The publication in 1591 of In artem analyticen isagoge by François Viète (1540–1603) constituted an important step forward in the development of a symbolic language. As Viète’s work came to prominence at the beginning of the 17th century, other authors, like Pietro Mengoli (1626/7–1686), also began to consider the utility of algebraic procedures for solving all kind of problems. Mengoli followed the algebraic research of Viète in order to construct geometry of species, Geometriae Speciosae Elementa (1659), which allowed him to use algebra in geometry in complementary ways to solve quadrature problems.

Mengoli, like Viète, considered his algebra as a technique in which symbols are used to represent not just numbers but also values of any abstract magnitudes. He dealt with species, forms, triangular tables, quasi ratios and logarithmic ratios. However, the most innovative aspect of his work was his use of letters to work directly with the algebraic expression of the geometric figure. On the one hand, he expressed a geometric figure by an algebraic expression, in which the ordinate of the curve that determines the figure related to the abscissa by means of a proportion, thus establishing the Euclidean theory of proportions as a link between algebra and geometry. On the other hand, he showed how algebraic expression could be used to construct geometrically the ordinate at any given point. This allowed him to study geometric figures via their algebraic expressions and at the same time through triangular tables and interpolated triangular tables to derive known and unknown values for the areas of a large class of geometric figures.

In my communication, I analyze Mengoli’s proof of maxima for a geometric figure in his Geometria, a proof that used Euclidean proportion theory and some properties of logarithms demonstrated by him. This work illustrates Mengoli’s mathematical ideas on the specific role of symbolic language as a means of expression and as an analytic tool.