Abstract
The brachistochrone problem of Johann Bernoulli is considered as the origin of the calculus of variations. The solutions presented by Johann and Jacob Bernoulli and by Newton and Leibniz were all different and highly original. Leibniz’ solution has received less attention than those of the Bernoullis, but I show here that his abstract idea was also general and powerful enough for a general theory, although the history of mathematics took a different path. In fact, his approach quite naturally emerges from his earlier treatment of the refraction of light by his then new calculus, i.e., his derivation of Fermat’s principle. I then analyze the development of his conceptions about the speed of light from that treatment through his work on the brachistochrone problem to his Nouveaux essais from 1706. From the work of the Bernoullis and Leibniz on variational problems, also an analogy between mechanical and optical problems emerged, and this naturally leads to Leibniz’ considerations on the physical concept of action and extremal principles. In contrast to later formulations of such a principle by Maupertuis and Euler, Leibniz devoted much effort to deriving more abstract principles based on considerations of symmetry and determination, as analyzed in De Risi (Geometry and monadology: Leibniz’s analysis situs and philosophy of space. Birkhäuser, Basel/Boston, 2007). Some of his corresponding ideas look surprisingly modern, for instance in the light of Feynman’s path integral approach to quantum mechanics. Leibniz’ ideas are put into the perspective of modern science in Jost (Leibniz und die moderne Naturwissenschaft. Springer, 2019).
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Notes
- 1.
In Gerhardt’s edition Leibniz (1855/1971) of Leibniz’ mathematical works, Leibniz’ derivation of the solution is given as a “Beilage” (pp. 290–295) to a letter written to Johann Bernoulli (letter XXIX, pp. 284–290), which, as Gerhardt explains on p. 117 of Leibniz (1855/1971), was found among Leibniz’ manuscripts.
- 2.
Leibniz expresses the transition to the limit as follows:”Quod si jam concipiamus viam polygonam facillimam ita continuari, ut constet ex angulis numero infinitis, qui incident in horizontales infinitesime distantes seu vicinissimas, habebimus Lineam facillimi descensus” (Leibniz 1855/1971, p. 292), that is, the brachistochrone, or as Leibniz calls it, the tachystoptote.
- 3.
- 4.
There are also several manuscripts of Leibniz (1906/1995) where he develops his ideas about refraction.
- 5.
Descartes had invoked the analogy of a ball losing speed when going through a medium with more friction. Leibniz criticizes this with the argument that a light particle regains its status when returning to the original medium: “cum tamen radius lucis ex medio magis resistente in medium minus resistens, primo simile, rursus ingressus, priorem statum recuperet, & posito duorum mediorum similium, primi & ultimi, superficies (illius emittentem, hujus recipientem) esse planas parallelas, Directionem recipiat per refractionem posteriorem illi parallelam, quam habuit ante priorem.” (Leibniz (1682/1985), cited after the text in the supplement of Leibniz (2011), p. 40). In an undated manuscript, Leibniz (1880/2008), pp. 304–309, about Descartes, Leibniz repeats the arguement against Descartes that light does not lose force by passing through a rare medium, because such a passage does not affect the angle of refraction:“Car s’il est vray que l’air à cause de sa flexibilité fait perdre une partie de la force comme le tapis celle du globule qui court là dessus, cette force perdue ne sera point rendue lorsque le rayon sort de l’air et retourne dans l’eau. Cependant nous voyons que le rayon y reprend la premiere inclinaison.”, Leibniz (1880/2008), p. 308.
- 6.
In a draft written in or before 1695, Leibniz (1880/2008), p. 472, Leibniz prefers Huygens’ theory of light: “j’avoue que ce que M.Hugens nous a donné sur la production de la lumiere et de la refraction paroisse plus vraisemblable que tout ce qu’on en a donné jusqu’icy.” In the letters he exchanged with Huygens himself, however, he is less committal and discusses the relation between light, gravity and magnetism, see for instance Leibniz (1849/1971), p. 182ff.
- 7.
It is, however, outside the scope of this contribution to investigate Newton’s views on the speed of light.
- 8.
Although it was mentioned by Huygens in his correspondence with Leibniz, see for instance Leibniz (1849/1971), p. 176, in a letter from 1694.
- 9.
Johann Bernoulli wrote brachystochrone instead of brachistochrone, the spelling now usually adopted. In fact, the Greek root is
, short, although its superlative is usually written as 
, shortest. - 10.
See also the German translations by P.Stäckel (1894/1976).
- 11.
Which was incorrect, see the analysis in Peiffer (1989).
- 12.
- 13.
I am working on a commented German edition of this fundamental text.
- 14.
Johann Bernoulli had sent the letter ot Leibniz in which he stated the brachystochrone problem on June 9, 1696, and the latter, in spite of ill health, replied to Bernoulli already on June 16. In his letter, Leibniz stated the differential equation, and in his manuscript (pp. 291–295 in Leibniz (1855/1971)), he derived the differential equation (7.17) for the solution, however, apparently without noticing that the solution is the cycloid (whose differential equation Leibniz himself had earlier derived), but only stating that he had already solved the problem in the past (“jam olim”). That the solution is the cycloid is pointed out by Johann Bernoulli, see Leibniz (1855/1971), p. 299, l.10-15, and Leibniz acknowledges this, ibid., p. 310, “Tu longius progressus cycloidem ipsam esse pulchre reperisti.”
- 15.
In Leibniz (1890/2008), it is undated, but Gerhardt suggests that it might have been written between 1690 and 1695. Since, however, Leibniz in this text speaks about the curve of swiftest descent (“la ligne de la plus courte descente entre deux points donnés”, p. 272), that is, the brachistochrone, it might have been written somewhat later, after 1696/7 when Leibniz and the Bernoullis had analyzed that problem.
- 16.
A mathematically more correct way to describe this is that such a circular that is longer than half an equator is a critical point of the variational integral, but not a minimum. In particular, it satisfies the corresponding Euler-Lagrange equation (for a geodesic on the sphere in modern terminology).
- 17.
There are considerable mathematical subtleties involved with the correct definition of such a functional integral. We do not enter that issue, but refer to Jost (2009) and the references provided there.
, short, although its superlative is usually written as 
, shortest.Bibliography
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Jost, J. (2019). Leibniz and the Calculus of Variations. In: De Risi, V. (eds) Leibniz and the Structure of Sciences. Boston Studies in the Philosophy and History of Science, vol 337. Springer, Cham. https://doi.org/10.1007/978-3-030-25572-5_7
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