Overview of Tabula Numerorum Arithmeticae Replicationis seu Figuratorum interpolata
This manuscript is a copy of the interpolation table found in Wallis' Arithmetica Infinitorum of 1656, at
proposition 135, p.165.
The precise dating of this manuscript is difficult, as it is undated and contains almost no references or mentions that could serve as a solid terminus a quo. There are three groups of manuscripts with which it can be associated. In the first, dated 1679, Leibniz explores the interpolation techniques of Pietro Mengoli and refers to the work of Wallis. The second, dated 1686, is concerned with formalising the Wallisian quadratures in the context of differential calculus. Finally, the third group is undated, but must be later than 1693, as they are working notes on a collection of Wallis' texts published that year. The writing and use of symbols makes this manuscript more like the 1686 and post-1693 groups than the first group of 1679. Nevertheless, the fundamental question of the expression of transcendental quantities is present in all three groups, which testifies to the non-linear character of Leibniz's thought, which weakens any dating hypothesis based solely on a genetic reconstruction of a reasoning.
This text presents two major interests. The first is the complete copy of the interpolation table. Leibniz does not leave out any terms in the table, which suggests that it is not only a matter of thinking during the concrete process of constructing the diagram but also of thinking from the completed diagram. Leibniz's main work on questions of interpolation can be found in Parisian texts or in the 1679 group. In all these texts, the interpolation tables are based on the work of Pietro Mengoli and not on Wallis. This is therefore the only Wallisian version of an interpolation table in Leibniz that we know of.
Given the difficulty of dating this manuscript, putting its contents into context is perilous. Indeed, such a work does not have the same significance if it dates from the late 1690s, when Leibniz started a correspondence with Wallis himself, or if it dates from the mid-1685s, before the trip to Italy, the writing of the Dynamica and the turning point of the 1690s with regard to Leibniz's conception of the ars Combinatoria and the Mathesis Universalis. Therefore, it is appropriate to focus on the content of the text only.
The second interest of this text lies in the fact that Leibniz confronts in it two mathematical approaches that are fundamental in his work: the formal approach and the tabular approach. When he sets out to generalise the quadrature formula from Wallis' work by a method inspired by Newton's binomial development, namely to find a formula that is infinite when the exponent is not integer, but finite when it is, Leibniz realises that he obtains different values from those obtained by Wallis' inductive interpolation process. Thus, the two approaches are not equivalent.
Nevertheless, Leibniz does not see this as a sign that the two approaches are incompatible, but rather that the solution to generalise the formula is not yet satisfactory. Leibniz's goal of achieving a correspondence between the formal properties of equations and the algorithmic processes on diagrams is particularly well illustrated in this text. Furthermore, this analysis could serve as a basis for comments on the place of harmony in the German philosopher's work.