The philosophy of mathematics · Auguste Comte
DESCRIPTIVE GEOMETRY.
All questions of geometry of three dimensions necessarily give rise, when we consider their graphical solution, to a common difficulty which is peculiar to them; that of substituting for the different constructions in relief, which are necessary to resolve them directly, and which it is almost always impossible to execute, simple equivalent plane constructions, by means of which we finally obtain the same results. Without this indispensable transformation, every solution of this kind would be evidently incomplete and really inapplicable in practice, although theoretically the constructions in space are usually preferable as being more direct. It was in order to furnish general means for always effecting such a transformation that _Descriptive Geometry_ was created, and formed into a distinct and homogeneous system, by the illustrious MONGE. He invented, in the first place, a uniform method of representing bodies by figures traced on a single plane, by the aid of _projections_ on two different planes, usually perpendicular to each other, and one of which is supposed to turn about their common intersection so as to coincide with the other produced; in this system, or in any other equivalent to it, it is sufficient to regard points and lines as being determined by their projections, and surfaces by the projections of their generating lines.
This being established, Monge--analyzing with profound sagacity the various partial labours of this kind which had before been executed by a number of incongruous procedures, and considering also, in a general and direct manner, in what any questions of that nature must consist--found that they could always be reduced to a very small number of invariable abstract problems, capable of being resolved separately, once for all, by uniform operations, relating essentially some to the contacts and others to the intersections of surfaces. Simple and entirely general methods for the graphical solution of these two orders of problems having been formed, all the geometrical questions which may arise in any of the various arts of construction--stone-cutting, carpentry, perspective, dialling, fortification, &c.--can henceforth be treated as simple particular cases of a single theory, the invariable application of which will always necessarily lead to an exact solution, which may be facilitated in practice by profiting by the peculiar circumstances of each case. * * * * * This important creation deserves in a remarkable degree to fix the attention of those philosophers who consider all that the human species has yet effected as a first step, and thus far the only really complete one, towards that general renovation of human labours, which must imprint upon all our arts a character of precision and of rationality, so necessary to their future progress. Such a revolution must, in fact, inevitably commence with that class of industrial labours, which is essentially connected with that science which is the most simple, the most perfect, and the most ancient. It cannot fail to extend hereafter, though with less facility, to all other practical operations. Indeed Monge himself, who conceived the true philosophy of the arts better than any one else, endeavoured to sketch out a corresponding system for the mechanical arts.
Essential as the conception of descriptive geometry really is, it is very important not to deceive ourselves with respect to its true destination, as did those who, in the excitement of its first discovery, saw in it a means of enlarging the general and abstract domain of rational geometry. The result has in no way answered to these mistaken hopes. And, indeed, is it not evident that descriptive geometry has no special value except as a science of application, and as forming the true special theory of the geometrical arts? Considered in its abstract relations, it could not introduce any truly distinct order of geometrical speculations. We must not forget that, in order that a geometrical question should fall within the peculiar domain of descriptive geometry, it must necessarily have been previously resolved by speculative geometry, the solutions of which then, as we have seen, always need to be prepared for practice in such a way as to supply the place of constructions in relief by plane constructions; a substitution which really constitutes the only characteristic function of descriptive geometry. It is proper, however, to remark here, that, with regard to intellectual education, the study of descriptive geometry possesses an important philosophical peculiarity, quite independent of its high industrial utility. This is the advantage which it so pre-eminently offers--in habituating the mind to consider very complicated geometrical combinations in space, and to follow with precision their continual correspondence with the figures which are actually traced--of thus exercising to the utmost, in the most certain and precise manner, that important faculty of the human mind which is properly called "imagination," and which consists, in its elementary and positive acceptation, in representing to ourselves, clearly and easily, a vast and variable collection of ideal objects, as if they were really before us.
Finally, to complete the indication of the general nature of descriptive geometry by determining its logical character, we have to observe that, while it belongs to the geometry of the ancients by the character of its solutions, on the other hand it approaches the geometry of the moderns by the nature of the questions which compose it. These questions are in fact eminently remarkable for that generality which, as we saw in the preceding chapter, constitutes the true fundamental character of modern geometry; for the methods used are always conceived as applicable to any figures whatever, the peculiarity of each having only a purely secondary influence. The solutions of descriptive geometry are then graphical, like most of those of the ancients, and at the same time general, like those of the moderns. * * * * * After this important digression, we will pursue the philosophical examination of _special_ geometry, always considered as reduced to its least possible development, as an indispensable introduction to _general_ geometry. We have now sufficiently considered the _graphical_ solution of the fundamental problem relating to the right line--that is, the determination of the different elements of any right-lined figure by means of one another--and have now to examine in a special manner the _algebraic_ solution.
Text from Project Gutenberg, public domain in the United States.