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mean time, our duty seems to require, that an account of the four volumes, which we possess, should no longer be withheld from the public.
Though the integral calculus, as it was left by the first inventors and their contemporaries, was a very powerful instrument of investigation, it required many improvements to fit it for extending the philosophy of Newton to its utmost limits. A brief enumeration of the principal improvements which it has actually received in the last seventy or eighty years, will very much assist us in appreciating the merit of the work which is now before us.
1. Descartes is celebrated for having applied algebra to geometry; and Euler hardly deserves. less credit for having applied the same science to trigonometry. Though we ascribe the invention of this calculus to Euler, we are aware that the first attempt toward it was made by a mathematician of far inferior note, Christian Mayer, who, in the Petersburgh Commentaries for 1727, published a paper on analytical trigonometry. In that memoir, the geometrical theorems, which serve as the basis of this new species of arithmetic, are pointed out; but the extension of the method, the introduction of a convenient notation, and of a peculiar algorithm, are the work of Euler. By means of these, the sines and cosines of arches are multiplied into one another, and raised to any power, with a simplicity unknown in any other part of algebra, being
expressed by the sines and cosines of multiple arches of one dimension only, or of no higher power than the first. It is incredible of how great advantage this method has proved in all the parts of the higher geometry, but more especially in the researches of physical astronomy. As what we observe in the heavens is nothing but angular position, so if we would compare the result of our reasonings concerning the action of the heavenly bodies, with observations made on the surface of the earth, we must express those results in terms of the angles observed, or the quantities dependent on them, such as sines, tangents, &c. It is evident that a calculus which teaches how this is to be accomplished, must be of the greatest value to the astronomer. Besides, the facility which this calculus gives to all the reasonings and computations into which it is introduced, from the elementary problems of geometry to the finding of fluents and the summing of series, makes it one of the most valuable resources in mathematical science. It is a method continually employed in the Mécanique Céleste.
2. An improvement in the integral calculus, made by D'Alembert, has doubled its power, and added to it a territory not inferior in extent to all that it before possessed. This is the method of partial differences, or, as we must call it, of partial fluxions. It was discovered by the geometer just named, when he was inquiring into the nature of
the figures successively assumed by a musical string during the time of its vibrations. When a variable quantity is a function of other two variable quantities, as the ordinates belonging to the different abscissæ in these curves must necessarily be, (for they are functions both of the abscissæ and of the time counted from the beginning of the vibrations,) it becomes convenient to consider how that quantity varies, while each of the other two varies singly, the remaining one being supposed constant. Without this simplification, it would, in most cases, be quite impossible to subject such complicated functions to any rules of reasoning whatsoever. The calculus of partial differences, therefore, is of great utility in all the more complicated problems both of pure and mixed mathematics: every thing relating to the motion of fluids that is not purely elementary, falls within its range; and in all the more difficult researches of physical astronomy, it has been introduced with great advantage. The first idea of this new method, and the first application of it, are due to D'Alembert: it is from Euler, however, that we derive the form and notation that have been generally adopted.
3. Another great addition made to the integral calculus, is the invention of Lagrange, and is known by the name of the Calculus variationum. The ordinary problems of determining the greatest and least states of a given function of one or more va
riable quantities, is easily reduced to the direct method of fluxions, or the differential calculus, and was indeed one of the first classes of questions to which those methods were applied. But when the function that is to be a maximum or a minimum, is not given in its form; or when the curve, expressing that function, is not known by any other property, but that, in certain circumstances, it is to be the greatest or least possible, the solution is infinitely more difficult, and science seems to have no hold of the question by which to reduce it to a mathematical investigation. The problem of the line of swiftest descent is of this nature; and though from some facilities which this and other particular instances afforded, they were resolved, by the ingenuity of mathematicians, before any method generally applicable to them was known, yet such a method could not but be regarded as a great desideratum in mathematical science. The genius of Euler had gone far to supply it, when Lagrange, taking a view entirely different, fell upon a method extremely convenient, and, considering the difficulty of the problem, the most simple that could be expected. The supposition it proceeds on is greatly more general than that of the fluxionary or differential calculus. In this last, the fluxions or changes of the variable quantities are restricted by certain laws. The fluxion of the ordinate, for example, has a relation to the fluxion of the abscissa
that is determined by the nature of the curve to which they both belong. But in the method of variations, the change of the ordinate may be any whatever; it may no longer be bounded by the original curve, but it may pass into another, having to the former no determinate relation. This is the calculus of Lagrange; and, though it was invented expressly with a view to the problems just mentioned, it has been found of great use in many physical questions with which those problems are not immediately connected.
4. Among the improvements of the higher geometry, besides those which, like the preceding, consisted of methods entirely new, the extension of the more ordinary methods to the integration of a vast number of formulas, the investigation of many new theorems concerning quadratures, and concerning the solution of fluxionary equations of all orders, had completely changed the appearance of the calculus; so that Newton or Leibnitz, had they returned to the world any time since the middle of the last century, would have been unable, without great study, to follow the discoveries which their disciples had made, by proceeding in the line which they themselves had pointed out. In this work, though a great number of ingenious men have been concerned, yet more is due to Euler than to any other individual. With indefatigable industry, and the resources of a most inventive mind, he devoted