Part 9 (1/2)
Finally, although, in the summary exposition which was the object of this chapter, I have had to exhibit the condition of extreme imperfection which still belongs to the integral calculus, the student would have a false idea of the general resources of the transcendental a.n.a.lysis if he gave that consideration too great an importance. It is with it, indeed, as with ordinary a.n.a.lysis, in which a very small amount of fundamental knowledge respecting the resolution of equations has been employed with an immense degree of utility. Little advanced as geometers really are as yet in the science of integrations, they have nevertheless obtained, from their scanty abstract conceptions, the solution of a mult.i.tude of questions of the first importance in geometry, in mechanics, in thermology, &c. The philosophical explanation of this double general fact results from the necessarily preponderating importance and grasp of _abstract_ branches of knowledge, the least of which is naturally found to correspond to a crowd of _concrete_ researches, man having no other resource for the successive extension of his intellectual means than in the consideration of ideas more and more abstract, and still positive.
In order to finish the complete exposition of the philosophical character of the transcendental a.n.a.lysis, there remains to be considered a final conception, by which the immortal Lagrange has rendered this a.n.a.lysis still better adapted to facilitate the establishment of equations in the most difficult problems, by considering a cla.s.s of equations still more _indirect_ than the ordinary differential equations. It is the _Calculus_, or, rather, the _Method of Variations_; the general appreciation of which will be our next subject.
CHAPTER V.
THE CALCULUS OF VARIATIONS.
In order to grasp with more ease the philosophical character of the _Method of Variations_, it will be well to begin by considering in a summary manner the special nature of the problems, the general resolution of which has rendered necessary the formation of this hyper-transcendental a.n.a.lysis. It is still too near its origin, and its applications have been too few, to allow us to obtain a sufficiently clear general idea of it from a purely abstract exposition of its fundamental theory.
PROBLEMS GIVING RISE TO IT.
The mathematical questions which have given birth to the _Calculus of Variations_ consist generally in the investigation of the _maxima_ and _minima_ of certain indeterminate integral formulas, which express the a.n.a.lytical law of such or such a phenomenon of geometry or mechanics, considered independently of any particular subject. Geometers for a long time designated all the questions of this character by the common name of _Isoperimetrical Problems_, which, however, is really suitable to only the smallest number of them.
_Ordinary Questions of Maxima and Minima._ In the common theory of _maxima_ and _minima_, it is proposed to discover, with reference to a given function of one or more variables, what particular values must be a.s.signed to these variables, in order that the corresponding value of the proposed function may be a _maximum_ or a _minimum_ with respect to those values which immediately precede and follow it; that is, properly speaking, we seek to know at what instant the function ceases to increase and commences to decrease, or reciprocally. The differential calculus is perfectly sufficient, as we know, for the general resolution of this cla.s.s of questions, by showing that the values of the different variables, which suit either the maximum or minimum, must always reduce to zero the different first derivatives of the given function, taken separately with reference to each independent variable, and by indicating, moreover, a suitable characteristic for distinguis.h.i.+ng the maximum from the minimum; consisting, in the case of a function of a single variable, for example, in the derived function of the second order taking a negative value for the maximum, and a positive value for the minimum. Such are the well-known fundamental conditions belonging to the greatest number of cases.
_A new Cla.s.s of Questions._ The construction of this general theory having necessarily destroyed the chief interest which questions of this kind had for geometers, they almost immediately rose to the consideration of a new order of problems, at once much more important and of much greater difficulty--those of _isoperimeters_. It is, then, no longer _the values of the variables_ belonging to the maximum or the minimum of a given function that it is required to determine. It is _the form of the function itself_ which is required to be discovered, from the condition of the maximum or of the minimum of a certain definite integral, merely indicated, which depends upon that function.
_Solid of least Resistance._ The oldest question of this nature is that of _the solid of least resistance_, treated by Newton in the second book of the Principia, in which he determines what ought to 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. But the course pursued by Newton, from the nature of his special method of transcendental a.n.a.lysis, had not a character sufficiently simple, sufficiently general, and especially sufficiently a.n.a.lytical, to attract geometers to this new order of problems. To effect this, the application of the infinitesimal method was needed; and this was done, in 1695, by John Bernouilli, in proposing the celebrated problem of the _Brachystochrone_.
This problem, which afterwards suggested such a long series of a.n.a.logous questions, consists in determining the curve which a heavy body must follow in order to descend from one point to another in the shortest possible time. Limiting the conditions to the simple fall in a vacuum, the only case which was at first considered, it is easily found that the required curve must be a reversed cycloid with a horizontal base, and with its origin at the highest point. But the question may become singularly complicated, either by taking into account the resistance of the medium, or the change in the intensity of gravity.
_Isoperimeters._ Although this new cla.s.s of problems was in the first place furnished by mechanics, it is in geometry that the princ.i.p.al investigations of this character were subsequently made. Thus it was proposed to discover which, among all the curves of the same contour traced between two given points, is that whose area is a maximum or minimum, whence has come the name of _Problem of Isoperimeters_; or it was required that the maximum or minimum should belong to the surface produced by the revolution of the required curve about an axis, or to the corresponding volume; in other cases, it was the vertical height of the center of gravity of the unknown curve, or of the surface and of the volume which it might generate, which was to become a maximum or minimum, &c. Finally, these problems were varied and complicated almost to infinity by the Bernouillis, by Taylor, and especially by Euler, before Lagrange reduced their solution to an abstract and entirely general method, the discovery of which has put a stop to the enthusiasm of geometers for such an order of inquiries. This is not the place for tracing the history of this subject. I have only enumerated some of the simplest princ.i.p.al questions, in order to render apparent the original general object of the method of variations.
_a.n.a.lytical Nature of these Problems._ We see that all these problems, considered in an a.n.a.lytical point of view, consist, by their nature, in determining what form a certain unknown function of one or more variables ought to have, in order that such or such an integral, dependent upon that function, shall have, within a.s.signed limits, a value which is a maximum or a minimum with respect to all those which it would take if the required function had any other form whatever.
Thus, for example, in the problem of the _brachystochrone_, it is well known that if _y_ = _f(z)_, _x_ = p(_z_), are the rectilinear equations of the required curve, supposing the axes of _x_ and of _y_ to be horizontal, and the axis of _z_ to be vertical, the time of the fall of a heavy body in that curve from the point whose ordinate is _z1_, to that whose ordinate is _z2_, is expressed in general terms by the definite integral
?_{_z2_}^{_z1_}v(1 + (_f'(z))_ + (p'(_z_))/(2_gz_))_dz._
It is, then, necessary to find what the two unknown functions _f_ and p must be, in order that this integral may be a minimum.
In the same way, to demand what is the curve among all plane isoperimetrical curves, which includes the greatest area, is the same thing as to propose to find, among all the functions _f(x)_ which can give a certain constant value to the integral
?_dx_v(1 + (_f'(x)_ )),
that one which renders the integral ?_f(x)dx_, taken between the same limits, a maximum. It is evidently always so in other questions of this cla.s.s.
_Methods of the older Geometers._ In the solutions which geometers before Lagrange gave of these problems, they proposed, in substance, to reduce them to the ordinary theory of maxima and minima. But the means employed to effect this transformation consisted in special simple artifices peculiar to each case, and the discovery of which did not admit of invariable and certain rules, so that every really new question constantly reproduced a.n.a.logous difficulties, without the solutions previously obtained being really of any essential aid, otherwise than by their discipline and training of the mind. In a word, this branch of mathematics presented, then, the necessary imperfection which always exists when the part common to all questions of the same cla.s.s has not yet been distinctly grasped in order to be treated in an abstract and thenceforth general manner.