would remain unfinished.
Such, then, are the various fundamental parts of rational geometry, arranged according to their natural dependence; the geometry of lines being first considered, beginning with the right line; then the geometry of surfaces, and, finally, that of solids.
INFINITE EXTENT OF ITS FIELD.
Having determined with precision the general and final object of geometrical inquiries, the science must now be considered with respect to the field embraced by each of its three fundamental sections.
Thus considered, geometry is evidently susceptible, by its nature, of an extension which is rigorously infinite; for the measurement of lines, of surfaces, or of volumes presents necessarily as many distinct questions as we can conceive different figures subjected to exact definitions; and their number is evidently infinite.
Geometers limited themselves at first to consider the most simple figures which were directly furnished them by nature, or which were deduced from these primitive elements by the least complicated combinations. But they have perceived, since Descartes, that, in order to constitute the science in the most philosophical manner, it was necessary to make it apply to all imaginable figures. This abstract geometry will then inevitably comprehend as particular cases all the different real figures which the exterior world could present. It is then a fundamental principle in truly rational geometry to consider, as far as possible, all figures which can be rigorously conceived.
The most superficial examination is enough to convince us that these figures present a variety which is quite infinite.
Infinity of Lines. With respect to curved lines, regarding them as generated by the motion of a point governed by a certain law, it is plain that we shall have, in general, as many different curves as we conceive different laws for