Taylor s Theorem is actually included as the 7th proposition of the book Methodus incrementorum directa et inversa by the English mathematician Brook Taylor (1685-1731), published in London in 1715. Taylor had already advised his teacher John Machin by letter of this result in 1712. Taylor s Theorem was so called for the first time in 1784 in the article Approximations in the French Encyclopaedia (Encycclop die m thodique) of Condorcet. In 1786 Simon Lhuilier also used this title in the book Exposition l mentaire des calculs sup rieure, honoured by an award by the Berlin Academy of Sciences (the thesis had been offered in a competition of the Academy). Since that time the theorem has entered all the handbooks of mathematical analysis and no one has called it anything else. We know nowadays, however, that the Scottish mathematician James Gregory already possessed it in the years 1671-72. Both Gregory and Taylor approached Taylor s Theorem starting from finite differences. At this point Taylor addressed himself directly to the problem of considering Newton s deliberately utterly vague explanation of his interpolation formulae. Newton had obtained his theorem by first allowing the independent variable to differ from zero by a (finite) increment and then - after a series of transformations - returning it to zero by dividing it into an infinitely large number of pieces . If we replace Taylor s extremely cumbersome notation by more modern notation, the proof appears as follows. Let y = f(x), where x is a variable which is varied, as he says, uniformly , that is, obtaining the successive values x, x + x, x + 2 x, ..., x + n x = x + h. And let the corresponding values of f(x) be y ( or y0), y1, y2, ..., yn. Let the successive differences (differences of the first order) between yk-1 and yk (k = 0, 1, ..., n - 1) be y, y1, ..., yn-1; the differences between these differences (differences of the second order) are y, y1, ..., yn-2; and so on. In order to visualise all this, let us write it in mathematic form: x + x x + 2 x x + 3 x ... x + n x y1 y2 y3 ... yn y y1 y2 ... yn-1 y y1 ... yn-2 y ... yn-3 .... ..... ... . It is then clear that: y1 = y + y, y2 = y1 + y1 , y1 = y + y , y3 = y2 + y2 , y2 = y1 + y1 , y1 = y + y , f(x + x) = y1 = y + y , f(x + 2 x) = y2 = (y + y) + ( y + y) = y + 2 y + y , f(x + 3 x) = y3 = (y + 2 y + y) + ( y + y) + ( y + y) = y + 3 y + 3 y + y , ............... .. ..... . Having observed the general regularity, Taylor concludes from this that: f(x + n x) = y + n y + (n(n - 1)/1 2) y + (n(n - 1) (n - 2))/1 2 3) y + ... + ny , (1) Setting n x = h (Taylor used v instead of h), we will have: n = h/ x , n - 1 = (h - x)/ x , n - 2 = (h - 2 x)/ x , ..., n - (n - 1) = (h - (n - 1) x)/ x . Substituting these values for n, (n - 1), (n - 2), ... into formula (1), Taylor obtained (in our notation): f(x + h) = y + h ( y/ x) + ((h(h - x))/1 2) ( y/ x ) + ((h(h - x) (h - 2 x))/1 2 3) ( y/ x ) + ..., (2) (h(h - x) (h - 2 x)...(h - (n - 1) x))/1 2... n) ( ny/ xn) He now assumed h to be fixed, n to be actually infinitely large, and x to be actually infinitely small ( zero ), inferring that this transformed y/ x into the first fluxion y. (dy/dx according to Leibnitz), y/ x into the second fluxion y.. (d y/dx according to Leibnitz), and so on. This transforms formula (2) into: f(x + h) y + y.h + y.. y /1 2 + y... h /1 2 3 + ..., Thus, even beginning with finite differences and only then removing them, Taylor still operated strictly in the style of Newton and Leibnitz, with actually infinitely large and actually infinitely small quantities and with the symbolic formulae of the calculus of fluxions, not wondering whether they had any real equivalent and not bothering to consider, of course, the convergence of the obtained series (even so the value of f(x + h)). One must note here that, although Taylor was an ardent adherent of Newton s in the quarrel with Leibnitz and therefore never used the latter s notation nor ever cited him, it is nonetheless no accident that Euler presented the proof*2 in the language of Leibnitz. As D.D. Mordukai-Boltovskoi notes, in essence Taylor addressed the Newtonian fluxions from the Leibnitzian, not the Newtonian, standpoint, namely from that of finite differences (see the Kommentarii cited in Yanovskaya, 1968, p.396). As for the history of MacLaurin s Theorem, it must be noted first of all that is was already present in Taylor in the form of a special case of his theorem at x = 0. It is true that, unlike MacLaurin, Taylor never used the MacLaurin series for the expansions already known at this time, for ax, sin x/a, cos x/a which are more easily obtained using this theorem. Furthermore, with respect to the manuscripts of Marx, who specifically mentioned that he borrowed the algebraic expansion directly from MacLaurin, it must be noted that the proofs of MacLaurin s Theorem (by the method of indeterminate coefficients) which were presented in Boucharlat s and Hind s textbooks actually belonged to MacLaurin himself. Such direct borrowing from the author whose name the theorem bears may also have taken place, of course, with reference to Taylor s Theorem. The bibliographic list which Marx compiled while preparing the historical sketch is apparent evidence that he had decided to become acquainted with Taylor s work in the original, although he did not succeed in carrying out this intention. 2) We find the same order in which Marx criticised the proof of Taylor s Theorem in manuscript 4302, in Boucharlat s textbook as well (J.-L. Boucharlat, El mens de calcul differ rentiel, 5th ed., Paris, 1838; Marx apparently had an English translation done from a different edition). Having stated the problem of successive differentiation in 30 (pp.19-20) - where, by the way, after having obtained 6a as the third derivative of ax he remarks (p.20), here it is no longer possible to differentiate since 6a is a constant - Boucharlat passes to MacLaurin s Theorem ( 31, pp.20-21), proving it by assuming the proof of Taylor s Theorem (later proved in 55-57, pp.34-37). As was already mentioned, Boucharlat proves MacLaurin s Theorem by following MacLaurin himself. He apparently did not read the latter s work, however. In fact, with respect to the title MacLaurin s Theorem , Boucharlat writes, this theorem, as G.Peacock has noted, was discovered by G. Stirling in 1717, consequently earlier than MacLaurin used it, although, as we have already mentioned, MacLaurin fully acknowledged that Taylor already had the theorem. Boucharlat s proof - which raises not a single question about the correctness of the assumptions made, not to mention the convergence of the series under consideration - we represent below in almost literal translation. (y) = A, (dy/dx) = B, (d y/dx ) = 2C, (d y/dx ) = 2 3D , In the following 32-34 (pp.21-22) expansions are found by means of MacLaurin s formula for y = 1/(a + x) , y = sqrt{a + bx}, y = (a + x)m . By this means the binomial theorem is derived from MacLaurin s theorem in the third example. In the first appendix to our 5th edition of Boucharlat s textbook entitled Proof of Newton s formulae by means of differential calculus , a direct derivation (by the same method of indeterminate coefficients) is given of Newton s binomial theorem (for positive integer powers) by means of successive differentiation. It appears as follows. Boucharlat begins with an expansion of (1 + z)m, from which the (?)required expansion for (a + x)m is obtained by the substitution of m (x/a). Assume, he says, (1 + z)m = A + Bz + Cz + Dz + Ez4 + ... (1) (1 + z)m = 1 + Bz + Cz + Dz + Ez4 + ... . m(1 + z)m - 1 = B + 2Cz + 3Dz + 4Ez + etc. m(m - 1) = 2C, C = (m(m - 1))/2 , (1 + z)m = 1 + mz + ((m(m - 1))/(1 2)) z + ((m(m - 1) (m - 2))/(1 2 3)) z + etc. 3) Boucharlat also demonstrates Taylor s Theorem by the method of indeterminate coefficients. In this case he not only assumes that an arbitrary function of many variables may be expanded into a series of powers of any of the variables, but he also considers this expansion unique; that is, that the coefficients of any two such expansions (in powers of one and the same variable) must be equal. This makes it possible to apply the method of indeterminate coefficients. In order to arrive at this possibility, that is, of comparing the coefficients of two expansions of one and the same function, Boucharlat begins with a lemma which asserts that the derivatives of f(x + h) with respect to x and to h are equal. Since Marx expresses dissatisfaction in manuscript 4302 (see Yanovskaya, 1968, p.540 [not translated]) with the demonstration of this lemma in Boucharlat s course-book, while it is impossible even to understand pp.41-42 (see Note 117 Yanovskaya, 1968 [not translated]) of manuscript 3888 without being acquainted with this proof, we present it here in full. Devoted to this is 55 (pp.34-35), in which we read: In the following 56 Boucharlat extends this lemma to derivatives of higher order and in 57 uses it to prove Taylor s Theorem. He begins this proof with the following words on what he considers - and as Marx calls it - his starting equation (37), applicable to any function: Let y1 be a function of x + h; let us assume that when we develop this function into powers of h we obtain y1 = y + Ah + Bh + Ch + etc. , (37) Differentiating equation (37) with respect to h and with respect to x, and having obtained by this means dy1/dh = A + 2Bh + 3Ch + etc. , dy1/dx = dy/dx + (dA/dx) h + (dB/dx) h etc. , The following 58-61 in Boucharlat s book contain examples of expansions of f(x + h) by Taylor s formula in the case of f(x) equals sqrt{x}, sin x, cos x, log x. Questions about the convergence of the series obtained are not even mentioned. Cases of inapplicability of the Taylor series are only considered in the very last paragraphs of the first part of the book (devoted to differential calculus) which are printed in small type. The concluding 62 of the section on Taylor s Theorem and its applications is devoted to a proof of MacLaurin s Theorem from Taylor s Theorem. Marx reproduces this proof in full in manuscript 3888 (see sheets 55-56; pp.51-52 in Marx s enumeration).
Marx's Mathematical Manuscripts 1881