A no less important difference between the two systems appears when we come to consider geometry in the concrete point of view. Indeed, we have already remarked that the relation of the abstract to the concrete in geometry can be founded upon rational bases only so far as the investigations are made to bear directly upon all imaginable figures. In studying lines, only one by one, whatever may be the number, always necessarily very small, of those which we shall have considered, the application of such theories to figures really existing in nature will never have any other than an essentially accidental character, since there is nothing to assure us that these figures can really be brought under the abstract types considered by geometers.
Thus, for example, there is certainly something fortuitous in the happy relation established between the speculations of the Greek geometers upon the conic sections and the determination of the true planetary orbits. In continuing geometrical researches upon the same plan, there was no good reason for hoping for similar coincidences; and it would have been possible, in these special studies, that the researches of geometers should have been directed to abstract figures entirely incapable of any application, while they neglected others, susceptible perhaps of an important and immediate application. It is clear, at least, that nothing positively guaranteed the necessary applicability of geometrical speculations. It is quite another thing in the modern geometry. From the single circumstance that in it we proceed by general questions relating to any figures whatever, we have in advance the evident certainty that the figures really existing in the external world could in no case escape the appropriate theory if the geometrical phenomenon which it considers presents itself in them.
From these different considerations, we see that the ancient system of geometry wears essentially the character of the infancy of the science, which did not begin to become completely rational till after the philosophical resolution produced by Descartes. But it is evident, on the other hand, that geometry could not be at first conceived except in this special manner. General geometry would not have been possible, and its necessity could not even have been