Lacroix and the Calculus pp 283-319 | Cite as

# The *Traité élémentaire*

Chapter

## Abstract

In 1802 Lacroix published a *Traité élémentaire du calcul différentiel et du calcul intégral* (Elementary treatise of differential and integral calculus) [Lacroix *1802a*]. According to the publisher’s list of elementary works by Lacroix, it was “tiré en partie”^{1} from the large *Traité* [Lacroix *1802a*, ii]. Indeed it is mostly an abridged version of the latter. It is divided into a “first part: differential calculus”, a “second part: integral calculus” and an “appendix: on differences and series”. The correspondence between these three parts and the three volumes of the large *Traité* is perfect.

## Keywords

Power Series Taylor Series Geometrical Construction Analytic Geometry Integral Calculus
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## References

- 2.The full title is “Réflexions sur la manière d’enseigner les Mathématiques, et d’apprécier dans les examens le savoir de ceux qui les ont étudiées” (“Reflexions on the manner of teaching Mathematics, and of evaluating in examinations the knowledge of those who have studied it”) [Lacroix
*1802a*, v–xxxii]. These “Refléxions” were afterwards included in [Lacroix*1805*] and therefore omitted from later editions of [Lacroix*1802a*].Google Scholar - 6.Or even just 1, 2, 4 and 6. Item 5 [Lacroix
*1795*] was not “essentiellement partie du cours élémentaire de Géométrie” (“essentially part of the elementary course of geometry”) [Lacroix*1805*, 346]. That minimal version of the*cours élémentaire*is the one that appears in the first edition of [Lacroix*Traité*, III] (in the usual advertisement for books by Lacroix), [Lacroix*1795*], [Lacroix*Traité*], and the*Complément des Elémens d’Algèbre*appearing apart. But it is not of much concern here whether [Lacroix*1795*] should be included in Lacroix’s*cours élémentaire*.Google Scholar - 9.At least it was not present within
*each*subject. Lacroix was an ardent supporter of the model of the*écoles centrales*, which offered a much wider range of subjects than either the pre-revolutionary*collèges*or the*lycées*that later replaced them. “[T]he avowed aim of [the*écoles centrales*] was a sound but encyclopedic education, covering all ‘positive’ knowledge” [Dhombres*1985*, 125]. Dhombres [*1985*, 130] seems to attribute an encyclopedic character also to each of Lacroix’s textbooks by extrapolating from the characteristics of [Lacroix*Traité*].Google Scholar - 13.The curriculum at each
*école centrale*was decided by a local commission. On mathematics the law only stipulated that at each*école centrale*there should be one teacher of that subject, placed at the “second section” (to which only pupils aged 14 and over were admitted). All subjects being optional for the students, the “special” character of some is doubtful. Moreover, transcendental mathematics might be taught in some*écoles centrales*but not in others. At the*École Centrale du Doubs*at Besançon, for instance, the most advanced topic seems to have been the application of algebra to geometry (no theory of series or calculus) [Troux*1926*, 167–170]. On the other hand, infinitesimal calculus (which would qualify as transcendental) was taught at the*école centrale*of Nantes; and yet, very few students from Nantes applied for the*École Polytechnique*[Lamandé*1988–1989*, 134–143]. The*lycées*, created by law in 1802, were on the contrary highly centralized. At each*lycée*there should be six “classes” of mathematics (two per year, giving a total of three years), taught by three teachers, plus two “classes” of “transcendental mathematics” (two years, one teacher). Transcendental mathematics included topics such as “application of differential [and integral] calculus to mechanics and to the theory of fluids” or “general principles of high physics, especially electricity and optics” [Lycées*1802*, 307].Google Scholar - 17.Dhombres [
*1987*, 95], on the other hand, suspects that the letter had been prepared by the predecessor of Lucien Bonaparte, Laplace.Google Scholar - 19.A very similar programme can be seen in [Éc. Pol.
*Concours 1802*] (1802, incidentally, is the year of publication of the first edition of Lacroix’s*Traité élémentaire du calcul*...).Google Scholar - 21.These were precisely the textbooks adopted in 1803 for the six normal “classes” of mathematics [Lycées
*1803*].Google Scholar - 22.This apparently meant a modest discount, as bought separately they would cost 29 fr. 50 c. But it may be a misprint, the
*Élémens d’algèbre*costing 4 fr., not 5 [Lacroix*1805*, iv, 391].Google Scholar - 26.It may also be relevant that the statutes of the University of Coimbra of 1772 established the regular teaching of differential and integral calculus in the second year of the new Faculty of Mathematics [Univ.Coimbra
*Estatutos 1772*, III, pt. 2] — for this teaching the calculus section in Bézout’s course was translated into Portuguese; even Belhoste acknowledges that Lagrange taught the calculus in an artillery school in Turin in 1758 and 1759, and that Euler appears to have done the same in St. Petersburg in the late 1720’s [*2003*, 477].Google Scholar - 28.Apart or in connection with these he also taught descriptive geometry, Euclidean geometry, statics, hydrostatics and dynamics [Fourier
*1796*, xv; Grattan-Guinness*1972*, 6–7]. But these subjects are not our concern here.Google Scholar - 31.Belhoste [
*2003*, 245], as well as Lorrain and Pepe [Fourier*1796*, xviii], associate Fourier’s use of finite differences in introducing the differential calculus to Prony’s lectures of year 3 [Prony*1795a*]. But Fourier uses finite differences in a traditional manner, similar to what Euler [*Differentialis*] and Cousin [*1777; 1796*] had done, and Bossut [*1798*] was about to do. Prony’s use of the calculus of finite differences instead of differential calculus is something quite different, and not necessary to explain Fourier’s short references.Google Scholar - 34.[Garnier
*1801*] is relatively common. But [Garnier*1800*] seems quite rare — no copies at the*École Polytechnique, Bibliothèque Nationale de France*, or British Library; oddly, there are copies in the Faculty of Science of Porto and Science Museum of Lisbon (with some differences between them — see the Bibliography below).Google Scholar - 37.Monge was in Egypt. Hachette published that year [Monge & Hachette
*1799*] to compensate for the lack of material on space curves in the first edition of [Monge*Feuilles*]Google Scholar - 39.But, according to an addition (“Note sur les numéros 6, 7, 8 et 9”) to [Garnier
*1800–1802*, II], in one of his courses Fourier used functional equations (“propriété[s] caractéristique[s]”) to obtain the differentials of transcendental functions, and then used these to arrive at their expansions.Google Scholar - 41.According to the
*Registre de Contrôle des Instituteurs et Agents*[Éc. Pol.*Arch*, X2c26], he had already fulfilled the duties of examiner in 1808 (seemingly in a temporary way), but was only appointed for the post in 1809.Google Scholar - 45.Lamandé has also compared [Lacroix
*1802a*] with [l’Hôpital*1696*], in [Lamandé*1998*]. A detail in the title of this paper is quite eloquent: “Une même mathématique?” (“The same mathematics?”). Still, there is a point in common between [Lacroix*1802a*] and [l’Hôpital*1696*]: both were*modern*when they were written; the same cannot be said of Bézout’s text.Google Scholar - 47.Actually, part of its influence may have been lost before 1815. From 1808, Ampère had introduced some developments of his own on the use of limits; and in 1812, limits were officially replaced by infinitesimals [Fourcy
*1828*, 303].Google Scholar - 58.Grabiner found these simple arguments “important because they exemplify translations of a verbal limit concept into algebraic language, however simple” [Grabiner
*1981*, 84]. That is true, but she appears to speak of them only as examples of a kind of argument that sometimes appeared around 1800; in other words, they are not major breakthroughs. For instance, the Portuguese mathematicians José Anastácio da Cunha and Francisco Garção Stockler had given more sophisticated arguments [Domingues*2004a*], as had l’Huilier.Google Scholar - 60.Actually, the fact that all the differential coefficients at
*x*= 0 of*e*^{−1/x}are zero had appeared in [Euler*Integralis*, I, § 327], in [Lacroix*Traité*, II, 149], and in the earlier editions of [Lacroix*1802a*] (in the context of the application of the improved version of Euler’s “general method” for approximation of integrals; it is at this point in the fourth edition that Lacroix includes Poisson’s explanation). But this did not seem to bother anyone, probably because they were used to the Taylor series to fail in particular points (see section 3.2.5). The*detail*added by Cauchy to the effect that this destroys what we would call the bijection between functions and Taylor series was certainly more disturbing.Google Scholar - 62.“If
*ε X dx*=*P*+*C*,*P*denoting the variable function immediately deduced by the process of integration,*C*the arbitrary constant, and if the integral ought to vanish for a value of*x*=*a*, which changes*P*into*A*; we shall then have the equation*A*+*C*= 0, from which we deduce*C*= −*A*, and*ε X dx*=*P − A*. Under this form the integral*ε X dx*is nothing more than the difference between the value of the function*P*, when*x*=*a*, and that which it acquires for every other value of the same variable. If, for example,*x*=*b*, changes*P*into*B*, there arises*ε X dx*=*B − A*.” [Lacroix*1816*, 271–272]Google Scholar - 63.It may be said to survive timidly in the passage giving the geometrical interpretation of the approximation method [Lacroix
*1802a*, 2nd ed, 310–312], and in the argument that because*Aα*+*A*_{1}*α*+*A*_{2}*α*...+*A*_{n−1}*α*<*A*_{m}*nα*=*A*_{m}(*b−a*), where*A*_{m}is the largest of*A*,*A*_{1},*A*_{2},...*A*_{n−1}and*a, b*are the limits of integration, then*ε X*d*x*<*M*(*b*−*a*), where*M*is the largest value of*X*between*x*=*a*and*x*=*b*(and similarly for*ε X*d*x*>*m*(*b*−*a*)) [Lacroix*1802a*, 2nd ed, 307]. But this argument might also be interpreted in terms of infinitesimals.Google Scholar - 65.In fact, Lacroix refers to a previous article, where Taylor series had been used to argue for the existence of solutions [Lacroix
*1802a*, 3rd ed, 402–404].Google Scholar - 66.Notice the dates: [Lacroix
*1802a*, 2nd ed] was also published in 1806.Google Scholar - 67.In spite of this, in the introduction to the section Lacroix keeps a distinction between particular solutions which are simply factors of the differential equation and others “intimately linked” to it [
*1802a*, 2nd ed, 429–430].Google Scholar - 70.And also the
*Tratado elementar de Arithmetica*(1810, translated by Francisco Cordeiro da Silva Torres e Alvim; I have not seen this book, but it is mentioned by Inocêncio [*DBP*, II, 367] and Circe M. S. Silva [*1996*, 82]), the*Elementos d’Algebra*(1811, translation of [Lacroix*1799*], also by Francisco Cordeiro da Silva Torres), the*Tratado Elementar de Applicação de Algebra á Geometria*(1812, partial translation of [Lacroix*1798b*], with an appendix on geometry in space, by José Victorino dos Santos e Souza), and the*Complemento dos Elementos d’Algebra*(1813; I have not seen this book, which is mentioned in [*NUC*, CCCX, 654]).Google Scholar - 72.“Fluxional coefficient” is of course evocative of Lacroix’s “differential coefficient”. Wallace gave a long list or works on the calculus, both British and Continental, [
*1815*, 388–389], but lamenting the absence of up-to-date books in English. Anyone wishing to study it “beyond its mere elements” should recur to Euler’s books, or French treatises — among the latter, he stressed Lacroix’s large*Traité*.Google Scholar - 77.
- 81.In Lithuanian: Zakarijas Niemčevskis [Venclova
*1981*; Yla*1981*] or Zacharijus Nemčevskis [Banionis*2001*, 43].Google Scholar - 82.He appears in Fourcy’s list of foreign students for 1804, as “Niemezewski” [Fourcy
*1828*, 387].Google Scholar - 83.Another translation of Biot’s
*géométrie analytique*, by Antoni Wyrwicz, was published in 1819 (by the same publisher, Marcinowski). Wyrwicz (or Virvichyus [Gyachyauskas*1979*, 170]) also taught at Vilnius University.Google Scholar - 84.He does not appear in Fourcy’s list of foreign students [Fourcy
*1828*, 387–389].Google Scholar

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