cos.(n + 1)x = cos.(n − 1)2 — 2sin.x sin. nx. ral way the application of the formulæ by which By these formulæ, if the first term sin.x be known, and the successive terms of a table may be computed, also the sines of (n-1)x and nx, the sine of (n + 1)x and the methods of checking the errors, whether may be computed. That is, if the first term of the of the computist or the printer, in those tables tables, and any two successive terms be known, all which have been already computed: the succeeding ones may be computed. By substiIt very rarely happens that the values of the sines tuting successively 1, 2, 3, .. for n, we ob or cosines of angles can be exactly expressed by integers or finite decimals. An approximation in sin. 2x = 2sin.x cos.x2 decimals can, however, be always obtained to any cos.2x = 1—2sin.°2 } degree of accuracy which may be required. The sin. 3x = sin.x + 2sin.x cos.2x 2 ordinary tables give the values continued to seven COS.3.3 sin.. — 2sin.x sin. 2x 5 places of decimals; but tables have been computed sin. 4x = sin.2x + 2sin.x cos.32 extending to ten, and even to fifteen decimal places. COS.4x = cos.2x — 2sin.x.sin. 3x 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 by which, if the first term of each series be known, necessary to extend the computation to twelve deci. all the others may be found. mal places, the radius being unity, To determine the first terms, we have If we suppose that the series of angles of which X7 the sines and cosines are to be tabulated, are in + 1 (5) (7) arithmetical progression the common difference, x6 and the number of places in the approximate values COS.X. = + will depend each upon the other. If A be an angle (6) of the table, and a the common difference, then As x is always small, these series converge suffi. several successive angles of the table will be ciently to determine sin.x, cos.X, with considerable A A + X, A + 2x, A + 3%. facility. Thus, if x be one degree = 3600", we have Now if upon calculation it be found that the sines 3600 240 of several of these successive angles agree in the first seven places, it is plain that seven places do 206265 13751' 240 not give a sufficient approximation to distinguish •::sin.x= (240) (240)5 + angles differing by so small a quantity as x. Let us 13751 (13751)3x6 (13751)3 X 120 suppose that the computed value of the sines of A, (240) (240) COS.X=I + A + x and A + 2x, were the same as far as seven (13751) X2'(13751) * X 24 places, but that the seventh place in sines (A + 3.0) 60 were different; it is obvious that in this case it If x be one minute, we have = 206265 13751 would be useless to tabulate sin. (A + x), sin. (A +2x), and that if the approximation be limited to in which case the convergence is still more rapid. In order that the values expressed in a given seven places, sin. (A + 3x) should succeed sin. A, and therefore that the common difference should be number of decimal places should be the nearest 3x; or if the common difference x be retained, the possible to the true value, the computation should approximation must be continued until the last be continued to a few additional places. The sines and cosines being found, the tangents, digit of sin. (A + x) differ from that of sin. A. It should also be observed, that the degree of approx- cotangents, secants, and cosecants, may be deter mined by imation necessary to distinguish angles, having a sin.x cos.2 given difference x, also depends on the values of the tan.23 cos. = cos.x' sin.x angles themselves, since the variation of the sine is very slow with respect to the variation of the arc, 23 X5 sin.x = X2 + (2) * 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. |