A Translation of Evangelista Torricelli's Quadratura Parabolae - Overview of the Translation

Andrew Leahy (Knox College)

Overview of the Translation

As noted above, the Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta is the second half of de Dimensione Parabolae. The original Latin version (see here or here) runs for 29 pages and consists of an introduction and Propositions 11 through 21 of de Dimensione Parabolae, interspersed with an additional 14 lemmas and several scholia and "Aliter" sections. Torricelli notes his focus at the outset of this section:

Until now, the matter about the measurement of the parabola has been related in the manner of the ancients. It remains that we should approach the same measurement of the parabola with a certain new but marvelous system--namely, by the aid of the Geometry of Indivisibles and with diverse methods in this manner.

There is no formal subdivision of the section, but the "diverse methods" that Torricelli cites can be roughly subdivided as follows:

Propositions 11-13: Solving the quadrature of the parabola through proportions from classical geometry. For Proposion 11, Torricelli uses Euclid's proportion (Euclid XII.10) between the cone and the cylinder. For Proposition 12, he uses Euclid's result (Euclid XII.2) that two circles are in the same ratio as the squares of the diameters (see Lemma 20 of Torricelli). For Proposition 13, he uses Archimedes' proportion (see [Heath, p. 43]) between the sphere and the cylinder.

Proposition 14: Solving the quadrature of the parabola using Archimedes' Equilibrium of Planes I, which determines the center of gravity of a triangle (See [Heath, p. 201]). He also makes use of a more recent result on the location of the center of equilibrium of a cone, which had been established by Federico Commandino (1506-1575) in his Liber de Centro Gravitatis Solidorum (1565).

Propositions 15-16: Solving the quadrature of the parabola by several approaches which generalize Archimedes' geometric summation method found in Quadrature of the Parabola to an infinite geometric series. This will be discussed in the section below.

Proposition 17: Solving the quadrature of the parabola using Archimedes' proportion between the first turn of a spiral and the circumscribing cylinder. Logically, this belongs with Propositions 11-13, but it requires a more general version of Cavalieri's Principle that Torricelli develops in Lemma 29.

Proposition 18-21: Solving the quadrature of the parabola using centers of gravity. Determining centers of gravity was a major area of research in the sixteenth and seventeenth centuries, and there were several major results extending the results of Archimedes (in Plane Equilibrium I and II), who determined the center of gravity of the triangle and the parabola. Proposition 18 finds the quadrature of the parabola with the center of gravity of a triangle and Proposition 19 finds the quadrature with the center of gravity of a parabola. It makes use of a very brief proof (Lemma 30) which shows how infinitesimals can demonstrate quite easily Archimedes' result on the center of gravity of a parabola found in On the Equilibrium of Planes II. Lemma 31 is the first published proof of the Pappus-Guldin Theorem, and it is used in Proposition 20 to establish a proof of the quadrature of the parabola using the Pappus-Guldin Theorem. Proposition 21 presents a quadrature of the parabola very similar in spirit to Proposition 1 of Archimedes' On The Method (which was unknown at the time), and Proposition 22 presents a quadrature proof using the center of gravity of a hemisphere, which Luca Valerio (1552-1618) had established in his work de Centro Gravitatis Solidorum libri tres (1603).