Aestimatio: Critical Reviews in the History of Science (Volume 2) 9781463222468

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AESTIMATIO Critical Reviews in the History of Science

Aestimatio Critical Reviews in the History of Science EDITOR Alan C. Bowen IRCPS CO-EDITOR Tracey E. Rihll

University of Wales, Swansea

PUBLICATIONS MANAGER Pamela A. Cooper

IRCPS

ADVISORY EDITORS Peter Barker

University of Oklahoma

Charles Burnett

The Warburg Institute

Serafina Cuomo

Birkbeck College, London

Bruce S. Eastwood

University of Kentucky

Gad Freudenthal

CNRS, Paris

Bernard R. Goldstein University of Pittsburgh Jens Høyrup

Roskilde University

Stephen Johnston

Oxford University

Richard Kieckhefer

Northwestern University

Daryn Lehoux

Queen’s University

William Newman

Indiana University

Vivian Nutton

University College London

Kim Plofker

Brown University

Eileen A. Reeves

Princeton University

Francesca Rochberg University of California, Berkeley Ken Saito

Osaka Prefecture University

Nancy Siraisi

Hunter College

John M. Steele

Brown University

Robert B. Todd

Toronto, Ontario, Canada

Christian Wildberg

Princeton University

AESTIMATIO Critical Reviews in the History of Science

Volume 2 2005 Edited by Alan C. Bowen and Tracey E. Rihll

A Publication of the Institute for Research in Classical Philosophy and Science Princeton, New Jersey

Co-Published by

C 2009 Institute for Research in Classical Philosophy and Science 

All rights reserved under International and Pan-American Copyright Conventions. No part of this volume may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without prior permission in writing from the publisher, the Institute for Research in Classical Philosophy and Science (Princeton), and the copublisher, Gorgias Press LLC. All inquiries should be addressed to the Institute for Research in Classical Philosophy and Science (Princeton). Co-Published by Gorgias Press LLC 180 Centennial Avenue Piscataway, NJ 08854 USA Internet: www.gorgiaspress.com Email: [email protected]

ISBN 978–1–60724–661–9 This volume is printed on acid-free paper that meets the American National Standard for Permanence of paper for Printed Library Materials. Printed in the United States of America

Aestimatio: Critical Reviews in the History of Science is distributed electronically free of charge at http://www.IRCPS.org/publications/aestimatio/aestimatio.htm. To receive automatic notices of reviews by email as they are posted on the Institute website, readers are invited to subscribe to Aestimatio-L by sending an email message to [email protected] containing in the body of the message the single command line ‘subscribe Aestimatio-L’. Alternatively, they may subscribe to the RSS feed on the Recent Reviews page (http://www.ircps.org/publications/ aestimatio/rreviews.htm). Though many of the reviews published in Aestimatio are solicited by special invitation, readers are welcome to volunteer to write a review by sending an email message to the Editor ([email protected]) that lists the title and author of the book they wish to review and gives a brief indication of their qualifications to undertake this review.

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . .

ix

Mauro Zonta on Maimonides: Medical Aphorisms edited by G. Bos . . . . .

1

John M. Steele on The Cults of Uruk and Babylon by M. J. H. Linssen . . . .

7

Sarah Symons on Architecture and Mathematics in Ancient Egypt by C. Rossi . . . . . . . . . . . . . . . . . . . . . . . 11 Joshua J. Reynolds on Hesiod’s Cosmos by J. S. Clay Sven Dupré on The Claude Glass by A. Maillet

. . . . . . . . . . . . . 16 . . . . . . . . . . . . 24

J. Brian Pitts on The Command of Light by G. K. Sweetnam Henry Mendell on Archimedes by S. Stein

. . . . . . . 33

. . . . . . . . . . . . . . . . 39

William Wians on The Female in Aristotle’s Biology by R. Mayhew . . . . . 42 Annette Imhausen on Ancient Mathematics by S. Cuomo . . . . . . . . . . . 49 Renzo Baldasso on Picturing Machines 1400--1700 edited by W. Lefèvre

. . . 58

Peter J. Ramberg on Methods and Styles in the Development of Chemisty by J. S. Fruton . . . . . . . . . . . . . . . . . . . . 64 Fernando Q. Gouvêa on Sherlock Holmes in Babylon edited by M. Anderson, V. Katz, and R. Wilson . . . . . . . . . . . . . . . . 67 Kim Plofker and Bernard R. Goldstein In memoriam: David Edwin Pingree (2 Jan 1933 -11 Nov 2005) . . . . . . . . . . . . . . . . . . . . 70

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C. Robert Phillips III on Heavenly Realms and Earthly Realities edited by R. S. Boustan and A. Y. Reed . . . . . . . . . . . . . . 72 Paolo Palmieri on Modelli idrostatici del moto da Aristotele a Galileo by M. Ugaglia . . . . . . . . . . . . . . . . . . . . . . 83 Noel G. Coley on Affinity, That Elusive Dream by M. G. Kim

. . . . . . . 88

Michael P. Fronda on A Critical History of Early Rome by G. Forsythe . . . . . 94 Mauro Zonta on Science in the Medieval Hebrew and Arabic Traditions by G. Freudenthal . . . . . . . . . . . . . . . . .

104

Massimo Raffa on Music and the Muses edited by P. Murray and P. Wilson . . . . . . . . . . . . . . . . . . . . .

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Ian Mueller on Simplicius: On Aristotle on the Heavens 1.5--9 translated by R. J. Hankinson . . . . . . . . . . . . . . . . .

119

Ronald Polanksy on The Midwife of Platonism by D. Sedley . . . . . . . .

127

Lee Ann Riccardi on Vitruvius by I. K. McEwen . . . . . . . . . . . . . .

135

Brooke Holmes on Lucretius on Creation and Evolution by G. Campbell . .

141

Peter Lautner on Pythagoras by C. Riedweg . . . . . . . . . . . . . .

162

Fabio Acerbi Archimedes and the Angel: Phantom Paths from Problems to Equations . . . . . . . . . . . . . . .

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George Gheverghese Joseph on Math through the Ages by W. P. Berlinghoff and F. Q. Gouvêa . . . . . . . . . . . . . . . . . . . .

226

John Peter Oleson on Frontinus by R. H. Rodgers

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. . . . . . . . . . . . .

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Hermann Hunger In memoriam: Erica Reiner (1924--2005) . . . . . . . .

236

Richard Lim on The Roman Empire at Bay, AD 180--395 by D. S. Potter

238

Books Received 2005 . . . . . . . . . . . . . . . . . . 244

Preface Aestimatio is founded on the premise that the finest reward for research and publication is constructive criticism from expert readers committed to the same enterprise. It therefore aims to provide timely assessments of books published in the history of what was called science from antiquity up to the early modern period in cultures ranging from Spain to India, and from Africa to northern Europe. By allowing reviewers the opportunity to address critically and fully both the results of recent research in the history of science and how these results are obtained, Aestimatio proposes to advance the study of pre-modern science and to support those who undertake this study. When we first began publication in 2004, the plan was to make the individual reviews in Aestimatio available primarily online as typeset files that could be read on screen in a web browser or downloaded and printed. But recently, we have arranged with Gorgias Press to publish all our annual volumes in print. We are very grateful to George Kiraz of Gorgias Press for his interest in Aestimatio and hope that this new mode of publication will enhance the utility of Aestimatio to its readers. Alan C. Bowen Tracey E. Rihll

Maimonides: Medical Aphorisms. Treatises 1--5. A Parallel ArabicEnglish Edition Edited, Translated, and Annotated by Gerrit Bos Medical Works of Moses Maimonides. Provo, UT: Brigham Young University Press, 2004. Pp. xxxii + 156 + 76 (in Arabic). ISBN 0-934893--75--6. Cloth $39.95

Reviewed by Mauro Zonta Università degli Studi di Roma “La Sapienza” [email protected] This is the second volume to appear in the new series The Medical Works of Moses Maimonides, edited by Gerrit Bos (chair of the Martin Buber Institute for Jewish Studies, University of Cologne). 1 Immediately after the publication of the present volume, there appeared a new biography of Maimonides by Herbert A. Davidson [2005]. Davidson’s work contains many innovative suggestions and conclusions about Maimonides, some of which [2005, 429--483 esp.] concern Maimonides’ work as a physician and should be taken into account here. Moses Maimonides (who, according to Davidson, was born between September 1136 and September 1138, and died on 12 December 1204) was a paramount figure in Medieval Hebrew culture: he practiced medicine full-time during the last decades of his life, while he was in Fustat (near Cairo). As Davidson argues, Maimonides, who wrote in Arabic, composed most of his medical works from 1191 onwards, when he had completed his writings as a jurist, a theologian, and a philosopher. Davidson lists ten authentic medical writings: these include short monographs on specific illnesses; systematic treatises on diet, hygiene, and pharmacology; as well as a number of commentaries on, or reworkings of, texts by Hippocrates and Galen. There are two other works of uncertain attribution on sexual intercourse and on the length of life.

1

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As I have already observed in reviewing Bos’ edition of Maimonides’ On Asthma, the dependence of Maimonides’ medical works upon ancient sources, and upon Galen in particular, is both evident and well known. It seems that Maimonides used to write down a number of texts in notebooks consisting of patchworks of quotations from Greek authorities (taken from Medieval Arabic translations of them) which were interspersed with his own brief observations usually introduced by the formula ‘Says Moses’. Among these notebooks, there are his unpublished compendia of 21 of Galen’s medical writings; and, in my opinion [see Zonta 2004b], a compendium of Aristotle’s zoological works, 2 the ascription of which to Maimonides may be confirmed by internal evidence, also belongs in this genre. Now, it should be noticed that very similar features, although not totally identical, are found in the medical aphorisms, mostly inspired by Galen’s works, which have now been published in Bos’ edition. The Medical Aphorisms, the longest of Maimonides’ medical works after the still unpublished compendia of Galen’s books, is comprised of 25 treatises of uncertain date. Bos [xx] suggests that treatises 1-24 were composed in an early period of Maimonides’ activity as physician and writer (up to 1185); Davidson [2005, 446], however, seems to suppose a later date, since he affirms that the author was working on them ‘by 1188’. In any case, treatise 25 seems to have been written in the latest period of Maimonides’ life. As affirmed by Maimonides himself in the introduction to his work [2--4], he intended to combine in this treatise short passages taken from approximately 90 of Galen’s texts. 3 These passages, which are sometimes quoted literally and sometimes paraphrased, are combined with a few quotations taken from later Arabic authors (e.g., the famous physicians Ibn Zuhr and Ibn Waf¯ıd) as well as with Maimonides’ own comments, and ordered according to their contents. In Davidson’s opinion, the Medical Aphorisms are a sort of ‘medical code’, which may be compared to the ‘law code’ Maimonides composed for Jewish religious tradition. Rather skeptically, Davidson [2005, 452] affirms that ‘despite its limitations, the Medical Aphorisms is a convenient distillation of Galen’s self-indulgently verbose writings on medicine’. In reality, however, it 2 3

This compendium, by the way, includes passages of medical interest. See pp. 131--139 to compare the list of passages from Galen’s books found in treatises 1--5 of the Medical Aphorisms.

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is an encyclopaedia of Galen’s medical thought, where each treatise deals with a part of it. Thus, treatises 1--3 cover anatomy, physiology, and pathology; treatises 4--6, symptomatology or the signs of illnesses; treatises 7--14, etiology or the causes of illnesses; treatise 15, surgery; treatise 16, gynecology; treatise 17, hygiene; treatises 18-20, diet; treatises 21--22, pharmacopoeia or the making of medicines; treatises 23--25 concern Galen’s works as such, studying some of their singularities and even criticizing some of their statements and doctrines [see Bos’ overview on p. xxi]. What is interesting about the quotations of Galen found in the Medical Aphorisms is that some of them have a remarkable utility in reconstructing some of his lost works, or in correcting misreadings found in the extant Arabic and Greek tradition of his writings. Bos, who is going to study these fragments in forthcoming articles, points out [xxi-xxii] that in treatises 7 and 16 of the Medical Aphorisms Maimonides transmits passages of two Galenic works, De motibus manifestis et obscuris and the commentary on Hippocrates’ De mulierum affectibus, which are not found in Greek; the same is true of a passage of Galen’s De instrumento odoratus [cf. 17, lines 19--25]. Notwithstanding Maimonides’ obvious dependence on Galen, it appears from some examples reported by Bos [xxii--xxv] that he proves himself to be an independent and critical physician who tries to eradicate prejudices and dictated dogmas in medicine, even if they originate with a physician as famous as Galen. [xxv] According to Bos [15], this is valid not only for the contents of treatise 25, which was especially devoted to a philosophical critique of Galen’s work, but for other treatises as well: for instance, Bos quotes treatise 1.34, in which Maimonides criticizes Galen for not giving a reason for, or resolving doubts about, voluntary actions (speaking, screaming, walking, and so forth) performed by men while they are asleep or being absentminded. There is no doubt that the success and popularity of the Medical Aphorisms in Late Medieval medicine in Europe, both among Christian and Jewish physicians, was due to Maimonides’ ability to adapt Galen’s medical thought to his readers’ requirements in didactic, concise, simple, and clear texts. As for the Latin tradition, Bos [xxv] reports the commonly accepted opinion that the Medical Aphorisms

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was translated into Latin in the 13th century; and that one version of this translation was from the hand of John of Capua, a well-known translator from Hebrew into Latin, who was active in Rome around 1300. However, it should be pointed out that, according to a very recent study of the Medieval tradition of Maimonides’ writings, the only Latin translation of the Medical Aphorisms made in the Middle Ages was not by John of Capua, but by an anonymous translator who worked in France (possibly in Paris) or in Italy around 1400 [see Hasselhoff 2004, 288--289, 327]. As for the Medieval Hebrew tradition, there were two translations: that by Zerah.ya H . en prepared in Rome in 1277, which was mostly read among Italian Jewish readers; and that by Natan ha-Me’ati, also written in Rome between 1279 and 1283, which won success in Spain and in France. (One would be led to suspect that there is a relationship between the two translations, since they were made in the same place and approximately in the same period; but so far as I know no scholar has seriously inquired into this up to now.) Gerrit Bos’ projected series of seven volumes devoted to the Medical Aphorisms (this is the first one of them) will contain the first complete critical edition of the Arabic text, and the first complete and annotated English translation based upon the original. This fact is very important, since until now the contents of the Medical Aphorisms were known only through Suessmann Muntner’s defective, non-critical edition in 1959 of Natan’s Hebrew translation, as well as through Muntner’s and Fred Rosner’s English translation of that edition in 1970 (revised by Rosner in 1989). 4 Bos’ edition is based mainly on the Arabic text found in the ms. Gotha (Erfurt-Gotha, Universitäts- und Forschungsbibliothek), Or. 1937, but the critical apparatus includes variant readings [xxix--xxxi] from the other five Arabic manuscripts held in European libraries. Here too, Bos’ scholarly competence and philological and editorial skill emerges from his work: as I have written in the case of On Asthma, ‘Bos’ work is a very valuable and indispensable tool for a better knowledge and understanding of Maimonides’ medical writings,’ and the same is true for the Medical Aphorisms. So I will conclude

4

See pp. xxviii--xxix, 79--88 for Bos’ observations on Muntner’s and Rosner’s errors in these works.

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with some circumscribed observations about the methods followed in producing this edition. Apparently, this edition does not include a detailed comparison of the Hebrew tradition (as happened in the edition of On Asthma), probably because in this case the original Arabic texts have been regarded as sufficiently well-transmitted. (The only exception is treatise 2.8, which is lost in the whole Arabic tradition and has been published by Bos [28--29] according to one manuscript of Zerah.ya’s Hebrew translation that has been compared in some points to Natan’s.) It should also be noticed that Galen’s quotations in Maimonides have been compared to the corresponding passages of the Arabic translations of Galen only in some cases (the quotations from De locis affectis, De usu partium and the commentary on Hippocrates’ De aere aquis et locis), probably because the manuscripts or critical editions of these works were easier to find. However, a comparison of all the writings of the Arabic Galen quoted by Maimonides (50 texts, mostly in manuscripts, for treatises 1--5) would have been almost impracticable; and the accurate comparison made by Bos is, in any case, a good ‘specimen’ of such an enormous work. Let us hope that in the future Bos’ edition will be supplemented by a ‘stemma’ of the mutual relationship between the Arabic manuscripts and the Medieval Latin and Hebrew translations. This would be useful for a still better understanding of the history of this important text, surely one of Maimonides’ most read writings in the Middle Ages. bibliography Bos, G. 2002. Maimonides: On Asthma. A Parallel Arabic-English Text Edited, Translated, and Annotated. Graeco-Arabic Sciences and Philosophy: Complete Medical Works of Moses Maimonides 1. Provo UT. Davidson, H. A. 2005. Moses Maimonides: The Man and His Works. Oxford/New York. Hasselhoff, G. K. 2004. Dicit Rabbi Moyses. Würzburg. Muntner, S. 1959. ed. Moses Maimonides: Aphorisms of Moses (in Hebrew). Jerusalem.

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Muntner, S. and Rosner, F. 1970. ed. and trans. Moses Maimonides: The Medical Aphorisms of Moses Maimonides. 2 vols. New York. Rosner, F. 1989. ed. and trans. Moses Maimonides: The Medical Aphorisms of Moses Maimonides. Haifa. Zonta, M. 2004a. rev. Bos 2002. Aestimatio 1:13--18. 2004b. ‘Maimonides as Zoologist? Some Remarks about a Summary of Aristotle’s Zoology Ascribed to Maimonides’. Pp. 83--94 in G. K. Hasselhoff and O. Fraisse edd. Moses Maimonides (1138--1204): His Religious, Scientific, and Philosophical Wirkungsgeschichte in Different Cultural Contexts. Würzburg.

The Cults of Uruk and Babylon: The Temple Ritual Texts as Evidence for Hellenistic Cult Practice by Marc J. H. Linssen Cuneiform Monographs 25. Leiden: Brill--Styx, 2004. Pp. xv + 343. ISBN 90--04--12402--0. Cloth ¤ 106.00

Reviewed by John M. Steele University of Durham [email protected] The study of Mesopotamian cultic practice in the Hellenistic period has largely been based upon a dozen or so temple ritual texts which prescribe the preparations and actions to be undertaken as part of the ceremonial process. However, these texts are copies of older original compositions, and therefore may not provide an accurate portrait of Hellenistic ritual practice. To address this potential problem, Marc Linssen has undertaken a detailed comparison of the activities prescribed in the temple ritual texts with information on ritual practice recorded in contemporary documents, in particular the so-called ‘astronomical diaries’ [see Sachs and Hunger 1988--1996]. In this interesting book, he has shown that, despite the changes in Mesopotamian society after the Greek conquest of Babylonia by Alexander the Great, the Babylonian cults continued to play an important and active part of life in Mesopotamia. Furthermore, the temple ritual texts apparently reflect actual cultic practice accurately. In Mesopotamian religion, many gods were represented by an anthropomorphic statue which was not only regarded as an image of the god, but also as an extension of the god’s personality, like a living being. Thus, many rituals involved the preparation of food for this living statue, the ritual clothing of the statue, and the procession of the statue into his temple. Basic rituals were performed every day; other more elaborate rituals were performed in monthly or annual cycles, or on special occasions such as during eclipses or at the rebuilding of a temple. Many of the rituals performed in Hellenistic Mesopotamia have at least a broad link to astronomy through the use of the lunisolar calendar throughout Babylonia. Perhaps the most important event C 2005 Institute for Research in Classical Philosophy and Science 

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in the cultic calendar was the New Year (ak¯ıtu) festival. In fact, there were two New Year festivals, one held in the first month Nisannu, the beginning of the civil year, the other in the seventh month Tašr¯ıtu, the beginning of the cultic year. In order to prepare for these festivals, foreknowledge of whether an intercalary month is to be inserted would be extremely helpful, as would any advance information on the day of first visibility of the lunar crescent which marked the beginning of the month. This demand of the ritual calendar may have been one of the reasons for the development and continued practice of astronomy within the temple environment. In the Hellenistic period we know, for example, that at least some of the astronomers were employed in the Esagila temple in Babylon and the Reš sanctuary in Uruk [see Rochberg 2000]. The eclipse-of-the-moon festival has a direct astronomical context. Lunar eclipses were traditionally seen as the most significant of celestial omens. In the Neo-Assyrian period, for example, we know of many occurrences of the so-called ‘substitute king (šar p¯ u hi) ritual’, whereby a substitute was placed on the throne during an ˘eclipse which portended the death of the king [see Parpola 1983, xxii--xxxii]. However, during the Hellenistic period, there was no indigenous king to be affected by this ritual. But we do find descriptions of other rituals, involving the playing of kettle-drums, the performance of lamentations, processions, and so forth, from this period. Some parts of the rituals were to be performed at different stages of the eclipse and, as Linssen and David Brown have noted in an earlier paper [Brown and Linssen 1997], there is a direct link between the terminology used to describe eclipses in astronomical texts and the terminology of the ritual texts. The first half of Linssen’s book contains a detailed description of the evidence for the various rituals attested at Babylon and Uruk in the Hellenistic period. When possible, he compares the rituals as prescribed in temple texts with references to rituals mentioned in the astronomical diaries. By and large he finds good agreement, implying that the ritual texts accurately reflect Hellenistic cultic practice. Of particular interest in this part of the book is a reconstruction of the cultic calendars of Uruk and Babylon [88--91]. It is interesting to note that no specific rituals are attested for the 30th day of the month; whether this is due to a deliberate avoidance of this day because only

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about half of the months of the lunar year will have a 30th day, or whether this is an accident of preservation, is not known, however. The remainder of the book contains new editions and translations of all known ritual texts from the Hellenistic period, including a few new examples identified by the author. Collecting all these texts in one place with consistent English translations makes this book extremely useful. Many of the texts contain colophons identifying the scribes and owners of these tablets. Several interesting conclusions concerning the intellectual activity of the scribes of the Hellenistic period can be made on the basis of this information. For example, many of the scribes are known also to have written or owned astronomical tablets, chronicle texts, and to have appeared as witnesses on legal and business documents. Anu-b¯elšunu, son of Nidintu-Anu of the Sîn-l¯eqe-unn¯ıni family, who owned the ritual tablets TU45 and TU46 (appendix F), is a particularly well known Uruk scribe who owned several astronomical tablets, including the interesting text A3405 which contains a collection of planetary and lunar phenomena calculated using the so-called ACT methods of mathematical astronomy [see Steele 2000], a mathematical text, and an illustrated astrological text [see Pearce and Doty 2000]; and he is one of the few identifiable natives for whom we possess his horoscope [see Beaulieu and Rochberg 1996]. In summary, this book makes an important contribution to our understanding of intellectual practice in Hellenistic Babylonia. It is lucidly written, carefully typeset, and an extremely useful resource for future study of this important period and region in both Mesopotamian and Classical history. bibliography Beaulieu, P.-A. and Rochberg, F. 1996. ‘The Horoscope of Anub¯elšunu’. Journal of Cuneiform Studies 48:89--94. Brown, D. and Linssen, M. J. H. 1997. ‘BM 134701=1966--10--14, 1 and the Hellenistic Period Eclipse Ritual from Uruk’. Revue d’Assyriologie 91:147--166. Parpola, S. 1983. ed. Letters from Assyrian Scholars to the Kings Esarhaddon and Assurbanipal: Part II. Commentary and Appendices. Kevelaer.

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Pearce, L. E. and Doty, L. T. 2000. ‘The Activities of Anu-b¯elšunu, Seleucid Scribe’. Pp. 331--341 in J. Marzahn and H. Neumann edd. Assyriologia et Semitica: Festschrift für Joachim Oelsner anläßich seines 65. Geburstages am 18. Februar 1977. Münster. Rochberg, F. 2000. ‘Scribes and Scholars: The t.upš¯ ar En¯ uma Anu Enlil’. Pp. 359--375 in J. Marzahn and H. Neumann edd. Assyriologia et Semitica: Festschrift für Joachim Oelsner anläßich seines 65. Geburstages am 18. Februar 1977. Münster. Sachs, A. J. and Hunger, H. 1988--1996. Astronomical Diaries and Related Texts from Babylonia: Diaries from 652 BC to 61 BC. 3 vols. Vienna. Steele, J. M. 2000. ‘A3405: An Unusual Astronomical Text from Uruk’. Archive for History of Exact Science 55:103--135.

Architecture and Mathematics in Ancient Egypt by Corinna Rossi Cambridge: Cambridge University Press, 2004. Pp. xxii + 280. ISBN 0--521--82954--2. Cloth £60.00

Reviewed by Sarah Symons University of Leicester, UK [email protected] Ancient Egyptian monuments include some of the most recognizable architectural elements in the history of design. We seem to acquire a familiarity with Egyptian style at an early age; and even people with little interest in ancient history recognize pyramids, obelisks, and temple pylons as quintessentially Egyptian. The more attuned eye, looking at tourist knick-knacks or the efforts of interior designers to produce an Egyptian-themed dining room in a suburban dwelling, can easily identify items displaying the ‘wrong’ proportions, or an intrusive Greek column, or hieroglyphs which are clearly gibberish. The West’s love affair with Egyptian style, which mushroomed after the discovery of the tomb of Tutankhamen, has produced a small but pervasive influence on today’s built environment. The style known as ‘Egyptian Revival’ influenced buildings from cinema façades to house porches (often introducing wholly fictitious elements such as coffin-shaped apertures and obelisks balanced on spheres) in homage to a civilization which was thought to embody in varying proportions the elegant, the exotic, and the enigmatic. The edifices on which these modern attempts are modelled have a particular and revered place in the canon of architectural styles, traditionally presented as the precursors to the great Greek orders. The Egyptian lotiform columns and massive pylons are often portrayed as the transition between the primitive, almost accidental, monolithic efforts of societies without the elevated talents of art and geometry, and the enlightened, graceful classical buildings with which the popular imagination associates the birth of democracy, science, and modern civilization. Egypt’s reputation for arcane knowledge and the desire of enthusiasts to place Egypt as the birthplace of proportion and spatial order are factors which, joined by scholarly interest in the early C 2005 Institute for Research in Classical Philosophy and Science 

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development of architecture and construction, have inspired vigorous investigation about the ‘rules’ on which the architecture was founded and the level of knowledge and insight that the ancient architects may have employed. Corinna Rossi’s book appears at an interesting juncture. The heady days of lone enthusiasts galloping all over Egypt measuring and hypothesizing are over; but, in the field of Egyptian architecture, the conversation about the significance of geometry within Egyptian monumental buildings in general, and a few famous individual structures in particular, is continuing. Rossi’s book reflects on the dialogue so far and offers a considered, even cautious, set of observations about the analyses put forward by her predecessors. Her approach, flavored by her background as an architect, is investigative, practical, and unemotional. The book treats buildings, blocks of stone, methods, rituals, and plans as items of evidence, viewed in a cultural context which never becomes overpoweringly symbolic: ‘one must never forget the weight of stone’, as the author warns in her conclusion. The book’s central issue, the relationship between architecture and mathematics, is discussed (following the structure of the book itself) with reference to proportion; Egyptian drawing, calculating, and religious practices; and those most mathematical of edifices, the pyramids. As the author observes in the case of the tomb KV2, for which a plan has survived, the outcome of the tension between mathematical theory, religious ideals, and construction methods seems often to have been that the final result appears to be a compromise between ritual ideas and practical considerations, which does not seem to leave room for the idea that dimensions could have been of specific, numerical interest. Part 1 discusses the search for a ‘rule’, a set of guidelines which the ancient architect or site foreman would have used to lay out and complete important buildings. The approach to identifying the rule is usually by trying to deduce it from (plans of) Egyptian structures. Rossi describes several historical attempts at finding geometrical systems based either on polygons (such as pentagons or particular types of triangles) or on series of numbers (most famously the Fibonacci series) which yield ratios or proportions for room plans, column heights

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and placing, and façades. The author details all the problems with such schemes—lack of accurate plans, questionable reconstructions of damaged or lost buildings, and the difficulties of working with tiny drawings where the width of a pencil line could lead to variations of several feet in real life—before concluding that Alexander Badawy’s identification [Badawy 1965] of a triangle with proportions 8:5 seems to have enough evidence of repeated use for Rossi to distinguish it as ‘the most successful among the geometrical constructions listed above, whereas all the others may be dismissed without losing any significant detail.’ Part 2, ‘Ancient Egyptian Sources’, moves away from analysis of modern plans in order to review the evidence contained within documents and contemporary plans for Egyptian perceptions of space and geometry. Surviving Egyptian plans are rare and fascinating documents which, while resembling only weakly the accurate and strictly planar modern architectural working drawings, contain information about dimensions, relationships between components, and aspects of both the plan and the elevations. Sources for Egyptian plans and architectural models are listed and analyzed. For example, comparison of three modern interpretations of a structure depicted on an Eighteenth Dynasty wooden board [Davies 1917] illustrates the ambiguities which the modern eye finds in the ancient style. Rossi infers techniques which the architect would have used (a cubit rod to draw straight lines) and constraints which he accepted when making the drawing (fitting it to the shape of the drawing surface). She notes that one was expected to ‘read the labels, not measure the lines’ and concludes that sources of this type ‘seem to have been just quick reminders of a few details of a building’. The details would have been supplied by ‘long consolidated building practice’, the tradition of craftsmanship and transmitted ideas which we expect from the conservative Egyptian civilization. The final section of part 2 is entitled ‘Foundation Rituals’, starting with the ‘stretching of the cord’ ceremony about which the author presents no new conclusions. The issue of astronomical alignment, which one would expect to have appeared by this point, is not discussed and is mentioned only parenthetically as a factor in establishing the outline of the new building. This is an indication of how focused the book is on the practical analysis of its central themes.

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The third and final part is devoted entirely to pyramids. Here, Rossi uses data from more than 80 pyramids for which dimensions are available or can be inferred. The data, complete with references, notes, and calculated ratios, is presented in the appendix. The range and distribution over time of the angle of slope (expressed in the Egyptian units, sekeds) used for pyramids is analyzed. Rossi suggests that some of the slopes may be related to triangles formed by using ‘Pythagorean triples’ (sets of integers a, b, and c which satisfy the relationship a2 + b2 = c2 ). The attestation of Pythagorean triples in ancient Egypt is generally accepted from the third century BC [Parker 1972] but is contentious before that date, even for the simplest triple, 3-4-5. Sekeds which can be related to triangles based on Pythagorean triples were used in the Sixth or even the Fourth Dynasty, and Rossi speculates that choices of pyramid angles lend weight to the argument that Pythagorean triples were known at that time. Rossi is herself cautious about drawing this conclusion; but it must also be noted that the way the seked unit is produced (being related to two side lengths of a triangle) will, within the range of sekeds such as we see in pyramids, produce some Pythagorean triple triangles fortuitously. Our own type of circular angular measurement would have been far less likely to result in the spontaneous appearance of Pythagorean triangles. In summary, this book provides a detailed examination of a relatively narrow subject area. There are no grand theories emerging, but instead, a reflective and practical methodology for analysis of past theories, and guidelines for thinking about present data. Rossi’s book will appeal to those looking for a coherent and reasoned explanation of controversial topics such as explicit knowledge and use in ancient times of abstract mathematical and geometrical concepts. The book also serves as an ideal counterweight to the proliferation of home-grown ‘temple theories’ available on the internet. Finally, this book should also find a readership among architects who are interested in the history of building design and construction. The section containing Egyptian plans and working drawings will in particular emphasize the antiquity and continuity of their profession. bibliography Badawy, A. 1965. Ancient Egyptian Architectural Design: A Study of the Harmonic System. Berkeley, CA.

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Davies, N. de G. 1917. ‘An Architect’s Plan from Thebes’. Journal of Egyptian Archaeology 4:194--199. Parker, R. A. 1972. Demotic Mathematical Papyri. London.

Hesiod’s Cosmos by Jenny Strauss Clay Cambridge: Cambridge University Press, 2003. Pp. xii + 202. ISBN 0-521--82392--7. Cloth $70.00

Reviewed by Joshua J. Reynolds Northwestern University [email protected] The aim of this book is to demonstrate that Hesiod’s Theogony and his Works and Days not only are self-consistent, but also present interrelated and complementary perspectives of the universe. After summarizing the main points of the author’s argument, the present review will focus on her account of Hesiod’s understanding of human knowledge. This account rests at the heart of her overall interpretation of Hesiod and has important implications for the poet’s place in early Greek intellectual history. The author admits in the introduction that Hesiod’s two poems exhibit ‘massive differences’ both in structure and content [5]. She goes on to point out that the tendency in past scholarship has been to explain these differences by means of a diachronic model, which assumes an evolution from ‘the more “traditional” Theogony to the more “individualistic” Works and Days’ [5]. Clay, on the other hand, takes what she calls a ‘synchronic view’, treating the two poems as ‘fundamentally complementary and interdependent’ [6]. Elaborating her approach, the author suggests that Hesiod intended the poems to be understood as two halves of a whole—a ‘diptych’—as he continually revised each poem in view of the other [6]. The first two chapters survey the content, structure, and movement of the Theogony and of the Works and Days. In the case of the former, Clay argues that Hesiod depicts the cosmos as ‘the product of a genealogical evolution and a process of individuation that finally leads to the formation of a stable cosmos and ultimately achieves its telos under the tutelage of Zeus’ [13]. In contrast with this ‘positive progression’, the movement of the Works and Days ends ‘on a far more pessimistic note’ [48]. This dynamic, Clay argues, involves a C 2005 Institute for Research in Classical Philosophy and Science 

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narrowing of focus as it moves not only from the larger political community to the farm, household, family, and individual human body, but also from the regular and predictable cycle of the seasons and months to the more obscure individual days [10, 48]. The author concludes that the poem exhibits a ‘progressive darkening of vision’ and describes a decrease in the human capacity for certainty [47--48]. The author’s surveys point to what she calls ‘irresolvable tensions’ between the divine and the human worlds [48]. In the subsequent chapters, she sets out to clarify these tensions by focusing on several of their specific manifestations. As part of this project, chapter 3 examines the proems of each work. Clay begins by considering the famous passage in the Theogony in which the Muses pre-authorize Hesiod’s account of the beginnings and evolution of the cosmos. In doing so, they remark rather obscurely, ‘We know how to compose many lies indistinguishable from things that are real; And we know, when we wish, to pronounce things that are true’ [58; Clay’s trans., Theog. 27--28]. The tendency in previous scholarship has been to identify an external source, such as Homer, as the target of this remark, on the grounds that Hesiod would not call into question the truth of his own message [58--59]. Clay, on the other hand, takes Hesiod’s claim here as a serious admission of the ambiguity of the Muses’ (and therefore of his own) words. Her interpretation is that Hesiod acknowledges an inability to guarantee the absolute truth of his account [63]. She explains: The unbridgeable gap between the Muses and their pupil is constituted by the difference between divine and human knowledge, more specifically, that knowledge, which is available to the gods alone, that can discern truths from falsehoods that masquerade as truths and human knowledge that cannot. [63--64] The author concludes that Hesiod introduces here an important, ‘but nonetheless qualified, skepticism’, even though he is not denying outright the veracity of poetry and human language [64]. In her analysis of the proem of the Works and Days, Clay seeks to highlight the distinctive perspective of that work. Although its proem also begins with an invocation of the Muses, it is quite brief in comparison to the invocation in the proem of the Theogony [72]. Furthermore, in the Works and Days, Hesiod does not invoke the

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Muses to authorize his knowledge of the subject matter. This is because, Clay suggests, the subject matter of the Works and Days is the realities ( τ τυμα) of human life—how to work and prosper, rather than the gods and their origins [72, 78]. Clay suggests that the different orientations of each proem show how the two poems complement one another. The proem of the Theogony, on the one hand, offers an Olympian perspective of the cosmos through the mediation of the Muses. Human beings cannot, therefore, distinguish its truths as such. The Works and Days, on the other hand, presents the human perspective of the cosmos under the order of Zeus. It may dispense with a divine intermediary, since common experience can confirm its veracity [78--80]. Chapter 4 continues the author’s examination of the particular tensions between the divine and human worlds. Here Clay is concerned with explaining how and why Hesiod’s accounts of the origins of mankind differ between the Theogony and the Works and Days. She suggests that in the myth of the five races in the latter poem the succession of ages constitutes a series of experiments to produce a race of inferior beings who would offer sacrifices and other honors to the gods [94]. Clay points out that in the Theogony human beings descend from the Giants, themselves a direct result of the blood from Uranus’ severed genitals dripping onto Gaia [97]. In this way, the poem attributes the origins of the human race to a fortuitous accident that occurred at the very beginning of the cosmos [98]. Clay explains the presence of these divergent stories of human origins in terms of the different perspectives of each poem: Olympus evidently regards mankind as a threat to divine supremacy, a threat that must be tamed and channeled into obedience; human beings look nostalgically to a golden age of happiness, which they set in an era before the reign of Zeus; over the course of time, they have become increasingly distant and subservient to the gods. [99] The author devotes chapter 5 to a discussion of Hesiod’s two versions of the Prometheus myth and the divergent perspectives on the relationship between mortals and immortals that each version offers. Clay argues that both versions represent the human condition as one of ambiguity, a mixture of good and evil, and as involving a ‘progressive estrangement of gods and men’ [101--102]. As an example, she

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points to the story of Pandora and her jar in the Works and Days. Pandora, she argues, is a ‘doublet’ of her jar: she has an external beauty that conceals the troubles within [103]. After Prometheus returns fire to mortals, Zeus substitutes Pandora for his initial attempt to take it away. He thus renders the technology a necessary evil by filling human life with the toil and misery of supporting a family [119--120]. Hope, Clay continues, is similarly ambiguous, as it ‘promises and seduces, but all too rarely delivers’ [103]. The author thus concludes that Hesiod regards Hope as an evil, ‘the ultimate kalon kakon’ which characterizes the human condition and situates it between ‘the ignorance of the beasts and the certain knowledge of the gods’ [103: cf. 124]. Clay goes on to consider the differences between the two versions of the Prometheus myth. She argues that both versions represent the history of the human race as proceeding in a negative direction, in contrast with the evolution of the gods and the ordering of the cosmos [126--127]. In the Theogony, however, the separation of mankind from the gods is the result of Zeus’ political efforts to secure his divine ascendancy. In this case, human beings are viewed externally from the divine perspective and as a potential threat to Zeus’ regime [116--117, 126--128]. In contrast, the Prometheus myth in the Works and Days represents the separation of mankind from the gods as the intended result of the gods’ need for the presence of inferior creatures to enjoy their own superiority [116]. In this way, the poem ‘presents mankind from an internal subjective human standpoint: the gods, who have deprived mankind of an earlier bliss, have filled human life with misery’ [128]. Clay concludes that ‘the full pathos of the human condition’ emerges only from combining the perspectives of both poems [128]. Chapter 6 continues the author’s overall project of demonstrating how the differences between the divine and human perspectives of Hesiod’s cosmos can be integrated into a larger whole. Here Clay elaborates the relationships between mortals and immortals by examining the role of human beings in the Theogony and the role of the gods in the Works and Days. In the former poem, Clay focuses on the ‘Hymn to Hecate’, in which humans play a prominent part. She draws attention to Hesiod’s description of Hecate’s powers over the lives of men: in particular, the goddess grants preeminence in council and victory in war to whomever she wishes; she assists kings

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and athletic competitors; and she grants success in fishing and raising sheep if she wills [133--134]. Clay observes here how Hecate’s goodwill is crucial to success, and concludes that the goddess represents the constant presence of chance in human affairs [135]. Just as in the case of Pindar’s Τ χα [Olympian 12], the arbitrary decisions of Hesiod’s Hecate explain why the gods only sometimes fulfill the prayers of men and why there is no truly reliable sign from the gods to mankind [135--137]. In this way, Clay explains, Hecate provides the ‘crucial intermediary’ between gods and men [138]. Next, the author examines the role of gods in the Works and Days by tracing the changing influence of Zeus. From the beginning of this poem, Clay argues, Zeus alone possesses the powers attributed to Hecate in the Theogony. This suggests that from the human perspective Zeus directly guides the fate of mankind [143, 149]. Clay continues by arguing that our certainty about Zeus’ power to determine (τεκμα ρεται) rewards and punishments diminishes as the poem narrows its focus [144]. In the Calendar, Zeus’ direct intervention has already begun to diminish, as the cycle of seasons functions on its own and provides mortals with useful astronomical signs [145]. Later in the poem, Zeus becomes simply the god of weather, which mortals cannot hope to foresee with the certainty that they had earlier attained in regard to the cycle of seasons and the life of justice. Clay concludes that Hesiod was ‘fully aware’ that uncertainty surrounds any human endeavor; the poem thus ends with a view of the human condition as ‘naked and vulnerable’ [148--149]. In chapter 7, Clay considers what the two types of hybrids (monsters and heroes) that violate the boundaries of Hesiod’s otherwise systematic cosmos might reveal about that system. She completes her study with a brief discussion of Hesiod’s place in the tradition of heroic epic. Clay concludes that instead of regarding his own project as a rejection of the heroic tradition, Hesiod considers it to be more universal and complete than Homer’s poetry. She explains, [h]is dual vision comprehends both the divine and the human cosmos and unites the traditions of theogonic poetry with those of ‘wisdom’ literature, the divine world of Being and the ephemeral human world of Becoming. The gulf Hesiod detects and illuminates between the divine and human perspective points forward to the philosophical endeavors of Empedocles, Parmenides, and Heraclitus. [181--182]

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In sum, Clay attributes to Hesiod a rather pessimistic outlook on the power of human knowledge. She emphasizes that Hesiod admits in the Theogony that he cannot guarantee the truth of his account, and that he thus embraces a ‘qualified skepticism’. She also attributes to the poet a view similar to the amechania of Pindar, who insists that the outcomes of human efforts depend entirely upon chance and that there is no such thing as a reliable sign (τ σαφ τ κμαρ) from the gods to mortals about the future. Clay likewise argues that in the Works and Days Hesiod characterizes the human condition by Hope and the inability to attain sure knowledge. Although Hesiod initially promises to reveal the realities ( τ τυμα) of human existence, she argues, the poem eventually gives way to uncertainty and darkness, and ends on a ‘pessimistic note’. Hesiod’s Cosmos offers a convincing and thorough explanation of how the poet’s often enigmatic claims about human knowledge can be interpreted consistently, both with one another and across each of his two poems. In fact, the book goes a long way towards showing that it is indeed meaningful to speak of such a thing as ‘Hesiod’s understanding of human knowledge’. With that said, however, Clay’s final assessment of Hesiod as ultimately skeptical and even pessimistic seems extreme. This is not to suggest that Hesiod believes that human beings are capable of divine knowledge, or that he denies that most cases of death, disease, and suffering are unpredictable. But it is clear that he tends to focus more on what human beings can know than what they cannot. For instance, it seems reasonable to say that the lesson of the Works and Days is that all sorts of important outcomes are gained not by chance, but as the result of a human agent’s understanding of a given situation. Whether or not the gods were responsible for establishing the system whereby the agent’s actions naturally led to the expected result is not relevant. The point, rather, is Hesiod’s insistence that human beings can in fact begin to understand that system and use their knowledge to their own advantage. The following considerations should support this assessment. In the Works and Days, Hesiod says that the gods have concealed the livelihood of men [Opera 42], but he does not mean that we cannot devise our own means of success; rather, he simply means that we must work to in order to succeed. Such work involves acting in accordance with our understanding of the world around us, or as

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Hesiod later puts it, thinking for ourselves about what course of action might be better in the end [τ λο , Op. 293--294]. Similarly, the entire Calendar section [Op. 383ff.] implies that there do exist in the heavens and in nature reliable signs of the appropriate times for effecting specific desirable results. It is on the assumption of the reliability of such indirect means of knowledge that Hesiod promises to reveal the realities of human life and how best to attain prosperity. With this said, Hesiod does still grant that the unexpected might occur. He explains, for example, that the will of Zeus is different at different times, and difficult ( ργαλ ο ) for mortals to know [Op. 479ff.]. But it is important to note that he does not say that Zeus’ will is impossible for men to know. He does admit that, even if one starts the winter ploughing too late, this is no guarantee of loss. But the possible gain here can still be known through rational calculation. For Hesiod explains that, in this case, if the call of the cuckoo should sound for the first time, and if Zeus should send a specified amount of rain on the third day, then the late-plougher will in fact prosper. Here, in laying out such detailed conditions for success, and in doing so in a positivistic tone characteristic of so many Hippocratic treatises (he calls his advice here a φ ρμακον, [Op. 485]), Hesiod suggests that mortals still have a reliable means to determine outcomes even when faced with the unexpected. Hesiod’s subsequent remarks bear out this suggestion. He insists that the lazy person who neglects the winter chores depends on ‘empty hope’, which leads to trouble [Op. 493 ff.]. Here the poet appears to be distinguishing Hope from another, more positive human mental faculty. A person who does not plan for the winter or engage in the appropriate winter work does so on the basis of ‘empty Hope’ to the extent that his tendency to wish for a desirable end without making the appropriate plans will not likely fulfill his needs. His consequent dearth, in fact, proves that Hope is no good [ λπ ο κ γαθ , Op. 500]. All this, however, suggests that there is a ‘good’ kind of expectation, one that a person who does plan for the future and work appropriately enjoys. His expectations are good because they will likely come to fruition in accordance with his understanding of the world around him. Although Hesiod would say that we usually have only the futile hope of foreseeing such outcomes as disease, misery, and death, this does not mean that he would characterize human beings by their possession of this weaker capacity. Nor would

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he likely say, as Clay puts it, that we are all, ‘in the final analysis, a thrall to Hope and ignorant of Zeus’s plans’ [124]. If the above sketch of Hesiod’s outlook on the potential of human knowledge is correct, then it need not invalidate Clay’s interpretation. For it, too, acknowledges the scepticism that Clay attributes to Hesiod. But it restricts this attitude to a few issues in life. For the most part, Hesiod would agree that we can be certain about how to prosper. Perhaps then, it is better to regard the poet as a forerunner of the more positivistic tradition which tends to defend man’s capacity for indirect knowledge—e.g., the Hippocratic authors, or the character Prometheus in Prometheus Bound, who insists that mortals do have available to them in astronomy and other sciences a reliable sign (τ κμαρ β βαιον) to determine the issues of their lives [cf. Prom. Vinct. 454--458, 486--487, 497--499]. Hesiod may believe that we are not able to know every decision of the gods on every occasion; but the fact that he focuses on what we can know, as opposed to, say, Pindar’s resignation, gives good reason to attribute to him a rather optimistic view of human intelligence.

The Claude Glass: Use and Meaning of the Black Mirror in Western Art by Arnaud Maillet. Translated by Jeff Fort New York: Zone Books, 2004. Pp. 300. ISBN 1--890951--47--1. Cloth $26.95

Reviewed by Sven Dupré Ghent University [email protected] When in the early 1980s a Parisian sculptor donated a Claude mirror to the Musée National des Arts et Traditions Populaires in Paris, this gift set in motion a complicated series of events. According to the French art historian Arnaud Maillet, a magnetizer who had come to examine it inserted some bits of paper inscribed with signs (for example, Solomon’s seal) between the backing and the glass and recommended that it be kept in charcoal, which is reputed to absorb evil forces. This mirror is therefore not exhibited, since someone who knows how to cast spells would be able to use it, even through a glass case. [31] Little surprise then that the black mirror fell into oblivion! In The Claude Glass, Maillet sets himself the task of rescuing it from eternal forgetfulness in an essay which—Maillet promises us—will be part of a doctoral thesis on painters and optical instruments since the second half of the 18th century. Maillet’s The Claude Glass is published by Zone Books, of which Columbia University’s art history professor, Jonathan Crary, is the founding editor. This is in itself significant, as Maillet is highly indebted to the project that Crary himself set out to undertake in his Techniques of the Observer [1990], both methodologically and in terms of the arguments that Maillet wishes to support. Convinced that a history of vision or perception ‘depends on far more than an account of shifts in representational practices’, Crary took as his problem the observer: Vision and its effects are always inseparable from the possiblities of an observing subject who is both the historical product C 2005 Institute for Research in Classical Philosophy and Science 

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Aestimatio and the site of certain practices, techniques, institutions, and procedures of subjectification. [Crary 1990, 5]

Crary’s basic argument was that the early 19th century saw the creation of a new kind of observer. He located in the 1810s and 1820s a rupture in the scopic regime between a geometric model of vision (in which vision was conceived as essentially passive, independent of the subject, and based on a radical distinction between interior and exterior) and a physiological model of vision (in which vision became subjective and the product of visual experience became located in the body of the observer). Crary developed his argument by contrasting two instruments— the camera obscura and the stereoscope—which he considered paradigmatic for his two models of vision respectively. In other words, the optical instruments are not just presented as Martin Kemp did in his ground-breaking and contemporaneous The Science of Art [1990], by detailing their material aspects or the diverse uses to which artists put these instruments in their representational practices. The optical devices in question, most significantly, are points of intersection where philosophical, scientific, and aesthetic discourses overlap with mechanical techniques, institutional requirements and socioeconomic forces. Each of them is understandable not simply as the material object in question, or as part of a history of technology, but for the way in which it is embedded in a much larger assemblage of events and powers. [Crary 1990, 8] Crary kept far from any underlying assumption that artists used optical instruments to arrive at photographical realistic images—an underlying assumption recently again brought into the spotlight by David Hockney’s Secret Knowledge [2001]. Many accounts of the camera obscura, particularly those dealing with the eighteenth century, tend to consider it exclusively in terms of its use by artists for copying, and as an aid in the making of paintings. There is often a presumption that artists were making do with an inadequate substitute for what they really wanted, and which would soon appear—that is, a photographic camera. [Crary 1990, 32] Maillet’s study of the Claude mirror is, therefore, not aimed at detailing the various uses to which artists put this instrument. Maillet’s

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goal is to contribute to such a history of vision as outlined by Crary. To that end he brings together insights from history of art, cultural history, literature and literary theory, philosophy and aesthetics. That this is Maillet’s aim the reader only learns by reading on, because the book lacks an introduction (which could have clearly set the problem) and, for that matter, also a conclusion. Otherwise, the book is well-organized in five sections. The first part sets out to define what kind of objects Claude mirrors are, to relate how they got their name, and to conjecture how they disappeared from the historical record. The shortest definition of a Claude mirror is that it is a convex tinted mirror. However, under that general label hide a variety of objects, as the choice of tint (not necessarily black), the convexity and the shape of the mirror (allowing it to be hand-held or not) can vary. Maillet insists that the Claude mirror is not to be confused with the Claude glass. The Claude glass is a filter made of colored glass. It is unfortunate, then, that the title of Maillet’s book contributes to the confusion, even if it is the case that in English ‘glass’ and ‘mirror’ may be used interchangeably. The Claude Glass is a book about the Claude mirror. The convex tinted mirror was baptized ‘Claude mirror’, not because the painter Claude Lorraine is known to have used one, but because this mirror gave the landscapes reflected in it the same somber light and golden tint associated with Lorraine’s paintings. Maillet tries to convince us that the convex mirror ‘refuses to conform to the rigid laws of optics’ [38] and, therefore, is generally conjured away in the historical records, such as the inventories of curiosity cabinets or opticians’ shops. The Claude mirror was, nevertheless, widely available in 18th-century curiosity cabinets, opticians’ shops and—last but not least—artistic circles. The second section is devoted to the occult associations of the black mirror in Western culture. On the one hand, mirrors were considered a source of errors and illusions. They were used (including their black variants) in necromancy and catoptromancy (divination with mirrors). Maillet argues that in the 18th century mirrors were almost systematically perceived as demonic, and that black mirrors were associated with death and other types of transgressions, from John Dee’s famous obsidian mirror to their re-emergence on websites today promoting sado-masochism. On the other hand, the wellknown Pauline mirror is a symbol of precision and clarity. Finally,

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the mirror attracts and fascinates the gaze. It even brings the observer into a light hypnotic state, Maillet claims. The third part of Maillet’s essay is something like the counterpart to the second section. Inasmuch as the second part wanders off in all directions suggested by the occult associations of the Claude mirror, so the third section of the book is concentrated on the almost sober description of the visual experience which the Claude mirror offers the observer. Drawing on Roger de Piles’ Principles of Painting (early 18th century), Maillet argues that the Claude mirror offers the observer a reduction of the visual field and of color (not unlike the reduction which a painting offers). This reduction allows a unification, Maillet argues. It unifies all objects into one glance of the eye and it reduces shadow, light, and colors to a tonal unity respectively. First, as concerns the visual field, Maillet’s arguments are actually about the convex mirror in general, not only about its black variant. On the basis of the recommendation of the convex mirror by the Flemish painter Gérard de Lairesse, Maillet claims that it served as a compositional aid because it brings, for example, a wide prospect within the mirror’s narrower field of view. Second, as for color, Leonardo and Leon Battista Alberti recommended the mirror as a means to judge the quality of paintings and the force of the colors. Again, in the 18th century, De Piles discussed the Claude mirror in this respect. Finally, the physiological optics developed by Hermann von Helmholtz in the 19th century gave a new momentum to the Claude mirror, as Helmholtz fully grasped the reason for ‘smoking’ the colors. The painter’s problem is that the colors on his palette cannot offer the infinite variety of tones for a single color on a scale from dark to light which reality presents. However, since the human eye is sensitive to the relations between different levels of brightness (rather than perceiving them absolutely), the painter’s task is to reproduce these relations (which involves ‘a translation into another scale of sensitiveness’ [118]). For Helmholtz, painting imitates ‘the action of lights upon the eye, and not merely the colors of bodies’ [119]. This was more easily accomplished with the aid of a Claude mirror, as Manet, Degas, and Matisse found out. The fourth section seems to be the most important part of the book conceptually. It discusses the heyday of the Claude mirror at the end of the 18th and the beginning of the 19th centuries. At

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that time the black mirror was lifted from obscurity by an aesthetics dominated by the picturesque en vogue in England. It was in those days that tourists visited the Lake District with a Claude mirror in their pockets, that English gardens were perceived and composed like paintings (particularly like those of Claude Lorraine), and that Coleridge and Wordsworth transformed the view in the Claude mirror into an ideal view. However, John Ruskin severely criticized the Claude mirror in the harshest words: It is easy to lower the tone of the picture by washing it over with gray or brown; and easy to see the effect of the landscape, when its colors are thus universally polluted with black, by using the black convex mirror, one of the most pestilent inventions for falsifying Nature and degrading art which was ever put into an artist’s hand. [148] Ruskin’s allergic reaction to the instrument was inspired both by his dislike of the somber luminosity typical of the aesthetics of Lorrain and by the mechanical aspect of the reflection in the Claude mirror. Maillet argues that the Claude mirror tends in the direction of a neo-classical theory of imitation which considers the reflection in a Claude mirror always lacking in relation to nature itself. Just like the camera obscura for Joshua Reynolds or Canaletto, the Claude mirror was appreciated in a role of comparison in the imitative process; but copying the mirror image or the image in the camera obscura was considered inferior to the production of a real work of art. Thus, for all its attraction, the image in the Claude mirror is ultimately ‘disappointing’, Maillet argues. He also considers the Claude mirror in Ruskin the emblematic instrument of monocular vision and, as such, opposed to the stereoscope (central to Crary’s argument), which emphasizes binocular vision. However, Maillet softens the rupture between the geometric and physiological scopic regimes described by Crary, as he argues that the Claude mirror has an ambiguous status, thus suggesting that Crary’s two scopic regimes co-existed for a while. Maillet claims that the use of the Claude mirror makes the observer already aware of his own body as an integral part of the reflection. Maillet shows that the Claude mirror offers a solution to physiological problems such as the already mentioned problem of brightness discussed by Helmholtz.

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In the fifth part, Maillet looks for the new meanings which artists gave to the black mirror after Ruskin’s critique. In his analysis of the use of the black mirror in the work of 20th-century artists such as Gerhard Richter or François Perrodin, the black mirror—devoid of Lorrain’s aesthetics—is associated with devaluation, the progressive loss of the image, and abstraction. This section of the book eventually evolves into a meditation on opacity, the blindness of the gaze and melancholy, all of which are especially brought out—Maillet claims—in these 20th-century works of art. In this context Maillet returns once more to Crary’s division of the history of perception into two periods and to his own claim that the black mirror belongs to both these periods. However, between these two periods the black mirror changed status. He echoes Crary’s view that ‘the relation between eye and optical apparatus becomes one of metonymy: both were now contiguous instruments on the same plane of operation’ [214; Crary 1990, 129]. This provoked ‘a second crisis of the gaze, that described by Crary, for while this instrument serves the eye, the eye also begins to serve this instrument’ [215]. Thus, just as Crary, Maillet is ultimately interested in the making of a new kind of observer. This quote from Crary’s Techniques of the observer nicely illustrates that the book’s heavy reliance on Crary’s work is not only a strength, but also one of its principal weaknesses. More often than not, Maillet fails to exemplify the theoretical insights he borrows. One of the most problematic aspects of Maillet’s indebtedness to Crary is that he takes over Crary’s division of history into geometric and physiological scopic regimes, even when Maillet is repeatedly obliged to point out that the Claude mirror has an ‘ambiguous status’, as it seems to belong to both these periods. In fact, Crary’s sudden transposition of vision inside the body in the early 19th century and his very clear-cut division of the history of observation is precisely one of the aspects that has repeatedly and justifiably come under attack [see, e.g., Summers 2001, Fiorentini 2004]. That Maillet wishes to hold to it anyway (and in light of the criticism of Crary’s account, one could ask why) comes at a considerable cost. The Claude mirror is thus also an instrument that participates in the transition between these two periods. And it can serve as a transitional element because it was used before and continued to be used after. The use of the black

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mirror is not limited to a period dictated by the fashions of tourism. Nor could it be buried by Ruskin. [152] Maillet rightly remarks, Otherwise, it would be difficult to explain how this mirror was used, for example, by Manet in 1861, by Matisse around 1900, and by Sutherland around 1946. These painters looked to the mirror less for visual characteristics—the proper element of the picturesque—than for a specific mode of vision, and they were interested more in sensation itself than in the objects of sensation. [152] But then what is this ‘specific mode of vision’? And is Maillet not begging the question? Maillet’s indebtedness to Crary’s categories (which he identifies as problematic) creates serious gaps in his own argumentation about the Claude mirror. The relevance for the history of science of Maillet’s approach to the Claude mirror—in line with Crary’s approach to the camera obscura or the stereoscope—is that it is an attempt to open up the history of observation. Maillet and Crary rightly question an approach to optical instruments which is limited to describing them solely in terms of their material characteristics and which relies on the kind of technological determinism in which the use of optical instruments is invariably associated with the conquest of an unproblematic realism (as in the famous Hockney-Falco thesis, which is criticized in Dupré 2005). A history of observational practice is in part that of instruments, buildings, and records; and in part that of less tangible cognitive and social practices. However, for all its good intentions, Maillet’s history of the Claude mirror ultimately fails to contribute to a history of observational practice. Notwithstanding the opening chapters, which show that a variety of objects match the definition of a Claude mirror, the Claude mirror of which Maillet wishes to write the history is an ideal type. Symptomatic of the missing materiality of Maillet’s black mirror is that the object itself disappears from view in Maillet’s approach. For example, in discussing Alberti’s and Leonardo’s use of mirrors, Maillet stretches the definition of a black mirror: Now, according to this experiment, if every mirror already absorbs and reduces the light it reflects, every reflected image

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will therefore be slightly tinted, because it is thus tainted. This means also that every mirror is already in some way a black mirror. [106] Indeed, most of Maillet’s discussion of the painterly use of mirrors is devoted to (convex) mirrors in general rather than to black mirrors. Maillet concludes his account of Alberti’s and Leonardo’s use of mirrors by writing that ‘it is in my view completely legitimate to ask whether a number of these mirrors, notably in Alberti’s case, might not be black mirrors’ [106]. On which historical basis does Maillet want to claim that the mirrors mentioned were black mirrors; and, in light of the above, does this matter to his argument? In the end there is no longer any object—let alone a historically specific object—that answers to Maillet’s black mirror. The black mirror thus ensures a breath for sight. As Jacques Derrida has written, the eye blink is nothing other than the breath of sight. It is the absolute speed of the moment, the critical moment par excellence, for then sight no longer sees; it is blinded. But in this critical moment, this suspension of perception, sight is realized and constituted. The black mirror, like the blinking of the eye, plunges the organ of sight into blindness, but this blindness is no less salutary for that. [213] What is the (historical) status of such claims? Here, and on many other occasions in the book, Maillet fails to give historical content to such theoretical statements. Maillet’s The Claude Glass has the merit of discussing a littleknown optical instrument. The book is to be applauded for the broad variety of discourses that it brings to bear on the Claude mirror. However, that the materiality of the objects and the historical specificity of the discourses ultimately vanish detracts from the book’s relevance for historians of science. bibliography Crary, J. 1990. Techniques of the Observer: On Vision and Modernity in the Nineteenth Century. Cambridge, MA/London. 1999. Suspensions of Perspection: Attention, Spectacle and Modern Culture. Cambridge, MA/London.

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Dupré, S. 2005. ed. Optics, Instruments and Painting, 1420--1720: Reflections on the Hockney-Falco Thesis. Special issue of Early Science and Medicine 10 (2). Leiden. Fiorentini, E. 2004. ‘Subjective Objective. The Camera Lucida and Protomodern Observers’. Bildwelten des Wissens: Kunsthistorisches Jahrbuch für Bildkritik 2:58--66. Hockney, D. 2001. Secret Knowledge: Rediscovering the Lost Techniques of the Old Masters. London. Kemp, M. 1990. The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat. New Haven/London. Summers, D. 2001. Review of Crary 1999. The Art Bulletin 83:157-161.

The Command of Light: Rowland’s School of Physics and the Spectrum by George Kean Sweetnam Philadelphia: American Philosophical Society, 2000. Pp. xxvi + 233. ISBN 0--81769--234--4. Cloth $25.00

Reviewed by J. Brian Pitts University of Notre Dame [email protected] This book discusses the work of Henry A. Rowland (1848--1901), professor of physics at Johns Hopkins University, careful experimenter, and inventor of the concave diffraction grating, along with the work of Rowland’s students and associates and their influence. This influence pertained to the invention of an important experimental apparatus, the institutionalization of American physics in the American Physical Society and elsewhere, the study of the solar spectrum, the improvement of this apparatus by eliminating misleading ‘ghost’ lines, the contribution of spectral data to atomic physics and quantum theory, and the establishment of the field of astrophysics involving stellar spectra and its institutionalization especially in observatories and the Astrophysical Journal. Sweetnam’s book is interesting, well written, and informative about its intended subject matter. It is the first to treat Rowland’s work in such detail. The main body of the text, which was the dissertation of the late G. K. Sweetnam, is augmented by a preface by Charles Gillispie and a detailed introduction by dissertation supervisor M. Norton Wise. Sweetnam succumbed suicidally to depression in 1997, so the publication of his work as a book was effected by others. Keeping track of the cast of dozens of figures is facilitated by the useful index. While one notices occasional errors in proofreading and limitations in the typesetting of equations, comprehension is not adversely affected. Concerning biographical details for Rowland, Sweetnam evidently feels no need to duplicate the sketches that are available. Thus, one must learn from elsewhere important facts about Rowland’s life such as that he married in 1890 and that his premature death was due to diabetes. Sweetnam aims to cover Rowland’s life just as it was relevant to the founding of a research school C 2005 Institute for Research in Classical Philosophy and Science 

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devoted to the study of light, as Gillispie’s preface observes. The author argues, persuasively to me, that Rowland formed a school devoted to the empirical study of light centered at Johns Hopkins University. The school displayed at least 12 of the 14 characteristics that Gerald Geison associates with research schools. Whereas an older tradition in the history of science could be internalist to the point of neglecting the spatio-temporal embodiment of scientific theories and experiments, and some more recent scholarship has attended so heavily to external social and institutional factors as to slight the scientific theories and data, Sweetnam strikes an attractive balance between externalism and internalism. Perhaps the heavily experimental flavor of the work of Rowland and his associates lends itself to such a balance. Close contact with apparatus and data leaves little room for factors other than the scientific ideas to play a dominant role, much as the 17th-19th century advocates of inductivism intended. Sweetnam makes use of Rowland’s and others’ papers and correspondence as sources. The list of archival sources [217] indicates that considerable travel and labor must have been invested to consult all the sources used. While some members of Sweetnam’s cast are unfamiliar to much of the contemporary physics community, their connection to important institutions and figures now remembered, especially in connection with atomic and quantum physics and astrophysics, is shown quite effectively. Two early examples come from Rowland’s own life. Rowland benefited from early recognition and publication assistance by James Clerk Maxwell. Before filling his duties as a new professor at the new Johns Hopkins University, Rowland was able to spend time in Helmholtz’ laboratory in Germany. One theme of Rowland’s work and that of his school was internationalism. It is not surprising that the physics community in a young and geographically isolated nation such as the United States of America in the late 19th century needed European connections to flourish. What is more noteworthy is that, in no small part due to Rowland and his school, American physicists, though not theoretically innovative at that time, made important contributions to physics through instrumentation and experimentation. Thus, the work of American physicists made necessary new theoretical ideas in atomic and quantum physics, though

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the generation of those ideas was typically left to Europeans. Rowland’s concave diffraction gratings made their way around the country and around the scientific world. The gratings were important in the study of solar and stellar spectra. The latter fact connected Rowland’s school with pioneers in American astrophysics, who founded astrophysical observatories to study stellar spectra rather than the positions of heavenly bodies. Sweetnam’s treatment of the relevant physics seems generally sure-footed. His willingness and ability to discuss scientific instrumentation and present a few formulas presumably are benefits of his undergraduate training and his work as a science journalist. He expects a tolerable acquaintance with classical electromagnetic theory, optics, and modern atomic physics from his reader for full comprehension. Given the importance of the diffraction grating to Sweetnam’s story, perhaps a brief explanation of diffraction and diffraction gratings would be useful here. Diffraction is an optical phenomenon that, unlike reflection by mirrors or refraction by lenses, is completely dependent on the wave nature of light. Diffraction gratings can be made out of many substances, but their key feature is a set of closely spaced lines that are made by a ruling engine and that affect light differently from the material on which the lines are made. A diffraction grating poses spatially periodic obstacles to the propagation of light, so the resulting pattern of alternately constructive and destructive interference produces, respectively, bright and dark patches at regular distances. The nature of the interference depends on the wavelength of the light. Unlike the toy mathematical problems in elementary physics texts, realistic physical light sources produce light that corresponds to the sum of light of various wavelengths, intensities, and polarizations. The mathematical breakdown of a mathematical function of space and time into components with a given wavelength is a Fourier transform from physical space to wavelength space. Diffraction gratings perform something like a Fourier transform on physical light sources, because the mathematical relationship between the grating spacing and the light-component wavelength dictates where constructive or destructive interference occurs for a given wavelength. Studying the resulting diffraction patterns reveals various wavelengths of light at various intensities. If a single element can be isolated and used as a source, then one can ascertain the spectrum of that atom. Such spectra revealed much about the energy levels

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and thus the atomic structure of elements, information important in constructing modern atomic theory. Getting good results from a diffraction grating—results that readily display nature rather than the construction of the grating, one wants to say—requires extreme mechanical precision in the construction and operation of the ruling engine. Thus, certain ruling engines and their stewards play a key role in Sweetnam’s story. Rowland was an early master at this work, as were a few others of his school later on. Sweetnam does not neglect the craftsmen on whom Rowland relied. While Sweetnam’s treatment of the scientific matters involving electromagnetism, optics, and atomic and quantum physics seems generally sound, his occasional brief forays into the special and general theories of relativity could use a bit more nuance. The claim that special relativity had ‘ruled out’ an electromagnetic ether [155], notwithstanding its retention in 1913 by Joseph Ames, is a remarkably strong claim that would have surprised the prominent H. A. Lorentz. Thus, Ames is faulted for theoretical conservatism, but perhaps not justly. A more usual and defensible view is that the ether had been stripped of most or all of the mechanical properties that made positing it seem worthwhile. Even this view itself might represent a later consensus available only once most grew accustomed to the idea that electromagnetic oscillations need not be the oscillations of any mechanical thing, a claim that is more readily accepted nowadays due to early and authoritative instruction of the young. Concerning general relativity, several interesting early (and in some cases now rejected) experiments touching the general theory of relativity are discussed. One wishes, however, that Sweetnam had been more explicit about which theories were in competition and on what grounds, in discussing the gravitational red shift in the 1910s [192]. Newtonian gravity was empirically adequate (apart from worries about Mercury’s perihelion precession) but imperiled theoretically by its instantaneous action at a distance, in contrast with electromagnetic retarded action by an intervening field or medium. Perhaps Newtonian gravity’s most natural successor was Gunnar Nordström’s scalar theory of gravity, which generalized Newton’s theory to a relativistic local field theory. Nordström’s theory was conceptually acceptable, but it did even less well than Newton’s theory regarding the anomalous precession of Mercury’s perihelion (though unseen matter could be invoked, not so unlike the dark matter and dark

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energy posited today for analogous difficulties). While observation of a gravitational red shift would confirm general relativity against Newtonian gravity, it would give no advantage over the more serious competition in Nordström’s theory. By contrast, gravitational light bending was predicted by Einstein’s general relativity but not by Nordström’s theory. Details about the confirmation of gravitational theories, though discussed imperfectly by Sweetnam, are admittedly peripheral to his project. They are worth mentioning because the issues are intrinsically interesting and not too widely understood. As Wise’s introduction and Sweetnam’s opening chapter mention, Rowland came from a line of Presbyterian ministers. This ancestry left its mark on his scientific work. Rowland’s interest in science was preceded by his father’s amateur scientific interest. Rowland the physicist found science to be a morally improving enterprise of diligent and disinterested search for truth by empirical investigation of the creation; this enterprise of pure science was distinguished from applications for the purpose of profit. Sweetnam observes, with little elaboration, that for Rowland the physicist there was no conflict between science and religion. While contemporary historians of science and religion such as David Lindberg, Ronald Numbers, John Hedley Brooke, David Livingstone and others have adequately refuted the Draper-White warfare thesis (that conflict between science and Christian theology has been the generic form of interaction) as a piece of polemical fiction, its lingering in the news media makes continued critique useful. That said, Sweetnam’s remarks are somewhat superficial in failing to sketch what form Rowland’s reconciliation of science and theology actually took. Probably, most intelligent and unbiased observers have thought that the Christian Scriptures, in addition to asserting core theological claims (the Triune nature of God, the Incarnation of God the Son as Jesus of Nazareth, and the atonement for human sin in the crucifixion of Jesus, his subsequent resurrection from the dead, and the like), also make some truth claims (whatever their detailed form) about the real history of the world’s creation, the fall into sin, the judgment in Noah’s flood, the election of Israel, the rescue of the Jews from Egypt, and so forth. It is not just obvious why it is a reasonable procedure to reject or significantly pare down the latter set of claims, while embracing the former at full strength, as Rowland (one is left to suppose) did. Perhaps Sweetnam, like so

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many, has conflated the acts of interpreting a text and believing it. If the credibility of the testifier is impugned on earthly matters, is the testifier still trustworthy on heavenly matters (as Jesus wonders in John 3:12)? It seems more consistent either to reject both or to accept both. If it is inadequate merely to announce that an apparent conflict is unreal, then it is disappointing that Sweetnam did not inquire into Rowland’s resolution. The meager light that Sweetnam sheds on Rowland’s views on science and theology is perhaps enhanced a bit by the revelation that the Rev. Henry A. Rowland (presumably the father of our physicist), who was ordained to the ministry ca 1830 and deceased ca 1860, sided with the New School (the more liberal side) in the 1830s in response to a Presbyterian church split. 1 Sweetnam occasionally quotes the minister-grandfather of Rowland the physicist and suggests parallels between the works of the two men, but the parallels sometimes seem largely metaphorical and the quotations largely decorative. Further elaboration could bring clarity here. bibliography Wallace, P. J. 2004. “The Bond of Union”: The Old School Presbyterian Church and the American Nation, 1837--1861. PhD thesis: University of Notre Dame, 2004. (Supervisor: James Turner).

1

So I am informed by Rev. Dr. Peter J. Wallace, historian of 19th-century Presbyterianism, who discovered this in undertaking his research for Wallace 2004.

Archimedes: What Did He Do Besides Cry Eureka? by Sherman Stein Washington, DC: The Mathematical Association of America, 1999. Pp. x + 155. ISBN 0--88385--718--9. Paper $27.95

Reviewed by Henry Mendell California State University, Los Angeles [email protected] There is much happening these days in the world of Archimedes. We may at last get a new, complete, literal, English translation; new examinations of the now available palimpsest have already yielded interesting insights; most of all, scholars have made deep advances in recovering the traditions of his works and in understanding Archimedes’ mathematical practice. For the rest of the world, perhaps this is more than is wanted. For them, Stein’s book might serve. Its goal is to introduce those with high-school mathematics to Archimedes’ accomplishments. It is traditional in that it makes full use of modern concepts and notations; it makes no attempt to put Archimedes in any mathematical context, nor does it seem aware of much recent scholarship. Though dedicated to Wilbur Knorr, it seems almost completely uninfluenced by his work. Inspired by the recent availability of the palimpsest, it nonetheless precedes the important work of Reviel Netz. So, with little exception, this is not a book for scholars. Is it then a book for the rest of us? The book is brief and deals exclusively with a few of Archimedes’ extant works. In are the Equilibrium of Planes, the Method (especially centers of gravity), the Quadrature of the Parabola, Floating Bodies (with a clear discussion on the stability of the paraboloid), the area of the spiral from On Spirals, the surface and volume of the sphere from On the Cylinder and Sphere, and the circumference of a circle from Measurement of the Circle. Out are other works and all non-extant works including the Cattle Problem, the traces of his astronomy, the construction of the regular heptagon, evidence for the book on the balance, trisection of the angle, and so forth. Given the purpose of the book, the only relevant question is whether it adequately introduces a reader to Archimedes, entertains her, and does not deceive her too much. C 2005 Institute for Research in Classical Philosophy and Science 

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It does often entertain. Stein reprints a delightfully romantic and silly announcement of the discovery of the palimpsest from the New York Times. His development of the basic properties of parabolas from affine geometry is amusing, even if it has little to do with Archimedes’ mathematics. Stein follows the more sceptical in his account of Archimedes’ life (thus, the title turns out to be a joke), and so tells us little about his world. That is a pity, since he could have given the reader a little of the history of Syracuse and life in the third century BC. Neverhetheless, Stein makes good pedagogical choices in the arrangement of mathematical problems and in the introduction of new material. One gets a good picture of some highlights of what Archimedes proved and often a good sense of the proof strategy, albeit without getting any sense of what it is like to work in Greek mathematics or with Greek proportions (Stein treats ratios and fractions indifferently) and so forth. Perhaps this is enough for the beginner. Even a specialist might luxuriate in seeing the strategy and structure of arguments with their motivation clearly presented in standard, modern notion without the details. The book comes with a rich collection of exercises without solutions, but occasionally followed by a discussion that gives a hint to the astute reader. With a few exceptions, the exercises are doable and enjoyable. The exceptions include one which Stein professes not to have solved, while some exercises contain trivial errors that the careless student might not notice: for example, on p. 108, the reader cannot show that π < 3.215, since this value has been rounded down from Archimedes’ estimate π < 1836/571; or on p. 70, the reader is required to show that one point is below another, which will not hold if the body is submerged. Only someone not versed in ancient mathematics can show [8] for two weights that W/w is rational (a fraction?) if and only if W and w are commensurable! Some exercises seem to require induction. As an intelligent amateur, Stein makes some significant errors. For example, he does not see that the mechanical technique used for the quadrature of the parabola in the Quadrature of the Parabola is very different from that of the Method [53--54] in its use of reductio by compression of areas. Stein [8] informs the reader that Archimedes would not multiply a weight by a length, but then falsely tells the

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reader that he would take the ratio of a weight to a length instead. Unfortunately, the reader unversed in ancient treatments of conic sections will find Stein’s account of the parablola difficult unless she reads the appendix on affine geometry first. And some less significant ones. A little illiterate Greek is more silly than offensive [15]; but Stein propagates two common, yet curious misunderstandings about translation. He evidently thinks that manuscript copies and editions in Greek are translations. He also does not realize that Heath and Dijksterhuis are abridgements that include much paraphrase, and that neither is properly a translation of the text [x, 6, 32, 75]. There are a few malicious typos among the numbers in the measurement of the circle [111, 116]. In sum, this is an old-fashioned book that is nonetheless readable and often gives a good intuition of the proof strategy, yet with errors and a lack of breadth in covering the work of Archimedes.

The Female in Aristotle’s Biology: Reason or Rationalization by Robert Mayhew Chicago: The University of Chicago Press, 2004. Pp. xi + 136. ISBN 0--226--51200--2. Cloth $28.00

Reviewed by William Wians Merrimack College [email protected] Robert Mayhew’s The Female in Aristotle’s Biology is devoted to a careful consideration of those passages in Aristotle’s biological writings that have become some of the more contentious parts of the surviving corpus. Considering that Aristotle’s biology remains relatively neglected, it is perhaps not surprising that the portions looked at by non-specialists should be those most relevant to current concerns. In particular, attention to the role and influence of gender in the history of philosophy has brought Aristotle’s remarks on females under scrutiny. The general conclusion has been that Aristotle’s biology of the female is not just factually mistaken (as are many of his scientific theories), but that it exhibits a marked degree of bias and prejudice reflecting the patriarchal ideology of ancient Greek society. Mayhew sets out to defend Aristotle the biologist against such charges. Aristotle’s biology may be mistaken; but this is the result of ‘honest’ science, not ideological rationalization. It is important to note at the outset that Mayhew’s analysis is deliberately limited to the biology. He does not offer sustained analysis of other parts of the corpus in which bias might be detected. Thus, remarks about women in the Ethics or Politics are mentioned only briefly. (This approach has important consequences for how the book as a whole is to be evaluated, to which I shall return later in the review.) Instead, the five central chapters of Mayhew’s monograph each take one area of the biology in which recent scholars have charged Aristotle with bias, and subject the relevant texts to a close reading, paying particular attention to the methods and arguments used by Aristotle in support of his conclusions. In most cases, Aristotle is exonerated (or at least found guilty of a lesser charge); often, his accusers are shown to have C 2005 Institute for Research in Classical Philosophy and Science 

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a distinct ideological bias of their own. In the process, one comes to a better understanding of what Aristotle’s views really were and the reasons—as opposed to alleged motives—he had for holding them. The touchstone against which claims of bias are tested comes in the book’s first chapter, ‘Aristotle and “Ideology”’. Mayhew gives precise criteria for judging ideological rationalization [7]. They require showing first that a given claim tends to promote a specific social agenda; and second that the claim exhibits arbitrary or implausible assumptions or is supported by conspicuously weak arguments, and that the claim conflicts with other assumptions that were fundamental to the thinker’s outlook. When these criteria are met, one has both a motive for bias and evidence of the influence that the bias exerted. When they are not, one cannot conclude that bias was present. Thus, even when a claim happens to support a specific social agenda such as the dominance of men over women in society, one cannot say that it is necessarily the result of bias until the strength of supporting assumptions or arguments is assessed and the claim is tested for consistency with a thinker’s fundamental principles. This sets the accuser’s burden of proof very high: To justify an accusation of ideological bias, we must show that the breach in logic is so obvious that. . . it is hard to imagine an intelligent person holding such a contradiction innocently or sincerely. [11] This implies that only malicious and intentionally held absurdities could count as instances of bias. Anything that is not patently illogical or willfully embraced would seem to escape being labeled ideological. In practice, Mayhew seldom needs to invoke such a restrictive standard to defend Aristotle’s biology against its critics. Often it is the critics who are inconsistent or even lazy, not bothering to read Aristotelian texts carefully enough to determine what Aristotle’s real position is. That certainly is the lesson of the book’s next two chapters. Chapter 2, ‘Entomology’, looks at Aristotle’s views on insects. These have been labeled sexist particularly because Aristotle calls the head of the beehive the king bee rather than the queen (the chapter also briefly discusses an insect’s sex in relation to its natural defenses and to its size and passivity in copulating). Mayhew demonstrates beyond question that Aristotle was not being dogmatic at all. What

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he in fact says is that the leader of the hive is neither male nor female; and he supports this perhaps surprising position with some very careful observations and inferences, hedged with admirable caution when observation is incomplete. There is nothing here to support claims of male hegemony, human or apian. The carelessness of Aristotle’s critics is even more evident in chapter 3, ‘Embryology’, which deals with the contributions of male and female to generation. Here Mayhew rebuts those who try to assimilate Aristotle’s position to the defense of Orestes given by Apollo in Aeschylus’ Eumenides, which notoriously reduced the female’s role to that of an incubator of the embryo with no independent contribution of its own. Quoting from critics like Eva Keuls, Mayhew shows that they go so far as to impute to Aristotle positions regarding generation that he in fact repudiates. Even more careful readers are shown to have made mistakes that should have been avoided, particularly regarding relationships between male and female, form and matter, and active and passive. Mayhew argues convincingly that the female makes her own active (though not decisive) contribution to the offspring, emitting a seed (σπ ρμα) that is unlike the male’s but still capable of imparting its own motions to the fetus. Given that Aristotle was working without a microscope, he reached a conclusion that was informed by sometimes ingenious inferences from phenomena such as wind eggs, not one based on ideology. Chapter 4, ‘Eunuchs and Women’, addresses what is perhaps the most notorious line in all of Aristotle’s biology, in which the Philosopher writes that the female is ‘as it were a mutilated male’ (De gen. an. 2.3.737a27--28). Mayhew begins by disarming the seemingly offensive ‘as it were’ ( σπερ), then devotes considerable care to explicating Aristotle’s comparison between women and eunuchs, who are indeed mutilated males. Though the chapter relies too much on quotations from other scholars with whom Mayhew agrees (a tendency evident elsewhere in the book), Mayhew argues persuasively that critics have reacted too quickly to the term ‘mutilated’ and so have not adequately considerd the empirical basis behind his claims. Chapter 5 examines a range of sometimes bizarre claims Aristotle makes about differences between male and female anatomy. Sexist assumptions have been blamed for his saying that women have smaller brains than men, that their skulls have fewer sutures, that

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their skin is paler, bones softer, and teeth fewer (the first three of these are specifically attributed to the human female, the last two are said to be shared by women and females of some other species). Mayhew’s general strategy in dealing with this somewhat disparate list is to look for observations which a Greek would probably have been able to make and which would tend to support each claim, or to argue that the claimed difference served no discernible ideological agenda. Given his criteria for ideological rationalization, observations even if mistaken or superficial by modern standards of evidence would tend to reduce the suspicion of ideological bias. Thus, the difference in brain size turns out to be independent of any difference in cognitive ability and so served no patriarchal goal; a single ‘circular’ suture could in fact have been observed in the skull of a pregnant woman; women kept indoors (as Greek women would have been) would certainly seem paler than Greek males, so that Aristotle may actually have been observing their less healthy complexion; and differences in diet could explain the perceived difference in the hardness and softness of bones. The case of women (and the females of goats, pigs, and sheep) having fewer teeth is harder to settle. Mayhew works through several possibilities, but is forced to conclude that it cannot be determined conclusively why Aristotle makes this puzzling claim. None of the possibilities, however, suggest any kind of ideological bias. The last chapter of substantive analysis seeks to understand why Aristotle held that females are temperamentally softer and less spirited than the males of most species, differences said to be most evident in humans. Mayhew first insists that the question be limited to the biological writings. Differences in the natural capacities of the souls of certain animals are not the same as a difference in virtues. Only human beings can become virtuous; and though the achievement of virtue may well depend on a man’s or woman’s psychological capacity (the ability to withstand pain or to control the impulses of spirit), that is a question for ethics and not biology and so is placed outside the scope of Mayhew’s study. With this restriction in place, Mayhew returns to the biology. Again, he conducts a careful survey of what Aristotle says regarding differences in cognitive and character traits. This time Mayhew concludes that Aristotle’s remarks satisfy the first criterion of ideological bias: they tend to justify the interests of men. But are they the result of conspicuously bad arguments or assumptions? Do they conflict with basic principles in his thought?

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Are they, he asks, ‘in general the result of rationalization rather than honest (but mistaken) science?’ [105]. Mayhew points to ‘observations’ Aristotle could have made of women in his culture, as well as to Greek attitudes toward reputedly ‘soft’ Scythians and their kings, that would certainly have supported his view of the differences between men and women. Mayhew’s defense of Aristotle is that in Aristotle’s culture, it would not take an ideological bias to reach his conclusions. It is unfair to criticize Aristotle for not freeing himself entirely from that context. Still, Mayhew concedes that Aristotle could have questioned some of his own claims or those of his culture more thoroughly. To the extent that he did not, Mayhew concludes, his position on the softer sex ‘is strongly tainted by ideological presuppositions, despite being based, in many ways, on observation and various degrees of plausible reasoning’ [113]. Despite the many strengths of the book and the almost willful misreadings committed by some of Aristotle’s critics, three general questions should be raised about Mayhew’s approach. First, is it reasonable to isolate Aristotle’s biology from the rest of his writings? While talk of Aristotle’s system is out of fashion, one can hardly deny that his outlook displays a high degree of coherence, so that the influence of one part of his thought is often felt in quite distant parts of the corpus. Mayhew’s decision to limit his study to the biological writings has the virtue of allowing for a close reading of a manageable range of texts, but it leaves unanswered the larger question of ideological influences exerted by other parts of Aristotle’s philosophy. Second, Mayhew tends to speak as if ‘empirical science’ or what he sometimes prefers to call ‘honest science’ is itself always objective, so that whenever it is shown that Aristotle based a conclusion on empirical observation, that conclusion could not be biased or ideologically motivated. Surely the history of science would make us question that. Science is filled with observations performed by scientists engaged in a ‘passionate search for passionless truth’ (J. H. Randall on Aristotle, quoted by Mayhew on p. 117) that nevertheless have been shaped, colored, or influenced (one must necessarily be vague) by bias and prejudice against women, non-white races, homosexuals, the mentally ill, and others. Though I agree with Mayhew that objectivity is not a myth and that science is our best means to pursue objective truth, the real progress achieved toward the truth has often

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been won only by shaking ourselves free of unquestioned assumptions that have masqueraded as objective truth, complete with appeals to ‘honest’ empirical findings to support them. Such assumptions may not constitute an ideology in the strict deterministic sense Mayhew rejects in the first chapter, since they often leave room for a concern for evidence and the avoidance of ‘pitifully weak arguments’ [3]. But they can exert a subtler though still pervasive influence on where one looks for evidence and on what portion of what one beholds captures the attention. In Aristotle’s case, such assumptions could have inclined him toward certain facts and away from others, and toward paying more heed to certain voices in his culture than to others—to certain Hippocratic treatises, for instance, rather than to the plays of Euripides. Mayhew is right to say that any historical figure must be judged against the background of his or her own culture. But the objective cannot be so neatly separated from the ideological. This leads to a final question. One can ask how the organization of the book affects its overall impact. Mayhew ends with the area in which he concludes bias is most likely, Aristotle’s view of women as softer and less spirited. Coming at the end of his study, it is as if the defense, largely effective through the preceding chapters, here concedes that the possibility of ideological bias remains in this one area. One wonders, however, how the argument of the book would have felt if the softer sex had been made the first substantive chapter rather than the last, with the conclusion quoted from p. 113 above serving as a lead-in to subsequent chapters rather than as a coda. Would their arguments have seemed as persuasive? Some would, no doubt. The carelessness of some critics would not be excused. But could Mayhew’s fine analysis of king bees and female menses have effaced the impression of a bias affecting the whole framework of Aristotle’s view of gender and the sexes? I am not so sure. These questions should not turn potential readers away from what is a very good book. It should be read by students of the biology, of course, for while it does not attempt to provide a comprehensive treatment of the biological treatises, it goes some way toward dispelling some highly influential critical myths about them. And for that reason it should be read by all those who have an interest in ancient philosophy and culture, for though the scope of its argument is narrow, its implications are broad. Mayhew shows convincingly that attacks on Aristotle are often so far removed from his texts that

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they most likely stem from ‘a fundamentally ideological motivation of their own’ [117]. While his arguments should not go unchallenged, they should not be ignored. Indeed, my greatest fear is that the book will simply be dismissed by the ideologically motivated critics whom Mayhew takes issue with. Like Aristotle’s biology, The Female in Aristotle’s Biology deserves to be read carefully by those who would disagree with it.

Ancient Mathematics by Serafina Cuomo London/New York: Routledge, 2001. Pp. xii+290. ISBN 0--415--16495-8. Paper $32.95

Reviewed by Annette Imhausen Trinity Hall, Cambridge University [email protected] Ancient Mathematics is an introduction to, and overview of, mathematical sources of various kinds from the Mediterranean region from the fifth century BC to the sixth century AD. Traditionally, the focus of the historiography of ancient mathematics has been on highlevel mathematics associated with names like ‘Euclid’, ‘Apollonius’, ‘Archimedes’, and others. Cuomo departs from this restrictive preselection and casts her net much wider—not only does she include ‘professional numeracy’ such as land-surveying and accounts, she also provides glimpses of mathematics presented in sources from poetry and politics. The result is an impressively rich picture of mathematics within its social and cultural context that conveys the importance and variety of mathematical practices in Greek and Roman culture. Cuomo’s sources can be assigned to one of the following three groups: 1) classical, high-level mathematical texts which are presented in excerpts and introduced with questions about their authenticity, transmission, and origin; 2) evidence from the practical uses of mathematics in daily life such as land-surveying, accounts, and others; 3) passages from literary and political documents reflecting on the perception of mathematics. In addition to textual evidence, material and pictorial evidence such as mathematical instruments, e.g., surveying instruments [68, 155], abaci [12, 147], and plans [7, 64, 156] is used to help the reader access and appreciate ancient mathematics. The period covered in just under 300 pages stretches over more than 1000 years. It is divided into four sections: ‘Early Greek Mathematics’, ‘Hellenistic Mathematics’, ‘Graeco-Roman Mathematics’ and ‘Late Ancient Mathematics’. Each of these sections is assigned two chapters, one for ‘the evidence’ and the other for ‘the questions’. C 2005 Institute for Research in Classical Philosophy and Science 

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The first chapter begins by pointing to the difficulties posed by the available source material: ‘fragmentary, scattered over time and place, or so concentrated in one place (Athens) as to make any generalization dangerous’ [5]. This sounds only too familiar to me, having worked on Egyptian material for some time, where we face the same problem. This situation has led to speculative interpretations that have since become ‘truths’, and Cuomo’s book goes a long way to correcting them by presenting the available sources and explaining the problems attached to them. There is no contemporary evidence of the mathematical achievements of Pythagoras and later statements, although made by ‘the Greeks themselves’, are proven to be unreliable. As in Egypt, Greek culture covers a long period of time; and Greek mathematics too should not be seen as homogeneous. How then, the reader of some older accounts of Greek mathematics may wonder, did previous authors gain so much knowledge about early Greek mathematics? The answer—Cuomo cites and discusses the original sources for these studies in chapter 2—is that there are passages about early Greek mathematicians and philosophers by authors like Diogenes Laertius, Proclus, and others. It is necessary to point out that these authors lived at a time when the early Greek mathematicians had already reached mythical status. There is no contemporary evidence of early Greek mathematics, nor is there a tradition of their knowledge and histories that was handed down faithfully. Rather, as Cuomo points out throughout the book, earlier material—when available—was subjected to reorganization and improvement according to the interests of later mathematicians. And it is with this caveat in mind that we have to study the Greek accounts of their predecessors. Chapter 2 starts by looking into the use of mathematics within politics as represented by land-division, commercial arithmetic, and accounts. Cuomo analyzes mathematical practices as an expression of democracy [40--41], but is also quick to point out that this is by no means the only form of state to use mathematics for its purposes. In chapters 3 and 4 Cuomo then discusses Hellenistic mathematics. The first paragraph of chapter 3 [62] sketches the geographical and temporal setting, and outlines the history of the empire of Alexander the Great in the third and second centuries BC. The evidence for this period (and all following periods) mostly comes from Egypt, which included, in Alexandria, an intellectual center where

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many scholars visited, lived, and worked, and which, by virtue of its comparatively dry climate, provided good conditions for the preservation of papyrus. Egypt at that time presents us with a variety of sources coming from at least two traditions, the native Egyptian one and the Greek tradition. In addition, the Persian rule of Egypt has left some influences on the Egyptian culture as well, e.g., on mathematical techniques (see below). Sources of the Egyptian tradition are written in Demotic, the stage of the Egyptian language before Coptic (its last stage). Sources of the Greek tradition are written in Greek. While the existence of these two traditions has been recognized for some time, the Demotic side of this period in Egypt has only now begun to be included. The main reason for the previous focus on Greek sources is the late start of Demotic studies within Egyptology due to the extremely cursive form of script which renders it intrinsically difficult to understand. However, recent publications have made it obvious that a focus on only one of the two traditions will result in an incomplete picture. Cuomo includes a couple of problems from the Demotic mathematical papyrus Cairo JE 89127--30 [71--72] cited in the translation of Richard Parker. This source is particularly interesting not only because of its mathematical content, but also because the other side of this papyrus contains a collection of laws. Based on paleographic criteria, it can be assumed that both texts were written around the same time (third century BC); and the combination of these two subjects on one papyrus is remarkable. 1 Apart from the monograph on Demotic mathematical papyri which is cited in the bibliography, several other publications of Demotic mathematical texts deserve to be mentioned, e.g., the edition of another mathematical papyrus containing a group of problems about trapezoid shaped fields [Parker 1975], as well as the publication of Demotic mathematical ostraca. 2 Cuomo outlines the features of the Demotic mathematical papyri: The problems in the demotic papyrus are solved not generally, but for specific causes, and, rather than a deductive proof, they contain a verification, or check step, introduced by the expression ‘to cause that you know it’. [72]

1 2

For an edition of the legal text, see Donker van Heel 1990. See the list of mathematical ostraca compiled by Jim Ritter [2000, 134n27].

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This is in accordance with the classic Egyptian tradition, as we can see in the earlier hieratic mathematical papyri, e.g., the Rhind, Moscow, and Lahun papyri. These date from more than 1000 years earlier than the Demotic sources. However, there are also changes to be noted in comparison to these earlier texts. Some problem types are ‘new’ for Egypt, but have been known from Mesopotamian sources for a long time [see Høyrup 2002]. Remarkable also is the appearance of multiplication tables, e.g., a multiplication table for 64 from 1 to 16 in P. British Museum 10520 [see Parker 1972, 64--65] which we do not find in the earlier sources. Indeed, the Egyptian technique of carrying out a multiplication documented in the Rhind papyrus, for example, makes multiplication tables like these obsolete. The questions Cuomo chose in chapter 4 for this period focus on the Greek side of the picture. In the first section ‘The problem of the real Euclid’ [126--135], Cuomo indicates the starting points for attempts to determine evidence for earlier Greek mathematics and the difficulties attached to them. Her scheme of the transmission of Euclid’s elements [127] illustrates well the many traditions involved. The second section ‘The problem of the birth of a mathematical community’ [135--141] discusses the situation of patronage and collaboration resulting in the emergence of a mathematical community. To this group belonged famous mathematicians like Euclid, Archimedes, and Apollonius and their close acquaintances whom we find mentioned in their works. Another question that could be raised in this chapter, resulting from the evidence Cuomo presented in the previous chapter, is that of the relation between Greek and Egyptian (i.e., Demotic) mathematics. To attempt to answer this question, a thorough analysis of the Demotic material is needed, and some of these sources have only become accessible after Cuomo’s book was published [see, e.g., Manning 2003]. Chapters 5 and 6 give an overview of mathematics in GrecoRoman times. Again Cuomo introduces the reader to a wealth of sources, e.g., mathematical papyri, financial documents, metrological texts, and planetary tables. Included also are mathematical instruments such as abaci [147], sundials [154], and sighting instruments like the groma [155]. Following the material evidence, the second part of chapter 5 introduces the reader to Vitruvius and Hero of Alexandria, followed by numerous ‘other Romans’ (Julius Sextus Frontinus, Hyginus Gromaticus, Marcus Junius Nipsus, Balbus, Celsus, Lucius

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Volusius Maecianus, Columella, Pliny the Younger, and his uncle Pliny the elder) and ‘other Greeks’ (Strabo, Philo of Alexandria, Nicomachus of Gerasa, Ptolemy, Sextus Empiricus, Alcinous, Theon of Smyrna, and Galen). Consequently, the two issues raised in chapter 6 are ‘the problem of Greek versus Roman mathematics’ [193--201] and ‘the problem of pure versus applied mathematics’ [201--210]. Cuomo concludes that the divides. . . were much more complicated than simple Greek/ Roman or pure/applied dichotomies. Those divides had a political significance, not just in a cross-national, but also in a cross-social-strata sense. [201] The two final chapters introduce the reader to Late Ancient mathematics (third to sixth century AD). The material evidence cited here includes accounts demonstrating the use of mathematics in everyday administrative practices, and school texts [214] showing the teaching of mathematical techniques used in calculating interest on loans or leases of land. Again, the second half of the ‘evidence chapter’ (chapter 7) presents mathematicians of this era and their works, e.g., Diophantus and his Arithmetic [218--223]; Pappus with his Mathematical Collection, commentaries, and Geography [223--231]; Eutocius [231--234]; and others. The last chapter focuses on two problems, those of ‘divine mathematics’ and of ancient histories of mathematics. Cuomo discusses the use of mathematics in Christianity, e.g., in time-keeping to regulate daily prayer and to establish the date of the Easter festival. The second section then analyzes the work of Pappus, Proclus, and Eutocius who ‘classified, defined and systematized’ [256] earlier works using ‘the past as it suited them and their present concerns.’ [261]. Throughout, this book is engagingly written; and it is a pleasure to entrust oneself to Cuomo’s choices as one is led through the various periods. There are many illustrations, all of good quality, which help one understand what the sources actually look like. The abundance of references given makes this book not only perfect for a beginner but offers valuable guidance into further reading. It is obvious that the author worked carefully, and the readers benefit from her hard work. I was rather frustrated, however, by the table of contents [v], which simply gives the skeleton outline of the division according to periods and evidence vs. problems. The table of contents given at the

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end of this review was established by including also the subheadings one finds in the individual chapters; maybe it will be helpful for some readers. Likewise, in addition to the figures of places mentioned [ix-xi] and the glossary [263--266], I would have welcomed an overview of all the people mentioned throughout the book. Some of them, but not all, can be found in the index [287--290]; so maybe a separate index of persons would be appropriate. The success of this book relies mainly on two elements. First, Cuomo has included both the ‘classic’ high-level mathematical texts as well as lesser known works. Thus, we find a variety in this book that may well represent the breadth of ancient Greek mathematical culture. It is only by taking into account the lower-level mathematics and the many uses that mathematics was put to, that one can appreciate Greek mathematics. It is thus deeply satisfying to read the account of classic mathematical texts within their historical and social context. The second outstanding achievement of this book is the extended use that Cuomo makes of her philological training. Previous accounts of Greek mathematics might not even mention the sources that the claims of their authors were based on. Cuomo goes a long way in rectifying this, by not only indicating the available source material but also by giving the reader a detailed introduction to the problems attached to them. This hopefully serves to help mathematicians understand what sorts of problems historians of mathematics face in their work. It also teaches a certain scepticism towards ‘long established truths’ about Greek mathematics which may well be ancient myths rather than realistic accounts. At the same time Cuomo does not discard all later Greek accounts about earlier mathematicians, but indicates that some of them—if read carefully—prove to be quite illuminating. I believe this book is apt to serve many types of readers, from the total beginner to those who have already read the classics; and with the detailed explanation given in each argument, it is usable by mathematicians, historians of mathematics, and general historians alike. Cuomo has clearly demonstrated that there is a wealth of sources available to provide information about ancient mathematics, a wealth that has not traditionally been included in historical studies.

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It left me with the wish that many of the aspects Cuomo touched upon will be explored in more detail in (her) further publications. bibliography Donker van Heel, K. 1990. The Legal Manual of Hermopolis. Leiden. Høyrup, J. 2002. Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and its Kin. New York. Manning, J. G. 2003. Land and Power in Ptolemaic Egypt. Cambridge. Parker, R. A. 1972. Demotic Mathematical Papyri. Providence, RI. 1975. ‘A Mathematical Exercise—P. Dem. Heidelberg 663’. Journal of Egyptian Archaeology 61:189--196. Ritter, J. 2000. ‘Egyptian Mathematics’. Pp. 115--136 in H. Selin ed. Mathematics across Cultures. Dordrecht.

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Ancient Mathematics Table of Contents

1

2

3

4

5

List of figures and tables List of abbreviations List of places mentioned Acknowledgements Introduction Early Greek Mathematics: The Evidence Material evidence: division of land, architecture, records of sales, fines, loans Historians, playwrights, and lawyers: Herodotus, Aeschylus, Aristophanes, Thucydides, Lysias, Demosthenes Plato Aristotle Early Greek Mathematics: The Questions The problem of political mathematics The problem of later early Greek mathematics Hellenistic Mathematics: The Evidence Material evidence: fortifications, machines, townplanning, land-surveying Non-mathematical authors: the rest of the world: Polybius Non-mathematical authors: the philosophers Little people: Autolycus of Pitane, Aristarchus of Samos, Theodosius, Aratus of Soli, Aristoxenus, Diocles, Eratosthenes, Biton, Philo of Byzantium Euclid Archimedes Apollonius Hellenistic Mathematics: The Questions The problem of the real Euclid (pre- and postEuclidean contributions to the Elements) The problem of the birth of a mathematical community (audience and patronage) Graeco-Roman Mathematics: The Evidence Material evidence: papyri from Egypt (mathematical tables, financial documents, metrological texts, planetary tables), sundials, evidence of land-surveying

vi vii ix xii 1 4 6 16

24 31 39 40 50 62 63 73 76 79

88 105 113 125 126 135 143 143

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Vitruvius Hero of Alexandria The other Romans (Corpus agrimensorum Romanorum, astronomical writings) The other Greeks (Strabo, Philo of Alexandria, Nicomachus of Gerasa, Ptolemy, Sextus Empiricus, Alcinous, Theon of Smyrna, Galen) 6 Graeco-Roman Mathematics: The Questions The problem of Greek vs Roman mathematics The problem of pure vs applied mathematics 7 Late Ancient Mathematics: The Evidence Material evidence: accounts, administration, surveying, legislation Diophantus Pappus Eutocius The philosophers The rest of the world 8 Late Ancient Mathematics: The Questions The problem of divine mathematics The problem of ancient histories of ancient mathematics Glossary Bibliography Index

159 161 169 178

192 193 241 212 212 218 223 231 234 241 192 241 256 263 267 287

Picturing Machines 1400--1700 edited by Wolfang Lefèvre Cambridge, MA: MIT Press, 2004. Pp. vi + 347. ISBN 0--262--12269-3. Cloth $40.00/£25.95

Reviewed by Renzo Baldasso Columbia University [email protected] Picturing Machines derives from a conference on Renaissance engineering drawings hosted by the Max Planck Institute for the History of Science in the summer of 2001. Divided into five sections, the volume’s nine contributions address specific drawings of machines as well as the modes and means of visual representation available to their creators. Rather than the technical subject of machines, the common denominator that binds the papers together is the drawings, which these authors consider not as mere illustrations of the verbal text but as ideas separated and partially independent of it. Following on the heels of The Power of Images in Early Modern Science— another collection of essays from a Berlin conference that Wolfang Lefèvre helped organize and edit [2003]—Picturing Machines is the latest contribution from the movement aiming to reconsider the role of images and visual representation within the intellectual and cultural dimensions of the history of science. Approaching drawings and printed figures from a variety of perspectives ranging from the social to the purely technical, these essays represent the effort of historians of science to mark their own territory within the emerging field of visual studies of early modern culture. In what follows I provide a brief summary of each contribution while allowing more space to those that I believe would be of most interest to the readers of this journal. Beyond an enlightening introduction which provides a useful overview of the volume as well as of the topic, Lefèvre also appends a short forward to part 1 in which he defines ‘machine drawings’. Different from the ubiquitous depictions of technical objects and representing only a limited subset of technical drawings, machine drawings are pragmatically defined as drawings ‘traced or used by technicians in C 2005 Institute for Research in Classical Philosophy and Science 

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the pursuit of their professional life or derived from such practitioners’ drawings’ [13]. Thus, rather than devise a definition based on visual and intellectual properties of machine drawings, Lefèvre privileges their makers’ social status and operative cultural space. In fact his definition reflects the conclusions of Marcus Popplow’s essay comprising part 1 and entitled ‘Why Draw Pictures of Machines? The Social Contexts of Early Modern Machine Drawings’. Arguing from later 16th- and early 17th-century examples, Popplow identifies four contexts in which machine drawings were employed during the early modern period. First, as is aptly exemplified by teatri di macchine, engineers used drawings of machines to present their devices to a broader, non-expert public. Second, the 16th century also saw the emergence of a distinction in social-standing between engineers and technicians; drawings were drawn by the former in order to provide ‘blue-prints’ for the latter who then built the actual machines. Third, engineers drew machines—theirs as well as their colleagues’— to keep a private record to serve as a reference and inspiration for future projects. Fourth, through drawings engineers analyzed the machines’ workings from a ‘theoretical’ perspective. Popplow’s categories adequately describe the contexts and, therefore, the audiences and purposes, in which machine drawings appeared during the Renaissance; they create a cultural identity for machine drawings that is unavailable for other coeval forms of visual representation of scientific and technical subjects, such as botanical, anatomical, or mathematical drawings. Undoubtedly, his ‘Linnean’ classificatory effort will provide valuable points of reference for those studying specific groups of drawings. In addition to Reiner Leng’s investigation of the pictorial language developed by German gunmakers to communicate with their colleagues and apprentices, part 2 includes an important essay by David McGee. In ‘The Origin of Early Modern Machine Design’, he presents four short case studies on the drawings of Villard de Honnecourt, Guido da Vigevano, Konrad Kyeser, and Mariano Taccola respectively, in order to support several general and noteworthy methodological and historiographic conclusions. The analysis of drawings by the first two engineers allows McGee to conclude that early Renaissance machine representations are neither naive nor incomplete. Instead, he recommends that we read and interpret these drawings primarily as an effective means of communication within a

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process involving two creative minds endowed with technical expertise. Building a machine required adjustments specific to the site and depended on the availability of materials in the field. Because of these contingent and unpredictable factors, drawings explicated crucial design principles rather than providing the equivalent of modern blueprints inclusive of measurements and specifics for all parts and materials. The visual language of these drawings functioned perfectly within the technical domain of those who built machines, a domain within which the coexistence of multiple viewpoints or inconsistencies of perspective within a drawing were clarifying and advantageous rather than confusing. However, as soon as princes and patrons became involved in the construction of machines—mainly when it came to funding—the visual representation of those machines changed, as is already evident in Kyeser’s ‘proto-perspectival’ renditions that include realistic backgrounds and human operators. Taccola’s sketches take this process a step further by dispensing with multiple viewpoints while also making the dimensions uniform, which helped to produce three-dimensionally coherent images of machines set in a natural, believable space. Yet it is noteworthy that this change was not limited to presentation drawings: Taccola also rendered his machines in this way in his personal sketches. Most importantly, his drawings clarify that a ‘realistic’ visual representation did not imply that machines were realistic or that their workings complied with real physical constraints. This latter point convinces McGee that Renaissance engineers should not be construed as ‘conceptual builders of the scientific revolution’ [84]. The essays by Pamela Long and Mary Henninger-Voss comprise part 3 and address aspects of the relationship between drawings and knowledge, knowledge created as well as assumed. In her ‘Picturing the Machine: Francesco di Giorgio and Leonardo da Vinci in the 1490s’, Long sheds light on how drawings of machines became a means for investigating natural philosophical problems. The figures in Francesco’s Trattato reveal his concerns with the investigation of power from a technical perspective, while Leonardo’s Madrid Codex 1 uses drawings to study the subject of natural motion as it pertained to the scientia de ponderibus. Most importantly, Long shows that in both cases texts and images function in symbiosis. While considering the readers’ perspective, Henninger-Voss’ study of fortification drawings reveals that the theoretical basis of the visual language shared

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by all of those involved with military architecture, from patrons and generals to architects and stone masons, was founded upon geometry and its applications in those sciences subalternated to mathematics. Training in these disciplines provided advanced skills in visual analysis and in reasoning by means of figures and diagrams. The three papers of part 4 shift the focus to the development of geometric techniques used for drawing machines. Filippo Camerota investigates the codification of the rules for technical drawings that, by the 18th century, became known as ‘descriptive geometry’. The progression he describes begins with 15th-century linear perspective, which is subsequently integrated with the orthographic, shadow, and double projections. Lefèvre’s own article in the volume instead details the introduction of the combined view, a technique first developed within artistic and architectural contexts at the beginning of the 16th century by Albrecht Dürer and Antonio da Sangallo the Younger. Finally, Jeanne Peiffer considers Dürer’s integration of optical laws of perception in his technical drawings. Although this element taken from the science of optics was not retained by technicians among Dürer’s immediate followers, it was incorporated in the high tradition from Daniele Barbaro onward, throughout the 17th century. The volume’s concluding section comprises Michael Mahoney’s ‘Drawing Mechanics’, an essay interesting for various reasons. In primis, while presenting Huygens’ notes and sketches to understand the scientist’s confrontation with the clockmaker Isaac Thuret, Mahoney offers exemplary analyses of drawings from the perspective of the history of science: he actually analyzes the visual evidence in detail, working through the drawings line by line. More importantly, in the opening pages Mahoney poses questions and underscores issues that should remain firmly in the mind of all scholars interested in the interaction between art and science in the early modern period. Even though the points that he makes relate strictly to the science of mechanics, they also expand and update the conclusions he presented long ago in the essay that doomed Samuel Edgerton’s argument for a pivotal role of Renaissance linear perspective and naturalism in the scientific revolution [Mahoney 1985, 198--220]. After demonstrating that by the later 17th century those concerned with the theoretical aspects of mechanics abandoned visual representation as irrelevant to a mathematical understanding, Mahoney rightly warns against reading

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into Renaissance machine drawings rudimentary theoretical principles of the science of mechanics that were developed only in the 17th and 18th centuries. Moreover, he is sceptical about treating the drawings of Renaissance engineers as genuine moments of scientific inquiry, and about their impact on the development of scientific theories. However, rather than questioning the potential import of the study of machine drawings for the history of the scientific revolution, Mahoney’s healthy scepticism should serve as a call to understand better the development and impact of the visual reasoning and visual thinking skills shared not only by those who created and read machine drawings, but also by those who drew and reasoned through images in pursuing natural philosophical issues. 1 Although the epistemological limitations of visual representation became apparent over the course of the 17th century, many Renaissance intellectuals truly hoped that images could serve as effective tools for understanding nature. In summary, this volume offers valuable insights and provides much food for thought not only to those interested in the history of machines and mechanics, but also to all scholars of early modern science and its interaction with art in the Renaissance. In spite of the recurring editorial glitches, MIT press should be praised for offering an important book at such an affordable price, and for ensuring that the volume contains excellent reproductions of all the images referred to by each contributor. bibliography Favaro, A. 1890--1909. ed. Galileo Galilei, Le Opere. 20 vols. Florence.

1

Mahoney cites a passage from a letter of Ludovico Cardi da Cigoli to Galileo Galilei in which the painter explains Christopher Clavius’ refusal to believe in the Moon’s roughness by citing the Jesuit’s lack of ‘disegnio’. (Such an accusation is obviously completely unfounded.) In the painter’s view— which he expects Galileo to share—mathematicians should be capable of reasoning from graphic, visual evidence. Moreover, given that the issue at stake concerned the properties of a celestial body, the ability to arrive at conclusions by means of visual analysis and demonstrations was probably expected also of natural philosophers. For the original letter, see Favaro 1890--1909, 11.167--169.

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Lefèvre, W.; Renn, J.; and Schoepflin, U. 2003. edd. The Power of Images in Early Modern Science. Berlin. Mahoney, M. 1985. ‘Diagrams and Dynamics: Mathematical Perspectives on Edgerton’s Thesis’. Pp. 198--220 in J. W. Shirley and F. D. Hoeniger edd. Science and the Arts in the Renaissance. London.

Methods and Styles in the Development of Chemisty by Joseph S. Fruton Philadelphia: American Philosophical Society, 2002. Pp. xviii + 332. ISBN 0--87169--245--7. Cloth $40.00

Reviewed by Peter J. Ramberg Truman State University [email protected] Joseph Fruton is a prominent biochemist who late in life found a second career as a historian of science. He is best known for his Pfizer Prize-winning Molecules and Life [1972], and for his extensive studies of the research laboratories led by the 19th century organic chemists Justus von Liebig, Adolf von Baeyer, and Emil Fischer and by the physiologist Franz Hofmeister [Fruton 1972, 1985, 1988, 1990]. In these works, Fruton succeeded admirably in comparing pedagogical and research styles in various research laboratories. With Methods and Styles, Fruton offers his first general history of chemistry; and he has largely succeeded in preparing a short, well written, and accessible introduction to the major figures and theories of chemistry. Methods and Styles is divided into nine chapters in a fairly standard manner. Of most importance to readers of Aestimatio are chapters 1--3. Chapter 1 offers a highly compressed tour of matter theory and alchemy from ancient Greece to the 17th century. Although this chapter manages to convey a large amount of material in a succinct manner, it is unavoidably superficial and takes slight notice of recent work on alchemy in the 17th century that has demonstrated far more continuity than discontinuity in chemistry before and after the Chemical Revolution [cf. Principe 1998, Newman 1994]. 1 As a result, Fruton continues the (not entirely unreasonable) assumption long held by historians of chemistry that ‘real’ chemistry began in the 18th century; and he divides the 18th century into Lavoisier (chapter 3) and everyone else (chapter 2). Despite this emphasis given 1

Two co-authored works, Newman and Principe 2001 and 2002, appeared at the same time as Methods and Styles and make more explicit the argument for continuities between alchemy and chemistry. C 2005 Institute for Research in Classical Philosophy and Science 

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to Lavoisier, in the conclusion Fruton expresses more sympathy and admiration for Priestley than for Lavoisier. The remaining chapters are organized around the emergence of Daltonian atomism, the development of radical and type theory in organic chemistry, the formulation of ideas of valence and molecular structure in the mid-19th century, and the appearance of stereochemistry and physical chemistry. The last chapter, ‘Electrons, Reaction Mechanisms and Organic Synthesis’, is somewhat unwieldy, as it seems to cover too much material in ranging from Faraday’s study of electricity and the discovery of the electron to the emergence of electronic theories of bonding and quantum mechanics in the 20th century and to the Nobel prize winner in chemistry (1965) R. B. Woodward and his work in organic synthesis. Most of this chapter is a welcome summary of the major developments in chemistry during the 20th century, although it does not cover colloidal, inorganic, and polymer chemistry. In each chapter, Fruton’s approach is primarily biographical, introducing the background and education of the relevant chemists before moving on to their ideas. Although Fruton has written an admirable, short account of the history of modern chemistry, the principal drawback of the book is the lack of a central theme or themes organizing the actual material in the text, thus making the title somewhat misleading. In the foreword, Fruton does discuss several possible meanings of ‘styles and methods in science’; and he is influenced by Alistair Crombie’s recent categorization of six styles of scientific reasoning [see Crombie 1994, 1995], which Fruton says can be applied profitably to the history of chemistry. Unfortunately, however, these six styles are largely ignored in the rest of the text, leaving the reader to assign each chemist to one of the six styles. Fruton returns to the styles in the conclusion but only in a vague way, never specifying how particular chemists match each type of style. For example, on page 227, Fruton quotes James Bryant Conant at length from the preface to his famous case studies; yet Fruton fails to make the connection explicit by comparing this pedagogical statement to Conant’s own chemistry. As a result, throughout the book, ‘style’ simply means what the reader draws from Fruton’s presentation about each individual chemist. It does not signfiy anything that is common to chemists at any given time or place. Similarly, there is no explicit discussion of ‘methods’ and what they might mean to chemists.

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Despite this reservation, and although Fruton’s analysis does not go far beyond what historians of chemistry have said elsewhere, Methods and Styles is a clearly written, succinct account of basic personalities and issues in the history of chemistry for the non-specialist. Fruton has made extensive use of recent secondary literature and his notes will lead a novice in the history of chemistry to much of the most important literature on each topic. bibliography Crombie, A. 1994. Styles of Scientific Thinking in the European Tradition. London. 1995. ‘Commitments and Styles of European Scientific Thinking’. History of Science 33:225--238. Fruton, J. 1972 Molecules and Life: Historical Essays on the Interplay of Chemistry and Biology. New York. 1985. ‘Contrasts in Scientific Style: Emil Fischer and Franz Hofmeister, Their Research Groups and Their Theory of Protein Structure’. Proceedings of the American Philosophical Society 129:313--370. ‘The Liebig Research Group: A Reappraisal’. Proceedings of the American Philosophical Society 132:1--66. 1990. ‘Contrasts in Scientific Style: Research Groups in Chemical and Biological Sciences’. Memoirs of the American Philosophical Society 191. Newman, W. R. 1994. Gehennical Fire: The Lives of George Starkey, an American Alchemist in the Scientific Revolution. Cambridge, MA. Newman, W. R. and Principe, L. M. 2002. Alchemy Tried in the Fire: Starkey, Boyle, and the Fate of Helmontian Chymistry. Chicago. Principe, L. M. 1998. The Aspiring Adept: Robert Boyle’s Alchemical Quest. Princeton. Principe, L. M. and Newman, W. R. 2001. ‘Some Problems with the Historiography of Alchemy’. Pp. 385--432 in W. R. Newman and A. Grafton edd. Secrets of Nature: Astrology and Alchemy in Early Modern Europe. Cambridge, MA.

Sherlock Holmes in Babylon and Other Tales of Mathematical History edited by Marlow Anderson, Victor Katz, and Robin Wilson Washington, DC: Mathematical Association of America, 2004. Pp. x+ 387. ISBN 0--88385--546--1. Cloth $49.95

Reviewed by Fernando Q. Gouvêa Colby College [email protected] Over their long history, the expository journals published by the Mathematical Association of America have included many articles on the history of mathematics. The editors of Sherlock Holmes in Babylon have put together a selection of these articles. The topics covered range from ancient mathematics to the 18th century. The authors decided not to include articles on 19th and 20th century mathematics, which is a pity, since there would have been many excellent ones from which to choose. Perhaps we can hope for a second volume. The earliest article (‘Number Systems of the North American Indians’, by W. C. Eells) appeared in the American Mathematical Monthly in 1913; the most recent ones are from 2002. Several of the articles included are well known classics: Judith Grabiner’s ‘The Changing Concept of Change’ and Frederick Rickey’s ‘Isaac Newton: Man, Myth, and Mathematics’, for example, have long been standard reading assignments in my history of mathematics classes. R. Creighton Buck’s ‘Sherlock Holmes in Babylon’, which lends its name to the whole collection, is a famous account of the Plympton 322 tablet and its connection to Pythagorean triples. The articles by William Dunham on the Harmonic Series and the Fundamental Theorem of Algebra are also familiar and worth reading. As a rule, historians of mathematics will not find anything new here; but they will find careful and readable expository accounts of important bits of the history of mathematics. Further, the book will be of great interest to mathematicians who want to learn a little history and to students who are beginning to learn the history of mathematics. In some ways, a collection like this one might actually be a better starting point than the standard compendia: reading C 2005 Institute for Research in Classical Philosophy and Science 

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these articles will give the student a little bit more of the experience of historical research and debate than a textbook could. The book suffers from some inevitable limitations. First of all, the editors were constrained to choose articles from the three expository journals published by the MAA. This, I am sure, was not a great burden. Second, and more serious, the editors decided to try to cover the major themes in the history of mathematics between Ancient Mesopotamia and the 18th century. This, at times, led to the inclusion of inferior articles. The most glaring example is the inclusion of Max Dehn’s long article on Greek mathematics (tellingly entitled ‘Mathematics, 600 BC -- 600 AD’). This is a routine recounting of the ‘standard story’ about Greek mathematics, written in the 1940s when that story was taken as settled fact. Much of what Dehn says has since been challenged by one or another historian, and the book gives little indication of the new ideas on Greek mathematics that have been dominant over the last few decades. Another example is Eells’ 1913 article, presumably included for the sake of covering Native American mathematics, but quite boring and not very insightful. In their introduction, the editors say that their goal was to select the best articles and then to ‘present them in the context of modern historical research’. I do not think that they quite succeed in this. Each section comes with a foreword and an afterword, but these do little more than set the stage and point to the literature. I would have liked to see a much fuller discussion of historiographical issues, particularly in the sections on ancient and medieval mathematics. At times, the pool of available articles itself provided the m eans to put the older articles in the context of current research. Buck’s ‘Sherlock Holmes in Babylon’ presents a historical interpretation of Plympton 322 which has since been challenged. Happily, the editors were able to balance Buck’s article with Eleanor Robson’s 2002 article, ‘Words and Pictures: New Light on Plimpton 322’. This creates an interesting case study in historical method which can lead to productive class discussions. (It would have been even more fun to include Robson’s ‘Neither Sherlock Holmes nor Babylon’, originally published in Historia Mathematica, which is a direct response to Buck. But HM is not an MAA journal.) Such pairings are particularly helpful for students, who need to understand how historians investigate and discuss the issues. The

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book includes a few other examples of this sort: a couple of articles on Descartes, several on Newton, several on Euler. None of these offer as direct a contrast as Buck/Robson, but they still offer different perspectives and thus open up possibilities for discussion and for deeper understanding. For me, these groups of articles are the high points of the collection. The book is well produced and presented. My only complaint on that end is about the decision to lay the text out in two columns. The resulting pages look much more like those of a textbook that one consults rather than like pages from a book meant for reading. These articles are meant to be read, so this layout sends the wrong message. Historians of mathematics will value Sherlock Holmes in Babylon because it provides easy access to some old favorites from the historical literature, many of which are well suited for class use. Mathematicians, students, and historians of science will find it a pleasant way to learn something about the history of mathematics, particularly if they are careful to note the original publication dates of the articles.

In memoriam David Edwin Pingree (2 Jan 1933 -- 11 Nov 2005) David Pingree, University Professor and Professor of the History of Mathematics and of Classics at Brown University, died in Providence, RI, at the Miriam Hospital, on Friday, 11 November 2005. He was born in New Haven, CT, son of the late Daniel and Elizabeth (Maconi) Pingree, and lived in Providence since 1971. In 1958 he went to India to study Sanskrit. Upon his return in 1960, he became a member of the Society of Fellows at Harvard where he learned Arabic and started to accumulate lists of relevant manuscripts on the exact sciences in Sanskrit. He received his Ph.D. from Harvard University in 1960. After working for several years at Harvard and at the Oriental Institute at the University of Chicago, he came to the Department of the History of Mathematics at Brown in 1971 as the successor to Otto Neugebauer, who founded the Department. He served as the Department’s Chair since 1986, and later as its sole regular faculty member, supervising approximately ten doctoral students in History of Mathematics and in Classics at Brown and other institutions; he had planned to retire at the end of the 2005/2006 academic year. David Pingree’s published work includes more than 30 monographs and well over a 100 book chapters and articles, on subjects relating to the exact sciences (notably mathematics and astronomy), astrology, and magic, in ancient Mesopotamia, classical Greece and Byzantium, Latin Europe, South Asia, the Islamic world, and several of the cultures that linked them, such as Sasanian Iran. He based this work on sources mainly in Akkadian, Greek, Latin, Sanskrit, Arabic, and Hebrew; and collaborated with eminent scholars in many fields such as Otto Neugebauer, Edward S. Kennedy, Charles Burnett, Erica Reiner, and Hermann Hunger, to name but a few. In the study of the exact sciences in India he excelled above all others, and the same can be said for astrology in ancient and medieval times in various cultural settings. He was devoted to his students and generous in spending time to help others on an extraordinary range of topics; indeed, he was regularly consulted by colleagues who greatly benefited from his vast erudition. In effect, he was at the center of an ‘invisible college’ of scholars who worked in many different disciplines. No C 2005 Institute for Research in Classical Philosophy and Science 

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one else in the recent past has combined his level of academic skills with such a breadth of interests and such an amazing productivity of scholarly books and articles, all of the highest quality. In addition, David Pingree served at various times on the editorial boards of several periodicals and publication series in the field, such as Historia Mathematica, Journal for the History of Astronomy, Journal for History of Arabic Science, and Islamic Philosophy, Theology and Science. His numerous academic honors include the Guggenheim and MacArthur Fellowships, and election to the American Academy of Arts and Sciences, the International Academy of the History of Science, and the American Philosophical Society; in 1992 he received an Honorary Degree of Doctor of Humane Letters from the University of Chicago. He was a member of the Institute for Advanced Study and was the co-founder of the Association of Members of the Institute for Advanced Study. He also established the American Committee for South Asian Manuscripts in 1994 with the goal of cataloguing all the Arabic, Persian, and Indian manuscripts in North America and elsewhere. In 2004 a Festschrift was published with contributions by many of his colleagues, entitled Studies in the History of the Exact Sciences in Honour of David Pingree. His pioneering research is fundamental to the cross-cultural study of the nature, intellectual context, and transmission of science in the pre-modern world. David Pingree will be sorely missed by all who knew him and all who read his work. 1 Kim Plofker Visiting Assistant Professor Mathematics Institute, University of Utrecht International Institute for Asian Studies, Leiden [email protected] Bernard R. Goldstein University Professor Emeritus University of Pittsburgh [email protected]

1

Some of this information came from an obituary that appeared in The Providence Journal, 16 November 2005.

Heavenly Realms and Earthly Realities in Late Antique Religions edited by Racanan S. Boustan and Annette Yoshiko Reed Cambridge: Cambridge University Press, 2004. Pp. xiv + 338. ISBN 0-521--83102--4. Cloth $80.00

Reviewed by C. Robert Phillips III Lehigh University [email protected] Warm praise for the genesis of this volume. From the editors’ preface [vii], it began in a ‘collaborative effort aimed at bringing together relatively unseasoned scholars—that is, graduate students—and their more experienced counterparts in an environment conducive to interdisciplinary research’. More precisely [vii--viii], in a weekly seminar culminating in a public symposium in January 2001. As a (relatively) senior scholar himself, the reviewer has experienced the enormous stimulation of youthful enthusiasm and willingness to question accepted traditions, and likewise his own enthusiasm at imparting the breadth of the long experience of living with those traditions. Prospective readers can be assured of no obvious differentiation in quality, however measured, between the two groups; put differently, one could never distinguish between ‘senior’ and ‘junior’ if the essays were anonymous. A caveat for readers and reviewers. The essays in toto range generously over Mediterranean antiquity. Few will possess all the languages and scholarship to attend equally to the details of all essays. That would require, at the least, Greek, Latin, Hebrew, and Syriac. A majority of the essays involve texts in Greek and Latin, but classicists should not be complacent—few of us could claim equal comfort levels with, say, the didactic poetry of Manilius, the Greek magical papyri, the Corpus Hermeticum, and Gregory Nazianzus. Thankfully, the essays’ generous quotations from the ancient texts appear with translations, enabling basic reading and comprehension for all. But those relying on the translations will consequently be unequipped to enter fully into critical dialogue with scholarship involving those languages. Since I am a classicist by training and occupation, my review C 2005 Institute for Research in Classical Philosophy and Science 

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will devolve principally on the languages and scholarship germane to that discipline. Begin with the jacket blurb: A poignant sense of the relevance of heavenly realms for earthly life can be found not only in Judaism and Christianity but also in Graeco-Roman religious, philosophical, scientific, and ‘magical’ traditions. First there are the H-words (‘heaven’, ‘heavenly’) and Greco-Roman polytheism. It is not idle pedantry to insist that these H-words did not exist inside that latter system, because neither did the concept; in addition, all know that the Judaeo-Christian tradition took matters in a rather different direction. That is precisely the point: the Hwords are explicitly Judaeo-Christianizing concepts with a significant contemporary semantic load and thus they can become misleading ‘background noise’ in the evaluation of the Greco-Roman traditions. 1 Fritz Graf in ‘The Bridge and the Ladder: Narrow Passages in Late Antique Visions’ [19--33] squarely confronts the H-word issue via his felicitously accurate and laudably non-judgmental phrase ‘the Beyond’. Indeed, even though his essay devolves on movement to the Beyond, its opening pages [19--21] merit everyone’s close rereading; none should ever again conflate the Greco-Roman Beyond with the Judaeo-Christian Beyond. I would offer the friendly addition of Achilles’ famous reply to Odysseus in the underworld book of Homer’s Odyssey. Achilles is in the Elysian fields; Odysseus marvels that here, just as in life, Achilles is a king [Od. 11.484--486], to which the ever suave Achilles ripostes [11.488--491] that he would rather be the most miserably poor mortal on earth than a king in the underworld. Graf also provides a valuable differentiation of the ways one got to the Beyond. In the Greco-Roman tradition one ‘simply walked from here to there’ [27], although that often meant some unusual transportation circumstances such as a journey to the edge of the earth (Odysseus) or a Sibyl as companion (Aeneas). But in the Christian tradition one simply got there [27]: ‘our world and the 1

For example, Liddell, Scott, and Jones 1968, s.v. ο ρ νιο and ο ραν , gives, respectively ‘heavenly, dwelling in heaven’ and ‘heaven: in Hom. and Hes.’ with ‘heaven, as the seat of the gods, outside or above this skyey vault, the portion of Zeus’. Thanks to my spouse, Linda Henry, for calling this to my attention.

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world beyond are much too different to share simple contiguity.’ Note in passing the radical contrast between Christian and non-Christian movement. Put baldly, for the former, good people go up; bad people go down. As for the latter, good people do not go up, with very rare exceptions such as Hercules; and note that even Homer seems unclear whether or not he ascended [Od. 11.601--604]. The new gods of the Roman imperial cult likewise seem not to have gone up: one will fruitlessly search for claims such as ‘the deceased Caesar, seated at the right hand of Jupiter’. The good, when their movement is discussed at all, descend, sometimes to the Elysian fields; but sometimes, as in the Greek hero cult, they seem to stay on earth or just under it (Oedipus at Colonus). 2 As for the pre-existing gods, all knew that Zeus inhabited Olympus; but Baucis and Philemon knew him as dinner company [Ovid, Met. 8.618--724], while Homer’s Poseidon can be missed at an assembly of the Olympians [Od. 1.19--27], and seems to spend far more time in the sea than on Olympus. Of course, the gods came to earth rather less frequently in historical Greco-Roman polytheism than in the ‘good old days’ of the mythic heroes [Od. 7.201-205; Vergil, Ecl. 4.15--16]. But come they did in recognizable physical form. By obvious contrast, in the Judaeo-Christian tradition, Divinity appears sporadically (Pentecost) and sometimes in rather non-anthropomorphic guise (Burning Bush). Like all of Greco-Roman religious knowledge, the location and population of the Beyond is of a different order than Judaeo-Christian religious knowledge, although only Graf and Johnston [infra] seem especially cognizant of it. Likewise the vexed issue of the relation between religion and ‘magic’. Moderns, and their immediate predecessors, consider this a non-issue. Magic for them is bad science or bad religion or both. Or, slightly more charitably as it was put in the 19th century, magic is where ‘primitives’ with their allegedly muddled childlike thinking begin, from which they ought to evolve either to religion or science. And, on that view, the religion was Christianity, usually the Protestant version. Scholarship in recent decades, my own included, has challenged those views; but it remains passing strange

2

There exists no clearer example than the hero cults of Attica [see Kearns 1989]. Likewise the many tombstone inscriptions which imply some manner of localized presence of, or concern by, the deceased.

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that many who ought to know better still hold those views, albeit expressing them more diplomatically. The contrary view would observe that there never existed any general legal definition of magic in classical antiquity, that anything theological you did which I disliked I could then call magic and invoke appropriate secular and sacred sanctions. Magic thus constituted a term of practical polemic, a relative and judgmental term in a Greco-Roman polytheistic world where any number of cults and theologies existed, all without any obligation on anyone to participate. In short, fluidity and permeability. Christianity, by contrast, set sharp boundaries as a strategy of self-definition, both from Judaism (the whole sad adversus Judaeos tradition) and also from polytheism: for Christians, the polytheists’ divinities were either demons, that is, connected with magic, or delusions. And in a Judaeo-Christian scholarly tradition that view became, and sadly remains, normative. I mention these issues at length because, although they are mercifully absent from Christopher Faraone’s ‘The Collapse of Celestial and Chthonic Realms in a Late Antique “Apollonian Invocation” (PGM I 262--347)’ [213-232], readers should be aware of them. Few know the evidence for ancient magic as well as Faraone, and none better. He powerfully shows the lack of boundaries inside Greco-Roman polytheism, using a text from the Papyri Graecae Magicae involving Apollo and necromancy. He examines the Olympian and the Chthonian, with special emphasis on the latter. Scholarship has tended either to throw up its collective hands in despair of ever plumbing the basic distinction, or else to take refuge in the facile equation of Chthonian with the underworld and magic. How could an Olympian divinity be involved in an underworld-based ritual? The explanation, briefly discussed supra, becomes ‘Easily. Magic is bad religion.’ Faraone provides valuable background to such considerations [214--224], while his discussion points out precisely that the necromantic ritual collapses the distinction. He invokes an excellent adunaton (impossibility) for the earlier Greek traditions, namely Helios’ considering sinking into the underworld in shame [Od. 12.382-3], which becomes possible once necromantic ritual redefines the boundaries of the two realms and makes the theologically impossible the possible. I only regret that his superb essay, while diplomatically and modestly shunning scholarly polemic, does not give a hint as to how much stale conceptual baggage it rightly consigns to the garbage heap.

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So for the ‘Heavenly Realms’ part of the book’s title. The ‘Earthly Realities’ part of the title gives pause. Earthly realities mean non-religious knowledge, the day-to-day secular knowledge which obviously plays an important role in constructing information about, and images of, the Beyond. We have already seen the role of travelers’ tales, but I focus here on the physically material. For example, today we call quantum physics a part of science; and yet the ancient figure associated strongly with atomism, Democritus, in his some sixty works wrote on topics as diverse as ‘Those in Hades’ and ‘Ethical Notes’. Compare the earliest known Greek philosopher, Thales, who reportedly had astronomical interests, apparently dabbled in practical mathematics [Diogenes Laertius, Vitae 1.27] and remarked that ‘all things are full of gods’ [Aristotle, De anima 411a7--8]. Aristotle might be taken as the pre-eminent ancient scientist; and yet we must balance his preserved biological treatises against his important writings on philosophy, logic, aesthetics, and political science. That is, the boundaries between ancient science and religion ran rather differently than they do in the modern world and, indeed, it is arguable whether the ancients even recognized such boundaries. This very diversity provides the answer to why Democritus’ atomism never took hold the way quantum physics, say, has today [see Milton 2002]. Material knowledge was fragmented. Those who investigated physical phenomena labored under what must seem today crushing burdens. First, their observations and theories could not be as widely propagated as they are now. Thus, for example, while there was something approaching agreement on the names and origins of the major winds, there existed a plethora of claims from various locations for individual local winds, claims examined by scholars as different as the philologist Callimachus and the scientist-philosopher Aristotle. 3 Second, there existed only one broad explanatory strategy, which treated religion and science as a continuum rather than as two intellectual endeavors lacking interpenetration, as many today, with varying amounts of correctness and error, suppose. Third, there was a strong agonistic component, where disputes among physical theorists kept their eyes from any alternative explanatory strategy, the more so because their disputes

3

Phillips 2003 gives a selection of references.

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lay with the particular rather than the general: one has only to consider the various medical schools and the understanding of the choice (hairesis) between them. 4 That is, an almost obsessive attention to particular doctrinal differences often precluded concern to delineate the theoretical boundaries between science and other activities [see esp. Lloyd 1983, 1987]. Fourth, an underlying mathematical basis would be required to produce any sort of unified alternative way of explaining things; and the ancient mathematicians, though there were a few notable exceptions, tended to constitute a closed group that was not interested in the implications of their work for those investigating physical phenomena [cf. Netz 2002, 200--201, 215--216]. Fifth, observation from signs had its ultimate roots in divination; and in the absence of wholly physical, causal accounts of material processes, there would always be a religious component: for all their physical and theoretical observations, the Stoics and Epicureans remained in divinity’s sway, albeit in some cases a distant and relatively marginalized divinity. Sixth, and finally, there is the issue of language; without a mathematical basis (fourth point, supra) investigators were inevitably mired in the inherent imprecision of ordinary speech, a point Gadamer has made both powerfully and evocatively: Greek knowledge. . . was so much within language, so exposed to its seductions, that its fight against the dunamis ton onomaton [‘power of words’] never led to the evolution of the ideal of a pure sign language, whose purpose would be to overcome entirely the power of language, as is the case with modern science and its orientation towards the domination of the existent. [Gadamer 1975, 413]. Let us consider some particular cases. The author of the pseudoAristotelian De mundo concludes with the assertion [400a] that the phenomena never reach the abode of divinity, buttressing this claim with a passage from Homer [Od. 6.42--45]. Further, compare Od. 4.561--569 with the aforementioned passage in the Odyssey about Achilles which locates the hero in the underworld and concerns a paradisical place that will receive Menelaus after his death. Homer merely locates it at the ends of the earth, while Hesiod [Op. 171] makes the place into the Isles of the Blessed; but it is impossible to 4

See von Staden 1982: the Hippocratic works On the Sacred Disease and Airs, Waters, Places address the issue of choice explicitly.

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determine why Achilles has gone to one place while Menelaus will go to a different place. Or consider Stoicism and Epicureanism. On the Stoic view, if various elements of traditional mythologies are reinterpreted cosmologically, then the concept of a divine abode (as humans understand ‘abode’) becomes conceptually liquidated. On some Epicurean views, the gods dwell in the empty spaces of the universe (intermundia: Cicero, Nat. deor. 1.18 with Obbink 1996, 8n1]; but while this sort of intermundial space preserves an abode for the gods, it does so on terms which do not admit physical demonstration, let alone conceptualization. Overall, the Greco-Roman mythological and philosophical systems offered knowledge both fluid and fragmented without any universally accepted empirical basis. Thus, Tartarus is as far below Hades as sky is from earth [Homer, Il. 8.13--17], a distance it took Hephaestus a day to cover in free fall [Il. 1.592--593], a year for a man [Hesiod, Theog. 740--743] but nine days for an anvil [Theog. 724--725]; whereas the heavenly city after the Last Judgment is 12,000 furlongs square [Rev. 21.16]. This information is either traditional or allegedly divinely revealed; in neither case is it empirical. Moreover, there existed no agreement on the precise denizens of Tartarus. For Hesiod, Tartarus contains the Titans and Hundred-Handers [Theog. 711--819]. Mere mortals guilty of hubris were not consigned to Tartarus; indeed, for Homer, Odysseus can glimpse Tityos, Tantalus, and Sisyphus [Od. 11.576--600]. But by the fifth century, Tartarus was conceived as a place for all the hybristic, giants and mortals alike [Aeschylus, Prometh. 152--159; Plato, Gorg. 523b]; and thus Odysseus would not have have been able to view Tityos and the others, as Hesiod and Homer would have it. Overall, then, there were neither the empirical means to calculate the location of the Beyond, nor was there agreement on who was where among the various parts of the Beyond. 5 Looming over all of this was the notion that Divinity was connected with geographic locations as mortals conceived such locations—the Epicurean intermundial spaces received scant currency outside that group. Epicureanism, like Stoicism, was a philosophical preoccupation of the socio-economic elite; and given the limited literacy in 5

I note as a point of interest that the greatest concern in antiquity devolved precisely on punishments for wrongdoers; there exists far less detailed evidence about the locations for the posthumous rewards of the virtuous.

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classical antiquity, its ideas propagated beyond the small circle of adepts unsystematically and only at the most general level. That is, the idea of a Beyond was fractioned among the conditions of literacy and the traditions of knowledge; and all represent parts of a single self-confirming system predicated on religious, that is antinaturalistic, postulates. There existed no uniform tradition, and no hope of one. Thus, I would supplement Gadamer’s observation in the previous paragraph with the remark that none of the ancient speculations could provide itself with empirical support qualitatively able to win converts to its view. This is not to say that every essay should delve into ancient science even at the modest length I have just indicated. But I must complain that no essay seems obviously aware of it. Ancients could debate the location of the Beyond, how one got there, what entitled one to get there, but always from the perspective of religion. Closest to awareness is Katharina Volk’s ‘“Heavenly Steps”: Manilius, Astron. 4.119--121 and Its Background’ [34--46]. As an examination of the imagery, as a piece of textual analysis, the essay is excellent; thus she uses, rather than kowtows to, Houseman’s famous edition. But in that Manilius employs astronomical information about the heavens, the passage cries out for consideration of the technical context, such as what Manilius could have known and what he appropriated from the possibilities. I would not single out an otherwise excellent essay but rather use it as example pars pro toto. Again, consider briefly a section in Susanna Elm’s ‘“O Paradoxical Fusion!”: Gregory of Nazianzus on Baptism and Cosmology (Orations 38--40)’ on Gregory’s ‘terminology of light’ [296--315, at 305--306]. What is the relation between Gregory’s ruminations on the relation of God and light to what could be known of light at the time? Certainly issues of light and optics had interested more than one ancient scientist— Euclid and Ptolemy come immediately to mind. Equally important, what was the status of Christian paideia at the time with regard to such work, and does Gregory agree with it? Finally, take the title’s quotation ‘O paradoxical fusion’ [Orat. 38.13]. What does Gregory mean by ‘paradoxical’ (παρ δοξο )? The ancient scientists never use παρ δοξο in the modern sense, but rather in the same sense as nontechnical authors, namely, as ‘remarkable’. A superior athlete can be παρ δοξο , and Apollonius of Perga uses it of geometric proofs in his Conica [Heiberg 1891--1893, 1.4.10], a usage that is clearly not in

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the modern sense! Rather, when the logicians had to express what moderns would call ‘paradoxical’, they used δ νατον (‘impossible’). Elm’s deftly close reading of her text becomes compromised by her apparent lack of attention to non-Christian Greek. Of course, ancient science constitutes a notoriously difficult subject for those classicists who have not specialized in it. But there exists another perspective, also absent, for which the excuse is less good, that of history. Graf’s aforementioned essay certainly considers it, and so do two others: Kirsti B. Copeland, ‘The Earthly Monastery and the Transformation of the Heavenly City in Late Antique Egypt’ [142--158] and Jan Bremmer, ‘Contextualizing Heaven in Third-Century North Africa’ [159--173]. Copeland’s subsection [152--158] on the connection between the monastery and heavenly Jerusalem raises the larger context of contemporary socio-historical events. She utilizes the History of the Monks in Egypt [152] aptly, but treats it as a disembodied document. Distortion ensues: it matters that the same work provides evidence of the monks’ destruction of the polytheists’ temples [History 5.2--4], part of a changing relation between Christianity and polytheism that was conditioned in no small part by historical circumstances; elsewhere there exists even more evidence for the monks’ destruction of polytheists’ temples. Bremmer’s use of unsubstantiated or wrong historical claims unfortunately undermines a well-conceived attempt to yoke the religious and the historical. He observes [160] that ‘Christian North Africa, compared with other areas of the Roman Empire, was unusually interested in visions. . . .’ Where is the evidence both for the claim and the comparison? To make such a claim is much like claiming that the fourth century ad was unusually prone to magic by citing the frequency of references to magic in Ammianus Marcellinus. Again, Bremmer rightly begins his essay with the unexceptionable and important assertion [159] that ‘heaven was no issue’ for the initial followers of Jesus. All expected the millennium in the very near future. But he then indicates that later in the first century matters became different. Jesus’ return and the millennium were delayed, ‘yet the persecutions required an elaboration of the afterlife’ [159] to compensate. It is true that heavenly visions appear in various martyr-acts, but that is after the fact. Put differently, there exists no evidence that visions such as Perpetua’s represented any sort of doctrine of heaven [169, quoting Passio Perpet. 8]; in any case, a far better example would have been

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Saturus’ vision [Passio 11] of a heavenly garden. But I must complain vigorously about the assertion that the persecutions caused this. There simply were not that many in the first century—Nero’s persection is provable and miniscule, Domitian’s is bogus—and Christian numbers were too few. 6 Finally, Sarah Johnston’s otherwise superb ‘Working Overtime in the Afterlife; or, No Rest for the Virtuous’ [85-100] tells us [89] that Alexander the Great had been declared a god. It is true that there exist three circumstances which indicate some interest in godhood: the Oracle of Zeus-Ammon, the affair of BactriaSogdiana, and the alleged ‘deification decree’. But even a cursory look at the voluminous Alexander scholarship reveals that the deification is far from proven, that many Alexander specialists indeed categorically reject it. That is, while I commend efforts to get outside of the text-based readings that many of these essays offer, those who do, more often than not, tend to rely on common misconceptions combined with inattention to detail. Only Graf and Faraone have done otherwise; would that their colleagues in this volume, junior and senior alike, had attended their example. For everyone except Graf and Faraone, Greco-Roman polytheism and Christianity are monoliths. For Christianity, there is no hint in any essay of the extraordinary variety of early Christian doctrines, the use of ‘heresy’ as a polemical term by the various competing groups, and the late and compromised arrival of the concept of orthodoxy. As for Greco-Roman polytheism, the impossibility of defining, say, Roman religio or Greek deisidaimonia, both usually and totally inadequately rendered in English as ‘religion’ and ‘superstition’, appears more honored in the breach than the observance. 7 In both cases, unquestioningly conceptualizing ancient knowledge as monolithic wholes variously compromises the many fine specific points made in each essay. This is not a bad book. Quite the contrary. It is good to have such a rich collection of uniformly strong essays so attentive to the texts of the diverse cultures and religions of classical antiquity. It is good, too, to promote intellectual interaction between junior and senior scholars. But it is not good to see widespread avoidance of ancient science, pace the jacket blurb, and the implications, or lack

6 7

See, respectively, Frend 1967, 156--162 and Hopkins 1998. See, respectively, Latte 1960, 38--41 and Martin 2004.

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thereof, of ancient science. It is right, of course, to attempt sympathetically to enter into the ancients’ mindset. But passim I receive the distinct impression of belles-lettres, of relentless analysis of the texts (fine) absent concern for the physical realities (not so fine) and conceptions of those realities in the world from which those texts originated—in short, that the heavenly realms are in place and the earthly realities are left to take care of themselves. Put less charitably, after reading the current volume it is all too easy to ask ‘So what?’ ‘Earthly Realities’ will not be banished so easily. bibliography Frend, W. 1967. Martyrdom and Persecution in the Early Church. New York. Gadamer, H.-G. 1975. Truth and Method. New York. Hopkins, K. 1998. ‘Christian Number and Its Implications’. Pp. 85-226 in Journal of Early Christian Studies 6.2. Kearns, E. 1989. The Heroes of Attica. London. Latte, K. 1960. Römische Religionsgeschichte. Munich. Liddell, H. G.; Scott, R.; and Jones, H. S. 1968. edd. A GreekEnglish Lexicon with a Supplement. Oxford. Lloyd, G. E. R. 1983. Science, Folklore and Ideology. Cambridge. 1987. The Revolutions of Wisdom. Berkeley. Martin, D. 2004. Inventing Superstition. Cambridge, MA. Milton, J. R. 2002. ‘The Limitations of Ancient Atomism’. Pp. 178-195 in Tuplin and Rihll 2002. Netz, R. 2002. ‘Greek Mathematicians: A Group Picture’. Pp. 196-216 in Tuplin and Rihll 2002. Obbink, D. 1996. Philodemus on Piety. Part 1. Oxford. Phillips, C. R. 2003. ‘Windkult’. Pp. 518--522 in H. Cancik and H. Schneider edd. Der Neue Pauly 12.2. Stuttgart. Tuplin, C. J. and Rihll, T. E. 2002. edd. Science and Mathematics in Ancient Greek Culture. Oxford. von Staden, H. 1982. ‘Hairesis and Heresy: The Case of the hairesis iatrikai’. Pp. 76--100 in B. F. Meyer and E. P. Sanders edd. Jewish and Christian Self-Definition. Vol 3. Philadelphia.

Modelli idrostatici del moto da Aristotele a Galileo by Monica Ugaglia Rome: Lateran University Press, 2004. Pp. 274. ISBN 88--465--0277--9. Paper ¤ 19.00

Reviewed by Paolo Palmieri University of Pittsburgh [email protected] This book is comprised of two main parts, an appendix, a name index, and an index locorum. There is no bibliography. In the first part, Monica Ugaglia argues that Aristotle’s physics has mostly been misinterpreted by generations of Aristotelian commentators who failed to understand the hydrostatic model of motion on which it was built. In particular, many commentators failed to realize the role played by the medium and the void. In the second part, the author argues that, while laboring under the delusion of overthrowing Aristotle’s physics, both Giovanni Battista Benedetti (1530--1590) and Galileo Galilei (1564--1642) actually reinstated its original theoretical core. Thus, in Ugaglia’s view, Benedetti’s and Galileo’s merit was not that of building a novel hydrostatic model of motion on the basis of Archimedes’ theory of flotation, but that of freeing motion theory from Aristotle’s hydrostatic model. The appendix is a brief essay on Benedetti’s theory of motion. So much for the thesis of this book. It is an ambitious project that raises expectations of fascinating insights. How does the author go about substantiating her thesis? Unfortunately, I must confess that I was disappointed by the superficiality of the arguments put forward by Ugaglia, and by the general paucity of historical and philosophical scholarship. I will give a few examples in order to illustrate my negative conclusions. Ugaglia starts by grandly asserting that la necessità di stabilire cosa sia da intedersi per natura è ovviamente all base di qualsiasi ricerca fisica, essendo la fisica per definizione lo studio della natura. [15] C 2005 Institute for Research in Classical Philosophy and Science 

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the necessity of establishing what one should mean by ‘nature’ is obviously the basis of any physical inquiry, since physics is by definition the study of nature. This is not exactly a crystal-clear statement by which to open a book. I do not know whether physics is by definition the study of nature. An abundance of literature in the philosophy of science has made me wary of such general assertions. Perhaps some qualifications are needed. Biologists interested in cancer research study the mechanisms of cell senescence, for instance, not physicists. Yet few would deny that cancers and cells belong to nature in some sense, and at the same time that biology is not physics. I also very much doubt that contemporary physicists and biologists base their inquiries on a preliminary agreement concerning the general meaning of ‘nature’. One would have hoped that such sloppiness was incidental. In fact it is rather common throughout the book. Worse, it appears to vitiate the author’s theses and conclusions, as the following considerations will make clear. At one crucial point the author claims that in Physics 228b26-229a1, Aristotle ‘unequivocally’ asserts the necessity of distinguishing between weight and specific weight. 1 Now the passage quoted in support of this claim is rather obscure (at least to me). Here is the text in the Barnes/Oxford edition: In some cases the motion is differentiated by quickness and slowness: thus if its velocity is uniform a motion is regular, if not it is irregular. So quickness and slowness are not species of motion nor do they constitute specific differences of motion, because this distinction occurs in connection with all the distinct species of motion. The same is true of heaviness and lightness when they refer to the same thing: e.g., they do not specifically distinguish earth from itself or fire from itself. [Barnes 1984, 1.386--387] I see no way of reading this passage as stating the necessity of distinguishing between weight and specific weight, let alone unequivocally.

1

‘Aristotele ribadisce in modo inequivocabile la necessità di distinguere tra peso e peso specifico’ [59].

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Yet this is the sort of evidence that Ugaglia relies on in order to persuade the reader that Aristotle’s motion theory was in its essence a hydrostatic theory. Dealing with specific weight, an essential ingredient of the hydrostatic theory of flotation, requires a mathematical treatment of magnitudes that is nowhere to be found in Aristotle. If you want to understand specific weight, you had better read Archimedes on floating bodies or Galileo’s De motu (ca 1590). It seems to me that Ugaglia is here carried away by her unquestioned presuppositions. This is all the more surprising in view of the fact that in the introduction Ugaglia warns the reader not to take references to modern terminology in the book as attributing modern concepts to Aristotle, but as a way of clarifying discussion [see Avvertimento importante, p. 13]. She should have been more guarded against falling into this anachronistic pitfall herself! As to the paucity of her historical scholarship, one is struck by the lack of evidence brought in support of the strong claim that Aristotle’s physics has mostly been misinterpreted by generations of Aristotelian commentators who failed to understand the hydrostatic model of motion on which it was built. The reason why Ugaglia fails to bring such evidence to bear is painfully obvious. It would have been a mammoth task to substantiate such a claim, a task beyond the capacity of a single scholar. 2 Forget ancient and medieval commentators for a moment, and just think about Renaissance ones. In this case, we have a splendid bibliographic monument which makes clear once and for all why such an enterprise could not possibly be achieved by one scholar. I am referring to Charles Lohr’s catalog of Renaissance commentaries on Aristotle [Lohr 1988--1995]. 3 Cast a glance at Lohr’s list and you will be convinced. The second part of the book is even more perplexing. The whole analysis of Galileo’s De motu is marred by the same sloppiness that

2

The only commentator discussed at length and referenced in the index locorum is John Philoponus [273]. Ugaglia relies mostly on secondary sources. 3 The first volume of Lohr 1988--1995 is a 500-page collection of material previously published. It contains a list in alphabetical order by the author’s name of commentaries on Aristotle from 1500 to 1650. The second volume is an index listing the opening (incipit) and closing (desinit) lines of each of the commentaries.

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we have already noted in the part on Aristotle. For example, we read sentences of the following tenor: L’ introduzione del vuoto come sistema di riferimento assoluto permette a Galileo di scomporre in modo univoco il peso apparente del corpo in peso reale. . . e spinta del mezzo. . . [221] The introduction of the void as an absolute frame of reference allows Galileo to decompose univocally the body’s apparent weight into real weight. . . and the medium’s thrust. . . Does Galileo speak in De motu of absolute frames of references? No. Can the void be a frame of reference or an ‘absolute’ frame of reference? I really wonder. Ugaglia concludes her analysis of Galileo’s De motu as follows: Così, la profondità e la coerenza estreme dell’ analisi con cui Aristotele nega l’ esistenza del vuoto sono state penalizzate dall’ averne l’ esperienza posteriore invalidato il risultato, mentre in base a quella stessa esperienza . . . vengono accettate (e spesso indicate ad esempio) le ingenuità filosofiche che stanno alla base delle argomentazioni di Galileo [235] Thus, the utmost depth and consistency of the analysis by which Aristotle denies the existence of the void have been penalized by the fact that subsequent experiments have voided it; whereas, on the basis of those experiments, the philosophical naiveté which is at the root of Galileo’s arguments is accepted and often brought forth as a model. So, for Ugaglia, the fact that subsequent experiments (not mentioned in more detail) have finally done away with the utmost consistency with which Aristotle denies the existence of the void raises no questions about the utmost consistency of that analysis. Further, it raises no questions about the supposed philosophical naiveté of Galileo’s arguments. Perhaps it is Ugaglia’s naiveté that mischievously shows up here. How about the thesis that Benedetti’s and Galileo’s merit was that of freeing motion theory from Aristotle’s hydrostatic model? In what sense is ‘merit’ used here? No discussion of this portion of Ugaglia’s thesis is to be found in the whole book. It looks as if it is an artifact hastily appended to the introduction ex post facto, since it neither guides her research nor receives supporting argument.

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To conclude this brief review, I think that the project of the book is fascinating and ambitious but that the author fails to carry it out in too many ways. bibliography Barnes, J. 1984. ed. The Complete Works of Aristotle. 2 vols. Princeton. Lohr, C. H. 1988--1995. Latin Aristotle Commentaries. 2 vols. Florence.

Affinity, That Elusive Dream: A Genealogy of the Chemical Revolution by Mi Gyung Kim Transformations: Studies in the History of Science and Technology. Cambridge, MA: MIT Press, 2003. Pp. xiv + 599. ISBN 0--262--11273-6. Cloth $55.00, ¤ 36.50

Reviewed by Noel G. Coley The Open University, UK [email protected] Theoretical chemistry had reached an impasse in the late 18th century, just as its horizons were expanding. New substances were being discovered, but the fundamental components of chemical substances and how they could be isolated had puzzled chemists for a long time. The ancient Greeks had imagined the four elements—earth, air, water, and fire—to account for the make-up of all matter and to these Paracelsus in the 16th century had added three further substances: sulfur, mercury, and salt. These basic principles, each of which was thought to convey certain chemical and physical properties, persisted through the 18th century, along with methods of analysis introduced by the alchemists, though sharp criticisms about their inadequacy for use by chemists had already been expressed in the 17th century by Robert Boyle. In practice, 18th-century chemistry was little more than a technical skill used in various industries or subordinated as a mere assistant to medicine. There were no ‘professional’ chemists and most of those who were interested in the subject pursued it as a hobby. In England, the 18th century saw the rise of pneumatic chemistry with the discovery of many new gases or ‘airs’, including Priestley’s ‘dephlogisticated air’ (oxygen), while in France chemical philosophy commanded more attention and fresh studies on chemical composition and the concept of affinity were made. There were also attempts to relate chemistry to physics and so develop a more quantitative structure. Lavoisier’s synthesis of these developments in the 1780s, his pragmatic definition of the chemical element, and his demonstrations that neither air nor water could be truly considered to be elementary led to the displacement of phlogiston by the oxygen theory of combustion; and together C 2005 Institute for Research in Classical Philosophy and Science 

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with the revision of chemical nomenclature and the introduction of new analytical techniques, these changes culminated in the Chemical Revolution. Subsequent developments eclipsed much of the earlier work. In recent years, historians have reassessed Lavoisier’s influence in changing the theoretical basis of chemistry; but we should not lose sight of the long history of the subject prior to the Chemical Revolution, nor should it be forgotten that the radical changes introduced by Lavoisier and his contemporaries grew out of chemistry as they found it. In what she modestly calls ‘an interpretative essay’, Mi Gyung Kim sets out to trace ideas about chemical composition in 18thcentury France. She discusses the nature of chemical principles (i.e., elements) and the role of affinity; ideas that shaped chemical theory in the 17th and 18th centuries. She compares the work of academic researchers at the Académie des Sciences and practical demonstrators at the Jardin du Roi where, influenced by Boyle, Nicholas Lemery introduced corpuscular explanations together with traditional chemical principles supposedly isolated during distillation analyses. Lemery expressed doubts about the validity of the five principles introduced by Estienne de Clave; 1 and like Boyle he preferred the gentler processes of solution analyses as did others including his contemporary at the Académie, Samuel Cottereau du Clos, and Etienne-François Geoffroy. To demonstrate the potential of solution analyses for elucidating chemical composition, Geoffroy constructed affinity tables; but distillation analysis, which was supposed to isolate the chemical principles, was not entirely displaced until Louis Lemery, son of Nicholas, demonstrated the inadequacy of distillation as an analytical tool. Thus, in the early 18th century, two fundamentally different approaches to chemical composition proceeded side-by-side. On the one hand, distillation analysis was thought to isolate the elementary principles of substances while, on the other, solution analysis, based on relative affinities, revealed the composition of the products. Kim explores the nature and meaning of Geoffroy’s affinity tables, comparing those who regarded them merely as a method of classifying collections of chemical observations with others who used them to explain chemical composition using crude notions of purity.

1

These were water or phlegm, earth, mercury or spirit, sulfur or oil, and salt.

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In solution analyses, displacement reactions were crucial to the separation of stable chemical compounds and, when Louis Lemery suggested that heat or fire might also act as a ‘solvent’, the concept of ‘displacement’ could be applied to those parts of the table that still depended upon distillation analysis. The German chemist Georg Ernst Stahl had first come to notice through his studies on salts, and Geoffroy used Stahl’s data from solution analysis as the basis for an ‘order of selectivity’ in chemical reactions. Geoffroy also discussed Stahl’s ideas about the transference of phlogiston between substances, specially those involving sulfur. It was through this application of fire in Geoffroy’s affinity tables that Stahl’s ‘phlogiston’ was introduced into French chemistry. In the 1720s, phlogiston was employed as the basis of a universal theory by Guillaume François Rouelle, a popular lecturer at the Jardin du Roi, and Pierre Joseph Macquer, his most famous pupil. Macquer wrote books promoting chemistry as a public science and seeking to situate it among the other sciences in a position of importance equal to medicine. By the mid-18th century, phlogiston was accepted as a principle that could be fixed in bodies, transferred from one body to another, or released as heat or fire. Thus, phlogiston figured along with the other principles as a component of chemical composition. Growing information on chemical reactions yielded ever more complex affinity tables during the 18th century, culminating in 1775 in the definitive affinity table of the Swedish master of chemical analysis, Torbern Bergman. This work showed experimental chemistry at its best and encouraged chemists to seek the laws of chemical combination. In the second quarter of the 18th century, French chemists were influenced by the philosophical chemistry of the Dutch physicianchemist Hermann Boerhaave. Rouelle introduced Boerhaave’s concept of ‘instruments’ of chemical change and held that the four ancient elements could each be both principle and instrument in turn, but that the most important instrument was phlogiston. About 1630, it was observed that when certain metals are calcined they gain in weight. This anomaly, overlooked by many chemists, became a problem for interested amateurs. In particular, it raised a difficulty for those who wished to bring chemistry into line with the quantitative physical methods introduced by Newton. Many fanciful attempts were made to account for this gain in weight, including the idea that phlogiston might show levity instead of gravity. The problem

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could not be solved satisfactorily, but the overall effect of such arguments was to introduce quantitative measurements into theoretical chemistry. In a lengthy discussion of Lavoisier’s work, Kim demonstrates his interest in affinity as well as his opposition to phlogiston. The ‘Arsenal Group’ led by Lavoisier included Claude Louis Berthollet, Antoine François de Foucroy, and Guyton de Morveau, the authors of the new chemical nomenclature. In addition to their support for the antiphlogistic theory, these four also sought to develop the notion of chemical affinities, to discover a means of quantifying this concept, and to establish a comprehensive chemical theory based on affinities and constitution rather than on principles. Among these four chemists, the joint authors of the Chemical Revolution, Kim singles out Berthollet as offering ‘the best guide to tracing its successes and failures’ [393]. She devotes her final chapter to a detailed investigation of Berthollet’s intellectual development, his success in turning the study of chemical composition away from the isolation of principles and towards the study of affinities, and his failure to complete his program in the absence of a satisfactory method of quantifying the concept of affinity. Berthollet, who began by studying the affinities of acids, alkalis, and salts following Macquer’s ideas on affinity, had a broader understanding of chemistry than Lavoisier. He was involved in many different aspects of the subject—industrial, experimental, and theoretical. He supported the phlogiston theory until 1785, when the decomposition of water made it impossible any longer to identify phlogiston with inflammable air. He then accepted Lavoisier’s system and began to argue as strongly in favour of it as he had formerly argued in support of phlogiston. Napoleon I sought to use Berthollet’s chemical expertise and his reputation grew until he was able to purchase a mansion furnished with a chemical laboratory at Arcueil, near Paris. Here he gathered a research group in his aim to establish a theory of chemical composition based on the joint action of affinity and heat. The Society of Arcueil became the center of French chemistry in Napoleonic France. However, Berthollet’s affinity still lacked quantification and his program failed to satisfy chemists. He disagreed with Bergman’s notion of ‘elective affinity’ that was based on total displacement reactions between salts in solution, as he thought that this introduced inaccuracies into the results of chemical analysis by

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suggesting that precipitants from reactions in solution were pure and uncontaminated. Berthollet, knowing that this was not true, also discredited affinity tables and set out to construct an improved model of chemical action more firmly based on the laws of mechanics. Believing that the mass of a given substance present in a reacting mixture must affect the direction in which the reaction would proceed, he introduced a form of ‘mass action’ in which the concept of affinity could play an important, though still unquantified, role. Kim places the Chemical Revolution in direct relation to all these ideas and argues that it was through attempts to quantify the affinity approach to chemical composition that the laws of chemical combination were seen to be essential: for, if the principles contained in bodies determined their properties, including affinities, it should be possible to conjure up an infinite variety of compounds with minute shades of difference in their composition. The specificity of affinities curtailed this realm of speculative possibilities. [436] The new theories introduced by the Chemical Revolution, including the corpuscular ideas introduced first by Boyle and later by Dalton, may then be seen as developments in a long continued endeavor to understand the complexities of chemical constitution: as Kim asserts, In order to elucidate the relationship between the chemical revolution and the chemical atomism, then, we must trace the revolutionaries’ articulation of the affinity program rather than their antiphlogistic path. [436] By concentrating upon the development of chemical ideas in 18th-century France, Kim has provided a detailed ‘genealogy’ for the French roots of the Chemical Revolution. However, she touches only lightly upon German chemistry, where the new ideas emanating from France were slow to take root. Swedish analytical chemistry of the period also receives scant attention, and she makes only passing reference to chemical advances in England and Scotland which also played an important part in the Chemical Revolution. Consequently, her claim to provide ‘a genealogy of the chemical revolution’ is only partially fulfilled. Nevertheless, her richly annotated discussion provides a fresh account of the rise of 18th-century chemical thought in France. She offers an important alternative context for studies of the

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Chemical Revolution in which the contributions of Lavoisier and his contemporaries are portrayed as evolving out of long-held ideas. Affinity in chemistry, like vital force in physiology, energy in physics, or natural selection in biology, was an ambiguous notion that changed constantly depending on the methods used to investigate it. Nevertheless, she argues that broad concepts such as these serve to define disciplines in general terms and deserve the attention of historians of science for that reason. They are concepts that serve a limited purpose and are discarded when they can no longer be reconciled with developments and cease to serve a useful purpose; nevertheless, Kim suggests that their life-cycles have much to teach us about the growth of scientific disciplines. Extensive bibliographical references, notes, and comments, located all together at the end of the book and covering over 70 pages, range widely across chemical and other relevant studies, and reveal the extent of the author’s research into this subject. Similarly, in a wide-ranging bibliography, all the important works with a bearing upon this study are cited. The list includes reference to many relevant original manuscripts, primary and secondary sources, books, and single articles; and it bears eloquent witness to the breadth and depth of scholarship which mark this important study. Future works on the Chemical Revolution will need to take account of such a detailed, well-documented study and of its seminal revisionist approach to the intellectual history of 18th century French chemistry. bibliography Boyle, R. 1661. The Skeptical Chymist. fac. edn. London: Dawsons of Pall Mall, 1965. Crosland, M. 1967. The Society of Arcueil: A View of French Science under Napoleon I. Cambridge, MA. Duncan, A. M. 1962. ‘Some Theoretical Aspects of EighteenthCentury Tables of Affinity’. Annals of Science 18:177--194, 217-232.

A Critical History of Early Rome: From Prehistory to the First Punic War by Gary Forsythe Berkeley: University of California Press, 2005. Pp. xiv + 400. ISBN 0-520--22651--8. Cloth $45.00, ¤ 29.95

Reviewed by Michael P. Fronda McGill University [email protected] As Gary Forsythe points out in the preface to A Critical History of Early Rome: From Prehistory to the First Punic War, since the 1990s there has been a ‘major reawakening of interest’ [4] in the study of early Roman history. Indeed, the last decade has witnessed among other works the publication of the first three volumes of Oakley’s magisterial Commentary on Livy Books VI--X [1997--2005] sandwiched between Forsythe’s new book and the T. J. Cornell’s Beginnings of Rome [1995]. In effect, A Critical History of Early Rome is Forsythe’s response to this last work. Compared to ‘hypercritics’ of the past generations, who assumed that ancient authors invented the bulk of early Roman history, Cornell adopted a relatively trusting attitude toward the ancient literary sources for early Roman history. Cornell argued that the ancient literary tradition contains a good deal of historical material, and that we must differentiate between the kernels of truth that form the narrative framework and the later layers of narrative detail and embellishment that have been superimposed. While Forsythe does not subscribe completely to the hypercritical school, he is far more doubtful about the historicity of the literary tradition. Thus, an overarching theme of A Critical History of Early Rome is how few details we know or can know about the early development of the city that would eventually govern one of the most successful empires in the world. Before continuing, the reviewer should state his own methodological biases. Despite advances in archaeological methods and the accumulation of material culture data, our understanding of early Roman history is still rooted in the interpretation of narrative sources. C 2005 Institute for Research in Classical Philosophy and Science 

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Cornell has shown that while the historian must not approach the sources uncritically, one can plausibly reconstruct events by using these challenging documents. This is not to say that Forsythe’s reconstruction of events is implausible; rather, by and large, the analysis and interpretations found in A Critical History of Early Rome are well argued and internally consistent. They are, however, based on a fundamentally different starting assumption: that the literary tradition cannot be trusted to provide even a reliable narrative framework, so the historian must be willing to throw out literary evidence and consider other explanations. Once free from the constraints of the narrative sources, Forsythe can propose an interesting and at times revisionist history of early Rome that is internally consistent, and in the absence of outside evidence can neither be proved nor disproved. The reviewer must admit that he finds Cornell’s approach more convincing in general, and that Forsythe’s systematic mistrust is rather too pessimistic and his more speculative reconstructions unnecessary. A brief introduction establishes that A Critical History of Early Rome will follow, broadly, the organization of the first ten books of Livy’s Ab urbe condita, our most important narrative source for the period in question. However, chapter 1 relies mostly on archaeological evidence to offer a general overview of Italian prehistory from the spread of agriculture to Italy (ca 5000 BC) to the rise of iron metallurgy on the peninsula in the tenth and ninth centuries BC. Here too Forsythe is pessimistic about his sources, emphasizing how archaeological material is preserved haphazardly and may not be representative of the culture that produced it. Moreover, material remains may shed light on some aspects of a culture but simply cannot answer other questions. Despite the gloomy assessment, a few interesting observations emerge, such as the fact that excavations of Neolithic settlements in Apulia have turned up obsidian fragments from Sicily, suggesting long distance trade at even this early date, while the bones of Forsythe, pigs, sheep, and the like imply the domestication of animals. There follows a fascinating discussion of a frozen man found in the Alps in 1991, whose corpse has been dated with Carbon 14 to about 3500--3000 BC. An analysis of the body and of the goods he carried suggests, among other things, trade across the Alps in prehistoric times. First, the working of bronze reached Italy from societies in central Europe and the Aegean, then iron metallurgy reached the peninsula by about 900 BC, and finally ‘regional

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differences begin to manifest themselves in the archaeological record with the coming of the Iron Age’ [26]. Chapter 2 examines the development of Etruscan and, eventually, Latin cultures in archaic Italy, 800--500 BC. Forsythe emphasizes how the Greeks and Phoenicians influenced indigenous Italian peoples. Phoenicians and Greeks traded with and colonized southern Italy, bringing with them important cultural artifacts such as the alphabet and the notion of the city-state (polis). In particular, Greeks traded heavily with the Etruscans, who in turn extended their cultural influence as far south as Campania. As for the communities of Latium, archaeological evidence suggests contact with the Greeks, either directly or with Etruscans as intermediaries. However, Forsythe argues against the traditional view that Latin communities were completely dominated by Etruscan culture. Rather, wealthy tombs at Castel di Decima and Praeneste suggest a Latin manifestation of a ‘larger aristocratic koine’ in Italy [58]. Even though Latium lacked the natural resources found in Etruria, especially deposits of metal ore, by the end of the sixth century, advancements in metallurgy, ceramic production, and agriculture, and the rise of local elites, and socially and politically differentiated populations transformed Latin villages into sophisticated city-states. Before turning his attention from Latium in general to Rome in particular, Forsythe provides a brief but useful survey of the literary sources for early Roman history [chapter 3]. Considering that A Critical History of Early Rome is meant, at least in part, as a response to Cornell’s Beginnings of Rome, it is not surprising that Forsythe repeatedly stresses the potential weaknesses of the literary sources for early Roman history. For example, Livy and Dionysius are the most important narrative sources; but they wrote centuries after the events they describe, and the annalistic sources they draw upon are often unreliable. Variant versions preserved by Diodorus Siculus do not represent an earlier (and, therefore, more reliable) source tradition. Forsythe expresses scepticism about not only the survival of early legal documents, but also about the ability of authors in late republic who cite them (such as Cicero) to understand the archaic Latin that the documents would have been written in, even if such documents did manage to survive. Forsythe cites both Cicero’s and Livy’s criticism that family histories were full of exaggeration. Finally, Forsythe assumes that many depictions of events in Roman

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history are patterned on Greek stories (or even later events in Roman history). Although Forsythe leaves open the possibility that a kernel of truth might exist in the sources, the withering assessment leaves the reader with the clear impression that he will not find that kernel beneath the heap of fabrications and exaggerations. Chapter 4 looks at Rome during the Regal Period, traditionally 753--509 BC. Predictably, Forsythe argues that there is likely little of historical value in the stories about Rome’s legendary seven kings. Forsythe agrees with Cornell’s assertion that Rome was not dominated by the Etruscans during the regal period, but does argue that there must have been cultural interaction between Romans and Etruscans. Forsythe concludes that Rome was a thriving city-state by the sixth century BC, and clearly the most important state in Latium by ca 500 BC. The picture of Rome as a city-state that was influenced by Etruscan culture is consistent with Forsythe’s general discussion of Latium in chapter 2. A brief discussion on archaic Roman religion comprises chapter 5. In large part, the chapter is an introductory survey, including a summary of various Roman deities and the calendar of religious festivals; it concludes that Roman religion should be seen as a local variant of the shared Italian cultural koine discussed in chapter 2. However, Forsythe does make one important argument that lays the groundwork for subsequent chapters. According to Forsythe, since early priesthoods were few in number and probably restricted to specific aristocratic families, they would have been highly valued offices. Moreover, the access to priesthoods would be critical in the self-definition of the patrician order. This last point is a key component to Forsythe’s interpretation of the so-called Struggle of the Orders, the two-century long conflict between the patrician and plebeian classes which according to the literary tradition dominated early Roman history, and which is analyzed in the central portion of A Critical History of Early Rome. Chapter 6 looks at the beginning of the Republic down to the middle of the fifth century BC. According to the literary tradition, the patrician class held a monopoly on high magistracies, especially

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the two annually elected consuls 1 who governed Rome after the fall of the monarchy, at least until the Sexto-Licinian Laws in 367 BC opened the consulship to plebeians. However, the consular fasti— a list of consuls preserved partly in literary sources and partly in inscriptions from the early imperial period—record a number of plebeian names as consul in the fifth century BC. Scholars who accept the literary tradition assume that these names are mistakes or later fabrications, thus privileging the literary accounts over the consular fasti. Forsythe argues that since the consular fasti were likely derived from pontifical records, they are probably more trustworthy than the literary accounts. Therefore, Forsythe rejects the tradition that the early consulship was limited to patricians, arguing instead that a patrician monopoly on the consulship emerged only in the late fifth century BC. Picking up on his discussion of priesthoods in chapter 2, Forsythe argues further that the patriciate was probably a hereditary priestly class, so that the patrician class was fundamentally religious not political in nature. Forsythe rejects as a fabrication the so-called First Succession of the Plebs (494 BC), which according to tradition yielded the creation of the office of plebeian tribune whose function was to protect the plebeians from overbearing patricians. Forsythe speculates that the tribunate was originally created merely as a domestic office to complement the foreign/military consuls, and had little to do with the supposed conflict between patricians and plebeians. There are three serious challenges to this revisionist account of the events traditionally related to the Struggle of the Orders. First, one must account for the ban on marriages between patricians and plebeians. A provision in the Twelve Tables, Rome’s first law code (traditionally dated 451--450 BC), reportedly prohibited patricianplebeian intermarriage. Since a Roman father had the legal authority to forbid his child from marrying someone of whom the father did not approve, there was no need for such a provision unless some patricians and plebeians did intermarry. The marriage ban was repealed by the Lex Canuleia (445 BC), and Livy portrays the passing of Lex Canuleia as bound up with the plebeian efforts to gain access to the 1

Forsythe argues that the two consuls were originally called Praetors, and that in 367 BC the office was changed to the Consulship while the title of Praetor was given to a newly created judicial office. The nomenclature is not important for the arguments that follow, so the reviewer will refer to the two magistrates as consuls.

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consulship. The ancient tradition makes perfect sense if we assume that wealthy plebeians sought access at least for their descendants to offices that were restricted to the patrician rank, that some patricians were obliging presumably so they could forge alliances with prominent plebeian families, and that some patrician families sought to preserve their political privileges. Second, the rather mysterious office of consular tribune must be explained. In most years between 444 and 367 BC, the Roman state was headed not by two consuls but by a board of consular tribunes, usually numbering between three and six. The consular tribunate was primarily a military office lacking a religious dimension, and it was open to plebeians. The creation of the office can be seen as a compromise wherein plebeians gained access to a high office, but left the consulship restricted to patricians. The fact that the use of consular tribunes became the norm rather than the exception by the beginning of the fourth century BC could reflect the increasing political influence of wealthy plebeians in the years leading up to the Sexto-Licinian Laws (367 BC). Third are the Sexto-Licinian laws themselves, which according to tradition opened the consulship to the plebeian rank, and after which the office of consular tribune disappears (presumably rendered unnecessary now that plebeians could be consul). These challenges are addressed in the following two chapters. Chapter 7 analyzes Roman society in light of what we know about the Twelve Tables, concluding that ‘a critical examination of the ancient historical tradition surrounding the codification of the Twelve Tables leaves very little worthy of credence’ [233]. The most controversial section of the chapter deals with the ban on patricianplebeian intermarriage. Forsythe argues that the marriage ban is probably not historical, and that references to a prohibition against patrician-plebeian intermarriage are likely the product of confusion on the part of later historians (such as Livy and Cicero) who misunderstood the exact meaning of the archaic and legalistic Latin in which the Twelve Tables were written. A key piece of Forsythe’s supporting evidence is Cicero’s comment [De leg. 2.59] that Sextus Aelius Paetus misunderstood the meaning of a word (lessum) in his redaction of the Twelve Tables. The argument thus relies on speculation that the learned jurist Cicero could point out the meaning of difficult archaic terms in the Twelve Tables that others misunderstood but basically misunderstood the nature of the marriage ban, one of the

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most notorious provisions of the Twelve Tables. And what about the Lex Canuleia, which repealed the marriage ban? Without a prohibition against patrician-plebeian intermarriage in the Twelve Tables, there would have been no need for a law to repeal it, so Forsythe assumes that the Lex Canuleia has also been misunderstood by Roman authors. 2 This is a perfect example of Forsythe’s methodological approach. The arguments are internally consistent, and there is no way to disprove them once one adopts such a highly skeptical view of the literary evidence we do possess. Thus, if one finds plausible that the marriage ban was unhistorical, then one will find equally compelling the suggestion that the relationship of the Lex Canuleia to the supposed ban is equally confused and should be rejected. Chapter 8 discusses the period from 444 to 367 BC, and thus deals with the consular tribunes and the Sexto-Licinian Laws. According to Forsythe, the explanation for the introduction of the consular tribunes is to be found not in a political struggle between patricians and plebeians, but in the immediate needs of a growing citystate. As the population of Rome grew, Roman society became more complex; and as Roman expansion brought Rome into more and more difficult wars with neighboring communities, two consuls were simply not enough to deal with the business of the state. This is plausible, though the same needs of the state could have been met by increasing the number of consuls or by creating additional, subordinate offices to relieve the administrative burden on the consuls. One suspects, therefore, that there must have been a political component to the creation of the consular tribunes, which would be explained by the patrician-plebeian dichotomy and a struggle on the part of plebeians to gain access to high office. As noted above, the fact that the consular tribunate ceased to exist once the Sexto-Licinian Laws opened the consulship to plebeians gives weight to the argument that the consular tribunate represented a political compromise between patricians and plebeians. Forsythe does accept the Sexto-Licinian Laws as basically historical. However, 2

Livy and Cicero mention the Lex Canuleia but Dionysius does not, and for Forsythe this reflects general confusion about the nature of the prohibition in the Twelve Tables. This strikes the reviewer as too sceptical, and the argument can be turned on its head by emphasizing that two out of three main sources mention the marriage ban.

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Forsythe cautions against reading these laws as the outcome of a patrician-plebeian conflict that endured since the early days of the republic. According to Forsythe, the Sexto-Licinian Laws were extremely important in the later development of a ‘plebeian mythology’, which retrojected back to the beginning of the republic a patrician monopoly of the consulship. Since there had been no patrician monopoly on the consulship, at least until the end of the fifth century, the opening of the consulship to plebeians in 367 BC should be seen as reflecting short-term social and political conditions, not as part of a centuries long conflict between Struggle of the Orders. In addition to reinstituting the consuls, the Romans also created three new magistrates for 366 BC, one praetor, and two curule aediles. Forsythe sees this as simply a reorganization of the government, with the typical board of six consular tribunes were replaced by five magistrates with differentiated and specialized powers. Thus, in these central chapters, Forsythe downplays the Struggle of the Orders in his account of Rome’s history from the beginning of the republic through the middle of the fourth century, and he rejects the bulk of narrative found in the ancient literary tradition. This does not mean that the reader is left with only negative conclusions. Forsythe emphasizes Rome as a developing city-state in the process of forming political, legal, and military institutions to respond to increasingly complex demands. For example, tribal assembly, organized by geography rather than property class, was created in the early fifth century since such a legislative body was a more convenient organ of government for a state with a growing population and territory. Forsythe postulates that priests probably monopolized most legal jurisdiction in archaic Rome. The codification of law in the Twelve Tables, therefore, was an important step in state formation that broke this priestly monopoly on legal jurisdiction. Likewise, the political reorganization resulting from the Sexto-Licinian Laws is consistent with the picture of Rome as a developing city-state, as the increased number of specialized civil and military magistrates bespeaks a more complex state structure. The last two chapters focus on the growth of Roman power in Italy. Chapter 9 covers the years 366--300 BC, dominated by the First and Second Samnite Wars. Chapter 10 discusses the final Roman conquest of Italy from 300 to 264 BC, including the Third Samnite War and the conflict with Pyrrhus, and ending at the outbreak of Rome’s

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first war with Carthage. Forsythe does, however, touch on internal political and social developments and he picks up on and develops a number of themes introduced in the previous chapters. First, although Forsythe places the Second Samnite War (326--304 BC) at the edge of Roman history, he continues to view with great scepticism many of the details provided by the literary sources. Forsythe repeatedly denies the historicity of episodes on the grounds that they are modeled on stories from Greek history or on later events in Roman history; or that they are chauvinistic fabrications or exaggerations aimed at glorifying Rome, balancing Roman defeats, or justifying Roman aggressions. Second, Forsythe sees Roman society becoming ‘increasingly secularized’ [320] as the number of priesthoods increased and became open to plebeians, further eroding any special patrician priestly privileges. Third, Roman institutions as a whole continued to be flexible and evolve, allowing the Romans to extend their hegemony over the peninsula. Thus, the Roman political system and mixed plebeian-patrician aristocracy that was oligarchic but not entirely closed encouraged competition and conquest. Likewise, the Roman military system was reformed by the late fourth century. For example, according to Forsythe, the election of 16 military tribunes in 311 BC corresponded to the adoption of the manipular legion, possibly in response to the disaster at Cadium. Finally, the political reorganization of Rome’s allies in 338 BC, the foundation of colonies, and the extension of military roads were important tools of empire and mechanisms for the gradual Romanization of Italy. Looking ahead, Forsythe sees the political and military institutions that Rome developed in the fourth century as paving the way for Rome’s eventual conquest of the Mediterranean. According to the foreword, A Critical History of Early Rome is aimed at the educated general reader, college undergraduates, and graduate students and scholars of classics and ancient history [2]. It is difficult to balance the needs of these different audiences, which is perhaps reflected in the unevenness of the prose. Thus, the advanced reader will certainly find unnecessary or even pedantic such inclusions as the description of T. R. S. Broughton’s Magistrates of the Roman Republic, which details among other things how it is divided into two volumes, the first covering the years 509--100 BC and the second covering 99--31 BC [155]. However, the less expert reader may have trouble following some of the denser and quite technical arguments.

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And certain readers will undoubtedly grow frustrated with Forsythe’s repeated dismissal of literary evidence in favor of his own hypotheses. This does not mean that A Critical History of Early Rome should be avoided. This book presents an engrossing and challenging analysis of early Roman history, and one that anyone seriously interested in the subject should read alongside Cornell’s more optimistic Beginnings of Rome. Even if he occasionally pushes his case too far, Forsythe reminds the reader that the literary sources for early Roman history must be approached with extreme caution. At the very least, A Critical History of Early Rome forces us to consider that Roman historians had a different understanding of historical truth and that they practiced their craft very differently from their modern counterparts. bibliography Broughton, T. R. S. 1984--1986. Magistrates of the Roman Republic. 3 vols. Atlanta, GA. Cornell, T. J. 1995. The Beginnings of Rome: Italy and Rome from the Bronze Age to the Punic Wars (c. 100--264 BC). London/ New York. Oakley, S. 1997--2005. A Commentary on Livy Books VI--X. 3 vols. Oxford.

Science in the Medieval Hebrew and Arabic Traditions by Gad Freudenthal Variorum Collected Studies Series 803. Aldershot, UK / Burlington, VT: Ashgate Publishing, 2005. Pp. xx + 350. ISBN 0--86078--952--7. Cloth $119.95

Reviewed by Mauro Zonta Università degli Studi di Roma “La Sapienza” [email protected] This book is a collection of 16 articles that appeared in the period 1986--2002 and were written by Gad Freudenthal, Senior Researcher of the Centre National de la Recherche Scientifique (CNRS). He has worked for 23 years in Paris-Villejuif at the Centre d’Histoire des Sciences et des Philosophies Arabes et Médiévales, once directed by the well known scholar of Medieval Arabic science Roshdi Rashed and now by Régis Morelon. Freudenthal is also the founding editor of Aleph: Historical Studies in Science and Judaism, the first journal devoted to all aspects of the history of Jewish science from Antiquity to the 20th century. 1 These articles collected in this volume have appeared in various languages: 11 in English, three in French, and two in Hebrew—the last two are here translated into English by the author himself. Some of the articles have been published in well known journals specifically devoted to the history of science (History of Science, Arabic Sciences and Philosophy, and Micrologus). Some appeared first in collections of essays mostly concerning the history of Medieval Jewish science: a volume on physics, cosmology, and astronomy from 1300 to 1700 [Unguru 1991]; a collection of studies on Levi ben Gershom edited by Freudenthal himself [1992]; a volume on Maimonides as a physician, scientist, and philosopher [Rosner and Kottek 1993]; and a book on Medieval Hebrew encyclopaedias of sciences and philosophy [Harvey 2000]. Other articles have been published in journals and volumes generally devoted to Jewish studies: the proceedings of the Tenth 1

The fifth volume [2005], published by the Hebrew University of Jerusalem and the Indiana University Press, has just appeared. C 2005 Institute for Research in Classical Philosophy and Science 

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World Congress of Jewish Studies held in Jerusalem in 1989 (published in 1990), the Israeli journal Qiryat sefer, and the French journal Revue des études juives (where the articles by Freudenthal were printed in the period 1989--1991). Finally, four articles have appeared previously in volumes and journals generally concerning philosophy and history of philosophy in general and do not involve directly either the history of science or Jewish culture: the proceedings of the Eighth International Congress of Medieval Philosophy, organized by the Société Internationale pour l’Étude de la Philosophie Médiévale (SIEPM) in Helsinki in 1987, the two-volume Routledge History of Islamic Philosophy [Nasr and Leaman 1996], and the journals Phronesis and Revue de métaphysique et de morale. The authors whose doctrines about science and, more generally, philosophy are mentioned and discussed in these articles occupy a wide range of space (from Iraq to Egypt, Greece, Spain, France, and even Germany) and time (from early Antiquity to the late Middle Ages). Among them there are, in approximately chronological order: the Greek philosopher Anaximander, who lived in the sixth century BC; 2 the Medieval Islamic philosophers Abu. Nas.r al-F¯ ar¯ ab¯ı (870--950), Avicenna (980--1037), and Averroes (1126--1198); the 10thcentury Jewish theologian Saadia Gaon (882--942) and the better known 12th-century Jewish philosopher Moses Maimonides (1138-1204); the 14th-century Provençal Jewish ‘philosopher-scientist’ Levi ben Gershom (Gersonides, 1288--1344), who is sometimes connected to a number of other minor Jewish authors active in the same period and milieu; the Spanish Jew Shem Tov Ibn Falaquera (1225--ca 1295) and the Provençal Jew Levi ben Abraham ben H . ayyim (1240-ca 1315), authors of some of the best known Medieval Hebrew encyclopaedias of science and philosophy; 3 two Castilian Jewish philosopher and scientists, Avner of Burgos (first half of the 14th century) and the lesser known Joseph Ibn Nah.mias (15th century) who was the author of an astrological work, Light of the World.

2

This article is the only one not devoted to the Medieval Arabic and Hebrew scientific traditions. 3 The section of Levi ben Abraham’s encyclopaedia devoted to cosmology and creation was published recently in an annotated critical edition by Howard Kreisel [2004].

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The various articles here collected by Freudenthal fall into three groups. The first two articles are devoted to ‘socio-cultural considerations’ about Medieval Jewish science: Why and how in the 13th to 15th centuries European Jews, active in Spain, Provence, and Italy, studied such sciences as logic, astronomy, mathematics, and physics, but—according to Freudenthal—were apparently not so interested in subjects like alchemy? Why did some Jewish authors, who saw themselves as the defenders of the Jewish religious tradition, oppose the study of sciences? In articles 3--10, the author concentrates on some of the above mentioned Medieval Jewish authors of scientific works, in particular on Gersonides: he has cleverly seen him as a ‘solitary genius’ of 14th-century Jewish philosophy, who gave to it a new scientific basis (physics, instead of theology) and anticipated some aspects of modern science—he even proposed a sort of proto-microscope, the first in European science [348]). Finally, articles 11--16 are devoted to an issue much studied by Freudenthal, the Greek theory of matter in pre-Socratic philosophy, in Aristotle’s works [see also Freudenthal 1995], and after Aristotle (e.g., in Stoicism), as this theory was interpreted in its various ‘reverberations’ in Medieval Arabic and Jewish authors such as Saadia Gaon, Avicenna, Averroes, and others. As should be clear from the above outline, this collection is a very impressive piece of learning; and I recommend it to all readers of Aestimatio interested in the history of science in general, and the study of different scientific themes in Medieval Arabic and Jewish sciences in particular. The articles published in this book are but a selection of the papers that Gad Freudenthal has devoted to these themes. Between 1976 and 2005, he has published 57 articles (in some cases in collaboration with other scholars and researchers), eight collections of essays, and two books, covering ca 1800 pages— to list only his works devoted to different aspects and authors of the history of science from ancient Greece to contemporary Europe. Of course, readers who find this volume of interest should also consult these papers—and this might also stimulate the author to prepare a new collection of his papers which do not appear here. bibliography Freudenthal, G. 1992. ed. Studies on Gersonides: A FourteenthCentury Jewish Philosopher-Scientist. Leiden/New York/Köln.

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Freudenthal, G. 1995. Aristotle’s Theory of Material Substance: Form and Soul, Heat and Pneuma. Oxford. Harvey, S. 2000. ed. The Medieval Hebrew Encyclopedias of Science and Philosophy. Amsterdam. Kreisel, H. 2004. ed. Levi ben Avraham: Livyat H . en, Book Six Part Three. The Work of Creation. Jerusalem. Nasr, S. H. and Leaman, O. 1996. edd. History of Islamic Philosophy. London/New York. Rosner, F. and Kottek, S. S. 1993. edd. Moses Maimonides: Physician, Scientist, and Philosopher. Northvale, NJ/London. Unguru, S. 1991. ed. Physics, Cosmology and Astronomy, 1300-1700: Tension and Accomodation. Dordrecht/Boston/London.

Music and the Muses: The Culture of ‘Mousike’ in the Classical Athenian City edited by Penelope Murray and Peter Wilson Oxford: Oxford University Press, 2004. Pp. xiv + 438. ISBN 0--19-92439--9. Cloth $115.00

Reviewed by Massimo Raffa Liceo Classico of Milazzo m.raff[email protected] It is well known that the meaning of the ancient Greek word μουσικ (mousik¯e) cannot be reduced sic et simpliciter either to ‘music’ nor to any parallel word in other modern languages (‘Musik’, ‘musique’, ‘musica’, etc.). The Greek concept of mousik¯e covers indeed a remarkably wide range of aspects regarding religion, education, politics, and even the art of war. As a consequence, it is unlikely that a comprehensive view of such a multifarious subject can be provided by any single scholar today. This is the reason, one might guess, why the most interesting books on this subject typically take the form of a series of contributions by some of the most authoritative scholars in various aspects of mousik¯e; 1 and this volume, which comes as a result of a colloquium held in Warwick in 1999, is no exception. The structure of the book seems designed to avoid the lack of connection and heterogeneity to which multi-authored works are prone. On one hand, the research is limited to a precise place, Athens, and to a precise period, the ‘Classical Age’ (fifth to fourth centuries (BC); on the other, the division of the essays into four parts (1. Mousik¯e and Religion; 2. Mousik¯e on Stage; 3. The Politics of Mousik¯e; 4. Mousik¯e and Paideia) allows the reader to find his bearings in this rich field. In the first essay, ‘Muses and Mysteries’ [11--37], Alex Hardie moves from the etymological relation established in several ancient sources between the Mousai and mousik¯e to explain the role played by music in mystery cults. One of the merits of Hardie’s work is to point out that music was not only a simple accompaniment of the 1

See, e.g., Gentili and Pretagostini 1988, Restani 1995, Gentili and Perusino 1995. C 2005 Institute for Research in Classical Philosophy and Science 

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rituals, but rather a way to convey privileged knowledge and wisdom to the initiated. Particular attention is paid to the idea that music could put men in touch with the gods, and ensure a sort of afterlife through the power of memory (not by chance were the Muses said to be daughters of Mn¯emosyn¯e). Barbara Kowalzig’s ‘Changing Choral Worlds: Song-Dance and Society in Athens and Beyond’ [39--65] deals with a singular contradiction in the development of choral performances from the archaic polis to classical Athens. The advent of Athenian democracy seems to have entailed a widening in the social background of the performers; but, in contrast, a remarkable reduction is to be noted as regards the variety of choral genera in favor of the Dionysiac choros. In such a context, Plato’s assertion of the need to have different genera for different gods can be interpreted as a conservative one. Some particular analyses, for instance, of Euripides’ Iphigenia in Tauris [61ff.] also show the attempt to bring non-Attic myths to Attica. This sort of ‘rewriting’ of myth was possible, in Kowalzig’s opinion, because choral performance, even in passing from an archaic to a democratic context, retained its aetiological power. This work, which combines both literary and sociological approaches, throws new light on crucial aspects of musical performance and is to be read as a continuation of other important studies [e.g., Musti 2000]. Choral music was also a fundamental part of political and religious cohesion not only inside the polis as a whole, but also between different poleis. By sending and receiving foreign choroi, Greek poleis asserted their own identity and, at the same time, strengthened their relationships to others. Ian Rutherford’s ‘Song-Dance and StatePilgrimage at Athens [67--90] provides an exhaustive outline of different typologies of the¯ oria in the Greek world as well as in non-Greek cultures both past and present, in order to sketch a possible setting for the Athenian state-pilgrimages to Apollo’s sanctuaries in Delphi and Delos. No evidence remains for Athenian choral performances in Delphi during the classical period, but Rutherford’s opinion that Pindar’s paeans [Snell and Maehler 1989, Frr. 52h and 52e] were meant to be sung in Delphi and in Delos respectively by Athenian performers is quite convincing. Xenophon’s expression χορ ε κ τ σδε τ π λεω (‘one chorus from this polis’) synthesizes the difference

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between the dithyrambic and the paean choros, the former being narrative and open to the innovation of the ‘new music’, the latter being more conservative both in literary and in musical structure. The first section of the book ends with Paola Ceccarelli’s ‘Dancing the Pyrrhich¯e in Athens’ [91--117], a discussion of pyrrich¯e or war-dance, which, although it did not necessarily involve a sung performance, belonged to the realm of mousik¯e as well. The performing of pyrrich¯e is widespread all over the ancient Greek world. Ceccarelli, who is not new to this subject [see Ceccarelli 1998], gives here a detailed account of the iconographic and epigraphic evidence for this dance and pays special attention to its significance in various contexts (in the Panathenaia, in the Tauropolia and Apatouria festivals, in theatrical and choral performances, and in association with the cult of Dionysos and with funerary rituals). Her opinion is that in the classical period the pyrrich¯e had a symbolic value and was no longer used for real military training. Her analysis is particularly subtle in pointing out the antiquity of the pyrrich¯e, whose chorus continues to be organized according to age groups, such as boys, adolescent males (eph¯eboi), and men, even after Cleisthenes’ wealth-based reforms. The first two papers of the second part (Eva Stehle’s ‘Choral Prayer in Greek Tragedy: Euphemia or Aischrologia?’ [121--155] and Claude Calame’s ‘Choral Forms in Aristophanic Comedy: Musical Mimesis and Dramatic Performance in Classical Athens’ [157--184]) are about choral performances in theatrical contexts and can be regarded as a diptych, as it were, of complementary points of view. The main problem that they deal with is whether—or to what extent—a choral song maintains its ritual meaning when performed on the stage in a tragedy or a comedy. Stehle begins by defining euph¯emia as a sort of limitation to which the language of a prayer was subjected in order to preserve its purity. For it was by reason of euph¯emia that prayers omitted, for example, any reference to violent death, pollution, lamentation, or any aischrologia (foul language or obscenity), and thus acknowledged the presence of the god. Then, she applies this to several passages of Greek tragic literature and shows how the rules of euph¯emia were violated every time in order to make the audience aware of the distance existing between the theatrical and the ritual performance of prayer. Such a distance becomes more evident when one considers that the complexity of rhythm and dance movements in tragic choruses, not to say the use of masks, was unlikely to

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make the audience identify themselves with the performers—which is fundamental in a ritual. As for aischrologia, while it is quite obvious to regard many Aristophanic choral passages as obscene, Stehle’s attempt to mark a deeper level of aischrology even in tragedy, especially in the representation of the ‘dead, sexuality, women’s power in reproduction and magic’ [155], is very convincing and intriguing. Calame lays a special stress on the theater as a ‘sanctuary of Dionysus’ [158, 161] and, consequently, on the ritual and cultic meaning of choral performances. In his view, choral song is a sort of place where the author, the audience, and even the performers can assume the authoritative role of ‘choral voice’ according to the circumstances. He focuses on the choral exodoi of the Thesmophoriazousai, Lysistrata, Peace, Birds, Ecclesiazousai, and applies that semiological analysis which seems to be his favorite approach to Greek poetry. 2 Aristophanes is also, as everybody knows, a fundamental source for those interested in music criticism in the last decades of the fifth century BC, when a new conception of musical composition, of metric and semantic relations between melos and logos, and even of the musician’s very social position, knocked the first nails into the coffin of the old aristocratic musical world. Andrew Barker’s ‘Transforming the Nightingale: Aspects of Athenian Musical Discourse in the Late Fifth Century’ [185--204] points out some problematic elements in the characterization of Nightingale in the Birds. This bird, which is supposed to remind the audience of the beautiful power of song and music, is represented here, in Barker’s opinion, as a female aulosplayer—such aul¯etrides were notoriously slaves and used also to act as prostitutes in banquets—who really plays the aulos. Barker is very acute, not to say ingenious, in using text-based evidence to show that Nightingale’s costume was designed to allow her to play on the stage [201]. So, if we agree that the very symbol of Music is here depicted as a hired whore, we can see how conservative and pessimistic Aristophanes’ position about ‘New Music’ was; indeed, as Barker points out, the only reason why the other characters on the stage, symbolizing the real audience, evidently love Nightingale’s performance, is that they and the audience itself are supposed to be as depraved as the music.

2

See also Calame 2005 and the useful review by Van Noorden [2005].

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The third section opens with Eric Csapo’s ‘The Politics of the New Music’ [207--248], the subject of which is closely linked to Barker’s contibution. Csapo focuses on economic, performative, and stylistic features of the ‘new music’; he then treats some other aspects of this new style which are symbolic and related to the ‘collective imaginary’. The mimetic and ‘expressionist’ nature of this music would have been impossible without technical improvement both in constructing and playing the aulos. As a consequence, the aulos-player stopped being an amateur and became a professional; so he was hired and paid no longer by the chorus-director (chor¯egos) as in the past, but directly by Athens itself. Such a separation between the aulosplayer and the choros is reflected in the very musical structure of choral composition (such as the dithyramb), the instrument acting as a rival of the chorus rather than providing a simple accompaniment. Csapo dedicates many pages to explaining the expressive potentialities of the aulos, such as playing the pitch continuum between two notes and providing an unbroken stream of sound by means of a particular ‘circular’ breathing technique, and so on. His attempt to link these features of the aulos with some phonetic, syntactic, and semantic aspects of the poetry of the same period is very interesting. The repetition of single words or groups of words, frequently occurring, for example, in Euripides’ late works, as well as the prolonging of the same syllable (even in the case of short syllables) on two or more different notes, are well known features of this new style; and Csapo appropriately quotes the famous passage of Dionysius of Halicarnassus [De comp. verb. 11] about the musical treatment of syllables and accents in some verses from Euripides’ Orestes. But it is in the analysis of semantics that the most important elements of novelty can be found, especially where the fragmentary style of some ‘new poets’, with rapid changes both in the syntactic structure and in the subject, is connected with the polyphony that is possible on the two pipes of the double aulos—and, one might add, with the frequency of modulations (metabolai) during the same piece, also made possible by new devices applied to the finger-holes. As for political implications, a semantic analysis of Greek words describing the new style shows that several nouns and adjectives had political or moral overtones reminiscent of the ideas of variability, instability, or slackening. No surprise, then, that the opposition to the new music assumed the tones of political and social conservatism, and that this music was

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credited with bad moral qualities such as effeminacy and a sort of non-Greek, eastern lasciviousness. On the other hand, the innovators did have their own Dionysiac—that is, genuinely Greek—tradition to refer to. Csapo’s conclusion is that the myth of a ‘pure’ old music was an a posteriori construction created to oppose the rising of new social classes due to the advent of democracy. 3 The treatment of the relationships between music and politics should not leave out Damon of Oa, who is supposed to be one—if not the most important—of Pericles’ advisers. Robert Wallace’s ‘Damon of Oa: A Music Theorist Ostracized?’ [249--267] gives here some hints of a discussion to be developed in a forthcoming work [249n1]. As Wallace has pointed out previously [1991], every serious study of Damon has to face the lack of sources and the difficulty of distinguishing what is to be credited to Damon himself and what was added by his successors (for example, by Plato). Wallace provides an outline of Damon’s thought, which he interestingly places in the context of sophistic philosophy, thus emphasizing on one hand Damon’s belief in the ‘ethical’ powers of music—which could have been the basis for Gorgias’ reflections on the analogous powers of speech in his Encomium of Helen—and on the other Damon’s inclination to make classifications of various rhythms and scales, a project which he could have derived from the sophist Prodicus, a friend of his. The tradition according to which Damon was ostracized, a claim some scholars have rejected, may indicate that he was a politician rather than merely a music theorist, for it seems strange that he deserved banishment only because of his musical ideas. Wallace describes the political setting of the last half of the 40s of the fifth century BC and shows how such a clever man could fall under suspicion of being an enemy of democracy, especially if his ostracism is to be placed—as the author convincingly argues—just after the ostracism of Thucydides, the son of Melesias, the oligarch leader who had been ostracized by Pericles in 444 or 443 BC. The ostracism of a music theorist appears less strange if one remembers the importance of music in its public and social context. Peter Wilson’s ‘Athenian Strings’ [269--306] is about the political and social perception of stringed instruments in classical Athens and 3

See also Meriani 2003a about a particularly instance of such an ‘idealization’ of the musical and political past.

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could be regarded, in one sense, as a work complementary to Csapo’s. According to a very common opinion, which has been reinforced by Nietzsche’s conception of the Apollonian and the Dionysiac, the lyre and the strings in general are supposed to symbolize in ancient Greek culture the realm of social order and ‘enlightened’ culture, while the aulos had to do with uncontrolled passions and social disorder (that is, democracy in the worst sense). Wilson warns against such a dichotomy. If in an early phase, there was an opposition between the lyre as the instrument of the elites and the aulos as belonging to the people (d¯emos), it is also true that it was neutralized in the late fifth century BC by the increase in professional players, some of whom, like Stratonicus, seem to have had a prominent role in music theory as well. 4 It is very interesting that a new version of the famous myth of Marsyas, the satyr skinned alive for challenging Apollo to play appears at the same time: the satyr, who in a earlier versions of the myth used to be represented as an aulos-player, now plays the lyre as well. The polemic against a music without order moves to a different field and addresses a new target, the instruments with too many strings; and Wilson opportunely cites Phrinis’ Cyclops and Pherecrates’ Chiron as the best examples that we have of the way in which the new perception of stringed instruments was elaborated by comic poets. 5 At this time, the old-fashioned lyre becomes a symbol of musical and political conservatism; and the possession or the lack of musical culture plays an important role in the evaluation of some prominent politicians (Themistocles, Cimon, Pericles himself, Cleon). Wilson lastly suggests a charming reading of Timotheus’ Persians as an attempt to provide an ‘inclusive’ and ‘democratic’ image of both new strings and new music. The fourth and last section is about the role of mousik¯e in education (paideia). Andrew Ford’s ‘Catharsis: The Power of Music

4

Stratonicus probably belonged to the group of the harmonikoi, who were trying to determine the smallest audible interval through acoustic experiments rather than by focusing on mathematical ratios. On these theorists, see Barker 1978 and Rocconi 1999, 96--97. For an interesting reconstruction of what the harmonikoi were doing, see Meriani 2003b, 106--110; and further remarks in Raffa 2006. 5 As for Pherecrates’ Chiron, one must remember, in addition to the studies cited by Wilson [287n46], the ground-breaking work by Restani [1983].

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in Aristotle’s Politics’ [309--336] addresses the problem of what Aristotelian catharsis is exactly and how it works. Ford opens with some clear and up-to-date pages on the status quaestionis [309--311]: an opposition has emerged between the conceptions of catharsis as a ‘purge’ or way of eliminating and neutralizing unnecessary emotions, and as an ‘intellectual “refinement”’ [310] of emotions and feelings which otherwise would remain out of control. The latter interpretation, which requires that the word mousik¯e as used by Aristotle be given the wide sense of ‘poetry’, is acutely questioned by Ford. Given that the philosopher focuses not only on the music with words, but, as he writes explicitly, even on ‘pure’ (ψιλ , without words) music [318], one should consider, according to Ford, the technical features of music (that is, the scales, rhythms, and so on) and the effects that they produce on the human soul. The music Aristotle imagines for the school seems to be quite different from that performed on the tragic and comic stage, the former being subjected to a series of limitations that the latter does not have. Ford’s opinion is that in Aristotle’s view theatrical music does not need any limitation because it is conceived for a wide and not necessarily ‘educated’ audience; such music, even if depraved, could not cause any harm to those who have received proper musical education. 6 (Obviously Aristotle, in his elitist view of culture and society, does not care about uneducated people.) On the contrary, young pupils must be educated through a much more constrained kind of music. So Aristotle, because of his idea that music cannot be socially harmful when considered outside an educational context, paradoxically seems to be less prescriptive than Plato in the matter of ‘public’ music . Victoria Wohl’s ‘Dirty Dancing: Xenophon’s Symposium’ [337-363] pays attention to the relationship between dance and philosophy in Xenophon’s dialogue. In her view, two different kinds of love (eros) are displayed here: on one hand, there is homoerotic love, 6

The idea that proper musical education can provide antibodies, as it were, against the disease of new bad music is also to be found in a passage from Aristoxenus, who was a pupil of Aristotle, that is cited by pseudo-Plutarch [De musica 1142b--c]. As a proof, we are told that a Theban composer called Telesias (otherwise unknown) was unable to compose music in Philoxenus’ modern style ‘because of the very good musical training he had received when he was young’ (γεγεν σθαι δ α τ αν τ ν κ παιδ καλλ στην γωγ ν). See also Meriani 2003c, 58--59 and Fongoni 2006.

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which ensures the continuity of moral values between successive generations and has to do with the static and imitation-based educational model of Plato (education consisting in conveying the whole moral system of the elders to the young); on the other, there is heterosexual love, which is regarded as a less refined but necessary for ensuring the continuity of the species and the very existence of the polis. This opposition is respectively symbolized in the two characters of Socrates, the philosopher of pure eros, and the Syracusan dancer, who is responsible for the dirty dance with which the dialogue ends. Wohl points out that, whereas Xenophon skilfully directs some of the charges historically brought against Socrates (for instance, that of corrupting the youth) to other targets—e.g., to Lykon (young Autolykos’ father), or to the Syracusan—he seems less definite about the possibility of separating the two kinds of love, given that even in such a high-minded assembly of philosophers and noble men, there is still room for staging a representation of heterosexual and ‘coarse’ love. Xenophon’s reader is thus left with an unsolved doubt. This book could hardly have had a better conclusion than Penelope Murray’s essay, ‘The Muses and Their Arts’ [365--389], the aim of which is to define the variations in the Muses’ field of influence throughout Greek literary history, from Homer up to the Second Sophistic (second century AD). A particular aspect of this intriguing excursus is the relationship between the Muses and the art of speech, that is, rhetoric. Murray points out that, after an isolated hint in Hesiod’s Theogony in which the sweet and persuading words of the prince are said to be a gift of the goddesses, such a connection seems to disappear. With the advent of sophistic culture on one hand, and of literacy on the other, the Muses’ role is limited to poetry: the activity of sophists, who proposed rhetoric as a set of teachable skills, seems not to need any guarantee from a god or goddess, while the reception of prose intended for private reading does not entail any specific time for performance and, as a consequence, does not call for any intervention of the Muse. No surprise, then, that we have a connection between Muses and rhetoric in the time of the Second Sophistic, when the art of speech is regarded as the center of paideia as a whole. Despite the different and sometimes opposing viewpoints of the authors, the volume has a considerable degree of inner cohesion, particularly as a result of the frequent references linking one essay to the others. This helps recreate the atmosphere of the colloquium for

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the benefit of those who, like the reviewer, did not have the good fortune to attend it. bibliography Barker, A. 1978 ‘ο καλο μενοι ρμονικο : The Predecessors of Aristoxenus’. Proceedings of the Cambridge Philological Society 204 (ns 24):5--21. Calame, C. 2005 Masks of Authority: Fiction and Pragmatics in Ancient Greek Poetics. P. M. Burk trans. Ithaca. Ceccarelli, P. 1998. La pirrica nell’antichità greco-romana. Pisa/ Roma. Fongoni, A. 2006. ‘L’educazione di Telesia di Tebe fra tradizione e innovazione (Ps.-Plut. Mus. 31, 1142bc)’. Proceedings of the VI Seminario Le musiche dei Greci: passato e presente. Valorizzazione di un patrimonio culturale. Ravenna. October 24--25, 2005 (forthcoming). Gentili, B. and Perusino, F. 1995. edd. Mousik¯e. Metrica ritmica e musica greca in memoria di Giovanni Comotti. Pisa/Roma. Gentili, B. and Pretagostini, R. 1988. edd. La musica in Grecia. Bari. Meriani, A. 2003. Sulla musica greca antica. Studi e ricerche. Napoli. 2003a. ‘Festa, musica, identità culturale. Il caso di Poseidonia (Aristox. fr. 124 Wehrli)’. Pp. 15--48 in Meriani 2003. 2003b. ‘Teoria musicale e antiempirismo nella Repubblica di Platone (Plat. Resp. VII 530b--531d)’. Pp. 83--113 in Meriani 2003. 2003c. ‘Tracce aristosseniche nel De musica pseudoplutarcheo’. Pp. 49--81 in Meriani 2003. Musti, D. 2000. ‘Musica greca tra aristocrazia e democrazia’. Pp. 9--55 in A. C. Cassio, D. Musti, and L. E. Rossi edd. Synaulìa. Cultura musicale in Grecia e contatti mediterranei. AION 5. Napoli. Raffa, M. 2006. Review of Meriani 2003. Il Saggiatore musicale (forthcoming).

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Restani, D. 1983. ‘Il Chirone di Ferecrate e la nuova musica greca’. Rivista italiana di musicologia 18:139--192. 1995. ed. Musica e mito nella Grecia antica. Bologna. 2005. ed. Etnomusicologia storica del mondo antico. Per Roberto Leydi. Ravenna (forthcoming). Rocconi, E. 1999. ‘Terminologia dello “spazio sonoro” negli Elementa Harmonica di Aristosseno di Taranto’. Quaderni Urbinati di Cultura Classica ns 61.1:93--103. Snell, B. and Maehler, E. 1989. edd. Pindari carmina cum fragmentis: Vol. 2. Fragmenta, indices. Leipzig. Van Noorden, H. 2005. Review of Calame 2005. Bryn Mawr Classical Review 2005.09.36. Wallace, R. W. 1991. ‘Damone di Oa ed i suoi successori. Un’analisi delle fonti’. Pp. 30--53 in R. W. Wallace and B. L. McLachlan edd. Harmonia Mundi: Music and Philosophy in the Ancient World. Rome.

Simplicius: On Aristotle on the Heavens 1.5--9 translated by R. J. Hankinson Ithaca, NY: Cornell University Press, 2004. Pp. x+181. ISBN 0--8014-4212--5. Cloth $69.95

Reviewed by Ian Mueller University of Chicago [email protected] The Greek texts of what are called the ancient commentaries on Aristotle were published in modern editions in the series Commentaria in Aristotelem Graeca (CAG) between 1882 and 1909, an enormous undertaking making available works produced mainly between the late second century and the mid sixth century AD. In 1987, the first volume of English translations from these commentaries appeared in the series Ancient Commentators on Aristotle under the general editorship of Richard Sorabji, a series which is now projected to run to over 100 volumes. 1 In the present volume, R. J. Hankinson offers an annotated English translation of the commentary (In de caelo) written by Simplicius of Cilicia (first half of the sixth century (AD) on the middle chapters of book 1 of Aristotle’s De caelo. The volume begins with a brief preface by Sorabji indicating some of the cosmological issues raised in the commentary and an introduction by Hankinson sketching Simplicius’ life and work, his philosophical attitude, and the textual situation; 2 and it concludes with a bibliography, an English-Greek glossary, a Greek-English index, and a subject index. In the first four chapters of In de caelo, Aristotle argues 3 that our ordered world (the cosmos) is made up from five simple bodies, 1

A good sense of the importance and motivation of the series can be gained from Gerson 2005. 2 The introduction is a somewhat shortened version of the introduction to Hankinson 2002. Readers interested in the complex textual situation concerning the commentary might consult the introduction to Mueller 2004. 3 A problem which frequently concerns Simplicius and should concern any philosophical commentator on the De caelo is the structure of Aristotle’s argumentation, which often seems to be circular. C 2005 Institute for Research in Classical Philosophy and Science 

(online) Aestimatio 2 (2005) 119--126

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ISSN 1549–4470

ISSN 1549–4489 (CD-ROM)

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each with a ‘natural’ place corresponding to its heaviness or lightness: earth, the heaviest naturally located in the lowest, then water, then air, then fire and outside these a fifth body, traditionally called ether, the stuff of the heavens. He also argues that when the sublunar simple bodies are not in their natural place, they tend to move there in a straight line; whereas ether, which does not interact with other bodies, has an eternal circular motion. In chapters 5--9, Aristotle is primarily concerned with the question of what is outside our cosmos. He argues that ether cannot be infinite [c. 5], that none of the other simple bodies is infinite [c. 6], that an infinite body is completely impossible [c. 7], that our cosmos is the only cosmos [c. 8], and that it is impossible for there to be another cosmos [c. 9]; and concludes with the amazing statement that ‘there is neither place nor emptiness (κεν ν) nor time outside ’ [279a17--18]. I and, I think, most contemporary readers do not find these chapters congenial. Throughout Aristotle takes for granted his now discredited doctrines of natural place and motion while arguing against opponents, the most prominent probably being the Greek atomists who reject or would reject them. Simplicius is completely faithful to Aristotle here, invoking his arguments to rebut post-Aristotelians who also reject Aristotle’s conception of the natural. Another discomfiting aspect of Aristotle’s reasoning is his handling of the infinite, e.g., his insistence that a part of something infinite cannot be infinite. Since the conceptual apparatus for dealing with this kind of issue is less than 200 years old, it is not surprising that Simplicius finds Aristotle’s argumentation conclusive; but the modern reader may not be impressed to see the same inadequate arguments driven home repeatedly. Simplicius is a Platonist or, as many would say, a Neoplatonist. But he wishes to distinguish himself from people whom he calls 4 friends (φ λοι) of Plato [276.10], people who stress the disagreements between Plato and Aristotle. For the most part, Simplicius is extremely respectful of Aristotle and stresses his agreement, or at least his non-disagreement, with Plato. But he is quite willing to invoke ideas from later Platonism, generally thought not to be either Platonic or Aristotelian, to support his interpretation of the text. The 4

I refer to Hankinson’s translation using the lineation of the CAG text which is indicated in the margins of the translation.

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commentary has a standard format in which a passage, presented in a ‘lemma’, 5 is discussed, then the next stretch of text is discussed, and so on. This breaking up of the text can impede understanding of its flow and overall structure, 6 particularly when the commentary brings in outside ideas and questions. Simplicius’ comments have a fairly standard, although not completely standardized, form. The lemma is first summarized in an expanded form, normally three or four times the length of the passage. Then questions are raised, often, in the case of this commentary, relating to interpretations offered by Alexander of Aphrodisias (ca 200 AD), a Peripatetic commentator of decidedly less Platonic inclinations than Simplicius, 7 but sometimes relating to those by other people who disagree with Aristotle. Simplicius does his best to provide answers to these questions; and, although he frequently prefaces his solutions with a ‘perhaps’, there is usually no question about what he thinks is the best answer. At the end of his discussion, Simplicius sometimes mentions textual points, alternative manuscript readings, proposed emendations, and so forth. Hankinson estimates the ratio of discussion to text discussed in In de caelo as 10 to 1. Simplicius’ prolixity is another obstacle to the modern reader, and makes the commentary of little value to the beginner wanting to gain access to the intricacies of Aristotle’s thought; such a person would do better with an annotated translation such as Guthrie 1939 or Leggatt 1995. In de caelo is of value, first, as a document in the history of the reception of Aristotle’s thought; second, as a treasure house of historical materials for which we often have no other source; and, third, as a basically word-by-word reading of the text from which even the most experienced Aristotelians can gain insights if they persevere. The existence of this first (and perhaps last) modern-language translation of a historically important text can only be welcomed.

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The lemmas in the CAG text contain only the first and last words of a passage; but Hankinson, wisely in my view, reproduces whole passages. However, it is important to realize that the words in a lemma are not a sound basis for inferring what text Simplicius read: contrast Hankinson’s note 36. 6 I remark that Hankinson’s division of Simplicius’ comments into short paragraphs is sometimes an obstacle to noticing connections. 7 The fragments of Alexander’s lost commentary on the De caelo are presented and discussed in Rescigno 2004.

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Scholars working on In de caelo will necessarily depend upon it. Hankinson’s extensive notes are primarily devoted to providing references for Simplicius’ statements about what Aristotle has said or says elsewhere, filling out his references to other authors or ideas, explaining Simplicius’ logical terminology, and assessing the philosophical merits (usually not high) of one or another argument canvassed. There are only a few proposed textual emendations, most of them relatively minor. The translation is serviceable, but it is not literal: a reader familiar with Simplicius’ not always transparent Greek would often have difficulty figuring out what Greek lay behind a given translation. In the remainder of this review, I am going to make some critical remarks of detail about translation and interpretation. I hope they will be seen as constructive and helpful and of use to readers of the commentary, since I am convinced that I am reviewing a valuable contribution to the study of ancient cosmology. ◦

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One of the difficulties in reading an ancient commentary is correlating what is said with what is in a lemma. Hankinson’s notes are helpful here, but his translations sometimes makes the connection more obscure than it has to be. For example, at 247.35, Simplicius quotes 276a22--23 exactly: but the translation of Simplicius differs from the translation in the lemma. Moreover, in the lemma for 275b6--11, ν τ ποι is translated (quite rightly) as ‘in place’; but in Simplicius’ discussion the translation becomes [e.g., at 236.16] ‘spatially located’, a phrase which might for some carry more conceptual baggage than the Greek original. 8 Note 48 says that 207.32--34 is somewhat garbled and might require wholesale alteration. But the sense is quite clear, if one understands α π το κ ντρου γ μεναι, ξ ν δι μετρο as the two radii making up a diameter. The standard Greek for ‘radii’ is α π το κ ντρου < γ μεναι γραμμα >, which Simplicius uses immediately after at 208.2. Note 49 is misleading in paraphrasing ‘if the lines are equal, the finite will be equal to the infinite’ [208.2] as ‘if they are equal they

I note also that Hankinson does not indicate which edition of the De caelo he takes as the basis of his translation. I found this problematic only in connection with 277a31--32, where, so far as I could tell, he was following Guthrie 1939.

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Aestimatio must be finite’, since Simplicius is arguing under the assumption that only one of the two radii is finite. Notes 81 and 82 suggest that Simplicius assigned to Aristotle the odd view that an infinite straight line is bounded because it is one-dimensional. But the interpretation is based on a mistranslation of π κε νο τ μ ρο , φ as ‘in respect of that part wherein’ and κατ τερον μ ρο as ‘in respect of another part’, overlooking the geometric use of μ ρο to mean something like ‘direction’ [cf. Heath 1926, 420]. 213.16--19 should be translated Just as a limited line, in so far as it is limited, is not infinite; or, if it is, it is so only in the direction in which it has length without limit; equally, planes, in so far as they are limited, cannot be infinite, even though they may sometimes be infinite in one direction. Note 84 misreports the manuscripts of Aristotle, all of which have ; the variations concern the text of Simplicius at 214.21, which almost certainly read . Similarly note 85 says that Aristotle wrote ‘EE’ at 272b27; but, in fact, he wrote ‘E’. At 215.7, the text of Simplicius has ‘Aristotle’; the translation, ‘Alexander’. At Phys. 6.10.241b6--7, Aristotle says ο δε λω τ δ νατον γεν σθαι γ γνεσθαι (‘. . . generally that that which cannot come to be should be in process of coming to be’), a passage which Simplicius cites at 218.3--4 as τ γ ρ δ νατον γεν σθαι ο δε τ ν ρχ ν γ νεται. Hankinson misses this reference, perhaps because he mistranslates Simplicius’ words as ‘what cannot have come to be is not even beginning to come to be’, overlooking the use of τ ν ρχ ν with a negative to mean ‘not at all’. A similar mistake occurs at 229.19, 21, and 24, at 235.34, and at 260.5. The phrase is correctly construed at 261.14, where the translation should be ‘What cannot have moved (κιν θηναι) to something cannot be moving (κινε ται) toward it in any way (τ ν ρχ ν),’ rather than ‘what cannot approach something cannot move toward it in any way’. The translation of 219.18 misconstrues κθεσι and στοιχε ον; it should read ‘he shows. . . by impossibility with a setting out of letters.’ At 274a8, ‘weight’ should be ‘time’.

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At 227.30, the translation describes a division as ‘negative’ with an explanatory note. But ντιφατικ means ‘contradictory’, and Simplicius’ point is that the alternatives in the division (‘finite or infinite’) exhaust the possibilities. 231.30 κινο μενα should be ‘things moved’, not ‘movers’. At 237.1, ο παλαιο (‘the early people’) is rendered ‘the early Stoics’. Prima facie this seems very unlikely, given the common use of ο νε τεροι (‘the more recent people’) to refer to the Stoics. The phrase in question is ‘the thema which ο παλαιο call third’. That these are Peripatetics is, I think, made likely by the fact that in his commentary on book 1 of the Prior Analytics Alexander of Aphrodisias credits Aristotle with discovering the third thema [see 274.19--21, 278.6--8]. In note 272, it is said that at 236.10 Simplicius describes certain arguments as both more concrete (παγματειωδ στεραι) and more general (καθολικ τεραι). But when Simplicius says ‘he once again shows by way of more general and more concrete demonstrations’, he is distinguishing between the concrete argument at 275b6--11 and the discussion which begins at 275b12 with the words, ‘It is possible to argue more formally (λογικ τερον) as follows.’ At 244.15, ν μ ν τ προσεχ ε ρημ ν is rendered ‘in the principal argument’, but Simplicius is just referring to what was just said by Aristotle. At 245.5, πλ θο is rendered ‘mass’, but it should be ‘number’ as in the Greek-English lexicon. At 249.4, χρ μενο το προληφθε σι is rendered ‘employing some earlier premisses’; but Simplicius’ reference is to the pair of ‘axioms’ which he has stated in his discussion of the preceding lemma and not to the passages cited in note 343. Immediately after, at 249.6, when Simplicius says that something κε ται, he is referring to the second of these two axioms. Consequently, κε ται should be rendered as ‘laid down’ not as ‘established’: cf. 249.11, 256.17. At 256.1, κε should be ‘there’, not ‘here’. στι, At 260.8, καστον το των (viz., heart, liver, bones), ν κα τ ν ξ ρχ σχε γ νεσιν should be rendered ‘each of them had its original genesis in the place in which it is’ (i.e., the heart

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Aestimatio is not a heart until it is located where the heart should be), not ‘each of them is in the place where it originally came to be.’ At 266.1, προκατασπωμ νου το βαρυτ ρου should be ‘the heavier side is dragged down first’, not ‘the heavier side drags it down first.’ In note 433, it should be mentioned that the introduction of the astronomical theory of epicycles predates Hipparchus; see, e.g., Toomer 1970. There is no reason to think that ‘or the snub’ at 278.15 ‘might be an intrusion’ [note 516]. Commentators often bring in a more standard example when Aristotle introduces an unusual one. Note 526 on 280.7 is too elaborate: all Simplicius means is that ‘natural body’ is a less general term than ‘substance’. At 285.25, τοπο ο ν π θεσι κα τ ζητο μενον προλαμβ νουσα τ φαντασ . . . should be ‘so the supposition is absurd and assumes in advance, using imagination, what is supposed to be proved. . . ,’ not ‘so the supposition which provides what is sought in the imaginary case. . . is absurd.’ The suggestion in note 588 that Simplicius thought that the Metaphysics preceded the physical works in some ordering is unlikely, and is not supported by the passage in the note. At 288.23, ο δε δ κα πρ το του ι να Αριστοτ λη ν τ Μετ τ φυσικ δ ναμιν ντα το πρ του παρ α τ νο . . . should be translated ‘and in the Metaphysics Aristotle has acknowledged an age (α ν) which is prior to this one , namely, the power of the mind which, according to Aristotle, is primary’. . . rather than ‘and Aristotle understood “age” prior to this, in the Metaphysics, as being the internal capacity of the primary mind. . . .’ At 290.15, ν ε πε should be ‘would he say’ rather than ‘does he say’. The emendation proposed for 291.1 is unnecessary for the reason stated in note 608; it has no real textual basis in that it is found only (as far as one can tell from the apparatus in the CAG) as a correction by Bessarion and in an 1865 printed edition of In de caelo.

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bibliography Gerson, L. P. 2005. Review of Sorabji 2005. Bryn Mawr Classical Review 2005.07.55. Gillispie, C. C. 1970--1990. ed. Dictionary of Scientific Biography. New York. Guthrie, W. K. C. 1939. ed. and trans. Aristotle: On the Heavens. London/Cambridge, MA. Hankinson, R. J. 2002. trans. Simplicius: On Aristotle on the Heavens 1.1--4 London. Heath, T. L. 1926. The Thirteen Books of Euclid’s Elements. vol. 1. Cambridge. Leggatt, S. 1995. ed. and trans. Aristotle: On the Heavens I and II. Warminster. Mueller, I. 2004. trans. Simplicius: On Aristotle on the Heavens 2.1-9. London. Rescigno, A. 2004. ed. Alessandro di Afrodisia, Commentario al De caelo di Aristotele. Frammenti del primo libro. Amsterdam. Sorabji, R, 2005. ed. The Philosophy of the Commentators, 200--600 AD: A Sourcebook. 3 vols. Ithaca, NY. Toomer, G. J. 1970. ‘Apollonius of Perga’. See Gillispie 1970--1990, 1.179--193.

The Midwife of Platonism: Text and Subtext in Plato’s Theaetetus by David Sedley Oxford: Clarendon Press, 2004. Pp. x + 201. ISBN 0--19--926703--0. Cloth $60.00

Reviewed by Ronald Polansky Duquesne University [email protected] As we might expect, this book displays impressive sophistication and learning. David Sedley acknowledges that there is already a good variety of secondary literature on Plato’s Theaetetus, but his aim is ‘to provide a corrective historical lens through which to read the dialogue’ [v]. On this reading, Plato is tracing how his own positions have Socratic inspiration. When Plato writes the Theaetetus toward the end of his middle period, according to Sedley, Plato has already gone beyond Socrates of the early dialogues and will continue to do so in later dialogues; but the Theaetetus provides an opportunity to show how Socrates is the ‘midwife’ of all this further enrichment of Platonism: By developing this implicit portrayal of Socrates as the midwife of Platonism, Plato aims to demonstrate, if not the identity, at any rate the profound continuity, between, on the one hand, his revered master’s historical contribution and, on the other, the Platonist truth. [8] This theme of the Socrates of the early dialogues giving birth to the later Platonism dominates the treatment of the Theaetetus. It is used to explain ‘why the dialogue so often takes a circuitous route’ [13]. This review calls into question Sedley’s case that Socrates of the early dialogues differs from later Platonism and that the later Plato supposes that he has arrived at truth surpassing the early dialogues, or that we should seek such explanations as Sedley proposes for what may appear to be the ‘circuitous route’ of the dialogue. Despite my doubts about Sedley’s interpretive strategy, those interested in the dialogue will certainly profit from many of Sedley’s insightful detailed analyses of particular arguments of the text. An especially impressive example is the treatment of the argument about flux in C 2005 Institute for Research in Classical Philosophy and Science 

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Theaet. 179c--183c [89--99]. Though his analyses are generally put in service of the main theme, many of Sedley’s points can be separated from this theme and employed in alternative interpretations. While Sedley holds that the main speaker in most Platonic dialogues speaks for Plato, this is not the case for the Theaetetus [6--7]. Here Plato has reverted to depicting the historical or semi-historical Socrates, so that a distinction opens between the speaker [Socrates] and the author [Plato]. Sedley says, The author is a Plato who has by this date developed a major metaphysical doctrine of obvious relevance to some of the dialogue’s central concerns; yet his speaker, Socrates, is to all appearances almost entirely innocent of that Platonic metaphysics. [7] The Socrates depicted in Plato’s early dialogues, and recreated in the Theaetetus, is ‘an open-minded critic and inquirer’ [9] who lacks the theory of Forms that emerges in Plato’s middle dialogues as well as the physics presented in the Timaeus. For Sedley, Socrates of the early dialogues is largely a moral thinker, as Gregory Vlastos suggested [18]. Yet Sedley removes the punch from Vlastos’ contention that Socrates and Plato differ as much from each other as from any third philosopher one might name [see Vlastos 1991, 46]: Sedley has the Socrates of the early period of Plato’s authorship serving as midwife for the later Platonic thought in metaphysics, physics, and psychology, on the ground that further reflection upon Socrates’ efforts and what these efforts entail leads more or less directly to Plato’s positions. Frequently, therefore, Sedley points out that something introduced into the Theaetetus appears there because it is characteristic of Socrates, even if Plato has good reason to deny it or modify it; or that something is introduced which goes beyond Socrates’ understanding in order to indicate to the seasoned reader how Plato’s own positions have supplemented Socrates’ more limited stance. Given Sedley’s interpretative interest, the key concern of the dialogue seems not so much the announced question, ‘What is knowledge ( πιστ μη)?’, but rather, ‘How do Socrates’ inquiries connect with more satisfactory Platonic answers?’ But does Plato really have such answers and, in particular, does he claim to know what knowledge is? Sedley holds that the midwife’s task ‘is not to hand the right answer to one’s interlocutors’, since the interlocutor has to work out

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the answer to the central question of the dialogue, yet ‘it by no means follows that Plato himself does not know it’ [11]. Presumably, then, we should find in the mature Plato a clear account of knowledge. In fact we do not: Sedley himself refers to ‘a definition [of knowledge] which Plato nowhere formulates in the dialogues, but leaves to his readers to work on’ [11], and asserts that the Platonist path that lies ahead (i.e., beyond the early dialogues) is one on which knowledge—although nowhere formally defined—will be recognized as a state of mind that differs far too radically from true judgement to be defined as a species of it. [179] Thus, Sedley concedes that Plato nowhere defines knowledge, though the recognition that knowledge differs from true opinion and has entirely different objects supposedly lies beyond Socrates’ ken. But if the mere acceptance of Forms and objects of knowledge beyond sensible objects does not provide a ready account of knowledge, why need we suppose that Plato knows what knowledge is or that the introduction of the Forms into the Theaetetus will facilitate defining what knowledge is? And is there any reason to assume that we shall arrive at an understanding of knowledge through any simple formula or definition, as Sedley seems to suggest? Another important issue in the Theaetetus, regarding which Sedley indicates that Plato has the truth and Socrates does not, is false judgment. Sedley thinks that the whole section on false belief [Theaet. 187a--200d] ‘turns out to contribute nothing to the definitional question at issue’ [13], but that it appears because Plato wishes to contrast dialectical argument with sophistry and to show ‘that Socrates had an understanding of cognitive psychology which went most of the way towards a solution’, the ‘definitive solution in the Sophist’ [119]. We might object immediately that false opinion needs to be considered at length in the Theaetetus not just because the sophist Protagoras depicted in the dialogue hides behind its denial, but because false opinion is really the contrary of knowledge; and that as much as the Sophist needs to consider both being and non-being, the Theaetetus’ investigation of knowledge should extend to its contrary, ignorance. And does Plato actually get any further with false judgment in the Sophist than in the Theaetetus? The Sophist clarifies how false statement is possible when what is other than the case is

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asserted; but it really does little beyond the Theaetetus to clarify why such false statement should be believed by anyone. Plato cannot in fact explain why anyone should be led by deceptive appearances to believe something untrue, i.e., to speak the untrue to himself in his own soul as the truth. I have called into question Sedley’s view that Plato possesses the truth about some key issues in the Theaetetus which he withholds so that he can display Socrates floundering fruitfully. Now we may also question that Socrates of any of the dialogues should be supposed naïve and ‘primitive’ [29]. Andrea Nightingale [1995, esp. 10--11] argues compellingly that Plato invents philosophy in opposition to other literary genres. In accord with this view, we find that nowhere in the dialogues is it suggested that any of the Presocratics with the exception of Parmenides is a philosopher. Apparently, for Plato, to be a philosopher one must accept some version of Forms, the only fully adequate sort of cause. It is Aristotle who seemingly invents the history of philosophy in which philosophy begins perhaps with Thales and continues through the Presocratics to Socrates, Plato, and Aristotle himself; and thus acknowledges as philosophers thinkers who in their search for causes and principles have not embraced the Forms. Now if this account of Plato is plausible and accords with the dialogues, is it not likely that Plato intends to present Socrates as the best conceivable human being, as a full-fledged philosopher, in order to surpass any of the competing figures from Greek literature and to provide the ultimate paradigm for the philosophical way of life? If so, would it still be appropriate to have Socrates miss crucial points which are clear to Platonist readers? And if he is a philosopher, will Socrates not have to entertain a theory of Forms? Yet Sedley rejects the claim that Plato depicts Socrates as the supreme human and as always having a theory of Forms. Sedley rather desperately urges, ‘there is no reason why even the reference to investigating “justice itself and likewise injustice” (in Theaetetus 175c) should necessarily imply a Platonic metaphysics’ [73]; and he similarly explains away such usages in early dialogues, e.g., Prot. 330d8--e1. He takes the Parmenides to be ‘the same kind of dramatic game’ as the Theaetetus: ‘the creation of a Socrates who cannot be straightforwardly identified as Plato’s mouthpiece’, but ‘a dumbeddown version of the “Socrates” who had been Plato’s middle-period spokesman’ [17]. But should we not read the Parmenides as Plato’s

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account of how the paradigmatic human, the Socrates depicted in almost the whole of Plato’s corpus, becomes what he is? In the Parmenides, Plato shows the young, brilliant Socrates still struggling with a theory of Forms that has too many difficulties to help him with his dialectical inquiries. It is the ‘exercise’ provided by Parmenides for the young Socrates, we are to understand, that suitably prepares him to be the Socrates capable of doing the divine labor of cross-examining others which we find in the other dialogues in which he participates. Though this is most likely inaccurate historically, it depicts the case in Plato’s fictive literary world; and the exercise as presented is intended to have similar impact upon Plato’s readers. Sedley proposes instead that we assume the position of Plato’s contemporaries who are supposedly reading the dialogues as soon as they appear and catch on to Plato’s intention to present an ‘autobiography’ of his own development; yet Sedley admits that although such a progression, from the semi-historical Socrates to a Socrates who voices Plato’s current thinking, may not have been evident to later generations of Platonists reading the Platonic corpus as a unity, it was presumably obvious to Plato’s contemporary readers. [10: cf. 17] Does Sedley hereby offer us a likely literary task for Plato in the Theaetetus? The evidence Sedley presents for Socrates’ limitations in the early Platonic dialogues is principally Aristotle’s account of Socrates in Meta. 13.4 [10]. What this text is purported to show is that Aristotle’s account of the historical Socrates is taken largely from Plato’s early dialogues and that Aristotle ‘make[s] a sharp philosophical distinction between Plato’s Socratic dialogues and those representing his mature work’ [15]. But is this so? In Meta. 13.4, Aristotle prepares to discuss the Ideas without yet viewing them as Form-Numbers and, therefore, he provides a brief account of the origin of the theory of Forms. As he sees it, this theory arose from combining the Heraclitean vision that sensible things are ceaselessly flowing, and so are unsuitable as objects of knowledge, with the quest by Socrates for universal definitions of moral virtues. To Socrates is attributed only inductive arguments and universal definitions, and it is asserted that Socrates did not make universals or definitions separate whereas some later thinkers did [see Meta. 1078b12--32]. Does Aristotle’s brief

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account of Socrates describe the historical Socrates or the Socrates to be found in Plato’s early dialogues? It is much more likely to present the historical Socrates than Sedley’s. After all, Aristotle sees Platonism as emerging from the confluence of Heracliteanism and Socrates rather than merely from Socrates. And Sedley seems to be in something of a question-begging position. Socrates is purely a moral thinker, as confirmed by the early dialogues and Aristotle’s testimony. In the Theaetetus, the semi-historical Socrates re-emerges; yet, when Socrates of the Theaetetus deals with what look like nonmoral issues such as the self-refutation of Protagorean relativism, Sedley resorts to the suggestion that Socrates can refute anything [61-62] and to a resemblance to arguments in the Euthydemus. Thus, the claim that Socrates is a limited moral thinker turns out to mean just that the theory of Forms is not made explicit; but this assumes that, if he possesses such a theory, Socrates has to introduce it prominently. Regarding the mathematics in Theaetetus 147c--148d, Sedley asserts that it is Plato, the author, who is in control here, and his speaker Socrates, in expressing approval for the mathematical paradigm, is unaware of the deeper philosophical significance which Platonically alert readers will be expected to spot. [28] But the ‘philosophical significance’ is, for Sedley, that it prefigures the mathematical education in the Republic. But surely, it is more pertinent to view the incommensurability of lengths and surds as suggesting the incommensurability of opinion and knowledge. Any Socratic awareness of mathematical incommensurability, however, risks exploding Socrates’ assumed obliviousness to such matters. Sedley clearly has a more nuanced than usual chronological interpretation of Plato. According to this interpretation, Plato on occasion, as in the Theaetetus, deliberately reverts to the Socrates of old rather than, as many chronological interpreters have supposed, eagerly and impulsively spilling his guts regarding his latest philosophical innovations so that one could trace a step-by-step progression in Plato’s thinking. And Sedley allows that Plato sometimes juxtaposes within the same text the semi-historical Socrates with the later Plato, as in the Meno and Republic. But if Sedley is so subtle regarding Platonic possibilities, why need we assume that Socrates in the early dialogues, were he to have had thoughts about a theory

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of Forms or physics, would need to spell them out? Would being explicit about the Forms be so clearly relevant to the early dialogues as they stand? Might the context and the interlocutor addressed be sufficient to explain what Socrates says, or refrains from saying, in any dialogue rather than a lack of sophistication? Sedley takes Socrates, in accord with the barrenness of midwifery, to lack doctrines, except those that connect with his very midwifery, such as the view that thought is silent internal dialogue [see 129--130, 109]. Consequently, it turns out that Plato and, hence, the Socrates whom Plato uses as ‘a mouthpiece for his own Platonic doctrines’ [9], have many doctrines. But this seems to misunderstand the Theaetetus’ account of barrenness and thought. Socrates declares himself barren in wisdom and opinion, but hardly barren in thought. Thought is not the same as belief or opinion. In Theaet. 189e--190a, Socrates treats opinion as the termination of thought, what one says to oneself internally when one ceases to have doubt. On this account of opinion, it is questionable whether Plato’s Socrates has any opinions at all though he is filled with thought. The ultimate support for Sedley’s sort of interpretation, then, is not special evidence for Platonic chronology but just what additional light this interpretation sheds on the Theaetetus. Sedley points particularly to his account of the Digression at Theaet. 172a--177c for confirmation of the value of his approach [see vi]. I will examine but one exemplary part of this account of the Digression. Sedley announces ‘the rarely noticed fact that in the Theaetetus piety is putting in a reappearance after a mysterious absence’ [82]. This absence was noticeable earlier, when in book IV of the Republic Plato reduces Socrates’ fivefold set of cardinal virtues to four, quietly dropping piety from the standard list. Its equally sudden reappearance is among the most significant Socratic features of the Digression. [82] But the other books of the Republic mention several more moral virtues [see, e.g., 368b8, 402c , 490c, and 615b]. Is it not plausible that in Republic 4 Socrates pares the list to four virtues in order to distribute them by process of elimination to three parts of city and soul rather than for the sort of reason suggested by Sedley? I reaffirm that Sedley offers a quite sophisticated treatment of Plato and the Theaetetus in a very engaging manner. As I have

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argued, however, his interpretive framework is questionable and may not advance our understanding of the dialogues. Yet his work offers many helpful interpretations of detailed arguments of the Theaetetus and of other dialogues that are separable from his global strategy for reading the dialogues. Those who approach Plato differently can well entertain and profit by the various points that he makes. bibliography Nightingale, A. 1995. Genres in Dialogue: Plato and the Construct of Philosophy. Cambridge. Vlastos, G. 1991. Socrates: Ironist and Moral Philosopher. Ithaca, NY.

Vitruvius: Writing the Body of Architecture by Indra Kagis McEwen Cambridge, MA: MIT Press, 2003. Pp. xiv + 493. ISBN 0--262--13415-20. Cloth $39.95

Reviewed by Lee Ann Riccardi The College of New Jersey [email protected] Vitruvius is one of the few ancient authors whose major work has survived virtually intact. Although sources tell us about other books on architectural or proportional theory that were written in classical antiquity, such as Iktinos’ treatise on the Parthenon or Polykleitos’ Canon, it is only Vitruvius’ De architectura (or Ten Books on Architecture, as we call it in English), written for the Roman Emperor Augustus, that is still extant and virtually complete. It is surprising then, that although art and architectural historians have been vitally interested in Vitruvius’ ideas, little of the secondary scholarship on him is by classicists and ancient historians. Although numerous translations of Vitruvius’ text exist, few have attempted to study it as a work of literature or, through what is perhaps an even more intriguing lens, as a work of Augustan propaganda. Augustan propaganda has been a hot topic in the last two decades. Particularly since the publication of Paul Zanker’s Jerome lectures at the University of Michigan in 1987 [Zanker 1987], and the English translation a year later [Zanker 1988], numerous scholars have turned their attention to this fascinating exploration; and a plethora of books, articles, and even museum exhibitions, have focused on decoding the propaganda of the first emperor. His patronage of the arts, building programs, numismatic choices, legislation, portraiture, and his final document, the Res gestae divi Augustae, have all been analyzed for their contributions to his program of propaganda and as ways to promote himself, his legacy, and his dynasty. 1

1

A small sampling includes Bartman 1999, Eck 2003, Galinsky 1996, Kienast 1999, Renucci 2003, and Wallace 2000. C 2005 Institute for Research in Classical Philosophy and Science 

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The major Augustan poets such as Vergil, Horace, and Ovid, who also wrote and dedicated their texts to Augustus, have been thoroughly examined in this light [see Barchiesi 1997, Kiernan 1999, Nappa 2005, Powell 1992, among others]. But perhaps because Vitruvius is not traditionally thought to have written a proper work of literature but rather a handbook of sorts for architects, his contributions as a writer and shaper of the themes of Augustan propaganda have thus far been overlooked. He has also often been regarded as a staid conservative focused on tradition and the past, and as out of touch with the revolution in Roman architecture made possible by concrete and vaulted forms. In Vitruvius: Writing the Body of Architecture, Indra McEwen focuses on Vitruvius’ own statement, repeated several times, that he is ‘writing the body of architecture’; and she therefore presents Vitruvius as an author with a mission and message very much in tune with his own times. She shows that Vitruvius is concerned with far more than buildings, materials, or engineering. In her view, he presents nothing less than a thorough guide for the development and spread of Roman civilization; and he provides for his emperor advice on how architecture was and could be used to establish Roman imperium. In Vitruvius’ mind, architecture was both the vehicle through which Roman domination was disseminated and the result of that domination. The development of the discipline of architecture is therefore ‘co-dependent on the Roman project of world domination’ [12]. McEwen shows that Vitruvius drew a constant parallel between architecture and the human body, specifically the body of Augustus. The word ‘corpus’, she notes, meaning ‘body of work’ postdates Augustus; and thus as Vitruvius uses it, he is inventing a new way of discussing architecture as a coherent and unified body of material assimilated to the body of the emperor [8--9]. McEwen’s chapters, ‘The Angelic Body’, ‘The Herculean Body’, ‘The Body Beautiful’, and ‘The Body of the King’, are each further subdivided into sections designed to explore the overall concept of the particular chapter. One of the advantages of McEwen’s organization is that most of these sections are chock full of densely articulated ideas and most could stand alone, even as they jointly contribute to her overall argument. The evidence she brings to bear in each of these sections is far-ranging in scope, and includes not only the evidence of specific works of architecture, city-planning, and engineering, but also

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evidence gathered from myth and literature, religious ritual, inscriptions, coins, and statues from Rome and the provinces. McEwen is comprehensive in her examination of the evidence used to develop her theory, meticulous in her referencing and widely read in her use of sources. With only a few missteps, she skillfully engages material disparate in media and in geographical origin in service to her theme. McEwen examines the purpose, structure, and audience for Vitruvius’ De architectura. Her interpretation is original. She begins by noting that in Vitruvius’ time, in order for something to be known, it had to be written [16], and that Vitruvius therefore wrote this commentary on architecture so that it could be known and remembered. This was at the very time when Augustus was publicly and emphatically restoring temples all over Rome. Vitruvius thought that this project, and architecture in general, would enhance and record Roman greatness worldwide; and his De architectura was intended to reveal how and why [38]. Public buildings provided visible auctoritas to power. Although he believed that architecture consisted naturally of three parts (buildings, gnomonice or the construction of clocks, and mechanics), he chose to write his commentary on 10 scrolls (now published as 10 books); and, as McEwen demonstrates, he arrived at this number by manipulating his subject, rather than because it was a logical division. To get to 10, he had to divide the section on temples into two books and to add another on water and aqueducts, a subject not part of his original tripartite scheme. But 10 was an important and perfect number, one believed to reveal universal order. Vitruvius understood architecture as coherent and unified [57], and 10 was the number required to show that. McEwen also addresses the issue of Vitruvius’ audience. She believes that he wrote specifically for Augustus, and that the De architectura was meant as both a gift and advice [69] so that the emperor might have access to orderly principles of architecture for his building programs [86--87]. Architecture and the De architectura were, therefore, a plan for and a record of Augustus’ achievements preserved for posterity. One of the major themes of McEwen’s book is the symbolic meaning of the Roman gods, and none are more central to her discussion than Hercules. In fact, chapter 2 (‘The Herculean Body’) begins with an anecdote about Alexander and his architect, Dinocrates, that Vitruvius relates in his preface to book 2 on building materials. The reason he includes this story, she suggests, is to draw a parallel

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between the relation of Alexander and Dinocrates and of Augustus and himself, and to use Hercules as a link between them all. In Vitruvius’ version of the story, Dinocrates is dressed as Hercules, a god also often chosen by Alexander as a model. Hercules was associated with brute strength but was regularly paired with Mercury, the god of communication (literally ‘running between’) and together they symbolized a common theme regarding the dual aspects of Roman power: force and speech, or strength tempered by reason [109]. In the anecdote, Dinocrates (and Vitruvius by extension) becomes the equivalent of Hercules, with the added knowledge of architecture. Alexander equals Augustus, and they all can be personified by Hercules. McEwen then introduces a story related by Dionysius of Halicarnassus [130], who wrote in the Augustan era. Dionysius told how Hercules civilized the world, mingling Greeks and barbarians as Alexander had. To do this Hercules needed architecture in the form of city-planning, heavy machinery to build roads and redirect rivers, and buildings. In this story, Hercules is Alexander and Augustus too, who, through architecture, brought culture to the uncultured, and thus benefited the whole world (or at least the world ruled by them). The cities they created were the seat of humanitas, where architecture brings together education and the circle of the world. And in the De architectura, Vitruvius makes clear that Rome is the city where civilization (and architecture) began [150]. Venus and the idea of beauty are also crucial to McEwen’s interpretation of Vitruvius’ aims. In chapter 3 (‘The Body Beautiful’), she explores the importance of beauty as revealed through geometry and proportion, particularly in regard to the foundation of cities and temples. It is this section, however, which is the most problematic, since she stretches her evidence farther than is reasonable. Although Vitruvius does not mention divination or augury in his discussion of how to found a city, McEwen insists that in both books 1 and 9 it is implicit. Her long section on the practices of augury and how it relates to geometry, symmetry, and proportion is fascinating; but it is not clear from the De architectura that Vitruvius actually took all this into account as he wrote his text. In regard to geometry, Vitruvius discusses the geometry of an ideal man, but he did not produce a drawing (as Leonardo finally did in the 15th century). He saw man as the source for geometry, not as a product of it [157]. Both a circle and a square can be traced around a man lying on his back. But

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in arguing that the true importance of these shapes rests in augury and how they were used by augurs to determine proper placement for cities, military camps, and temples, McEwen pushes Vitruvius’ intentions beyond what his words actually allow. Yet this chapter also contains one of her most intriguing sections, on the origins and meaning of the Corinthian order, the primary architectural style used by Augustus. In order to guarantee proper relations with the gods and fix political chaos in the aftermath of the civil war that brought him to power, Augustus had to demonstrate proper piety. This he did by the extensive and expensive project of restoring the crumbling temples in Rome. He made them whole and beautiful, and in doing so made extensive use of the Corinthian order. McEwen argues effectively that for Vitruvius (and probably Augustus too), its use is really about Rome and her civilizing mission [220]. For art historians who have long pondered the Roman preference for the previously little used Corinthian order, this is a particularly enlightening explanation. In chapter 4 (‘The Body of the King’), McEwen again introduces an argument based on assumptions that may not be valid and reveals that, despite the overall breadth of her knowledge, some issues are beyond her, in this case, those concerning Roman copies. The statue of Augustus from Primaporta postdates Vitruvius; and while it probably reflects an earlier version in bronze, the similarity of its appearance to the original is unknown. Romans were capable of making nearly exact versions of sculptures, but it was time-consuming and difficult to take so many precise measurements; and if they did, it was with the expectation that they could make multiple versions of the statue for maximum profit. But the Primaporta is unique. It is, therefore, highly unlikely that it closely reflected the original. Yet McEwen assumes that it did; and she uses the statue as the culminating demonstration of the principles of the De architectura, an impossibility for Vitruvius since the statue came after his text. Despite the occasional misstep, however, McEwen has still managed a remarkable feat of scholarship. She has presented a highly informative, comprehensive, fascinating, and original interpretation of a well-known text; and in doing so, she has demonstrated how much richer it is than classicists and ancient historians had realized. No one who reads Vitruvius: Writing the Body of Architecture will ever again

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think that the De architectura was simply a manual for architects or a chronicle of the history of architecture. It can no longer be doubted that Vitruvius’ purpose was much more grandiose, and was no less than to link forever Augustus, Rome, civilization, and architecture and to provide a guide for the establishment of Roman imperium. Vitruvius, like the major Augustan poets, contributed to the overall shaping of the image and message of the first emperor of Rome. bibliography Barchiesi, A. 1997. Poeta e il principe: Ovid and Augustan Discourse. Berkeley, CA. Bartman, E. 1999. Portraits of Livia: Imaging the Imperial Woman in Augustan Rome. Cambridge. Eck, W. 2003. Augustus und seine Zeit. Malden, MA. Galinsky, K. 1996. Augustan Culture: An Interpretive Introduction. Princeton, NJ. Kienast, D. 1999. Augustus. Prinzeps und Monarch. Darmstadt. Kiernan, V. G. 1999. Horace: Poetics and Politics. New York. Nappa, C. 2005. Reading after Actium: Vergil’s Georgics, Octavian, and Rome. Ann Arbor, MI. Powell, A. 1992. ed. Roman Poetry and Propaganda in the Age of Augustus. London. Renucci, P. 2003. Auguste le révolutionnaire. Paris. Wallace, R. 2000. ed. Augustus, Emperor of Rome, 63 B.C.--14 A.D., Res gestae divi Augusti as Recorded in the Monumentum Ancyranum and the Monumentum Antiochenum. Wauconda, IL. Zanker, P. 1987. Augustus und die Macht der Bilder. Munich. 1988. The Power of Images in the Age of Augustus. A. Shapiro trans. Ann Arbor, MI.

Lucretius on Creation and Evolution: A Commentary on De rerum natura 5.772--1104 by Gordon Campbell Oxford/New York: Oxford University Press, 2003. Pp. xii+385. ISBN 0--19--926396--5. Cloth $114.80

Reviewed by Brooke Holmes University of North Carolina, Chapel Hill [email protected] If survival of the fittest is a principle as relevant to ideas as it is to species, Darwinism has proved to be a hardy breed, especially if we judge provocation to be a sign of life. Last November alone, voters in Pennsylvania ousted school board members who had instated policies that gave Intelligent Design a hearing alongside Darwin in ninthgrade biology, while the Kansas Board of Education removed obstacles to teaching both perspectives. The Austrian Cardinal Christoph Schoenborn felt obliged to go on the record once again about the Church’s views on the debate over evolution in America. And an ambitious exhibition simply entitled ‘Darwin’ opened at the American Museum of Natural History in New York. One reviewer [Rothstein 2005], citing an 1844 letter in which Darwin says that writing about his ideas was ‘like confessing a murder’, takes the curator to task for domesticating a ‘bizarre’ and ‘shocking’ idea. What is so unnerving about Darwinism? The Cardinal is blunt: ‘It’s all about materialism, that’s the key issue’ [Heneghan 2005]. It is the idea that matter is the only reality. Epicureanism is Antiquity’s most infamous promoter of materialism, as well as the attendant horrors outlined by the Kansan Board—secular humanism, atheism, and the idea that life is accidental, both in the everyday and in its genesis. 1 Nowhere, perhaps, is the defense of these tenets more vivid and calculated than in the fifth book of Lucretius’ De rerum natura, in which our gaze is shifted from the primordia rerum, the imperishable first-beginnings of things, to 1

Of course, strictly speaking, the Epicureans are not atheists; but their gods inhabit another world and care not a whit for humankind, which is to say that their brand of humanism is aggressively secular. C 2005 Institute for Research in Classical Philosophy and Science 

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the contingent origins of our mortal cosmos and the organisms that inhabit it. Lucretius has an axe to grind in this book with Antiquity’s teleologists—the Platonists, the Aristotelians, and, most of all, the Stoics. His stated aim is to disabuse his reader of the idea that the nature of the universe is owed to any demiurgic blueprint. From the outset, the most visible cost of this disenchantment—which for Lucretius, of course, is its greatest gain—is the uncoupling of the human from any cosmic master plan. The critique of teleology is, then, at its heart, a critique of the idea that the world was created hominum causa, for the sake of people: cosmology for Epicurus and his followers always entails a payoff for ethics. Human exceptionalism is, in fact, recast as the particular hostility of the earth towards these creatures. This is best represented by the helplessness of the human infant, which is contrasted to the ease with which the other animals are at home in the world. Of course, birth trauma also matters to a teleologist like Plato. But whereas the Timaeus sees education as a realignment of our true nature with the divine order of the universe, in Epicureanism, human development is a history of creating defenses, good and bad, of which philosophy is the most noble and effective. This phylogenesis is then restaged, at least in part, as ontogenesis. The second half of book 5 of De rer. nat. takes up the task of accounting for human nature in a cosmos short on divine solicitude and partial to all the other animals, by investigating how anthropogony parts ways with zoogony under the force of circumstances. That is, it sets out to explain how the human (scil. us) is produced in time through the interaction of organism and environment, rather than to describe the pet project of a benevolent creator. Lucretius’ is a complex story in book 5: he tries to explain the spontaneous generation of life, speculates about human prehistory, gives a description of early communities that includes laconic explanations of the origins of justice and language, and traces the development of cities and civilization. For years, the mixture of apparently dystopic and utopic elements at both the early and the late stages of this story puzzled scholars, who split on whether Lucretius was a ‘Primitivist’ or a ‘Progressivist,’ i.e., whether he idealized the past or the present. The past few decades, however, have seen an increasing dissatisfaction with this either/or opposition and an attempt to

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engage more carefully with the text and its competing movements. 2 This trend has opened up a space for a new set of questions for readers of book 5: What constitutes the distance between humans and other creatures—viable life forms and monsters? What is the relationship between human vulnerability and the pursuit of ars? What is the role of necessity in producing the human, and where or when does it give way to a form of self-fashioning? What ensures the survival of humans in a quasi-Darwinian world of species competition? How do the stories that Lucretius tells about the spontaneous origins of life or human exceptionalism relate to our own dominant, albeit hotly contested, evolutionary narrative? Lucretius on Creation and Evolution, Gordon Campbell’s new commentary on some of the most interesting lines in Lucretius’ story (5.772--1104), stakes out the ground of some of these issues. Campbell has written a thoughtful and timely reassessment of Lucretius’ engagement with his teleological opponents, as well as with Presocratic zoogony—good use is made of the recently published Strasbourg fragments of Empedocles—Golden Age myths, and, of course, the elusive master text of Epicurus himself. Yet, if all roads lead to Lucretius, they approach him both from the periods prior to the poem and from our own recent, and sometimes very recent past. Lucretius’ poem was one of the most prominent explanations of the creation of life in mechanistic and non-teleological terms from the early Renaissance until the publication of The Origin of the Species in 1859, and Darwin’s theories bear some striking similarities to those found in the ancient tradition of thinking about the beginnings of life. Besides pursuing its natural task of Quellenforschung, then, Campbell’s commentary wagers that De rer. nat. 5 intersects with our present set of questions and anxieties about evolution in interesting ways, and sets out to map these points of intersection. ‘Map’ may be the wrong word here, however, since it is in his joining of past and present that Campbell is at his most creative. In fact, it would have been helpful if a better map of Campbell’s own work had been made available to the reader. No attempt is made in the introduction to integrate the lines chosen into the poem as a whole or book 5. Ready familiarity with more general 2

E.g., recently, Blickman 1989, Nussbaum 1994, Asmis 1996, and Holmes 2005.

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questions of background and context for the poem is assumed, as is a strong grounding in Epicureanism. Most of the Greek is translated, while, unsurprisingly, the Latin is not, although Campbell offers, in addition to the Latin text, a lucid translation of his excerpt. 3 Still, a reader working in translation on key topics, such as the origins of language or failed species, could glean much from the series of section introductions which form the backbone of the commentary proper. These range from brief transitioning remarks to condensed versions of working papers that Campbell has published elsewhere [2002a, 2002b]. Campbell has also added two long appendices—a ‘Table of Themes in Accounts of Creation, Zoogony, and Anthropogony’ and a ‘Table of Themes in Prehistories and Accounts of the Golden Age’ (including Isles of the Blessed, Ideal States, Noble Savages, and so forth)—which, while not exhaustive, offer a wealth of data. The index itself is unfortunately short and rather arbitrary. But despite limitations on the text’s accessibility to non-specialists, both students of Epicureanism and those interested in the history of thinking about the origins of life and the human in the Western tradition now have a stimulating guide to what is fascinating, if difficult, Lucretian terrain. The luxury of painstaking attention to the text is the great joy of a commentary. Such attentiveness in a commentary like this one, where it is combined with a deep sensitivity to larger issues, often rewards. For centuries, scholars have debated the complex relationship between poetry and science in the epic masterpiece of a philosophical tradition ostensibly hostile to myth and poetry. One of the strengths of Campbell’s commentary is its deft negotiation of Lucretius’ poetic and philosophical strategies, which allows one to observe the details of this dynamic. Lucretius has recourse to atoms and void as explanatory mechanisms in book 5 less often than elsewhere in the poem. This is not to say that we lose sight of the atomic underpinnings of the visible world, but we do spend more time at the macrophysical level. As a result, book 5 is rich in topoi on the formation of life, accounts of the Golden Age, and the development of human civilization, making it an excellent place to observe Lucretius’ ‘remorseless appropriation and recontextualization’ [138] of his nonEpicurean predecessors in action. 3

On Campbell’s text, see Volk 2004.

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Central to Campbell’s treatment of Lucretius’ strategy is his use of Richard Dawkins’ theory of memes. Memes are described as the sort of generally accepted background ideas whose origins are untraceable. . . that tend to exist and evolve as if they have a life of their own independent of any writer. [180--181] Campbell speaks, for example, of the ‘Darwin meme’, later corrected to ‘the pack of memes surrounding Darwinism that make up our Bildungsgut of prehistory’ [183]. More commonly, however, memes are not attached to a proper name. This presumably justifies Campbell’s decision to introduce them only ad lin. 925--1010, i.e., over halfway through the commentary, rather than in the context of his discussion of the Presocratic background to Lucretius’ zoogony. What appears key to the definition of a meme, then, is that it is a critically unexamined concept which travels easily beyond its initial context: the meme ‘survival of the fittest’, whose peregrination in the 19th and 20th centuries has been problematic, is an excellent example. Thus, it makes sense that Campbell appropriates memes for his discussion of how Lucretius integrates material from other sources and shapes it to suit his purposes, all the while working, whether actively or more subtly, to head his reader off from the kinds of incorrect inferences to which those sources fell prey. Memes, for Dawkins and Campbell, function like viruses against which one may need to be ‘vaccinated’, although they may still be useful. Campbell sees the Golden Age ‘as an integral part of Lucretius’ prehistory, and as the very material out of which he builds it. Certain themes’, he claims, ‘are rationalized and debunked, while others are allowed to remain untouched and to do their work of vaccination simply by their recontextualization in Lucretius’ account’ [184]. Analyzing a descriptive passage on the streams that satisfied the thirst of early humans, Campbell argues, for example, that Lucretius appropriates pastoral poetry in order to advance the principle of ‘cultural gradualism’ against the myth of divine beneficence. He then goes on to suggest that Lucretius also uses this idyllic picture ‘to both legitimate and illustrate Epicurean ethics’ [206], by showing that the body’s necessary and natural needs are easily met in a world still uncontaminated by luxury goods. The utility of an intermediary stage of village life between the erramento ferino (wandering in the wild) and the formation of cities, which is unique to Epicurean prehistory,

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thus becomes clear. For it offers a picture of a simpler time when Epicurean justice could emerge in rudimentary form, without, for all that, resorting to a Golden Age: ‘in prehistory we find the Epicurean theory at its most powerful, stripped of the accretions of culture and civilization, saving the human race itself from extinction’ [254]. Much of the Golden Age imagery that one does find in the account of spontaneous generation, when people were born from great wombs rooted to the earth, can be seen as helping Lucretius answer the question of how these beings survived without mothers or τ χναι (artes): although the image of the Earth as mother, with its overtones of Stoic allegory, has him ‘skating on thin ideological ice’ [60], he needs the Earth to give forth a milk-like juice if these first creatures are to be nourished. Campbell’s attention to Lucretius’ tendency to stress positive or negative aspects of early human life according to what he is trying to do at a given moment places him firmly within the trend of moving beyond the Primitivist-Progressivist opposition. This sensitivity to Lucretius’ strategies extends from Campbell’s handling of broad themes to his comments on single words and phrases. He is alert to how Lucretius manipulates language so as to downplay external agency (e.g., on exclusae, ad 802 [71]), or conversely, how it slips into a teleological idiom (e.g., on crerint, ad 782 [47]). This makes him an ideal reader of Lucretius, that is, someone whom it is ideal to read with. In this sense, Campbell’s is a more satisfying commentary than Costa 1984, which covers book 5 in its entirety. The two commentaries, in fact, work nicely together, since Costa attends more to questions of grammar, but is more reticent and conservative vis-à-vis the big issues. But despite the benefits of working with the text in this format, it is precisely the strength and cohesiveness of Campbell’s interpretation that makes one begin to wonder why he chose to prepare a commentary rather than a monograph. Lucretius on Creation and Evolution occasionally feels like one of the hybrid creatures that Lucretius describes in the zoogony. It can seem as though an argument is being carried on in footnotes [see below]. On the other hand, while individual lemmata often give rise to wide-ranging and imaginative discussions, the commentary can begin to resemble a cabinet of curiosities. It is at these moments that one especially feels that the format allows Campbell to accumulate information without properly sorting it out, or simply to wander off. Commentaries, it is true,

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are built through the accretion of comparanda. But sometimes this parataxis can be confusing, if not misleading, especially given that Lucretius’ dense blend of argumentation and never-innocent illustration requires careful untangling. Take, for example, Campbell’s treatment of Lucretius’ denial of the existence of Centaurs. Lucretius moves from the argument that men and horses age differently to pointing out that they neither ‘burn with the same passion of Venus, nor come together with a single lifestyle, nor find the same things pleasant for the bodies’ [De rer. nat. 897--898, Campbell’s translation]. This last claim opens onto the observation that hemlock is great for goats, poisonous to humans. Lucretius is building on the theory of perception outlined in book 4 whereby different bodies have different pores that determine what they experience as painful or pleasurable. In his notes on this [154], Campbell responds to this shift from sexual desire to food with an excursus on the association between food and sex in Lucretius, which he claims is underwritten by Greek biological thinking on human seed. Now it is perfectly true, as Campbell explains, that Aristotle and probably earlier Presocratic writers understood seed as a residue of concocted food, and that elsewhere Lucretius directly relates diet to the quality of seed [De rer. nat. 4.1260ff]. Yet this has little relevance here, where Lucretius’ easy transition from sexual desire to food hinges on the word iucunda (pleasures). As Campbell points out ad 897, Lucretius thinks that seed is stimulated through seeing, and that the only catalytic objects of vision are members of one’s own species. In a similar way, certain species gain pleasure from certain foods, although it should be said that those whose bodies are incompatible with the food in question may be harmed by it, whereas the seed of a person looking at Black Beauty is presumably simply indifferent to equine loveliness. It is not that Campbell is wrong about ancient ideas about the role of food in the production of seed. But his digression on this topic obscures what really matters here, namely, the Epicurean lock-and-key perception theories that can explain both species-specific desire and some species’ pleasure in foods that are poison for other species. As it stands, the notes on food and seed seem better suited to the material on diet and seed in book 4, or, even better, to the discussion of food (pabula) on page117, where, in fact, they would have been quite helpful. It is not that valuable information is missing: the relevant comments on seed—that there

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is no viable means of reproduction without visual stimulation—are made ad 897, while the explanation of the lock-and-key theory of perception is available ad 899--900. It is simply that there is so much unorganized information that the notes no longer clarify Lucretius’ argument. I should say that Campbell’s notes are chock-full of interesting information, and that they are occasionally very funny. Indeed, one is often content just to wander with him. It is simply that at times the line between erramento and error seems a bit too fine. While, in principle, Campbell is looking backwards and forwards equally, from Lucretius’ predecessors to his Nachlëben and our own conceptual habits, in practice, the fact that these habits are shaped by modern evolutionary biology means that they have a special claim to truth. As a result, the motivation for introducing modern evidence is often ambiguous. Here again one feels that the commentary format is problematic in that it allows Campbell to remain less than forthcoming about his own agenda, especially vis-à-vis Lucretius’ anthropology. Campbell reasonably argues that clarifying our own, heavily Darwinian ideas about what an anti-teleological story of the creation of the animate world should look like enables us to understand better the specific mechanisms of Lucretius’ system. Lucretius, for example, does not accept mutation at the genetic level and seems to accommodate the inheritance of acquired characteristics, at least in humans [7--8]. At the same time, Lucretius’ difference from Darwin sometimes seems presented in such a way as to make him seem more cutting-edge: his view of species as fixed and bounded entities, for example, allies him with post-Darwinian notions of species stasis [124-125]. In such a context, the use of modern parallels appears to be less about crystallizing our own preconceptions and more about vindicating Lucretius. On page 221, for example, Neolithic archeology is enlisted as support for Lucretius’ picture of moderate violence among early humans over and against Moschion’s more lurid account of cannibalism. Ancient bones again work to bolster Campbell’s reading of the role of cooperation in human evolution [280--281] (‘L. is quite correct to place pity for the weak in prehistory’), although it should perhaps be noted that the solidity of the bones cannot be extended to the inferences drawn from them that are used to prove Lucretius correct. Different things appear to be at stake in different comparisons undertaken either casually or at some length in the commentary, without

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these stakes always being made clear. Moreover, the interest in showing that Lucretius anticipated the work of modern science can dull Campbell’s often incisive readings of the dense nodes of myth, Epicurean philosophy and rhetoric that are so characteristic of book 5. In the second half of this review, I want to take a look at the most ambitious part of Campbell’s interpretation of Lucretius’ evolutionary narrative. Campbell argues that Lucretius’ claims that humans are the only species to have evolved; that they evolved into cooperative beings; and that, in doing so, they ensured the survival of the species and the definitive break between the human and the animal. How does this work? Lucretius’ account at 5.1011--1027 of the ‘softening’ of human nature and the formation of the first communities is notoriously elliptical. In recent years, much attention has been paid to how his apparent citation of the principle of Epicurean justice—neither to harm nor be harmed (nec laedere nec violari)—participates in the story being told here. Campbell sees the entire passage as a turning point in the evolution of the species: whatever changes human nature undergoes here are, henceforth, passed down as inherited characteristics, making Lucretius a Lamarckian. Yet, in part because Campbell’s argument is carried on disjointedly across lemmata, it is unclear how the external environment provokes these changes to human nature and what these changes are. Some of this vagueness is due to Lucretius himself. At 5.1011, without obvious motivation, Lucretius introduces a new stage of human development in which humans have houses, clothing, fire, and nuclear families. From this point on, they begin to soften (mollescere). Spending time indoors, their bodies can no longer bear exposure to the elements. Venus has a hand in this softening, and children break the arrogant natures of their parents with their ‘winning ways’. Somehow this process creates the conditions for humans to establish pacts with one another neither to harm nor to be harmed, and to set up the principle of pitying the weak, which Campbell calls altruism. It is this ‘somehow’, of course, that matters. Also of crucial importance is how we understand what has been gained in this development. Campbell knows what he wants to show, namely, that ‘instead of being a woolly-minded pipedream, the Epicurean theory is the most pragmatic and realistic approach to justice’ [281], and that Lucretius

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anticipates post-Darwinian work on the evolutionary benefits of cooperation: ‘our co-operative ability is thus the feature that defines our humanity, and enables us to survive’ [262]. Lucretius’ presentation of the Epicurean theory, as Campbell understands it, accords with recent refinements to the model of the prisoner’s dilemma which was first analyzed in an Epicurean context by Nicholas Denyer [1983]. The basic form of the prisoner’s dilemma takes two prisoners charged with the same offense. Each is offered a deal by the police chief: if both cooperate, they each get a year in jail; if one defects by confessing and betraying the other, he gets off scot-free, while his counterpart (the sucker) receives 10 years. 4 Conventional wisdom once held that the rational choice would be defection. When the dilemma was translated into Darwinian terms, it became hard to see how any form of cooperation could be plausible in the theater of ‘survival of the fittest’. As a result, cooperation could only be explained as selfsacrifice for the good of the community. Campbell introduces the challenges to these assumptions posed by the research of Robert Axelrod, who worked with computer models of the prisoner’s dilemma in the 1980s. Axelrod’s research revealed that it is not competition but cooperation that proves more advantageous when the game is repeated. More specifically, a strategy called ‘tit-for-tat’, which always reciprocates the behavior of its opponents—it betrays when betrayed, cooperates with those who have cooperated with it—emerged as the strongest. Moreover, after some time playing with tit-for-tat, the ‘suckers’ began to thrive and the defectors became nearly extinct. Campbell uses these results to reformulate Denyer’s conclusions: ‘the Epicurean model would achieve the best result both for the individual and for the group, and the individual gives up no direct advantage by sticking to the friendship/non-aggression/mutual aid pacts of the Epicurean theory, but receives a direct personal advantage by doing so’ [258]. This appears a valid application of Axelrod’s work. In casting early humans’ negotiation of non-aggression pacts (amicitia) in a utilitarian light, Campbell keeps Lucretius firmly in the realm of Epicureanism, where ‘natural’ action is always motivated by a desire to secure the individual’s pleasure.

4

Campbell omits a third scenario in which both defect and receive five years apiece.

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However, there are two problems with what Campbell does from here. First, after applying Axelrod’s work to Epicurean mutual nonaggression pacts, Campbell immediately rephrases his claim thus: ‘it is a powerful individual survival strategy that all should pity the weak’ [258]. He presents such pity as ‘learned behavior’ and gives it pride of place in human evolution: ‘now that the human race has evolved, they are able to make the conceptual leap from their previous conviction [i.e., aggressive competition was the only survival technique] to “it is fair that all pity the weak”.’ But is this pity the same thing as amicitia? What I find troubling about this easy conflation of the two ideas is that the prisoner’s dilemma assumes a community of equals; the tit-for-tat strategy relies on the opportunity for future retribution in the case of defection. If it is just a question of the weaker versus the stronger from the outset, game theory seems irrelevant: we would seem to be in the world of Plato’s Gorgias, where the only answer to Callicles’ law of the stronger would be the one that Plato gives, namely, that doing injustice is worse than suffering it. Campbell might claim that in a community which includes defectors, the weak (i.e., the suckers), and tit-for-tat players, altruism (always cooperating) still emerges as the second-best strategy. Indeed, he seems to suggest this on pages 277 and 280. But even if a player is programmed to cooperate always within the game, the rules of the game still assume the conditions of total reciprocity, that is, that the weak, at least in theory, have the power, say, to lessen my prison sentence. In any kind of pragmatic situation, if a player is by definition weaker, the stronger has no reason not to cooperate with his equals and dominate the weaker. 5 Thus, asserting that the principle ‘pity the weak’ is strictly utilitarian hardly seems valid. I stress this because, although Campbell acknowledges that ‘there does seem to be a huge conceptual gap between the pacts “neither to harm nor be harmed” and “it is fair that all pity the weak” ’ [277], he nevertheless groups amicitia and pity for the weak together as cooperation. To

5

It is worth pointing out that Axelrod seems only to admit strict altruism in cases of kin-relations, where self-sacrifice can be understood in terms of propagation of the gene pool [1984, 88--89 and 134--135]. At the same time, he does speculate that it is cooperation within kin groups that leads to the adoption of cooperative strategies outside of kin relationships, a move that Campbell reproduces [see above].

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classify pity for the weak as cooperation is a bold move, since cooperation is the human trait that Campbell argues protects the species from extinction, but it does not appear justified by the argument. Yet it is clear that there must be some connection between the softening of humans, the formation of friendship pacts, and pity for the weak. The second problem with Campbell’s account is that he is not explicit about how this connection works in Lucretius. Nevertheless, with a little work, his reconstruction of Lucretius’ argument can, I think, be discerned. Campbell recognizes that any evolutionary change must come from the environment. This means that the softening of human nature in response to key aspects of the domestic environment—the warmth of the fire, shelter, marriage, and childcare—enjoys some claim to priority as a cause for all future changes. What gives rise to this shift towards domesticity is not evident; Lucretius may be silently assuming that the vulnerability of humans to other beasts (or at least some humans to other beasts) could have driven them to extinction, had they not formed families and communities. In any case, Campbell argues that it is because men become softer in their relationships with their wives and their children that they behave favorably to their neighbors: ‘the amicitia does not seem to arise from utility as in Vat. 23 but from a more spontaneous and more nearly altruistic motive. The results are pragmatic’ [273]. Thus, he understands the softening of human nature as leading directly to the gain of reasoning (λογισμ ): the argument would be that once the benefits of cooperation, initiated not for utilitarian reasons but as a result of the softening of human nature, were seen within the family, humans were then moved to forge relationships of amicitia with their neighbors. This would explain why family life is so important to the account of justice: it generates the conditions under which humans learn the benefits of cooperation, whatever we take these to be—again, Campbell is vague on this point, as well as on how these benefits map onto the benefits of cooperation within the prisoner’s dilemma model. The final stage would be something that looked like altruism: the first humans ‘are pictured extrapolating from the lessons they have learned at home, and applying the results to the women and children of other families in a positive development of the negative nec laedere nec violari’ [277]. In this version of the argument, pity for non-kin weak thus gets a more complex explanation than Campbell puts forth in his original

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continuous version of the argument [see 252--261], where pity is regularly just substituted for amicitia. Or rather, it gets two and perhaps even three explanations. On the one hand, Campbell recognizes the difficulties posed by his earlier inclusion of women and children in the prisoner’s dilemma: ‘the weak cannot strictly engage in the first part of the pact nec laedere (do not harm), and so this transcends the basic non-aggression pacts’ [277]. His solution is to make the softening of human nature the cause of both the non-aggression pacts and pity for non-kin weak. In other words, pity for non-kin weak bears no direct connection to the non-aggression pacts, although both have their roots in the formation of nuclear families. Such a position makes Campbell ’s vagueness about the ‘lessons. . . learned at home’ all the more troubling, since these lessons can no longer been seen as derived from the utilitarian benefits associated with the non-aggression pacts. It seems likely that Campbell is close to the Lucretian position when he says that the softening of human nature ‘is not strictly driven by utility, and the value of such a psychological change becomes clear only later’ [272]. In fact, Campbell has not shown the place of utility at all at this stage of evolution: the utility associated with his re-reading of the prisoner’s dilemma only becomes clear when non-aggression pacts are formed. What this means for his argument is that, first, some fundamental psychological change unrelated to the intellectual perception of benefit is the only explanation for the development of amicitia and any utilitarian justification comes later. It is thus unclear, if ‘the first friendship pacts had to be learned intellectually’ [278], what experience the first humans are learning from. Second, since at this stage of the argument Campbell no longer relates pity for non-kin weak directly to the non-aggression pacts and the benefits of cooperation that they reveal, the relationship between utilitarianism and altruism falls apart. Pity for non-kin weak may be an extension of intrafamilial cooperation, but we do not know what that is. We are left to conclude that pity is an acquired characteristic derived from the softening of human nature that never enters into the utilitarian calculus. But on the other hand, Campbell is clearly attached to the idea that altruism for Lucretius holds an evolutionary advantage and is an important part of the cooperation that saves the species. Having recognized that women and children are technically out of place in the prisoner’s dilemma since they do not have the power to harm,

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he nevertheless reintroduces the prisoner’s dilemma in an attempt to reestablish the utilitarian pedigree of altruism. This return to the earlier arguments seems to offer an explanation of pity that is incompatible with the one in which altruism and amicitia are parallel developments arising from changes to the psychological makeup of human beings. It may be that Campbell understands both altruism and amicitia as independently revealing their utility over time. But then we are back where we started: how can we evaluate the utility of altruism if its benefits are not those derived from cooperation within a community of equals? If this utility lies elsewhere, why introduce the prisoner’s dilemma to explain altruism? There is still a third explanation of pity lurking in Campbell’s text, one that suggests yet again the underlying problems with the utilitarian reading of altruism that he offers. On page 278, Campbell shifts gears and embeds altruism once again in the development of amicitia, rather than allowing it to develop directly from the nuclear family. On this view, pity is the extension of the non-aggression pacts to unequal power relationships with other men’s women and children. 6 Here, it is as if having learned the benefits of cooperation in the quasi-political sphere, men no longer want to exploit their power advantage even when reciprocity is out of the question although, if domination is taken to be a good thing in itself, pace Plato, aggression would yield more gain than cooperation with the weaker. Campbell gets around the problem of why men stop dominating the weak even when there is no advantage in their restraint by reintroducing the theory of acquired characteristics. Cooperation is thus ‘learned behavior’ which is ‘passed down to offspring as an instinctive response to women and children’ [278]. That Campbell offers this interpretation would seem to be a tacit acknowledgement that it is difficult to construe pity for the weak in utilitarian terms, or at least utilitarianism as it appears within game theory. One also wonders whether pity is the result of a characteristic acquired during the process of domestication or if it comes after the development of amicitia. Most importantly, while the position that pity for non-kin weak is an ‘instinctive response’ solves the problem of fitting altruism into the prisoner’s dilemma model, it moves decisively away from Lucretius’ 6

This is sometimes called the ‘associationist’ argument, which is treated at length in the context of Epicurean friendship by Philip Mitsis [1988].

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text, where amicitia and what Campbell calls altruism are closely linked. As a result, altruism disappears from the species-saving cooperation that Campbell thinks is so important to Lucretius’ concept of the human. Campbell’s overall argument has its merits, and one wishes it had been laid out more systematically at some point. His practice of referring the question of pity for the weak to Axelrod’s work on the prisoner’s dilemma seems to reflect his own desire to make Lucretius’ definition of the human conform to what one suspects is his own. Unfortunately, his use of game theory distracts him from the text itself. It would seem that Lucretius is telling two stories in these lines. One certainly has to do with justice. Equally important, however, is the fixation on children, which cannot be fully explained by staying within the parameters of the debate about justice. As Campbell himself observes, when Lucretius talks about pity for the weak, he ‘deliberately go[es] beyond what Epicurus would consider allowable in giving only one possible cause of justice’ [276]. Excessive, too, is 5.1027, a verse on which Campbell is uncharacteristically silent. Lucretius asserts that the human race would have been destroyed without concordia, then adds that ‘nor would the offspring have been able to prolong the race to this day.’ Another oddity is the expression ‘muliebreque saeclum’ (‘female race’) at 5.1021, found nowhere else in Latin literature. Campbell insists that Lucretius does not see women as a separate race or species. He relates ‘muliebreque saeclum’ to similar phrases (e.g., ‘muliebre secus’) in other authors, although ad 853 he has a long note on ‘saeclum’ where it, with ‘genus’, is clearly Lucretius’ word for species, whether human or animal [e.g., 5.791, 1169, 1238]. And ‘muliebre genus’ is found at 5.1355, in a discussion of the first forms of clothing. Lucretius tells us there that Natura forced men to weave first, since the genus virile excelled in ars and was far more clever than the race of women. Not a separate species, then, but also, perhaps, not entirely at home in the genus humanum. One begins to wonder whether all the talk of human nature in both Lucretius and Campbell forgets a difference embedded in the text that needs to be acknowledged if these lines are to seem less opaque. My own suspicion is that there is a bit of a conceptual lacuna in the erramento ferino, which exerts pressure on this subsequent

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phase. Lucretius goes to great pains, as we have seen, to account for how the first humans survived by positing a surreally maternal earth. But this earth becomes hard and brutal after the early days of spontaneous generation. We know too that care of the young is a pressing object of concern for Lucretius: as I noted above, earlier in book 5, he argues against teleological cosmogonies by enumerating the earth’s faults, foremost among which is its hostility to the human infant cast out from its mother’s womb. So what is going on in the period when the earth has turned cruel but families have not yet formed? At 5.1011ff, children re-enter the picture because men and women join in marriage. A line is probably lost between 5.1012 and 1013, and it might have shed light on the implications of this move towards the nuclear family. It is intriguing that the manuscripts have cognita sunt—although Campbell accepts Lachmann’s emendation (coniubium) and argues against the lacuna—which anticipates the intellectual act implied by the next words: ‘and they saw the children created from them.’ ‘They’, as Campbell rightly notes, must be the fathers, an indication that this ‘human’ story is, in fact, being told from the perspective of one sex, whose acquisition of reason may rely more on the inference of paternity than on the perception of any intrafamilial cooperation. For I think Campbell is right to understand the verb ‘to see’ (videre) to mean that the fathers realize that the children are their own, that is, that they are created ex se (from themselves). This recognition, combined with fire and love, prepares the way for amicitia and pity for the weak. But how? In recent years, it has been argued [see van der Waerdt 1988] that later Epicureans modified Epicurus’ denial that there could be any natural affection between humans—even between children and parents 7 —by adopting some version of the Stoic concept of ο κε ωσι , which Long and Sedley [1987 I.351] translate as ‘affectionate ownership’. Campbell accepts these arguments in order to make ο κε ωσι a factor in the development of pity for non-kin weak [277--278] in the third explanation which I sketched above. Yet it would make more sense to locate ο κε ωσι at the moment when fathers recognize their children as their own and take on wives, especially since Stoic

7

Epicurus advised the sage to stay single, marriage being full of ‘inconveniences’.

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ο κε ωσι begins with, in addition to a desire for self-preservation, love of offspring. If we understand the protection of wives and children as now implicated in the well-being of the autonomous male individual [cf. Homer, Od. 9.114--115], we can see that pity for the weak may fall under the logic of a mutual non-aggression pact without being simply a knee-jerk extension of it or a parallel development. It is not completely clear whether the weak are threatened by wild beasts or other humans (already in the erramento ferino, sex is marked by either mutual desire or male violence, and perhaps by the idea, too, that women need to trade sex for food), or by both. If it is other humans that are the problem, men may entrust (commendarunt) women and children, presumably to one another, in these pacts on the basis of a shared vulnerability—I will not harm your family if you will not harm mine—thus restoring the conditions of reciprocity and the promise of retribution required by game theory. This may explain Lucretius’ emphasis on the importance of man-to-man communication in the passage on altruism (‘signing with cries and halting gestures that it is right for all to pity the weak’). But if we at least entertain the possibility that the danger lies with wild beasts, pity then encompassing a commitment to the protection of wives and children from these beasts, we might get a sense of why Lucretius’ conclusion to this passage is so apocalyptic. If Lucretius so readily groups the female race with children as imbecilli (the weak and the helpless), it is not just a question of how this ‘race’ ever survived, but how children managed to. For, presumably, women were solely responsible for childcare before. Acknowledging this tension between the need to offer protection to women in the first communities and the earlier assumption of gender-neutral selfsufficiency sheds some light on 5.1027, then, where the worry is not only that the human race will be wiped out, but that the offspring will not be able to lead the species (saecla) into the future: to reproduce, they have to survive their traumatic and vulnerable early years. There may be a sense here that in such a hostile world, women and children need men, that the future of the species required fathers once Natura herself turned cruel. The Garden, Epicurus’ philosophical school, was revolutionary in its acceptance of women, and Epicureanism may, as a philosophical system, leave more room for ‘feminine principals’, as Campbell’s

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mentor, the late Don Fowler [1996], has argued. 8 Campbell suggests that the very process of softening may be taken as the ‘feminization’ of the human race. But this only reminds us that what matters here is the evolution of men [see 267--268 ad 5.1014], whose softening creates the impetus for ars, undertaken on both their own behalf and that of a genus that is far less clever. Lucretius, it is clear, excludes the saeclum muliebre from justice and the political at the origins of these institutions, and, as I have suggested, inscribes the male protection of women and the paternal oversight of the family line into the order of necessity. This order of necessity becomes, for Campbell, the very conditions under which the ‘human’ emerges as such. At the very least, it is worth observing that when Campbell talks about the need for humans to learn cooperation to survive, he, like Lucretius, is talking about a community of men. Thus, the awareness of difference within the genus humanum is not just some politically correct orthodoxy. Nor does it mean that we have to reinstate a binary opposition, this time between Lucretius the misogynist and Lucretius the feminist; for like Primitivism and Progressivism, these terms conceal more than they reveal. Rather, it is a question of marking complications that seem to disrupt the text itself. In mimicking Lucretius’ own conflation of human nature and male nature by reducing everything here to the prisoner’s dilemma and equating political cooperation with what is human, Campbell’s reading falls short of full engagement with a complex text. In short, I rather like the idea that Lucretius could have submitted the winning entry in Axelrod’s first iterated prisoner’s dilemma competition. My concern is that if we focus on whether Lucretius’ idea of human nature, and particularly that which makes us human, is validated by modern views of utilitarianism in an evolutionary context, we may miss how his story is implicated in the dynamics of the poem and its assumptions, something to which Campbell is sensitive at other points. Moreover, such a strategy fails to recognize that even if the conclusions of modern science and Lucretius coincide, it still matters how they arrived at those conclusions: surface similarities betray fundamental differences in the production of knowledge. 8

While Campbell’s citations are often extensive, in noting Fowler on the ‘feminine principal’, he fails to cite the article that sparked it, Nugent 1994, which presents a less positive picture of gender in Lucretius.

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Finally, and perhaps most importantly, Campbell seems to take the evidence of science—evolutionary biology, game theory, prehistoric archaeology—as the gold standard for determining what it is that makes us human, that is, where it is that Lucretius gets it right. I want to stress that I find Campbell’s revision of Denyer’s account of Epicurean justice in many ways persuasive. It has changed my understanding of this passage and challenged me to rethink it; I have no doubt it will stimulate further discussion of these lines. Thus, it may seem perverse to want to restore sexual disequilibrium or a politics of dominance to a reading of Lucretian prehistory that celebrates the social utility of cooperation and the ethics of altruism. But it is precisely the use of modern sources to legitimate Lucretius’ humanism (and, perhaps, vice versa) that makes a reading which recognizes the specificity of the saeclum muliebre end up looking like support for a socio-biological theory of sex roles and, more insidiously, like an endorsement of such a theory as a blueprint for defining the human. That is, the reading which I have adumbrated starts looking strangely prescriptive. And so, it is not just that I think that Campbell’s fidelity to game theory sets an unnecessary limitation on his interpretation of these lines. I also find the resulting naturalization of Lucretius’ story troubling. Lucretius on Creation and Evolution offers a bold and sophisticated attempt to come to terms with Lucretius’ arguments on evolution in the spirit of the poem’s most ambitious commentators. It deserves not only consultation but active perusal. I could not agree more with Campbell’s commitment to putting Lucretius and Epicureanism into conversation with the present and with our own attempts to figure out where humans belong in a world of chance and impersonal necessity. But reading book 5, it seems to me, is an exercise in tracing the contingency of anthropologies and anthropogonies. We can locate its contemporary salience in the interaction visible within it between materialism and the stories which Lucretius tells of what nature ‘forces’ [e.g., 5.1028, 1354] us to be or become, and in Lucretius’ interpretation of the imperatives which he believes he finds inscribed into us. As Benjamin writes, ‘It is true that men (Menschen) as a species completed their evolution thousands of years ago; but mankind (Menschheit) as a species is just beginning his’ [1996,

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I.487]. 9 In this chronology, the equation of mankind and humankind remains the least evolved thing of all. 10 bibliography Asmis, E. 1996. ‘Lucretius on the Growth of Ideas’. See Giannantoni and Gigante 1996, 763--778. Axelrod, R. 1984. The Evolution of Co-operation. New York. Benjamin, W. 1996. ‘Einbahnstraße’. Pp. 83--148 in T. Rexroth ed. Walter Benjamin. Gesammelte Schriften IV.1. Frankfurt am Main. See Jephcott 1996, 444--488. Blickman, D. R. 1989. ‘Lucretius, Epicurus and Prehistory’. Harvard Studies in Classical Philology 92:157--191. Campbell, G. 2002a. ‘Lucretius 5.1011--27: The Origins of Justice and the Prisoner’s Dilemma’. Leeds International Classical Studies 1.3:1--12 (http://www.leeds.ac.uk/classics/lics/2002/200203. pdf). 2002b. ‘Lucretius and the Memes of Prehistory’. Discussion Paper 1, November 2002. Leeds International Classical Studies (http://www.leeds.ac.uk/classics/lics/discussion/ 2002dp1.pdf). Costa, C. D. N. 1984. ed. Lucretius: De rerum natura V. Oxford. Denyer, N. 1983. ‘The Origins of Justice’. Pp. 133--152 in ΣΥΖΗΤΗΣΙΣ. Studi sull’ epicureismo Greco e romano offerti a Marcello Gigante. Naples. Fowler, D. 1996. ‘The Feminine Principle: Gender in the De rerum natura’. See Giannantoni and Gigante 1996, 813--22. Reprinted in Fowler 2002, 444--452.

9

If Benjamin’s own Menschheit, like Lucretius’ genus humanum, functions as the umbrella term of sexual difference in order to name a single species, there is a certain honesty to Edmund Jephcott’s politically incorrect English translation, ‘mankind,’ which lays bare what is concealed by the ostensibly neutral German and Latin terms. 10 Many thanks to Jim Porter, Gerry Passannante, and Miles Nelligan for helpful criticisms.

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Fowler, D. 2002. Lucretius on Atomic Motion: A Commentary on Lucretius, De rerum natura 2:1--332. Oxford. Giannantoni, G. and Gigante, M. edd. Epicureismo Greco e romano: Atti del Congresso Internazionale, Napoli 19--26 maggio 1993. Naples. Heneghan, T. 2005. ‘Vienna Cardinal Draws Lines in Intelligent Design Row.’New York Times Nov 20. Holmes, B. 2005. ‘Daedala Lingua: Crafted Speech in De Rerum Natura’. American Journal of Philology 126:527--585. Jephcott, E. 1996. ‘One-Way Street’. Pp. 444--488 in M. Bullock and M. W. Jennings edd. Walter Benjamin: Selected Writings. Vol. 1: 1913--1926. Cambridge. Long, A. A. and Sedley, D. N. 1987. The Hellenistic Philosophers. 2 vols. Cambridge. Mitsis, P. 1988. Epicurus’ Ethical Theory: The Pleasures of Invulnerability. Ithaca, NY. Nugent, S. G. 1994. ‘Master Matters: The Female in Lucretius’ De rerum natura’. Colby Quarterly 30:179--205 Nussbaum, M. 1994. Therapy of Desire. Princeton. Rothstein, E. 2005. ‘Enough to Make an Iguana Turn Green: Darwin’s Ideas’. New York Times Nov 18. van der Waerdt, P. A. 1988. ‘Hermarchus and the Epicurean Genealogy of Morals’. Transactions of the American Philological Association 118:87--106. Volk, K. 2004. rev. Campbell 2003. Bryn Mawr Classical Review 2005.06.26.

Pythagoras: His Life, Teaching, and Influence by Christoph Riedweg. Translated from the German by Steven Rendall in Collaboration with Christoph Riedweg and Andreas Schatzmann Ithaca, NY/London: Cornell University Press, 2005. Pp. xiv + 184. ISBN 0--8014--4240--0. Cloth $29.95

Reviewed by Peter Lautner Pázmány Péter Catholic University, Budapest-Piliscsaba [email protected] This book is a translation of Pythagoras. Leben, Lehre, Nachwirkung. Eine Einführung [Munich: C. H. Beck, 2002]. It is furnished with a rich and adequate bibliography, and a general index. However, it is not graced with the map which is on page 182 of the German original, and it also lacks the representations of Pythagoras on pages 81 and 83. These representations include a bronze coin from Samos from the second century BC, a bronze bust from the Villa dei Papiri in Herculaneum, and a devotional relief from Sparta from the fourth century BC. This latter is particularly interesting and raises the question of the nature of the interest in Pythagoras in Sparta. The general aim of the book is to present us with reliable information about Pythagoras’ life and, more importantly, about the influence of his teachings on posterity, both ancient and early modern. The book divides into four chapters. The first describes the ancient narratives about Pythagoras. As one would expect, these narratives show remarkable differences and are sometimes contradictory. We find reports on Pythagoras as the coach of Milon of Croton, the wrestler who ate a lot of meat, and on Pythagoras’ advocating vegetarianism. Vegetarianism also created the problem of whether to participate in the religious feasts of the polis involving sacrifices. On one account, in sacrificing to Zeus in a cave, Pythagoras wore black wool—which contravenes the Orphic-Pythagorean taboo on burying the dead in woolen clothing. Moreover, some of the reports may not have been derived from a direct knowledge of the sources concerning Pythagoras himself. A good example is Ovid, whose account in the Metamorphoses is heavily shaped by Empedocles [see Hardie 1995, which is not in Riedweg’s bibliography]. C 2005 Institute for Research in Classical Philosophy and Science 

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In the second chapter, Riedweg attempts to recover the historical figure of Pythagoras. He distances himself from two extreme positions. The one is held by Walter Burkert, who treats Pythagoras as a purely religious thinker and argues that the views concerning scientific issues were falsely attributed to him. The other is held by Leonid Zhmud, who is very much tempted to deprive Pythagoras of all the ritualistic elements. On Riedweg’s view, nothing rules out the possibility that Pythagoras possessed both an extraordinary charisma, manifested in religious practices, and a scientific authority due to his own activity in certain sciences. For Reidweg, even if the sources talk about Pythagoreans and not about Pythagoras, we can still trace those scientific doctrines back to Pythagoras or his innermost circle. Perhaps, this is the context for interpreting Pythagoras’ famous statement he is not a wise man (σοφ ) but a lover of wisdom (φιλ σοφο ). Riedweg offers a new investigation into the original meaning of the term. He comes to the conclusion [97] that it was not modesty that caused Pythagoras to describe himself as φιλ σοφο : the decisive consideration may have been the need to distinguish his superior insight from the many other skills—skill being the original meaning of ‘σοφ ’. The question he confronted is this: ‘In what art (τ χνη) are you skilled (σοφ )?’—to which he answered that he was not σοφ in any τ χνη, but rather a φιλ -σοφο . That may also have been accompanied by the effort to distinguish himself from earlier sages. In any case, the term ‘φιλ σοφο ’ does not thus denote something less important than does the term ‘σοφ ’, but rather may signify something of greater importance. Examination of the relevant testimony [Cicero, Tusc. disp. 5.8 = Wehrli 1953, fr. 88] also shows that the context of the statement is religious as well in that the text may refer us to the Pan-Hellenic festivals in Olympia [92]. Riedweg’s reconstruction of Pythagorean cosmology is based on Plato’s Philebus. Plato claims to derive the knowledge of the fundamental distinction between ‘unlimited’ and ‘limited’ from ancient sources [Phil. 16c, 23c]. Riedweg takes this as a reference to the Pythagoreans, as have some others, although this interpretation has been severely criticized by Dorothea Frede [1997, 130--131]. As Riedweg sees it, Philolaos had a view similar to Plato’s, though Philolaus was talking about ‘limiters’ (περα νοντα) and ‘unlimited things’ ( πειρα). For Philolaus, the cosmos depends on harmony that fits the two elements together and determines its structure; and the origin of the

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world is the central fire which may be equated with the ‘one’ ( ν) or ‘unity’ (μον ) [85]. The problem with this picture is twofold. First, there is a difference between Plato’s ‘limit’ (π ρα ) and Philolaus’ ‘limiters’ in so far as the latter are active, whereas Plato introduces the intellect to establish an active force in the universe. Furthermore, there is a difficulty in accounting for the pairs of opposites on this basis. In Meta. 986a21--26, Aristotle lists ten pairs of opposites, which he calls principles. He attributes the theory to certain Pythagoreans. He does not say that they originate from one another; they are principles and as such they do not depend on one another. If this is the case, then we have to think of a fundamentally dualistic structure in which the positive elements are in no sense responsible for the existence of the negative ones. The right does not generate left, neither does the limit create the unlimited. If Philolaus’ central fire has any role in cosmogony, it may be in the arrangements of the physical elements. But, on the theory based on the pairs of opposites, the One does not generate multiplicity. In short, the theory which Aristotle recounts does not show the sign of a Platonizing transformation of the Pythagorean doctrine that made the One the origin of all that is. In the third chapter, we find a thorough discussion of the Pythagorean societies. The author emphasizes the secrecy that characterized these societies. Such secrecy led to deviations from social norms. This, and a certain presumption of superiority, apparently aggravated the tensions with the non-Pythagoreans in the polis. The ancient sources diverge as to how many revolts against the Pythagoreans and their rule actually took place, and where. Aristoxenus writes about two, both in Croton. Other sources mention anti-Pythagorean attacks throughout Magna Grecia. As a result of theses revolts, the Pythagorean communities lost many of their distinguished members and had to withdraw from political activity. That may not mean that they stopped doing scientific research, and in fact it is in this period that the μαθηματικο separated from those who took political roles. Riedweg also offers a short and useful prosopography of the most important members. The fourth chapter is devoted to the influence of Pythagoras and his pupils. While discussing the Presocratics, particularly Xenophanes and Heraclitus, Reidweg includes Parmenides as well, which may be too generous. To the best of my knowledge, we do not have any clear evidence for the thesis that Parmenides was indebted to

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Pythagoras in the crucial points of his philosophy. There is nothing in Parmenides’ mythical account that could not be derived from Orphic sources equally well; and doctrinal similarities, in any case, do not always amount to direct influence. For instance, the distinction between truth and appearance is very common in that age. If it originated in Pythagorean doctrines, then I am sure one could say that Sophocles was also under the influence of Pythagoras. On the other hand, Pythagoras’ influence on Plato and his Academy is undeniable. The most outstanding result is the Timaeus. But the theory of principles in the ‘unwritten doctrines’ is remarkably different, even if Aristotle pointed out some similarities. Riedweg rightly stresses [119] that the differences between Plato and the Pythagoreans became blurred with Theophrastus, who in his Physical Opinions treated the two philosophies as almost overlapping. Of the numerous Neopythagorean writings, Riedweg discusses at length the Golden Verses and the role of Nigidius Figulus (ca 100--45 BC), the naturalist and grammarian, in reinvigorating Pythagorean teachings—Iamblichus is examined in this context but Porphyry is not, though he was mentioned earlier in a different context. Riedweg also discusses early modern authors such as Reuchlin, who was the Christian Kabbalist, Copernicus, and Kepler, who was particularly influenced by such ideas as the harmony of the world and the music of the spheres. The last paragraph of the chapter mentions Shakespeare’s The Merchant of Venice and modern authors such as Dannie Abse. Riedweg appends a chronological table of the main stages and persons of the Pythagorean movement, and of those who were influenced by it [135--140]. The only item I missed here is the De natura mundi et animae attributed to Timaeus of Locri (although it is most certainly a forgery). The date of its composition is not known but the work was included in the Platonic corpus, which ensured its survival. As is made clear by the German title, the book is an introduction; and as such it is aimed at a general audience. This may explain some of the features of the presentation that may seem distinctly idiosyncratic. For instance, Riedweg asks whether Pythagoras can be called a guru [60ff., 73], or whether his circle can be considered a sect in the modern sense of ‘an exclusive group that markedly stood out in various respects from the surrounding society’ [99]. The first of these questions seems quite out of place. The description of the

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charismatic leader with which Riedweg operates is that his followers attribute to him special, extraordinary abilities, whilst ‘outsiders usually reject him more or less brusquely’ [60]. No one could remain indifferent toward him. But do we know anything about the reaction of Parmenides? Is there any allusion in his work to the person of Pythagoras? If there is not, can it be the sign of Parmenides’ indifference? The problem is that there is no way to show that some of Pythagoras’ contemporaries neglected him, except by arguing from their silence. As for the question whether the Pythagoreans can be considered a sect, we are in an even more delicate situation. If we stick to the definition of sect quoted above, then we have to apply it to all sort of social groups (from the metics to Plato’s Academy and the Lyceum), since they in some way or another distinguished themselves, or were distinguished by others, from the surrounding society. To put it otherwise, the definition is all too general to reveal the specific nature of the Pythagorean societies. Still, the author does offer meticulous analyses of the ancient reports on the social structure of the early Pythagorean circles. His sociological analysis centers around six features [99--104]. ◦ ◦ ◦ ◦ ◦ ◦

The sect is a minority group having a somewhat strained relation to the majority. It recognizes a charismatic leader. It has a clearly recognizable form of organization. It is characterized by a high degree of spiritual integration. It regulates the life of the members in a way that deviates from the way of life that the majority follows. The members may also see themselves as an elite in society.

The only problem with this approach is that the twofold organization of the Pythagorean societies (μαθηματικο and κουσματικο ) makes the contrast with the rest of the society less sharp. It seems as if there was a more smooth transition between the Pythagorean circles and the rest of the society than the modern characteristics of a sect may suggest. There is an interesting suggestion on page 85 that in a certain sense some of the Pythagoreans may be considered forerunners of modern structuralism, ‘which is primarily concerned with hierarchical chains of binary oppositions in texts and other objects of analysis’. One might raise doubts about the extent to which these suggestions are justified. At first glance, at least, the role of

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binary divisions in Pythagorean philosophy seems to differ from the one we may find in modern approaches. I have found only a few slips, the most serious being the date of Alexander of Aphrodisias [81]. He flourished around 200 AD, not 200 BC (the German original says rightly ‘200 n. Chr.’). Likewise on page 123, it is said that the date of composition of the Pythagorean Golden Verses is somewhere between the second half of the fourth century BC (Thom) and fourth century BC (Nauck): Nauck, however, suggested the fourth century AD, as the German original says (‘n. Chr.’) on page 161. The author also seems to follow Sir David Ross’ translation of διακ σμησι in Aristotle’s Meta. 986a5--6 by ‘arrangement of the heavens’, which is not correct to my mind. The text is not about the generation of the uttermost sphere, rather it deals with the construction of the whole cosmos. The translation is reliable; it follows the German original closely. On occasions, one might say, it follows it too closely by mirroring the grammatical structure of the original, which is quite unnecessary and sometimes results in garbled sentences. Sometimes, it takes considerable effort to find the structure of the sentence. But in sum, these considerations apart, Riedweg has written a good introductory work on the subject. bibliography Frede, D. 1997. Platon, Werke, Übersetzung und Kommentar. Band III 2: Philebos. Göttingen. Hardie, P. 1995. ‘The Speech of Pythagoras in Ovid, Metamorphoses 15: Empedoclean epos’. Classical Quarterly 45:204--214. Wehrli, F. 1953. ed. Herakleides Pontikos. Die Schule des Aristoteles: Texte und Kommentar 7. Basel.

Archimedes and the Angel: Phantom Paths from Problems to Equations Fabio Acerbi CNRS, UMR 8163 ‘Savoirs, textes, langage’, Lille [email protected]

Introduction This paper is a critical review of Reviel Netz’ The Transformation of Mathematics in the Early Mediterranean World. 1 Its aim is to show that Netz’ methods of inquiry are too often unsatisfactory and to argue briefly for a substantially different interpretation of the evidence which he adduces. These two aims are pursued in parallel throughout the review and can hardly be disjoined. The review is uncommonly long and uncommonly direct, at times perhaps even a trifle vehement, in criticizing the methods and conclusions displayed in the book. There are two reasons for this. First, the transformation of Academic scholarship into a branch of the editorial business and, very recently, into an expanding division of the media-driven star-system, has dramatically reduced the time left to study primary sources, to get properly informed of works by other scholars, 2 and even just to read more than once what was written as a first draft. Second, at the same time, it is increasingly difficult to find serious reviews. Though the number of reviews and book-notices is exploding, it is clear that most reviews are written just to get a copy of an otherwise too expensive book. At any rate, the current policy is to keep sharp criticisms, if any, hidden under a seemingly gentle stylistic surface. The present paper is organized as follows. In the first section, I present Archimedes’ problem; in the next, I report the contents 1

R. Netz, The Transformation of Mathematics in the Early Mediterranean World: From Problems to Equations. Cambridge: Cambridge University Press, 2004. Pp. ix + 198 (20 figures). ISBN 0--521--82996--8. Cloth £50.00, $70.00. 2 See the lamentatio of the 85-year old P. O. Kristeller [1996, 567--583] for a lost way to scholarship in his 1990 Charles Homer Haskins lecture ‘A Life of Learning’. C 2005 Institute for Research in Classical Philosophy and Science 

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and aims of Netz’ book. The third section introduces some general remarks on the book. Then, a very detailed analysis of Netz’ arguments is expounded in the fourth and fifth sections; it is here that I adduce new evidence bearing on the main theme of the book. The closing parts of the fifth section contain, in very rough outline, an alternative assessment of all the evidence. (A warning to the reader: these last two sections often require having Netz’ book at hand.) In the final section, I discuss some methodological issues. Archimedes’ problem The book studies the transformation from Greek to Arabic mathematics of a very particular problem—we shall call it the original problem—that was left unsolved in Archimedes’ De sphaera et cylindro 2.4, which proposes ‘to cut a given sphere so that its segments have to each other a given ratio’. This problem is reduced so that one is required ‘to cut a given ΔZ at X and to bring about that as XZ is to a given , so the given is to the on ΔX’. 3 We shall call this the unconstrained problem. In this form, the problem is not always solvable; indeed it gives rise to what is called a διορισμ , namely, a specification of the conditions of existence of the solution. If the constraints implied by the διορισμ are embodied in the problem, then the problem is, of course, always solvable. The constrained problem is ‘given two straight lines BΔ, BZ where BΔ is double of BZ and a point Θ on BZ, to cut ΔB at X and to bring about that as the on BΔ is to the one on ΔX, XZ is to ZΘ’ [cf. 14--15]. 4 To cut the sphere, Archimedes takes the two problems as solved and promises that ‘both of them will be analyzed and synthesized at the end’ [Heiberg 1910--1915, 1.192.5--6]. No such appendix has been found in any surviving manuscript of De sph. et cyl. It was clearly unavailable to Dionysodorus and Diocles (both near contemporaries of Apollonius) who invented alternative approaches to the primary 3

Heiberg 1910--1915, 1.190.22--25. I have skipped the two expressions rightly bracketed by Heiberg. Translations are mine unless otherwise stated. 4 Heiberg 1910--1915, 1.192.1--5. B is therefore a well-defined point on the basic line ΔZ of the unconstrained problem. The differences between the three problems are somewhat hidden in Netz’ treatment, who conflates them to represent ‘the Archimedean problem’. Greater care was in order.

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construction, and it remained missing up to the time when Eutocius (sixth century ad) wrote his commentary on the Archimedean treatise. Indeed, it is with a real coup de théâtre 5 that Eutocius claims, while commenting on De sph. et cyl. 2.4, that he found Archimedes’ solution ‘in a certain old roll’. 6 The copy, Eutocius writes, was marred with errors both in the text and in the diagrams but was still recognizable because the text preserved in part the ‘beloved Doric dialect’. 7 Eutocius goes on to propose Archimedes’ solution as recovered from the ‘old roll’. The commentator reports in succession Archimedes’ analysis of the unconstrained problem, the synthesis of the same, and the discussion of the related διορισμ which is postponed until after the synthesis. 8 Further expansions of the proof by Eutocius follow. 9 (Apparently, no solution of the constrained problem was contained in the putatively recovered appendix.) After that, Eutocius reports the alternative solution by Dionysodorus, 10 who was not able to attack

5 6

7 8

9

10

Recall that the commentary on De sph. et cyl. is Eutocius’ first work as a commentator, and that he presents it to his teacher Ammonius for judgment. Usually translated ‘book’. If the book was really as old as Eutocius claims, we can safely assume that it was a roll. That Eutocius writes on codices and consults them also when drawing from others’ commentaries is clear from his references to writing his notes in the margins: see his commentaries to Apollonius’ Conica [e.g., at Heiberg 1891--1893, 2.176.19--22 and 2.354.7--8]. Heiberg 1910--1915, 3.130.29--132.6. The extant text of De sph. et cyl. bears no traces of Doric dialect. Heiberg 1910--1915, 3.132.19--136.13, 3.136.14--140.20, 3.140.21--146.28, respectively. Heiberg regarded the whole of 3.148.1--27 as Eutocius’ addition to Archimedes’ incomplete discussion of the διορισμ . As we shall see, Netz places the beginning of the addition earlier at 3.144.31. Eutocius turns his attention to the relationships between the two forms of the reduced problem. He first shows that the constrained problem comes in fact from embodying the condition of the διορισμ in the enunciation of the unconstrained one [3.150.1--22]. Second, he puts emphasis on the fact that two points actually solve the problem, whenever this is solvable; but that only one of them effects the original construction and is in fact always contained in line ΔB [3.150.23--152.14]. Heiberg 1910--1915, 3.152.27--160.2. A part of the proof is postponed as a lemma [3.158.13--160.2]. Dionysodorus’ solution is not framed in the format of analysis and synthesis.

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the problems left unsolved by Archimedes 11 and found an alternative way of solving the main problem directly. Eutocius then expounds the solution proposed by Diocles, who reduced the construction in a simpler way than Archimedes did and solved the problem resulting from this reduction (which is slightly different from Archimedes’) through analysis and synthesis. 12 Of the latter problem, Eutocius presents both the analysis and the synthesis, while the analogous text preserved in the Arabic tradition of On Burning Mirrors leaves out the synthesis on the grounds that it is ‘clear’. 13 This rather complex state of affairs constitutes the Greek evidence that Netz deals with in his book. Contents and aims of Netz’ book After introducing the original problem, the first chapter (‘The problem in the world of Archimedes’) presents in succession Archimedes’ synthesis and a part of the corresponding analysis [section 1.2] (we will say more below on this choice), the core of Dionysodorus’ solution [1.4]—apart from a lemma which Netz judges irrelevant, though it is actually not so—and Diocles’ reduction and his analysis of the reduced problem [1.5]. These sections consist mainly of translations and very detailed paraphrases of the several Greek texts. Two further sections [1.3, 1.6] underline the geometrical nature of Archimedes’ problem as opposed to treating it as a search for solutions of a cubic equation. 14 In these sections, we also find a characterization of the ideology of Greek geometers, who worked in a social context where

11

Eutocius points out this fact twice: see Heiberg 1910--1915, 3.130.19--22 and 3.152.16--21. 12 Heiberg 1910--1915, 3.160.3--162.16 (presentation of the construction and reduction), 3.162.17--168.25 (analysis), 3.168.26--174.4 (synthesis), 3.174.5-176.5 (construction of cutting the sphere). 13 Editions in Toomer 1976, 77--87 and Rashed 2000, 119--125. 14 Netz [25--26] takes as representative of the latter interpretation a nonsensical patchwork of sentences that results from his cut-and-paste adaptation of Heath’s three-page account of Archimedes’ solution [Heath 1921, 43--45]. Using such a method, one might as well take the seven-line sentence on pages 91--92 of Netz’ book (four of which are equations) as representative of his reading of the limits of solvability.

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agonism and self-promotion were the norm, 15 that relies on the concept of ‘aura’ in W. Benjamin’s sense: the ancient author aimed at providing his work with an aura— with a sense of uniqueness that defies subsumption under any general heading. [59] The second chapter (‘From Archimedes to Eutocius’) deals with the transformation and appropriation of Archimedes’ solution by Eutocius. In sections 2.1 and 2.2, the Archimedean discussion of the limits of solvability is first presented, followed by a detailed analysis of what is originally Archimedean and what is Eutocius’ addition. (Recall that the entire text is contained solely in Eutocius’ commentary). Netz argues that the addition starts one full page before the line which Heiberg proposed in his edition. The strictly geometrical character of Archimedes’ proof is then pointed out in section 2.3. Eutocius’ contribution, which amounts to adding a missing case to the Archimedean proof and to commenting thereon, is analyzed in section 2.4 and declared to be original in two respects: First, [Eutocius] describes the systematic relation holding in the line: the symmetry around the point E. Second, he has an explicit concept of a functional relation between mathematical objects. [94] a claim that can hold true only if Netz is right to displace the beginning of Eutocius’ addition as he has. 16 Section 2.5 is probably the core of the book. Here we find a very detailed discussion of a particular expression denoting a solid as ‘a surface epi line’. The expression comes from the analysis of the unconstrained problem in the following way. The requirement of cutting ΔZ at X entails considering, for a given point Σ on ΔZ, 15

The obvious reference is to Lloyd 1996, cited once in Netz’ book but not for this reason! Netz carefully takes his examples from Hellenistic mathematics. The explanation, however, does not work for every period of Greek mathematics, as is shown, e.g., by the Eutocian list of solutions of the problem of finding two mean proportionals between given lines, several of which are almost identical. Such are the proofs of Hero, Philo, and Apollonius, as well as those of Diocles and Pappus. The whole report is in Heiberg 1910--1915, 2.54.27--106.24. The hypothesis that later mathematicians were unaware of their predecessors’ solutions is patently untenable. 16 These two points of originality are related in the following way:

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the solids whose base is the square on ΔΣ and whose height is ΣZ. These are shown to be equal to a solid built from a base surface and a height that are among the data of the problem; the διορισμ comes exactly from the fact that there is a maximum to such ΔΣZ-solids at a point called B such that BZ is 1/3 of ΔZ. The constrained form of the problem takes this into account: the line ΔZ is already cut at the point B, and X is sought on ΔB (thus, producing a solid less than the solid ΔBZ) such that a certain proportion holds. Every such solid is denoted thereafter [cf. Heiberg 1910--1915, 3.136.7ff.] by the phrase ‘surface epi line’. The epi expression lies, therefore, at the core of the whole proof. It is also to be found in De sph. et cyl. 2.8 aliter and in a lemma by Eutocius to it, as well as in several authors between Archimedes and Eutocius, even if it is not attested in ‘mainstream’ authors. 17 The phrase ‘surface epi line’ may have both a geometric and an arithmetic connotation: on the one hand, a perpendicular is said to be epi a line or a plane; on the other, ‘epi’ denotes multiplication between numbers. 18 The interest of such a register-crossing expression is best introduced in Netz’ words: Archimedes employs it since ‘he wishes to mark a piece of text, to endow it with its own distinctive aura. He therefore makes it different—and this difference leads on to the possibility of mathematical change’ [114]. 19 In other terms, and considering also Eutocius’ later appropriation: But Eutocius also says how one point in the argument relates to another point in the argument [this is Netz’ remark on the symmetry of the line], and therefore the relation between the points in the line becomes for him more like the relation between the points in the argument [this is Netz’ concept of the functional relation between points]. [95] 17 The reader would have appreciated a list of the occurrences in the Archimedean corpus and in Eutocius’ commentaries. As we shall see, Netz’ census of the occurrences of the epi phrase in other Greek mathematicians is also at best partial. 18 In all instances, the form is π with the accusative. 19 On page 111, it is asserted that all the occurrences of the special epi we study here appear in a continuous stretch of text. They appear either in (what is now) the penultimate proposition of Sphere and Cylinder II, or in the appendix to that book.

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But there is also a major way in which Archimedes’ text, very surprisingly, makes a deliberate choice to deal with objects as if they were quantitative in nature. This choice, more than any other feature of Archimedes’ text, points forwards towards a more algebraic understanding of the problem. Its later appropriation by Eutocius, in particular, would make Eutocius’ text appear truly algebraic. [98] All the above features of Eutocius’ approach to mathematics find an explanation [section 2.6] within the category of deuteronomic text, introduced by Netz in a paper of 1998 20 —note that the introduction of ‘aura’ is an addition to the argument of this paper. In a nutshell: Archimedes and Diocles aimed at the individual aura; Eutocius aims at contextualization, which is the removal of aura. Hence, Eutocius’ mathematics has concepts that are different from those of Archimedes and Diocles, and are different in a The claim is obviously false, owing to the presence of the final proposition, unless a new notion of ‘continuous’ is admitted. Netz’ tendentious specification ‘(what is now)’ seems to point to arguments suggesting that the final proposition is spurious (no such arguments exist) and to the unwarranted assumption that there was in fact an Archimedean appendix: all of this forges the fiction of a ‘continuous stretch of text’. There is more: The second book of Sphere and Cylinder is a very complex combination of proportion theory [sic] and solid geometry. Towards its end, it gets more and more complicated. The alternative proof for the penultimate proposition of the book has a unique structure, effectively a theorem for which only the analysis is given. Then the appendix may be the most complex piece of mathematics of the entire corpus. [111] The last inference is ineffable, and no reader acquainted with the second book of De corporibus fluitantibus, for example, would agree with the resulting, totally subjective claim. Netz’ goal is to make plausible the immediately following assertion: ‘Our epi appears as a unique expression, perhaps intentionally employed to mark a unique context.’ The reader should have already understood at this point where the subsequent discussion will lead. 20 In the present book, the notion is presented this way: We see then that a large part of intellectual activity in Late Antiquity was involved not with writing about things, but in writing about books. This is writing which is essentially dependent upon some previous writing—what I call a deuteronomic text. [121]

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The third chapter (‘From Archimedes to Khayyam’) begins with a quick survey of ‘the history of Archimedes’ problem in the Arab world up to Khayyam’ [129]. After an overview of the Arabic textual tradition of Archimedes’ works and Eutocius’ commentary, the contributions of al-M¯ ah¯ an¯ı, Abu al J¯ ud, al-Kh¯ azin, and Ibn al-Haytham are briefly presented, whereas the solution proposed by al-Q¯ uh¯ı is analyzed in greater detail. Section 3.2 offers some generalities about al-Khw¯ arizm¯ı’s algebra. Khayy¯ am’s treatise Algebra is presented in section 3.3. The solution of Archimedes’ construction appears again there, in a very disguised form, as one of the cases in a classification of cubic equations, ‘A cube and a number equal a square’, even if the solution is worked out in a geometrical setting by intersection of conic sections. Three principles underlie Khayy¯ am’s treatise; they are, as usual, best summarized in Netz’ words: The first was an inter-penetration of the introduction, and the treatise proper: the treatise was a direct continuation of the introduction, since the treatise was simultaneously, in algebra, and about algebra. The second was the strongly articulate, systematic nature of the treatise: it constantly arranges itself in various divisions and lists. Finally, we saw how the two features are connected through the principle of exhaustive lists. The interest of the treatise is in arranging claims—and objects—into systematic orders. . . . [154--155] 22 Section 3.4 offers the translation of Khayy¯ am’s solution of the construction of cutting the sphere, and this is subsequently compared 21

Of course, from the perspective of a ‘practice that naturally gives rise to [new] mathematical objects’ [emphasis mine], it is difficult to explain why this does not happen to every commentator working on every text. 22 All italics in the quotations here and below are in the original, unless otherwise stated.

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with Archimedes’ in section 3.5. Here Netz discusses, mainly with linguistic arguments, Khayy¯ am’s proof, which is shown to have a certain duality (in a sense, continuous with Archimedes himself)—conjuring non-geometrical possibilities, while manifesting a sustained geometrical conception of the problem. [161] The main difference between the approaches taken by Archimedes and Khayy¯ am lies in the obvious prominence that Khayy¯ am gives to the study of cases and in his foregrounding of equalities over proportions, and in the fact that Archimedes does the opposite. Section 3.6 is devoted to discussing Khayy¯ am’s criticisms of Abu al J¯ ud’s solution and to comparing Khayy¯ am’s polemical style with that of Dionysodorus and of Diocles. The section ends with a discussion of Sharraf al-D¯ın al-T¯ usı’s ‘Copernican revolution’ of the ordering principle of Khayy¯ am’s classification: while the latter ‘divided various equations into kinds, primarily, according to the geometrical tools they required’ [179], the former looked at a systematization of the limits of solvability. In this way, Archimedes’ original construction has finally become just an entry in a classification of equations. In section 3.7, such a difference, the main germs of which are in Khayy¯ am’s work, is shown to result ‘from Khayyam’s cultural practices which, like those of Eutocius, were deuteronomic—he was the author of texts essentially dependent upon previous texts’ [129]. The aim of the book is to show that ‘mathematics has a history’ [1] by following step-by-step the transformation summarized above of Archimedes’ problem into an equation. Two major historiographical turning points are under Netz’ polemical focus. Sabetai Unguru’s sharp criticisms [1975--1976, 1979] of the now abandoned treatment of some branches of Greek mathematics as ‘geometrical algebra’ are criticized in turn for having a-historically led him away from studying the dynamics of the transformation from the ancient to the modern. Unguru’s premise was that of a great divide separating ancient from modern thinking. The assumption of a great divide, in itself, is not conducive to the study of the dynamics leading from one side of the divide to the other. [4] This is unfair, since it is undeniable that much of Unguru’s later work fits very well his original program, namely, to understand ancient Greek mathematics in its own terms. The fons et origo of Unguru’s

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premise is in Klein’s Greek Mathematical Thought and the Origin of Algebra [1968]: 23 Ancient mathematics (and science in general) was, according to Klein, based on a first-order ontology, modern mathematics (and science in general) is based on a second-order ontology. [5] and they are separated by a great conceptual divide. Netz’ goal is to refute the second horn of Klein’s thesis while corroborating the first, 24 by showing how second-order thought came out of the practice of commenting and systematizing earlier texts, specifically, the treatises of the great mathematicians of the first Hellenistic period whose works were raised to the status of canonical texts in late antiquity. 25 Scholarship Fascinating and deep theses, brilliant argumentation, an unmistakably flamboyant style, lucid and rhetorically very effective expositions of the difficult proofs presented, wide-ranging interpretative perspectives, and refined tools of analysis are the highlights of this very ambitious study. Yet the book is utterly disappointing. The point is not even whether Netz’ approach should be labeled as history of mathematics, or whether, more likely, he is inventing a new genre, and whether this border-crossing will disturb hard-nosed and 23

This fundamental study is mis-cited by Netz in two ways: the ‘Origin’ in the title is everywhere written ‘Origins’; and, though the reference in the bibliography is ‘Klein, J. 1934--6/1968’, the exact reference to the original study in German is nowhere to be found. 24 References to Klein’s theses are scattered throughout the book. 25 In the author’s own words: To anticipate, my claim, in a nutshell, is that Late Antiquity and the Middle Ages were characterized by a culture of books-referringto-other-books (what I call a deuteronomic culture). This emphasized ordering and arranging previously given science: that is, it emphasized the systematic features of science. Early Greek mathematics, on the other hand, was more interested in the unique properties of isolated problems. The emphasis on the systematic led to an emphasis on the relations between concepts, giving rise to the features we associate with ‘algebra’. [8]

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narrow-minded historians of mathematics as the present reviewer is. Netz’ book simply raises serious problems of methods: it seems as though the traditional tools and careful approach of classical scholarship had to be wiped out in order to make the disruptive charge of the book manifest in full. Yet the shortcomings of the book can be described in wholly traditional terms: unfalsifiable conjectures about the aims and intentions of ancient mathematicians, unsatisfactory command and highly tendentious use of primary sources and of secondary literature, unwarranted resort to over-generalization and to modern mathematical concepts, taking conjectures as established facts about ancient mathematics—and treating the received texts and diagrams as if they were what their alleged author wrote in the first place (excepting, of course, the cases where the opposite stance supports the author’s thesis). The main thesis itself is a true masterpiece of interpretative insight; yet, I believe, it is simply not supported by the textual evidence adduced. The only conclusion one draws after reading Netz’ book is that the Greek tradition is a dead end, and that Arabic mathematicians reconsidered the whole issue on entirely new grounds. Despite the author’s efforts, there is no continuous trajectory from problems to equations, and Klein’s thesis is in fact confirmed in its strongest form. The reader will look in vain for a discussion of possible influences on Arabic authors of Eutocius’ alleged innovations. The only point of contact between the approaches of late Greek and Arabic mathematicians, their deuteronomic character, is by and large a historiographical artifact that is alleged to be supported by the very same evidence it was devised to explain. Moreover, Netz’ willingness to prolong the bridge as far as Eutocius forces the author, as I shall show, to propose strained and at bottom misleading readings of the evidence itself. The attentive reader might have guessed the overall strategy already on page 16, where one reads, with undifferentiated reference to Archimedes’ nest of problems: ‘The trajectory, from problems to equations, is to a certain extent implied within the problem itself.’ All of this is made to pass almost unperceived by virtue of the author’s fascinating style, as he drives the reader away from where the real problems lie. A minimal but significant example of Netz’ masterly command of words is his use of the verb ‘re-deploy’ [ix] to inform the reader that the book contains material coming from two studies previously

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published as journal articles. 26 Such a practice is common among scholars, but should be kept within reasonable limits. In Netz’ book, this ‘re-deploying’ amounts to a massive verbatim reproduction of long stretches of text coming from those papers. Moreover, all translations from Greek in the book are also contained in the first volume of the translation of the works of Archimedes by the same author [Netz 2004]: Netz transfers them lock, stock, and barrel from the latter book to the one under review. The result is a detailed but fairly useless apparatus of references to propositions in the Elements, 27 a series of remarks of no subsequent use, and, most notably, a translation encumbered with a pedantic numbering of the steps of the proof. 28 The latter device finds so infrequent use in the book that a more economical way to mark the handful of steps to be referred to would have been worth an afternoon’s work. If one adds that the paraphrastic expositions of the several proofs translated take at least as many pages as the translations themselves, 29 the outcome is 26

Netz 1999b, 2002a. Moreover, section 2.6 is an abrégé of Netz 1998. Embarrassing consequences include, for example, presenting almost identical notes (to almost identical formulaic phrases): see 22n40, 31n47, 67n4. 28 Other drawbacks of the proposed translation include the orgiastic use of angular brackets: their presence can hardly be avoided in English in the case of understood words, a feature typical of mathematical prose style in ancient Greek, but it might be reduced to a reasonable minimum. For example, take the introduction of ‘’ in expressions such as ‘cone having. . . ZA height’ (κ νο χων. . . ψο δ τ ν ZA: see, e.g., passim on page 32 in the translation of Dionysodorus’ text [Heiberg 1910--1915, 3.154--156]. This can be correct, since in the Greek text ψο has no definite article and, hence, may be read as an appositive, the object being τ ν ZA. But in English an idiomatic way to mark an appositive noun is to insert ‘as’ in front of it without the brackets. (But why not translate this more simply by ‘cone having height ZA’?) At any rate, here and elsewhere, Netz is transgressing his own principles of translation, since he should have translated τ ν ZA regularly by ‘the ZA’. Other problems with the translation will be pointed out in what follows, but this is rather a job for a reviewer of Netz 2004 [see, e.g., Sidoli 2005]. 29 In fact, Netz’ explanations of the several proofs translated attempt to clarify them by proposing a possible meta-analysis designed to reconstruct the original path of thought, very much in the style of Knorr 1986. Netz’ analyses are more discursive, heuristic, and reader-friendly than Knorr’s; but in both cases, the analyses are nothing but conjectures, and they tell more about the ingenuity of their proposers than about Greek mathematics. 27

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a book that adds nothing new to what the author had already said in previous publications. One might well wonder whether any real need was felt for such a book, beyond that of adding an item to the list of the author’s academic publications. I will illustrate the above points in what follows, as I try to uncover Netz’ highly strategical presentation of the evidence and to indicate where a modicum of time devoted to the study of primary sources and secondary literature might have enriched the argumentative fabric of the book and helped to avoid some of its shortcomings. Most importantly, I will try to show that what appears to be the crucial step in the whole book is grounded on a very tendentious reading of the sources. I am not competent to judge the third chapter. As the journal Farhang is not easily accessible, I only list the pages in the book where Netz reproduces word for word his previous analysis of Khayy¯ am’s approach: pp. 145--160 [cf. Netz 2002a, 230--245], pp. 161--171 [Netz 2002a, 245--254], pp. 182--185 [Netz 2002a, 255--258]. The reader is invited to do a similar collation between chapter 2 of the present book and the corresponding paper in Archive for History of Exact Sciences [Netz 1999b] as an instructive exercise. 30 Minor characters Dionysodorus Netz’ presentation of Dionysodorus’ proof is misleading in that a completely artificial splitting of the proof is introduced at a step [Heiberg 1910--1915, 3.156.8--9] where a proportion identical with the one in Archimedes’ unconstrained problem is reached. Such a step is presented as ‘the goal of Dionysodorus’ argument’ [37], after which he, ‘in effect, recapitulates Archimedes’ argument as available to him, presenting it as his own’ [38]. Neither claim is supported by the text: the step is not marked in any special way and what follows it in Dionysodorus’ argument is fairly different from Archimedes’, most notably through the introduction of an auxiliary cone.

30

Especially striking is a passage on page 77 produced by a masterful pasting of what in Netz 1999b, 28 were originally two separate sections: the title itself of the section is embodied in the text through the skillful introduction of a single clause!

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The equivalence of the auxiliary cone with the spherical segment is proved in a lemma which Netz does not report, and whose importance he clearly did not appreciate. 31 Indeed, he underrates the importance of the condition AZ = radius of the sphere [Heiberg 1910-1915, 3.154.5--6], 32 which is in fact a key assumption in the lemma: 33 he states that ‘no use is made [of the equality], inside the solution itself’ [36]. But the lemma is an integral part of the solution; and it is stated as a lemma simply to make the proof not too cumbersome, a typical move among Greek geometers. Another misleading feature of Netz’ exposition, a feature that the reader encounters throughout the book, is the assumption that using a proportion implies an understanding that is quantitative, hence abstract, and hence algebraic. In the case of Dionysodorus, Netz intuits (this is the right word given the absence of any argument) ‘from the way in which Dionysodorus makes his conic sections appear inside the proposition, that he conceives of them, in fact, in a more purely quantitative way than Archimedes did in his solution’ [36]. Once this unwarranted and tendentious assumption is accepted—and I cannot see how the very short but still verbose argument on pages 36--37 can be thought to ‘account for’ the ‘apparent paradox that Dionysodorus’ basic setting is more geometrical, while his approach in the solution itself is more abstract’—such conclusions naturally follow as ‘We begin to perceive a dialectical relation between the “geometrical” and the “abstract” (which we may even refer to as the “algebraic”)’ [37]. 34 Yet, Dionysodorus’ proof is nothing but the usual mixing of geometrical constructions and manipulations of equalities and proportions that is the distinctive feature of every non-elementary proof in Greek geometry. This whole nest of subtle misconceptions is devised to

31

In 32n61, the lemma is said ‘not [to] touch on our main theme’; but this is true only in Netz’ very partial reading of Dionysodorus’ approach. 32 Another rash assertion is that ‘[Dionysodorus’] diagram was made to include an inert circle, AΠB, which does not participate in the solution’ [36], while in fact the circle AΠB represents the sphere to be cut and does indeed take part in the solution! 33 Most notably, such an assumption is what makes the application of De sph. et cyl. 2.2 possible at Heiberg 1910--1915, 3.158.17--19. 34 Cf. ‘[Dionysodorus’ proof], where the conic sections and the very approach to the problem were rather quasi-algebraic’ [53].

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make Dionysodorus’ proof fit the requirements of placing it in a preconceived interpretative scheme, one that looks at the ‘algebraic’ potentialities of Archimedes’ problem. But the only honest answer to the question ‘Where is [Dionysodorus’s proof] in the trajectory leading from problems to equations?’ [35] is ‘Nowhere’—and in a strong sense, since the question itself is historiographically meaningful only in a teleological perspective: otherwise, the trajectory simply does not exist. Diocles ‘We do not know the chronological relation between Dionysodorus and Diocles’ [39]. Of course, this is true in exact temporal terms; but Netz seems unaware of an interesting series of connections suggesting that Dionysodorus and Diocles were also chronological contemporaries and not only ‘mathematical contemporaries’ [39]. 35 In the prefatory letter to Conica 2, which is addressed to Eudemus, Apollonius [Heiberg 1891--1893, 1.192.8--11] says: ‘And Philonides the geometer, whom I introduced to you in Ephesus, if ever he is in Pergamum, acquaint him with it [scil. the second book] too.’ Now, a papyrus from Herculaneum happens to contain a β ο Φιλον δου, 36 an Epicurean philosopher with strong mathematical bent. 37 In this papyrus, we learn that Philonides had Eudemus and Dionysodorus of Caunos as teachers, that he collected and edited their lectures, and that he was acquainted with some Zenodorus. 38 An obvious conjecture, still unchallenged since Crönert’s first proposal, is that the latter two are the well-known mathematicians. On the other hand, in the prefatory letter to the extant version of On Burning Mirrors, Diocles refers to an astronomer who proposed that he find a surface concentrating the solar rays at one point. The Arabic name of the astronomer has been emended to a name compatible with Zenodorus in the original Greek by G. J. Toomer, though W. R. Knorr and R. Rashed read an otherwise unknown Hippodamus. As can be seen, the whole series of connections is by and large conjectural; but it fits well the extant evidence and in any event makes Netz’ assertion rash.

35

See also Toomer 1976, 2. See Crönert 1900 for the editio princeps. Gallo’s new edition [1980] incorporates the detailed remarks offered in Capasso et alii 1976. 37 This and other testimonies contradict the widespread belief that the Epicureans had no interest in mathematics. 38 Fragments 7, 25, 32 and 31, 34, respectively. 36

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Diocles’ Arabic text and its relationship with Eutocius’ version are not discussed at all; and Netz’ rejection of the synthesis proposed by Eutocius as an addition is given but a terse remark: ‘this was in all likelihood Eutocius’ own contribution (since, in the Arabic version, Diocles explicitly ignores the synthesis as trivial)’ [44]. The only discussion of the issue is in a short paragraph [95], where we read that ‘it is clear that the Arabic text may be closer, in some ways, to Diocles’ text, than Eutocius’ version is’ and that ‘it appears that Eutocius had interfered in Diocles’ text in a way directly comparable with his interference in Archimedes’ text.’ 39 The former sentence may well be true—and in such generic terms and with such modal qualifications it cannot be false—but it still calls for more argument. The Arabic Diocles is in fact already a translation of a Greek compilation: it is definitely not, as Netz seems to believe [39], the ‘Arabic translation of Diocles’ original treatise’. Consequently, the Arabic text of Diocles’ solution should not be regarded as superior to the one we find in Eutocius on the sole basis of its being included in a work with the right title and author. But even taking this for granted, let us look for a moment at Eutocius’ interferences in Diocles’ analysis. They are massive and radical: Diocles’ introductory considerations and the reduction are almost identical, but the construction and the proof of the reduced problem are entirely rewritten and not just supplemented with additions: it is enough to compare steps 154--161 in Toomer’s edition of the Arabic text with Heiberg 1910--1915, 2.164.4--19, where proportions are consistently replaced by equalities. Most notably, the whole proof in the Arabic text is formatted in the language of the ‘givens’, whereas Eutocius retains it at convenient places only. Therefore, the two versions should not be said to correspond ‘very closely, though not exactly’ [39n64], and the changes should not be described simply as ‘Eutocius provid[ing], in his text, several very elementary arguments that are omitted in the Arabic version, besides including the synthesis of the problem’ [95], as if the changes were only additions. Looking at the game the other way around, one is led to infer from this that Eutocius heavily reworked the text of the Archimedean appendix too. Since what will concern Netz are linguistic changes

39

There is a passing reference to a further discussion in 39n64.

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and not additions of whole portions of text, this would have serious consequences for the main thesis of the book. As for Netz’ analysis of the Greek proof, it is artificial to give prominence to the seemingly more complicated conditions in Diocles’ problem (two proportions involving lines only instead of one proportion involving lines and areas): Netz speaks of ‘the extreme artificiality of the conditions in Diocles’ problem’ [53: cf. 46], whose ‘operating urge seems to be to distinguish the problem, as sharply as possible, from Archimedes’ [46]. The artificiality is in the eyes of the interpreter. Very simply, Diocles had found a different way to solve the original problem—his way appears to be sufficiently different from Archimedes’ and there was no reason not to mark it as such. In the same vein, it is tendentious to treat the addition to step 17 without mentioning the fact that the whole text has been heavily reworked so as to give this addition more prominence. Step 17 in Diocles’ proof reads as follows in Netz’ translation [42]: ‘So through this, whenever P falls between A, Z, then Σ falls outside H, and vice versa’ [Heiberg 1910--1915, 3.164.17--19]. The Arabic text is similar (the only similar step in a long stretch of text otherwise completely reworked), but without the ‘vice versa’. 40 A minimal interpretation is that, when he came to step 17, Eutocius simply did his job, and did it well: when he perceived a missing step, a missing case, or an incomplete discussion, he supplied it. Eutocius did the same in his additions to Archimedes’ διορισμ . Instead, Netz sets the two interventions in parallel to support his thesis that Eutocius, qua deuteronomic author, introduced germs of change into Archimedes’ text: the addition of the case changes the meaning of the argument: instead of a special observation on a special configuration, the text, transformed by Eutocius, sets out the constant relationships between possible configurations. [96] All of this is at best unwarranted in Archimedes’ case as well as in Diocles’. Hero The presentation of the achievements of Hero is completely misleading, and relies on an imperfect knowledge of the Heronian 40

Netz’ interpretation of the ‘vice versa’ as ‘also when Σ falls between B, H, then P falls outside Z’ [96] is wholly arbitrary, since this is not the only possibility. An exact converse would be ‘when Σ falls outside H, then P falls between A, Z.’

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corpus. Netz takes Hero as the main source of ‘a geometry organized around calculation and not. . . around proofs’ [112]. Opposed to Hero is Archimedes, whose surviving texts are ‘very clearly geared towards proof, and not towards calculation.. . . Rigor is the whole point of the discussion’ [113]. This is but a partial image of the mathematical works of Hero. 41 A cursory reading of the Metrica is enough to make one realize this fact: in addition to the calculations exemplifying the proposed formulas, rigorous proofs of the same formulas are provided in perfect ‘Euclidean style’. In the same way, a number of full-fledged alternative proofs to Euclid’s Elements are known to come from Hero’s commentary, maybe the first one in the genre [see Vitrac 2004, esp. 30--34], and some of them, as the extant 3.12, found their way into the main Greek text. All of this makes meaningless Netz’ claim that ‘The Heronian register is in general defined relative to the Euclidean style. It is a variation on the Euclidean style’ [113]. But there was no well-defined ‘Heronian register’, just as there was no ‘Euclidean style’ before Euclid became a canonical author in late antiquity. Referring to important studies by Jens Høyrup, 42 Hero is made a champion of an approach rooted in ‘oral calculation puzzles’: Hero, in particular, seems to employ even a language reminiscent of such calculation puzzles: we recall his treatment of geometrical relations in terms of multiplication, and Archimedes’ deliberate exploitation of this tradition. [141] One should bear in mind that Høyrup’s researches are mainly, if not exclusively, focused on the Geometrica and that this work, 43 although included in the Heronian corpus, is definitely apocryphal. Moreover, one cannot rightly speak of ‘deliberate exploitation’ of the tradition 41

The whole discussion on pages 113--114 is vitiated by the embarrassing fact that Hero comes well after Archimedes, whereas Netz’ reconstruction would have greatly benefited from the opposite. Netz remarkably allows: ‘Not that Hero is a perfect antecedent [scil. of Archimedes]’ [113]. 42 And not ‘Hoyrup’ as we find throughout the book. Notice also that the paper cited by Netz does not contain any analysis of the Greek tradition apart from a short discussion of Elem. 2. A better reference would have been Høyrup 1997. 43 Note that the use of the singular is actually unwarranted, since the work is ‘a modern conglomerate of two (indeed more) ancient conglomerates’ [Høyrup 1997, 73].

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by Archimedes: there is not the slightest information about when and where the tradition passed from the Ancient Near East to Greece. In the same vein, to assert after a half-page resumé of Høyrup’s findings that ‘In the Ancient Greek world, the two forms—literate geometry, and oral calculation puzzles—subsisted separately, 44 with occasional contacts [. . . ]’ [141] is to state a dichotomy that is false in two fundamental respects. First, the two forms did not exhaust the ancient mathematical field, as Netz’ formulation suggests: neither of the authors mentioned in the same page, Hero and Diophantus, exclusively practiced either of them, and neither the metrical tradition nor ancient number theory (not even indeterminate analysis à la Diophantus) should be defined simply as ‘oral calculation puzzles’. Second, no mathematical tradition subsisted separately from the others, as is shown by the very same authors, and the contacts were more than simply occasional. In the following section, I will discuss Netz’ use of the evidence for the epi phrase as it appears in Hero’s Metrica. The core of the argument The main task of the present paper is to discuss the crucial issues of the first two chapters, namely Netz’ discussion of the διορισμ and the relevance of the epi phrase. I will argue that the textual evidence indicates that the epi expression was not in Archimedes’ original texts; that Netz’ presentation of this evidence is tendentiously incomplete; and that Eutocius introduces no change, since in fact the use of the expression in geometrical and mixed settings was a matter of routine well before him. The extent of Eutocius’ reworking of the appendix Eutocius expressly says that he rewrote the entire Archimedean tract. Netz believes the opposite: ‘Eutocius promises to transcribe this text “as it has been written” ’ [72]. Netz is drawing primarily on Heiberg 1915-1910, 3.132.12, but there Eutocius only claims to have ‘studied the text as it has been written’. In fact, Eutocius immediately explains that what he will do with the text is to ‘write the ideas as far as possible in a language common and clear’ [3.132.14--15]. The sentence is paraphrased by Netz [72], who goes beyond the text in making 44

The problem here lies in asserting the oral character of a tradition about which we are (of course) acquainted through written sources only.

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the references explicit and specific, as ‘[to] re-write the proof with modern terminology and in the dominant dialect’, thereby excluding any serious reworking of the text. Netz’ reference is to the fact, mentioned by Eutocius a few lines before, that the recovered text employed the Doric dialect and an obsolete terminology: exactly these features made Eutocius suspect that he had hit upon Archimedes’ appendix [3.132.5--11]. Yet, the clause at 3.132.14--15 is preceded by a description of the defects of the recovered text which strongly suggests that Eutocius’ rewriting went well beyond a few cosmetic changes: ‘since we found difficult, because, as has been said, of the number of mistakes, after having stripped off the ideas one by one we write as far as possible in a language common and clear’ [3.132.12--15]. 45 Maybe Netz was misled by his own earlier mistranslation of the clause at 3.132.14--15: 46 ‘We write [the content] down as far as possible, word-for-word (but in a language that is more widely used, and clearer)’ [Netz 2004, 318]. The Greek text actually reads κοινοτ ρ κα σαφεστ ρ κατ τ δυνατ ν λ ξει γρ φομεν: hence, in Netz’ translation the preposition κατ is erroneously distributed between δυνατ ν and λ ξει. His mistake may come from his reading κατ λ ξιν which does mean ‘word-for-word’. 47 On this crucial error rests Netz’ belief that we may confidently read the text of the recovered appendix as faithfully Archimedean. Eutocius gives us no examples of theorems by pre-Apollonian mathematicians for which he did not heavily rework the sources. As we have seen, the text of Diocles’ solution is a first instance. Eutocius also asserts that he has ‘corrected’ Dionysodorus’ solution of 45

A detailed mention of the poor status of the text in the ‘old roll’ was already made at 3.132.1--4: ‘we read theorems written there which were obscure in parts because of the errors and which were mistaken in a variety of ways about the diagrams.’ 46 The right translation is provided in Decorps-Foulquier 2000, 73n52, for example, in the context of a very detailed analysis of Eutocius’ editorial procedures. The book is cited once by Netz [18n20]; however, the reference is very generic, and, most importantly, it is in a note to the translation of the alleged Archimedean synthesis. Therefore, the reference belongs in the material that has been lifted from Netz 2004. 47 There is a well-known occurrence of this phrase in Simplicius’ statement [Diels 1882, 60.27] that he will transcribe κατ λ ξιν Eudemus’ report of Hippocrates’ quadrature of lunules.

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Archimedes’ original construction because ‘it too’ was marred with scribal mistakes: the verbal form employed (διορθωσ μενοι) would normally entail more than slight interventions [see Heiberg 1910-1915, 3.152.15--26, esp. 152.22]. Other obvious cases in point are found among the several solutions of the problem of finding two mean proportionals which Eutocius offers in his long excursus in the commentary to De sph. et cyl. 2.1: radical interventions can be detected, for instance, in the solutions by Diocles, Menaechmus, and Archytas apud Eudemus [Heiberg 1910--1915, 3.66.8--70.5, 78.13--84.7, 84.12-88.2, resp.]. 48 Eutocius also modifies extensively the Apollonian proof of the locus-theorem which he reports in his commentary to the Conica, as is clear from the Arabic sources. 49 Eutocius’ editorial principles are briefly expounded in his commentary to Apollonius’ Conica [Heiberg 1891--1893, 2.176.17--22 and 354.5--7]. 50 Most notably, in sending Anthemius the revised fourth book of the Conica, Eutocius writes that Anthemius will find that it ‘is satisfying and clear for the readers, most notably in [his own] edition’ [2.354.5--7]. Such exigencies of ‘clarity’, often referred to in his commentaries when editorial choices are to be justified, 51 would naturally have induced Eutocius to rewrite the text of the ‘old roll’; and this has to be taken as the most likely hypothesis, unless compelling evidence to the contrary is adduced. Several recent works have shown that Eutocius affected the very structure of the Apollonian treatise, 52 even if there has been no study yet of the possibility of his intrusions at a linguistic level. Partial results in this direction include, for example, the proposal that the exact references to theorems in Apollonius’ Conica which are found in the commentary to De sph. et cyl. 2.4 are in fact post-Eutocian additions, and that there was later editing of the Eutocian text of the Conica. 53

48 49 50 51 52 53

See the analyses in Knorr 1989, esp. 81--87, 94--110, 225--245. This fundamental book is never cited by Netz. Heiberg 1891--1893, 2.180.11--184.20, to be compared with the evidence presented in Hogendijk 1985, 213--218. See Decorps-Foulquier 1998 and 2000, esp. 67--97. See Decorps-Foulquier 1998. One should add Knorr 1982 at least to the studies cited in the preceding notes. See Decorps-Foulquier 2000, 82n92, 128--134.

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It is plain that, on these grounds, no inference can be drawn from lexical peculiarities of the Archimedean appendix. The safest assumption is that the language in which it is written is Eutocius’ language. Of this crucial point, Netz offers a discussion that is entirely unsatisfactory: the argument takes less than one page and its outcome is the following assumption: Other than this [scil. the terminology of conic sections] Greek mathematical terminology hardly changed through the centuries between Archimedes and Apollonius. . . . These three transformations—critical corrections, dialect translation, and terminological standardization—are all innocuous. [72] But this is exactly what is to be proven, namely, that the presence of an extremely peculiar mathematical phrase in an alleged Archimedean text reported by Eutocius is not a consequence of Eutocius’ reworking. The rest of the discussion, which extends as far as page 84, take this assumption as an acquired fact and proceeds to a remarkable attempt at ‘distinguishing Archimedes from Eutocius’ (as in the title of the section), i.e., to segregate which portions of the text (in fact of the διορισμ ) are Eutocius’ additions to the recovered Archimedean appendix. Thus, Netz offers a black-and-white reconstruction: from a certain Heiberg-line onwards, there are Eutocius’ additions, in particular, what will subsequently be called ‘the second part of the proof’—see immediately below for an account of the relations between the two parts—before that line there is Archimedes’ text of the lost appendix, affected only by trifling, cosmetic changes in the process of editing. Other interpretations are ruled out a priori and Netz even seems to regard more nuanced accounts as a product of ‘philological paranoia’ [76]. As is clear, the whole discussion is conducted in a tendentious way, since the reader is diverted from the real point, namely, whether there are conspicuous linguistic changes introduced by Eutocius in the text, to the secondary issues of whether the text recovered by Eutocius was really Archimedean or where Eutocius’ supplement to the Archimedean διορισμ really begins. The text of the διορισμ But even the discussion of these secondary issues is in fact unsatisfactory. As a preliminary caveat, notice that the goal of sifting out Eutocius’ additions (in Archimedes’ as well as in Diocles’ case) is undertaken in order to show that Eutocius had an explicit concept of a functional relation between objects [94:

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Figure 1. The diagram of the διορισμ

[71]

quoted on p. 172, above]. This is a side issue in relation to Netz’ main thesis, the transition from problems to equations through the introduction of elements leading towards an algebraic approach. But it is strategically crucial in making more credible the general portrait of Eutocius as injecting germs of later mathematics. The idea of a functional relation between points comes out in the following way. Consider the description of the διορισμ given above. The first part of the διορισμ proves that any BΣA-solid is less than the BEA-solid when an arbitrary point Σ is taken between E and B; the second part shows that the same holds true even when the arbitrary point Σ (now called ) is taken between E and A. (The reader must not be bewildered by the change in the lettering. It occurs in the Eutocian appendix too: the line to be cut

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is now AB, point E is where the maximum is reached, EB is the diameter of the sphere and AE is equal to its radius, which means that AE is 1/3 of AB. 54) Now, is found by means of the very point Σ already displayed in the first part of the proof. The trick is to set the B A-solid equal to the BΣA-solid, even if this equality is a only a consequence of two other equalities stated in the text. 55 Eutocius recognizes the trick at the very end of the διορισμ , and adds further considerations [Heiberg 1910--1915, 3.148.7--27] which Netz takes to entail that Eutocius has an explicit concept of a functional relation between mathematical objects. I will provide further details below, but it is mandatory to discuss the textual issues first. That Eutocius added the second part of the proof is very likely, simply because Archimedes was not interested, from the very outset, in what was going to happen between A and E: EB is the diameter of the sphere and the point useful for cutting the sphere itself must, therefore, fall between E and B. As we have seen for Diocles, missing (and very often useless) cases and steps are exactly what a wise commentator supplies. The thesis is, therefore, highly plausible (even if, I repeat, its strategic character must be borne in mind); but the discussion [75--85] supporting the new location of the beginning of Eutocius’ addition to the διορισμ is based on very questionable arguments. I will discuss each of them in succession. Netz’ first argument [76] is ex silentio and it is the only one offered to rule out the possibility that the whole διορισμ is Eutocius’ forgery: ‘had [Eutocius] invented anything as original as [the διορισμ ] he would have been wild of pride. . . we would be certain to hear much more of this, had Eutocius been creative at such a scale.’ This is plausible even if the inference is far from cogent, as Netz himself notes. Still there are other possibilities. In fact, it is odd that Eutocius dismisses the issue of authenticity in so few words: he alludes 54

In Netz’ book, the correspondence between the different letterings has to be deduced from the texts alone; Netz offers his reader no explanation. The correspondence is Z → A, Δ → B, B → E. 55 Actually, the consequence is immediate: both the B A-solid and the BΣAsolid are equal to ‘the by HΩ epi A’. This is proven and the statements of the equalities are in steps 35 and 52 [Heiberg 1910--1915, 3.144.25--26 and 3.146.19--20, resp.]. Eutocius simply states the equality, adding ‘as is manifest from the preceding proofs’ [Heiberg 1910-1915, 3.148.17--18].

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only to the Doric dialect and to the archaic terminology for the conic sections. But Eutocius had already read Archimedes’ De sph. et cyl. without Doric as his lemmata show; and the idea that one can date a text on the basis of its ‘archaic’ terminology for conic sections is a historiographical fiction perpetuated from the times of Eutocius at least. 56 He may well have had this as a firm belief, but we are not forced to share his convictions. Notice also that Eutocius’ description of the ‘old roll’ appears to entail that the Archimedean text was not contained there as an appendix to the main treatise but as an isolated text, of course, without any indications of authorship. (It might also have been the only text contained in the roll, but more likely the roll was a miscellany). Moreover, the standards and means of a critique of authenticity had been in development since the times of the first Alexandrian scholars; and by the sixth century ad, a century of heated religious controversy, there were in hand some comparatively refined tools available to uncover forgeries. 57 One would, accordingly, have expected a more detailed discussion from Eutocius. The second argument [77--81] is a detailed analysis of the presence in the diagram of a phantom parabola whose defining relations are set out in the first part of the proof, but which is identified and used only in the second part, in a passage already recognized as Eutocian by Heiberg himself [1910--1915, 3.148.7--10]. A first remark is that Netz prints a single diagram for the whole διορισμ , whereas Heiberg has two. Heiberg chose to split the diagram into two for clarity’s sake. 58 Neither in the book under review nor in his earlier 56

The spreading of the Apollonian terminology was very likely far slower than usually believed. Most notably, one should explain away the several occurrences of archaic terminology in fragments from Geminus—e.g., in his optical fragment (see, e.g., pseudo-Hero, Definitiones 135.13 [Heiberg 1912, 108.1]) and in the classification of lines reported by Proclus [Friedlein 1873, 111.7--8]—and in the material collected in the pseudo-Heronian Def. n. 94 [Heiberg 1912, 60.1--5]. See also Toomer 1976, 9--10, on the presence of the ‘archaic’ designations of conic sections in Diocles’ work. Other alleged instances of ‘archaic’ terminology have recently been recognized as historiographical fictions as well: on the use of complex expressions to denote points A instead of the ‘canonical’ τ A), see Vitrac in a diagram (e.g., τ φ 2002, esp. 245--255. 57 For an introductory assessment, see, e.g., Wilson 1996, 53 ff. 58 Heiberg 1910--1915, 3.142 in apparatu.

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translation [Netz 2004] is there any reference to this discrepancy. A second remark of greater importance is that there is no need to follow Netz in reading what he calls steps v and b as ‘indirectly defin[ing] a parabola around the axis ZH, passing through T, P . . . ’, 59 which entails as a consequence that ‘the diagram includes a meaningful line which is not mentioned by the text as we have it’ [77]. 60 The two steps may indeed be read as giving the definitional relation of a parabola passing through T and P , respectively, and with latus rectum HΩ; but this is not the role they play in the proof. Here they serve as intermediate steps in a long chain of equalities involving squares and rectangles. In fact, step v is necessary for achieving the first part of the proof and step b for achieving the second. As for the remark that the parabola is in the diagram from the very outset, this is naïve (had Eutocius to produce a multi-layered diagram?) The simple fact is that Eutocius was used to redrawing all diagrams and compressing all information contained in a proof into a single diagram, even if the proof had several cases. 61 So, it is not surprising that the same diagram serves both for the alleged Archimedean διορισμ and for Eutocius’ remarks immediately following it. It is not the case, then, that we are here facing a further example of Netz’ tenet that ‘it often happens that the specification of objects in Greek mathematics is left for the diagram, so that the textual specification is a subset of the

59

In Netz’ translation [69--70], step v: ‘So let the on T X come to be equal to the by XHΩ’ [Heiberg 1910--1915, 3.144.19--20]; step b : ‘Let the on P B  come to be equal to the by B  HΩ’ [Heiberg 1910--1915, 3.146.14--15]. 60 In fact, ‘the proof as we have it’, namely, the whole proof of the διορισμ , obviously contains the introduction of the parabola—unless one has already decided what belongs to Archimedes and what does not, that is, unless one takes for granted, as Netz does already in this second argument, exactly what one has set out to prove. 61 This is explained in detail in Decorps-Foulquier 1999; 2000, 94--97. The poor condition of the diagrams was pointed out by Eutocius in his initial description of the recovered appendix, as we have seen.

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diagrammatic specification’ [78]; 62 and I do not see how it may be said that in the present instance ‘the specification of the text and the specification of the diagram clash’ [79]. There is no contradiction at all between text and diagram. 63 But Netz assumes that there is and is then led to discuss the location in the proof of what he calls step p, where point P , one of the points on the phantom parabola, 64 is apparently first introduced as a further point on a hyperbola. The discussion is confused and fallacious. In the first place, step p [Heiberg 1910--1915, 3.144.7--8] is translated as follows [68--69]: ‘So let the hyperbola, produced, as towards P , be imagined as well.’ This is not a good translation. It should read: ‘So, let the hyperbola also be conceived as produced as far as P .’ 65 The difference is crucial. The first translation might be the basis for asserting that ‘[t]he text of Step p seems to assume that the hyperbola of the diagram ends at K, and that there is a freefloating point P indicating the location of the continuation of the hyperbola’ [79]. (Recall once more that this single diagram is to serve for the whole διορισμ as well as for Eutocius’ commentaries.) One might even infer that ‘[s]uch a free-floating point is in itself a bold innovation’ [80], were it not for the fact that the point is firmly placed on the hyperbola. However, as is clear from the second and correct translation, the Greek text simply conveys the sense that one is invited to conceive the hyperbola produced as far as a convenient point, called P . Introducing points whose specification is provided later in the proof, a move that is standard in Greek mathematics, 66 is simply dictated by the reasonable aim of keeping the number of labeled points to a strict minimum.

62

63

64 65 66

Reference is made to Netz 1999a, chapter 1. This is a paradigmatic example of the author’s eagerness to take his own interpretations as well-established facts. I completely disagree with the thesis of the chapter referred to, if not because the small sample of evidence there analyzed makes a general statement as the one just quoted unwarranted. A little later on the same page, ‘another clash between text and diagram’ is discovered in step p, when a hyperbola is ‘imagined’ to be produced as far as a point called P . We will discuss this step presently. But this will be implicit in step b which of course comes after step p. The use of π in this sense is standard. There are several examples (not referring to hyperbolas) in Euclid, Elem. 1.

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Next, the introduction of point P early in the proof is explained by the need to ‘complet[e] the argument that the parabola contains the hyperbola’ [80]. The argument to be ‘completed’, while necessary of course, is in fact not even sketched in the proof. Point P on the hyperbola beyond K is instead explicitly used in the second part of the proof, and its introduction appears thus motivated by this very requirement: the key point is step b , whose stipulation is eventually equivalent, as Eutocius will explain later [Heiberg 1910--1915, 3.148.7--18], to the B A-solid’s being set equal to the BΣA-solid. Netz aims instead to show that step p is originally Archimedean, in order to corroborate his thesis that Eutocius ‘simply added an adaptation of the proof to the case that is not explicitly covered by the first part’ [81], and, by implication, that he did not touch the rest of the proof where the crucial feature (namely, the epi phrase) is located. On the contrary, Netz’ analysis itself supports the view that the whole διορισμ was rewritten by Eutocius, who inserted in the first part constructions of certain objects which were only of use in the second part. The fact that P is inserted in a strict alphabetic sequence in the first part of the proof, as noted by Netz [79n58], simply confirms this thesis. Further, contrary to what is asserted on page 80 [quoted above], step 23 ‘therefore the parabola is tangent to the hyperbola at K’ [Heiberg 1910--1915, 3.144.6--7] is enough to secure that the hyperbola is above, that is, contained in, the parabola, even if its role as such is not made explicit in the proof. Moreover, step p simply cannot give any information about the location of P and, a fortiori about the relative position of the parabola and the hyperbola. 67 The upper part of the hyperbola is, therefore, totally useless so far as the first part of the proof is concerned; and the problems arising from the absence of any argument showing that the whole hyperbola is contained in the parabola arise from a reworking of the original proof so as to make it fit the second part. Finally, the whole ‘phantom parabola’ argument relies on a logic which, I must confess, escapes me. Let us start with the conclusion. After recalling that ‘[t]he way to make sense of the location of Step 67

The problem was recognized and discussed to some extent in Netz 1999b, 39, even though on page 30 of that paper one finds just the assertion quoted above from page 80 of the book under review. Of course, the analysis of 1999b, 39, which cast strong doubts on the assertion of page 30, has disappeared from the book.

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p is to understand it as a constituent in an argument in which the second part of the proof is redundant,’ Netz writes, ‘But we already saw, through the verb “imagine”, that the second part of the proof was probably written by a different hand to that of Step p’ [81]. In the subsequent line the ‘probabilities’ are declared to ‘begin to accumulate’ (actually just one has been offered until now), and then the heap reaches certitude in the subsequent line when Netz writes: ‘it is time to replace the cumbersome expression “the author of the second part of the proof”, by the simpler name “Eutocius”.’ (This entails, of course, that it was already well established that the author was not Archimedes, exactly what Netz has set out to prove.) If one looks back for discussion of the verb ‘imagine’, one finds first that it ‘is used when the object to be “imagined” is not visible in the diagram, either because it is not an object a diagram can represent directly. . . , or because it simply is not drawn’ [79]. 68 And then, after the remark about the ‘free-floating point P indicating the location of the continuation of the hyperbola’, one finds the conclusion: The diagram is different, and actually extends the hyperbola to P . It is probable that whoever drew the hyperbola as far as P , was not the one who wrote the text of Step p. But notice that without drawing the hyperbola as far as P , the second part of the proof is impossible. Therefore the same probable argument seems to show that the second part of the proof could not have been written by the author of Step p. [80] I cannot see any argument at all here, not even a probable one, except for the totally fictitious distinction between who drew the hyperbola and who wrote step p. Yet, even in this form, the argument is grounded, as we have seen, on a misunderstanding of the text. The third argument [81--82] is that the second part of the proof begins with the λλ δ at Heiberg 1910--1915, 3.144.31,

68

These are very basic facts about Greek mathematical style; but a reference is nonetheless provided to Netz 1999a, chapter 1, where allegedly the author had shown that ‘imagination. . . , an established operation in Greek mathematics. . . often has a precise signification’ [79]. The ‘imagination’ is simply the technical use of the verb νοε ν. If this is not playing with words!

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Aestimatio an expression which is common in Greek discursive prose but is very rare in the special discourse of Greek mathematical proofs. It is used only once in the Archimedean corpus (of about 100,000 words). For Eutocius, the transition into a formulation that he considered non-Archimedean was sufficient to mark off the remaining text. [82]

This is no argument. To assert that a phrase is ‘very rare’, one has to produce data. The figure of 100,000 words is there to impress the reader and cannot at all be considered relevant from a statistical point of view. 69 The reasons why Eutocius considered the formulation with λλ δ as non-Archimedean will of course remain a mystery. Notice, moreover, that here Netz keeps silent about the explanation of an editorial choice of his, related to the present issue and made just at the beginning of his translation of Eutocius’ remarks to the second part of the διορισμ . The passage, already regarded as Eutocian by Heiberg, begins at Heiberg 1910--1915, 3.148.1 with πιστ σαι δ (‘now one must understand’). This is the reading of the manuscripts; but Heiberg corrects δ to δ . Netz’ choice to follow the manuscripts is motivated by the fact that ‘δ is a more natural connector inside a stretch of discourse, whereas δ is a more natural connector at the beginning of a new stretch of discourse’ [93n70]—which serves to make his proposal of an early beginning of Eutocius’ addition more plausible. 70 My personal experience is that the contrary is true; 71 but more important is that a claim like the above must be supported by data, and that it obviously clashes with the just mentioned contention that λλ δ at 3.144.31 opens the Eutocian addition. Moreover, William of Moerbeke’s translation has autem in its translation of the line at 3.148.1, a canonical, even if not exclusive, way of translating δ . 72 λλ δ is rendered as at vero, while the only occurrence of autem 69

A census of the connecting phrases that one meets in the Archimedean corpus and their relative frequencies would have been much to the point. The occurrence of λλ δ is in De sph. et cyl. 1.11 [Heiberg 1910--1915, 1.42.23]. 70 Netz takes his proposal as an acquired fact and employs it as a supporting argument for his choice to follow the manuscripts. 71 It is enough to recall the position of the clause introduced by δ inside μ ν. . . δ . . . . 72 Clagett 1976, 41rG, 262. Netz asserts that δ ‘may have also been read by the Latin translator’ [93n70]. The ‘may’ deserved a short discussion.

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as a translation of δ is marked as doubtful even by Clagett, and is very likely William’s mistake. 73 Thus, it is most likely that one has to keep δ and to read Netz’ note with δ and δ interchanged. (All of this supports his proposal for the new location of the beginning of the Eutocian addition, of course.) The fourth argument [82--83] focuses on the absence of ‘scholiastic material’ (mainly, exact references to Apollonius’ Conica) in the second part of the proof. Netz should have discussed and refuted Decorps-Foulquier’s well-founded remark [2000, 82n92] that the presence of similar material in other parts of the proof is the result of a post-Eutocian intervention. The last argument [83--85] comes from the use in the second part of the proof of primed and uncommon letters of the Greek alphabet to denote points. Netz takes them to be numerals: uncommon letters such as , already are numerals; common letters become numerals when primed—this usage is very early. The reason for this, according to Netz, is that ‘objects labeled by numerals are thereby strongly marked.’ More simply and more plausibly, the reason is that the author had exhausted the alphabet at that point of the proof. 74 Anyone, Archimedes’ included, 75 would have used new symbols when arriving at Ω, most probably primed letters or uncommon letters. 76 Even with the above drawbacks, the conclusion of the discussion should not just be that Eutocius ‘simply added an adaptation of the proof to the case that is not explicitly covered by the first part’ [81]. Rather, it has to be that Eutocius reshaped the whole proof in order 73

Clagett 1976, 36vB, 238: cf. Heiberg 1910--1915, 3.56.10. Clagett’s reservations [1976, 645, 661] are expressed in the indices. 74 Netz [83] asserts that ‘it is not completely clear whether we should read in our text or Γ  .’ Nothing hangs on it, and I have followed Heiberg’s Γ  . where In Netz 2004, 328, the claim is sharper: ‘I think I might see a Heiberg (whom I follow) prints a Γ  .’ In fact, Γ  is Heiberg’s emendation: in the manuscripts, as he says on Heiberg 1910--1915, 3.147 he too read in apparatu (the apparatus in Heiberg 1910--1915 is very often placed on the right-hand page). 75 Netz is aware of this, but suggests that finishing the first part of the proof with all letters of the alphabet employed ‘probably was Archimedes’ intention’ [84n61]. But why ‘probably’ and to what purpose? 76 There are several examples in the Archimedean corpus, and a further instance occurs in Euclid, Elem. 13.16.

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to make it fit the addition of the second part. Eutocius was unable or simply unwilling to write the second part of the proof in a way that was independent from the first one. The best he could do was to make the fictitious chance point on AE depend on the original chance point Σ on EB. Thus, by virtue of the very constraints of the proof Eutocius was proposing, points and Σ are subordinate the one to the other, a fact that entails problems of generality, since is not in fact a true chance point. Eutocius tried to cope with these problems in the very last part of the διορισμ [Heiberg 1910--1915, 3.148.18--27], where Netz sees the explicit concept of a functional relation. But Eutocius only remarks that, once a suitable parabola (that is, the phantom parabola) is drawn, both and Σ are obtained by dropping perpendiculars to the main line AB from the points where the parabola intersects the hyperbola: the intersection point T will find (ε ρ σει) Σ, whereas the intersection point P will find . To quote Netz again, Eutocius is original in two ways: ‘First, he describes the systematic relation holding in the line: the symmetry around the point E. Second, he has an explicit concept of a functional relation between mathematical objects’ [94], that is, between the points of intersection and the points on the main line. 77 A clue to the efforts Eutocius had to make in proposing such a concept are found in the use of the non-technical verb ε ρ σκειν [94]—and the discussion ends there! The apodeictic statement just quoted is simply unfounded. First, as for Eutocius’ describing symmetrical configurations around point E, no one acquainted with Apollonius’ De sectione rationis, for instance, will see any originality in it. But there is a more serious

77

By composition of the two functional relations just cited (orthogonal projections) and of the one deriving from the fact that T and P are on the same parabola, Netz might even have extended his remark to stating a direct functional relation between Σ and . I wonder why Netz did not make the connection explicit, even if hints of it can be found on page 94. In its stead, we find a very aphoristic conclusion of the whole argument: ‘Eutocius’ conic section is an arena for equalities between points: it is thus, we may say, a sum of points, defined quantitatively. Thus it has become akin to the conic section of analytic geometry’ [95]. What is ‘an arena for equalities between points’? What is the σ μπτωμα of a conic section if not a quantitative definition of it? What has analytic geometry to do here?

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problem: the text does not support Netz’ interpretation, 78 to which the concept of a functional relation is eventually reduced [94]. In fact, Eutocius introduces the suitable parabola in the following way. He asserts first that one can take any of the two points solving the problem, either the one between E and B or the one between E and A. He then says that if one wants to take the point between E and B, then, as has been said, if the parabola through the points H, T is drawn, cutting the hyperbola at two points, the one closer to H, namely, to the axis of the parabola, will find the between E, B, as here T finds Σ, whereas the one farther off the between E, A, as here P finds . [Heiberg 1910-1915, 3.148.22--27: cf. Netz 93--94] But T is already on the hyperbola [steps r and s: Heiberg 1910-1915, 3.144.9--11]! Hence, an independently conceived parabola is not drawn that determines both points of intersection on a par: one of those points is instead needed in order to draw the parabola itself. There is no symmetry at all here. We must conclude that Eutocius did not describe any symmetry around E, and a fortiori that he was unable to assess the problems of generality raised by his proof. What could possibly have driven Eutocius to such a lapse I am unable to say; but Netz was perhaps misled by the fact that, in his diagram [71], the phantom parabola does not pass through T , as it should do and as it is made to pass in Heiberg’s first diagram [Heiberg 1910-1915, 3.143]. 79 The epi phrase A by-product of the above analysis is that it would be inappropriate to ascribe the introduction of the crucial epi phrase to Archimedes rather than to Eutocius. The argument offered in support of the claim that the epi phrase was originally Archimedean (the ‘explanation’ in section 2.2 [see 72--76] has been discussed above) is as follows:

78

But almost surely the text is corrupt and does not correspond to what Eutocius wrote. As Netz remarks [93n76], the entire sentence at 3.148.21-27 is introduced by a μ ν that finds no correlative δ . Netz ignores the μ ν in his translation. 79 There is no information on this feature in the book under review or in Netz 1999b, 2004.

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Aestimatio As explained in section 2.2 above, the source used directly by Eutocius was probably very close indeed to an original Archimedean text, and while Eutocius had certainly transformed this source in several ways, he would have had no motivation for introducing the epi locution (a suitably elliptic prism-based locution would be easily understood by Eutocius, and would not have been more cumbersome than the epi locution). I add that we have the epi locution used repeatedly in Archimedes’ Sphere and Cylinder II, the alternative proof to proposition 8 (the penultimate proposition of that book). I personally believe that this alternative proof is by Archimedes himself (it is radically original in many ways, which may explain why Archimedes would have been interested in offering such an alternative proof in addition to a more ‘standard’ proof—while it is difficult to see who else was capable of and interested in producing such a proof, only to leave it as a gloss in the text of Archimedes!). At any rate, the expression was certainly used in this geometrical context, if not by Archimedes himself, then by some other highly competent Greek mathematician. [103]

As should be clear to everyone, this is no argument at all. It is a good approximation of a circular argument; and it even employs the historians’ ultimate resource, the principle of sufficient reason. 80 The connection between the last and the penultimate sentence is ineffable. What is worse, it omits any mention of parallel passages in later authors (some of the passages will be mentioned some 15 pages later, when the conclusion of the above ‘argument’ has long been taken for granted), and neglects any obligation to supply precise references to Archimedes’ idiosyncrasies as a writer, and so forth. In a word, what is missing is the honest and low-flying apparatus of standard scholarship; in its stead we find an expression of personal belief. Other assertions in the section are equally questionable. First, the discussion leading to the conclusion that the epi phrase must be read as A (epi B), namely, that it ‘is not a single object, but is a composite clause, with a noun—the figure—modified by the adverbial expression “epi line” ’ [106], is grounded on an unsatisfactory

80

This is an argument of the kind sharply criticized in Netz 2002b.

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linguistic argument. It relies on the position of the copula in a statement of equalities of ratios contained in De sph. et cyl. 2.8 aliter: the copula is inside the antecedent of the second ratio, placed exactly between A and epi in the above formula. 81 Netz claims that ‘[t]he most natural position for this copula in Greek is immediately following the first object of the second part’ [106]. But it is one thing to assert that the position is the ‘most natural’ and another that it is ‘the only possible’. In the first case, this is no argument, since if there are exceptions to the rule, the present occurrence might well be one of them. In the second case, the burden falls on Netz to produce some evidence in which such usage is corrected or criticized. Second, the discussion of associativity is unsatisfactory. We are informed at the beginning of the discussion whose conclusion is that the epi phrase must be read as A (epi B) that ‘the following [discussion] may hold for the arithmetical case as well, but my evidence derives from the geometrical case’ [105]. But when associativity is introduced, it is stated that ‘[i]n both the geometrical and the arithmetical cases, the question cannot even be raised, whether epi is associative or not’ [106]. The first sentence quoted is amazing: had Netz no time to check the arithmetical case? And when he wrote the second sentence, did his sample get enlarged without notice? Apart from this, the question of associativity of epi in arithmetical cases can, of course, be raised meaningfully, at least to the same extent that the question of commutativity of epi can [see 104]. Third, a problem of method. On pages 104--106, given the fact that some expressions are unattested, Netz infers the actual impossibility of their having been produced by Greek mathematicians. But, in truth, no such inference is warranted from the absence of expressions like ‘A epi B epi C’ or ‘line epi figure’. Even if Netz recasts his case with greater care, his discussion of commutativity and associativity appear to rely more on an impossibility argument than on actual evidence. That such an approach is untenable has been shown by Netz himself [2002b]. Fourth, in this and in the subsequent section, it is stated that the epi phrase is exported solely from arithmetical contexts: we never have an expression of the form. . . Line epi line. [104] 81

Far better evidence is available, as we shall see below.

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Aestimatio [t]he remarkable thing is that [Archimedes] chose to import an expression from the domain of arithmetical calculations; we would have expected him to import from a nearer domain. [114]

In fact, perpendiculars are canonically said to be drawn epi a straight line or a plane, and they have exactly the function required in the Eutocian epi phrase. It is absurd to dismiss this fact with the following, rather uncorrelated, argument: the preposition epi does not have the function required. It does not serve as a static description of the three-dimensional object resulting from the plane and the line; rather, it serves dynamically, to lead the act of drawing. [101] 82 Fifth, the absence of an expression for a solid such as ‘the prism having the area A as base, and the line B as height’ instead of the epi phrase is tentatively explained in terms of ‘the territorial imperative’ of another expression [111]. 83 I cannot attach any sense to this claim, the author’s reference to Netz 1999a notwithstanding. What is striking in the discussion of the epi phrase are several and conspicuous omissions. These omissions are of two sorts: the first concerns evidence from Hero’s works and the second, the presentation of Archimedes’ solution. References to the epi phrase in Hero are limited to a minor occurrence in the preface to Metr. 2 [Schöne 1903, 94.29]. 84 This passage is of a calculative nature and the multiplicative connotation of the epi phrase is plain [114]. But Netz had at his disposal also Metr. 1.7-9 [Schöne 1903, 18.8--11, 22.15--19, 24.10--13, 26.13--22]. 85 The first and third passages are also calculative and refer explicitly to numbers. But 1.8 is definitely a geometrical proof and appears to contradict 82

Netz discusses the perpendicular to a plane only. It is astounding to follow him as he tells us what the Greeks really had in their minds. 83 On the very top of page 110, there is a misprint which makes the argument difficult to follow: the formula marked *(30) should be marked *(28). Moreover, the formula has a line that should not belong to it. In general, the book would have benefited from a further editing. 84 Incidentally, the description of the generation of certain solids in this very preface appears to be a good geometrical translation of the epi operation. 85 See also Metr. 3.4 [Schöne 1903, 148.16--25], where one finds both a metrical and a geometrical connotation of the epi phrase.

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at least two statements in this and the preceding sections. The first is that, ‘[t]he reason Hero allows himself such a language [scil. multiplying an area epi a line] is clear, namely, he simply chooses to ignore the phenomenon of irrationality’ [110]. This is false in view of the very claim by Hero that some of his methods are able to cope with ‘non expressible’ lines [Schöne 1903, 26.3], and of the fact that the triangle in 1.9, for example, is expressly chosen to have the area and hence the height irrational. The second is that ‘[i]n fact, this epi is never used in any other context besides “figure epi line” ’ [104]. In Metr. 1.8, one repeatedly finds ‘figure epi figure’. What is more, Pappus refers to ‘predecessors who express nothing at all coherent when they say the contained by these epi the square on this or epi the by these ’. 86 Sure, the sentence quoted from page 104 appears to refer to the occurrences in Archimedes and Eutocius. But the whole section is intended to state properties of the epi phrase which are valid in general, as its title ‘Is the expression completely algebraic?’ [104] shows. 87 As for the title itself, the expression could not be completely algebraic even in the case ‘number epi number’: this is not algebra, it is arithmetic. The translation of Archimedes’ analysis in section 1.3 ends with three suspension dots. The omitted text is left uncommented by Netz and may be divided into two parts. The first part [Heiberg 1910--1915, 3.134.13--29] concludes, after a series of standard steps concerning some basic properties of conics, that a certain point K is given. Here is the text of the second part [Heiberg 1910--1915, 3.134.29--136.13]; the translation is taken from Netz [2004, 320] without modifications, apart from omitting his numbering of the deductive steps:

86

Cf. Pappus, Collectio 7.39 [Jones 1986, 123.8--14 = Hultsch 1876--1878, 680. 15--19]. This crucial, and indisputably geometrical, remark by Pappus serves Netz not as a testimony that the epi phrase was used well before Eutocius in the same way as he himself did, but to support the portrait of Pappus as a mathematical Atticist who reacted to the proposal by advocating and practicing a systematic use of compound ratios: ‘Pappus is thus a witness to an avenue leading to algebra—not a participant in this movement’ [116]. But then, we should not be interested in Pappus but in his ‘predecessors’! 87 One would have expected that general properties of the epi phrase be discussed after a presentation of all the evidence, not before.

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δ δοται ρα τ K. κα στιν π α το κ θετο KE π θ σει δεδομ νην τ ν AB · δ δοται ρα τ E. πε ο ν στιν, EA πρ τ ν δοθε σαν τ ν AΓ , ο τω δοθ ν τ Δ πρ τ π EB, δ ο στερε ν, ν β σει τ π EB κα τ Δ, ψη δ α EA, AΓ , ντιπεπ νθασιν α β σει το ψεσιν· στε σα στ τ στερε · τ ρα π EB π τ ν EA σον στ τ δ θεντι τ Δ π δοθε σαν τ ν Γ A. λλ τ π BE π τ ν EA μ γιστ ν στι π ντων τ ν μο ω λαμβανομ νων π τ BA, ταν διπλασ α BE τ EA, δειχθ σεται· δε ρα τ δοθ ν π τ ν δοθε σαν μ με ζον ε ναι το π τ BE π τ ν EA. Therefore K is given. And KE is a perpendicular drawn from it to a given in position, to AB; therefore E is given. Now since it is: as EA to the given AΓ , so the given Δ to the on EB; two solids, whose bases are the on EB and the Δ, and whose heights are EA, AΓ , have the bases reciprocal to the heights; so the solids are equal; therefore the on EB, on EA is equal to the given Δ, on the given Γ A .* But the square on EB 88 on EA is the greatest of all the similarly taken on BA, when BE is twice EA, as shall be proved; therefore the given on the given must be not greater than the on BE on EA . But why does Netz cut the text off at this point? Yet, the clause τ ρα π EB π τ ν EA σον στ τ δοθ ντι τ Δ π δοθε σαν τ ν GA is the best possible evidence that the epi phrase ‘is not a single object’: the dative τ δοθ ντι τ Δ makes it clear that the objects to be equated are the base surfaces, not the whole solids. This feature, which is obscured in Netz’ translation by his misleading introduction of the phrases , is argued solely, as we have seen on pp. 200--201 above, on the basis 88

For mysterious reasons, the comma in the formula ‘square on A, on B’ is missing from here on.

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of the single, doubtful example taken from De sph. et cyl. 2.8 aliter. Further, the first time the reader meets the epi phrase in Netz’ book is at the very beginning of Archimedes’ synthesis [16]; whereas in Eutocius’ text, the passage reported above comes first, as is natural since it is the end of the analysis. Now, it is not the case that the above passage makes all Netz’ elaborations on the epi phrase fall—even if it is inconvenient for Netz to read Eutocius-Archimedes speaking about solids when referring to ‘areas epi lines’ 89 —but at the very least it deserved a detailed discussion. Why is there none? Where I have placed an asterisk, one finds the following note in Netz’ translation of Archimedes [2004, 320n353]: The expression ‘plane on line’ has here a geometrical significance, yet it can also be interpreted as the multiplicative ‘on’ used in the examples of calculation earlier, where we had ‘number on number’. For this ambiguity of meaning, see Netz (forthcoming b). This is the only place in the whole translation of Eutocius’ commentary in Netz 2004 where the epi phrase is mentioned. 90 One would, therefore, expect that this passage be a major issue for discussion in ‘Netz (forthcoming b)’ which, as it turns out, is the book currently under review. Border-crossing: a modest proposal ‘As later readers broke down the borders between registers, geometry became algebrized’ [120]. I strongly doubt that geometry ever became algebraized in the hands of the Greeks, but we can reasonably ask who broke the borders. As we have seen, Netz completely misconceives Hero’s contribution and wrongly rules him out as an early and deliberate border-crosser. Moreover, further uses of the epi phrase by mathematical authors who are undeniably earlier than Eutocius have either been ignored, as we have seen with the passages in Metr. 1, or explained away by Netz: the obvious strategy in the latter case was to deem the examples as belonging to minor streams of Greeks mathematics—as if this could be an acceptable historiographical category, given the 89

We find a similar expression, although less explicit, also at the beginning of the synthesis [Heiberg 1910--1915, 3.136.20--28]. 90 The issue is very briefly introduced on page 233, within the ‘General Comments’ to De sph. et cyl. 2.8 aliter.

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extant evidence. Actually, their obvious relevance to the problem at issue is not even discussed, and the examples are no more than very quickly surveyed on pages 112--116. 91 The occurrences of the epi expression in the Eutocian text, instead, have for Netz the virtue of being contained in a lemma which ‘is in pure proportion theory: not some unique isolated point in the outskirts of geometry, but at the very hearth of the Euclidean geometrical discourse’ [116--117]. But Eutocius’ lemma is not in pure proportion theory: it belongs in the rather exotic and very special theory of the composition of ratios, a subject which appears to have been regarded as worthy of some systematization only in post-Hellenistic authors. Eutocius’ lemma does, however, involve manipulations of proportions. But if this is Netz’ criterion, most of Greek geometry is ‘in pure proportion theory’. Actually, in the whole ancient mathematical corpus, there is just one place where general proportion theory is addressed: Euclid, Elem. 5. Thus, proportion theory is by no means ‘at the very hearth of the Euclidean geometrical discourse’. Manipulations of proportion are, of course; but this is another matter. Indeed, such confusion between proportion theory and applications of proportions in geometry is one of the pervasive misconceptions in Netz’ book. The analysis itself of Eutocius’ lemma is disconcerting. Step 9 π το Γ π τ ν Δ [Heiberg in the proof reads O ρα στ ν 1910--1915, 3.200.4--5] (‘therefore O is the on Γ , on Δ’ in Netz’ translation [117]). The masculine article instead of the neuter τ permits identifying the result of the operation on the term Γ as a ‘square term’, not merely as a ‘square’. This is not the only occurrence in the proof: there are four others in [Heiberg 1910--1915. 3.200.19--24]. Netz comments that there are two essential ways in which this breaks new ground relative to Classical Greek geometry. First, all objects, regardless of 91

Contrary to what is suggested on page ix, no ‘interpretation’ is offered of the examples. We find instead on page 115 the remarkable, aphoristic claim: ‘The measurement of solids is never defined in mainstream Greek geometry.’ What does ‘to define’ mean here? What is mainstream Greek geometry? Is ‘never’ relative to the extant evidence or to Greek geometry without qualification? Did Archimedes really not ‘measure’ solids or is he outside mainstream Greek geometry? Is it necessary to put numbers to a geometrical figure in order to ‘measure’ it?

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their dimensionality, are considered on a par (everything is a ‘magnitude’). Second, objects are directly produced from each other through multiplication (O is the on Γ , on Δ), and are not just merely related to each other by their geometrical configuration.. . . Things, in the universe of Eutocius’ lemma, are defined by the multiplications and equalities that give rise to them.. . . Step 9—as well as Eutocius’ lemma as a whole—firmly belong to the world of algebraic equations. [119] The first claim of originality is false: 92 ρο was a technical term in proportion theory since Aristotle, and already in Elem. 5 all objects are considered on a par (everything is a ‘magnitude’). 93 The second claim is vitiated, as we shall presently see, by an unsatisfactory discussion of the textual evidence; and it conveys the strange belief that every proof in Greek mathematics has to refer to some geometrical configuration. But any object in any proof in Greek number theory is ‘defined by the multiplications and equalities that give rise to them’. Now the textual evidence. The critical apparatus in Heiberg’s edition (printed in this case on the page facing that of the Greek text) casts doubt on the assertion that the use of the masculine article is ‘a remarkable result of the semiotic eclecticism of this text’ [117n105]. All of the five occurrences of the masculine article in the proof are recast to neuter pronouns in William of Moerbeke’s translation (as well as to neuter articles by modern editors such as Torelli): id quod is consistently used everywhere and the masculine articles before other designations of ‘terms’ are consistently not translated. 94 92

In the same page we read, ‘Thus the main innovation of Eutocius’ text— referring throughout to the object “term”—is determined by Archimedes’ [epi] expression.’ 93 Much of the alleged novelty of Eutocius’ lemma disappears once one realizes that Eutocius simply singles out an abstract step in need of proof, that terms and numbers are regularly represented by line segments, and that products and squares of numbers are canonically expressed in the same terms as their geometrical counterparts [cf., e.g., Elem. 9.15] as in any proof in number theory. 94 The weight of William’s translation as a testimony is relative, since it appears that the translation of Eutocius’ commentaries was made on Valla’s now lost codex A. Nevertheless, Heiberg takes it as an independent witness in his apparatus.

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Certainly, it is not unlikely, as Clagett [1976, 43vF--I, 277--278] points out, that William had actually misread the neuter relative pronoun for the masculine article . But we know from a subscription to codex G that Valla’s A was written using many compendia, and that accents and breathings were not marked [Clagett 1976, 520]. 95 This makes the choice of versus a matter which a modern editor must make arbitrarily (even if we might have expected to find a neuter article, not a neuter pronoun in the expressions at issue). None of this makes Netz’ interpretation impossible. But it does suggest that the passage may have caused troubles to other copyists too, and so serves as a warning against taking the received texts as they have been established by modern editors to be faithful representatives of the originals. On these grounds, no clear-cut inference from such a text should be drawn; and, at any rate, a discussion of the textual matters was mandatory. Let us return now to the issue of border-crossing. Consider the following proposition, taken from Diophantus’ De polygonis numeris. Here ‘proportion theory’ (in Netz’ sense) is undoubtedly at work; border-crossing is patent; and no one would dare to locate Diophantus at an ‘isolated point in the outskirts’ of Greek mathematics.

Ε ν δ θ μεν τ π συναμφοτ ρου HΘ. κα το KB σον τ ν N ξ ριθμ ν, σται κα π συναμφοτ ρου το HΘ.ΘM ο π τ ν π το KB ον σο τ π το Nξ  . . .

Εστω συναμφοτ ρ τ HΘ.ΘM σο A, τ δ KB σο δ συναμφοτ ρου το HΘ.ΘM κα το KB σο Γ · λ γω τι κα π συναμφοτ ρου το HΘ.ΘM (τουτ στιν π το A), π τ ν π το KB (τουτ στιν π τ ν π το B) σ. τ π το Γ . Κε σθω το A, B σοι π ε θε α ο ΔE, EZ, κα ναγεγρ φθω π α τ ν τετρ γωνα τ ΔΘ, EΛ, κα συμπεπληρ σθω τ ΘZ παραλληλ γραμμον. Ω ρα ΔE πρ τ EZ, ο τω τ ΔΘ πρ τ ZΘ παραλληλ γραμμον· δ ΘE πρ EK, ο τω τ ΘZ παραλληλ γραμμον πρ EΛ· τ ρα ΘZ παραλληλ γραμμον μ σον ν λογ ν στι τ ν ΔΘ.KZ ων · τ ρα π τ ν ΔΘ.ZK B, τ

95

See Heiberg 1910--1915, 3.x--xi.

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ων σ. τ ΔΘ σον τ

π το ΘZ παραλληλογρ μμου· κα στι τ μ ν π συναμγοτ ρου το HΘ.ΘM , τ δ ZK ον σον τ π το KB, τ δ ΘZ παραλληλ γραμμον σον τ N ξ. κα τ ρα π συναμφοτ ρου το HΘ.ΘM ον π τ π το KB ον σ. τ π το N ξ τετραγ ν . [Tannery

1903--1905, 1.466.1--4 and 1.466.21--468.13] 96 If we set the number N ξ equal to the by HΘ ΘM taken together and KB, also the square on HΘ ΘM taken together epi the square on KB will be equal to the square on N ξ. . . Let A be equal to HΘ ΘM taken together, B be equal to KB, Γ to the by HΘ ΘM taken together and KB: I say that, in addition, the on HΘ ΘM taken together (i.e., the one on A) epi the one on KB (i.e., epi the one on B) is equal to the one on Γ . Let ΔE, EZ, equal to A, B, be put in a straight and let squares ΔΘ, EΛ be described on them, and let the parallelogram ΘZ be completed. Therefore, as ΔE is to EZ, so ΔΘ is to parallelogram ZΘ; and as ΘE is to EK, so parallelogram ΘZ is to EΛ. Therefore, the parallelogram ΘZ is the mean proportional of the squares ΔΘ ZK. Therefore, the by the squares ΔΘ ZK is equal to the one on the parallelogram ΘZ; and ΔΘ is equal to the on HΘ ΘM taken together, the square ZK equal to the one on KB, the parallelogram ΘZ equal to N ξ. Therefore, in addition, the square on HΘ ΘM taken together epi the square on KB is equal to the square on N ξ. The figure accompanying the text actually displays squares and rectangles; the proof is definitely geometrical. This very aspect should not by any means be dismissed: Diophantus and Hero with him, while not acting as commentators either in the Metrica or in the Arithmetica, obviously broke borders between registers; and they did this well before Eutocius. 97 Therefore, the deuteronomic character of 96 97

I keep the format of Tannery’s text. Recall, moreover, that al-B¯ır¯ uni [Suter 1910--1911, 39] ascribes the procedure for calculating the area of a triangle proved in Metr. 1.8 to Archimedes, and that there we find expressions such as ‘figure epi figure’ [see p. 203,

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the latter’s approach cannot be considered the only force (if there exist any such forces) that could and did drive the historical phenomenon of register-crossing. 98 An effective and, I would say, prior contribution came also from the ‘metric tradition’, which Netz regards as a minor, segregated stream of Greek mathematics, thereby greatly undervaluing it as an important engine of change. A less tendentious appraisal of the sources suggests instead a trajectory of the epi phrase that coincides with the mainstream Greek mathematics of the post-Hellenistic period: 99 Hero → Ptolemy → Diophantus → Pappus → other commentators → Eutocius. The sources also suggest that the Hero-Diophantus usage of the epi phrase was a matter of course, and that it found a natural application in the composition of ratios, a domain which in its turn was later systematized by Eutocius. 100 The passages from Diophantus, Pappus, and above]. Incidentally, one may also wonder why the mixed proportion between magnitudes and numbers introduced in Elem. 10, and actually going back at least to Theaetetus, is not considered a proper form of registercrossing. 98 The testimony in Pappus, Coll. 7.39 [see 204n86, above] shows decisively that the expression was also introduced in advanced geometrical research. 99 My use of the term ‘mainstream’ is, of course, a provocation, since it is a convenient and very often used commodity in Netz’ black-and-white world. The concept is, in truth, empty and dangerous: it simply obliterates the selectivity of the textual tradition. 100 Eutocius provides us with two accounts, one in his commentary to De sph. et cyl. 2.4 itself [Heiberg 1910--1915, 3.120.16--126.20], the other in the commentary to Apollonius’ Conica [Heiberg 1891--1893, 2.218.3--220.25]. In the first passage, Eutocius mentions Pappus, Theon, one Arcadius, Nicomachus, and one Heronas as preceding authorities on the subject. All of their treatments are deemed as unsatisfactory, on the ground that they relied on inductive proofs only (i.e., proofs based on numerical examples); and Eutocius expressly claims that he is the first to give a satisfactory exposition of the subject. In the second passage, Eutocius asserts that he has provided a third treatment in his ‘commentary on the first book of Ptolemy’s Syntaxis’. Theon’s account is in his commentary to Almagest 1.13 [Rome 1936, 532.1-535.9]; and it is likely that the exposition by Pappus to which Eutocius refers was contained in a commentary to Alm. 1. One further treatment is contained in the so-called Prolegomena to the Almagest, which Knorr [1989, 155--177] identifies as Arcadius’. An inductive exposition constitutes the whole of a short tract by Domninus of Larissa [see Knorr 1989, 201-207]. Knorr [1989, 157 and n17] takes ‘Heronas in the commentary to the

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other authors entail that the epi phrase was a standard way to refer to complex objects, solids as well as ‘multidimensional’ figures, when they were considered simply qua magnitudes. Register-crossing was largely practiced well before Eutocius; and the epi phrase in mixed contexts was employed well before him too. The conclusion is simply that Eutocius is merely a late actor in a long-standing tradition. In this context, I should regard it as more likely, and at least as the safest historiographical stance, to take as a working hypothesis that Eutocius, on looking at the extant De sph. et cyl. 2.8 aliter, appropriated the epi phrase from a context well known to him—at a time when a still live tradition supported his choice—in his rewriting of Archimedes’ appendix, thereby producing a very effective and concise expression. Netz should adduce more compelling evidence, if he wishes to show that the epi phrase was in the Archimedean original. The occurrences in De sph. et cyl. 2.8 aliter itself are not decisive, as the balance is more contra its authenticity than pro (and, at any rate, Netz should have discussed the issue rather than simply offering his personal beliefs [103: quoted on p. 201, above]). I will just sketch a few of the arguments contra. Eutocius knows the proposition at the place where we too find it, but Archimedes did not typically write alternative proofs. Apart from the one at issue, the Archimedean corpus contains alternative proofs to De sph. et cyl. 1.7 and to De planorum aequilibriis 1.10 and 13. The former is trivial and clearly spurious; the reasons for which the latter two are unanimously regarded as spurious need not detain us here. The second proof of De sph. et cyl. 2.2, although far from trivial, is suspect because it is preceded by a porisma and is inserted into the text in a rather casual way. It is true that our text is the same as that read by Eutocius, but he worked on an Archimedean corpus which had already received conspicuous editorial care. 101 So, Arithmetic Introduction’ [Heiberg 1910--1915, 3.120.22--23] as referring to an otherwise unknown commentator on Nicomachus’ extant tract. But I would suggest a connection with the author of the ‘preliminaries to the arithmetic στοιχε ωσι ’: cf. Knorr 1993, where it is proposed that the author of the Heronian Def., as well as of the lost Preliminaries, was Diophantus. 101 Here a careful study of Knorr 1989 is absolutely mandatory. The case of the Dimensio circuli is blatant, but the De sph. et cyl. was also edited: for instance, Eutocius’ lemmata in his commentary show that he already read a text deprived of Doric forms. Moreover, I am not convinced that the two

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the occurrence of but two alternative proofs in the whole corpus, both in the same treatise, is certainly grounds for suspicion. More generally, the presence of authorial alternative proofs in advanced works must be argued positively, the presumption being always in favor of their being late additions. De sph. et cyl. 2.8 aliter is not a full-fledged proof, but a raw reduction, almost a heuristic preliminary to an analysis. 102 Again, one must argue positively in favor of Archimedes’ willingness to present a proof in such a poor state of elaboration, when making it canonical would have required only a minimal work, and against the claim that it was provided by some later scholiast. (It is highly unlikely that the proof in its actual form is the result of large-scale textual corruptions. 103 ) Moreover, it would be the first instance of a theorematic analysis, a strange beast which was not fashionable form of argument before Hero. 104 On the other hand, several features of the proof remind us of Pappus’ lemmata in Coll. 7: the exclusive and straightforward use of a compound ratio in the first part, 105 the relatively uncon-

102

103

104

105

mentions by Pappus of an Archimedean σ νταγμα refer solely to the Dim. circ. and not to some corpus of writings: cf. Pappus, Coll. 5.6 [Hultsch 1876-1878, 1.314.2] and In Alm. 6.7 [Rome 1931, 254.1]. Eutocius supplies the syntheses of the two parts in which the proposition is divided; they are trivial restatements of the analyses framed as searches for preconditions. A nice feature of the text are the two apparently backwardslooking steps [Heiberg 1910--1915, 1.220.16.18] introduced in the majority of the manuscripts by δι τι. Heiberg emends them to δε , τι as iotacisms. Regularly introduced by τι are the steps at 1.220.21.25 and 222.3. Contrary to what is stated in Netz 2004, 231, the backward-looking character of the steps is not changed by the emendations. Netz [2004, 230] offers only the usual argument based on the principle of sufficient reason: ‘Arguably, no one but the author would dare to introduce such a radical, massive interpolation.’ But any alternative proof is a radical, massive interpolation. In Knorr 1986, 356--360, it is vigorously argued that the very concept of theoretical analysis is Pappus’. The first known instances of complete analysis/synthesis of this sort (e.g., the five alternative proofs to 13.1--5 in the Greek tradition of the Elements) are arguably due to Hero [see Heiberg 1903, 59; Vitrac 2001, 399--400 and 2004, 30--34, 40]. In the Collectio, Pappus offers several alternative proofs that hinge upon repeated application of compound ratios, clearly suggesting that such a systematic exploitation of the method was his own and that he was proud of

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strained language in the repeated backward-looking statements, 106 and the very presence of many such statements. 107 It is difficult not to take De sph. et cyl. 2.8 aliter as a scholastic by-product. Methodological issues Statistics One problem with Netz’ argumentation is his use of statistics. 108 If I am not wrong, statistically-based arguments are used on pages 166 and 168--169 only. They serve in comparing certain features of Archimedes’ and Khayy¯ am’s proofs. Thus, Netz writes: Another major difference [between the proofs] has to do with the technical tools used to achieve those aims, especially ratios and proportions.. . . Archimedes’ solution has many more proportion statements than Khayyam’s. Of Archimedes’ 40 steps, 16 assert proportions (40 percent); of Khayyam’s 35 steps, only 4 assert proportions (11 percent). Instead of proportions, Khayyam more often asserts equalities, and he asserts 8 equalities in his argument.. . . Archimedes’ 16 proportions compare with 9 or 11 equalities: Khayyam’s 8 equalities compare with 4 proportions. [168--169]

it [see, e.g., Coll. 7.68, 74--75, 84, 86, 194, 197, 210, 246, 253, 255, 272]. Compound ratios in the Archimedean corpus are found in De conoidibus et sphaeroidibus 10, 23, 24 [Heiberg 1910--1915, 1.304.13,17; 1.364.12,14,24 and 366.7; 1.368.26 and 370.6, resp.]: these are all standard references to properties of cones well known from the Elements. De con. et sphaer. 31 [Heiberg 1910--1915, 1.432.1 10], De sph. et cyl. 2.4 [Heiberg 1910--1915, 1.190.4, 8, 15, 17], De corp. fluit. 2.10 [Heiberg 1910--1915, 2.388.13 and 390.2] and of course 2.8 aliter [Heiberg 1910--1915, 1.216.15.24] are of a different character, and the last occurrence in De corp. fluit. states that the compound ratio of 2:5 and 5:1 is 2:1. It is difficult to draw any clear-cut conclusions from such disparate data, but only 2.8 aliter is wholly grounded on a clever application of compound ratios. In the extant Greek text of Apollonius’ Conica, compound ratios feature in 25 enunciations; but it is not said that they are all original. 106 Interesting in this respect is the occurrence twice of τι without governing verb at Heiberg 1910--1915, 1.218.3 and 11. 107 Backward-looking statements are a typical mark of editorial intervention: see Vitrac 2001, 41--69. 108 Statistically-based arguments had far greater prominence in Netz 1999a.

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This would be unexceptionable by itself, but we read the following as a conclusion to a number of increasingly generalizing reformulations of the same idea: Within geometrical properties, Archimedes foregrounds proportions, backgrounds equalities, Khayyam foregrounds equalities, backgrounds proportions.. . . It is this inverse ordering of foreground and background that makes the proofs so different, which finally makes us feel that Khayyam’s proof ‘just couldn’t be Greek’—that it is, indeed, already algebra. [171] 109 I will not discuss either the questionable belief that proportions and equalities are ‘geometrical properties’, or the idea that the complement of what looks like Greek mathematics is algebra—beyond observing that a similar ‘statistical assessment’ of the occurrences of equalities and proportions in the first four books of the Elements would probably lead us to conclude that they are pure algebra. More seriously, I offer the following arguments against Netz’ use of statistical argument. ◦ ◦

109

The statistical sample is too restricted: some 40 steps are too few for significant analysis; and only one proposition in an entire treatise is being analyzed. This notwithstanding, a clear-cut pattern (foregrounding versus backgrounding) is inferred from data that at best suggest only a marked tendency. This raises serious epistemological problems. That there are 16 steps in a certain Archimedean proof which assert proportions is a fact about the text which has come down to us, a text which may or may not be the same as Archimedes’ original. But, in any case, this fact should not be used to imply anything about the existence of such a phenomenon as ‘foregrounding’. The latter is at best an interpretation-dependent and statistically based factoid. Even worse, such a factoid requires assuming as an explanatory factor the mental state of a long-deceased mathematician. Such a mental state is forever hidden to us and, at any rate, it can hardly be a matter of inquiry for

Notice that, if we take for granted that Diocles’ original proof is the one in the Arabic text, the proto-algebraist Eutocius changed the text by foregrounding equalities over proportions (contrary to Netz’ belief that he did not rework the proof).

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the historian. In the physical sciences, correlations have replaced causal links, and the existence, for example, of a particle, that is, a fact about the real world, is identified with the existence of a peak in suitable energy spectra. This raises very serious epistemological problems which the practicing physicist simply ignores. In the medical sciences, alleged facts inferred from statistical correlations constitute most of the published material. They are regularly disproved by subsequent statistically-based studies. Informed scholars such as historians of science should handle statistics more carefully. The argument fails to account for the fact that the two proofs are neither statistically homogeneous nor mathematically homologous, as Netz himself had just shown [166]. Khayy¯ am’s proof has 10 steps of preliminary study of cases; Archimedes’ has 6 or 3 (3 steps are already part of the solution), very much as the same proof has ‘9 or 11 equalities’. Such disparities already indicate the fallaciousness of Netz’ statistical approach. Thanks to Elem. 6.16, the borderline between proportions and equalities is permeable enough. Moreover, equalities and proportions obtained from one another by means of 6.16 are not uncorrelated occurrences, and their statistical significance is thereby lowered. The same happens, for instance, when an equality is deduced by transitivity from two other equalities. The third equality is not independent from the first two, and its occurrence does not have the same statistical weight. The same holds for proportions obtained from other proportions by the standard modifications (alternation, ex aequali and so forth). The ‘step’ is somewhat arbitrary as a unit of measurement, even if the high frequency of connectives with sharply defined logical meaning in the prose of Greek geometry makes this choice unambiguous in mo1 st cases. Yet, problems may arise with expressions resuming entire chains of steps in a single clause. A case in point is just the ‘vice versa’ [quoted on p. 184 above] which Netz includes in step 17 of Diocles’ proof. It is plain that the ‘vice versa’ is at least one new step in the proof, so that Netz errs in his reckoning here. But how many steps really are entailed by the ‘vice versa’? Just one step or as many steps as there are in the unfolded deductive chain? Similar problems arise in the case of steps that are understood in the text.

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Anachronisms and over-generalizations One surprising feature of Netz’ book is the author’s polemical stance against the geometrical algebra, in contrast to his unconstrained use of modern conceptions in describing the achievements of ancient mathematicians. Maybe the underlying ideology is that displaying formulas is anathema, 110 whereas speaking in the natural language of Eutocius as an unaware proto-algebraist is no anachronism. In fact, if there is a disturbing feature in the book, it is the continuous search for stretches of text where the practice of Greek or Arabic mathematicians ‘begins to suggest our own modern algebra’ [129]. This perspective is ahistorical and teleologically driven. It amounts to reviving a form of the oldfashioned, long-discredited search for precursors, and, most notably, to maintaining that we are entitled to see more or less concealed forms of algebra in the ancient Greek corpus—the very thesis that Unguru showed to be untenable. A few examples. Archimedes’ choice of using the epi phrase is qualified this way: This choice, more than any other feature of Archimedes’ text, points forwards towards a more algebraic understanding of the problem. Its later appropriation by Eutocius, in particular, would make Eutocius’ text appear truly algebraic. [98] As we have seen, Netz’ willingness to see germs of later developments in ancient authors extends as far as claiming that ‘[Eutocius] has an explicit concept of a functional relation between mathematical objects’ [94], even if the claim is immediately recast less sharply as ‘The sense of a functional relation between points reflects an awareness of [a] symmetry [around the point E], no more.’ The reason given is that Eutocius is still not completely modern. For instance, while he notes one structural property—namely the symmetry around E— he does not note another, namely the monotonic arrangement of the solutions. [94]

110

But the completely algebraic transcription of Diocles’ proof in Toomer 1976, 209--212 is described as ‘a very valuable discussion’ [39n64].

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The painstakingly detailed analysis of the monotonic arrangement of the solutions in the discussion of the διορισμο is one of the most striking features of Apollonius’ De sectione rationis. Is Apollonius completely modern? Two pages later the main theme is resumed without qualifications in this way: ‘In the case of Diocles’ Step 17, Eutocius notices a constant [sic] functional relationship between two areas of a given diagram’ and ‘we see Eutocius stumbling, as it were, across the idea of the function.’ About the same step, a note to the translation says: ‘The author of this passage is aware both of topological considerations, and of a functional relation between variables’ [42n78]. Since our proto-algebraist is Eutocius, Archimedes cannot be said to be aware of anything; but he nevertheless still spreads productive seeds: Besides each [scil. of two conic sections] serving its own specific geometrical function, they also happen to be defined relative to the same lines so that one can—if one wishes to— describe them as functionally interrelated. [28] and ‘he [Archimedes] also happens to produce them in such a way that they can be defined in terms of a functional relation uniting them’ [29]. A move which Eutocius will make, of course. All of this is sheer anachronism; and if this is the way in which mathematics is asserted to have a history, I must confess to having a less forwardlooking conception of what history is. Walter Benjamin once told a beautiful story about the Angel of History: 111 Es gibt ein Bild von Klee, das Angelus Novus heißt. Ein Engel ist darauf dargestellt, der aussieht, als wäre er im Begriff, sich von etwas zu entfernen, worauf er starrt. Seine Augen sind aufgerissen, sein Mund steht offen und seine Flügel sind ausgespannt. Der Engel der Geschichte muß so aussehen. Er hat das Antlitz der Vergangenheit zugewendet. Wo eine Kette von Begebenheiten vor uns erscheint, da sieht er eine einzige Katastrophe, die unablässig Trümmer auf Trümmer haüft und sie ihm vor die Füße schleudert.

111

W. Benjamin, Über den Begriff der Geschichte, thesis ix. See Tiedemann and Schweppenhäuser 1974--1989, 1.697.

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Aestimatio A Klee painting named ‘Angelus Novus’ shows an angel looking as though he is about to move away from something he is fixedly contemplating. His eyes are staring, his mouth is open, his wings are spread. This is how one pictures the angel of history. His face is turned toward the past. Where we perceive a chain of events, he sees one single catastrophe which keeps piling wreckage and hurls it in front of his feet. [Arendt 1969, 257]

The task of the historian is to show that in many instances the Angel is wrong, and that what we perceive as chains of events are really so. But, hélas, there are cases in which such chains remain figments of our perceptive system; and we are left with loose ends scattered in a landscape of rubble. Nor is it permitted to a historian to reconstruct any such chain by taking for granted from the very outset what is its last link: if the Angel itself is looking backwards and bears the vision of such an awful landscape, why should a mere historian, sitting among the ruins, be allowed to interpret the past by looking forwards over the Angel’s shoulders? There is nothing behind the Angel. Overall, in Netz’ view, there are general forces that drive historical change, even if his analysis of these forces is grounded on their concrete, material setting such as the practice of working mathematicians. As he sees it, these general forces get instantiated into particular, even minimal, features, from which the general driving principles are nevertheless recoverable. Conversely, the textual evidence demands interpretation in the light of those forces. What is more, in postulating the reality of these general forces, Netz takes certain seeds or elements of change for granted and as necessary, though one can at best argue (as Netz does not) that they they actually are at variance with canonical practices. Thus, he introduces the historiographic category of the ‘aura’, which enables him to describe the world of Hellenistic mathematics in a way that clashes with the extant evidence, that gives instead obvious prominence to the systematic works of authors like Euclid, Apollonius, and Diophantus. 112 112

Netz might, for instance, have argued about the differences between this kind of systematics and the one at work in Arabic authors. (It is superfluous to point out how such an emphasis on isolated problems is dependent on Knorr 1986, a book regularly omitted in the bibliography.) To the remark of the prominence of systematic works, Netz replies [10] that the prominence

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Netz’ notion of an aura is employed to answer a question which makes sense only from a teleological perspective—and the answer has neither cogency nor historical meaning: If so, we can explain, historically [sic], why Greek mathematics produced problems, and not equations [this question is meaningless outside a teleological perspective]. Seen inside the context of polemical mathematical practice, it is natural [here is where necessity first appears] that Greek mathematical works should [necessity again] aim [notice the language of intentions applied to inanimate objects] to posses an individual aura, in the sense developed above. Mathematical solutions possessing an aura would naturally [necessity again] have the characteristics we have seen in this chapter: solutions that involve configurations of specific lines that have to be brought into a particular order, everything possessing a mathematical meaning through an individual diagram, created especially for the particular solution. Such solutions strike us as ‘problems’ in a real geometrical sense, rather than ‘equations’. [63] So, in Greek hands, the particularization in mathematics of the general aim to provide human creations with an aura produced Greek geometry most naturally and necessarily exactly as it was. That is a charming conceit, but it should hardly be termed an explanation, to say nothing of the presumption that it is a historical explanation. Few historians will be persuaded that such general forces exist—at least not on the strength of anything Netz offers—and many will

given to Euclid (he does not mention either Apollonius or Diophantus) is a consequence of a selection made in Late Antiquity, and amounts to a distortion caused by the pedagogical aims of the late editors. But this, while being a commonly held belief of any historian of ancient mathematics, is already an interpretation of the extant evidence and not a fact about Greek mathematics. The other remark, I believe the work [scil. the Elements] as we know it today may be more systematic than it originally was, due to a Late Ancient and Medieval transformation including, e.g., the addition of proposition numbering, titles such as “definitions” etc. [10] is so naïve as not to deserve any comment.

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conclude that Netz is describing a world that exists more in his mind than in the extant sources. Most importantly, Netz’ idealistic, fundamentally ahistorical perspective leads him to propose clear-cut over-generalizations in order to capture the essence of whole ages of (mathematical) thought. 113 The title of the book makes the author’s penchant for such overgeneralization clear: a case study is taken to be representative of a whole current of thought and the entire building has a minimal linguistic splinter (the epi phrase) as its foundation. In fact, Netz needs something as the central pier of the bridge which he is going to build from Archimedes to Khayy¯ am. Otherwise the gap between the two would simply have been too wide, thereby supporting Klein’s sharp thesis of the great divide. This role is to be played by Eutocius. But we have seen that either Netz’ reconstruction is unwarranted or that the role he assigns to Eutocius is greatly exaggerated (and that in its stead a yet largely unexplored net of connections comes to the fore). It is plain that no historian can do without interpretation. But it is also plain that any inquiry which is not to result in fantasy must at least be conducted in accordance with the shared rules and wellestablished practices of scholarship. It certainly should not proceed by way of an unsatisfactory analysis of the available evidence to grand, unfalsifiable conclusions. bibliography Arendt, H. 1969. ed. and trans. Walter Benjamin, Illuminations: Essays and Reflections. New York. Capasso, M.; Cappelluzzo, M. G.; Concolino Mancini, A.; Falcone, N.; and Longo Auricchio, F. 1976. ‘In margine alla vita di Filonide’. Cronache Ercolanesi 6:55--59. Clagett, M. 1976. Archimedes in the Middle Ages. Volume 2: The Translations from the Greek by William of Moerbeke. Memoirs of the American Philosophical Society 117 (3 parts in 2 tomes). Philadelphia.

113

Over-generalization is a programmatic stance in Netz’ historiographical perspective: see his discussion on pages 188--189.

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Crönert, W. 1900. ‘Der Epikureer Philonides’. Sitzungsberichte der königlichen preussichen Akademie der Wissenschaften zu Berlin 2:942--959. Decorps-Foulquier, M. 1998. ‘Eutocius d’Ascalon éditeur du traité des Coniques d’Apollonios de Pergé et l’exigence de « clarté ». Un exemple des pratiques exégétiques et critiques des héritiers de la science alexandrine’. Pp. 87--101 in G. Argoud, J. Y. Guillaumin edd. Sciences exactes et sciences appliquées à Alexandrie. Saint-Étienne. 1999. ‘Sur les figures du traité des Coniques d’Apollonios de Pergé édité par Eutocius d’Ascalon’. Revue d’histoire des mathématiques 5:61--82. 2000. Recherches sur les Coniques d’Apollonios de Pergé et leurs commentateurs grecs. Paris. Diels, H. 1882. ed. Simplicii in Aristotelis physicorum libros quattuor priores commentaria. Commentaria in Aristotelem Graeca 9. Berlin. Friedlein, G. 1873. ed. Procli Diadochi in primum Euclidis elementorum librum commentarii. Leipzig. Gallo, I. 1980. ‘Vita di Filonide epicureo (PHerc. 1044)’. Pp. 23--166 in I. Gallo ed. Frammenti biografici da papiri II. Rome. Heath, T. L. 1921. A History of Greek Mathematics. 2 vols. Oxford. Heiberg, J. L. 1891--1893. ed. Apollonii Pergaei quae graece exstant cum commentariis antiquis. 2 vols. Leipzig. 1903. ‘Paralipomena zu Euklid’. Hermes 38:46--74, 161-201, 321--356. 1910--1915. ed. Archimedis opera omnia cum commentariis Eutocii. 3 vols. Leipzig. 1912. ed. Heronis Alexandrini opera quae supersunt omnia. Vol. IV: Heronis definitiones cum variis collectionibus, Heronis quae feruntur geometrica. Leipzig. Hogendijk, J. P. 1986. ‘Arabic Traces of Lost Works of Apollonius’. Archive for History of Exact Sciences 35:187--253.

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Høyrup, J. 1997. ‘Hero, Ps.-Hero, and Near Eastern Practical Geometry: An Investigation of Metrica, Geometrica, and Other Treatises’. Pp. 67--93 in K. Döring, B. Herzhoff, and G. Wöhrle edd. Antike Naturwissenschaft und ihre Rezeption. vol. 7. Trier. Hultsch, F. 1876--1878. ed. Pappi Alexandrini Collectionis quae supersunt. 3 vols. Berlin. Jones, A. 1986. ed. Pappus of Alexandria: Book 7 of the Collection. 2 vols. New York/Berlin. Klein, J. 1968. Greek Mathematical Thought and the Origin of Algebra. Cambridge, MA. (English translation of ‘Die griechische Logistik und die Entstehung der Algebra’. Quellen und Studien zur Geschichte der Mathematik, Astronomie, und Physik B3 1934--1936:18--105, 122--235.) Knorr, W. R. 1982. ‘The Hyperbola-Construction in the Conics, Book II: Ancient Variations on a Theorem of Apollonius’. Centaurus 25:253--291. 1986. The Ancient Tradition of Geometric Problems. Boston/Basel. 1989. Textual Studies in Ancient and Medieval Geometry. Boston/Basel. Kristeller, P. O. 1996. Studies in Renaissance Thought and Letters IV. Storia e letteratura: raccolta di studi e testi 193. Roma. Lloyd, G. E. R. 1996. Adversaries and Authorities. Cambridge. Netz, R. 1998. ‘Deuteronomic Texts: Late Antiquity and the History of Mathematics’. Revue d’histoire des mathématiques 4:261-288. 1999a. The Shaping of Deduction in Greek Mathematics. Cambridge. 1999b. ‘Archimedes Transformed: The Case of a Result Stating a Maximum for a Cubic Equation’. Archive for History of Exact Sciences 54:1--47. 2002a. ‘Omar Khayy¯ am and Archimedes: How does a Geometrical Problem Become a Cubic Equation?’ Farhang 14 (39-40):221--259.

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Netz, R. 2002b. ‘It’s not that they couldn’t’. Revue d’histoire des mathématiques 8:263--289. 2004. The Works of Archimedes Translated into English, together with Eutocius’ Commentaries, with Commentary, and Critical Edition of the Diagrams. Vol. 1: The Two Books On the Sphere and the Cylinder. Cambridge. Rashed, R. 2000. ed. Dioclès. Pp. 1--151 in R. Rashed ed. Les catoptriciens grecs. I: Les miroirs ardents. Paris. Rome, A. 1931. ed. Commentaires de Pappus et de Théon d’Alexandrie sur l’Almageste. Tome I: Pappus d’Alexandrie, Commentaire sur les livres 5 et 6 de l’Almageste. Studi e Testi 54. Rome. 1936. ed. Commentaires de Pappus et de Théon d’Alexandrie sur l’Almageste. Tome II: Théon d’Alexandrie, Commentaire sur les livres 1 et 2 de l’Almageste. Studi e Testi 72. Città del Vaticano. Schöne, H. 1903. ed. Heronis Alexandrini opera quae supersunt omnia. Vol. III: Rationes dimetiendi et Commentatio dioptrica. Leipzig. Sidoli, N. 2004. rev. Netz 2004. Aestimatio 1:148--162. Suter, H. 1910--1911. ‘Das Buch der Auffindung der Sehnen im Kreise von Ab¯ u’l Raih¯ an Muhammad-al-B¯ır¯ un¯ı’. Bibliotheca Mathematica III Folge 11:11--72. Tannery, P. 1903--1905. ed. Diophanti Alexandrini opera omnia cum graecis commentariis. 2 vols. Leipzig. Tiedemann, R. and Schweppenhäuser, H. 1974--1989. ed. Walter Benjamin. Gesammelte Schriften. 7 vols. Frankfurt am Main. Toomer, G. J. 1976. ed. Diocles: On Burning Mirrors. Berlin/New York. Unguru, S. 1975--1976. ‘On the Need to Rewrite the History of Greek Mathematics’. Archive for History of Exact Sciences 15:67--113. 1979. ‘History of Ancient Mathematics: Some Reflections on the State of the Art’. Isis 70:555--565. Vitrac, B. 2001. trans. and comm. Euclide. Les Éléments. Vol. 4: Livres XI à XIII. Paris.

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Vitrac, B. 2002. ‘Note textuelle sur un (problème de) lieu géométrique dans les Météorologiques d’Aristote (III.5, 375b16-376b22)’. Archive for History of Exact Sciences 56:239--283. 2004. ‘À Propos des démonstrations alternatives et autres substitutions de preuves dans les Éléments d’Euclide’. Archive for History of Exact Sciences 59:1--44. Wilson, N. G. 1996. Scholars of Byzantium. rev. edn. London.

Math through the Ages: A Gentle History for Teachers and Others by William P. Berlinghoff and Fernando Q. Gouvêa Washington, DC: Oxton House Publishers/The Mathematical Association, 2004. Pp. xii + 275. ISBN 0--88385--736--7. Cloth $39.95

Reviewed by George Gheverghese Joseph University of Manchester [email protected] It was in response to the demands of mathematics teachers to use historical materials from different cultures in their teaching that led to our writing Multicultural Mathematics: Teaching Mathematics from a Global Perspective [see Nelson et alii, 1993]. In that book we attempted to combine three elements: a non-Eurocentric account of the development of mathematics, a justification for a historical and multicultural approach to the teaching and learning of school mathematics, and some suggestions and lesson plans for the teachers. Berlinghoff and Gouvêa have similar objectives. However, their approach is different: history in about 60 pages sprinkled with anecdotes and biographies, followed by 25 sketches, usually between four and six pages long, which explore the development of mathematical concepts and notations that include perennial subjects such as zero, negative numbers, and pi as well as useful surveys of quadratic and cubic equations and the development of geometry. Topical subjects such as Fermat’s Last Theorem and electronic computers are also covered. The book concludes with a useful list of reference texts as well as information from the Internet. It is clear that the task that the authors set themselves, especially in their historical account in the early section of their book, is quite formidable. And the fact that they do succeed in providing a lucid and comprehensive account is the great strength of the book. The multicultural dimension is present in this historical account, but somewhat sparse after the 12th century. This is borne out by the total neglect of Kerala Mathematics of Medieval India in the period between the 14th and 16th centuries. Its importance may be gauged from the fact that it is the first occurrence of what may C 2005 Institute for Research in Classical Philosophy and Science 

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be described as a ‘passage to infinity’ that heralded the emergence of modern mathematics. 1 There are also errors of commission and omission when Indian mathematics is mentioned in the sketches. For 2 ¯ example, Aryabhat . a (oddly described as a ‘Hindu’ mathematician) is supposed to have obtained an implicit value for pi of 62832/20000 around AD 530 [108--109] when the correct date is AD 499, the year ¯ of composition of his masterpiece Aryabhat .¯ıya in which this value first occurs. A similar neglect is also evident in the lack of discussion of Arab mathematics (described as ‘Arabic Mathematics’) after the 12th century. The contributions of Chinese mathematics are understated, especially in the historical section. The underlying approach of the book is to concentrate on the internal development of mathematics and this is reflected in the items that are included in the readings and bibliography. Of course, as a result, the book avoids the historical pitfalls of retrospective privileging, but also at the same time does not emphasize the social-cultural context in which mathematics developed in different societies. The question of cross-cultural mathematical transmissions is also underplayed as result. Thus, on page 14 one reads: Were there contacts between civilizations and did the mathematics of one influence the other(s). For this period (i.e., the BC period) we don’t know. We certainly know more than the authors imply. bibliography Nelson, D; Joseph, G. G.; and Williams, J. 1993. Multicultural Mathematics: Teaching Mathematics from a Global Perspective. Oxford.

1

For a short summary of Kerala mathematics, see pages 286--294 and pages 406--415 of item 78 of their bibliography. 2 To the reviewer, an irritating aspect of this book is the somewhat indiscriminate use of the term ‘Hindu’: on page 88, the Jaina mathematician, Mah¯ av¯ıra, is referred to as a ‘Hindu mathematician’.

Frontinus: De aquaeductu urbis Romae by Robert H. Rodgers Cambridge Classical Texts and Commentaries 42. Cambridge: Cambridge University Press, 2004. Pp. xv + 431. 11 Tables. ISBN 0--521-83251--9. Cloth $130.00

Reviewed by John Peter Oleson University of Victoria [email protected] In AD 97, Sextus Julius Frontinus, one of the most distinguished and influential men in late first century Rome, accepted from Nerva the post of curator aquarum for the city. Not coincidentally, at the same time, he served on a senatorial commission looking for ways to cut the costs of administering Rome and the Empire. In 98, Frontinus was part of the small group of senators who held the constitutional reins of power until the arrival of Trajan, and he may have continued in office as curator aquarum until 100, or even until his death in 103/4. After a life of these and other accomplishments, with ironic modesty, he declared that a funerary monument would be superfluous: inpensa monumenti supervacua est; memoria nostri durabit, si vita meruimus 1 [Pliny, Ep. 9.19.6--8]. Sometime around 98, Frontinus prepared a booklet that may have been entitled De aquaeductu urbis Romae. This commentarius on the water-supply system of Rome is unique among the surviving works of Latin literature, and—although relatively brief (about 12,750 words in length)—it has spawned a bulky modern bibliography. Rodgers has meticulously prepared a critical edition of the text and a commentary that synthesizes all this previous work, and supersedes previous editions and commentaries. It is a shame that Rodgers’ elegant and precise translation, which has now appeared with notes for undergraduate readers in Rodgers 2005, was not included with the edition. 2 The issues involved in the study of the De aquaeductu are many and varied: the text, the form of the booklet, its intended audience 1

‘The expense of a monument is superfluous. My memory will endure if my life has merited it.’ 2 The text is now online at http://www.uvm.edu/~rrodgers/Frontinus.html. C 2005 Institute for Research in Classical Philosophy and Science 

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and purpose, and its utility as a guide to the topography, technology, and administration of the water-supply system of Rome. In consequence, any thorough consideration of the De aquaeductu has to marshal evidence from paleography, Latin literature, Roman history, law, epigraphy, topography, architecture, and hydraulic technology. Perhaps for this reason, there has been no ‘full commentary’ on the text since that of Giovanni Poleni in 1722 [xii]. Despite the challenge, Rodgers orchestrates all these sources in a masterful manner, although he gratefully farmed out to Brunn discussion of the vexing problem of the exact value of the quinaria measure [appendix C]. The first half of the introduction [1--61] provides a concise but thorough introduction to the life and career of Frontinus; the date, content, form, audience, and purpose of the De aquaeductu; and the administrative role of the curator aquarum, along with discussion of the sources, language and style of the booklet. The second half provides a full discussion of the textual tradition and evaluation of the modern editions and commentaries. The text and apparatus occupy 53 printed pages [64--117], while the dense commentary extends to 215 pages [121--336]. There are three short appendices: ‘A. Poggio’s Use of the De Aquaeductu’ [337--338], ‘B. Inscriptions Pertinent to Frontinus’ Text’ [339--341], and ‘C. The Impossibility of Reaching an Exact Value for the Roman Quinaria Measure’ by Christer Brunn [342--346]. Three overly schematic maps display the routes of the aqueducts outside and inside the city [347--349], and 11 Tables [350-359] marshal evidence regarding the lengths of the aqueducts, Roman mathematical fractions, small adjutages relative to the quinaria, pipe sizes, quinariae assigned to the various aqueducts, categories of distribution, castella and distributions, distributions of water outside and inside the city, distributions by regions, and known curatores aquarum up to the time of Frontinus. There is a lengthy bibliography [360--403], followed by indices of literary and epigraphical citations [404--412] and of terms and names [413--431]. The latter index includes Latin words from the text which are discussed in the commentary. As with the other titles in this series, the book has been very carefully edited and nicely produced. I did not notice any typographical errors or incorrect index entries. I found that one bibliographical reference appears to be missing: ‘Wikstrand 2000’, cited as the source of Lewis 2000.

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Rodger’s book is in many ways the labour of a lifetime, undertaken in 1978 [xii], but even then based on a decade of experience with the manuscripts of Peter the Deacon at the Abbey of Monte Cassino. There was something of a renaissance in Frontinus studies over this same period, presenting Rodgers with ongoing challenges, but in the end allowing him to produce a comprehensive and convincing text and commentary. Two editions of the text appeared during these decades—those of Kunderewicz in 1973 and Gonzáles Rólan in 1985—and at least seven translations: Hainzmann (1979), Kühne (1982), Pace (1983), Gonzáles Rólan (1985), Hansen (1986), Evans (1994), and Galli (1997). In addition, during the 1990s, F. Del Chicca was at work on an edition, translation, and commentary on the De aquaeductu, which appeared only a few months after the publication of Rodgers’ book [see De Chicca 2004]. Finally, L’année philologique (online) lists 71 articles, books, and chapters concerned with Frontinus and the De aquaeductu published between 1973 and 2003. I have not noted any substantive omissions from the bibliography prior to 2002, after which, apparently, the book went into production. In fact, other than Del Chicca 2004 and Peachin 2005, no major publications relevant to Rodgers’ topic have appeared since 2002. Rodgers’ Latin text of the De aquaeductu, of course, is the foundation for the rest of the book. It is the first text since Krone’s Teubner edition (1922) to be based on ‘the single authoritative witness’ [xii], the Codex Casinensis 361 (labeled ‘C’). This manuscript, probably based on a Carolingian original, was copied around 1130 at Monte Cassino by Peter the Deacon. Since the eccentric personality of this individual has had some effect on the text of C, Rodgers reconstructs Peter’s life and provides a brief but fascinating portrait of the man [34--44]. He concludes that an editor of Frontinus. . . ought not to ignore the dangers of placing undue confidence in the authority of a manuscript written by a man whose attitude and purposes are always questionable and whose concern for exactitude is never conspicuous. [44] Although the absence of a second independent manuscript tradition simplifies some editorial problems, the archetype itself presents difficulties: errors of transcription, blank spaces, and dreadful handwriting. The humanist scholar Poggio Bracciolini, who hunted this

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codex down in the abbey library in person in 1429, described it as ‘mendosus et pessimis litteris adeo ut vix queam legere’ 3 [33]. Poggio made a copy, which apparently is lost or unidentified; but Rodgers argues convincingly that all 11 surviving 15th-century manuscripts are descendants of C [44--52]. Del Chicca has arrived at the same conclusion. In the end, Rodgers constructs a conservative edition of the text while recording instructive conjectures in the apparatus. We sometimes learn most from those whose views are different, and at risk of being scorned for lack of judgment, I have piously recorded suggestions, implausible in themselves, which have helped me understand the author I study. [61] The preface and the first half of the introduction present in concise form Rodgers’ conclusions about the major issues surrounding Frontinus and his De aquaeductu. The preface provides a stark summary of the issues: Our author sketches the history of Rome’s aqueducts, furnishes a wealth of technical data on supply and delivery, quotes verbatim from legal documents and touches on a variety of other topics incidental to his administrator’s viewpoint. Yet he is not composing a treatise on the engineering of aqueducts, he barely concerns himself with fiscal aspects of management, nor does he compile what might comprise a comprehensive administrative manual of use to a successor. In plain truth we do not surely understand what purpose he might have intended for the De Aquaeductu and the work remains something of an enigma. Nothing quite like it is known, let alone survives, from the ancient world. [xi-xii] Given Frontinus’ elevated social position, active military and political careers, and literary friends such as Pliny the Younger and Martial, there is significant literary and epigraphical evidence for his life [1-5]. Frontinus himself dates his assumption of the office of curator aquarum to 97 [De aqu. c. 102.17], and historical events at the end of the first and beginning of the second century suggest to Rodgers that he continued in office until at least his third consulship in 100, and possibly until his death in 103/4 [7]. Self-referential comments 3

‘full of faults, and written in such dreadful script that I could scarcely read it’.

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about the purpose and utility of the booklet do not conflict with an assumption that it was composed while Frontinus was still in office. Rodgers highlights the ambiguity concerning the intended contents and the literary form of the De aquaeductu [c. 2.3]. In his prologue, Frontinus himself describes the work as a commentarius [c. 2.2--3], a genre not easily defined since it applied ‘to notes and records of many sorts, some of which might remain in the form of data such as lists or compendia, while others might. . . be polished for wider circulation’ [10]. In his ‘Table of Contents’ [c. 3], Frontinus promises to provide data on individual aqueducts [cc. 5--22], data on distribution [cc. 23--86], and legal matters pertinent to water rights, upkeep, damage [cc. 94--130]. Frontinus states that the work originated as a collection of material for self-instruction and personal reference [c. 2.2--3], and he goes well beyond his declared subject matter in providing here and there a ‘critical review of the data he has collected and his administrative analysis of the system’ [9], e.g., chapters 64--76 and 87--93. Rodgers concludes that such a combination is not the rule for a commentarius and, given the rather abstruse subject-matter of water-conduits and water-rights, oversight and upkeep, the De aquaeductu is in fact unique as a specimen of Roman literature, and even perhaps of the ancient world as a whole. [11] He concurs with recent scholarly opinion that Frontinus should not be considered ‘a technical writer’ simply because his booklet included some technical discussions, and that the work is certainly not a manual for construction, maintenance, or even for administration of the water-supply system of Rome. The intended audience was apparently the senatorial class as a whole and the new princeps, Trajan [13]. While the purpose of the De aquaeductu must remain the object of conjecture, Rodgers cites various recent proposals and concludes with an appealing theory by Michael Peachin. 4 The De aquaeductu should be described as a pamphlet, perhaps originally delivered as a speech, addressed to fellow senators and to commercial consumers of public water, announcing the restoration of policies and penalties that had been overlooked for some time by the responsible officials [14].

4

Now published in Peachin 2005.

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Finally, Frontinus’ frequent emphasis on the emperor as both the source of his administrative authority and a colleague in the oversight of the enormous aqueduct system indicate to Rodgers an interest in restoring senatorial claims to this kind of administrative post. After Marcus Agrippa, the administration of the water system had gradually passed in large part to imperial slaves and the emperor’s freedmen, a system parallel to the senatorial curatorial post. The subsequent administrative confusion, in Frontinus’ eyes, had given rise to the abuses he undertook to correct [14--19]. The sources of Frontinus’ commentarius seem clear enough: ‘documents found in the archives of his own and closely related bureaux’ [20]. At several points, Frontinus cites commentarii principum as a source of data [cc. 31.2, 64.1], and he refers to commentarii on the water-supply system kept by Marcus Agrippa [cc. 25.1, 99.3--4]. Legal texts also figured among his sources [20--21]. This mixed bag of source material, along with the diverse ostensible and tendentious motives for composition of the booklet, prescribed a variety of prose styles, from the tabular and formulaic to the rhetorical. Rodgers provides excellent documentation and discussion of these styles, both in the introduction [21--29] and throughout the commentary. Although the introduction will serve for many years as a reliable and comprehensive summation of the major issues surrounding Frontinus and the De aquaeductu, the meat of Rodgers’ book naturally is the commentary. This is also the sort of material that cannot be reviewed in detail, and I must pass over many fascinating discussions. In short, Rodgers does justice to all aspects of this complex text. Although the commentary is not intended to serve as a guide to the surviving remains of the aqueduct system, there is frequent reference to topographical information where it is relevant to interpretation or reconstruction of the text. For example, he discusses the many problems concerning the location of the intake for the Anio Vetus and of the point where it entered Rome [153--156]. Definition of water distribution points is important to an understanding of the system, and there is discussion of this question at pages 199--200 and elsewhere. Rodgers also carefully explicates Frontinus’ occasional forays into rhetorical embellishment [e.g., 121--122, 188--189, 335--336]. Legal and administrative issues naturally constitute an important focus: the availability of statistics and maps [190--191], the provision of water grants [283--284, 288--289], personnel in the curator’s office [173,

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298--302], and verbatim recording of important legal documents and inscriptions [257, 263--264, 318--335]. I suspect, however, that most readers will come to Rodgers’ book in search of information concerning the technological details of the water-supply system, so it is reassuring to find that he provides a reliable account of this aspect of the De aquaeductu. Such topics include the use of arcades as opposed to underground channels [133, 183], the function of distribution tanks [castella: 135--136, 193--194, 219, 289], removal of calcium carbonate deposits from channels [sinter: 252, 309], the purpose of continuous night-time flow (to justify an interpolation: 281--282], the manufacture and classification of pipes [211, 267, and Brunn’s appendix], pressurized pipe systems or inverted siphons [93--94], the setting of an off-take pipe [modulus, calix: 220-221], the nature of gauges to measure flow or volume [mensurae: 197, 228, 231], construction materials [183, 310--311, 315--316], and the composition of construction crews [299--300]. One can occasionally quibble about details. For example, water in a pipeline, particularly a long one, will not rise of its own accord ‘up to the level at which it first issued’ [194], since the coefficient of friction impedes the flow. At other points, Rodgers seems aware of this problem of ‘hydraulic gradient’ [e.g., 219]. Brunn’s appendix provides a clear summary of the evidence that ‘the Romans were not capable of calculating exactly the volume of flowing water’ [346]; and that therefore they (and we) cannot reach an exact value for the quinaria measure. In summary, this long-awaited book does not disappoint in any way. The text is judicious; the introduction and commentary, thorough and engaging. Rodger’s scholarship will appeal to a wide and varied audience and will undoubtedly serve as a firm foundation for future research concerning Frontinus, the water-supply system of ancient Rome, and Roman municipal administration. The dedication to Herbert Bloch, and several warm references to Rodgers’ ‘beloved master’, will strike a chord with other of Bloch’s former students, including this reviewer. bibliography Del Chicca, F. 2004. Frontino, De aquae ductu Urbis Romae. Introduzione, testo critico, traduzione e commento. Rome.

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Peachin, M. 2005. Frontinus and the curae of the curator aquarum. Stuttgart. Rodgers, R. H. 2005. ‘A New Translation of Sextus Iulius Frontinus, De Aquaeductu’. Journal of Roman Archaeology 18:514--534. See also http://www.uvm.edu/~rrodgers/Frontinus.html.

In memoriam Erica Reiner (1924--2005) Assyriology has suffered a great loss with the death of Erica Reiner on 31 December 2005. She was mostly known as editor-in-charge of the Assyrian Dictionary, produced at the Oriental Institute of the University of Chicago, a task which she fulfilled from 1973 to 1996, after having been involved with the project since 1956. Even after her resignation from the editorship, she continued to contribute to the Dictionary in various ways and was anxious to ensure that it would be successfully finished. (In point of fact, the manuscript of the three volumes remaining to be published is completed, and just needs to go through the several stages of printing.) From her personal experience, Erica Reiner wrote a small book on the launching of the printed form of the Assyrian Dictionary, which she entitled ‘An Adventure of Great Dimension’ (2002) after a quotation from Benno Landsberger. The Dictionary does not just list possible translations for each Akkadian word; as is appropriate for a language belonging to a culture of the remote past, it also serves to some extent as an encyclopedia of Assyro-Babylonian culture. Erica Reiner combined expertise in linguistics and a wide knowledge of cuneiform texts not only in her person but also in the entries she wrote or edited for the Dictionary. When editing dictionary articles, she took care to arrange the manuscript so that the various meanings that can be attributed to the word in question were clearly visible to the reader. She published Linguistic Analysis of Akkadian (1966) which applied the current methods of linguistics to this earliest Semitic language for the first time. The Elamite language, still only partially understood, was treated in in her contribution to the Handbuch der Orientalistik (1969). Literature was one of her favorite topics. Apart from general surveys in handbooks, she also wrote a book on Akkadian poetry, Your Thwarts in Pieces, Your Mooring Rope Cut (1985), using a line from an elegiac poem as the title. Here she carefully analyzed poetic texts from different times and places of origin in order to provide examples both of Akkadian literature and of the way she considered appropriate to explain it. C 2005 Institute for Research in Classical Philosophy and Science 

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History of science in the widest sense became a topic of several of her publications when she began to work on an edition of Babylonian omens from events in the sky. In order to evaluate the astronomical significance of these texts, which in general would not have been considered ‘scientific’ by modern scientists, she got the help of David Pingree, a specialist in ancient astronomy (who passed away a short time before her). 1 In the course of time, there appeared four volumes of Babylonian Planetary Omens (1975--2005), the last only a few months before her death. Apart from her own books on the subject, she suggested and supervised the dissertation by Francesca Rochberg on Babylonian lunar eclipse omens (1988). This attraction of the Babylonians to the sky was the topic of another book of hers, Astral Magic in Babylonia (1995), in which she collected the numerous but widely spread attestations in cuneiform texts of attempts to engage celestial forces for earthly purposes. This is only a small selection from the many contributions of Erica Reiner to our knowledge of the ancient Near East. All who knew her will always gratefully remember her. Hermann Hunger Institut für Orientalistik University of Vienna [email protected]

1

See the memorial notice by Kim Plofker and Bernard R. Goldstein in Aestimatio 2 (2005) 70–71.

The Roman Empire at Bay, AD 180--395 by David S. Potter London/New York: Routledge, 2004. Pp. xxii + 762. ISBN 0--415-10058--5. Paper $40.00

Reviewed by Richard Lim Smith College [email protected] Gracing the cover of this erudite ook is an image of the Roman emperor Valerian (ca 200--ca 260) submitting to his captor, the Sassanian King-of-kings Sapor I. Carved into the living rock, this massive relief from Naqsh-i-Rustam in Iran stands as an apt illustration for a book that sets out to examine the critical two centuries during which the Roman world came to be transformed into a notionally postclassical society. Professor Potter asks virtually the same question as that which motivated Edward Gibbon to write his Decline and Fall of the Roman Empire more than two centuries ago, albeit without the latter’s investment in the paradigm of ‘decline and fall’: Why did the Roman imperial state come to be weaker, being less capable of exerting influence abroad and commanding the loyalty of its citizens at home, in AD 395 than it had been 200 years previously? Just as with Gibbon’s magisterial work, The Roman Empire at Bay interweaves discussions of political and military events with a broader survey of developments in the social, cultural, and religious spheres. As the status of the Roman imperial state still constitutes the major topic under consideration, the book gives the most weight to a narrative based on l’histoire des événements as well as to the underpinning structures of Roman imperial administration, economy, and military organization. The material is set out chronologically, with the main chapters in each of the five parts focusing on the major political events and changes in the organization of the Roman imperial state as traced through successive imperial reigns. These chapters serve as a general framework into which the author inserts a series of well-seen remarks regarding the religious, cultural, and intellectual currents of the time. C 2005 Institute for Research in Classical Philosophy and Science 

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Part 1 (chapters 1--2) provides an overall survey of the Roman Empire’s social and political order, its physical and human ecologies, and the beliefs and religious traditions of its peoples. Part 2 (chapters 3--5) lays out the phenomena traditionally referred to as the Crisis of the Third Century, the analysis of which has long served as the gateway to the understanding of the many transformations that took place in late antiquity. Chapter 3 samples important intellectual trends of the early third century: Philostratus’ Lives of Philosophers and Sophists and his Life of Apollonius of Tyana illustrate the appropriation of the classical literary heritage by sophists and grammarians who creatively invoked the past for their contemporary audiences even as they underscored their own self-appointed roles as cultural intermediaries. The second section of the same chapter highlights the importance of Platonist philosophy and its impact on Christian intellectuals such as Origen of Alexandria and—somewhat surprisingly—Hippolytus of Rome. Readers with an especial interest in cultural and intellectual history will find these discussions full of insight but may also rue the absence of any treatment of the period’s distinctive Aristotelianizing and Stoicizing philosophical trends and, more generally, the phenomenon of cultural eclecticism of which they were a part. Part 3 (chapters 6--7) examines the third century proper and commences with the Severan emperors’ efforts at stabilization and reform and with the gradual emergence of a much more centralized Roman state under Diocletian, whose wide-ranging reforms contributed to the creation of the later Roman Empire. A central theme in this section is Rome’s relations with neighboring Sassanian Persia which emerged as its most challenging rival in the east from the third century to the rise of Islam in the seventh. Part 4 (chapters 8--11) focuses on the changes that took place during the reign of Constantine I and his heirs, including the emperor’s political reforms and the impact he had on the institution of Christianity, of which he became an important patron and to which he famously converted in 312. A particularly welcome contribution is to be found in chapter 8, ‘Alternative Narratives: Manichaeans, Christians and Neoplatonists’, which outlines how religious and philosophical figures such as Mani, Eusebius of Caesarea, Lactantius, Plotinus, Porphyry, and Iamblichus constructed their own places in the world through the lenses of their distinctive cosmologies. The author thus seeks to integrate

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a variety of religious and cultural perspectives into the underlying historical narrative through reminding his readers that plural philosophical and religious senses of self and group existed and how such all-encompassing world-views might at times have served as alternative sources of identities for certain inhabitants of the Roman Empire. In his earlier study of the mid-third century Thirteenth Sibylline Oracle, a work of political prophecy that exemplifies a particular local reception of the events that constituted the Third Century Crisis, Potter [1990] demonstrates a fine appreciation of the importance of giving voice to such local narratives, particularly as they stand in some tension with the normative imperial representations of life and social order that historians often rely on to construct their own historical narratives. In the current book, the author’s discussion of ‘alternative narratives’ serves to underpin his broader proposition that the progressive decline in Roman state power was, at least in part, attributable to the ascendancy of competing (religious) constructions of reality that caused individuals and groups to regard themselves less as Roman citizens or subjects than as members of distinctive religious ethnicities. This point is very much in accord with the current scholarly emphasis on the varieties of religious and cultural identities in the Roman world and their importance to how social and other forms of relationships were structured in late antiquity [see Lieu 2004]. Part 5 (chapters 12--14), intriguingly entitled ‘Losing Control’, offers the chief reasons why the author regards the late Roman state as one that effectively became so weak that it was no longer able to decide its own fate nor define what it meant to be ‘Roman’. Using this particular criterion for evaluating the vitality of the Roman state helpfully introduces into the debate the issue of ‘cultural identity’, now a reigning paradigm in scholarly conversations regarding the nature of Roman imperial society. The final chapter, ‘The End of Hegemony, 367--95’, treats conventional topics such as the impact of the battles of Adrianople and Frigidus, the roles played by emperors and imperial courts, and relations between emperors and bishops leading up to the death of Theodosius I. The argument advanced in this book for the progressive decline of Roman state power is ultimately not unlike that which Gibbon

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made in his magnum opus, minus the subtext of Enlightenment anticlericalism. Two factors are highlighted here as the most critical contributory causes of Roman decline: the loss of control over the meaning of what ‘being Roman’ entailed, and the increasing dominance of the Roman state by ‘groups with special interests that did not necessarily benefit society as a whole’ [581]. Yet another consideration is a growing disjuncture between an increasingly centralized state and ‘realities of diversity’ that existed on the ground, a feature that ‘would mold the evolution of the Roman state through the next [fourth] century’ [298]. For its serious efforts to bring in religious and cultural topics, The Roman Empire at Bay remains fundamentally a traditional historical account that takes on the Roman imperial state as its central topic for investigation. While it effectively relates this core aspect of the book to other historiographical objects, the marriage of traditional political history and the history of events with the newer modalities of cultural history and rhetorical analysis still remains an unequal one. The choice of AD 395 as the terminus for the book well illustrates this point. The death of Theodosius I has served many fine works as a reasonable ending point for their narratives and yet many of the social and cultural trends discussed in the current book continued to play out through the following century and beyond. For instance, the suggestion that the rise of ‘alternative narratives’ served to disrupt the unifying sense of Roman cultural identity needs to be substantiated more fully through an examination of the evolution of Christian religious controversies in the fifth century (and indeed beyond), during which a rhetoric of righteous religious violence came to be deployed in a manner that challenged the Roman state’s longheld monopoly over the exercise of legitimate violence [see Gaddis 2005]. That the story of how religious ethnicities came to the fore in the course of conflict and rivalry is not more fully traced, in a book that hints at how alternative identities undermined Roman imperial coherence, subtracts from its overall explanatory power. Still, this is a highly intelligent and well-presented account and the nuanced and sophisticated discussion in its pages do its accomplished author credit. Of particular value to specialist readers are Potter’s thoughtful disquisitions on the operations of historical memory and historiography. Throughout the book, the author helps his readers navigate the intricate scholarly debates over the nature of the

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(mainly literary) evidence [see Potter 1999]. Here one finds useful discussions on the practice of damnatio memoriae, a well-practiced imperial art, whereby opponents and rivals were not just defeated but also written out of history itself. Using the post-Constantinian rewritings of the history of the Tetrarchy, for example, Potter is able to demonstrate that historical revisionism was part of the normative operations of cultural memory in any imperial regime. At the same time, the skillfully constructed narrative and the highly engaging vignettes and examples that have been chosen to accompany it ease novice readers into a vast subject without ever losing their attention. Potter also does not patronize them with seductive generalizations but brings them fully into the enterprise by providing substantial quotations that they can analyze themselves. As the Roman Empire and its fate has increasingly become a foil for expressing contemporary concerns regarding the state of our world both in popular culture and in academe, The Roman Empire at Bay speaks to the wisdom of seeing a state’s power as more than its military and economical might. Any imperial state can be successful in the long term only if it is able also to offer a compelling ‘cultural identity’ that commands not just the loyalty of its inhabitants but also a sense of common enterprise on their part. For assessing the state of health of any imperial society, the traditional political and military narrative is no longer adequate to the task. What is required is a nuanced and varied interdisciplinary approach that takes on all relevant factors, from the seemingly mundane to those that appear hauntingly ‘spiritual’. This impressive book has sought to accomplish just that and its broad scope belies the seemingly straightforward visual image on its cover. The defeat and submission of Valerian to the Sassanian King-of-kings, or any given military reversal for that matter, is less the main explanation of historical change but rather the starting point for posing a searching range of questions such as one finds in this book. bibliography Gaddis, M. 2005. There Is No Crime for Those Who Have Christ: Religious Violence in the Christian Roman Empire. The Transformation of the Classical Heritage 39. Berkeley.

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Lieu, J. M. 2004. Christian Identity in the Jewish and GraecoRoman World. Oxford. Potter, D. S. 1990. Prophecy and History in the Crisis of the Roman Empire: A Historical Commentary on the Thirteenth Sibylline Oracle. Oxford. 1999. Literary Texts and the Roman Historian. London.

books received Aestimatio 2 (2005)

Michael J. B. Allen and James Hankins with William R. Bowen, Marsilio Ficino: Platonic Theology. Volume 6: Books XVII-XVIII. Cambridge, MA: Harvard University Press, 2006. Robert Bork ed. with Scott Montgomery, Carol Neuman de Vegvar, Ellen Shortell, and Steven Walton, De re metallica: The Uses of Metal in the Middle Ages. Aldershot/Burlington, VT: Ashgate, 2005. Alan C. Bowen and Robert B. Todd, Cleomedes’ Lectures on Astronomy: A Translation of ‘The Heavens’ with an Introduction and Commentary. Berkeley/Los Angeles/London: University of California Press, 2004. Gordon Campbell, Lucretius on Creation and Evolution: A Commentary on De rerum natura 5.772--1104. Oxford/New York: Oxford University Press, 2003. Maria Silvana Celentano ed., Ars/Techne. Il manuale tecnico nelle civiltá greca e romana. Atti del Convegno Internazionale Uni˙ Annunzio’ di Chieti-Pescara, 29--30 ottobre 2001. versitá ‘Gd’ Alessandria: Edizioni dell’ Orsi, 2003. Timothy Chappell, Reading Plato’s Theaetetus. Sankt Augustin: Academia Verlag, 2004. Jean Christianidis, Classics in the History of Greek Mathematics. Boston Studies in the Philosophy of Science 2. Dordrecht/ Boston/London: Kluwer Academic Publishers, 2004. Salvo De Meis, Eclipses: An Astronomical Introduction for Humanists. Rome: Istituto Italiano per l’Africa e l’Oriente, 2002. Gary Forsythe, A Critical History of Early Rome: From Prehistory to the First Punic War. Berkeley/Los Angeles/London: University of California Press, 2005. Gad Freudenthal, Science in the Medieval Hebrew and Arabic Traditions. Burlington, VT: Ashgate, 2005. C 2005 Institute for Research in Classical Philosophy and Science 

(online) Aestimatio 2 (2005) 244--247

ISSN 1549–4497

All rights reserved (print)

ISSN 1549–4470

ISSN 1549–4489 (CD-ROM)

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Jim Grant, Sam Gorin, and Neil Fleming edd., The Archaeology Coursebook: An Introduction to Study Skills, Topics and Methods. 2nd edn. London/New York: Routledge, 2005. Bo Gräsland, Early Humans and Their World. London/New York: Routledge, 2005. R. J. Hankinson, Simplicius on Aristotle’s On the Heavens 1.5--9. Ithaca, NY: Cornell University Press, 2004. Robert Hannah, Greek and Roman Calendars: Constructions of Time in the Classical World. London: Duckworth, 2005. Sandra Herbert, Charles Darwin, Geologist. Ithaca, NY: Cornell University Press, 2005. Carl A. Huffman, Archytas of Tarentum: Pythagorean, Philosopher and Mathematician King. Cambridge: Cambridge University Press, 2005. Marguerite Johnson and Terry Ryan edd., Sexuality in Greek and Roman Society and Literature: A Sourcebook. London/New York: Routledge, 2005. Christiane L. Joost-Gaugier, Measuring Heaven: Pythagoras and His Influence on Thought and Art in Antiquity and the Middle Ages. Ithaca, NY/London: Cornell University Press, 2006. Helen King ed., Health in Antiquity. London/New York: Routledge, 2005. Philippa Lang ed., Re-Inventions: Essays on Hellenistic and Early Roman Science. Kelowna, BC: Academic Printing and Publishing, 2005. Christophe Lécuyer, Making Silicon Valley: Innovation and the Growth of High Tech, 1930--1970. Cambridge, MA: MIT Press, 2005. Stanton J. Linden ed., The Alchemy Reader: From Hermes Trismegistus to Isaac Newton. Cambridge: Cambridge University Press, 2003. G. E. R. Lloyd, Ancient Worlds, Modern Reflections: Philosophical Perspectives on Greek and Chinese Science and Culture. Oxford: Clarendon Press, 2004.

BOOKS RECEIVED 2005

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Arnaud Macé ed., Science et philosophie dans l’antiquité. Archives de Philosophie 68.2. Paris: Centre Sèvres/Facultés jésuites de Paris, 2005. Trevor Murphy, Pliny the Elder’s Natural History: The Empire in the Encyclopedia. Oxford: Oxford University Press, 2004. Penelope Murray and Peter Wilson edd., Music and the Muses: The Culture of ‘Mousike’ in the Classical Athenian City. Oxford: Oxford University Press, 2004. Andrea Wilson Nightingale, Spectacles of Truth in Classical Greek Philosophy: Theoria in its Cultural Context. Cambridge: Cambridge University Press, 2004. Gianluigi Pasquale, Aristotle and the Principle of Non-Contradiction. Sankt Augustin: Academia Verlag, 2005. Gianna Pomata and Nancy G. Siraisi edd., Historia: Empiricism and Erudition in Early Modern Europe. Cambridge, MA: MIT Press, 2005. Peter Hanns Reill, Vitalizing Nature in the Enlightenment. Berkeley/Los Angeles/London: University of California Press, 2005. Helen Rhee, Early Christian Literature: Christ and Culture in the Second and Third Centuries. London/New York: Routledge, 2005. Christoph Riedweg, Pythagoras: His Life, Teaching, and Influence. Translated from the German by Steven Rendall. Ithaca, NY/London: Cornell University Press, 2005. Eleonora Rocconi, Le parole delle muse. La formazione del lessico tecnico musicale nella Grecia antica. Rome: Edizioni Quasar, 2003. R. H. Rodgers, Frontinus. De aquaeductu urbis Romae. Edited with Introduction and Commentary. Cambridge: Cambridge University Press, 2004. David Sedley, The Midwife of Platonism: Text and Subtext in Plato’s Theaetetus. Oxford: Clarendon Press, 2004. Robert W. Sharples ed., Philosophy and the Sciences in Antiquity. Aldershot, UK/Burlington, VT: Ashgate, 2005.

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David Shotter, Augustus Caesar. London/New York: Routledge, 2005. David Sider, The Fragments of Anaxagoras: Introduction, Text, and Commentary. Sankt Augustin: Academia Verlag, 2nd edn. 2005. Andrew Smith, Philosophy in Late Antiquity. London/New York: Routledge, 2004. Wolfgang Smith, The Wisdom of Ancient Cosmology: Contemporary Science in Light of Tradition. Foreword by Jean Borella. Oakton, VA: Foundation for Traditional Studies, 2003. Ira Spar and W. G. Lambert edd., Cuneiform Texts in The Metropolitan Museum of Art: Volume 2. Literary and Scholastic Texts of the First Millennium B.C.. New York: Metropolitan Museum of Art/Brepols Publishers, 2005. Brett D. Steele and Tamera Dorland edd., The Heirs of Archimedes: Science and the Art of War through the Age of Enlightenment. Cambridge, MA/London: MIT Press, 2005. Charis Thompson, The Ontological Choreography of Reproductive Technologies. Cambridge, MA/London: MIT Press, 2005. Monica Ugaglia, Modelli idrostatici del moto da Aristotele a Galileo. Rome: Lateran University Press, 2004. Cristina Viano ed., L’alchimie et ses racines philosophiques. La tradition grecque et la tradition arabe. Paris: J. Vrin, 2005. Dorothy Watts, Boudicca’s Heirs: Women in Early Britain. London/New York: Routledge, 2005. Herwig Wolfram, The Roman Empire and Its Germanic Peoples. Translated by Thomas Dunlap. Berkeley/Los Angeles/London: University of California Press, 2005.