Readers intent on the issues that are central to Ptolemy's methodology can be forgiven for ignoring some of the present chapter's more detailed ruminations in favour of their own, less pernickety reading of 1.3. My remarks in this chapter amount to a partial commentary on that stretch of text, and on little else. As I remarked earlier, the phase of the investigation conducted here, which I labelled as Stage (i), is only a preliminary. It is designed to establish the proposition that pitch is a quantitative attribute of sound, and to identify the causal factors responsible for its variations. Ptolemy treats these questions as closely interconnected. The proposition about pitch cannot be established without a study of the causes; and in practice the two issues are pursued simultaneously. Both have important roles in the sequel. The first will legitimise Ptolemy's policy of expressing pitch relations as ratios of numbers, in accordance with their mathematical forms rather than with the corresponding pathē. The second will serve as a basis for correlating the one mode of description with the other; it will also provide an account of the principles underlying the construction of experimental instruments, and the groundwork for an understanding of their use. The main purposes of my project would be served well enough by a bare sketch of the arguments in this passage.

Nevertheless, the details are of some interest, and I shall spend a little time on them.