Abstract Summary
The Pythagorean Archytas at the beginning of his work on music theory (fr. 1 Huffman) established a division of the mathematical sciences in astronomy, geometry, harmonics, and arithmetic which became famous thanks to Plato’s adaptation in Republic VII. Socrates praises geometry (way over the other sciences) for its abstraction from reality and for its value as a means to comtemplate the forms, in contast with Archytas himself, who had argued for the superiority of arithmetic on the grounds of its attachment to real-world concerns, and because it allegedly contained the ultimate first principles of the other sciences (fr. 3). From Plato’s and Ptolemy’s discussion, it is clear that proportion (i.e. ratio), a basic tool of arithmetic, underlies the Archytan division of the mathematical sciences, in the sense that arithmetic and geometry are both to be seen as playing the same structural role in relation with harmonics and astronomy respectively (i.e. harmonics is to arithmetic as astronomy is to geometry). It is from this perspective, I propose, that we are to understand the intriguing statement that “our predecessors made good distinctions in the nature of wholes, and therefore they were likely to see well how things are in their parts” (Archytas fr. 1): a good harmony in the division of the sciences is the appropriate basis for the development of these very sciences. Then I will show (and attempt to interpret) that Ptolemy adapted Archytas’ concept of well-proportionate division of the sciences in two ways: first, by imitating the text of Archytas fr. 1 in the beginning of the Almagest, where he discusses his predecessors’ divisions of philosophy, including mathematics as a contributor to physics and theology in the theoretical part. Secondly, some of Ptolemy’s works, in particular his Harmonics, are divided in two sections, the last of which consisting of an “application” of one department of knowledge to another, and bearing what seems to be a fixed proportion, in length, with the main part.
