Abstract
The conceptualization and codification of infinitesimal calculus during the 17th century were not able to elucidate their fundamental notions. The criticisms and controversies surrounding the foundation of calculus finally led to important scientific activity which gave rise to the rigorous foundation of analysis. Nevertheless, before the final period of its arithmetization in the 19th century, there was a turning point in the 18th century when algebrization transformed the metaphysical face of analysis.
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Notes
J. L. Lagrange, “Manuscrit” of his lectures (1797–1798). Notes by the citizen Le Gentil. M.S.S. 1323. Bibliothèque de l’École des Ponts et Chaussées, Paris.
Cf. his monumental work: Theory of analytic functions,containing the principles of the differential calculus, free from any consideration of infinitely small quantities or evanescents, of limits or of fluxions and reduced to the algebraic analysis of finite quantities, Paris 1797. J. L. Lagrange, Oeuvres, t. X, p. 7. The framework of the lagrangian theory appeared in 1772, cf. his mémoire: “Sur une nouvelle espèce de calcul relatif à la différentiation et à l’intégration des quantités variables”
Nouveaux Mémoires de l’Académie Royale des Sciences et Belles Lettres de Berlin, pp. 185–221.
Newton gave lectures from 1673 to 1683. His lectures were published in 1707, cf.
Arithmetica Universalis, sive de compositione et resolutione arithmetica liber, Cantabrigiae. Typis Academicis and Londini. Benjamin Tooke.
I. Newton, op. cit.,p. 1.
For more details, cf. the paper of H. G. Green and H. J. J. Winter: “John Landen F.R.S. (1719–1790), mathematician”, ISIS 35 (1944) pp. 6–10.
“An investigation of a general theorem for finding the length of any arc, of any conic hyperbola, by means of two elliptic arcs, with some other new and useful theorems deduced therefrom”, Phil. Trans 1775 pp. 283–289.
Cf. J. E. Montucla: Histoire des Mathématiques,Vol. III, Paris. A. Blanchard, 2e éd. 1968, p. 240.; Cantor, Vorlesungen über Geschichte der Mathematik, Vol. III, Leipsig 1894, pp. 842–847. For more details, cf. G. Mittag-Leffler, “An Introduction to the theory of the elliptic functions”, Annals of Mathematics 1922–23; Ch. Houzel, “Fonctions elliptiques et intégrales abeliennes”, Abrégé d’Histoire des Mathématiques, Paris, Hermann 1978, pp. 1–113.
G. Vivanti, Il Concetto d’infinitessimo et la sua applicazione dela matematica,Mantova 1894, p. 57.
Animadversions on Mr. Simpson Fluxions by Z. Tertius. Truth Triumphant: or, Fluxions for the Ladies. Shewing the cause before the Effect, and different from it; that space is not speed, nor Magnitude Motion, with a Philosophic Vision, most humbly dedicated to his illustrious Public, by X, Y, and Z who are not of the family of x, y, z but near relations of x’, y’ and z’ ... London printed for W. Owen, 1752.
London, J. Nource.
J. Landen, op. cit., pp. 2–5.
Idem, pp. 6–7.
C. B. Boyer, The History of the Calculus and Its Conceptual Development, New York, Dover 1959, p. 237.
It is only an anonymous letter signed with his initials A.B., addressed to the authors of the Monthly Review.
June 1759.
. L. Lagrange, Theory of analytic functions... Paris 1797, p. 4.
“First, the title of Residual Analysis is no more than a new term given to Sir Isaac Newton’s method of differences, and therefore is no new branch of algebraic art: since it has been known, and treated of by many, in a more easy and familiar manner than by Mr. Landen; especially, besides the inventor, by Brook Taylor; by Cotes in his Harmonia Mensurarum; by Stirling in his book called Methodus differentialis; and occasionally by many others. But Mr. Landen surely dares not say this theorem is of his own invention, or that it was taken notice of before. He may, perhaps, imagine he has so disguised it by a new form, as to make it pass for his own, amongst credulous and ignorant readers. But to show that this curious invention is no new one, Mr. Laurin says, in his Algebra, page 109, art. 118. Generally, if you multiply a”’—b“ by a”-m+a“-2’”b“’+a”-4nib3’“+ etc. continued till the terms is in number equal to n/m, the product will be a”—b“. The author cannot sure plead ignorance, and say he has not read Mr Laurin’s work. This would look ridiculous for one who cites in his works, L’Hospital, Bernoulli, Act. Erud. Leips. Archimedes etc. to pretend not to know the authors of his own country, and in his mother tongue”, Monthly Review, p. 560.
Landen in his reply in the July issue does not hide his annoyance of this “scientist”: “He objects to prime number,function etc. as terms never heard before. Alas! how egregiously does he betray his ignorance”.
After the publication of his Discourse Landen who continued to work on Residual Analysis published only a paper about the summation of series. Cf. “A new method of computing the sum of certain series”, Phil. Trans. 1760 pp. 67–118.
London, L. Hawes, W. Clarke and R. Collins, 1764.
John Landen, op. cit. pp. 3–4.
However, many years later, Landen uses the method of fluxions, cf. “A disquisition concerning certain fluents, which are assignable by the arcs of the conic sections; where are investigated some new and useful theorems of computing the fluents”, Phil. Trans. LXI 1771, pp. 298–309.
Cf. the first pages of the Theory of analytic functions...
Cf. also the book of Maseres: Scriptores logarithmici, 1791, Vol. 2, p. 170.
John Landen, op. cit., p. 2.
Idem.
“By the method of computation which is bounded on these borrowed principles, may be done, as well by another method founded on the anciently-received principles of algebra”, John Landen: A Discourse concerning Residual Analysis: A new branch of the Algebraic Art, London 1758, pp. 4–5.
John Landen, The Residual Analysis, a new branch of the Algebraic Art, of very extensive use,both in Pure Mathematics and Natural Philosophy, London 1764, p. 3.
Idem, p. 4.
Idem,p. 5.
Idem, p. 6.
For more details about function’s concept, cf. Ch. Phili’s conference at the Ecole Normale Supérieure, Seminar of Philosophy and Mathematics, Paris 1974, pp. 1–24.
Mém. de l’Academie des Science, Paris 1718, p. 100, Opera omnia, t. II, Lausannae et Genevae 1742.
ntroductio in Analysin Infinitorum, Lausanne 1748, liv. II, ch. I.
With the sign x Landen expresses the similar function “if, in any given expression or function of x, wherein x is not concerned, x be substituted instead of x, the given expression and that which results from such substitution are called similar functions of x and xrespectively”, J. Landen, op. cit., p. 5.
“The said quotient, which algebraists commonly denote by (x—x) or yy=y, — xx=xi we shall for brevity sake, sometimes denote by [x/y]; and the special value thereof we shall express by [x=y]” op. cit., p. 5.
C. B. Boyer, The History of the Calculus and Its Conceptual Development, New York, Dover, 1949, p. 236.
Lagrange continued in the same way. The second part of his Theory of analytic functions contains geometrical applications.
Landen, op. cit., p. 2.
Idem, p. 6.
We follow the original notation of the book, naturally these “brackets” play the role of parentheses.
Idem, p. 6.
Several commentators at the end of the eighteenth century called this formula “Landen’s formula”, cf. Maseres, op. cit.,p. 170.
Cf. note 70.
About Landen’s demonstration, cf. chapter VIII “John Landen et les démonstrations algébriques” of M. Pensivy Thesis: Jalons historiques pour une epistémologie de la serie infinie du binôme. Thèse de 3ème cycle, Université de Nantes, Nantes 1986.
Landen, op. cit., p. 10.
dem, p. II.
De Morgan says that in Landen’s residual quotient exists the notion of the limit of D’Alembert: “It is the limit of D’Alembert supposed to be attained instead of being a terminus which can be attained as near we please. A little difference of algebraic suppositions makes a fallacious difference of forme: and though the residual analysis draws less upon the disputable part of algebra than the method of Lagrange, the sole reason of this is that the former does not go so far into the subject as the latter”. Penny Cyclopaedia Art: Differential Calculus.
J. L. Lagrange, Theory of Analytic functions..., Paris 1797, pp. 4–5.
A. Clairaut, Eléments d’Algèbre,Paris 1797, Tom. I, p. 9.
A. Clairaut, op. cit.,Tom. II, p. 90.
A. Clairaut, Eléments d’Algèbre, 6th ed. Paris 1801, p. 325.
Cf. J. Leslie’s Dissertation on the progress of mathematical and physical science; John Leslie, Dissertation Fourth. Encyclopedia Britannica, 7`h ed. vol. i. 1842, p. 601.
“Therefore, having shewn how x”—x“ may be divided by x— x, it will now be proper to shew how n=—n= may also be divided by the same divisor (x — x). In doing we shall first assign the value of n in a certain series of terms of n and x, wherein the exponents of the several powers of these quantities shall be invariable: by the help of which series we shall be enable readily to obtain the desired quotient of nj—n’ divided by x—x”. J. Landen, op. cit., p. 30.
Landen’s method is the same as the one which Euler uses in his Introductio in Analysin Infinitorum.
J. Landen, op. cit., p. 31.
“It may sometimes be of use to observe, that, when y is very large number, the Log of 1+ÿ will be = ÿ nearly the value of the series -2- — - + etc. being there so very small that is may be neglected”. Idem p. 33.
Idem,p. 46.
The titles of the ten chapters of his book are the following: “Terms and Characters explained”; “Of the inventions of Rules necessary to facilitate computations in this analysis”; “Of exponentials and logarithms”; “Of the properties of certain algebraic expressions”; “Of the tangents of curve lines”; “Of the investigations of useful theorems, by finding the nature of a curve from a given property of its tangents”; “Of the evolution and curvature of lines, with some inferences relating to the focuses of reflected and refracted rays and the curves called caustics”; “Of the greatest and least ordinates, the points of contrary flexions, and reflexion, and the double and triple etc. points of curve lines”; “Of the asympotes of curve lines”; “Of the diameters and centers of curve lines”.
J. Landen, op. cit., p. 9.
Idem, p. 46.
dem, pp. 50–51.
“Certainly when we suppose the increments to vanish, we must suppose their proportions, their expressions, and everything else derived from the supposition of their existence, to vanish with them”, G. Berkeley, The Analyst, or a Discource addressed to an Infidel Mathematician. Wherein it is examined whether the Object, Principles, and Inferences of the Modern Analysis are more distinctly conceived, or more evidently deduced, than religious Mysteries and Points of Faith. London 1734, § 13.
“Infinite parve concipimus, non ut nihila simpliciter et absolute, sed ut nihila respectiva..., id est ut evanescentia quidem in nihilum, retinentia tamen characterem ejus quod evanescit”, G. W. Leibniz, Lettre à Grandi, 6 Sept. 1713, M. IV, p. 218.; cf. also: “une quantité infiniment petite n’est rien d’autre qu’une quantité évanouissante, et c’est pourquoi en réalité elle sera égale à 0.” L. Euler, Institutions Calculi differentialis, St. Pétersbourg 1755, p. 77.
P. Mansion: Resumé du cours d’analyse infinitésimale à l’Université de Gand, 1887, p. 213. For more details, cf. G. Vivanti, op. cit.
J. Landen, op. cit., p. 10.
Idem,p. 3.
Landen’s profession was land surveyor.
Cf. the anonymous letter in Monthly Review, notes 14 and 15.
Cf. his paper on fluxions, cf. note 23; moreover, from his former publications we can establish that Landen never returned to his theory. Cf. Mathematical Memoirs, Vol. I, II, London 1780.
Lagrange also had abandoned his own theory, seeing that his paper in 1772, made no impression and he had followed Leibniz’s conception in his Mécanique Analitique.
“The Residual Analysis rests on a process purely algebraical: but the want of simplicity... is a very great objection to it”, Monthly Review Vol XXVIII, 1799. Appendix Review of Lagrange’s Theory of functions...
Cf. Landen’s foreword; cf also “I can also mention a method which Landen gave in 1758 to avoid consideration of infinity, of motion or of fluxions... he is perhaps the only English mathematician, who has acknowledged in inconvenience of the method of fluxions”, Review of Lacroix’s Calcul différentiel. Monthly Review, London 1800, p. 497.
“Landen, I believe, first considered and proposed to the treat of fluxionary calculus merely as a branch of Algebra.” R. Woodhouse, Principles of Analytical Calculation, Cambridge 1803, p. xviii. His homogeneous method was also appreciated by the philosophers Compte, Hegel, Cohen etc. Marx in his Mathematical Manuscrits stresses the importance of Landen’s uniformity, cf. The Mathematical Manuscrits of Karl Marx, New Park Publication, New York 1983, pp. 33, 75, 113, 139.
Moritz Cantor in his History of Mathematics criticizes severly Landen’s Residual Analysis: “die residual Division, geschieht aber selbstverständlich durch Grenzübergang, so dass die Landenschen Methode nichts wesentlich Neues darbietet”, M. Cantor, Vorlesungen über Geschichte der Mathematik, Band III, Leipsig, 1894, p. 661.
J. L. Lagrange, “Discours sur l’objet de la théorie des fonctions analytiques”, Journal de l’Ecole Polytechnique, 6• Cahier, t. II, Thermidor (Juillet-Août) An VII, p. 232.
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Phili, C. (1994). John Landen: First Attempt for the Algebrization of Infinitesimal Calculus. In: Gavroglu, K., Christianidis, J., Nicolaidis, E. (eds) Trends in the Historiography of Science. Boston Studies in the Philosophy of Science, vol 151. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3596-4_21
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