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The work of Tschirnhaus, La Hire and Leibniz on catacaustics and the birth of the envelopes of lines in the 17th century. (English) Zbl 1076.01016

The theory of caustic curves was developed towards the end of the 17th century by various mathematicians. It was partly based on geometrical considerations and partly created by application of infinitesimal methods. A central problem was the determination of the envelope of secondary light rays, reflected by a polished surface (catacaustic) or refracted by a given surface (diacaustic). The rectification and quadrature of such caustic curves presented additional challenges to mathematicians.
The aim of the present paper is “to examine the way in which Tschirnhaus introduced the concept of catacaustic, his initially erroneous construction of the curve, the subsequent emendations he himself made, and those made by Johann Bernoulli and Philippe de La Hire” and also “Leibniz’s demonstration of the rectificability of the catacaustic”. The authors first derive the equation of the catacaustic, then describe how Ehrenfried Walter von Tschirnhaus (1651–1708) had first studied catacaustics in 1681/82 (he mentioned it in letters to Leibniz and published an article in Acta Eruditorum). Soon Johann Bernoulli corrected an error of Tschirnhaus and included his relevant research in the lectures written for G. de l’Hospital. Later, in 1692/93, his brother Jakob Bernoulli also sent three papers about the topic to the Acta Eruditorum. Meanwhile Tschirnhaus had published two further articles in the same journal in 1690, containing a corrected diagram, further contributions to the theory and additional examples. In a review of Tschirnhaus’ first paper, written some time after 1687 but published only posthumously in 1730, the French geometer de La Hire added further results in six theorems. These are also presented in the present article, as are Leibniz’s contributions, especially his most important paper, containing the treatment of envelopes with the help of differentials.
In their conclusion the authors announce a further study that will be devoted to the investigations of Jakob and Johann Bernoulli on caustics.

MSC:

01A45 History of mathematics in the 17th century
26-03 History of real functions
78-03 History of optics and electromagnetic theory
78A05 Geometric optics
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