See a translation for the manuscript
Overview of Quaestio de jure negligendi quantitates infiniti parvas (> 1702)
This manuscript has been already transcribed by Pasini (App., [Pasini 1985, 40-47).
One of the main interests of Quaestio de Jure is that it contains most of the arguments put forward by Leibniz to legitimize his calculus between 1701 and 1702. Moreover, Leibniz makes an allusion to “ingeniosissimos viros” just before developing examples, which are based on series and which illustrate the computational intractability of zero and infinity when considered as absolutes. These developments are extremely close to those exchanged in October 1702 with Jenisch [LH 35, 7, 17, 2ro-5ro] and Naudé [LH 35, 7, 17, 7ro] (transcribed both in App. [Pasini 1985, 19-31] and [AIII 9, 216-226]. This provides a terminus ad quo for the dating of our text.
The main criticism of Leibniz’ calculus is the question of the elimination of infinitely small quantities of any order during the computing process. This criticism is crucial because it is intimately linked to the status of differentials and in particular to the existence of higher-order differentials. This subject, already raised during the confrontation with the Dutchman Bernard Nieuwentijt, was taken up forcefully by opponents of calculus at the Académie royale des sciences (Michel Rolle, Jean Gallois and Thomas Gouye). It is also mentioned at the beginning of the Defense du calcul des differences (available here) as one of the main criticisms addressed to Leibnizian calculus.
Quaestio is divided into two parts. In the first, Leibniz clarifies the status of differentials and their operations. In the second he shows, through an application to numerical series, the impossibility of dealing with absolute zero or absolute infinity, and then uses this last explanation as an additional argument to argue in favor of the fruitful use of infinitesimals.
Before starting these two developments, Leibniz points out that mathematics should not depend on metaphysics. It does not matter whether the infinitesimals are real or not: in practice it is enough – and it is even necessary – that they are considered and manipulated as are the imaginary roots in solving algebraic equations. This compendium of thought is legitimized by the possibility of reducing it to a demonstration ad absurdum, by substituting the idea of infinitely small quantity by that of quantity determined but less than a given assignable quantity [commutando indefinite parvam, in parvam definite, sive assignata minus].
The safeguard constituted by this compendium allows flexibility in the way of considering the infinitesimal as incomparable, so that Leibniz allows himself two analogies with finite quantities (the diameter of a grain of sand compared to the diameter of the Earth or the latter in relation to the distance from the fixed stars). Leibniz knows that these analogies are not without danger if they are introduced without precaution. Such was the case during his intervention in the Journal de Trévoux in November-December 1701 [Leibniz 1701b] which shocked both opponents and defenders of differential calculus (See also here).
The force of his argument is that it makes it possible to consider the infinitely small or infinite compared to finite quantities but also infinitely infinitely small. The Lemmas on Incomparables [Leibniz 1689] evolved over time from a simple practical prescription to a key concept: by forging a new kind of law of homogeneity, Leibniz endowed transcendental analysis with a more general rule than the one introduced by François Viète for the dimensions of finite quantities.
Leibniz evokes another utility of differentials: they make it possible to state universal propositions. In fact, this argument is not new and Leibniz here revives an idea he had developed in 1674 in “Méthode de l’universalité” [A VII, 7, 102-105]: it is possible to consider the equation of a parabola, a triangle or even a point as special cases of the general equation of a conic by introducing infinite and infinitely small quantities. But while he described this general equation as “the key of all the harmonies and differences between things” [clef de toutes les harmonies et différences de choses], Leibniz now insists on linking this kind of situation as falling under the Law of Continuity.
The perfect adequacy of infinite quantities to express continuous phenomena shows that, despite their imaginary character, they are quam maxime reales.
It is not the same for zero and absolute infinity because their analytical processing leads to difficulties, even to impossibilities. In a certain sense, the notions of absolute zero and infinity would therefore be completely useless in mathematics and would instead fall under metaphysics.