Page:Popular Science Monthly Volume 66.djvu/421

Rh imaginaries, which supplies the place of the principle of continuity and furnishes demonstrations as general as-those of analytic geometry; (3) the simultaneous demonstration of propositions which are correlative, that is to say, which correspond in virtue of the principle of duality.

Chasles studies indeed in his work homography and correlation; but he avoids systematically in his exposition the employment of transformations of figures, which, he thinks, can not take the place of direct demonstrations since they mask the origin and the true nature of the properties obtained by their means.

There is truth in this judgment, but the advance itself of the science permits us to declare it too severe. If it happens often that, employed without discernment, transformations multiply uselessly the number of theorems, it must be recognized that they often aid us to better understand the nature of the propositions even to which they have been applied. Is it not the employment of Poncelet's projection which has led to the so fruitful distinction between projective properties and metric properties, which has taught us also the high importance of that cross ratio whose essential property is found already in Pappus, and of which the fundamental role has begun to appear after fifteen centuries only in the researches of modern geometry?

The introduction of the principle of signs was not as new as Chasles supposed at the time he wrote his 'Traité de Géometrie superiéure.'

Moebius, in his barycentrische Calcul, had already given issue to a desideratum of Carnot, and employed the signs in a way the largest and most precise, defining for the first time the sign of a segment and even that of an area.

Later he succeeded in extending the use of signs to lengths not laid off on the same straight and to angles not formed about the same point.

Besides Grassmann, whose mind has so much analogy to that of Moebius, had necessarily employed the principle of signs in the definitions which serve as basis for his methods, so original, of studying the properties of space.

The second characteristic which Chasles assigns to his system of geometry is the employment of imaginaries. Here, his method was really new and he illustrates it by examples of high interest. One will always admire the beautiful theories he has left us on homofocal surfaces of the second degree, where all the known properties and others new, as varied as elegant, flow from the general principle that they are inscribed in the same developable circumscribed to the circle at infinity.

But Chasles introduced imaginaries only by their symmetric