Descartes’ general construction of solid problems c. 1625

Abstract

While in Paris in 1625–1626 Descartes communicated to some mathematicians a construction (without proof) of two mean proportionals by means of a parabola and a circle.¹ One of the recipients was Mydorge, who devised a proof of the construction and showed it to Descartes. Later, in 1632, Mersenne sent Descartes another proof, provided by Roberval.² Mersenne published Descartes’ construction (without mentioning his name) and Roberval’s proof in his Universal harmony in 1636.³ Meanwhile Descartes had met Beeckman again in 1628. At that occasion Descartes showed him the construction, the proof by Mydorge, and a general construction of the roots of any third- or fourth-degree equation also by means of a parabola and a circle. Beeckman copied these in his Journal.⁴ About the general construction he noted:

Mr Descartes values this invention so much that he avows never to have found anything more outstanding, indeed that nothing more outstanding has been found by anybody.⁵

Cf. especially [Mersenne 1933–1986] vol. 1 pp. 256–259 and [Descartes 1964–1974] vol. 10 pp. 651–659.

Cf. Descartes to Mersenne, end June 1632, [Descartes 1964–1974] vol. 1, pp. 254–257.

[Mersenne 1636] Livre IV, pp. 407–412; the Latin edition of the book contained a less elaborated version ([Mersenne 1636b], Liber IV, Prop. II, pp. 146–147). The French text of the passage is published in the sources mentioned in Note 1 above.

[Beeckman 1939–1953] pp. 136–139, the passage is also published in [Descartes 1964–1974] vol. 10, pp. 342–346.

[Beeckman 1939–1953] p. 139, cf. [Descartes 1964–1974] vol. 10 p. 346: “Hanc inventionem tanti facit D. des Chartes, ut fateatur se nihil unquam preastantiùs invenisse, imò a nemine unquam praestantiùs quid inventurn.”