The philosophy of mathematics · Auguste Comte
PROPERTIES OF LINES AND SURFACES.
When we examine with attention the geometrical investigations which do not seem to relate to the _measurement_ of extent, we find that they consist essentially in the study of the different _properties_ of each line or of each surface; that is, in the knowledge of the different modes of generation, or at least of definition, peculiar to each figure considered. Now we can easily establish in the most general manner the necessary relation of such a study to the question of _measurement_, for which the most complete knowledge of the properties of each form is an indispensable preliminary. This is concurrently proven by two considerations, equally fundamental, although quite distinct in their nature. NECESSITY OF THEIR STUDY: 1. _To find the most suitable Property._ The _first_, purely scientific, consists in remarking that, if we did not know any other characteristic property of each line or surface than that one according to which geometers had first conceived it, in most cases it would be impossible to succeed in the solution of questions relating to its _measurement_. In fact, it is easy to understand that the different definitions which each figure admits of are not all equally suitable for such an object, and that they even present the most complete oppositions in that respect. Besides, since the primitive definition of each figure was evidently not chosen with this condition in view, it is clear that we must not expect, in general, to find it the most suitable; whence results the necessity of discovering others, that is, of studying as far as is possible the _properties_ of the proposed figure. Let us suppose, for example, that the circle is defined to be "the curve which, with the same contour, contains the greatest area." This is certainly a very characteristic property, but we would evidently find insurmountable difficulties in trying to deduce from such a starting point the solution of the fundamental questions relating to the rectification or to the quadrature of this curve.
It is clear, in advance, that the property of having all its points equally distant from a fixed point must evidently be much better adapted to inquiries of this nature, even though it be not precisely the most suitable. In like manner, would Archimedes ever have been able to discover the quadrature of the parabola if he had known no other property of that curve than that it was the section of a cone with a circular base, by a plane parallel to its generatrix? The purely speculative labours of preceding geometers, in transforming this first definition, were evidently indispensable preliminaries to the direct solution of such a question. The same is true, in a still greater degree, with respect to surfaces. To form a just idea of this, we need only compare, as to the question of cubature or quadrature, the common definition of the sphere with that one, no less characteristic certainly, which would consist in regarding a spherical body, as that one which, with the same area, contains the greatest volume. No more examples are needed to show the necessity of knowing, so far as is possible, all the properties of each line or of each surface, in order to facilitate the investigation of rectifications, of quadratures, and of cubatures, which constitutes the final object of geometry. We may even say that the principal difficulty of questions of this kind consists in employing in each case the property which is best adapted to the nature of the proposed problem. Thus, while we continue to indicate, for more precision, the measurement of extension as the general destination of geometry, this first consideration, which goes to the very bottom of the subject, shows clearly the necessity of including in it the study, as thorough as possible, of the different generations or definitions belonging to the same form.
2. _To pass from the Concrete to the Abstract._ A second consideration, of at least equal importance, consists in such a study being indispensable for organizing in a rational manner the relation of the abstract to the concrete in geometry. The science of geometry having to consider all imaginable figures which admit of an exact definition, it necessarily results from this, as we have remarked, that questions relating to any figures presented by nature are always implicitly comprised in this abstract geometry, supposed to have attained its perfection. But when it is necessary to actually pass to concrete geometry, we constantly meet with a fundamental difficulty, that of knowing to which of the different abstract types we are to refer, with sufficient approximation, the real lines or surfaces which we have to study. Now it is for the purpose of establishing such a relation that it is particularly indispensable to know the greatest possible number of properties of each figure considered in geometry.
In fact, if we always confined ourselves to the single primitive definition of a line or of a surface, supposing even that we could then _measure_ it (which, according to the first order of considerations, would generally be impossible), this knowledge would remain almost necessarily barren in the application, since we should not ordinarily know how to recognize that figure in nature when it presented itself there; to ensure that, it would be necessary that the single characteristic, according to which geometers had conceived it, should be precisely that one whose verification external circumstances would admit: a coincidence which would be purely fortuitous, and on which we could not count, although it might sometimes take place. It is, then, only by multiplying as much as possible the characteristic properties of each abstract figure, that we can be assured, in advance, of recognizing it in the concrete state, and of thus turning to account all our rational labours, by verifying in each case the definition which is susceptible of being directly proven. This definition is almost always the only one in given circumstances, and varies, on the other hand, for the same figure, with different circumstances; a double reason for its previous determination.
_Illustration: Orbits of the Planets._ The geometry of the heavens furnishes us with a very memorable example in this matter, well suited to show the general necessity of such a study. We know that the ellipse was discovered by Kepler to be the curve which the planets describe about the sun, and the satellites about their planets. Now would this fundamental discovery, which re-created astronomy, ever have been possible, if geometers had been always confined to conceiving the ellipse only as the oblique section of a circular cone by a plane? No such definition, it is evident, would admit of such a verification. The most general property of the ellipse, that the sum of the distances from any of its points to two fixed points is a constant quantity, is undoubtedly much more susceptible, by its nature, of causing the curve to be recognized in this case, but still is not directly suitable. The only characteristic which can here be immediately verified is that which is derived from the relation which exists in the ellipse between the length of the focal distances and their direction; the only relation which admits of an astronomical interpretation, as expressing the law which connects the distance from the planet to the sun, with the time elapsed since the beginning of its revolution. It was, then, necessary that the purely speculative labours of the Greek geometers on the properties of the conic sections should have previously presented their generation under a multitude of different points of view, before Kepler could thus pass from the abstract to the concrete, in choosing from among all these different characteristics that one which could be most easily proven for the planetary orbits.
_Illustration: Figure of the Earth._ Another example of the same order, but relating to surfaces, occurs in considering the important question of the figure of the earth. If we had never known any other property of the sphere than its primitive character of having all its points equally distant from an interior point, how would we ever have been able to discover that the surface of the earth was spherical? For this, it was necessary previously to deduce from this definition of the sphere some properties capable of being verified by observations made upon the surface alone, such as the constant ratio which exists between the length of the path traversed in the direction of any meridian of a sphere going towards a pole, and the angular height of this pole above the horizon at each point. Another example, but involving a much longer series of preliminary speculations, is the subsequent proof that the earth is not rigorously spherical, but that its form is that of an ellipsoid of revolution. After such examples, it would be needless to give any others, which any one besides may easily multiply. All of them prove that, without a very extended knowledge of the different properties of each figure, the relation of the abstract to the concrete, in geometry, would be purely accidental, and that the science would consequently want one of its most essential foundations.
Such, then, are two general considerations which fully demonstrate the necessity of introducing into geometry a great number of investigations which have not the _measurement_ of extension for their direct object; while we continue, however, to conceive such a measurement as being the final destination of all geometrical science. In this way we can retain the philosophical advantages of the clearness and precision of this definition, and still include in it, in a very logical though indirect manner, all known geometrical researches, in considering those which do not seem to relate to the measurement of extension, as intended either to prepare for the solution of the final questions, or to render possible the application of the solutions obtained. 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.
Text from Project Gutenberg, public domain in the United States.