felt, if a long series of special labours on the most simple figures had not previously furnished bases for the conception of Descartes, and rendered apparent the impossibility of persisting indefinitely in the primitive geometrical philosophy.
The Ancient the Base of the Modern. From this last consideration we must infer that, although the geometry which I have called general must be now regarded as the only true dogmatical geometry, and that to which we shall chiefly confine ourselves, the other having no longer much more than an historical interest, nevertheless it is not possible to entirely dispense with special geometry in a rational exposition of the science. We undoubtedly need not borrow directly from ancient geometry all the results which it has furnished; but, from the very nature of the subject, it is necessarily impossible entirely to dispense with the ancient method, which will always serve as the preliminary basis of the science, dogmatically as well as historically. The reason of this is easy to understand. In fact, general geometry being essentially founded, as we shall soon establish, upon the employment of the calculus in the transformation of geometrical into analytical considerations, such a manner of proceeding could not take possession of the subject immediately at its origin. We know that the application of mathematical analysis, from its nature, can never commence any science whatever, since evidently it cannot be employed until the science has already been sufficiently cultivated to establish, with respect to the phenomena considered, some equations which can serve as starting points for the analytical operations. These fundamental equations being once discovered, analysis will enable us to deduce from them a multitude of consequences which it would have been previously impossible even to suspect; it will perfect the science to an immense degree, both with respect to the generality of its conceptions and to the complete co-ordination established between them. But mere mathematical analysis could never be sufficient to form the bases of any natural science, not even to demonstrate them anew when they have once been established. Nothing can dispense with the direct study of the subject, pursued up to the point of the discovery of precise relations.