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Published online by Cambridge University Press: 20 January 2009
Chasles in his Aperçu, Historique sur l'origine et le développement des Méthodes en Géométrie (seconde édition, 1875, pp. 214–215) makes the following statement:
“Essays of the same kind as the geometry of the rule and that of the compasses, and which hold, so to speak, the mean between the two, had long previously engaged the attention of famous mathematicians. Cardan first of all in his book De Subtilitate had resolved several of Euclid's problems by the straight line and a single aperture of the compasses, as if one had in practice only a rule and invariable compasses. Tartalea was not long in following his rival on this field, and extended this mode of treatment to some new problems. (General trattato di numeri et misure; 5ta parte, libra terzo; in-fol. Venise, 1560). Finally, a learned Piedmontese geometer, J.–B. de Benedictis, made it the object of a treatise entitled: Resolutio omnium Euclidis problematum, aliorumque ad hoc necessario inventorum, una tantummodo circini data apertura; in-4°. Venise, 1553.”
* This is the 25th proposition of Euclid's sixth book. Tartaglia in his translation of Euclid's Elements into Italian (Venice, 1543) now and then changes the order of Euclid's propositions.
* Apollonius proves that this construction will give a tangent to the hyperbola, the ellipse, and the circle, but Prop. 35 shows that he knew how to modify it to obtain a tangent to the parabola.
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