A Translation of Evangelista Torricelli's Quadratura Parabolae - Translation Notes

Andrew Leahy (Knox College)

Translation Notes

  1. A word about typesetting: Torricelli uses a combination of italics and Roman type, depending on the type of claim he is making (e.g., "Lemma" or "Proposition"). We have followed him in our typesetting of the translation, so, e.g., a figure ABCD may sometimes be referred to as ABCD or ABCD depending on the context.
  2. A trilineum is a generalization of a triangle in which one or more of the sides is curved instead of rectilinear. The terminology goes back at least to Commandino.
  3. Here Torricelli is using Euclid XII.10, which establishes that a cone is 1/3 of the cylinder in the same base with the same height.
  4. In Proposition XXII of his Liber de Centro Gravitatis Solidorum (1565), Federico Commandino established that the center of gravity of a cone--i.e., the point at which it balances at equilibrum--lies at the point on its axis which is 1/4 of the way from the base to the vertex. In other words, the distance from the center of gravity to the vertex is three times the distance from the center of gravity to the base.
  5. Proposition 14 of Archimedes' On the Equilibrum of Planes I establishes that the center of gravity of a triangle lies at the intersection of any two median lines. It follows from this that the center of gravity lies at a point on a given median line that is 2/3 of the way from the vertex to the opposite side. A well-known property of centers of gravity is that a figure will be suspended at equilibrium from a point on its boundary if and only if the line connecting that point to the center of gravity is vertical, which means parallel to FA in this case. Since the triangle is right, FI must be twice IC by similarity.
  6. The Latin version of this sentence reads: Dico in huiusmodi flexilineo esse omnes, & singulos ad unguem terminos qui sunt in progressione proportionis AC ad DE. in infinitum continuatae. "Ad unguem" means literally "to a fingernail" or "exactly".
  7. This result is the main conclusion of Archimedes' On the Equilibrium of Planes II (see Proposition 8 of the work), which was established with much effort on Archimedes' part. Torricelli is demonstrating how much easier it is to prove this result using indivisibles.
  8. Giannantonio Rocca (1607-1656) was a mathematical correspondent of Cavalieri and Galileo, among others. His demonstration was apparently already established by 1628, two years before the publication of Guldin's result. (See p. 83 of Seventeenth-Century Indivisibles Revisited, ed by Vincent Julien.) He was instrumental in helping Cavalieri formulate a response to Guldin's criticism of his work in Cavalieri's Exercitationes. (See pp.154-155 of Infinitesimal: How a Dangerous Mathematical Theory Shapred the Modern World by Amir Alexander.) Letters between Cavalieri and Rocca can be found in Lettere d' uomini illustri del secolo XVII a Giannantonio Rocca filosofo e matematico Reggiano con alcune del Rocca a' medesimi (Moderno 1765).
  9. For a discussion of what a "compounded" ratio is, see the discussion of Euclid Definition V.9 on pages 132-133 of Heath, Thomas trans., Euclid: The Thirteen Books of the Elements, Vol. 2. Dover (1956). Also, the definition of a moment stems from Archimedes' Law of the Lever: Given two masses m and M, the masses will balance on a lever when placed distances d and D (respectively) from a fulcrum, where d and D satisfy the reciprocal proportion m/M = D/d. By cross-multiplfying, we have m d = M D. This product is called the moment of the system. For, say, the line segment DH the center of gravity is the midpoint L. Thus, the moment is (algebraically speaking) \(DH\cdot\frac{1}{2}DH=\frac{1}{2}DH^2\).
  10. When taking the ratio of the moment of DH to the moment of HF, the "1/2" terms will cancel. Hence the ratio of the moments of the line segments is the same as the ratios of the squares. Note that the conclusion of this Lemma is a version of the Pappus-Guldin Theorem.
  11. This follows from Proposition 21 of Archimedes' On Conoids and Spheroids, which states (in the Heath edition) that "Any segment of a paraboloid of revolution is half as large as the cone or segment of a cone which has the same base and the same axis", after applying Euclid XII.10.
  12. In Proposition XXXI of De Centro Gravitatis solidorum libri tres (1603), Luca Valerio established that the center of gravity of a hemisphere was at the point on the axis which is 3/8 of the way from the center to the circumference--i.e., which divides the axis in a ratio of 5 to 3.