The philosophy of mathematics Part 13

CHAPTER II.

ANCIENT OR SYNTHETIC GEOMETRY.

The geometrical method of the ancients necessarily const.i.tuting a preliminary department in the dogmatical system of geometry, designed to furnish _general_ geometry with indispensable foundations, it is now proper to begin with determining wherein strictly consists this preliminary function of _special_ geometry, thus reduced to the narrowest possible limits.

ITS PROPER EXTENT.

_Lines; Polygons; Polyhedrons._ In considering it under this point of view, it is easy to recognize that we might restrict it to the study of the right line alone for what concerns the geometry of _lines_; to the _quadrature_ of rectilinear plane areas; and, lastly, to the _cubature_ of bodies terminated by plane faces. The elementary propositions relating to these three fundamental questions form, in fact, the necessary starting point of all geometrical inquiries; they alone cannot be obtained except by a direct study of the subject; while, on the contrary, the complete theory of all other figures, even that of the circle, and of the surfaces and volumes which are connected with it, may at the present day be completely comprehended in the domain of _general_ or _a.n.a.lytical_ geometry; these primitive elements at once furnis.h.i.+ng _equations_ which are sufficient to allow of the application of the calculus to geometrical questions, which would not have been possible without this previous condition.

It results from this consideration that, in common practice, we give to _elementary_ geometry more extent than would be rigorously necessary to it; since, besides the right line, polygons, and polyhedrons, we also include in it the circle and the "round" bodies; the study of which might, however, be as purely a.n.a.lytical as that, for example, of the conic sections. An unreflecting veneration for antiquity contributes to maintain this defect in method; but the best reason which can be given for it is the serious inconvenience for ordinary instruction which there would be in postponing, to so distant an epoch of mathematical education, the solution of several essential questions, which are susceptible of a direct and continual application to a great number of important uses. In fact, to proceed in the most rational manner, we should employ the integral calculus in obtaining the interesting results relating to the length or the area of the circle, or to the quadrature of the sphere, &c., which have been determined by the ancients from extremely simple considerations. This inconvenience would be of little importance with regard to the persons destined to study the whole of mathematical science, and the advantage of proceeding in a perfectly logical order would have a much greater comparative value. But the contrary case being the more frequent, theories so essential have necessarily been retained in elementary geometry. Perhaps the conic sections, the cycloid, &c., might be advantageously added in such cases.

_Not to be farther restricted._ While this preliminary portion of geometry, which cannot be founded on the application of the calculus, is reduced by its nature to a very limited series of fundamental researches, relating to the right line, polygonal areas, and polyhedrons, it is certain, on the other hand, that we cannot restrict it any more; although, by a veritable abuse of the spirit of a.n.a.lysis, it has been recently attempted to present the establishment of the princ.i.p.al theorems of elementary geometry under an algebraical point of view. Thus some have pretended to demonstrate, by simple abstract considerations of mathematical a.n.a.lysis, the constant relation which exists between the three angles of a rectilinear triangle, the fundamental proposition of the theory of similar triangles, that of parallelopipedons, &c.; in a word, precisely the only geometrical propositions which cannot be obtained except by a direct study of the subject, without the calculus being susceptible of having any part in it. Such aberrations are the unreflecting exaggerations of that natural and philosophical tendency which leads us to extend farther and farther the influence of a.n.a.lysis in mathematical studies. In mechanics, the pretended a.n.a.lytical demonstrations of the parallelogram of forces are of similar character.

The viciousness of such a manner of proceeding follows from the principles previously presented. We have already, in fact, recognized that, since the calculus is not, and cannot be, any thing but a means of deduction, it would indicate a radically false idea of it to wish to employ it in establis.h.i.+ng the elementary foundations of any science whatever; for on what would the a.n.a.lytical reasonings in such an operation repose? A labour of this nature, very far from really perfecting the philosophical character of a science, would const.i.tute a return towards the metaphysical age, in presenting real facts as mere logical abstractions.

When we examine in themselves these pretended a.n.a.lytical demonstrations of the fundamental propositions of elementary geometry, we easily verify their necessary want of meaning. They are all founded on a vicious manner of conceiving the principle of _h.o.m.ogeneity_, the true general idea of which was explained in the second chapter of the preceding book.

These demonstrations suppose that this principle does not allow us to admit the coexistence in the same equation of numbers obtained by different concrete comparisons, which is evidently false, and contrary to the constant practice of geometers. Thus it is easy to recognize that, by employing the law of h.o.m.ogeneity in this arbitrary and illegitimate acceptation, we could succeed in "demonstrating," with quite as much apparent rigour, propositions whose absurdity is manifest at the first glance. In examining attentively, for example, the procedure by the aid of which it has been attempted to prove a.n.a.lytically that the sum of the three angles of any rectilinear triangle is constantly equal to two right angles, we see that it is founded on this preliminary principle that, if two triangles have two of their angles respectively equal, the third angle of the one will necessarily be equal to the third angle of the other. This first point being granted, the proposed relation is immediately deduced from it in a very exact and simple manner. Now the a.n.a.lytical consideration by which this previous proposition has been attempted to be established, is of such a nature that, if it could be correct, we could rigorously deduce from it, in reproducing it conversely, this palpable absurdity, that two sides of a triangle are sufficient, without any angle, for the entire determination of the third side. We may make a.n.a.logous remarks on all the demonstrations of this sort, the sophisms of which will be thus verified in a perfectly apparent manner.

The more reason that we have here to consider geometry as being at the present day essentially a.n.a.lytical, the more necessary was it to guard against this abusive exaggeration of mathematical a.n.a.lysis, according to which all geometrical observation would be dispensed with, in establis.h.i.+ng upon pure algebraical abstractions the very foundations of this natural science.

_Attempted Demonstrations of Axioms, &c._ Another indication that geometers have too much overlooked the character of a natural science which is necessarily inherent in geometry, appears from their vain attempts, so long made, to _demonstrate_ rigorously, not by the aid of the calculus, but by means of certain constructions, several fundamental propositions of elementary geometry. Whatever may be effected, it will evidently be impossible to avoid sometimes recurring to simple and direct observation in geometry as a means of establis.h.i.+ng various results. While, in this science, the phenomena which are considered are, by virtue of their extreme simplicity, much more closely connected with one another than those relating to any other physical science, some must still be found which cannot be deduced, and which, on the contrary, serve as starting points. It may be admitted that the greatest logical perfection of the science is to reduce these to the smallest number possible, but it would be absurd to pretend to make them completely disappear. I avow, moreover, that I find fewer real inconveniences in extending, a little beyond what would be strictly necessary, the number of these geometrical notions thus established by direct observation, provided they are sufficiently simple, than in making them the subjects of complicated and indirect demonstrations, even when these demonstrations may be logically irreproachable.

The true dogmatic destination of the geometry of the ancients, reduced to its least possible indispensable developments, having thus been characterized as exactly as possible, it is proper to consider summarily each of the princ.i.p.al parts of which it must be composed. I think that I may here limit myself to considering the first and the most extensive of these parts, that which has for its object the study of _the right line_; the two other sections, namely, the _quadrature of polygons_ and the _cubature of polyhedrons_, from their limited extent, not being capable of giving rise to any philosophical consideration of any importance, distinct from those indicated in the preceding chapter with respect to the measure of areas and of volumes in general.

GEOMETRY OF THE RIGHT LINE.

The final question which we always have in view in the study of the right line, properly consists in determining, by means of one another, the different elements of any right-lined figure whatever; which enables us always to know indirectly the length and position of a right line, in whatever circ.u.mstances it may be placed. This fundamental problem is susceptible of two general solutions, the nature of which is quite distinct, the one _graphical_, the other _algebraic_. The first, though very imperfect, is that which must be first considered, because it is spontaneously derived from the direct study of the subject; the second, much more perfect in the most important respects, cannot be studied till afterwards, because it is founded upon the previous knowledge of the other.

GRAPHICAL SOLUTIONS.

The graphical solution consists in constructing at will the proposed figure, either with the same dimensions, or, more usually, with dimensions changed in any ratio whatever. The first mode need merely be mentioned as being the most simple and the one which would first occur to the mind, for it is evidently, by its nature, almost entirely incapable of application. The second is, on the contrary, susceptible of being most extensively and most usefully applied. We still make an important and continual use of it at the present day, not only to represent with exactness the forms of bodies and their relative positions, but even for the actual determination of geometrical magnitudes, when we do not need great precision. The ancients, in consequence of the imperfection of their geometrical knowledge, employed this procedure in a much more extensive manner, since it was for a long time the only one which they could apply, even in the most important precise determinations. It was thus, for example, that Aristarchus of Samos estimated the relative distance from the sun and from the moon to the earth, by making measurements on a triangle constructed as exactly as possible, so as to be similar to the right-angled triangle formed by the three bodies at the instant when the moon is in quadrature, and when an observation of the angle at the earth would consequently be sufficient to define the triangle. Archimedes himself, although he was the first to introduce calculated determinations into geometry, several times employed similar means. The formation of trigonometry did not cause this method to be entirely abandoned, although it greatly diminished its use; the Greeks and the Arabians continued to employ it for a great number of researches, in which we now regard the use of the calculus as indispensable.

This exact reproduction of any figure whatever on a different scale cannot present any great theoretical difficulty when all the parts of the proposed figure lie in the same plane. But if we suppose, as most frequently happens, that they are situated in different planes, we see, then, a new order of geometrical considerations arise. The artificial figure, which is constantly plane, not being capable, in that case, of being a perfectly faithful image of the real figure, it is necessary previously to fix with precision the mode of representation, which gives rise to different systems of _Projection_.

It then remains to be determined according to what laws the geometrical phenomena correspond in the two figures. This consideration generates a new series of geometrical investigations, the final object of which is properly to discover how we can replace constructions in relief by plane constructions. The ancients had to resolve several elementary questions of this kind for various cases in which we now employ spherical trigonometry, princ.i.p.ally for different problems relating to the celestial sphere. Such was the object of their _a.n.a.lemmas_, and of the other plane figures which for a long time supplied the place of the calculus. We see by this that the ancients really knew the elements of what we now name _Descriptive Geometry_, although they did not conceive it in a distinct and general manner.

I think it proper briefly to indicate in this place the true philosophical character of this "Descriptive Geometry;" although, being essentially a science of application, it ought not to be included within the proper domain of this work.

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 subst.i.tuting 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 s.p.a.ce 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 h.o.m.ogeneous system, by the ill.u.s.trious 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--a.n.a.lyzing 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 circ.u.mstances 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 cla.s.s 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 subst.i.tution which really const.i.tutes 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 s.p.a.ce, 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, const.i.tutes 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.

ALGEBRAIC SOLUTIONS.

This kind of solution, the evident superiority of which need not here be dwelt upon, belongs necessarily, by the very nature of the question, to the system of the ancient geometry, although the logical method which is employed causes it to be generally, but very improperly, separated from it. We have thus an opportunity of verifying, in a very important respect, what was established generally in the preceding chapter, that it is not by the employment of the calculus that the modern geometry is essentially to be distinguished from the ancient. The ancients are in fact the true inventors of the present trigonometry, spherical as well as rectilinear; it being only much less perfect in their hands, on account of the extreme inferiority of their algebraical knowledge. It is, then, really in this chapter, and not, as it might at first be thought, in those which we shall afterwards devote to the philosophical examination of _general_ geometry, that it is proper to consider the character of this important preliminary theory, which is usually, though improperly, included in what is called _a.n.a.lytical geometry_, but which is really only a complement of _elementary geometry_ properly so called.

Since all right-lined figures can be decomposed into triangles, it is evidently sufficient to know how to determine the different elements of a triangle by means of one another, which reduces _polygonometry_ to simple _trigonometry_.

TRIGONOMETRY.

The difficulty in resolving algebraically such a question as the above, consists essentially in forming, between the angles and the sides of a triangle, three distinct equations; which, when once obtained, will evidently reduce all trigonometrical problems to mere questions of a.n.a.lysis.

_How to introduce Angles._ In considering the establishment of these equations in the most general manner, we immediately meet with a fundamental distinction with respect to the manner of introducing the angles into the calculation, according as they are made to enter _directly_, by themselves or by the circular arcs which are proportional to them; or _indirectly_, by the chords of these arcs, which are hence called their _trigonometrical lines_. Of these two systems of trigonometry the second was of necessity the only one originally adopted, as being the only practicable one, since the condition of geometry made it easy enough to find exact relations between the sides of the triangles and the trigonometrical lines which represent the angles, while it would have been absolutely impossible at that epoch to establish equations between the sides and the angles themselves.