Sir Isaac Newton (1642-1727) undoubtedly arrived at the principles and practice of a method equivalent to the -infinitesimal calculus generally much earlier than Leibnitz, and, like Roberval, his conceptions were obtained from the dynamics of Galileo. He considered curves to be described by moving points. An "arc" thus became the "fluent" of the velocity of the point with which it is supposed to be described. Newton's notation (it must be admitted) for the "inverse method of fluxions" was far clum sier, even, and far inferior to Leibnitz's "LI the long "s." And it was owing to the long acrimonious dispute between Newton and Leib nitz, mixed up with insinuations anent what is sometimes mistakenly called "patriotism," that for considerably more than a century British mathematicians failed to perceive the great su periority of the "notation" of Leibnitz. In fact tt was not until the beginning of the 19th cen tury that there was formed, at Cambridge, a society to introduce and spread the use of Leib nitz's notation among British mathematicians.
"It may be said" (we quote Sophus Lie) "that the conceptions of differential quotient and integral, which in their origin certainly go back to Archimedes, were introduced into modern science by the investigations of Kepler, Des cartes, Cavalicri, Fermat and Wallace. . . . The capital discovery that differentiation and integration are inverse operations belongs to Newton and Leibnitz" (in Leipsiger Berichte XLVII, 1895, Math.-phys. Classe, p. 53). And indeed in the opinion of one who has con tributed much to the advancement of Ameri can mathematics as well as of Britain (indeed, of the world!), J. J. Sylvester, *it seems to be expected of every pilgrim up the slopes of the mathematical Parnassus, that he will at some invent or other of his journey sit down and Invent a definite inte gral or two toward the increase of the com mon stock.* (See CALCULUS, INFINMESIMAL and consult his 'Notes to the Meditation of Poncel let's Theorem,' Mathematical Papers, Vol II, p.
214). Consult also Archimedes of Syracuse 'On with an introduction by . David Eugene Smith and an English translation by Lydia G. Robinson (Chicago) ; Cavalieri, 'Geo metria Indivisi'bilium) (1635) ; Cheyne, (Fluxi onum Methodus Inversa> (1703) i Condorcet, 'Du Calcul Craig, 'De Cal culo Fluentium) (1718) ; Descartes, 'Letters,) 'Geometry,' etc. (1637); Euler,. 'Institutiones Calculi Integralie (3 vols., Saint Petersburg 1768-70) • Fermat, 'Opera Varia ; liymer's 'Integral Calculus) (1844) ; Johnson's W. W., 'Elementary Treatise on the Integral Calculus' ; and his larger work 'Inte gral (2 vols. in one: New York 1915); Kepler, 'Nova Stereometria Doliorum Vinanorum) (1615) ; La Place, 'L'Usage du Calcul aux differences partielles,) Men. de l'Acad. (1777) ; Leibnitz, 'Quadrature of the Ord& (in Leips. Acts and Phil. Trans., 1682); id., 'Nova Methodus pro et minimis itemque (in Lams. Acts, 1684); id., 'Correspondence with Newton); Mascheroni, (Annotationo_ ad Cal. Integ. Ru le& (1790)1 Monge, (Sur le Calcul Integral des equat. aux dif. partielles,) Mem. de l'Acad. (1784); Newton, 'De Analysi per 2Equationes numero terminorum Infinitas) (circulated in MS. in 1669, and printed in (Commercium Epis tolicum,) 1712; it is also contained in a vol ume edited by W. Jones, entitled rAnalysis per Quantitum Series, Fluruones ac Differentiae,) 1723) ; Newton, 'Principia) (esp. lib. ii, sect. iii, lemma 2; 1687) ' • id., 'Tractatus de Quadra tura (published with 'Optics,' 1704) ; id., 'The Method of Fluxions) (1736) ; Price's 'Infinitesimal Calculus) (2 vols., 1854) ; Picard, Emile, 'Traits d'analyse) (4 vols., 1891); perhaps the most extensive and one of the most advanced treatises that has as yet ap peared; Roberval, 'Traits des Indivisibles,) Mem. de l'Acad. des Sciences (1666) ; Tod hunter, Isaac, 'A Treatise on the Integral Cal culus) (Cambridge 1868) ; Wallis, (Anthmetica Infinitorum) (1655).