TAYLOR'S THEOREM, so called from its discoverer, Dr. Brook Taylor (q.v.), is a general method for the algebraic development of a function of a quantity, x, in powers of its increment h, and may be thus briefly explained and illustrated: Let fix + h) denote any function of x + h (subject to the limitations below), then fix + h) = f(x) + f (x)12, + f f "(x) 1.2.3 + ...., where f(x) is the same function of x, as f(x +h) is of x+ h, and f(x),if(x), etc., are the first, second, etc., differential coefficients of Ax). By a supplementary theorem, due to Lagrange, who was the first to appreciate to the full the value of Taylor's discovery, it was shown that the sum of all the terms of the series after n terms, could be represented by p(x +Oh) , where 0 is some positive 1.2...n fraction less than unity. The theorem supposes that between certain limits, indicated by h = o, and It = some finite quantity, neither f(x) nor any of its derived functions vanish, or all of them do not become infinite; and the cases in 'which these conditions are not satisfied are often spoken of as instances of the " failure of Taylor's theorem." An important particular case of this theorem, known as Maclaurin's, or (more properly) Stirling's Theorem, was independently discovered; it is that case of the general theorem in which the various functions of x are made functions of zero, and is written f(o + h) = f(o) + f (o)h + f'(o) +, etc. The best illustrations of these theorems are the binominal, exponential, logarithmic, and circular series; thus if the function be (x+ then fix) = f (x) = 72.V. (x) = n(n — 1)x° "" etc.; and by substitution of these
values we obtain Newton's binomial theorem; if the function be ar Taylor's series 2 1.2 gives us as its equivalent + h. log. a + —+ and Maclaurin's gives ah = 1 + h . log. a + (log, + .....which latter is the exponential theorem, and may be obtained from Taylor's series by division: if the function be log.(1 +x+ h), log.
z +h being one of the cases in which Taylor's theorem fails), then Maclaurin's series ha gives the logarithmic theorem, log. (1 + h) = h — 1,2 —, etc.; and the same theorem gives the various series expressing the values of sin. it, cos. it, sin. etc., etc. The history of this celebrated theorem is remarkable. On the first publication of the Methodus Incrementorurn, it was entirely neglected by Leibnitz, who, in ignorance of its value, severely criticised the whole work; while the bitter hostility of John Bernouilli to British men of science, blinded him to the existence of any merit in any part of the work. The theorem never appeared in any of the works on the calculus published before D'Alembert's Recherch,es, and after that was given only in the French Ency elopcedia; but neither D'Alembert nor Condorcet seems to have known that it was Taylor's, or to have fully appreciated its importance; and it was not till Lagrange, in the Berlin Memoirs for 1,772, gave the name of its true author, and proposed to make it the foundation of the differential calculus, that it assumed that important position which it deserved to hold.