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Original Articles

Convergence and Formal Manipulation of Series from the Origins of Calculus to About 1730

Pages 179-199 | Published online: 05 Nov 2010

  • 1998 . Until now the following articles have been published: 'Some Aspects of Euler's Theory of Series. Inexplicable Functions and the Euler-Maclaurin Summation Formula' . Historia Mathematica , 25 : 290 – 317 .
  • 1999 . The First Modem Definition of the Sum of a Divergent Series. An Aspect of the Rise of the 20th Century Mathematics' . Archive for History of Exact Sciences , 54 : 101 – 35 .
  • 2000 . The Value of an Infinite Sum. Some Observations on the Eulerian Theory of Series' . Sciences et techniques en perspective , 4 : 73 – 113 .
  • Ferraro , G. 2000 . The Value of an Infinite Sum' . Sciences et techniques en perspective , 4 : 73 – 104 .
  • Panza , Marco . 1992 . La forma della quantità. Cahiers d'historié et de philosophie des sciences Vols 38 and 39 , 215 – 36 . Nantes where some of the topics I treat in this article are also tackled
  • Bernoulli , Jacob . 1689-1704 . Positiones arithmeticae de seriebus infinitis, eananque summa finita Basic , reprinted in Jacobi Bernoulli basileensis Opera, 2 vols (Geneva, 1744), Vol. I, 375-402 and S17-S42, Vol. II, 745-67, 849-67 and 955-75
  • Bernoulli , Jacob . 1713 . Ars conjectandi. Opus posthumum. Accedit tractatus de seriebus infinitis et epistola gallicè scripta de ludo pilae reticularis Basic
  • 2000 . 'Geometrical and Analytical Aspects in Leibniz's Theory of Series' . Studio Leibnitiana , 30 : 123 – 56 . See, for example, Newton's approach in next section. I have treated in detail Leibniz's ideas on series in
  • I disregard the fact that the quantitative notion of the sum was usually interpreted in the sense that the limit of the partial sums had to be effectively achieved. For instance, Leibniz often stated that a series is convergent if it can be continued to the extent that it differs from a certain finite quantity less than any given quantity; on the other hand, he repeatedly affirmed that a quantity is exactly equal to a series only when the whole series is taken into consideration (see G.W. Leibniz, 'De vera proportione circuits quadratum circumscriptum in numeris rationalibus', ActaEruditorum (1682),41-6, reprinted in Leibnizes Mathematische Scriften, edited by C.I. Gerhardt (Berlin and Halle, 1849-63), Vol. Ill, 985). Using a different and more Newtonian terminology, one may state that 5 was conceived as the sum of Zm_j a. if 5 was the ultimate value of the series.
  • Leibniz , G.G. 1993 . “ De quadratura arithmetica circuit ellipseos et hyperbolae cujus corollarium est trigonometria sine tabulis ” . In Abhandlung der Akademie der Wissenschaften der Gottigen Edited by: Knobloch , E. Vol. 43 , 83 – 84 . The reference is to
  • Later, in his Miscellanea analytica de seriebus et quadratwls (London, 1730), 129, this condition was explicitly affirmed by A. de Moivre, who termed this method 'bernoullian', asserting that the summability of series is of no importance provided that the nth term converges to zero.
  • Euler , L. 1911-75 . “ 'Consideratio progressionis cuiusdam ad circuit quadraturam inveniendam idoneae' ” . In Leonhardi Euleri opera omnia. Series I: Opera mathematica Vol. 14 , 350 – 65 . Berne
  • Engelsman , Steven B. 1984 . Families of Curves and Origins of Partial Differentiation , Amsterdam : Elsevier .
  • Newton , I. 1769-1785 . “ 'Recensio libri qui inscriptus est Commercium Epistolicum CoIlMi el aliorum de Analisi Promote' ” . In Opera quae exstant omnia commentariis ilustrabat Samuel Horsley 5 vols , London Vol. IV, 450-97. In particular, see p. 459
  • FoF instance, it is worth recalling Newton's words: '[W]hatever common analysis performs by equations made up of a finite number of terms (whenever it may be possible), this method may always perform by infinite equations' (De analysis per aequationes numéro terminorum infinitas, ibid.. Vol. I, 257-83).
  • For instance, P. Varignon was explicit in his rejection of divergent series. In his 'Precautions à prendre dans l'usage des suites ou séries infinies resultantens, tant da la division infinie des fractions, que du développement à l'infini des puissance d'exposants négatifs entiers'. Mémoires de l'Académie Royal des Sciences, Mémoires mathématiques et physique (1715), 203-25, he observed that if m < n (< O) then the equality (1) is true; if (O 0) and the denominator is the difference m - n then equality (1) is simply the equality °° =°° : it is true but does not provide information; if m =n and the denominator is the sum m + n then equality ( 1 ) is false.
  • Euler , L. 1911-75 . “ 'Delucidationes in capita postrema calculi mei differentialis de functionibus inexplicabilibus' ” . In Leonhardi Euleri Opera omnia. Series I: Opera mathematica Vol. 16 , Berne THe method actually consists in adding fictive terms to a given series. This idea probably gave the cue for, part 1, 1-33
  • Newton . 1769-85 . “ Anis analytica specimena vel gêometria analytica ” . In Opera quae exstant OiYOUa commentarus illustrate t Samuel Horsley 5 vols , London See, for instance, Vol. I, 391-519 (chs 1 and 3) and Newton (note 24), 257
  • This method is found in the 1692 version of De quadratura curvarum (see The Mathematical Papers of Isaac Newton, ed. by D.T. Whiteside, 8 vols (Cambridge, 1967-81) Vol. VII, 94-96). It is also formulated in Ex Epistola Newtoni ad Wallisium, de radicibus aequationum fluxionalium extrahendix, in Opera quae exstant omnia commentarus illustrabat Samuel Horsley, 5 vols (London, 1769-85), Vol. I, 293-97. A different method is in Newton (note 32), Vol. I, 413-28.
  • Newton, Ex Epistola Newtoni ad Wallisium (note 33), Vol. I, 294.
  • The Mathematical Papers of Isaac Newton, ed. by D.T. Whiteside, 8 vols, (Cambridge, 1967-81), Vol. VII, 93. It was precisely to solve this problem that Newton expounded the above illustrated method for finding the solution of a differential equation and, in a corollary, stated '[A]Il curves can be quadrated by means of indeterminate convergent series.' (ibid., 96). Newton explained that, given a curve F(x,y)=Q, it is sufficient to express the ordinale y=y(x) by a series using the above method and integrating it term by term.
  • NeWtOn, Artis analytica (note 32), Vol. I, 418.
  • Bos , H.J.M. 1974 . 'Differentials, Higher-Order Differentials and the Derivatives in the Leibnizian Calculus' . Archive for History of Exact Science , 14 : 1 – 90 .
  • Collins , J. 1712 . Commercium Epistolicum D. Johamis Collinis etaliorum de Analisi Promota 84 – 85 . London
  • Celeberr , Virorum . 1745 . Gothofredi Guillelmi Leibnitii et Johm. Bernoulli Commercium philosophicum et mathematician 2 vols , 213 Lausanne Vol. I
  • Ibid., 219. An example of the use of power series in summing numerical series is due to de Moivre and is interesting for my purpose since it concerns the Leibnizian method illustrated in Section 2. de Moivre rethought this method, by associating O?_ñ á. with the power series O?_ñ á,÷*. This allowed him to generalize the method (showing the greater power of function series). He indeed considered an appropriate polynomial p(x) and derived a new series p(x) O_ñ á,÷* =O_ñ b,x*. If X0 is one of the roots of p(x), then O?_ñ e.xjl =O and - ba is the sum of Z?_j6.xj>TFor/»(jt) =jt - 1 and á.=l/n de Moivre, like Leibniz, obtained O !/«(«+ 1)=l (de Moivre (note 18), 130-31).
  • Feigenbaum , L. 1985 . 'Brook Taylor and the Method of Increments' . Archive for History of Exact Sciences , 34 : 1 – 140 . For these I refer to numerous published articles by various authors. In particular, I point out
  • Stirling , J. 1730 . Methodus differentialis sive tractatus de summations et interpolations serierum infinitarum 135 – 37 . London
  • Moivre , A. de . 1718 . The Doctrine of Chances or a Method of Calculating the Probabilities of Events in Play London
  • 1722 . 'De fractionibus algebraicis radicalitate immunibus ad fractionibus simpliciores reducendis, deque summandis terminis quarundam serierum aequali intervallo a se distantibus' . Philosophical Transactions , 32 : 162 – 78 .
  • 1730 . Miscellanea analytica de seriebus et quadraturis London
  • C. Goldbach, 'De terminis generalibus serierum' (note 27); Daniel Bernoulli Observation« de seriebus quae formantur ex additione vel subtractione quacunque terminorum se mutuo consequentium' (note 27). See also P.H. Fuss (note 27), Vol. II, 273-75.
  • A. de Moivre, The Doctrine of Chances (note 63), 133. The term recurrent was only introduced in 1722, in de Moivre, 'De fractionibus algebraicis' (note 63), 175-76. In the second edition of The Doctrine of Chances (London, 1738), 193, de Moivre gave the following definition: call that a recurring series which is so constituted, that having taken at pleasure any number of its terms, each following term shall be related to the same number of preceding terms, according to a constant law of relation'.
  • ThC first signs of an evolution towards the merely combinatorial aspect can be found in Newton's theorem of the inversion of series discussed earlier and subsequently in de Moivre's generalization. The derivation of Bemouli's series (see Johann Bernoulli, 'Additamentum effectionis omnium quadraturam et rectificationum curvarum per seriem quandam generalissimam', Acta Eruditorum (1694), 437-441) and Leibniz's analogy (see G.G. Leibniz, 'Symbolismus memorabilis calculi algebraic! et infinitesimalis, in comparatione potentiarum et differentiarum, et de lege homogeneorum transcendental!'. Miscellanea Berolinensia (1710), 160-65) also gave considerable support to this approach to series.
  • Fraser , Craig G. 1989 . 'The Calculus as Algebraic Analysis: Some Observations on Mathematical Analysis in the 18th Century" . Archive for History of Exact Sciences , 39 : 317 – 35 . 330 See, for instance
  • Bernoulli , D. 1771 . 'De summationibus serierum quarunduam incongrue veris earumque interpretatione atque usu' . Novi commentarii academiae scientiarum imperialis Petropolitanae , 16 : 71 – 90 . 84 This is the base of Euler's conception, which Daniel Bernoulli expressed speaking of a sum false in concrete but true in abstracto

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