دانلود کتاب Convolutions in French Mathematics, 1800–1840 From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics

excellent treatment of Ampère's mathematical works, esp. ch. 14 pp. 917ff. (PDF pp. 908ff.): “[The Entry of Physicist Ampère: Electricity and Magnetism, Especially Their Connections, 1820–1827](https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/GUH34JXZ/Grattan-Guinness%20-%201990%20-%20The%20entry%20of%20physicist%20Amp%c3%a8re%20electricity%20and%20mag.pdf).”
This article is a sort of synopsis:
Grattan-Guinness, Ivor. “[Lines of Mathematical Thought in the Electrodynamics of Ampère](https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/28JF7FBX/Grattan-Guinness%20-%201991%20-%20Lines%20of%20mathematical%20thought%20in%20the%20electrodynami.pdf).” *Physis* 28, no. 1 (1991): 115–29.


§3.2.5 (p. 135 // PDF p. 135) is on analysis vs. synthesis
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Almost all major developments of the mathematical aspects of the mathematization of physics during the period 1800–1830 were made in France. The purpose of this book is to describe in great detail (principal books and papers, but also archival studies) the development of calculus in tandem with its applications to an ever wider class of physical phenomena (mechanics, heat diffusion, electricity, magnetism, and optics). Education, essentially centralised in Paris, played a significant role in these developments since, after the Revolution of 1789, the founding of the École Polytechnique signalled the creation of other influential schools. In particular, because of the need to supply engineers for public works and military applications, engineering turned out to be a major preoccupation of French science; it gave rise to significant progress in earth pressure theory and hydrodynamics.
The first volume deals with the period 1800–1815. Research begins around 1800, with Lagrange's authoritative view of the calculus, in which functions were treated as algebraic formulae. The various ways in which his views were extended by others include foundational points of view and a theory of operators. Lagrange was also the dominant figure in the theory of equations in relation to mechanics. Lagrange expressed the principles of mechanics in variational form, an expression which was an extension of his algebraic approach to the calculus. In celestial and planetary topics, Laplace took over prime position with a change of philosophy and scientific strategy. Laplace's celestial mechanics (effects of perturbing forces; stability of the solar system) needed geometrical thought and differential modelling to achieve its results, casting aside variational formulations. Generally speaking, the distinctive feature of French celestial mechanics at that time was that the French sought exact, though complicated, rather than approximate solutions to astronomical problems. Around 1805 Laplace launched a research program in physics, according to which all phenomena of nature should be reduced to the action at a distance between molecules. The program was especially successful in optics—light was understood as a molecular phenomenon. As the Laplacian program continued to develop, however, physicists such as Biot and Poisson made use of fluids to supplement the activity of the molecules, especially magnetic and electrical.
Volume II of this study focuses on the period 1815–1830. The Bourbon Restoration brought several institutional changes, and with them calculus and mechanics were disturbed by waves of new innovations. Fourier's work on heat diffusion, culminating in a new integral solution, is especially significant as first mathematical treatment of a phenomenon outside rational mechanics. Cauchy is the dominant figure in the emergence of mathematical analysis, resulting from fierce competition between him and Poisson; Cauchy's intellectual journey goes from his first ideas on complex variables to the elevation of the theory of limits as basis of the theory of functions. In physical optics, Fresnel used a wave theory to examine diffraction in explicit opposition to the Laplacians; he moved from a theory of longitudinal vibrations of the particles of the aether to a transversal theory. In his efforts toward a mathematization of electrodynamics and electromagnetism, Ampère assumed that all phenomena are electrical, so that magnetism for instance would appear as a particular form of electricity; he was able to deduce an inverse square law between the strength of action and the distance of separation between element and wire. Elasticity theory was developed by Cauchy on the basis of a stress-strain model, in reaction to an initial breakthrough made by Navier. As for engineering, a new general approach to energy mechanics was made possible through the concept of work (Coriolis, Poncelet). These various developments in physics indicate that the Laplacian tradition was well into decline by 1825. By 1830 all the major changes had taken place. There followed a period of consolidation in the next decade, but also fragmentation: thus the scientific community divided by its research interests into two groups, those involved in general applications and pure mathematics and those involved in engineering questions.
This history points to the following philosophical issues: (i) each new development testifies to the similarity of structure between mathematical and physical theories, and yet a wide range of opinions existed in regard to how this similarity should be interpreted; the issue is particularly pressing in the case of differentials representing very small objects within the Laplacian framework, since the Laplacians did not confine theorising to experiential categories; (ii) the unification of the various branches of mathematical physics: the fall of Laplacians meant that the hopes for unification had to be abandoned, at least for a time; (iii) the issue of whether physical theories are true or hypothetical is a constant theme underlying these developments: significantly enough, Laplace will speak of the "veritable hypotheses of nature''; (iv) mathematical research was carried out in at least three "styles of thinking'', evolving from the algebraic to the analytical, the geometrical style being maintained throughout the period as an aid to the discovery of new results; these styles involve both epistemological and methodological aspects of each work considered in this study.
Information and data of various kinds are collected in the third volume: a series of manuscripts and obscurely published passages related to the whole development of mathematical analysis and mathematical physics; a comprehensive chronological table of events; a full bibliography of both primary and secondary sources.
Reviewed by [Pierre Kerszberg](http://ams.rice.edu/mathscinet/search/author.html?mrauthid=100610)