 Archytas B1 is the beginning of his book on harmonics, and most of it is devoted to the basic principles of his theory of acoustics and, in particular, to his theory of pitch described in section 2.2 above. In the first five lines, however, Archytas provides a proem on the value of the sciences (mathêmata) in general. There are several important features of this proem. First, Archytas identifies a set of four sciences: astronomy, geometry, “logistic” (arithmetic) and music. B1 is thus probably the earliest text to identify the set of sciences that became known as the quadrivium in the middle ages and that constitute four of the seven liberal arts. Second, Archytas does not present this classification of sciences as his own discovery but instead begins with praise of his predecessors who have worked in these fields. Some scholars argue that, when he praises “those concerned with the sciences,” he is thinking only of the Pythagoreans (e.g., Zhmud 1997, 198 and Lasserre 1954, 36), but this is wrongly to assume that all early Greek mathematics is Pythagorean. Archytas gives no hint that he is limiting his remarks to Pythagoreans, and, in areas where we can identify those who influenced him most, these figures are not limited to Pythagoreans (e.g., Hippocrates of Chios in geometry, see section 2.1). He praises his predecessors in the sciences, because, “having discerned well about the nature of wholes, they were likely also to see well how things are in their parts” and to “have correct understanding about individual things as they are.” It is here that Archytas is putting forth his own understanding of the nature and value of the sciences; because of the brevity of the passage, much remains unclear. Archytas appears to be praising those concerned with the sciences for their discernment, their ability to make distinctions (diagignôskein). He argues that they begin by distinguishing the nature of wholes, the universal concepts of a science, and, because they do this well, they are able to understand particular objects (the parts). Archytas appears to follow exactly this procedure in his Harmonics. He begins by defining the most universal concept of the science, sound, and explains it in terms of other concepts such as impact, before going on to distinguish between audible and inaudible sounds and sounds of high and low pitch. The goal of the science is not the making of these distinctions concerning universal concepts, however, but knowledge of the true nature of individual things. Thus, Archytas’ harmonics ends with the mathematical description of the musical intervals that we hear practicing musicians use (see section 2.2 above). Astronomy will end with a mathematical description of the periods, risings and settings of the planets. One way to understand Archytas’ project is to see him as working out the program suggested by his predecessor in the Pythagorean tradition, Philolaus. One of Philolaus’ central theses was that we only gain knowledge of things insofar as we can give an account of them in terms of numbers (DK 44 B4). While Philolaus only took the first steps in this project, Archytas is much more successful in giving an account of individual things in the phenomenal world in terms of numbers, as his description of the musical intervals shows.