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# Le Poivre, Jacques-François

(fl. France, early eighteenth century)

mathematics.

A minor figure in the early history of projective geometry, Le Poivre is known only by his short treatise, Traité des sections du cylindrie et du cône considérées dans le solide et dans le plan, avec des démonstrations simples & nouvelles (Paris, 1704). According to the review of this work in the Journal des sçavans, he lived in Mons and worked on the treatise for three years; nothing more is known about him.

Aimed both at presenting the conic sections in a form readily understandable to the novice and at offering new results to specialists, the Traité is divided into two parts. The first examines the ellipse by means of the parallel projection of a circle from one plane to another and, by inversion of the projection, within the same plane. Although primarily interested here in the tangent properties of the curve, Le Poivre also proves several theorems concerning its conjugate axes in a much simpler way than had been achieved earlier.

Part 2 is more interesting for its greater generality. Here Le Poivre generates the conic sections by means of the central projection of a circle. Taking a circle as base and a point S (at first assumed to be off the plane of the circle) as summit, he draws two parallel lines ab and DE in the plane of the circle (see Fig.1).

He then draws through any point f on the circle an arbitrary line intersecting DE at E and ab at a. Having drawn Sa, he draws EF parallel to it and intersecting Sf at F, he then shows, lies on a conic section, the precise nature of which depends on the location of line ab: if ab lies wholly outside the circle, the section is an ellipse; if ab is tangent to the circle, the section is a parabola. if ab cuts the circle, the section is a hyperbola. The theorems that he goes on to prove are nevertheless independent of the specific position of ab and hence apply to conic sections in general.

If one views ab and DE as the intersections of two parallel planes with the plane of the circle, and Sf as an element of the cone determined by point S and the circle, then Le Poivre’s construction reduces to that of Apollonius. But the construction itself does not rely on that visualization of a solid, nor does it require that S in fact be off the plane of the circle. Hence the theorems also hold for central projections in the same plane.

In reviewing Le Poivre’s Traité for the Acta erduitorum, Christian Wolff spoke warmly of its elegance and originality. The anonymous reviewer of the Journal des sçavans,however, insisted on Le Poivre ’s omission of several important conic properties, in particular the focal properties, and on the extent to which he had failed to go beyond (or even to equal) the methods of Philippe de La Hire. In fact, Le Poivre ’s method of central projection is essentially that employed by La Hire in Part 2 of his Nouvelle méthode en géométrie, pour les sections des superficies coniques et cylindriques (Paris, 1673). Whether independent of La Hire or not, Le Poivre’s work was apparently lost in his shadow, not to be brought to light again until Michel Chasles’s survey of the history of geometry in 1837.

## BIBLIOGRAPHY

Le Poivre ’s Traité is quite rare. The above description of it is taken from the reviews in the Acta eruditorum (Mar. 1707), 132-133 (the identification of Wolff as reviewer is from a contemporary marginal in the Princeton copy); and the Journal des sçavans,32 (1704), 649-658. Chasles, who felt that Le Poivre belonged in a class with Desargues, Pascal, and La Hire, discusses his work in pars. 31-34 of ch. 3 of the Aperçu historique sur l’ origine et le dévelopment des méthodes en géométric, particulièrement de celles qui se rapporent à la géométrie moderne (Brussels, 1837); German trans. by L. A. Sohncke (Halle, 1839; repr. Wiesbaden, 1968). The entry for Le Poivre in the Nouvelle biographic générade, XXX (Paris, 1862), 852, stems entirely from Chasles.

Michael S. Mahoney