The construction of this general theory having destroyed the chief interests of geometers in this kind of questions, they rose almost immediately to the consideration of a new order of problems, at once more important and more difficult,— those of isoperimeters. It was then no longer the values of the variables proper to the maximum or the minimum of a given function that had to be determined. It was the form of the function itself that had to be discovered, accoi'ding to the condition of the maximum or minimum of a certain definite integral, merely indicated, which depended on that fvmction. We cannot here follow the history of these problems, the oldest of which is that of the solid of least resistance, ti'eated by Newton in the second book of the ' Principia,' in which he determines what must be the meridian curve of a solid of revolution, in order that the resistance experienced by that body in the direction of its axis may be the least possible. Mechanics first furnished this new class of problems; but it was from geometry that the subjects of the principal investigations were afterwards derived. They were varied and complicated almost infinitely by the labours of the best geometers, when Lagrange reduced their solution to an abstract and entirely general method, the discovery of which has checked the eagerness of geometers about such an order of researches.
It is evident that these problems, considered analytically, consist in determining what ought to be the form of a certain unknown function of one or more variables, in order that such or such an integral, dependent on that function, may have, within assigned limits, a value which may be a maximum or a minimum, with regard to all those which it would take if the required function had any other form whatever. In treating these problems, the predecessors of Lagrange proposed, in substance, to reduce them to the ordinary theory of maxima and minima. But they proceeded by applying special simple artifices to each case, not reducible to certain rules; so that every new question reproduced analogous difficulties, without the solutions SOME OF ITS APPLICATIONS. 89 previously obtained being of any essential aid. The part common to all questions of tbis class had not been discovered; and no abstract and general treatment was therefore provided. In his endeavours to bring all isoperimetrical problems to depend on a common analysis, Lagrange was led to the conception of a new kind of diif erentiation; and to these new Differentials he gave the name of Variations. They consist of the infinitely small increments which the integrals receive, not in virtue of analogous increments on the part of the corresponding variables, as in the common transcendental analysis, but by supposing that the form of the function placed under the sign of integration undergoes an infinitely small change. This abstract conception once formed, Lagrange was able to reduce with ease, and in the most general manner, all the problems of isoperimeters to the simple common theory of maxima and minima.
Important as is this great and happy ti'ans-,.
formation, and though the Method of Varia- ^[qi^^ ' tions had at first no other object than the rational and general resolution of isoperimetrical problems, we should form a very inadequate estimate of this beautiful analysis if we supposed it restricted to this application. 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 for all which present, for any reason whatever, two different ways of making the same magnitudes vary. Lagrange himself made an immense and all-important applicatiom of his Calculus of Variations, in his ' Analytical Mechanics,' by employing it to distinguish the two sorts of changes, naturally presented by questions of rational Mechanics for the different points we have to consider, according as we compare the successive positions 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 the common differentials; the other occasions variations which are, there as elsewhere, only differentials taken from a new point of view. It is in such a general acceptation as this that we must conceive of the Calculus of Variations, to appreciate fitly the importance of this admirable logical instrument; the most powerful as yet constructed by the human mind.
This Method being only an immense extension of the general trancendental analysis, there is no need of proof that it admits of being considered under the different primary points of view allowed by the calculus of indirect functions, as a whole. Lagrange invented the calculus of variations in accordance with the infinitesimal conception, properly so called, and even some time before he undertook the general reconstruction of the transcendental analysis. When he had effected that important reform, he easily showed how applicable it was to the calculus of variations, which he exhibited with all suitable development, according to his theory of derived functions. But the more difficult in the use the method of variations is found to be, on account of the higher degree of abstraction of the ideas considered, the more important it is to husband the powers of our minds in its application, by adopting the most direct and rapid analytical conception, which is, as we know, that of Leibnitz. Lagrange himself therefore constantly preferred it in the important use which he made of the calculus of variations in his * Analytical Mechanics.' There is not, in fact, the slightest hesitation about this among geometers. Relation to the III the section on the Litegral Calculus, I ordinary Cal- noticed D'Alembert's creation of the Calcvlvs culns. Qj' partial differences, in which Lagrange recognized a new calculus. This new elementary idea in transcendental analysis, — the iiotion of two kinds of increments, distinct and independent of each other, which a function of two variables may receive in virtue of the change of each variable separately, — seems to me to establish a natural and necessary transition between the common infinitesimal calculus and the calculus of variations. D'Alembert's view appears to me to approximate, by its nature, very nearly to that which serves as a general basis for the Method of Variations. This last has, in fact, done nothing more than transfer to the independent variables themselves the view already adopted for the functions of those variables; a process which has remarkably extended its lase. A recognition of sucli a derivation as this for the method of variations may exhibit its i^hilosophical character CONSIDERATIONS ON THE METHOD OF VARIATIONS. 91 more clearly and simply; and this is my reason for the reference.
The Method of Variations presents itself to us as the highest degree of perfection which the analysis of indirect functions has yet attained. We had before, in that analysis, a powerful instrument for the mathematical study of natural phenomena, inasmuch as it introduced the consideration of auxiliary magnitudes, so chosen as that their relations were necessarily more simple and easy to obtain than those of the direct magnitudes. But we had not any general and abstract rules for the formation of these differential equations; nor were such supposed to be possible. Now, the Analysis of Variations brings the actual establishment of the differential equations within the reach of the Calculus; for such is the general effect, in a great number of important and difficult questions, of the varied equations, which, still more indirect than the simple differential equations, as regards the special objects of the inquiry, are more easy to form: and, by invariable and complete analytical methods, employed to eliminate the new order of auxiliary infinitesimals introduced, we may deduce those ordinary differential equations which we might not have been able to establish dii'ectly. The Method of Variations forms, then, the most sublime part of that vast system of mathematical analysis, which, setting out from the simplest elements of algebra, organizes, by an uninterrujited succession of ideas, genei'al methods more and more potent for the investigation of natural j)hilosophv. This is incomparably the noblest and most unquestionable testimony to the scope of the human intellect. If, at the same time, we bear in mind that the employment of this method exacts 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 I'estingplaces, and in which signs are of scarcely any assistance, we shall understand how it may be that so little use has been made of such a conception by any philosophers but Lagrange.
We have now reviewed Mathematical analysis, in its bases and in its divisions, very briefly, but from a philosophical point of view, neglecting those conceptions only which are not organized with the great whole, or which, if nrged to their limit, would be found to merge in some which have been examined. I must next offer a similar outline of Concrete Mathematics. My particular task will 1)6 to show how, — supposing the general science of the Calculus to be in a perfect state, — it has been possible to reduce, by invariable procedures, to pure questions of analysis, all the problems of Geometry and Mechanics; and thus to invest these two great bases of natural philosophy with that precision and unity which can only thus be attained, and which constitute high perfection.
W GENERAL VIEW OF GEOMETRY.
E have seen that Geometry is a true ^ natural science; — only more simple, ' ^^* and therefore more perfect than any other. We must not suppose that, because it admits the application of mathematical analysis, it is therefore a purely logical science, independent of observation. Every body studied by geometers presents some primitive phenomena which, not being discoverable by reasoning, must be due to observation alone.
The scientific eminence of Geometry arises from the extreme generality and simplicity of its phenomena. If all the parts of the universe wei'e regarded as immovable, geometry would still exist; whereas, for the j^henomena of Mechanics, motion is required. Thus Geometry is the more general of the two. It is also the more simple, for its phenomena are independent of those of Mechanics, while mechanical phenomena are always complicated with those of geometry. The same is true in the comparison of abstract thermology with geometry. For these reasons, geometry holds the first place under the head of Conci'ete Mathematics.
Instead of adopting the inadequate ordi- d r >• nary account of Geometry, that it is the science of extension, I am disposed to give, as a general description of it, that it is the science of the measurement of extension. Even this does not include all the operations of geometry, for there are many investigations which do not appear to have for their object the measurement of extension. But regarding the science in its leading questions as a whole, we may accurately say that the measurement of lines, of surfaces, and of volumes, is the invariable aim, — sometimes direct, though ofteiier indirect, — of geometrical labours.
T, r o The rational study of geometry could never have begun if we must have regarded at once and together all the physical properties of bodies, together with their magnitude and form. By the character of our minds Ave are- able to think of the dimensions and figure of a body in an abstract way. After observation has shown us, for instance, the impi'ession left by a body on a fluid in which it has been placed, we are able to retain an image of the impression, which becomes a ground of geometrical reasoning. We thus obtain, — apart from all metaphysical fancies, — an idea of Space. This abstraction, now so familiar to us that we cannot conceive the state we should be in withoiat it, is perhaps the earliest ])hilosophical creation of the human mind.
,..,. There is another abstraction which must tension "^ made beiore we can enter on geometrical science. We must conceive of three kinds of extension, and leani to conceive of them separately. We cannot conceive of any space, filled by any object, which has not at once volume, surface, and Ime. Yet geometrical ([uestions often relate to only two of these; frequently only to one. Even when all three are to be finally considered, it is often necessary, in order to avoid complication, to take only one at a time. This is the second abstraction which it is indispensable for us to practise, — to think of surface and line apart from volume; and again, of line apart from surface. We effect this by thinking of volume as becoming thinner and thinner, till surface appears as the thinnest possible layer or film: and again, we think of this surface becoming narrower and narrower till it is reduced to the finest imaginable thread; and then we have the idea of a line. Though we cannot speak of a point as a dimension, we must have the abstract idea of that too; and it is obtained by reducing the line from one end or both, till the smallest conceivable portion of it is left. This point indicates, not extension, of course, but position, or the place of extension. Surfaces have clearly the property of circumscribing volumes; lines, again, circumscribe surfaces; and lines, once more, are limited by points.
METHODS OF MEASUREMENT. 95 The Mathematical meaning of measurevient is simply the finding of the value of measurement^ the ratios between any homogeneous magnitudes: but geometrically, the measurement is always indirect. The comparison of two lines is direct; that of two surfaces or two volumes can never be direct. One line may be conceived to be laid upon another: but one volume cannot be conceived of as laid upon another, nor one surface upon another, with any convenience or exactness. The question is, then, how to measure surfaces and volumes.
Whatever be the form of a body, there Measurement must always be lines, the length of which of surfaces and will define the magnitude of the surface or volumes, volume. It is the business of geometry to use these lines, directly measurable as they are, for the ascertainment of the ratio of the surface to the unity of surface, or of the volume to the unity of volume, as either may be sought. In brief, the object is to reduce all comparisons of surfaces or of volumes to simple comparisons of lines. Extending the process, we find the possibility of reducing to questions of lines all questions relating to surfaces and volumes, regarded in relation to their magnitude. It is true that when the rational method becomes too complicated and difiicult, direct comparisons of surfaces and volumes are employed: but the procedure is not geometrical. In the same way, the consideration of weight is sometimes brought in, to determine volume, or even surface; but this device is derived from mechanics, and has nothing to do with rational geometry.
In speaking of the direct measurement of.
lines, it is clear that right lines are meant. lines" ^^ When we consider curved lines, it is evident that their ineasurement must be indirect, since we cannot conceive of curved lines being laid upon each other with any precision or certainty. The pi'ocedure is first to reduce the measurement of curved to that of right lines; and consequently to reduce to simj^le questions of right lines all questions relating to the magnitude of any curves whatever. In every curve, there always exist certain right lines, the length of which must determine that of the curve; as the length of the radius of a circle gives us that of the circumfei'ence; and again, as the length of an ellipse depends on that of its two axes.
Thus, the science of G-eometry has for its object the final reduction of the comparisons of all kinds of extent to comparisons of right lines, which alone are capable of direct comparison, and are, moreover, eminently easy to manage.
I must just notice that there is a primary distinct branch of Geometry, exclusively devoted to the right line, on account of occasionable insurmountable difficulties in making the direct comparison; its object is to determine certain right lines from others by means of the relations proper to the figures resulting from their assemblage. The importance of this is clear, as no question could be solved if the measurement of right lines, on which every other depends, were left, in any case, uncertain. The natural order of the parts of rational geometry is therefore, first the geometry of line, beginning with the right line; then the geometry of surfaces; and, finally, that of volumes.
. The field of geometrical science is absofield * l"*^^y unbounded. There may be as many questions as there are conceivable figures; and the variety of conceivable figvires is infinite. As to curved Lines, if we regard them as generated by the motion of a point governed by a certain law, we cannot limit their number, as the variety of distinct conditions is nothing short of infinite; each generating new ones, and those again others. Surfaces, again, are conceived of as motions of lines; and they not only partake of the variety of lines, but have another of their own, arising from the possible change of nature in the line. There can be nothing like this in lines, as points cannot describe a figure. Thus, there is a double set of conditions under which the figures of surfaces may vary: and Ave may say that if lines have one infinity of possible change, surfaces have two. As for Volumes, they are distingi;ished from one another only by the surfaces which bound them; so that they partake of the variety of surfaces, and need no special consideration under this head. If we add the one further remark, that sui'faces themselves furnish a new means of conceiving of new curves, as every curve may be regarded as SCOPE OF GEOMETRICAL SCIENCE. 97 The connection between abstract and con- Properties of Crete geometry is established by the study lines and surof the properties of lines and surfaces. With- faces, out multiplying in this way our means of recognition, we should not know, except by accident, how to find in nature the figure we desire to verify. Astronomy was recreated by Kepler's discovery that the ellipse was the cuiwe which the planets describe about the sun, and the satellites about their planet. This discovery could never have been made if geometers had known no more of the ellipse than as the oblique section of a circular cone by a plane. All the properties of the conic sections brought out by the speculative labours of the G-reek geometers, were needed as preparation for this discovery, that Kepler might select from them the characteristic which was the true key to the planetary orbit. In the same way, the sjiherical figure of the earth could not have been discovered if the primitive character of the sphere had been the only one known; — viz. the equidistance of all its points from an interior point. Certain properties of surfaces Avere the means used for connecting the abstract reasoning with the concrete fact. And others, again, were required to prove that tlie earth is not absolutely spherical, and how much otherwise. The pursuit of these labours does not interfere with the definition of Geometry given above, as they tend indirectly to the measurement of extension. The great body of geometrical researches relates to the properties of lines and surfaces; and the study of the proj^erties of the same figure is so extensive, that the labours of geometers for twenty centuries have not exhausted the study of conic sections. Since the time of Descartes, it has become less important; but it appears as far as ever from being fi.nished. And here opens another infinity. We had before the infinite scope of lines, and the double infinity of surfaces: and now we see that not only is the variety of figures inexhaustible, but also the diversity of the j)oints of view from which each figure may be regarded.
I. H There are two general Methods of treating Methods geometi'ieal questions. These are commonly called Synthetical Geometry and Analytical Geometry. I shall jirefer the historical titles of Geometry of the Ancients and Geometry of the Moderns. But it is, in my view, better still to call them Sj^ecial Geometry and General Geometry, by w^hich their nature is most accurately conveyed.
Special or "^^^ Calculus was not, as some suppose, ancient, and unknown to the ancients, as we perceive by general or their applications of the theory of propormodern Geo- tions. The difference between them and us me ly. ^^ ^^^ ^^ much in the instrument of deduction as in the nature of the questions considered. The ancients studied geometry with reference to the hodies under notice, or specially: the moderns study it with reference to the phenomena to be considered, or generally. The ancients extracted all they could out of one line or surface, before passing to another; and each inquiry gave little or no assistance in the next. The moderns, since Descartes, employ themselves on questions which relate to any figure whatever. They abstract, to treat by itself, every question relating to the same geometrical phenomenon, in whatever bodies it may be considered. Geometers can thus rise to the study of new geometrical conceptions, which, applied to the curves investigated by the ancients, have brought out new properties never suspected by them. The superiority of the modern method is obvious at a glance.
The time formerly spent, and the sagacity and effort employed, in the path of detail, are inconceivably economized liy the general method \ised since the great revolution Tinder Descartes. The benefit to Concrete Geometry is no less than to the Abstract; for the recognition of geometrical figures in nature was merely embarrassed by the study of lines in detail; and the application of the contemplated figure to the existing body could be only accidental, and within a limited or doubtful range: whereas, by the general method, no existing figure can escape application to its true theory, as soon as its geometrical features are ascertained. Still, the ancient method was natural; and it was necessary that it should precede the SPECIAL AND GENERAL GEOMETRY. 99 modern. The experience of tlae ancients, and the materials they accumulated by their special method, were indispensable to suggest the conception of Descartes, and to furnish a basis for the general procediu-e. It is evident that the Calculus cannot originate any science. Equations must exist as a starting-point for analytical operations. No other beginning can be made than the direct study of the object, pursued up to the point of the discovery of precise relations.
We must briefly survey the geometry of the ^, ^ r .. (TponiPii'V or ancients, in its chai-acter of an indispensable ^j^^ ancients introduction to that of the moderns. The one, special and preliminary, must have its relation made clear to the other, — the general and definitive geometry, which now constitutes the science that goes by that name.
We have seen that Geometry is a science founded upon observation, though the materials furnished by observation eive few and simple, and the structure of reasoniug erected upon them vast and complex. The only elementary materials, obtainable by direct study alone, are those wliich relate to the right line for the geometry of lines; to the quadrature of rectilinear plane areas; and to the cuhature of bodies terminated by plane faces. The beginning of geometry must be from the observation of lines, of flat surfaces angularly bounded, and of bodies which have more or less bulk, also angularly bounded. These are all; for all other figures, even the circle, and the figures belonging to it, now come under the head of analytical geometry. The three elements just mentioned allow a sufficiency of equations for the calculus to proceed upon. More are not needed; and we cannot do with less. Some have endeavoured to extend analysis so as to dispense with a portion of these facts; but to do so is merely to return to metaphysical practices, in presenting actual facts as logical abstractions. The more we perceive Geometry to be, in our day, essentially analytical, the more cai-eful we must lie not to lose sight of the basis of observation on which all geometrical science is founded. When we observe people attempting to demonstrate axioms and the like, we may avow that it is better to admit more than may be quite necessary of materials derived from observation, than to carry logical demonstration into a region where direct observation "vvill serve us better.
There are two ways of studying the right thTri'^'h/Hne ^^"® — ^^^® graphic and the algebraic. The thing to be done is to ascertain, by means of one another, the different elements of any right line whatever, so as to understand, indirectly, a right line, under any „ circumstances whatever. The way to do this solutions ^^' fi^^t' t^ study the figure, by constructing it, or otherwise directly investigating it; and then, to reason from that observation. The ancients, in the early days of the science, made great use of the graphic method, even in the form of Construction; as when Aristarchus of Samos estimated the distance of the sun and moon from the earth on a triangle constructed as nearly as possible in resemblance to the right-angled triangle formed by the three bodies at the instant when the moon is in quadrature, and when therefore an observation of the angle at the earth would define the triangle. Archimedes himself, though he was the first to introduce calculated determinations into geometry, frequently used the same means. The introduction of trigonometry lessened the practice; but did not abolish it. The Greeks and Arabians emjiloyed it still for a great number of investigations for which we now consider the use of the Calculus indispensable.
While the grajjhic or constructive method answers well when all the parts of the proposed figure lie in the same plane, it must receive additions before it can be applied to figures whose parts lie in different planes. Hence arises a new series of considerations, and different systems of Projections. Where we now employ sj^herical trigonometry, especially for problems relating to the celestial sphere, the ancients had to consider how they could replace constrnctions in relief by plane constructions. This was the object of their analemvnas, and of the other plane figures which long su])plied the place of the Calculus. They were acquainted with the elements of what we call Descriptive Geometry, though they did not conceive of it in a distinct and general manner.
Digressing here for a moment into the region of ai)pli- DESCRIPTIVE GEOMETRY. 101 cation, I may observe that Descriptive G-eo-.
metry, formed into a distinct system by Geometry^ Monge, practically meets the difficulty just stated, but does not warrant the expectations of its first admirers, that it would enlarge the domain of rational geometry. Its grand use is in its application to the industrial arts; — its few abstract problems, capable of invariable solution, relating essentially to the contacts and intersections of surfaces; so that all the geometrical questions which may arise in any of the various arts of construction, — as stone-cutting, carpentry, perspective, dialling, fortification, etc., — can always be treated as simple individual cases of a single theory, the solution being certainly obtainable through the particular circumstances of each case. This creation must be very important in the eyes of philosophers who think that all human achievement, thus far, is only a first step towai-ds a philosop>hical renovation of the labours of mankind; towards that precision and logical character which can alone ensure the future progression of all arts. Such a revolution must inevitably begin with that class of arts which bears a relation to the simplest, the most perfect, and the most ancient of the sciences. It must extend, in time, though less readily, to all other industrial operations. Monge, who understood the philosophy of the arts better than any one else, himself indeed endeavoui'ed to sketch out a philosophical system of mechanical arts, and at least succeeded iu pointing out the direction in which the object must be pursued. Of Descriptive Geometry, it may further be said that it usefully exercises the students' faculty of Imagination, — of conceiving of complicated geometrical combinations in space; and that, while it belongs to the geometry of the ancients by the character of its solutions, it approaches to the geometry of the moderns by the nature of the questions which compose it. Consisting, as we have said, of a few abstract problems, obtained through Projections, and relating to the contacts and intersections of surfaces, the invariable solutions of these problems are at once graphical, like those of the ancients, and general, like those of the modei'ns. Yet, as destined to an industrial aj^plication, Descriptive Geometry has