Overview of Untitled (Tables sur la formule de Heron)
This manuscript is dated March 1678. It is part of a corpus of texts in which Leibniz works on diophantine problems, in particular triangles in numbers, i.e. triangles whose sides all have integer ratios to each other. Here, Leibniz thus engages in a systematic study of this problem, developing a method for generating all possible triangles in numbers. However, isolating this manuscript from the arithmetical context in which it appears to be set, Leibniz's presentation of it places it more within the framework of geometry, and more particularly geodesy. In mentioning this discipline, Leibniz refers directly to Heron of Alexandria, whose famous formula for determining the area of any triangle from its three sides is being studied. Thus, this manuscript must be situated at the crossroads of research on practical mathematics and applied geometry and those interested in number theoretical problems.
The mathematical aspects of this text are very valuable, because here Leibniz pushes the method of finite differences to a level of complexity that is not found elsewhere. As a reminder, at the beginning of his stay in Paris, Leibniz developed a method for analysing progression by successive differences in the form of a
numerical table: the difference triangle. At first he saw this as a real field of theoretical research, but he quickly turned away from this position. Thus, the difference triangles quickly became an essential heuristic tool for the young philosopher, but at the same time they acquired a standard, very stable form, no longer being themselves the object of questioning but a tool for answer. This text therefore shows an extremely original practice of the triangle of differences, which has no equivalent in the corpus as far as we know, and which is extremely late, since in 1678 the practice had been stabilised for more than three years.
This text thus testifies to the fact that tabular practice retains a central place in the Leibnizian method. And despite an apparent local stabilisation of certain practices, the philosopher continues to see it as a tool that can be further developed for the progress of the art of inventing.
More concretely, after a change of variables produced by purely combinatorial reasoning, Leibniz makes a complete list of the triplets of numbers that can produce a triangle. This list depends on three parameters which are the three differences. Therefore, by listing all the areas of all the triangles, Leibniz does not obtain a classical unidimensional progression, but a quantity that progresses in three directions. Thus, the method of difference triangles, based on the relation between a term and its neighbour, no longer operates, since a term has several neighbours. Leibniz therefore develops here a three-dimensional difference triangle, which establishes the differences of terms in three steps. First the triangle table gives the difference triangles for the variable f. Then the table of members gives the triangles for the variable e and finally the table of heads gives those for the variable d. Each step uses the previous step. The
Leibniz procedure is not commutative and gives asymmetric roles to the three variables. It is therefore more accurate to say that this table has several layers than to say that it has several dimensions. Indeed, the differences are always made in a fixed order and Leibniz does not mention the fact that one can
obtain the same values by following other paths. For this reason, it would be unreasonable to see in this procedure a discrete version of partial derivatives.
Nevertheless, the values obtained in the head table do correspond to the coefficients of the partial derivatives of Heron's formula (taken with Leibniz's parameterization). Moreover, it is clear that this text shows a differential method to analyse quantities that depend on several parameters.
Finally, we add to this manuscript a fragment that is obviously related to it. The short paragraph of text in this fragment confirms Leibniz' position on the use of difference triangles. The aim is to obtain coefficients that reduce the calculation of areas to a simple sequence of addition and subtraction, in order to relieve the
mind of any calculation.