## Mémoire sur la Dispersion de la Lumière. [As published in Nouveaux Exercices de Mathématiques].

Prague: J.G. Calve, 1835-36.

First edition of the memoir in which Cauchy explained the dispersion of light from the undulatory theory, and first derived the equation, named after him, relating the refractive index and wavelength of light for a particular transparent material. This memoir was issued as eight parts of his periodical ‘Nouveaux Exercices de Mathématiques’ (see below). “The dispersion of light when passing through matter without getting fully absorbed, i.e., the variation of the refractive index *n* (which denotes the ratio of the velocity of light in vacuo to the velocity of light in matter) with the wavelength, always played an important role in the history of optics since the times of Isaac Newton. After the undulatory theory of light won over the corpuscular view in the beginning of the nineteenth century, Augustin-Louis Cauchy proposed an explanation of the dispersion phenomenon, based on the assumption that matter consisted of molecules having finite distances from each other [the offered work]. According to his theory the refractive index *n* increased with decreasing wavelength *λ* of the light – *n* was approximately proportional to *λ*^{-2} – in general agreement with the data on what was later called ‘normal’ dispersion” (Mehra & Rechenberg, p. 630). In his publication Cauchy nowhere referred for inspiration to Augustin-Jean Fresnel, the originator in France of wave optics. Instead, he wrote that Gustave-Gaspard Coriolis, having read Cauchy’s earlier work on the equations of motion that govern a system of material points, suggested that terms which Cauchy had there neglected might account for dispersion. “Cauchy’s mathematics for dispersion set a programme of research that was pursued in France, Britain, and Germany during the 1830s and (in Germany and France) into the 1850s. During the 1830s, in fact, optical theory became for a time nearly synonymous with Cauchy’s ether dynamics” (Buchwald, p. 461). Cauchy’s dispersion formula works reasonably well for normally dispersive bodies and was only replaced towards the end of the 19th century following the discovery of anomalous dispersion by the Danish physicist Christian Christiansen in 1870 and consequent changes in theory by Wolfgang Sellmeier and Hermann von Helmholtz in Germany. The present memoir is a continuation and expansion of his earlier work, of the same title, published in Paris in 1830; the earlier work did not contain Cauchy’s dispersion formula.

In the early 1820s Fresnel developed a mechanical theory of the ether, the all-pervasive medium through which light was believed to propagate. It was a system of point masses that exert central, repulsive forces upon one another. “In 1823 Fresnel advanced a qualitative explanation of dispersion, based on ether dynamics, that became immensely influential in the 1830s, particularly for Cauchy. Dispersion, according to Fresnel, depended on the spacing and forces between the mutually repelling particles of the ether. The clear implication of Fresnel’s remarks was that theory had to address these two factors (spacing and force) in order to deal quantitatively with dispersion. This was precisely what Cauchy had begun to analyse, in a different context, in 1827, the very year that Fresnel died …

“Cauchy’s early ether dynamics dates to 1830. By that time he was well prepared to see the possibilities in Fresnel’s suggestions, and in particular immediately to correct the great lacuna in Fresnel’s dynamics: namely the assumption that the ether lattice remains essentially rigid even when one of its elements is displaced. Assuming only that the displacement is small in comparison with the distance between the points, Cauchy was able to generate a differential equation in finite differences for the motion of an arbitrary lattice element as a function of the differences between its displacement and that of every other element of the lattice. The equation became so common in optics articles and texts during the ensuing decade that it should be called canonical. To produce from these intricate expressions a theory of dispersion whose constants reflected ether properties was no easy task, as the almost impenetrable mound of computations and approximations that Cauchy eventually published in 1836 would seem to show. In essence, Cauchy first imposed symmetry conditions on the lattice and then calculated the differences in the displacements by means of a Fourier series. After a very great deal of tedious work these elephantine calculations birthed … a series for the refractive index that has since become known eponymously as ‘Cauchy’s series’ … the series did seem to work empirically … there can be no doubt that many of Cauchy’s contemporaries, including William Rowan Hamilton, were deeply impressed by his ability to obtain a dispersion formula. Articles on Cauchy’s theory poured forth, particularly in Britain, and a very great deal of thought was devoted to it” (Buchwald, pp. 460-462).

“The first livraisons [of Cauchy’s memoir] appeared successively during the year 1835, at Prague. The first is a repetition of the ‘Memoire sur la dispersion’ [Paris, 1830]. In the second, (besides other discussions) is contained the actual deduction of the formula for dispersion. In September, 1835, [Cauchy] circulated a lithographed memoire on interpolation, in which he gives a calculation by that method, of Fraunhofer’s indices for one kind of flint glass, but without any theoretical explanation. In the ‘Nouveaux Exercices’ for 1836, he gives in detail his most elaborate and exact method of computation, and applies it with perfect success to all the indices determined by Fraunhofer” (Baden Powell, pp. xlvi-xlvii).

“In 1826 Cauchy began to publish his ‘Exercises de Mathematiques’, which was essentially a mathematical periodical consisting entirely of papers written by himself; it appeared at approximately monthly intervals until 1830. He used similar series as vehicles for his publications at later stages of his career; the ‘Nouveaux Exercices de Mathématiques’ were published in Prague in 1835-36 and the ‘Exercices d'Analyse et de Physique Mathématique’ appeared at intervals from 1840 to 1853” (Smithies, p. 113).

Baden Powell, *A General and Elementary View of the Undulatory Theory, as Applied to the Dispersion of Light*, 1851. Buchwald, ‘Optics in the nineteenth century,’ in *The Oxford Handbook of the History of Physics* (Buchwald & Fox, eds.), 2013. Mehra & Rechenberg, *The Historical Development of Quantum Theory*, Vol. 1, Part 2, 1982. Smithies, *Cauchy and the Creation of Complex Function Theory*, 1997.

4to (259 x 202 mm), pp. [iv, title and foreword of the Exercices], iv [title and foreword of the Mémoire], 236. Contemporary half-calf over marbled boards, marbled end papers, engraved bookplate to front paste down (hinges worn but still strong, head and tale of spine professionally repaired), in all a very nice copy.

Item #2354

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Price:
$1,500.00
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