Institute and Museum of History of Science, Florence, ITALY
Evangelista Torricelli
3.1 Cubature of the hyperboloid of revolution (acute hyperbolic solid) 
The geometrical method of indivisibles developed by Torricelli constituted an important innovation with respect to Cavalieri's method. "Our method [...]", Torricelli explained, "will proceed with curved indivisibles without following the example of any predecessor [...]". [Torricelli, Opere, edited by G. Loria and G. Vassura, Faenza, 1919, vol. I (1), p. 174] This innovation was used by Torricelli to demonstrate the theorem related to the "acute hyperbolic solid", which allowed him to establish the equivalence between the solid infinitely long generated by a hyperboloid rotating about its own axis, and a cylinder of finite height. [ibid., I (1), pp. 193194]. The demonstration is based on 5 lemmas: 


First lemma Given a hyperbole with asympotes AB, AC, if the figure is made to rotate about the axis AB one obtains the "acute hyperbolic solid" which is infinitely long in the direction of B (fig. 1). Consider, then, within the solid defined in this way a rectangle passing through AB, for example, the rectangle DEFG. Let AH be the semiaxis of the hyperbole. In this way, one demonstrates that the square constructed on AH has the same area as every rectangle DEFG [ibid., p. 191]. 
fig. 2 
Second lemma It is shown that all of the cylinders inscribed in the "acute hyperbolic solid" around the common axis AB (fig. 2) are "isoperimetric" (i.e. the lateral surfaces are equal) [ibid., pp. 191192]. 
Third lemma It is shown that all these isoperimetric cylinders have volume proportional to the diameter of their base [ibid., p. 192]. 

fig. 3 
Fourth lemma It is shown that the lateral surface of each cylinder GIHL fig. 3) is 1/4 of the surface of the sphere AEFC [ibid., pp. 1923].

Fifth lemma It is shown that the lateral surface of each cylinder GHIL described in the acute solid as in the previous figure, is equivalent to a circle of radius DF [ibid., p. 193].


fig. 4 
Theorem It is shown that the infinitely long solid FEBDC made up of the acute hyperbolic solid EBD and its cylinder, base FEDC, is equivalent to the cylinder ACGH of height AC (fig. 4) [ibid., pp. 193194]. 
Illustration taken from L'Oeuvre de Torricelli: science galiléenne et nouvelle géométrie, publications de la faculté des lettres et sciences humaines de Nice, diff. Les Belles Lettres, Paris, 1987 
Some observations
relating to the demonstration Torricelli's demonstration is based on the "Fifth lemma": the surface area of the side of each "inscribed cylinder", such as, for example, GIHL, is equal to the "circle of radius DF" (fig. 3). This conclusioin is decisive for "constructing" the cylinder ACGH (fig.5), which Torricelli considers as the aggregate of an infinite number of circles. In figure 5, the surface of the side of cylinder OILN is equal to the circle passing through the point I. This conclusion is true for any inscribed cylinder, to each of which corresponds a circle (of constant radius DF) passing through one of the infinite points of the segment AC. 


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