Aspects of differences and series

Part of the Science Networks. Historical Studies book series (SNHS, volume 35)


This section is concerned with the indexed (subscript) notation for sequences or series, as in u0+u1+. . .+u n +. . . This may seem a rather trivial subject, but the dedication of a section to it is justified for three reasons: 1 — the standard reference on the history of notations pays very little attention to it [Cajori 1928–1929, II, 265–266], making it only a detail in the notations on finite differences, and seemingly giving priority to Lagrange in 1792 (while it had been used nearly twenty years before by Laplace); 2 — the question of “;when the subscript notation arose”; has been asked very recently, at the end of a paper that illustrates the importance of unifying terminologies and powerful notations [Sandifer 2007, 299]; and finally 3 — its use by Lacroix has caused some confusion, its creation or its introduction in France being misattributed to him. Thus, Dhombres [1986, 156], quoting [Lacroix Traité, 2nd ed, I, 33], remarks that “;c’est à cette occasion que Lacroix introduit la notation indexée A0 + A1x + A2x2 + A3x3 + . . .”1. While Schubring [2005, 386] gives a lengthy footnote on the subject, which is worth quoting in full (citations of Lacroix have been adapted):

“Standard French textbooks up to about 1800 do not give sequences of quantities or variables with a notation identifying the single term of a sequence as part of a generally labeled sequence, for example, a3 as part of a sequence (a n ) with the general term a n . Lagrange used letters in alphabetic order to label elements as part of a sequence, for example, the function terms in developing it into a series as P,Q,R, and so forth or coefficients with A,B,C, and so forth. With such an unspecific approach, he was not able to label the last term of a sequence or a general term. It is notable that Crelle shifted to indexed series in the sections he added to his translation of the Théorie des fonctions analytiques, for example: B1,B2. . ., B n or P1, P2, P3 with P n as general term (Lagrange 1823, Vol. 2, 332 ff.). Lacroix had already used general indexed quantities a1, a2 . . ., a n in both 1798 and 1802, but only in a narrowly restricted field of calculus: within integral calculus to operate with the sequence of approximate values in using approximation to determine integral values [Lacroix Traité, II, 135 ff.; 1802a, 285 ff.]. Lacroix, who had studied the contemporary literature intensively, may have been encouraged to introduce this usage—even though very partial— by the publications of the German school of combinatorics, which used indexed quantities as one of their everyday tools.”


General Term Singular Integral Alphabetic Order Roman Numeral Integral Calculus 
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  1. 2.
    It is not completely clear whether Dhombres, in the sentence quoted above, means that Lacroix introduces the indexed notation absolutely (as in, say, “;in his first article on differential calculus, Leibniz introduces the d notation”;), or only in the context of his book (“;in [1696] l’Hôital introduces differentials as infinitely small differences”;). If the latter is the case, he was not mistaken. But the former seems much more likely (a few pages earlier he had remarked that Euler had not used that notation [Dhombres 1986, 153]). In [1988, 19] Dhombres and Pensivy were more cautious, speaking only of Lacroix having diffused the modern indexed notation.Google Scholar
  2. 5.
    On the German Combinatorial School, see for example [Jahnke 1993].Google Scholar
  3. 10.
    They did the same for [Lagrange 1759c]: compare (my emphases) “;si l’exposant de y exprime toujours la place qui tient la particule qui parcourt l’espace y, en comptant depuis la premiére F”; in [Lagrange 1759c, 1st ed, 9], with “;si l’indice de y exprime toujours la place qui tient la particule qui parcourt l’espace y, en comptant depuis la premiére F”; in [Lagrange OEuvres, I, 55].Google Scholar
  4. 14.
    For instance, in [Laplace 1773a, 57] we see px and 1px, meaning p and 1p raised to the xth power.Google Scholar
  5. 15.
    Euler had already used generating functions, but Laplace “;was perhaps the first to exploit fully”; this concept [Goldstine 1977, 127, 185].Google Scholar
  6. 17.
    Towards the final lectures, Prony also wrote z0, z1, z11, &c... z(n) [1795a, IV, 544].Google Scholar
  7. 19.
    Around the same time, Fourier, in his lectures at the École Polytechnique [1796, 54–55], gave a similar proof for the expansion of ax, with two differences: 1 — he did not use indices, but rather the alphabetical order A,B,C,...; 2 — instead of ax × au = ax+u he used the property a2x = (ax)2, which makes calculations much easier, and indices dispensable. In the second edition of his Traité, Lacroix mentioned this approach in a footnote, but he preferred ax × au = ax+u for being more general and expressing the most extensive definition of ax [Traité, 2nd ed, I, 35].Google Scholar
  8. 23.
    In [Domingues 2005, 289] I said that (7.3) come from [Lagrange 1772a]. I was wrong: [Lagrange 1772a] gives analogies between powers and higher differences and derivatives (like (7.4)), and it is the inspiration for [Lorgna 1786–87 ], but (7.3) are not found there. Incidentally, (7.1) is, but with un referred to only verbally [Lagrange 1772a, § 17].Google Scholar
  9. 25.
    Charles [1785b, 560] has “;V = 0, & V = 0”;, which is clearly a typo.Google Scholar
  10. 28.
    Oddly, Grattan-Guinness [1990, I, 224] has said that in the first edition Lacroix “;mentioned”; Biot’s memoir, “;and gave a lengthy account of it in the second edition”;. In fact, the lengthy account of the second edition is virtually identical to that of the first edition [Lacroix Traité, 1st ed, III, 237–247; 2nd ed, III, 250–260]. He has also said [1990, I, 227] that Lacroix used Biot’s paper on mixed difference equations [Biot 1799] only in the second edition of his Traité, but as we will see below that already happened in the first edition. Grattan-Guinness must have underestimated the degree of collaboration between Lacroix and Biot.Google Scholar
  11. 33.
    This is better explained in [Biot 1797, 184–185] than in [Lacroix Traité, III, 238].Google Scholar
  12. 34.
    Apparently it was Poisson who first made it [1800, 180].Google Scholar
  13. 36.
    [Biot 1797, 183] has Fx, a, yxa = 0, Fx′, a′, yx a′ = 0, Fx′, a″, yxa′ = 0, which must be a triple typo for Fx, a, yxa = 0, Fx, a′, yxa′ = 0, Fx, a″, yxa″ = 0.Google Scholar
  14. 40.
    [Charles 1785a] is probably the result of combining several memoirs submitted to the Paris Academy in 1779 and 1780, and possibly one submitted in May 1785 [Hahn 1981, 84]. [Charles 1785b], as we have noticed above, was read in November 1785.Google Scholar
  15. 42.
    Condorcet’s underestimation of the variety of transcendental functions is one of the biggest problems with his “;general”; theory of integration [Gilain 1988, 93].Google Scholar
  16. 44.
    The procès-verbal says that the reporters were Lavoisier, Cadet and Darcet, which must be a mistake (these were all chemists). According to Hahn [1981, 84] the reporters were Cousin and Condorcet.Google Scholar
  17. 45.
    Another contribution, immediately preceding that one, is the article “Intégral (Calcul intégral des équations en différences finies)”[Charles 1785c], more than half of which is also reproduced from [Charles 1785a, 574–579]Google Scholar
  18. 46.
    Both Frankel [1978, 41] and Grattan-Guinness [1990, I, 227] attribute it to Lacroix, and there is no reason to question this attribution; on the contrary — its terminology (“differences”-instead of “finite differences”; “partial differentials” instead of “partial differences”; “indirect integrals”; “differential coefficients”) points to Lacroix, and so does a reference to Fontaine’s authorship of the “important remark” that a differential equation is the result of elimination of constants between a “primitive equation” and its differentials.Google Scholar
  19. 47.
    Biot was an associé-correspondant of the Société Philomatique. Although this summary has an indication “;Institut Nat.”; on the side, the report of the activities of the Société states that Biot also read the memoir to its members [Soc. Phil. Rapp, IV, 14].Google Scholar
  20. 48.
    As has been remarked above, Lacroix does mention Condorcet, but he does not say anything about the contents of [Condorcet 1771 ], nor establishes any relation between Condorcet’s and Biot’s theories.Google Scholar
  21. 49.
    Well, mostly constants. In some cases, more or less obvious, what is intended is elimination of functions of period Δx (constant for values of Δx that differ by Δx). One reference by Biot [1799, 297] to the possibility of a “;more general”; characterization may be an allusion to this issue.Google Scholar
  22. 53.
    The fact that in the table of contents Lacroix only mentions [Charles 1785d] is the final indication that he did not really know, or did not pay attention to, [Charles 1785a].Google Scholar
  23. 54.
    François-Joseph Français (1768–1810), who was for some time a teacher in Colmar.Google Scholar

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