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Continuation. Progress of the methods for resolving differential equations. New step in the problem of isoperimetrical figures. The integral calculus with partial differences.
THE knowledge of some general property, capable of directing the method of resolution, was still wanting to the theory of differential equations. Since the problem of catenary curves, whence this theory began to assume a solid form, a great number of differential equations of all orders had been resolved; but for each particular case a particular method was employed; and frequently the object was attained only by a kind of tentative process, which might lead us to admire the genius and sagacity of the analyst, but after all gave no opening to problems of another kind. Geometricians therefore wanted some sign, or mark, by which they might know, whether an equation were directly resolvable in the state in which it offered itself to them, or required some preparation to render it so.
That many useless attempts must be spared by such a knowledge is obvious; and the honour of having made this noble discovery for differential equations of the first order is shared between Germany and France. Euler, Fontaine, and Clairaut, each separately arrived at it about the same time, or
at least without having derived any assistance from each other. Justice however forbids me to conceal, that Euler took the first step. In his Mechanics, published in 1736, he employs an equation dependent on this theory; but the demonstration did not make appearance till the year 1740, when it was given in the Memoirs of the Academy of Petersburg for 1734. Now the researches of Fontaine and Clairaut bear date in 1739, so that these mathematicians could not have seen those of Euler.
Euler having afterward discovered the conditions, under which differential equations of higher orders are resolvable, transmitted them to Condorcet, but without adding the demonstrations. The french geometrician not only discovered these in a very direct and simple way, but gave a farther extension to the theory. This was the first essay of great talents for analysis, to which it is to be for ever regretted, that the author did not entirely devote himself, both for his own happiness, and for the advancement of science. All the World knows, that Condorcet, having involved himself in the political dissensions of the french revolution, was driven to commit suicide, as the only means of avoiding the scaffold.
The isoperimetrical problem, which had been so much agitated between James and John Bernoulli, reappeared again occasionally on the stage, in consequence either of new applications of it, or attempts to simplify it's general solutions. Among those, who turned their attention to it, Euler is principally to be distinguished. Passing over his first essays, printed among the Memoirs of the Academy of Petersburgh, I
shall come at once to his celebrated book, Methodus inveniendi Lineas curcas, maximi minimize Proprietate gaudentes, published in 1744.
In this work the author distinguishes two sorts of maxima and minima, the one absolute, the other relative. The maxima or minima are absolute, when a curve possesses, without restriction, a certain property of a maximum or minimum, among all the curves corresponding to the same absciss. Such is the curve of swiftest descent. The maxima or minima are relative, when the curve, which is to possess a certain property of a maximum or minimum, must likewise fulfil another condition; such, for instance, as of being equal in length to all the curves terminated with it by two given points; as the circle, which has the property of including the greatest area of all curves of equal circumference. The methods employed by Euler for resolving all these problems are very simple, and as general as can be required. He first invented and demonstrated a theorem of the highest importance on this subject; which was, that problems of the second class might always be reduced to the first, by multiplying the two terms expressing the two conditions of the curve by constant coefficients, adding together the two products, and supposing, that the sum of them forms a maximum or minimum. Euler's work contains a number of curious applications, in which we every where perceive the profoundest science in the calculations, and the greatest elegance in the solutions. Considered in this view, it has all the perfection possible in the present state of analysis; but the general solutions have been still farther simplified, and subjected to easy calculations,
by means of the method of variations, which Euler himself afterward adopted, and which he has developed in several particular memoirs, as well as in an appendix to the third volume of his treatise on the Integral Calculus. Finally, he has reduced this kind of calculus to the common integral calculus.
About the middle of the last century a new discovery was made in analysis, the extent and applications of which are without bounds. For this we are indebted, at least in part, to the illustrious d'Alembert, one of the men who have done most honour to France as a geometrician of the first order, and it may be added as author of the elegant preface to the Encyclopédie. I speak of that branch of the integral calculus, which is at present called the integral calculus with partial differences.
The nature of my work does not allow me here to give a very clear idea of it to the reader; accordingly I must content myself with saying, that the object of this species of calculus is, to find a function of several variable quantities, when we know the relation of the coefficients, which affect the differentials of the variable quantities of which this function is composed. Let us suppose, for instance, a differential equation of the first order, with three variable quantities: in the problems of the common integral calculus, the differential coefficients are given directly by the conditions of the question; and then the business is, to resolve the equation, either exactly, when this can be done, or by multiplying it by a factor, or by separating the indeterminate quantities, or lastly by the methods of approximation: by
any one of these means we arrive at a finite equation, which includes an arbitrary constant quantity. But if, in the differential equation proposed, the differential coefficients be originally given, the method, that must be employed to find the finite equation, belongs to the integral calculus with partial differences. This equation includes an arbitrary function of one of the three variable quantities, and may contain likewise an arbitrary constant quantity comprised in the function. There would be arbitrary functions of two variable quantities, if the primitive differential equation were of the second order. In general, the operations of the integral calculus with partial differences bring out arbitrary functions, in the same manner, and in like number, as those of the common integral calculus do arbitrary constant quantities.
Some vestiges of this new kind of calculus may be found in a paper of Euler's, already mentioned, in the Petersburgh Transactions for 1734. The work of d'Alembert, On the general Cause of Winds,' contains somewhat fuller notions of it; and this geomctrician was the first, who employed it in an explicit manner, though subjected a little too much to the common integral calculus, in the general solution of the problem of vibrating cords.
Dr. Taylor, in his Methodus Incrementorum, had determined the curve formed by a vibrating cord, stretched by a given weight, supposing, 1st, that the cord, in it's greatest excursions, moves but little out of the rectilinear direction of the axis; 2dly, that all it's points arrive at the axis in the same time.