Cauchy's Work Research Articles

19th-century real analysis, forward and backward

19th century real analysis received a major impetus from Cauchy's work. Cauchy mentions variable quantities, limits, and infinitesimals, but the meaning he attached to these terms is not identical to their modern meaning. Some Cauchy historians work in a conceptual scheme dominated by an assumption of a teleological nature of the evolution of real analysis toward a preordained outcome. Thus, Gilain and Siegmund-Schultze assume that references to limite in Cauchy's work necessarily imply that Cauchy was working with an Archi-medean continuum, whereas infinitesimals were merely a convenient figure of speech, for which Cauchy had in mind a complete justification in terms of Archimedean limits. However, there is another formalisation of Cauchy's procedures exploiting his limite, more consistent with Cauchy's ubiquitous use of infinitesimals, in terms of the standard part principle of modern infinitesimal analysis. We challenge a misconception according to which Cauchy was allegedly forced to teach infinitesimals at the Ecole Polytechnique. We show that the debate there concerned mainly the issue of rigor, a separate one from infinitesimals. A critique of Cauchy's approach by his contemporary de Prony sheds light on the meaning of rigor to Cauchy and his contemporaries. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis, and indicates that he was a pioneer of infinitesimal techniques as much as a harbinger of the Epsilontik.

Cauchy's Work on Integral Geometry, Centers of Curvature, and Other Applications of Infinitesimals

Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy applied infinitesimals in an 1826 work in differential geometry where infinitesimals are used neither as variable quantities nor as sequences but rather as numbers. He also applied infinitesimals in an 1832 article on integral geometry, similarly as numbers. We explore these and other applications of Cauchy's infinitesimals as used in his textbooks and research articles. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis and geometry. We demonstrate the viability of Cauchy's infinitesimal techniques in fields as diverse as geometric probability, differential geometry, elasticity, Dirac delta functions, continuity and convergence. Keywords: Cauchy--Crofton formula; center of curvature; continuity; infinitesimals; integral geometry; limite; standard part; de Prony; Poisson

Euler, the clothoid and

One of the many definite integrals that Euler was the first to evaluate was(1)He did this, almost as an afterthought, at the end of his short, seven-page paper catalogued as E675 in [1] and with the matter-of-fact title, On the values of integrals from x = 0 to x = ∞. It is a beautiful Euler miniature which neatly illustrates the unexpected twists and turns in the history of mathematics. For Euler's derivation of (1) emerges as the by-product of a solution to a problem in differential geometry concerning the clothoid curve which he had first encountered nearly forty years earlier in his paper E65, [1]. As highlighted in the recent Gazette article [2], E675 is notable for Euler's use of a complex number substitution to evaluate a real-variable integral. He used this technique in about a dozen of the papers written in the last decade of his life. The rationale for this manoeuvre caused much debate among later mathematicians such as Laplace and Poisson and the technique was only put on a secure footing by the work of Cauchy from 1814 onwards on the foundations of complex function theory, [3, Chapter 1]. Euler's justification was essentially pragmatic (in agreement with numerical evidence) and by what Dunham in [4, p. 68] characterises as his informal credo, ‘Follow the formulas, and they will lead to the truth.’ Smithies, [3, p. 187], contextualises Euler's approach by noting that, at that time, ‘a function was usually thought of as being defined by an analytic expression; by the principle of the generality of analysis, which was widely and often tacitly accepted, such an expression was expected to be valid for all values, real or complex, of the independent variable’. In this article, we examine E675 closely. We have tweaked notation and condensed the working in places to reflect modern usage. At the end, we outline what is, with hindsight, needed to make Euler's arguments watertight: it is worth noting that all of his conclusions survive intact and that the intermediate functions of one and two variables that he introduces in E675 remain the key ingredients for much subsequent work on these integrals.

THE CONCEPT OF ‘VARIABLE’ IN NINETEENTH CENTURY ANALYSIS

Previous articleNext article No AccessDiscussionsTHE CONCEPT OF ‘VARIABLE’ IN NINETEENTH CENTURY ANALYSISJOHN P. CLEAVEJOHN P. CLEAVEUniversity of Bristol Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by The British Journal for the Philosophy of Science Volume 30, Number 3September 1979 Society: The British Society for the Philosophy of Science Views: 4Total views on this site Citations: 1Citations are reported from Crossref Views: 4Total views on this site Citations: 1Citations are reported from Crossref Article DOIhttps://doi.org/10.1093/bjps/30.3.266 Views: 4Total views on this site Citations: 1Citations are reported from Crossref © 1979 by The Author. All rights reserved.PDF download Crossref reports the following articles citing this article:NIGEL CUTLAND, CHRISTOPH KESSLER, EKKEHARD KOPP, and DAVID ROSS On Cauchy's Notion of Infinitesimal, The British Journal for the Philosophy of Science 39, no.33 (Dec 2020): 375–378.https://doi.org/10.1093/bjps/39.3.375

Cauchy and the spectral theory of matrices

It is well-known that Cauchy (1829) provided the first general proof that the eigenvalues of a symmetric matrix are real. Furthermore, Cauchy's paper initiated the developments that resulted in the creation of a substantial spectral theory of matrices by the early 1870's. The following essay relates Cauchy's work and its historical significance to the consideration of eigenvalue problems during the 18th century.

Notes on the History of the Uses of Analyticity in Operator Theory

provided that d(X) #0. Heref and g are members of the function class C[[a, b]. The function d(X) is an entire analytic function of the complex parameter X and D(s, t; X), which is a continuous function of (s, t), is also an entire analytic function of X. The display of analyticity in the solution of the Fredholm equation of the second kind is an early signal of the important role which analyticity was destined to play in spectral theory. Anticipations of spectral theory itself can be traced at least as far back as 1836, which was the date of publication of the work of Sturm [38], [39] and Liouville [28], [29] on second-order ordinary differential equations and the associated eigenvalue and expansion theorems. In 1904 Hilbert [20] showed how to transform a differential boundary value problem into a problem of solving an integral equation. Moreover, the Green's function of the differential problem turned out, under certain conditions, to be a kernel of Hilbert-Schmidt type, from which followed a great many things of interest. The utility of analytic functions and the calculus of residues in connection with expansions in series of special functions has been known since the work of Cauchy. In 1894 Poincare [34] dealt with the problem of the vibrating membrane and related the eigenvalues and eigenfunction expansions to the meromorphic character of the Green's function's dependence on the parameter. In 1908 Birkhoff made explicit the use of the meromorphic behavior of the Green's

POWER SERIES SOLUTION, BY MACHINE, OF A NONLINEAR PROBLEM IN TWO‐DIMENSIONAL FLUID FLOW*

SummaryThe solution of a partial differential equation, with Cauchy data given on an initial surface, can be obtained as a power series expansion in the independent variables about a point on the surface if the functions appearing in the differential equations, the equation for the initial surface, and the Cauchy data are all analytic, as has been known since the work of Cauchy and Kowalewski. This power series and its analytic continuations can be used in principle for calculating the solution. The practical application of these ideas to a certain class of problems, with the use of a modern computing machine, can be achieved by means of techniques that have been developed at New York University. Typical of the class is the problem of determining the flow of air past an axially symmetric blunt body moving at supersonic speed. This flow contains a detached shock ahead of the body and, aft of the shock, a subsonic region near the nose and a supersonic region farther downstream. Normally heat conduction and viscosity can be ignored, except in a boundary layer of negligible thickness on the surface of the body. If the problem is inverted–if the shock is assumed to lie on a given analytic surface and the Rankine‐Hugoniot jump conditions at the shock are used to provide Cauchy data from which the flow aft of the shock can be obtained, and from which, in turn, the body shape required to produce the assumed shock and resulting flow can be determined, there results a problem of the kind described, with a fourth‐order system of nonlinear partial differential equations in two independent variables. The techniques described, which yield a powerful computational procedure for problems of this kind, are (1) algorithms that enable the machine to perform the needed algebraic as well as the numerical work; (2) special floating‐point subroutines, utilizing the so‐called significance arithmetic; (3) methods for obtaining analytic extensions of the functions describing the flow. These techniques and typical results are described.