Having thus recognized, as a general principle, the close and necessary connexion of the study of the properties of lines and surfaces with those researches which constitute the final object of geometry, it is evident that geometers, in the progress of their labours, must by no means constrain themselves to keep such a connexion always in view. Knowing, once for all, how important it is to vary as much as possible the manner of conceiving each figure, they should pursue that study, without considering of what immediate use such or such a special property may be for rectifications, quadratures, and cubatures. They would uselessly fetter their inquiries by attaching a puerile importance to the continued establishment of that co-ordination.
This general exposition of the general object of geometry is so much the more indispensable, since, by the very nature of the subject, this study of the different properties of each line and of each surface necessarily composes by far the greater part of the whole body of geometrical researches. Indeed, the questions directly relating to rectifications, to quadratures, and to cubatures, are evidently, by themselves, very few in number for each figure considered. On the other hand, the study of the properties of the same figure presents an unlimited field to the activity of the human mind, in which it may always hope to make new discoveries. Thus, although geometers have occupied themselves for twenty centuries, with more or less activity undoubtedly, but without any real interruption, in the study of the conic sections, they are far from regarding that so simple subject as being exhausted; and it is certain, indeed, that in continuing to devote themselves to it, they would not fail to find still unknown properties of those different curves. If labours of this kind have slackened considerably for a century past, it is not because they are completed, but only, as will be presently explained, because the philosophical revolution in geometry, brought about by Descartes, has singularly diminished the importance of such researches.
It results from the preceding considerations that not only is the field of geometry necessarily infinite because of the variety of figures to be considered, but also in virtue of the diversity of the points of view under the same figure may be regarded. This last conception is, indeed, that which gives the broadest and most complete idea of the whole body of geometrical researches. We see that studies of this kind consist essentially, for each line or for each surface, in connecting all the geometrical phenomena which it can present, with a single fundamental phenomenon, regarded as the primitive definition.
THE TWO GENERAL METHODS OF GEOMETRY.
Having now explained in a general and yet precise manner the final object of geometry, and shown how the science, thus defined, comprehends a very extensive class of researches which did not at first appear necessarily to belong to it, there remains to be considered the method to be followed for the formation of this science. This discussion is indispensable to complete this first sketch of the philosophical character of geometry. I shall here confine myself to indicating the most general consideration in this matter, developing and summing up this important fundamental idea in the following chapters.
Geometrical questions may be treated according to two methods so different, that there result from them two sorts of geometry, so to say, the philosophical character of which does not seem to me to have yet been properly apprehended. The expressions of Synthetical Geometry and Analytical Geometry, habitually employed to designate them, give a very false idea of them. I would much prefer the purely historical denominations of Geometry of the Ancients and Geometry of the Moderns, which have at least the advantage of not causing their true character to be misunderstood. But I propose to employ henceforth the regular expressions of Special Geometry and General Geometry, which seem to me suited to characterize with precision the veritable nature of the two methods.