Grice ed Angeli: la parola infinita

 The concept of infinitesimal was beset by controversy from its beginnings. The idea makes an early appearance in the mathematics of the Greek atomist philosopher Democritus c. 450 B.C.E., only to be banished c. 350 B.C.E. by Eudoxus in what was to become official “Euclidean” mathematics. We have noted their reappearance as indivisibles in the sixteenth and seventeenth centuries: in this form they were systematically employed by Kepler, Galileo's student Cavalieri, the Bernoulli clan, and a number of other mathematicians. It was Galileo's pupil and colleague Bonaventura Cavalieri (1598–1647) who refined the use of indivisibles into a reliable mathematical tool (see Boyer [1959]); indeed the “method of indivisibles” remains associated with his name to the present day. Cavalieri nowhere explains precisely what he understands by the word “indivisible”, but it is apparent that he conceived of a surface as composed of a multitude of equispaced parallel lines and of a volume as composed of equispaced parallel planes, these being termed the indivisibles of the surface and the volume respectively. While Cavalieri recognized that these “multitudes” of indivisibles must be unboundedly large, indeed was prepared to regard them as being actually infinite, he avoided following Galileo into ensnarement in the coils of infinity by grasping that, for the “method of indivisibles” to work, the precise “number” of indivisibles involved did not matter. Indeed, the essence of Cavalieri's method was the establishing of a correspondence between the indivisibles of two “similar” configurations, and in the cases Cavalieri considers it is evident that the correspondence is suggested on solely geometric grounds, rendering it quite independent of number. The very statement of Cavalieri's principle embodies this idea: if plane figures are included between a pair of parallel lines, and if their intercepts on any line parallel to the including lines are in a fixed ratio, then the areas of the figures are in the same ratio. (An analogous principle holds for solids.) Cavalieri's method is in essence that of reduction of dimension: solids are reduced to planes with comparable areas and planes to lines with comparable lengths. While this method suffices for the computation of areas or volumes, it cannot be applied to rectify curves, since the reduction in this case would be to points, and no meaning can be attached to the “ratio” of two points. For rectification a curve has, it was later realized, to be regarded as the sum, not of indivisibles, that is, points, but rather of infinitesimal straight lines, its microsegments.