See a transcription for the manuscript
Overview of Justification du Calcul des infinitésimales par celuy de l’Algèbre ordinaire (1702)
There are three versions of the text we present, which we refer to as V1, V2 and V3 (respectively, LBr. 951 Bl. 14, LBr. 951 Bl. 12-13, LBr. 951 Bl. 15-16). V3 has already been edited by Gerhardt [GM IV, 104-106]. We offer a new transcription, integrating the variants from V1 and V2 which seem significant to us.
This text is to be situated among a series of responses that Leibniz formulated particularly for the attention of the members of the Académie des sciences during la querelle des infiniment petits. Indeed, barely one year earlier, in a short article in the Journal de Trévoux [Leibniz 1701b], Leibniz had explained that the practice of his calculation did not require taking the infinitely small in a rigorous way, but that it was sufficient to consider quantities as small as needed. This way of proceeding differs from the manner of the Ancients, he argued, only in the “expressions”. Using the examples illustrated by his “Lemma of Incomparables” [Leibniz 1689], he maintained that it is thus possible to consider that the ratio between a ball and the diameter of the Earth or that of the Earth and the distance from the fixed stars as infinitely small, or that the ratio between the ball and the distance from the fixed stars as infinitely infinitely small. Its development, however, based on a comparison with finite quantities, caused confusion amongst the members of the Académie des sciences, whether be they opponents and defenders of differential calculus. In particular, Varignon asked Leibniz for clarification. On 2 February 1702, Leibniz hastily wrote a response to his friend, who made it public in the Journal des Sçavans in March [Leibniz 1702]. In this response, Leibniz first tried to clarify what he meant by “to explain the infinite by the incomparable” [expliquer l’infini par l’incomparable] and what is meant by “incomparable”. This intervention, according to Varignon, calmed those attacking differential calculus, in particular Father Gouye. But Leibniz wanted to be sure that he had been understood and it was probably partly for this reason that he adopted a new strategy for his calculus and sent it on 21 April to Varignon, wishing it to be published (but this was not the case). In this later text, Leibniz intended to show that infinitesimals were not the privilege of differential calculus but had been used since algebra had been applied to geometry, so that “we found in the calculus of ordinary Algebra traces of the transcendental calculus of differences” [on trouve dans le calcul de l’Algebre ordinaire les traces du calcul transcendant des differences]. To justify this point Leibniz essentially relied on a geometric configuration, which he chose for its simplicity and in which a triangle vanishes while keeping the same shape. This example can be found almost identically (except for the notation of points) in another manuscript, written at the same period and entitled Defense du Calcul des différences (available here) (Note that Leibniz had chosen to entitle the first version V1 “Défense des infinitésimales par le calcul de l’Algèbre ordinaire suivant son usage conforme à la Loy de la continuité”).
To understand this geometrical situation using calculus in all its generality, that is to say until the triangle vanishes completely, it must be accepted that the lengths of the sides never become absolutely zero, but that ‘”at the last moment” they are infinitesimal. Thus, thanks to the Law of Continuity, they keep among themselves the essential property of magnitudes which is to have a ratio. It is the existence of this “last ratio” that allows us to understand the geometrical situation in all its generality. Conversely, Leibniz shows by a reasoning ad absurdum – which, in passing, is not conclusive – that such a conclusion would not hold considering that the sides become absolutely zero.
The interest of this text among the other texts of our section Justification of differential calculus is that Leibniz adopts a strategy of justification, different from the the call to the argument of the incomparable, and that it is based, for this purpose, on the principle of continuity applied to algebraic calculus.