Aristotle's Theory Unity of Science

27 Genus, Abstraction, and Commensurabilityquantities as well as rational, and in order to accomplish this he had tomodify some definitions fundamentally. Take, for example,V. def. 6: Let magnitudes which have the same ratio be called proportional,where being in the same ratio is defined asV. def. 5: Magnitudes are said to be in the same ratio, the first to thesecond and the third to the fourth, when, if any equimultiples whateverbe taken of the first and third, and any equimultiples whatever of thesecond and fourth, the former equimultiples alike exceed, are alike equalto, or alike fall short of, the latter equimultiples respectively taken incorresponding order.We are instructed to multiply the first and third quantity by any equalfactor, and multiply the second and the fourth quantity by any other equalfactor. If the quantities stand in proportion, then, no matter what factorswe use, the first and third will always be likewise smaller than, equalto, or larger than the second and the fourth respectively. This definitionis expressly formulated in terms of equimultiples in order to overcomethe challenge of incommensurability, which arises in the environmentof irrational quantities. For this reason the definition requires neither arational relationship between the first and the second term, and the thirdand the fourth, nor between the two pairs of magnitudes. Compare thiswithVII. def. 20: numbers are proportional when the first is the same multipleor the same part, or the same parts of the second that the third is of thefourth,which assumes that all the members of the proportion are commensurablerational numbers.For the single theorem Aristotle is referring to, that if A:B::C:D, thenA:C::B:D, Euclid's proofs are parallel in broad outline, both depending onfinding some means of measuring the first quantity by the third. Theyaccomplish this in different ways: V. prop. 16, the general proof, usesequimultiples to establish the proportion between A and C, and Band 0,while the theorem for discrete number, VII. prop. 9, can show this moredirectly, since A is the same part of B that C is of 0 , and wholes with thesame number of parts have their parts proportional with their wholes.