Aristotle's Theory Unity of Science

25 Genus, Abstraction, and Commensurabilityless-developed work on proportion into two books of the Elements, book Von general magnitude and book VII specifically on number.18 As a resultwe can compare the general treatment of alternating proportion with oneof the specific treatments, and make observations regarding the appropriateper se and qua predications.We find that the error in proving a theorem on the specific level ratherthan the general is not so clearly an error in this case as it was in the caseof 2R proved of isosceles rather than of triangle. This is because there arelegitimate proofs both on the general and the specific levels, and as a resultthe predicate, proportionals alternate, belongs at both qua-levels, thoughin different ways. This case provides a good introduction to situations inwhich qua-levels of predicates are not perfectly demarcated.In his introductory definitions to the two books Euclid provides both ageneral and a specific definition for 'part':V. def. 1: A magnitude is a part of a magnitude, the less of the greater,when it measures the greater.VII. def. 3: A number is a part of a number, the less of the greater, whenit measures the greater.The term 'part' is defined in different ways in each case. If we applyAristotle's language of necessity to these definitions, the terms in thedefinition are related to the definiendum, part, by the per se (1) relationship.As a result, in V def. 1 a part is magnitudinal, and in VII def. 3 a part isnumerical. 19 It is dear, then, that the per se relationships of the generaland the specific sciences are different, and although Euclid has preservedfor us the specific science governing only discrete quantity, it is easy to seehow the definition can be modified to be appropriate for time and so on:A time is a part of a time, the less of the greater, when it measures thegreater.The subject-genus is different in each of the specific sciences, and theterms that enter into the demonstrations concerning the subject-genus18 See Heath 1956, II, 112-13; Mueller 1970a. For the general proofs of Eudoxus, seeProdus, In primum Ellclidis 67, 3- 5: 7rPWTM TWV Ka86.\ov Ka.\ov}J.EvWV 8€wpruuiTwv TO7J'.\i180S' 'YJU{'YJfT€v.19 So also V. def. 2: 'The greater is-a multiple of the less when it is measured by theless,' and VII. def. 5: 'The greater number is a multiple of the less when it is measured·by the less.' Translations from Heath 1956.