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B. Simon: Mind and Madness in Ancient Greece: the classical roots of modern psychiatry . Pp. 336. Ithaca and London: Cornell University Press, 1978. $17.50.

Though Thekla Horowitz's work begins and ends with the warning to those unversed in geometry that, according to Tzetzes, was inscribed over the doors of the Academy, it is devoid of mathematical diagrams and almost devoid of formulae, and the general reader familiar with German will have no difficulty in reading it. The subject of ratio and proportion is the main topic of current interest in the study of Greek mathematics, and the title raises expectations that are not likely to be satisfied. The book is, in fact, not so much the Gescbichte as the Ur-Gescbicbte of a mathematical term, and the real history begins only where the author breaks off. Beginning with the root of \6yo<: in the Indo-Germanic languages and its use in Homer and Hesiod, and likewise the use of hvd. both separately and in combination as h.va\oyia, the author studies the usage of Heraclitus, Parmenides, Archytas, and other natural philosophers, thinkers, and poets, and the meaing of the terms in finance and daily life, until she arrives at Plato, who receives a fairly full treatment. It is only with the school of Plato that the mathematical interest of the term gets hot. Nearly half a century after the death of Plato Euclid was still at a loss how to define \6yos and weakly asserted that it was 'a sort of relation in respect of size between two magnitudes of the same kind'; but on this basis that it was a fundamental, undefinable notion he, following in the steps of Eudoxus, erected a magnificently logical general theory of proportion. We now know that, before Eudoxus, there was a general theory of ratio and proportion based on hvdixpaifieoL^ — the taking away in turn of one number in a ratio from the other until the process terminates (in the case of commensurables) or it is realized that it will not terminate. This was equivalent to the modern theory of continued fractions. Horowitz's book, however, gives no hint of these exciting speculations. The seminal paper of O. Becker (193 3) is included in the bibliography, but as the list of works virtually stops at 1970, when this study was in its essentials presented for a doctorate, there is naturally no reference to such books as The Evolution of the Euclidean Elements by W. R. Knorr or the several papers by D. H. Fowler in which the Greek notions of ratio and proportionality are now being explored in depth.