A more general consideration.This essential consideration is only the final complement of a more general and more important consideration relative to the variations of different independent variables. If these variables are really independent of one another, as when we compare together all the imaginable curves susceptible of being traced between two points, it will be the same with their variations, and, consequently, the termsrelating to each of these variations will have to be separately equal to zero in the general equation which expresses the maximum or the minimum. But if, on the contrary, we suppose the variables to be subjected to any fixed conditions, it will be necessary to take notice of the resulting relation between their variations, so that the number of the equations into which this general equation is then decomposed is always equal to only the number of the variables which remain truly independent. It is thus, for example, that instead of seeking for the shortest path between any two points, in choosing it from among all possible ones, it may be proposed to find only what is the shortest among all those which may be taken on any given surface; a question the general solution of which forms certainly one of the most beautiful applications of the method of variations.
Questions of the second Class.Problems in which such modifying conditions are considered approach very nearly, in their nature, to the second general class of applications of the method of variations, characterized above as consisting in the investigation ofrelativemaxima and minima. There is, however, this essential difference between the two cases, that in this last the modification is expressed by an integral which depends upon the function sought, while in the other it is designated by a finite equation which is immediately given. It is hence apparent that the investigation ofrelativemaxima and minima is constantly and necessarily more complicated than that ofabsolutemaxima and minima. Luckily, a very important general theory, discovered by the genius of the great Euler before the invention of the Calculus of Variations, gives a uniform and verysimple means of making one of these two classes of questions dependent on the other. It consists in this, that if we add to the integral which is to be a maximum or a minimum, a constant and indeterminate multiple of that one which, by the nature of the problem, is to remain constant, it will be sufficient to seek, by the general method of Lagrange above indicated, theabsolutemaximum or minimum of this whole expression. It can be easily conceived, indeed, that the part of the complete variation which would proceed from the last integral must be equal to zero (because of the constant character of this last) as well as the portion due to the first integral, which disappears by virtue of the maximum or minimum state. These two conditions evidently unite to produce, in that respect, effects exactly alike.
Such is a sketch of the general manner in which the method of variation is applied to all the different questions which compose what is called theTheory of Isoperimeters. It will undoubtedly have been remarked in this summary exposition how much use has been made in this new analysis of the second fundamental property of the transcendental analysis noticed in the third chapter, namely, the generality of the infinitesimal expressions for the representation of the same geometrical or mechanical phenomenon, in whatever body it may be considered. Upon this generality, indeed, are founded, by their nature, all the solutions due to the method of variations. If a single formula could not express the length or the area of any curve whatever; if another fixed formula could not designate the time of the fall of a heavy body, according to whatever line it may descend,&c., how would it have been possible to resolve questions which unavoidably require, by their nature, the simultaneous consideration of all the cases which can be determined in each phenomenon by the different subjects which exhibit it.
Other Applications of this Method.Notwithstanding the extreme importance of the theory of isoperimeters, and though the method of variations had at first no other object than the logical and general solution of this order of problems, we should still have but an incomplete idea of this beautiful analysis if we limited its destination to this. In fact, the abstract conception of two distinct natures of differentiation is evidently applicable not only to the cases for which it was created, but also to all those which present, for any reason whatever, two different manners of making the same magnitudes vary. It is in this way that Lagrange himself has made, in his "Méchanique Analytique," an extensive and important application of his calculus of variations, by employing it to distinguish the two sorts of changes which are naturally presented by the questions of rational mechanics for the different points which are considered, according as we compare the successive positions which are occupied, in virtue of its motion, by the same point of each body in two consecutive instants, or as we pass from one point of the body to another in the same instant. One of these comparisons produces ordinary differentials; the other gives rise tovariations, which, there as every where, are only differentials taken under a new point of view. Such is the general acceptation in which we should conceive the Calculus of Variations, in order suitably to appreciate the importance of this admirable logicalinstrument, the most powerful that the human mind has as yet constructed.
The method of variations being only an immense extension of the general transcendental analysis, I have no need of proving specially that it is susceptible of being considered under the different fundamental points of view which the calculus of indirect functions, considered as a whole, admits of. Lagrange invented the Calculus of Variations in accordance with the infinitesimal conception, and, indeed, long before he undertook the general reconstruction of the transcendental analysis. When he had executed this important reformation, he easily showed how it could also be applied to the Calculus of Variations, which he expounded with all the proper development, according to his theory of derivative functions. But the more that the use of the method of variations is difficult of comprehension, because of the higher degree of abstraction of the ideas considered, the more necessary is it, in its application, to economize the exertions of the mind, by adopting the most direct and rapid analytical conception, namely, that of Leibnitz. Accordingly, Lagrange himself has constantly preferred it in the important use which he has made of the Calculus of Variations in his "Analytical Mechanics." In fact, there does not exist the least hesitation in this respect among geometers.
In order to make as clear as possible the philosophical character of the Calculus of Variations, I think that I should, in conclusion, briefly indicate a consideration which seems to me important, and by which I can approachit to the ordinary transcendental analysis in a higher degree than Lagrange seems to me to have done.[12]
We noticed in the preceding chapter the formation of thecalculus of partial differences, created by D'Alembert, as having introduced into the transcendental analysis a new elementary idea; the notion of two kinds of increments, distinct and independent of one another, which a function of two variables may receive by virtue of the change of each variable separately. It is thus that the vertical ordinate of a surface, or any other magnitude which is referred to it, varies in two manners which are quite distinct, and which may follow the most different laws, according as we increase either the one or the other of the two horizontal co-ordinates. Now such a consideration seems to me very nearly allied, by its nature, to that which serves as the general basis of the method of variations. This last, indeed, has in reality done nothing but transfer to the independent variables themselves the peculiar conception which had been already adopted for the functions of these variables; a modification which has remarkably enlarged its use. I think, therefore, that so far as regards merely the fundamental conceptions, we may consider the calculus created by D'Alembert as having established a natural and necessary transition between the ordinary infinitesimal calculus and the calculus of variations; such a derivation of which seems to be adapted to make the general notion more clear and simple.
According to the different considerations indicated in this chapter, the method of variations presents itself as the highest degree of perfection which the analysis of indirect functions has yet attained. In its primitive state, this last analysis presented itself as a powerful general means of facilitating the mathematical study of natural phenomena, by introducing, for the expression of their laws, the consideration of auxiliary magnitudes, chosen in such a manner that their relations are necessarily more simple and more easy to obtain than those of the direct magnitudes. But the formation of these differential equations was not supposed to admit of any general and abstract rules. Now the Analysis of Variations, considered in the most philosophical point of view, may be regarded as essentially destined, by its nature, to bring within the reach of the calculus the actual establishment of the differential equations; for, in a great number of important and difficult questions, such is the general effect of thevariedequations, which, still moreindirectthan the simple differential equations with respect to the special objects of the investigation, are also much more easy to form, and from which we may then, by invariable and complete analytical methods, the object of which is to eliminate the new order of auxiliary infinitesimals which have been introduced, deduce those ordinary differential equations which it would often have been impossible to establish directly. The method of variations forms, then, the most sublime part of that vast system of mathematical analysis, which, setting out from the most simple elements of algebra, organizes, by an uninterrupted succession of ideas, general methods more and more powerful, for the study of natural philosophy, andwhich, in its whole, presents the most incomparably imposing and unequivocal monument of the power of the human intellect.
We must, however, also admit that the conceptions which are habitually considered in the method of variations being, by their nature, more indirect, more general, and especially more abstract than all others, the employment of such a method exacts necessarily and continuously the highest known degree of intellectual exertion, in order never to lose sight of the precise object of the investigation, in following reasonings which offer to the mind such uncertain resting-places, and in which signs are of scarcely any assistance. We must undoubtedly attribute in a great degree to this difficulty the little real use which geometers, with the exception of Lagrange, have as yet made of such an admirable conception.
The different fundamental considerations indicated in the five preceding chapters constitute, in reality, all the essential bases of a complete exposition of mathematical analysis, regarded in the philosophical point of view. Nevertheless, in order not to neglect any truly important general conception relating to this analysis, I think that I should here very summarily explain the veritable character of a kind of calculus which is very extended, and which, though at bottom it really belongs to ordinary analysis, is still regarded as being of an essentially distinct nature. I refer to theCalculus of Finite Differences, which will be the special subject of this chapter.
Its general Character.This calculus, created by Taylor, in his celebrated work entitledMethodus Incrementorum, consists essentially in the consideration of the finite increments which functions receive as a consequence of analogous increments on the part of the corresponding variables. These increments ordifferences, which take the characteristic Δ, to distinguish them fromdifferentials, or infinitely small increments, may be in their turn regarded as new functions, and become the subject of a second similar consideration, and so on; from which results the notion of differences of various successive orders, analogous, at least in appearance, to the consecutive orders of differentials. Such a calculus evidentlypresents, like the calculus of indirect functions, two general classes of questions:
1°. To determine the successive differences of all the various analytical functions of one or more variables, as the result of a definite manner of increase of the independent variables, which are generally supposed to augment in arithmetical progression.
2°. Reciprocally, to start from these differences, or, more generally, from any equations established between them, and go back to the primitive functions themselves, or to their corresponding relations.
Hence follows the decomposition of this calculus into two distinct ones, to which are usually given the names of theDirect, and theInverse Calculus of Finite Differences, the latter being also sometimes called theIntegral Calculus of Finite Differences. Each of these would, also, evidently admit of a logical distribution similar to that given in the fourth chapter for the differential and the integral calculus.
Its true Nature.There is no doubt that Taylor thought that by such a conception he had founded a calculus of an entirely new nature, absolutely distinct from ordinary analysis, and more general than the calculus of Leibnitz, although resting on an analogous consideration. It is in this way, also, that almost all geometers have viewed the analysis of Taylor; but Lagrange, with his usual profundity, clearly perceived that these properties belonged much more to the forms and to the notations employed by Taylor than to the substance of his theory. In fact, that which constitutes the peculiar character of the analysis of Leibnitz, and makes of it a truly distinct and superior calculus, is the circumstance that the derivedfunctions are in general of an entirely different nature from the primitive functions, so that they may give rise to more simple and more easily formed relations: whence result the admirable fundamental properties of the transcendental analysis, which have been already explained. But it is not so with thedifferencesconsidered by Taylor; for these differences are, by their nature, functions essentially similar to those which have produced them, a circumstance which renders them unsuitable to facilitate the establishment of equations, and prevents their leading to more general relations. Every equation of finite differences is truly, at bottom, an equation directly relating to the very magnitudes whose successive states are compared. The scaffolding of new signs, which produce an illusion respecting the true character of these equations, disguises it, however, in a very imperfect manner, since it could always be easily made apparent by replacing thedifferencesby the equivalent combinations of the primitive magnitudes, of which they are really only the abridged designations. Thus the calculus of Taylor never has offered, and never can offer, in any question of geometry or of mechanics, that powerful general aid which we have seen to result necessarily from the analysis of Leibnitz. Lagrange has, moreover, very clearly proven that the pretended analogy observed between the calculus of differences and the infinitesimal calculus was radically vicious, in this way, that the formulas belonging to the former calculus can never furnish, as particular cases, those which belong to the latter, the nature of which is essentially distinct.
From these considerations I am led to think that the calculus of finite differences is, in general, improperlyclassed with the transcendental analysis proper, that is, with the calculus of indirect functions. I consider it, on the contrary, in accordance with the views of Lagrange, to be only a very extensive and very important branch of ordinary analysis, that is to say, of that which I have named the calculus of direct functions, the equations which it considers being always, in spite of the notation, simpledirectequations.
To sum up as briefly as possible the preceding explanation, the calculus of Taylor ought to be regarded as having constantly for its true object the general theory ofSeries, the most simple cases of which had alone been considered before that illustrious geometer. I ought, properly, to have mentioned this important theory in treating, in the second chapter, of Algebra proper, of which it is such an extensive branch. But, in order to avoid a double reference to it, I have preferred to notice it only in the consideration of the calculus of finite differences, which, reduced to its most simple general expression, is nothing but a complete logical study of questions relating toseries.
EverySeries, or succession of numbers deduced from one another according to any constant law, necessarily gives rise to these two fundamental questions:
1°. The law of the series being supposed known, to find the expression for its general term, so as to be able to calculate immediately any term whatever without being obliged to form successively all the preceding terms.
2°. In the same circumstances, to determine thesumof any number of terms of the series by means of theirplaces, so that it can be known without the necessity of continually adding these terms together.
These two fundamental questions being considered to be resolved, it may be proposed, reciprocally, to find the law of a series from the form of its general term, or the expression of the sum. Each of these different problems has so much the more extent and difficulty, as there can be conceived a greater number of differentlawsfor the series, according to the number of preceding terms on which each term directly depends, and according to the function which expresses that dependence. We may even consider series with several variable indices, as Laplace has done in his "Analytical Theory of Probabilities," by the analysis to which he has given the name ofTheory of Generating Functions, although it is really only a new and higher branch of the calculus of finite differences or of the general theory of series.
These general views which I have indicated give only an imperfect idea of the truly infinite extent and variety of the questions to which geometers have risen by means of this single consideration of series, so simple in appearance and so limited in its origin. It necessarily presents as many different cases as the algebraic resolution of equations, considered in its whole extent; and it is, by its nature, much more complicated, so much, indeed, that it always needs this last to conduct it to a complete solution. We may, therefore, anticipate what must still be its extreme imperfection, in spite of the successive labours of several geometers of the first order. We do not, indeed, possess as yet the complete and logical solution of any but the most simple questions of this nature.
Its identity with this Calculus.It is now easy to conceive the necessary and perfect identity, which has been already announced, between the calculus of finite differences and the theory of series considered in all its bearings. In fact, every differentiation after the manner of Taylor evidently amounts to finding thelawof formation of a series with one or with several variable indices, from the expression of its general term; in the same way, every analogous integration may be regarded as having for its object the summation of a series, the general term of which would be expressed by the proposed difference. In this point of view, the various problems of the calculus of differences, direct or inverse, resolved by Taylor and his successors, have really a very great value, as treating of important questions relating to series. But it is very doubtful if the form and the notation introduced by Taylor really give any essential facility in the solution of questions of this kind. It would be, perhaps, more advantageous for most cases, and certainly more logical, to replace thedifferencesby the terms themselves, certain combinations of which they represent. As the calculus of Taylor does not rest on a truly distinct fundamental idea, and has nothing peculiar to it but its system of signs, there could never really be any important advantage in considering it as detached from ordinary analysis, of which it is, in reality, only an immense branch. This consideration ofdifferences, most generally useless, even if it does not cause complication, seems to me to retain the character of an epoch in which, analytical ideas not being sufficiently familiar to geometers, they were naturally led to prefer the special forms suitable for simple numerical comparisons.
However that may be, I must not finish this general appreciation of the calculus of finite differences without noticing a new conception to which it has given birth, and which has since acquired a great importance. It is the consideration of those periodic or discontinuous functions which preserve the same value for an infinite series of values of the corresponding variables, subjected to a certain law, and which must be necessarily added to the integrals of the equations of finite differences in order to render them sufficiently general, as simple arbitrary constants are added to all quadratures in order to complete their generality. This idea, primitively introduced by Euler, has since been the subject of extended investigation by M. Fourier, who has made new and important applications of it in his mathematical theory of heat.
Series.Among the principal general applications which have been made of the calculus of finite differences, it would be proper to place in the first rank, as the most extended and the most important, the solution of questions relating to series; if, as has been shown, the general theory of series ought not to be considered as constituting, by its nature, the actual foundation of the calculus of Taylor.
Interpolations.This great class of problems being then set aside, the most essential of the veritable applications of the analysis of Taylor is, undoubtedly, thus far, the general method ofinterpolations, so frequently and so usefully employed in the investigation of the empiricallaws of natural phenomena. The question consists, as is well known, in intercalating between certain given numbers other intermediate numbers, subjected to the same law which we suppose to exist between the first. We can abundantly verify, in this principal application of the calculus of Taylor, how truly foreign and often inconvenient is the consideration ofdifferenceswith respect to the questions which depend on that analysis. Indeed, Lagrange has replaced the formulas of interpolation, deduced from the ordinary algorithm of the calculus of finite differences, by much simpler general formulas, which are now almost always preferred, and which have been found directly, without making any use of the notion ofdifferences, which only complicates the question.
Approximate Rectification, &c.A last important class of applications of the calculus of finite differences, which deserves to be distinguished from the preceding, consists in the eminently useful employment made of it in geometry for determining by approximation the length and the area of any curve, and in the same way the cubature of a body of any form whatever. This procedure (which may besides be conceived abstractly as depending on the same analytical investigation as the question of interpolation) frequently offers a valuable supplement to the entirely logical geometrical methods which often lead to integrations, which we do not yet know how to effect, or to calculations of very complicated execution.
Such are the various principal considerations to be noticed with respect to the calculus of finite differences. This examination completes the proposed philosophical outline ofabstract Mathematics.
Concrete Mathematicswill now be the subject of a similar labour. In it we shall particularly devote ourselves to examining how it has been possible (supposing the general science of the calculus to be perfect), by invariable procedures, to reduce to pure questions of analysis all the problems which can be presented byGeometryandMechanics, and thus to impress on these two fundamental bases of natural philosophy a degree of precision and especially of unity; in a word, a character of high perfection, which could be communicated to them by such a course alone.
BOOK II.
GEOMETRY.
BOOK II.
Its true Nature.After the general exposition of the philosophical character of concrete mathematics, compared with that of abstract mathematics, given in the introductory chapter, it need not here be shown in a special manner that geometry must be considered as a true natural science, only much more simple, and therefore much more perfect, than any other. This necessary perfection of geometry, obtained essentially by the application of mathematical analysis, which it so eminently admits, is apt to produce erroneous views of the real nature of this fundamental science, which most minds at present conceive to be a purely logical science quite independent of observation. It is nevertheless evident, to any one who examines with attention the character of geometrical reasonings, even in the present state of abstract geometry, that, although the facts which are considered in it are much more closely united than those relating to any other science, still there always exists, with respect to every body studied by geometers, a certain number of primitive phenomena, which, since they are not established by anyreasoning, must be founded on observation alone, and which form the necessary basis of all the deductions.
The scientific superiority of geometry arises from the phenomena which it considers being necessarily the most universal and the most simple of all. Not only may all the bodies of nature give rise to geometrical inquiries, as well as mechanical ones, but still farther, geometrical phenomena would still exist, even though all the parts of the universe should be considered as immovable. Geometry is then, by its nature, more general than mechanics. At the same time, its phenomena are more simple, for they are evidently independent of mechanical phenomena, while these latter are always complicated with the former. The same relations hold good in comparing geometry with abstract thermology.
For these reasons, in our classification we have made geometry the first part of concrete mathematics; that part the study of which, in addition to its own importance, serves as the indispensable basis of all the rest.
Before considering directly the philosophical study of the different orders of inquiries which constitute our present geometry, we should obtain a clear and exact idea of the general destination of that science, viewed in all its bearings. Such is the object of this chapter.
Definition.Geometry is commonly defined in a very vague and entirely improper manner, as beingthe science of extension. An improvement on this would be to say that geometry has for its object themeasurementof extension; but such an explanation would be very insufficient, although at bottom correct, and would be far from giving any idea of the true general character of geometrical science.
To do this, I think that I should first explaintwo fundamental ideas, which, very simple in themselves, have been singularly obscured by the employment of metaphysical considerations.
The Idea of Space.The first is that ofSpace. This conception properly consists simply in this, that, instead of considering extension in the bodies themselves, we view it in an indefinite medium, which we regard as containing all the bodies of the universe. This notion is naturally suggested to us by observation, when we think of theimpressionwhich a body would leave in a fluid in which it had been placed. It is clear, in fact, that, as regards its geometrical relations, such animpressionmay be substituted for the body itself, without altering the reasonings respecting it. As to the physical nature of this indefinitespace, we are spontaneously led to represent it to ourselves, as being entirely analogous to the actual medium in which we live; so that if this medium was liquid instead of gaseous, our geometricalspacewould undoubtedly be conceived as liquid also. This circumstance is, moreover, only very secondary, the essential object of such a conception being only to make us view extension separately from the bodies which manifest it to us. We can easily understand in advance the importance of this fundamental image, since it permits us to study geometrical phenomena in themselves, abstraction being made of all the other phenomena which constantly accompany them in real bodies, without, however, exerting any influence over them. The regular establishment of this general abstraction must be regarded as the first step which has been made in the rational study of geometry, which would have been impossible ifit had been necessary to consider, together with the form and the magnitude of bodies, all their other physical properties. The use of such an hypothesis, which is perhaps the most ancient philosophical conception created by the human mind, has now become so familiar to us, that we have difficulty in exactly estimating its importance, by trying to appreciate the consequences which would result from its suppression.
Different Kinds of Extension.The second preliminary geometrical conception which we have to examine is that of the different kinds of extension, designated by the wordsvolume,surface,line, and evenpoint, and of which the ordinary explanation is so unsatisfactory.[13]
Although it is evidently impossible to conceive any extension absolutely deprived of any one of the three fundamental dimensions, it is no less incontestable that, in a great number of occasions, even of immediate utility, geometrical questions depend on only two dimensions, considered separately from the third, or on a single dimension, considered separately from the two others. Again, independently of this direct motive, the study of extension with a single dimension, and afterwards with two, clearly presents itself as an indispensable preliminary for facilitating the study of complete bodies of three dimensions, the immediate theory of which would be too complicated.Such are the two general motives which oblige geometers to consider separately extension with regard to one or to two dimensions, as well as relatively to all three together.
The general notions ofsurfaceand oflinehave been formed by the human mind, in order that it may be able to think, in a permanent manner, of extension in two directions, or in one only. The hyperbolical expressions habitually employed by geometers to define these notions tend to convey false ideas of them; but, examined in themselves, they have no other object than to permit us to reason with facility respecting these two kinds of extension, making complete abstraction of that which ought not to be taken into consideration. Now for this it is sufficient to conceive the dimension which we wish to eliminate as becoming gradually smaller and smaller, the two others remaining the same, until it arrives at such a degree of tenuity that it can no longer fix the attention. It is thus that we naturally acquire the real idea of asurface, and, by a second analogous operation, the idea of aline, by repeating for breadth what we had at first done for thickness. Finally, if we again repeat the same operation, we arrive at the idea of apoint, or of an extension considered only with reference to its place, abstraction being made of all magnitude, and designed consequently to determine positions.
Surfacesevidently have, moreover, the general property of exactly circumscribing volumes; and in the same way,lines, in their turn, circumscribesurfacesand are limited bypoints. But this consideration, to which too much importance is often given, is only a secondary one.
Surfaces and lines are, then, in reality, always conceived with three dimensions; it would be, in fact, impossible to represent to one's self a surface otherwise than as an extremely thin plate, and a line otherwise than as an infinitely fine thread. It is even plain that the degree of tenuity attributed by each individual to the dimensions of which he wishes to make abstraction is not constantly identical, for it must depend on the degree of subtilty of his habitual geometrical observations. This want of uniformity has, besides, no real inconvenience, since it is sufficient, in order that the ideas of surface and of line should satisfy the essential condition of their destination, for each one to represent to himself the dimensions which are to be neglected as being smaller than all those whose magnitude his daily experience gives him occasion to appreciate.
We hence see how devoid of all meaning are the fantastic discussions of metaphysicians upon the foundations of geometry. It should also be remarked that these primordial ideas are habitually presented by geometers in an unphilosophical manner, since, for example, they explain the notions of the different sorts of extent in an order absolutely the inverse of their natural dependence, which often produces the most serious inconveniences in elementary instruction.
These preliminaries being established, we can proceed directly to the general definition of geometry, continuing to conceive this science as having for its final object themeasurementof extension.
It is necessary in this matter to go into a thoroughexplanation, founded on the distinction of the three kinds of extension, since the notion ofmeasurementis not exactly the same with reference to surfaces and volumes as to lines.
Nature of Geometrical Measurement.If we take the wordmeasurementin its direct and general mathematical acceptation, which signifies simply the determination of the value of theratiosbetween any homogeneous magnitudes, we must consider, in geometry, that themeasurementof surfaces and of volumes, unlike that of lines, is never conceived, even in the most simple and the most favourable cases, as being effected directly. The comparison of two lines is regarded as direct; that of two surfaces or of two volumes is, on the contrary, always indirect. Thus we conceive that two lines may be superposed; but the superposition of two surfaces, or, still more so, of two volumes, is evidently impossible in most cases; and, even when it becomes rigorously practicable, such a comparison is never either convenient or exact. It is, then, very necessary to explain wherein properly consists the truly geometrical measurement of a surface or of a volume.
Measurement of Surfaces and of Volumes.For this we must consider that, whatever may be the form of a body, there always exists a certain number of lines, more or less easy to be assigned, the length of which is sufficient to define exactly the magnitude of its surface or of its volume. Geometry, regarding these lines as alone susceptible of being directly measured, proposes to deduce, from the simple determination of them, the ratio of the surface or of the volume sought, to the unity of surface, or to the unity of volume. Thus the general object ofgeometry, with respect to surfaces and to volumes, is properly to reduce all comparisons of surfaces or of volumes to simple comparisons of lines.
Besides the very great facility which such a transformation evidently offers for the measurement of volumes and of surfaces, there results from it, in considering it in a more extended and more scientific manner, the general possibility of reducing to questions of lines all questions relating to volumes and to surfaces, considered with reference to their magnitude. Such is often the most important use of the geometrical expressions which determine surfaces and volumes in functions of the corresponding lines.
It is true that direct comparisons between surfaces or between volumes are sometimes employed; but such measurements are not regarded as geometrical, but only as a supplement sometimes necessary, although too rarely applicable, to the insufficiency or to the difficulty of truly rational methods. It is thus that we often determine the volume of a body, and in certain cases its surface, by means of its weight. In the same way, on other occasions, when we can substitute for the proposed volume an equivalent liquid volume, we establish directly the comparison of the two volumes, by profiting by the property possessed by liquid masses, of assuming any desired form. But all means of this nature are purely mechanical, and rational geometry necessarily rejects them.
To render more sensible the difference between these modes of determination and true geometrical measurements, I will cite a single very remarkable example; the manner in which Galileo determined the ratio of the ordinary cycloid to that of the generating circle. Thegeometry of his time was as yet insufficient for the rational solution of such a problem. Galileo conceived the idea of discovering that ratio by a direct experiment. Having weighed as exactly as possible two plates of the same material and of equal thickness, one of them having the form of a circle and the other that of the generated cycloid, he found the weight of the latter always triple that of the former; whence he inferred that the area of the cycloid is triple that of the generating circle, a result agreeing with the veritable solution subsequently obtained by Pascal and Wallis. Such a success evidently depends on the extreme simplicity of the ratio sought; and we can understand the necessary insufficiency of such expedients, even when they are actually practicable.
We see clearly, from what precedes, the nature of that part of geometry relating tovolumesand that relating tosurfaces. But the character of the geometry oflinesis not so apparent, since, in order to simplify the exposition, we have considered the measurement of lines as being made directly. There is, therefore, needed a complementary explanation with respect to them.
Measurement of curved Lines.For this purpose, it is sufficient to distinguish between the right line and curved lines, the measurement of the first being alone regarded as direct, and that of the other as always indirect. Although superposition is sometimes strictly practicable for curved lines, it is nevertheless evident that truly rational geometry must necessarily reject it, as not admitting of any precision, even when it is possible. The geometry of lines has, then, for its general object, to reduce in every case the measurement of curved lines tothat of right lines; and consequently, in the most extended point of view, to reduce to simple questions of right lines all questions relating to the magnitude of any curves whatever. To understand the possibility of such a transformation, we must remark, that in every curve there always exist certain right lines, the length of which must be sufficient to determine that of the curve. Thus, in a circle, it is evident that from the length of the radius we must be able to deduce that of the circumference; in the same way, the length of an ellipse depends on that of its two axes; the length of a cycloid upon the diameter of the generating circle, &c.; and if, instead of considering the whole of each curve, we demand, more generally, the length of any arc, it will be sufficient to add to the different rectilinear parameters, which determine the whole curve, the chord of the proposed arc, or the co-ordinates of its extremities. To discover the relation which exists between the length of a curved line and that of similar right lines, is the general problem of the part of geometry which relates to the study of lines.
Combining this consideration with those previously suggested with respect to volumes and to surfaces, we may form a very clear idea of the science of geometry, conceived in all its parts, by assigning to it, for its general object, the final reduction of the comparisons of all kinds of extent, volumes, surfaces, or lines, to simple comparisons of right lines, the only comparisons regarded as capable of being made directly, and which indeed could not be reduced to any others more easy to effect. Such a conception, at the same time, indicates clearly the veritable character of geometry, and seems suited to show at a single glance its utility and its perfection.
Measurement of right Lines.In order to complete this fundamental explanation, I have yet to show how there can be, in geometry, a special section relating to the right line, which seems at first incompatible with the principle that the measurement of this class of lines must always be regarded as direct.
It is so, in fact, as compared with that of curved lines, and of all the other objects which geometry considers. But it is evident that the estimation of a right line cannot be viewed as direct except so far as the linear unit can be applied to it. Now this often presents insurmountable difficulties, as I had occasion to show, for another reason, in the introductory chapter. We must, then, make the measurement of the proposed right line depend on other analogous measurements capable of being effected directly. There is, then, necessarily a primary distinct branch of geometry, exclusively devoted to the right line; its object is to determine certain right lines from others by means of the relations belonging to the figures resulting from their assemblage. This preliminary part of geometry, which is almost imperceptible in viewing the whole of the science, is nevertheless susceptible of a great development. It is evidently of especial importance, since all other geometrical measurements are referred to those of right lines, and if they could not be determined, the solution of every question would remain unfinished.
Such, then, are the various fundamental parts of rational geometry, arranged according to their natural dependence; the geometry oflinesbeing first considered, beginning with the right line; then the geometry ofsurfaces, and, finally, that ofsolids.
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 curvedlines, regarding them as generated by the motion of a point governed by a certain law, it is plain that we shall have, ingeneral, 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 acircle; if it is the sum or the difference of its distances from two fixed points which remains constant, the curve described will be anellipseor anhyperbola; 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 aparabola; if it revolves on a circle at the same time that this circle rolls along a straight line, we shall have acycloid; 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 calledspirals, 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 tosurfaces, 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 ofthe 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 tovolumes, 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 instarting 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.
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 definitionabove 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 assign as the object of geometry themeasurementof 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 constitute our present geometry do not at all appear to have for their object themeasurementof 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, sometimesdirect, though most oftenindirect, 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.
When we examine with attention the geometrical investigations which do not seem to relate to themeasurementof extent, we find that they consist essentially in the study of the differentpropertiesof 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 ofmeasurement, 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.Thefirst, 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 itsmeasurement. 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 thepropertiesof the proposed figure. Let us suppose, for example, that thecircle 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 whichis 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 thenmeasureit (which, according to the first order of considerations, would generally be impossible), this knowledge would remain almost necessarilybarren 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 undoubtedlymuch 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 longerseries 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 themeasurementof 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 eachfigure, 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 becauseof 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.
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 class of researches which did not at first appear necessarily to belong to it, there remains to be considered themethodto 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 totwo methodsso 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 ofSynthetical GeometryandAnalytical Geometry, habitually employed to designate them, give a very false idea of them. I would much prefer the purely historical denominations ofGeometry of the AncientsandGeometry of the Moderns, which have at least the advantage of not causing their true characterto be misunderstood. But I propose to employ henceforth the regular expressions ofSpecial GeometryandGeneral 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 ofdeterminate. 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 analysis 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 principal 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 analogy 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 thebodiesstudied or to thephenomenato 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 principal characteristics of the ancient geometry is that they studied, one by one, the different lines and the different surfaces, not passing 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 assistance, 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 assured; so that they could not be certain, in advance, of obtaining any solution whatever, however analogous 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 assistance 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.