Leibniz's proof of the impossibility of squaring the hyperbola

While the problem of squaring the circle dates back to Greek antiquity, the squaring of the hyperbola became known only during the second half of the seventeenth century, mainly in connection with the study of logarithms. By the second half of the seventeenth century, the practical significance of the problem had become evident: if one had a method of computing the areas of the hyperbolic sectors, one would also have a means of computing the logarithms of the corresponding base-segments. Brouncker, Mercator and Wallis all made efforts to solve the quadrature of an equilateral hyperbola by means of the newly introduced technique of infinite series, and, in this way, to provide a satisfactory solution to the problem of computing the logarithms of arbitrary positive quantities. In his manuscript notes from the years 1673–1676, Leibniz repeatedly stressed the continuity between the arithmetical quadrature of the hyperbola obtained by the above-mentioned authors, and his own efforts to solve the quadrature of the conic sections, culminated in the writing of his De quadratura arithmetica circuli ellipseos et hyperbolae cujus corollarium est trigonometria sine tabulis (completed in 1676). In the present talk, I shall consider specifically Leibniz’s attempt to prove the impossibility of the universal quadrature of the hyperbola in the final proposition of the treatise. This proof follows a parallel structure to that of the impossibility of the universal quadrature of the circle. But as the latter relied on the alleged impossibility of expressing the problem of dividing an angle into an increasing (prime) number of equal parts via an explicit algebraic equation of a fixed, finite degree, so the former relies on the analogous impossibility of expressing, via an analogous algebraic equation, the problem of inserting an arbitrary number of mean proportionals between two given quantities. Both the angular-division problem and the mean-proportional problem had a methodological function in order to set the limits of Cartesian geometry. While the origin of the former problem can be found in Viete’s work, the latter is not to be found in the mathematical literature of the time. In this talk I shall clarify the content and meaning of the problem, and trace it back to Descartes’ second book of the Geometrie.