Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/22

 12 INFINITESIMAL CALCULUS judgment, but recommended Rolle to conform more strictly to the rules of the Academy, and Sauriri to forgive the proceedings of his adversary. Rolle afterwards did justice to the calculus by acknowledging his error in op posing it, and admitted that he had been urged forward by malevolent persons, one of whom was the Abb6 Gallois. Mathematicians have differed as to the best way of expounding the principles of the calculus. Newton, as has already been stated, employed the theory of motion as the means of connecting its doctrines with the principles of ordinary analysis. Leibnitz, again, with the same view, conceived quantity as passing from one degree of magni tude to another by the continual addition of infinitely small parts. The mind finds no great difficulty in dis tinctly apprehending the subject in either way. Objec tions have, however, been taken to both, and attempts made to substitute a better. Euler considered the infinitely small quantities of Leibnitz as absolutely zeros, that have to each other ratios derived from those of the vanishing quantities which they replace. D Alembert proposed to make the basis of the calculus the consideration of the ratios of the limits of quantities. This method, as was indeed stated by D Alembert, does not differ in any material respect from Newton s prime and ultimate ratios. An English mathematician, Landen, substituted for the Newtonian method of fluxions another purely algebraical. His views are contained in a work entitled The Residual Analysis, a new branch of the Algebraic Art La- (1764). Lagrange, too, in the Memoirs of the Berlin grange. Academy for 1772, proposed to base the calculus alto gether on the expansion of functions, and thus to establish it on algebraical principles merely. He subsequently developed his method in his TJieorie des Fonctions Analytiques (1797), and in his Lemons sur le Calcid des Fonctions (1806). Lagrange, however, adopted the infini tesimal method as the basis of his most important work, viz., the Mecanique Analytique. He states in his preface to its second edition (1811) that &quot;when we have properly conceived the spirit of the infinitesimal method, and are convinced of the exactness of its results by the geometrical method of prime and ultimate ratios, or by the analytical method of derived functions, we may employ infinitely small quantities as a sure and valuable means of abridging and simplifying our demonstrations.&quot; We shall close this introduction with a list of works on the subject. Principal Works bearing on the Infinitesimal Method before the Invention of the Calculus. Kepler, Nova Stcreometria Doliorum Vinariorum, 1615; Cavalieri, Geometria Indivisibilium, 1635; Id., Exercitationcs Gcomctricse Sex, 1647; Descartes, Geometric, 1637; Torricelli, De Sphtera et Solidis Sphszralibus, 1644 ; Gregoire St Vincent, Da Quadratura Circuli, 1647 ; Huygens, Tlicoremata de Quadratura, 1647; Id., Horologium Oscillatorium, 1673; Wallis, Arithmetica Infinitorum, 1655; Id., Opera Mathcmatica, 3 vols., 1693-99; Fermat, Opera Varia Mathcmatica, 1679; Mer- cator, Logarithmotechnia, 1668 ; James Gregory, Vera Circuli et Hyperbolae Quadratura, 1668; Barrow, Lectioncs Geometries, 1670; Slusius, &quot;Tangents to all Geometrical Curves,&quot; Phil. Trans., 1672; Wren, &quot;Rectification of the Cycloid,&quot; Phil. Trans., 1673; Bullialdns, Arithmetica Infinitorurn, 1682. List of some of the Principal Works on the Calculus. Newton, De Analyst per ^Equationcs numero terminorum infinitas, circulated in MS. in!669 (extracts from this memoir appeared in the 2d vol. of Wallis s works, 1693, which comprehends the first publication to the world of the method of fluxions); Id., Principia, 1687; Id., Tr r tatus de Quadratura Curvarum, published with his Optics, 1704; Id. , Mctlwdus Differcntialis, 1711; Leibnitz, &quot; Nova Methodus pro maximis ct minimis, itcnique tangentibus,&quot; Acta Erud., 1684; Leibnitz et Bernoulli, Commcr. Epis. Ph. et Math., 1745; John Bernoulli, &quot; Inventio Linese Brachistochronrc/ ^ctofiViwl, 1696; Id. Analysis Problematis Isoperimetrici, 1697; Id., Opera Omnia,174:2; James Bernoulli, Opera, 1744 ; De 1 Hopital, Analyse des infinimcnt Pdits, 1696; Cheyne, Fluxionum Methodus Invcrsa, 1703; Hayes, Treatise mi Fluxions, 1704; Manfrcdi, De Construe. JEquai. Diff. Primi Gradus, 1707; Taylor, Methodus Incrementorum, 1715; Stirling, Lin. Tcrt. Ordin. Ncwtoni, 1717; Hermann, &quot; DC Construe. Equat. Diff.,&quot; Comm. Petrop., 1726 ; Fontenellc, EUmcns de la Geo metric de I Infmi, 1727; Clairaut, &quot; Detenninatio Curvse ejusdem DilF. ,&quot; Acta Erud., 1729; Do Moivre, Miscellanea Analytica, 1730; Hodgson, Fluxions, 1736; Simpson, Fluxions, 1737; Mac- laurin, Fluxions, 1742 ; Donna Agnesi, Instituzioni Analitichc, 1748; Euler, Mcth. invcn. Lin. Curv. max. vel min. prop, gaud., 1744; Id., Introd. Analy. Infin., 2 vols., 1748; Id., Institut. Cal. Diff., 2 vols., 1755; Id., Institut. Cal. Intcg., 3 vols., 1768^70 (the titles of Euler s numerous memoirs on the Differential and Integral Calculus are given in the edition of his Differential Cal culus published at Pavia iu 1787); Walmesly, Analyse des Mesurcs, des Rapports, ct des Angles, 1750; Stirling, Methodus Differentialis, 1753; Bougainville, Traite du Calcul Integral, 1754; Landen, Mathematical Lucubrations, 1755; Id., Residual Analysis, 1764; Id. , Mathematical Memoirs, 1780; Saunderson, Method of Fluxions, 1756; Kiistner, Scparatio Indctcrminat. in sEquat. Diff., 1756; D Alambert, Opuscules Mathematiqucs, 1761-80 ; Robins, Mathe matical Tracts, 1761; Waring, Miscellanea Analytica, 1762; Id., Meditationcs Aiialyticse, 1776; Condorcet, Du Calcul Integral, 1765; Le Sour et Jacquier, EUmcns du Calcul Integral, 1768; Lexell, &quot; Methodus integrand! YEq. Diff.,&quot; Comm. Petrop., 1769; Fontaine, Traite du Calcul Diff. ct Integral, 1770; Gianella, DC Fluxionibus et earum Usu, 1771; Cousin, Traite du Calcul Differ - entiel et Integral, 1776; Laplace, &quot; L Usnge du Calcul. aux Diff. part.,&quot; Mem. de I Acad., 1777; Condorcet, &quot; De Intcg. cujusdam ^Equationis,&quot; Comm. deBonon., 1783; Paoli, Mcmoriasull cquazionc a diffcrenze finite e parziali, 1784; Monge, &quot;Sur le Cal. Int. des Equat. aux Diff. part.,&quot;3/^m. dcTAcad., 1784; Charles, &quot;Rechcrches sur le Calcul Integral,&quot; Mem. de I Acad., 1784; L Huillier, Ejywsi- tion des Principcs des Calculs Supiricurs, 1786 ; Id., Princip. Calculi Diff. et Intcg., 1795; Mascheroni, Annotationcs ad Cal. Integ. Eulcri, 1790; Tabiescen, Principia atquc Historia Calculi Diff.ctlnteg.necnon Mcthodi Fluxionum, $%, Lagrange, &quot; Calcul des Variations,&quot; Misc. Taur., vols. ii. and iv., 1760-69; id., Thcoria des Fonctions Analy tiqtics, 1797; Id., Lemons sur le Calcul des Fonctions, 2d ed. , 1806 ; Id., separate Memoirs, edited under the caro of Serret, 7 vols., 1867-77 (the remainder of his works are in course of republication in the same series) ; Vince, Principles of Fluxions, 1797; Carnot, Reflexions sur la Metaphysiquc du Calcul Infinitesimal, 1797 ; Lacroix, Traitt du Calcul Differentiel et du Calcul Integral, 1797; Arbogast, Calcul des Derivations, 1800; Legendre, Exercices de Calcul Integral, 3 vols., 1811-19; Id., Traite des Fonctions Elliptiqucs, 3 vols., 1825-28; Cauchy, Cours d Analyse, 1821; Id., Appl. Geom. du Cal. Infin., 1823; Id., Mem. sur les int. def. prises entre des limites imag., 1825 ; Id., Lemons sur le Calcul Differentiel, 1829; Ohm, M., System dcr Matlumatik, 9 vols., 1822-52; Id., Lehrbuch f. d. gesammte Hoh. Math., 2 vols., 1839; Magnus, Sammlung von Aufgabcn d. Analyt. Geom., 1833; Navier, Lemons d Analyse dc I Ec. Polyt., 1840; Moigno, Lemons de Cal. Diff. ct de Cal. Int., 2 vols., 1840-44; Id., Calcul des Variations, 1861; Duhamel, Cours d Analyse dc VEc. Polyt., 2 vols., 1840-41; 3d ed. by Bertrand, 2 vols., 1874-75; Cournot, Theoric des Fonctions ct du Calcul Infinitesimal, 1841 ; Gregory, Examples on the Diff. and Int. Calculus, 1841 ; De Morgan, Differential and Integral Calculus, 1842; Hymers, Integral Cal culus, 1844; Schlb miich, Handbuch dcr Differential- und Integral- rechnung, 1847; Id., Compendium dcr Hohcren Analysis, 2 vols., 1874; Minding, Sammlung von Integraltafcln, 1849; Meyer, Expose Elem. de la Theoric des Int. Def., 1851; Todlmnter, Differ ential and Integral Calculus, 2 vols., 1852; Id., On Functions of Laplace, Lame, and Bcssel, 1875; Price, Infinitesimal Calculus, 2 vols., 1854; Bierens De Haan, Tables dintegralcs definics, 1858; Id., Expose dc la theoricdcs integrates dtfinics, 1862; Boole, Differ ential Equations, 1859; Id., Calculus of Finite Differences, 1860; Grassmann, Die Ausdchnungslchre, 1862; Bertrand, Traite dc Cal. Diff. etde Cal. Int., 2 vols., 1864-70; Meyer, G. F., Vorlcs. ii. d. Theoric d. bcstimmten Integrale, 1871 ; Williamson, Differential and Integral Calculus, 1872-74; Hermite, Cours d Analyse, 1873; Durege, Theoric d. Funktionen ciner complexen verandcrl, Grossc, 2d ed., 1873; Folkierski, Principles of Diff. and Int. Calc. (Polish), Paris, 2 vols. , 1873; Rubini, Elementi di Calcolo infinitcsimalc, 2 vols., 1874-75; Serret, Cours de Calc. Diff. et Int., 2d ed., 2 vols., 1878-79 (the 8th edition of Lacroix s Traite Elemcntairc, by Serret . and Hermite, contains in the notes many valuable additions) ; Riemann, Gcsam. Math. Wcrkc, 2d ed., 1876; Id., Particlle Diffcr- cntialglcichungcn, 2d ed., 1876; Lipschitz, Lehrbuch dcr Analysis, 2 vols., 1877-80; Hoiiol, Cours dc Calcul Infinitesimal, 3 vols., 1878-79 ; Boucharlat, El. de Calc. Diff. et Int., 8th ed. by Laurent, 1879; Stegemann, Differential- und Intcgralrechnuny, 2 vols., 3d ed., 1880. The preceding list contains the names of some of the most im portant existing treatises on the calculus. It makes no pretence to completeness ; in fact, many of the most valuable contributions to the subject are published in the numerous mathematical journals, and in the transactions of learned societies. In treating of elliptic and hyperolliptic functions we shall give a short list of the chief works on that great branch of the calculus.