Abstract
Abstract<title> SUMMARY </title>The classical definitions of ratio (otherwise: λγo&sigmav;, 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.
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