Up to the end of the fourth century, all our direct evidence about mathematical harmonics comes in the form of scraps. There are the fragments of Philolaus and Archytas, together with a few brief later reports on which we can reasonably rely; there are Plato's comments in the Republic and his psycho-musical construction in the Timaeus; there is a scattering of allusions and discussions in Aristotle and a couple of acid comments in Aristoxenus. Apart from the critique mounted by Theophrastus, which we shall consider in Chapter 15, there is very little else.

In earlier chapters I have tried to extract as much enlightenment from these bits and pieces as they can yield, and the amount is not negligible. But in some respects the absence of any complete treatise in the field leaves serious gaps in our knowledge. Quite apart from the loss of theories and arguments, we simply do not know what a ‘complete treatise’ of that sort would have looked like. We have nothing that unambiguously reveals the aims of such a work, the list of items that we might formulate as its table of contents, the way in which its propositions were combined with one another and integrated into a systematic whole (if indeed they were), or the style of presentation it adopted.