Page:Popular Science Monthly Volume 66.djvu/419

Rh propositions in juxtaposition; and he had the idea of substituting for Poncelet's demonstrations, which required an intermediary curve or surface of the second degree, the famous 'principle of duality,' of which the signification, a little vague at first, was sufficiently cleared up by the discussions which took place on this subject between Gergonne, Poncelet and Pluecker.

Bobillier, Chasles, Steiner, Lamé, Sturm, and many others whose names escape me, were, at the same time as Pluecker and Poncelet, assiduous collaborators of the Annales de Mathmétiques. Gergonne having beconle rector of the Acadamy of Montpellier, was forced to suspend in 1831 the publication of his journal. But the success it had obtained, the taste for research it had contributed to develop, had commenced to bear their fruit. Quetelet had established in Belgium the Correspondance mathématique et physique. Crelle, from 1826, brought out at Berlin the first sheets of his celebrated journal, where he published the memoirs of Abel, of Jacobi, of Steiner.

A great number of separate works began also to appear, wherein the principles of modern geometry were powerfully expounded and developed.

First came in 1827 the 'barycentrische Calcul' of Moebius, a work truly original, remarkable for the profundity of its conceptions, the elegance and the rigor of its exposition; then in 1828 the 'Analytisch-geometrische Entwickelungen' of Pluecker, of which the second part appeared in 1831 and which was soon followed by the 'System der analytischen Geometrie' of the same author published at Berlin in 1835.

In 1832 Steiner brought out at Berlin his great work: 'Systematische Entwickelung der Abhaengigkeit der geometrischen Gestalten von einander,' and, the following year, 'Die geometrischen Konstruktionen ausgefuehrt mittels der geraden Linie und eines festen Kreises,' where was confirmed by the most elegant examples a proposition of Poncelet's relative to the employment of a single circle for the geometric constructions.

Finally, in 1830, Chasles sent to the Academy of Brussels, which happily inspired had offered a prize for a study of the principles of modern geometry, his celebrated 'Aperçu historique sur l'origine et le développement des methodes en geometrie,' followed by 'Mémoire sur deux principes généraux de la science: la dualité et 1'homographie' which was published only in 1837.

Time would fail us to give a worthy appreciation of these beautiful works and to apportion the share of each. Moreover, to what would such a study conduct us, but to a new verification of the general laws of the development of science. When the times are ripe, when the fundamental principles have been recognized and enunciated, nothing