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.


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
Full Text: DOI


[1] Adam, C. and Tannery, P. (edits.) 1897-1913: Oeuvres de Descartes publiées par Charles Adam and Paul Tannery sous les auspices de Ministère de l?Instruction publique. Paris.
[2] Bernoulli, Jakob. (1692a): Additamentum ad solutionem curvæ causticæ, una cum meditatione de natura evolutarum & variis osculationum generibus. Acta eruditorum, 110-116.
[3] Bernoulli, Jakob. (1692b): Lineæ cycloidales, evolutæ, ant-evolutæ, causticæ, anti-causticæ, peri-caustica spira mirabilis & c. Acta eruditorum, 207-212.
[4] Bernoulli, Jakob. (1693): Curvæ diacausticæ, earum relatio ad evolutas, aliaque nova his affinia & c. Acta eruditorum, 244-255.
[5] Bernoulli, Johann. (1692): Solutio curvæ causticæ per vulgarem geometriam cartesianam; aliaque. Acta eruditorum, 30-35.
[6] Bernoulli, Johann. (1742): Lectiones Mathematicæ de Methodo Integralium. Lectio XXVII. Lectio XXVIII. Opera Omnia, II?III. Paris, 109-112, 471-472.
[7] Gerhardt, C. I. (ed.) (1971): Briefwechsel zwischen Leibniz und dem Freiherrn von Tschirnhaus. In Leibniz, G.W. Mathematische Schriften, band IV George Olms Verlag.
[8] Huygens, C. (1962): Treatise on light. In which are explained the causes of that which occurs in Reflexion, & in Refraction. And particularly in the strange refraction of Iceland Crystal, Rendered into English by Silvanus P. Thompson. New York: Dover Publications. Reprint of London: Macmillan and Co., 1912. · JFM 43.0078.01
[9] La Hire, P. de (1730): Examen de la Courbe formée par les rayons reflechis dans un quart de cercle. Mémoires de l?Académie Royal des Sciences. vol. IX, 448-471.
[10] Leibniz, G. W. (1682): Unicum opticæ, catoptricæ et dioptricæ principium. Acta eruditorum, 185.
[11] Leibniz, G. W. 1684: Nova Methodus pro maximis itemque tangentibus, quæ nec fracta, nec irrationales quantitates moratur et singulare pro illis calculi genus. Acta eruditorum, 467-473.
[12] Leibniz, G. W. (1689): De lineis opticis et alia. Acta eruditorum, 36-38.
[13] Leibniz, G. W. (1694): Nova Calculi differentialis applicatio et usus ad multiplicem linearum constructionem ex data tangentium conditione. Acta eruditorum, 311-314.
[14] L?Hospital, G. F. A. DE. (1696): Analyse des infiniment petits. Paris, 109-112.
[15] Maclaurin, C. (1742): A treatise of fluxions in two books. Edinburgh.
[16] Matthes, C. J. (1837): Dissertatio mathematica de invenienda æ quatione causticarum (A mathematical dissertation on the discovery of the equation of caustics). Lugdunum Batavorum (Leiden).
[17] Montucla, J. E. (1802): Histoire des Mathématiques. 2, Quatriène Partie, Livre Sixième. Paris, 387-390.
[18] Tschirnhaus, E. W. (1682): Inventa nova exhibita Parisiis Societati Regiae Scientiarum. Acta eruditorum, 364-365.
[19] Tschirnhaus, E. W. (1690a): Methodus Curvas determinandi, quæ formantur a radiis reflexis, quorum incidentes ut paralleli considerantur. Acta eruditorum, 68-73.
[20] Tschirnhaus, E. W. (1690b): Curva Geometrica, quæ seipsam sui evolutione describit, aliasque insignes proprietates obtinet, inventa a D. T. Acta Eruditorum, 169-172.
[21] Tschirnhaus, E. W. (1691): Singularia effecta vitri caustici bipedalis quod omnia magno sumtu hactenus constructa specula ustoria virtute superat. Acta eruditorum, 517-520.
[22] Alonso, M. and Finn, E. (1992): Physics. A. Wesley, 860.
[23] Kracht, M. and Kreyszig, E. E. W. (1990): Tschirnhaus: His Role in Early Calculus and His Work and Impact on Algebra, Historia Mathematica, 16-35. · Zbl 0696.01005
[24] Shapiro, A. E. (1984): The optical papers of Isaac Newton. Cambridge University Press.
[25] Shapiro, A. E. (1973): Kinematic optics: a study of wave theory of light in the seventeenth century, Archive for history of exact sciences, 134-266. · Zbl 0274.01018
[26] Shapiro, A. E. (1990): The Optical Lectures and the foundations of the theory of optical imagery, in Before Newton: The Life and Times of Isaac Barrow, ed. Mordechai Feingold (Cambridge University Press, 105-178.
[27] Gomes Teixeira, F. (1971): Traité des courbes spéciales remarquables planes et gauches, Tome III. Chelsea Publishing Company, 353-355.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.