18 INFINITESIMAL CALCULUS Again, the infinitesimals da, db, dA, dB arc connected by the equation da dY _ dA db tan a tan 13 tan A tan b This follows immediately from the equation sin a sin B = sin A sin I. 36. These and the analogous formula;, when we adopt small differ ences instead of differentials, are of importance in astronomy in determining the errors in a computed distance arising from small errors in observation. They also enable us to determine the cir cumstances under which the most favourable observations are made, viz., those for which small errors in observation produce the least error in the required result. The relations between the variations in the sides and angles of plane and spherical triangles were first treated of by Cotes, in his Estimatio Errorum in mixta Mathcsi (1722). (1) The values of and , when x, it. z are connected by two dx * dx equations of the form/(#,i/,2) = 0, $(x,y,z} = (), are found to be df d(J) df d<p df d<p df d(f> dy dx dz dz dx dz _dy dx dx dy dx df d<f> df d<j> > dx df d<f> df d<f> dz dy dy dz dz dy dy dz (2) If /(w)="0(v), where u and v are each functions of x and y, it is easily shown that du dv du dv dx dy dy dx (3) In a spherical triangle, if sin C the relations sin c be constant, and equal da db do
cos A cos B cos C and cos A da + cos B db + cos C dc = K*d (sin a sin I sin c) can be readily established. (4) More generally, it may be shown that, if K also be supposed to vary, da db dc _, / 1
^ + - ^ = tanAtan BtanC^ -- ,
/ and
^ - ^ =
cos A cos B cos C cos A da + cos B db + cos C dc = xd(it sin a sin b sin c) . (5) If u be a function of t=x H , show that y du du , du and =- du , du ..du dx , du du 7J7 """ 2/7F7 du "Tr,} Taylor s Theorem and Development of Functions. 37. We have already noticed that the development of functions by infinite series was a branch of analysis that rose into prominence during the latter portion of the 17th century. The first series thus published were that of Nicholas Mercator in his Logarithmotechma (1668) for the expansion of log (1 + a;), or what was then styled the area of an hyperbola (this he arrived at by the aid of "Wallis s method of quadratures) ; and that of James Gregory, in a letter to J. Collins, 1671, for the expansion of an arc in terms of its tangent. About the same time the first efforts of Newton s genius were directed to this subject ; and, as we have already seen, he thus arrived at his binomial theorem, and other general expansions, such as those of sin x, cos x, e x , &c. It was not, however, until many years after these discoveries that it was found that all such expansions may be regarded as particular cases of one general theorem. This theorem was discovered by Dr Brook Taylor, and published by him in 1715 in his Methodus Incrementorum. 38. Before proceeding to a consideration of this important series it should be observed that, in 1694, John Bernoulli published, in the Acta Erudilorum, his well-known expansion under the title Additamcntum cffectionis omnium quadraturarum et rcctificationum curvarum per scriem quandam general issimam. This series may be written as follows, slightly altering Bernoulli s notation : I-. <a dx 1 . 2 . o ax Bernoulli obtained this result immediately by differentiation, by which process it can be easily verified. /* , a 2 dy I ydx = xy - -r- - J 1.2 dx This is the first general theorem on series that was discovered ; and it was easily shown by its author that the ordinary series, such as the expansions of log (1 +x), of sin x, and others, can be deduced from it. This theorem of Bernoulli, however, is but a particular case of Taylor s, as will be shown subsequently. 39. Taylor arrived at his theorem as a particular case of another in finite differences, a branch of the calculus treated of for the first time in his Mcth. Incrcm. Introducing the modern notation, Taylor s proof, with some modifications, is as follows. Let/(x) be any function of x, and supposes changed successively into x + Ax, x + 2Ax, x + 2Ax, ... x + nAx; and let the functions f(x) , f(x + Ax), f(x + 2Ax) , . . . f(x + nAx) be represented by y , 2/1 , 2/2 . y> Then we have 2/!-7/=A2/, 2/,,-2/^A?/!, . . . 7/-2/,.-i = A7/ n -i, Ay 1 -Ay = A 2 y, Ay.,- Ay i = A 2 The final result consists in expressing y n in terms of y A 2/ A 2 ;/ , . . . A";/ . "We have In like manner, substituting y n -3 + Ay n -3 for y n -2, we get 2/ = 2/ - s + 3 Ay n - 3 + 3 A 8 i/,, - 3 + A 3 ?/,, - 3 , the coefficients being the same as those in the expansion of (a + &) 3 . Nmv, if we assume that the same law holds for any value n, it is readily seen by the method of mathematical induction, of which we have given an example in 27, that it holds for the value immedi ately superior ; and we thus get y n =y+nAy + - 1.2 1.2.3 40. This result can be readily established also by the principles* of the symbolic calculus, a branch of the subject to which a short space will be devoted subsequently. "We shall anticipate the con sideration of that method by giving an application of it to the de termination of the preceding result. Regarding A as a symbol of operation, the equation y n = yn-i + Ay n - 1 may be written y n = (1 + A)y n -i . In like manner, y,,-i also 41. If we suppose the equation becomes y n = (l + A)*y n -3 ; and in general n.(n-l) -A 2 + . . . +A ~ ^n. or n= Ax Ay If now, h being regarded as constant, we suppose n to increase, and consequently Ax to diminish, indefinitely, we obtain, on pro ceeding to the limit, This is called Taylor s series. 42. In order to complete the investigation, it will be necessary to examine into the convergency or divergency of the series, and to obtain an expression for the remainder in it after any number of terms ; this we shall immediately proceed to consider. 43. It may be observed that Taylor does not seem f o have been aware of the great importance of his theorem, nor did he give any examples of its application. This probably accounts for the fact that so long a time elapsed before its real value was dis covered ; and, although Stirling introduced a particular case of it in his Methodus Differcntialis (1717), it was not noticed in any of the English treatises on the calculus such as Simpson s Fluxions (1737), Emerson s Fluxions (1743), Landen s liesidual Analysis (1764), nor is it mentioned in the first edition of Montucla s Hist. dcs Math., 1758. The theorem is to be found in Euler s Cal. I) if. (1755) ; but, although Euler makes extensive use of it, he made no reference to Taylor s name in connexion with the series, and would appear to have given the theorem as his own, or rather perhaps to have connected it with Bernoulli s series.
