Evaluating between the extremes 0 and is we get Beruouilli's Series (with Remainder); s +/10'(h) - 1te2 - 0"(h) - 13 r'(h) - • • • ± To avoid the alterna 1" o tion in sign we take fie(a-u)du and proceed as before: then on putting a iss 0(24-Fh)="0(wo)-Fh9 + L. 0"(xo)-F • • • in + —fune+.(x+h—u)du, o< It < R.
Such is the swiftest, directest, nearest-lying deduction of the fundamental Taylor's Series, by which the value off at (so + k) is built up out of the value of 0 and its D's at So. The ROI is here yielded as a definite integral, from which form the other forms, as Lagrange's, Cauchy's, Schldmilch's, come at once on apply ing the Maximum-Minimum Theorem. This development holds under the two necessary and sufficient conditions (Pringsheim) : 1. That0(x) possess everywhere
definite finite differential coefficients of every finite order ; 1 2. That Lim. ,-0(0)(xo+h). kn converge uni formly on 0 (for n= m , for all pairs (h, k) for which 0
The Infinitesimal Analysis or Method of Limits is very highly developed and is applica ble to almost every subject of exact thought, often asserting itself in the most surprising fashion, as in the Theories of Numbers and of Knots, to which it might seem wholly alien, suddenly unlocking and laying wide open secret passages utterly unsuspected. In particular the Integral Calculus shows itself amazingly and unendingly fertile in the generation of new notions. As other and still other fields are exposed to investigative thought, the Calculus will receive more and more applications, and there seems to be no limit to the subtlety and refinement of its processes, to the keenness and penetration that may be given to this two edged sword of the spirit, the strongest, sharp est and most flexible ever fashioned or wielded by the mind of man.
Historical Sketch.- Passing by anticipa tions, especially of Integration, that reach back at least to Archimedes (287-212 p.c.), we come to Barrow's 'Lectiones optics: et geometries' (1669-70), on which Newton collaborated, how much no one knows. Barrow used the Differential Triangle even in 1664 (indefinite parvum, . . . ob indefinitam curvm parvitu dinem), calling dy a and .1 x e (as Fermat used A and E 1638). Newton was busied with
Series at Cambridge, 1665-66 (eo tempore pestis ingruens coegit me dine fugere—in his famous letter, filling 30 pages in the Opuscula, to Oldenburg, secretary of the Royal Society, 24 Oct. 1676). His MS. We analyst per aequationes numero terminorum infinitas> (partially published first in Wallis's Works, Vol. H, 1693) was shown to Barrow, Collins, Lord Brouncker in 1669, wherein he used o for a magnitude ultimately vanishing, as had James Gregory already in his (Geometric: pars universalis' (1667, Venice). He treated Recti fication, Cubature and Mass-Centre determi nations as reducible to Quadrature and to be solved by introducing the notion of (Momen tum)=instantaneous change, thus going beyond Barrow. Newton's Fluxi onum et Serierum infinitarum' was ready for the press before 1672, but not printed till 1736. In it he proposes, (1) to find the velocity at any instant from the space trav ersed up to each instant, (2) to find the latter (space) from the former (velocity)—the two problems of Derivation and Integration con ceived kinematically. The equicrescent mag nitude x, as a space, is called fluens (Cavalieri fluens, 1639, Napier flusus, 1614, Clavius fluere, 1574) ; the velocity he writes s and calls fluzio Derivative (as to the time t). Momentum varies as fluxion, is written so, and corresponds to our Differential x (in crementa indefinite parva). This treatise seems to have been revised after 1673, hence does not clearly attest Newton's knowledge in 1671. Leibnitz wrote, 26 Oct. 1675 (follow ing Cavalieri), Omnia w, etc.; but 29 Oct. 1675, Utile exit scribifpro omn. ut f 1 pro omn. 1 id est summa ipsorum 1; again, the same day, nempe utf ougebit, ita d minuet dimensiones. significat summam, d differentiam. There and then was born the °Algorithm of the and Integral 6 Calculus' Under date of 11 Nov. 1673, Leibnitz wrote f ydy-- — )4 ' but the 3 was originally 5. His 2 °Characteristic Triangle,' equivalent to Dif ferential Triangle, he took not from Barrow but from Pascal. All attempts to show any real dependence of Leibnitz on Newton have failed. The germs of the new Method were abroad in the air.