The method of Partial Diferences, one of the greatest improvements in the calculus, was the invention of D'Alem bert, who found it when he was inquiring into the figure which a musical string assumes during its vibrations. The germ of the discovery may, however, be traced to a me moir of Euler's in the Petersburg Commentaries for the year 1734 ; and to him we are also indebted for the form of the theory and its notation.
Newton, in explaining the method of fluxions, consider ed all quantities as generated by motion, a line by the mo tion of a point, a surface by the motion of a line, a solid by the motion of a surface, &c. Leibnitz, on the other hand, employed the consideration of infinitely little increments, (injiniment petite). Against both these methods, objections have been urged. It has been said, that motion is an idea quite foreign to pure mathematics, and therefore it ought not to be employed in establishing its doctrines ; and still stronger objections have been urged against the introduc tion of infinitely small quantities into mathematics. Our ingenious countryman, Mr Landen, proposed to lay aside the consideration of motion in explaining fluxions, and, in stead of the Newtonian theory, he proposed to substitute another, which he called the Residual Analysis : this was about the year 1760. His method has not been followed, but his candor in getting the better of national prejudice in favour of Newton's method, has procured him the approba tion of foreign mathematicians.
Lagrange, in the Berlin Memoirs for 1772, proposed to shew, that the theory of the development of functions into series, contained the true principles of the differential cal culus, independently of the consideration of infinitely small quantities or limits ; and he demonstrated by this theory the theorem of Taylor, which he regarded as the fundamental principle of the calculus, and which had only been demon strated by the help of the calculus itself, or else by the con sideration of infinitely little quantities. This is the view of the subject which he has given in his Theorie des Fonctions Analytiques, published in 1792, and, at a later period, in his Leyons sur le calcul des Fonctions. These works have been much and justly admired, on account of the luminous views they present of many important points in the calculus. In
explaining his method, the author has employed a new mode of notation; but although sonic of the best foreign writers have adopted his principles, they have generally adhered to the old notation, considering it more expressive and commodious than that which was proposed.
Among the British mathematicians of later times, who have cultivated this calculus, Dr Waring is conspicuous. His writings are the only mathematical works published in this country, until of late years, that have kept pace with the improvements made in this science on the conti nent. We fear they are less read than they deserve, per haps in consequence of their peculiar style, and being com posed in the Latin tongue.
In concluding this introduction, we mention, with re gret, the fact, that there is not a book in the English lan guage from which any thing like a tolerable knowledge of the fluxional or differential calculus, in its present im proved state, can be obtained. Such as wish to study this science beyond its mere elements, must have recourse to the writings of Euler, or to the French Treatises. La Croix composed a work in three quarto volumes, in which he professed to have collected, into one point of view, all the improvements contained in Euler's writings and in acade mical memoirs. A second edition, in which the work is somewhat enlarged, is now printing, and two volumes have already come to this country. There are other smaller works of great merit also in the French tongue ; these, as well as some English treatises, which our limits will not admit of our noticing more particularly, will be found in the following list of writers on this subject ; Newton, De Analysi per equationes nonzero terminorum infinitas, published in the Commercium Epistolicum in 1712, but circulated among his friends in MS. as early as 1669.
Newton, Tractatus de quadratura curvarunz, published along with his Optics in 1704 ; also together with the trea ties De Analysi, Ste. by Jones, 1711.
Newton, Princi/zia. Lib. II. Sect. II. Lem. 2. 1687. Newton, The Method of Fluxions and Infinite Series, 1736.