Apollonius of Perga's \textit{Conica}: text, context, subtext (Q2740903)

Apollonius' great work \textit{Conica} was written in about 200--190 BC. It consisted, according to Apollonius' own introduction, of eight Books. Of these, four have survived in Greek, and Books V--VII in an Arabic translation. There are various translations into Latin and modern languages, but up to the present there has been no English translation of Book IV. This has now been provided by M. Fried, the younger of the two authors, in the present volume. (In the sequel, A. stands for Apollonius, F\&U for the authors.) NEWLINENEWLINENEWLINEIn strong contrast to Euclid's \textit{Elements}, which permit glances at various earlier authors and stages of developments, the \textit{Conica} is a unified whole written by a great mathematician, who absorbed the works of his predecessors and transformed them, together with his own considerable contributions, into a homogeneous treatise. Therefore one has to take the text of the \textit{Conica} as the single basis for any interpretation. This is what the authors do, and they do it in the context of earlier interpretations, notably the most influential one by \textit{H. G. Zeuthen} [Die Lehre von den Kegelschnitten im Altertum. Kopenhagen: Höst \& Son (1886; JFM 17.0029.03)]. Further, they are looking for a subtext in A.'s text by trying to discover the overall plan of his work and the motivations for individual definitions, theorems and proofs. NEWLINENEWLINENEWLINEThe text. In spite of the subtitle, the text of the \textit{Conica}, except for Book IV, is not reproduced in its entirety. However, great parts of the text are quoted in their original form in the course of the discussion of various aspects of the material, so that the reader will get an adequate picture of A.'s way of thinking. This manner of thinking is geometric, and hence the diagrams are an inseparable part of the discourse. Unfortunately, it was (obviously) the publishers decision to move the diagrams away from their proper places and collect them at the ends of the respective chapters, a most lamentable distortion of the authors intentions. (In this respect, foldout diagrams, like, e.g., those in Balsam's 1861 edition, are a much more user-friendly solution.) In an ancient text, there are, naturally, many concepts unfamiliar to the modern reader, all the more so because only a few readers will be at home in the classical theory of conic sections. It would have been helpful if the authors had collected the most important concepts and presented and explained them in appropriate diagrams. For instance, the important notion of ``the figure'' (eidos) belonging to a conic can only be located with the help of the index in some footnote. NEWLINENEWLINENEWLINEThe context. ``The purpose of this book is, then, to present a new interpretation of the \textit{Conica}, an alternative to Zeuthen's reading of the text.'' (p. 1) Zeuthen transformed A.'s concepts and proofs for a great part into algebraic formulae of modern coordinate geometry and calculated in the modern algebraic way with variables where A. used proportions and the like in the Greek manner. He even asserts (p. 459 of his ``Lehre \dots '') that the first Greeks who studied conics ``did not conceive it as their problem to study plane sections of cones, but on the contrary one was looking for representations of curves that were known beforehand''. This same statement is repeated by van der Waerden (Science Awakening) in his otherwise very useful introductory chapter on Apollonius. Typical is van der Waerden's remark (echoing M. Cantor, F\&U p. 393) that ``Apollonius is expertly hiding'' his originally algebraic way of thinking and ``expressing it clumsily in geometric terms''. F\&U succeed admirably in refuting this ahistorical position. They take A.'s text and study it as it is, all the proportions and geometric manipulations as they are, and give the reader a convincing impression of the style of the \textit{Conica}. Their arguments start with showing how the very beginning of Book I sets the stage for sectioning the cone, and how the sections themselves remain the theme throughout the book until the end. The so-called symptomata, equivalent to modern equations, may be used (and are used by A.) to characterize the sections as plane curves, but they always remain subordinate to the original definition as a section of a solid. (F\&U 92-97) NEWLINENEWLINENEWLINEBy the subtext of the \textit{Conica} the authors mean -- at least as far as this reviewer is able to see -- the intentions and general considerations of A., the motivation of the organization of the work, from the sequences of theorems to the structure of the whole book. In his introductory letter to the first Book, A. says that he is in the course of reworking his first draft. However, F\&U still locate rather many inconsistencies in the positions of several theorems and obviously A.'s work is a research monograph aimed at the expert geometer, not at the beginning student. As typical for F\&U's treatment of the subtext one may take their discussion of Book VI of the \textit{Conica}, which is concerned with equality and similarity of conic sections. They compare A.'s treatment of these topics with Euclid's treatment of the same topics for polygons in his \textit{Elements}, stressing, for instance, that ``Equality and similarity represent, for Apollonius, two radically different approaches to the study of shape'' (p. 259). (Just as, one might add, for Euclid a square is not a rectangle and for A. a circle is not an ellipse.) NEWLINENEWLINENEWLINEOn Historiography. Framing the main text of the book are two additional chapters, (1) and (9), presumably mainly by the senior author, Unguru, of F\&U, on general principles of the historiography of mathematics. Ch.(1) is specifically directed against the algebraic interpretation of Greek geometrical texts, the so-called geometrical algebra. It traces the first faint origins of this concept back to Heron and many others but mainly describes (p. 40--48) the position of Zeuthen (1886) as the principal opponent in the discussion. Chapter (9) resumes the historical review, but this time more specifically discussing authors who wrote about the \textit{Conica}. It looks, so to speak, (except for one Saul turned Paul,) largely like calling up and fighting the spirits of the deceased. NEWLINENEWLINENEWLINEIntermingled in these two chapters are some highly interesting remarks about contents and form (of expression) of mathematical texts and the role of various kinds of translations (pages 35--38, 49--51, 404--406). ``The substance of the text, its content, is always embedded in a certain form, which is the idiosyncratic manner in which the content presents itself \dots It follows, then, that every mathematical text is a structural unity, which should not be broken into violently by the careless translator (interpretator) who may find its peculiar form awkward and antiquated.'' The consequence of this is that geometrical algebra, even if it is a possible interpretation of a Greek text, is in fact a wrong way of reading the text. ``It is \dots ahistorical by definition''. For a mathematician, who celebrates the discovery of the identity of objects presented originally in different forms, this sounds alien. But it is justified with respect to the exaggerations of the proponents of geometric algebra. For the historian, who is trained to look at the individuality of cultural products, it is natural. -- But even the historian of today is a human being of our times and most probably not able to look into the head of Apollonius and his contemporaries. Van der Waerden thought he could, with respect to mathematics, when he occasionally spoke about ``our colleagues from Athens''. F\&U take, with respect to these problems, a rather extreme position: ``The immunity from cultural specificity that mathematical truths command stems from the prevailing view that their outward appearance, their packaging, as it were, and their purely mathematical content, the packaged merchandise, as it were, are neutral, unrelated, and mutually independent items. It is a calamitous view and the root of all evil in the historiography of mathematics.'' (p. 405) NEWLINENEWLINENEWLINEFor this reviewer, a more moderate position would be a discussion of how far ``translations'' could go without distorting the original too much in each special case. Then let a thousand different flowers bloom in the same field. If it were not for Zeuthen and his followers, the general mathematical public of today would take little notice of Apollonius' work. Only specialists are ready to delve into the intricacies of A.'s formalism. For these latter, however, Fried and Unguru's book will for many years remain an indispensable tool for any further study of Apollonius' \textit{Conica}.