"To anticipate the book’s conclusions on this issue, I believe that a very strong case can be made in Ptolemy’s favour, and I shall do my best to provide it. If it can indeed be shown that when he wrote the Harmonics, Ptolemy not only had a well honed understanding of experimental methods but was also seriously committed to their use, that fact should obviously provoke the question whether this treatise is merely a freakish anticipation of later concepts of science, or whether once these methods have been drawn to our notice, we shall be able to find convincing traces of comparable procedures in other Greek works in the ‘exact’ or ‘mathematical’ sciences. Such questions have of course been asked before; but it may be that a starting-point in the Harmonics, where the issues are brought so insistently to our attention, will place them in a fresh perspective. My business in this book is only to provide the necessary point of departure. No doubt the wider questions are the more important, but they must be reserved for a different book and probably for a different writer. Here I intend to keep the focus as sharp as possible, restricting myself to an examination of this single text, without drawing elaborate comparisons or attempting to generate large conclusions about Greek science in general.
"Ptolemy’s treatment of the strictly ‘rational’ or ‘theoretical’ phases of his enquiry also raises issues relevant to the other sciences, particularly, perhaps, to astronomy. He proposes that what the ear perceives as musically admirable relations between pitched sounds are manifestations of mathematically intelligible and elegant form; and their complex and various structures can be derived, through orderly mathematical procedures, from principles of ‘reason’ whose credentials are accessible to the mind. This is all very fine and inspiring. But given that our initial data are simply patterns of sound which are perceived as musically satisfactory, we must plainly ask, first, how we are to represent them in ways that express their mathematical form and make them amenable to ‘rational’ (that is, mathematical) manipulation.We must also ask how we are to move from our initial perceptions to a grasp on the principles which govern their orderly relationships; why it is that these rational principles and no others are the appropriate ones for the task; why it is that some mathematically describable patterns are ‘better’ and correspond to ‘finer’ musical relations than others; by what procedures well formed musical systems are to be derived from the initial principles, and why (since different methods of derivation will yield different results); and so on. All these questions have their counterparts in astronomy, at least as Ptolemy conceived it, and the answers offered in the Harmonics may shed some light on the character of his reasoning in the Syntaxis, perhaps on that of other ancient astronomers too. But those issues, once again, will not be addressed in this book.
"From time to time in the course of this study it will be necessary to examine rather closely some of the finer details of Ptolemy’s arguments, partly for the light they shed on the nature and application of his method, and partly for their own intrinsic interest. The procedure of the Harmonics depends to a high degree on rigorous reasoning, and its sophisticated intricacies can on occasion provide matter for serious philosophical reflection. I shall also suggest, on the other hand, that some of his arguments fail to pass muster by the standards he purports to accept. Some of his constructive strategies seem to break down in their applications; and his criticisms of his predecessors are sometimes more rhetorically than rationally persuasive. It is not always possible to judge whether Ptolemy is merely being too hasty, or whether on some occasions he is deliberately seeking to mislead. This book is a discussion of procedures and the principles governing them, not primarily of the substance of Ptolemy’s conclusions or those of other writers in this field; but it would hardly be possible to explore the issues at which I have gestured, at least in any depth, without introducing some musicological technicalities. I shall expound them, however, only to the extent that seems necessary for my main purposes, and a good deal of detail will be ignored. I shall also do my best to introduce them in ways that will be accessible to readers unversed in the conundrums of Greek harmonics, and digestible by those for whom the subject is not itself of special interest."