Lacroix and the Calculus pp 343-352 | Cite as

# Final remarks

## Abstract

*Traité*Lacroix made a declaration of modesty:

“Parmi beaucoup de choses extraites des ouvrages des grands Géomètres de nos jours, il se trouvera peut-être quelques détails qui m’appartiendront; mais je ne disputerai pas là-dessus, et je me contenterai de ce qu’on voudra bien me laisser.”

^{1}[Traité, I, xxviii] We have seen that he did not keep this promise entirely. In the Preface of the second edition, and in his “Compte rendu [...] des progrès que les mathématiques ont faits depuis 1789 [...]” (appendix B) he claimed priority for some details: his use of indices in proving the power-series expansions of transcendental functions (section 7.1.2); the change of independent variable without consideration of constant differentials (section 3.2.4); a proof of Newton’s theorem on the sums of powers of the roots of an equation [Traité, I, 283–286]; remarks on limitations in the number of arbitrary functions in integrals of higher-order partial differential equations (section 6.2.2.3); and the analytical theory of the different kinds of integral of total differential equations in three variables that do not satisfy the conditions of integrability (section 6.2.4.2). To this, we can also add the section on the “development of curves traced on surfaces” in the second edition, adapted from Lacroix’s 1790 memoir (appendix A.2).

## Keywords

Final Remark Late 18th Century Analytic Geometry Relevant Point Geometrical Application## Preview

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## References

- 3.Lützen [
*1990*, 779] had already noticed this misattribution, while classifying [Ross*1977*] as an “excellent paper” [Lützen*1990*, 306].Google Scholar - 8.With the exception of the construction argument for the admissability of discontinuous functions in integrals of partial differential equations. But once again, a situation in which Lagrange himself referred to Monge [Truesdell
*1960*, 295].Google Scholar - 11.There is the occasional flaw: in the first edition, the chapter on the calculus of variations omits the power-series approach; in [
*Fonctions*] Lagrange had already tried to use it for variations [Fraser*1985*, 181–182].Google Scholar - 12.It is not only in reference to Lacroix that Schubring misses uses of infinitesimals: he claims that the discussion on curves of double curvature in [Monge
*Feuilles*] has “absolutely no reference to the*infiniment petits*” [Schubring*2005*, 379]. Compare with [Monge*Feuilles*, n^{o}32; 3rd ed, 343–344]: “Par un point*A*de cette courbe, soit mené un plan*MNOP*perpendiculaire à la tangente en*A*; par le point*a*infiniment proche, soit pareillement mené un plan*mnOP*perpendiculaire à la tangente en*a*[...] tous les points de l’arc infiniment petit*Aa*[...]” (“Let a plane*MNOP*be drawn through a point*A*of that curve, and perpendicular to the tangent in*A*; let a plane*mnOP*be similarly drawn through the infinitely close point*a*, and perpendicular to the tangent in*a*[...] all the points in the infinitely small arc*Aa*[...]”).Google Scholar