The philosophy of mathematics Part 12

_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 circ.u.mstance which has nothing a.n.a.logous in curves, since the points which describe them cannot have any distinct figure. Two cla.s.ses 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 cla.s.s 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.

_a.n.a.lytical invention of Curves, &c._ Finally, all the various direct means for the invention of figures have scarcely any farther importance, since rational geometry has a.s.sumed 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 a.n.a.lytical 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 a.n.a.lytically 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.

EXPANSION OF ORIGINAL DEFINITION.

To complete the formation of an exact and sufficiently extended idea of the nature of geometrical inquiries, it is now indispensable to return to the general definition above given, in order to present it under a new point of view, without which the complete science would be only very imperfectly conceived.

When we a.s.sign as the object of geometry the _measurement_ of all sorts of lines, surfaces, and volumes, that is, as has been explained, the reduction of all geometrical comparisons to simple comparisons of right lines, we have evidently the advantage of indicating a general destination very precise and very easy to comprehend. But if we set aside every definition, and examine the actual composition of the science of geometry, we will at first be induced to regard the preceding definition as much too narrow; for it is certain that the greater part of the investigations which const.i.tute our present geometry do not at all appear to have for their object the _measurement_ of extension. In spite of this fundamental objection, I will persist in retaining this definition; for, in fact, if, instead of confining ourselves to considering the different questions of geometry isolatedly, we endeavour to grasp the leading questions, in comparison with which all others, however important they may be, must be regarded as only secondary, we will finally recognize that the measurement of lines, of surfaces, and of volumes, is the invariable object, sometimes _direct_, though most often _indirect_, of all geometrical labours.

This general proposition being fundamental, since it can alone give our definition all its value, it is indispensable to enter into some developments upon this subject.

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 const.i.tutes the final object of geometry. We may even say that the princ.i.p.al 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 pa.s.s 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 pa.s.s 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 establis.h.i.+ng 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 circ.u.mstances 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 a.s.sured, 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 circ.u.mstances, and varies, on the other hand, for the same figure, with different circ.u.mstances; a double reason for its previous determination.

_Ill.u.s.tration: 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 quant.i.ty, 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 mult.i.tude of different points of view, before Kepler could thus pa.s.s 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.

_Ill.u.s.tration: 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 const.i.tute 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.

THE TWO GENERAL METHODS OF GEOMETRY.

Having now explained in a general and yet precise manner the final object of geometry, and shown how the science, thus defined, comprehends a very extensive cla.s.s of researches which did not at first appear necessarily to belong to it, there remains to be considered the _method_ to be followed for the formation of this science. This discussion is indispensable to complete this first sketch of the philosophical character of geometry. I shall here confine myself to indicating the most general consideration in this matter, developing and summing up this important fundamental idea in the following chapters.

Geometrical questions may be treated according to _two methods_ so different, that there result from them two sorts of geometry, so to say, the philosophical character of which does not seem to me to have yet been properly apprehended. The expressions of _Synthetical Geometry_ and _a.n.a.lytical Geometry_, habitually employed to designate them, give a very false idea of them. I would much prefer the purely historical denominations of _Geometry of the Ancients_ and _Geometry of the Moderns_, which have at least the advantage of not causing their true character to be misunderstood. But I propose to employ henceforth the regular expressions of _Special Geometry_ and _General Geometry_, which seem to me suited to characterize with precision the veritable nature of the two methods.

_Their fundamental Difference._ The fundamental difference between the manner in which we conceive Geometry since Descartes, and the manner in which the geometers of antiquity treated geometrical questions, is not the use of the Calculus (or Algebra), as is commonly thought to be the case. On the one hand, it is certain that the use of the calculus was not entirely unknown to the ancient geometers, since they used to make continual and very extensive applications of the theory of proportions, which was for them, as a means of deduction, a sort of real, though very imperfect and especially extremely limited equivalent for our present algebra. The calculus may even be employed in a much more complete manner than they have used it, in order to obtain certain geometrical solutions, which will still retain all the essential character of the ancient geometry; this occurs very frequently with respect to those problems of geometry of two or of three dimensions, which are commonly designated under the name of _determinate_. On the other hand, important as is the influence of the calculus in our modern geometry, various solutions obtained without algebra may sometimes manifest the peculiar character which distinguishes it from the ancient geometry, although a.n.a.lysis is generally indispensable. I will cite, as an example, the method of Roberval for tangents, the nature of which is essentially modern, and which, however, leads in certain cases to complete solutions, without any aid from the calculus. It is not, then, the instrument of deduction employed which is the princ.i.p.al distinction between the two courses which the human mind can take in geometry.

The real fundamental difference, as yet imperfectly apprehended, seems to me to consist in the very nature of the questions considered. In truth, geometry, viewed as a whole, and supposed to have attained entire perfection, must, as we have seen on the one hand, embrace all imaginable figures, and, on the other, discover all the properties of each figure. It admits, from this double consideration, of being treated according to two essentially distinct plans; either, 1, by grouping together all the questions, however different they may be, which relate to the same figure, and isolating those relating to different bodies, whatever a.n.a.logy there may exist between them; or, 2, on the contrary, by uniting under one point of view all similar inquiries, to whatever different figures they may relate, and separating the questions relating to the really different properties of the same body. In a word, the whole body of geometry may be essentially arranged either with reference to the _bodies_ studied or to the _phenomena_ to be considered. The first plan, which is the most natural, was that of the ancients; the second, infinitely more rational, is that of the moderns since Descartes.

_Geometry of the Ancients._ Indeed, the princ.i.p.al characteristics of the ancient geometry is that they studied, one by one, the different lines and the different surfaces, not pa.s.sing to the examination of a new figure till they thought they had exhausted all that there was interesting in the figures already known. In this way of proceeding, when they undertook the study of a new curve, the whole of the labour bestowed on the preceding ones could not offer directly any essential a.s.sistance, otherwise than by the geometrical practice to which it had trained the mind. Whatever might be the real similarity of the questions proposed as to two different figures, the complete knowledge acquired for the one could not at all dispense with taking up again the whole of the investigation for the other. Thus the progress of the mind was never a.s.sured; so that they could not be certain, in advance, of obtaining any solution whatever, however a.n.a.logous the proposed problem might be to questions which had been already resolved. Thus, for example, the determination of the tangents to the three conic sections did not furnish any rational a.s.sistance for drawing the tangent to any other new curve, such as the conchoid, the cissoid, &c. In a word, the geometry of the ancients was, according to the expression proposed above, essentially special.

_Geometry of the Moderns._ In the system of the moderns, geometry is, on the contrary, eminently _general_, that is to say, relating to any figures whatever. It is easy to understand, in the first place, that all geometrical expressions of any interest may be proposed with reference to all imaginable figures. This is seen directly in the fundamental problems--of rectifications, quadratures, and cubatures--which const.i.tute, as has been shown, the final object of geometry. But this remark is no less incontestable, even for investigations which relate to the different _properties_ of lines and of surfaces, and of which the most essential, such as the question of tangents or of tangent planes, the theory of curvatures, &c., are evidently common to all figures whatever. The very few investigations which are truly peculiar to particular figures have only an extremely secondary importance. This being understood, modern geometry consists essentially in abstracting, in order to treat it by itself, in an entirely general manner, every question relating to the same geometrical phenomenon, in whatever bodies it may be considered. The application of the universal theories thus constructed to the special determination of the phenomenon which is treated of in each particular body, is now regarded as only a subaltern labour, to be executed according to invariable rules, and the success of which is certain in advance. This labour is, in a word, of the same character as the numerical calculation of an a.n.a.lytical formula. There can be no other merit in it than that of presenting in each case the solution which is necessarily furnished by the general method, with all the simplicity and elegance which the line or the surface considered can admit of. But no real importance is attached to any thing but the conception and the complete solution of a new question belonging to any figure whatever. Labours of this kind are alone regarded as producing any real advance in science. The attention of geometers, thus relieved from the examination of the peculiarities of different figures, and wholly directed towards general questions, has been thereby able to elevate itself to the consideration of new geometrical conceptions, which, applied to the curves studied by the ancients, have led to the discovery of important properties which they had not before even suspected. Such is geometry, since the radical revolution produced by Descartes in the general system of the science.

_The Superiority of the modern Geometry._ The mere indication of the fundamental character of each of the two geometries is undoubtedly sufficient to make apparent the immense necessary superiority of modern geometry. We may even say that, before the great conception of Descartes, rational geometry was not truly const.i.tuted upon definitive bases, whether in its abstract or concrete relations. In fact, as regards science, considered speculatively, it is clear that, in continuing indefinitely to follow the course of the ancients, as did the moderns before Descartes, and even for a little while afterwards, by adding some new curves to the small number of those which they had studied, the progress thus made, however rapid it might have been, would still be found, after a long series of ages, to be very inconsiderable in comparison with the general system of geometry, seeing the infinite variety of the forms which would still have remained to be studied. On the contrary, at each question resolved according to the method of the moderns, the number of geometrical problems to be resolved is then, once for all, diminished by so much with respect to all possible bodies.

Another consideration is, that it resulted, from their complete want of general methods, that the ancient geometers, in all their investigations, were entirely abandoned to their own strength, without ever having the certainty of obtaining, sooner or later, any solution whatever. Though this imperfection of the science was eminently suited to call forth all their admirable sagacity, it necessarily rendered their progress extremely slow; we can form some idea of this by the considerable time which they employed in the study of the conic sections. Modern geometry, making the progress of our mind certain, permits us, on the contrary, to make the greatest possible use of the forces of our intelligence, which the ancients were often obliged to waste on very unimportant questions.

A no less important difference between the two systems appears when we come to consider geometry in the concrete point of view. Indeed, we have already remarked that the relation of the abstract to the concrete in geometry can be founded upon rational bases only so far as the investigations are made to bear directly upon all imaginable figures. In studying lines, only one by one, whatever may be the number, always necessarily very small, of those which we shall have considered, the application of such theories to figures really existing in nature will never have any other than an essentially accidental character, since there is nothing to a.s.sure us that these figures can really be brought under the abstract types considered by geometers.

Thus, for example, there is certainly something fortuitous in the happy relation established between the speculations of the Greek geometers upon the conic sections and the determination of the true planetary orbits. In continuing geometrical researches upon the same plan, there was no good reason for hoping for similar coincidences; and it would have been possible, in these special studies, that the researches of geometers should have been directed to abstract figures entirely incapable of any application, while they neglected others, susceptible perhaps of an important and immediate application. It is clear, at least, that nothing positively guaranteed the necessary applicability of geometrical speculations. It is quite another thing in the modern geometry. From the single circ.u.mstance that in it we proceed by general questions relating to any figures whatever, we have in advance the evident certainty that the figures really existing in the external world could in no case escape the appropriate theory if the geometrical phenomenon which it considers presents itself in them.

From these different considerations, we see that the ancient system of geometry wears essentially the character of the infancy of the science, which did not begin to become completely rational till after the philosophical resolution produced by Descartes. But it is evident, on the other hand, that geometry could not be at first conceived except in this _special_ manner. _General_ geometry would not have been possible, and its necessity could not even have been felt, if a long series of special labours on the most simple figures had not previously furnished bases for the conception of Descartes, and rendered apparent the impossibility of persisting indefinitely in the primitive geometrical philosophy.

_The Ancient the Base of the Modern._ From this last consideration we must infer that, although the geometry which I have called _general_ must be now regarded as the only true dogmatical geometry, and that to which we shall chiefly confine ourselves, the other having no longer much more than an historical interest, nevertheless it is not possible to entirely dispense with _special_ geometry in a rational exposition of the science. We undoubtedly need not borrow directly from ancient geometry all the results which it has furnished; but, from the very nature of the subject, it is necessarily impossible entirely to dispense with the ancient method, which will always serve as the preliminary basis of the science, dogmatically as well as historically. The reason of this is easy to understand. In fact, _general_ geometry being essentially founded, as we shall soon establish, upon the employment of the calculus in the transformation of geometrical into a.n.a.lytical considerations, such a manner of proceeding could not take possession of the subject immediately at its origin. We know that the application of mathematical a.n.a.lysis, from its nature, can never commence any science whatever, since evidently it cannot be employed until the science has already been sufficiently cultivated to establish, with respect to the phenomena considered, some _equations_ which can serve as starting points for the a.n.a.lytical operations. These fundamental equations being once discovered, a.n.a.lysis will enable us to deduce from them a mult.i.tude of consequences which it would have been previously impossible even to suspect; it will perfect the science to an immense degree, both with respect to the generality of its conceptions and to the complete co-ordination established between them. But mere mathematical a.n.a.lysis could never be sufficient to form the bases of any natural science, not even to demonstrate them anew when they have once been established.

Nothing can dispense with the direct study of the subject, pursued up to the point of the discovery of precise relations.

We thus see that the geometry of the ancients will always have, by its nature, a primary part, absolutely necessary and more or less extensive, in the complete system of geometrical knowledge. It forms a rigorously indispensable introduction to _general_ geometry. But it is to this that it must be limited in a completely dogmatic exposition. I will consider, then, directly, in the following chapter, this _special_ or _preliminary_ geometry restricted to exactly its necessary limits, in order to occupy myself thenceforth only with the philosophical examination of _general_ or _definitive_ geometry, the only one which is truly rational, and which at present essentially composes the science.