The Garden of Archimedes  A Museum for Mathematics

# Squaring Methods: from antiquity to the Seventeenth Century

Works in the section

The problem of calculus of the area and volumes and the problem of constructing tangents represent the two typical debated questions, solved with the birth of calculus.

However the first problem was approached since antiquity employing the so called "method of exhaustion" with some remarkable results. This method, traditionally attributed to Eudosso and employed by Euclid, was brought to its highest refinement by Archimedes (287-212 b.C.) Evidence of this is found in the discussion of the parabola, the circle, the sphere, the cone and the cylinder in the writings On the Quadrature of the parabola, On the Measure of the circle, On the sphere and the cylinder.

The procedure through exhaustion allows to demonstrate the results with exactitude, but doesn't provide indications of the path to follow to reach them. Therefore, during the Renaissance the conviction spread that Archimedes had a secret method. This conviction was partly confirmed by the finding, which happened only in 1906, of a palinsest, containing the so called Method in the form of a letter to Erastotene.

From the 2nd half of the 16th century onwards, the problem of "worshipping" the presumed method and finding a short cut to the complications that exhaustion presents as the generality of the results increases, accompany the rediscovery and the reemplacement of the classics. The geometric and mechanical work by Archimedes, brought back to the light and studied by mathematicians, represents, more than any other, the yardstick of comparison and inspiration until the birth of calculus. The name of Archimedes, which - with Euclid's one - mathematicians refers to as assured exactitude, became associated with a variety of topics and methods that inspired by his works, engendered new results.

This is the case of the calculus of the barycentre of figures Simon Stevin (1548-1620) wrote about or of the contributions published in De centro gravitatis solidorum libri tres (1604) and in the Quadratura parabolae per simplex falsum (1606) by Luca Valerio (1552-1628), defined by Galileo "the new Archimedes of our times".

Johannes Kepler (1571-1630) ventures into a more autonomous direction of research with the Nova stereometria doliorum (1615). In relation to the practical problem of the construction of barrels, he uses infinitesimal considerations to demonstrate classical and original results.

Representing an attempt at organic and coherent exposition of a new theory we find the work by Bonaventura Cavalieri (1598-1647) Geometria indivisibilibus continuorum nova quadam ratione promota, which was printed in 1635. Here, the well known "principle of Cavalieri" can be found, as source of many subsequent applications.

The versatility of the method of indivisibles, and object of severe criticism by numerous contemporaries who felt it lacked exactitude, was supported and employed by Evangelista Torricelli (1608-1647). Confronting the classical techniques, he uses the technique of Cavalieri for the study of new curves, like the "curved" indivisibles. He calculates, for example, the volume of the hyperboloid of revolution. (On the measure of the parabola and of the hyperbolic solid with appendix on the measure of the cycloid).

Analogous subjects were dealt with in France during the same period by Pierre De Fermat, who found the quadrature of the parabolas of higher level, and Gilles Personne de Roberval who wrote a treatise entitled De indivisibilibus.

A rather daring use of the infinitesimal quantity is made in Arithmetica infinitorum by the English John Wallis (1616-1703) who got to know the geometry of indivisibles through Torricelli.

Panels of the exhibition