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MONGE, Gaspard. Application de l'Analyse à la Géométrie, à l'usage de l'École Impériale Polytechnique; par M. Monge, membre de L'Institut. 

Paris, Vve Bernard, 1809.

Fourth edition, enlarged with an article on the "construction de l'équation des cordes vibrantes " of this important work on analitic geometry.
The first edition appeared in 1795 under the title 'Feuilles d'analyse appliquée à la géométrie'. In this work, Monge "assembled, along with general considerations regarding the theory of surfaces and the geometric interpretation of partial differential equations, monographs on about twenty families of surfaces defined by their mode of generation." (DSB [IX p. 476]).
In 1802, working with Hachette, he prepared a brief exposition, of analytic geometry that was designed to replace, the few remarks on the subject contained in the Feuilles. Entitled 'Application de l’algèbre à l’analyse', it was published separately in 1805; in 1807 edition and also for the fourth 1809 edition, it became, the first part of the final version of 'Feuilles d’analyse', now entitled 'Application de l'Analyse à la Géométrie.'
It's an, important work, in which, "The authors show that every plane section of a second degree surface is a second degree curve, and that parallel planes cut out similar and similarly placed curves. These results parallel Archimedes' geometric theorems. The authors also show that the hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces, that is, each can be generated in two different ways by the motion of a line or each surfiace is formed by two systems of lines. The result on the one-sheeted hyperboloid was known by 1669 to Christopher Wren, who said that this figure could be Senerated by revolving a line about another not in the same plane. With the work of Euler, Lagrange, and Monge, analytic geometry became an independent and full-fledged branch of mathematics." (Kline in. Mathematical ... p. 547).

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