The philosophy of mathematics · Auguste Comte
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 this motion, which may evidently be determined by an infinity of distinct conditions; although it may sometimes accidentally happen that new generations produce curves which have been already obtained. Thus, among plane curves, if a point moves so as to remain constantly at the same distance from a fixed point, it will generate a _circle_; if it is the sum or the difference of its distances from two fixed points which remains constant, the curve described will be an _ellipse_ or an _hyperbola_; if it is their product, we shall have an entirely different curve; if the point departs equally from a fixed point and from a fixed line, it will describe a _parabola_; if it revolves on a circle at the same time that this circle rolls along a straight line, we shall have a _cycloid_; if it advances along a straight line, while this line, fixed at one of its extremities, turns in any manner whatever, there will result what in general terms are called _spirals_, which of themselves evidently present as many perfectly distinct curves as we can suppose different relations between these two motions of translation and of rotation, &c. Each of these different curves may then furnish new ones, by the different general constructions which geometers have imagined, and which give rise to evolutes, to epicycloids, to caustics, &c. Finally, there exists a still greater variety among curves of double curvature.
_Infinity of Surfaces._ As to _surfaces_, the figures are necessarily more different still, considering them as generated by the motion of lines. Indeed, the figure may then vary, not only in considering, as in curves, the different infinitely numerous laws to which the motion of the generating line may be subjected, but also in supposing that this line itself may change its nature; a circumstance which has nothing analogous in curves, since the points which describe them cannot have any distinct figure. Two classes of very different conditions may then cause the figures of surfaces to vary, while there exists only one for lines. It is useless to cite examples of this doubly infinite multiplicity of surfaces. It would be sufficient to consider the extreme variety of the single group of surfaces which may be generated by a right line, and which comprehends the whole family of cylindrical surfaces, that of conical surfaces, the most general class of developable surfaces, &c. _Infinity of Volumes._ With respect to _volumes_, there is no occasion for any special consideration, since they are distinguished from each other only by the surfaces which bound them.
In order to complete this sketch, it should be added that surfaces themselves furnish a new general means of conceiving new curves, since every curve may be regarded as produced by the intersection of two surfaces. It is in this way, indeed, that the first lines which we may regard as having been truly invented by geometers were obtained, since nature gave directly the straight line and the circle. We know that the ellipse, the parabola, and the hyperbola, the only curves completely studied by the ancients, were in their origin conceived only as resulting from the intersection of a cone with circular base by a plane in different positions. It is evident that, by the combined employment of these different general means for the formation of lines and of surfaces, we could produce a rigorously infinitely series of distinct forms in starting from only a very small number of figures directly furnished by observation.
_Analytical invention of Curves, &c._ Finally, all the various direct means for the invention of figures have scarcely any farther importance, since rational geometry has assumed its final character in the hands of Descartes. Indeed, as we shall see more fully in chapter iii., the invention of figures is now reduced to the invention of equations, so that nothing is more easy than to conceive new lines and new surfaces, by changing at will the functions introduced into the equations. This simple abstract procedure is, in this respect, infinitely more fruitful than all the direct resources of geometry, developed by the most powerful imagination, which should devote itself exclusively to that order of conceptions. It also explains, in the most general and the most striking manner, the necessarily infinite variety of geometrical forms, which thus corresponds to the diversity of analytical functions. Lastly, it shows no less clearly that the different forms of surfaces must be still more numerous than those of lines, since lines are represented analytically by equations with two variables, while surfaces give rise to equations with three variables, which necessarily present a greater diversity. The preceding considerations are sufficient to show clearly the rigorously infinite extent of each of the three general sections of geometry.
Text from Project Gutenberg, public domain in the United States.