Sulla Teoria Delle Proporzioni Nel Seicento

Abstract

Abstract<title> SUMMARY </title>The classical definitions of ratio (otherwise: λγoς, proportio, respect, reason) and proportion (otherwise: αναλoγία, rationes similes, rationes aequales, rationes eaedem, proportionalitas and also proportio, analogy, proportionality), defined respectively by the third and the fifth definitions of the V Book of Euclid's Elements (see I. L. Heiberg, Elementa, vol. II, pp. 2-3), were subjected to a rigorous examination in the Seventeenth century: Among the critics and revisers of those definitions, in this paper we deal with Giovanni Alfonso Borelli and Andre Tacquet (whose definition of «equal reasons» has inspired generations of mathematicians: see Appendixes). Borelli and Tacquet devised refined procedures to figure out the «equality of reasons» by approximation.

The development of Euclidean axiomatics

“…In Book III, axiom R17 on the existence of the fourth proportional is split into two different axioms. The system of principles in Book III, completed with a set of new definitions for equiproportionality, is intended to radically reform the Eudoxian theory of proportions (see Palladino 1991; Giusti 1993). Axioms I13, I14 and I15 are extensions of the usual axioms on the straight line, while I8 is explicitly referred to Elements XI, 1, where it is implicitly assumed by Euclid; I16 is referred to Elements XI, 2.…”