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the aberration of different stars, and, on comparing the results with his observations, the coincidence appeared almost perfect, so that no doubt remained concerning the truth of the principle on which he had founded his calculations.
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 its great importance. 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, 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 à 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 simplieity 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 theresearches of physical astronomy. 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.
An improvement in the integral calculus, made by M. D'ALEM. BERT, 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 par. tial differences, or, as we must call it, of partial fuxions. It was diecovered 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 mixt mathe.
matics: every thing relating to the motion of fluids that is not purely clementary, 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 generally been adopted.
Another great addition made to the integral calculus, is the invention of La Grange, and is known by the name of the Calculus varia. tionum. The ordinary problems of determining the greatest and least states of a given function of one or more variable quantities, is easily reduced to the direct method of Auxions, 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 marimum 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 desideratur in mathematical science. The genius of Euler had gone far to supply it, when La Grange, 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 Auxionary or differential calculus. In this last, the fluxions or changes of the variable quantities are restricted by certain laws. The fluxion abscissa of the ordinate, for example, has a relation to the fluxion of the 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 La Grange; 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.
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 a long life entirely to the pursuits of science. Besides producing many works on all the different branches of the higher mathematics, he continued, for more than fifty years during his life, and for no less than twenty after his death to enrich the memoirs of Berlin, or of Petersburgh, with papers that bear, in every page, the marks of originality and invention. Such, indeed, has been the industry of this incomparable man, that his works, were they collected into one, notwithstanding that they are full of novelty, and are written in the most concise language by which human thought can be expressed, might vie in magnitude with the most trite and verbose compilations.
The additions we have enumerated were made to the pure mathea matics; that which we are going to mention belongs to the mixt. It is the mechanical principle, discovered by D'Alembert, which reduces every question concerning the motion of bodies, to a case of equilibrium.' It consists in this: If the motions, which the particles of a moving body, or a system of moving bodies, have at any instant, be resolved each into two, one of which is the motion which the particle had in the preceding instant, then the sum of all these third motions must be such, that they are in equilibrium with one another. Though this principle is, in fact, nothing else thar. the equality of action and reaction, properly explained, and traced into the secret process which takes place on the communication of motion, it has operated on science like one entirely new, and deserves to be considered as an important discovery. The consequence of it has been, that as the theory of equilibrium is perfectly understood, all problems whatever, concernining the motion of bodies, can be so far subjected to mathematical computation, that they can be expressed in Auxionary or differential equations, and the solution of them reduced to the integration of those equations. The full value of the proposition, however, was not understood, till La Grange published his Méchanique Analytique ; the principle is there reduced to still greater simplicity; and the connexion between the pure and the mixt mathematics, in this quarter, may be considered as complete.
Furnished with a part, or with the whole of these resoursces, according to the period at which they arose, the mathematicians who followed Newton in the career of physical astronomy, were enabled to add much to his discoveries, and at last to complete the work which he so happily began. Out of the number who embarked in this undertaking, and to whora science has many great obligations, five may be regarded as the leaders, and as distinguished above the rest, by the greatness of their achievements. These are CLAIRAUT, EULER, D'ALEMBERT, La GRANGE, and La Place. By their efforts it was found, that, at the close of the last century, there did not remain a single phenomenon in the celestial motions, that was not explained on the principle of Gravitation as was taught by Newton, that forces propagated through an elastic medium diminish as the square of the distance, whether the cause be motion, as has been recently maintained.
The work which unites the application of all these discoveries is the Traité de Méchanique Céleste of La Place. The reasoning employed is every where algebraical ; and the various parts of the higher mathematics, the integral calculus, the method of partial differences and of variations, are from the first outset introduced, whenever they can enable the author to abbreviate or to generalize his investigations. No diagrams or geometrical figures are employed; and the reader must converse with the objects presented to him by the language of arbitrary symbols alone. The perfection of Algebra tends to the banishment of diagrams, and of all reference to them. La Grange, in his treatise of Analytical Mechanics, has no reference to figures, notwithstanding the great number of mechanical problems which he resolves. The resolution of all the forces that act on any point, into three forces, in the direction of three axes at right angles to one another, enables one to express their relations very distinctly, without representing them by a figure, or expressing them by any other than algebraic symbols. Thus Algebra, which was first introduced for the mere purpose of assisting geometry, and supplying its defects, has ended, as many auxiliaries have done, with discarding that science (or at least its peculiar methods) almost entirely. In those abstruse branches of mathematics, it must be confessed, that the French ma
nematicians have for years taken the lead of the English, but it may be hoped that the recent translation of Lacroix's work on the differential calculus by Messrs. Peacock and Herschel, and the writings of Messrs. Woodhouse and Ivory, will soon render these subjects familiar in England, and that, as in many former instances, we shall greatly improve on the invention of our neighbours.
ELEMENTS OF MATHEMATICS.
DEFINITIONS. 1. ALGEBRA is the science of Analysis, which, reasoning apon quantity or number by symbols, examines, in general, all the different methods and cases that can exist in the doctrine and calculation of numbers.
2. The symbols or characters adopted are the letters of the alphabet, which are called algebraic quantities, as a, b,
Notes.-1. These letters stand for numbers, and can, therefore, be applies to any thing to which numbers can be applied.
%. They differ from figures, because euch figure expresses a determinate nomber ; but each of the letters stands for any number whatever, and, consequently, any thing which can be proved respecting any of the letters or symbols, is applicable to any number whatever.
3. Numbers are connected with the algebraic symbols in two different ways; as, 7x, or *®; signifying 7 times x, or the second power of x.
4. When the figure is put before the algebraic quantity, it is called the co-efficient, and shows how often the quantity is taken; as 30, or 7X.
5. When the figure is put at the right hand corner of the algebraic quantity, it is called an index, or exponent, and dea potes its power; as xe, a', 2.", X“, &c. where x shows the square, or second power of x; a' its cube, or third power ; In the mth power of w; 2* the nth power, &c.
Nole. The mth or nth power of any quantity is a general expression for ai y power whatever, 6. If the figure be a fraction, as at, xt, ruti
, x*, it represents the root; thus, acte denotes the square root of x; st its cube root; its mth root, * its nth root, &c.
Note. The mth or nth root of any quantity is a general expression for any root whatever.
7. Like quantities are composed of the same letters with the same indices, as a, 6a, 7ab, or 7b'x", 9box".
-8. Unlike quantities consist of different letters, or of the same letters with different indices; as a and b, or 2a and a', or 3a*b*, 1abc.