Leibniz's notation
In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively.
Related to this is the integral in which the infinitesimal increments are summed (e.g. to compute lengths, areas and volumes as sums of tiny pieces), for which Leibniz also supplied a closely related notation involving the same differentials, a notation whose efficiency proved decisive in the development of continental European mathematics.
However, in many instances, the symbol did seem to act as an actual quotient would and its usefulness kept it popular even in the face of several competing notations.
However, this requires that derivative and integral first be defined by other means, and as such expresses the self-consistency and computational efficacy of the Leibniz notation rather than giving it a new foundation.
While Newton worked with fluxions and fluents, Leibniz based his approach on generalizations of sums and differences.
[2] Leibniz was fastidious about notation, having spent years experimenting, adjusting, rejecting and corresponding with other mathematicians about them.
[5] His integral sign first appeared publicly in the article "De Geometria Recondita et analysi indivisibilium atque infinitorum" ("On a hidden geometry and analysis of indivisibles and infinites"), published in Acta Eruditorum in June 1686,[6][7] but he had been using it in private manuscripts at least since 1675.
[8][9][10] Leibniz first used dx in the article "Nova Methodus pro Maximis et Minimis" also published in Acta Eruditorum in 1684.
At the end of the 19th century, Weierstrass's followers ceased to take Leibniz's notation for derivatives and integrals literally.
Although the notation need not be taken literally, it is usually simpler than alternatives when the technique of separation of variables is used in the solution of differential equations.
[14] This notation owes its longevity to the fact that it seems to reach to the very heart of the geometrical and mechanical applications of the calculus.
[18] One reason that Leibniz's notations in calculus have endured so long is that they permit the easy recall of the appropriate formulas used for differentiation and integration.
In each of these instances the Leibniz notation for a derivative appears to act like a fraction, even though, in its modern interpretation, it isn't one.
In the 1960s, building upon earlier work by Edwin Hewitt and Jerzy Łoś, Abraham Robinson developed mathematical explanations for Leibniz's infinitesimals that were acceptable by contemporary standards of rigor, and developed nonstandard analysis based on these ideas.
From the viewpoint of nonstandard analysis, it is correct to view the integral as the standard part of such an infinite sum.
In a letter to l'Hôpital in 1693 he says:[23] One of the secrets of analysis consists in the characteristic, that is, in the art of skilful employment of the available signs, and you will observe, Sir, by the small enclosure [on determinants] that Vieta and Descartes have not known all the mysteries.
"[24] For instance, in his early works he heavily used a vinculum to indicate grouping of symbols, but later he introduced the idea of using pairs of parentheses for this purpose, thus appeasing the typesetters who no longer had to widen the spaces between lines on a page and making the pages look more attractive.
