Overview of Untitled, sur l’interpolation de Mengoli
This text is part of a wider investigation by Leibniz in 1679 into the question of transcendence. On this occasion, he reopened a file that he had probably closed at the end of his stay in Paris in 1676: the interpolation techniques of Wallis and Mengoli for squaring the circle.
On the front are two triangular diagrams derived from Mengoli's triangular tables, which are probably copies of the diagrams that Leibniz wrote in his 1676 notes on Mengoli's Circolo. In the left-hand margin of the triangular tables, there is a geometric description of the curves represented by each term of the triangle. In the right-hand margin, Leibniz compares the advantages of this tabular method with those of Newton's formula.
This first part is devoted to the search for a method of determining a transcendental expression for squaring the circle, based on the tabular method of Mengoli and Wallis. Leibniz notes that one of the terms of the triangle must be expressed differently, depending on whether it is considered to be the term of a diagonal or horizontal sequence in the table. For him, this can be explained by the fact that the common denominator of all the terms of the table is transcendental. This search for a transcendental expression for quadratic curves such as the circle must therefore involve thinking about exponents. Therefore, Leibniz thinks that he can express the circle with a transcendental but finite expression.
Overleaf, Leibniz emphasises the role of differential calculus in these quadratic problems. The relationship between the two linked variable abscissas and residue can be of two kinds: $r+a$ is a constant or $r-a$ is a constant. In the first case, the quadratures depend on that of the circle, while in the other they depend on that of the hyperbola. Leibniz wants to know how these two very similar types of relationship could be articulated: on the one hand, the relationships induced by the triangular tables of Mengoli or Wallis and, on the other, those deduced by the differential algorithm.
Finally, a marginal paragraph describes a geometric figure that represents the relationships obtained by applying the differential algorithm. The diagram shows that the two curves $f(a)$ and $f(r)$ are congruent, and Leibniz wanted to find a way of obtaining the elements he deduced from the diagram and its symmetry by calculation alone.
This text shows in a remarkable way how the resolutely formalist attitude of the German philosopher only implies a mistrust of geometric diagrams, and yet remains largely based on another diagrammatic practice: that of tables.