A Translation of Torricelli's Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta

Introduction

One of the most important threads in seventeenth century mathematics was the introduction of infinitesimal methods for solving area and volume problems. Foremost among these newfound techniques was Cavalieri's Principle, a creation of Bonaventura Cavalieri (1598-1647) that used Eudoxus' theory of proportions (as found in Book V and Book VI of Euclid's Elements) to study infinitesimal slices of objects and derive a proportion between the objects themselves. Ironically, when mathematicians in the seventeenth century wanted to understand Cavalieri's method, often they didn't turn to the work of Cavalieri himself. Instead, they turned to the work of his contemporary Evangelista Torricelli (1608-1647). Torricelli's sole published work, his Opera Geometrica (1644), was a lengthy and wide-ranging tome, but one of the most important sections for understanding Cavalieri's method was de Dimensione Parabolae.

De Dimensione Parabolae consisted of two parts, both of which were devoted to alternate proofs of the classical result from Quadrature of the Parabola by Archimedes of Syracuse (287-212 BCE). The first part of de Dimensione Parabolae was Quadratura Parabolae Pluribus modis per duplicem positionem, more antiquorum, absolutae ("The Quadrature of the Parabola solved with many methods through a two-fold placing in the manner of the ancients"). As the name implies, this section was devoted to finding the area of a segment of the parabola using Archimedean techniques; i.e., the method of exhaustion. However, in the second section, Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta ("The Quadrature of the Parabola solved with many methods through the new geometry of indivisibles"), Torricelli presented 11 Propositions which gave unique and wide-ranging approaches to solving the quadrature of the parabola, all of which involve Cavalieri's method or similar infinitesimal methods. As Torricelli himself noted, in the process he demonstrated that

it is certain that this wonderful geometry is a shortcut for invention, and that it confirms countless almost inscrutable theorems with brief, direct, and affirming demonstrations, which is certainly not able to be done easily through the ancient teaching. To be sure, this is truly the Royal Road in the mathematical thorn hedges, that Cavalieri, creator of these wonderful inventions, first among everyone opened up and made public for the common good [Torricelli, p. 56].

The purpose of this paper is to give a translation of Torricelli's Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta. To put the work in context, we will first look briefly at the 17th century mathematical figures who provided a milieu for its development. Then, following an overview of the results in Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta, we will present a slightly modernized introduction to Archimedes' Quadrature of the Parabola, the work that was the focus of Torricelli's de Dimensione. With an eye toward understanding how Torricelli's work might be used in a classroom today, we also look closely at an example from the translation which, while not strictly an application of Cavalieri's method, demonstrates just how conceptually different these new infinitesimal methods were from the traditional approach presented in Quadrature of the Parabola that had been the gold standard for mathematical rigor for centuries.