X.routledge.ancient.mathematics.sep.2001 [614606]

iANCIENT MATHEMATICS
The theorem of Pythagoras, Euclid’s Elements , and Archimedes’ method to
find the volume of a sphere are all parts of the invaluable legacy of ancientmathematics. Its discoveries and insights continue to amaze and fascinate
the modern reader. But ancient mathematics was also about counting and
measuring, surveying land and attributing mystical significance to thenumber six.
This volume offers the first accessible survey of the discipline in all its
variety and diversity of practices. The period covered ranges from the fifthcentury
BC to the sixth century AD, with the focus on the Mediterranean
region. Topics include:
• mathematics and politics in classical Greece
• the formation of mathematical traditions
• the self-image of mathematicians in the Graeco-Roman period• mathematics and Christianity
• the use of the mathematical past in late antiquity
There are also segments on historiographical issues – such as the nature
of the evidence on early Greek mathematics, or the problem of the authentic
text of the Elements – as well as individual sections on ancient mathematicians
from Plato and Aristotle to Pappus and Eutocius. Fully illustrated with
plates, drawings and diagrams, and with an extensive bibliography, Ancient
Mathematics will be a valuable reference tool for non-specialists, as well as
essential reading for those studying the history of science.
S. Cuomo is a lecturer at the Centre for the History of Science, Technology
and Medicine, Imperial College, London. She is the author of a book on
Pappus, and of articles on Hero and Frontinus.

iiSCIENCES OF ANTIQUITY
Series Editor: Roger French
Director, Wellcome Unit for the History of Medicine,
University of Cambridge
Sciences of Antiquity is a series designed to cover the subject matter of what
we call science. The volumes discuss how the ancients saw, interpreted and
handled the natural world, from the elements to the most complex of living
things. Their discussions on these matters formed a resource for those wholater worked on the same topics, including scientists. The intention of this
series is to show what it was in the aims, expectations, problems and
circumstances of the ancient writers that formed the nature of what theywrote. A consequent purpose is to provide historians with an understanding
of the materials out of which later writers, rather than passively receiving
and transmitting ancient ‘ideas’, constructed their own world view.
ANCIENT ASTROLOGY
T amsyn Barton
ANCIENT NATURAL HISTORY
Histories of nature
Roger French
COSMOLOGY IN ANTIQUITY
M.R. Wright
ANCIENT MATHEMATICS
S. Cuomo

iiiANCIENT
MATHEMATICS
S. Cuomo
London and New York

ivFirst published in 2001
by Routledge
11 New Fetter Lane, London EC4P 4EE
Simultaneously published in the USA and Canada
by Routledge
29 West 35th Street, New York, NY 10001
Routledge is an imprint of the T aylor & Francis Group
© S. Cuomo
All rights reserved. No part of this book may be reprinted or reproduced or
utilised in any form or by any electronic, mechanical, or other means, now
known or hereafter invented, including photocopying and recording, or in
any information storage or retrieval system, without permission in writing
from the publishers.
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging in Publication Data
Cuomo, S. (Serafina)
Ancient mathematics / S. Cuomo.
p. cm. – (Sciences of antiquity)
Includes bibliographical references and index.
1 Mathematics, Ancient. I. Title. II Series.
QA22 .C85 2001
510′.93–dc21 2001019499
ISBN 0–415–16494–X (hbk)
ISBN 0–415–16495–8 (pbk)This edition published in the Taylor and Francis e-Library, 2005.
“To purchase your own copy of this or any of Taylor & Francis or Routledge’s
collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.”
ISBN 0-203-99573-2 Master e-book ISBN

vCONTENTS
List of figures and tables vi
List of abbreviations vii
List of places mentioned ix
Acknowledgements xii
INTRODUCTION 1
1 EARLY GREEK MATHEMATICS: THE EVIDENCE 42 EARLY GREEK MATHEMATICS: THE QUESTIONS 393 HELLENISTIC MATHEMATICS: THE EVIDENCE 62
4 HELLENISTIC MATHEMATICS: THE QUESTIONS 125
5 GRAECO-ROMAN MATHEMATICS: THE EVIDENCE 1436 GRAECO-ROMAN MATHEMATICS: THE QUESTIONS 1927 LATE ANCIENT MATHEMATICS: THE EVIDENCE 2128 LATE ANCIENT MATHEMATICS: THE QUESTIONS 249
Glossary 263
Bibliography 267
Index 287

viFIGURES AND TABLES
Figures
1.1 Plan of Metapontum and surrounding area 7
1.2 The Eupalinus tunnel 101.3 The so-called Salamis abacus 12
1.4 The so-called Darius vase 13
3.1 Doura Europus with city plan and fortifications
in evidence 64
3.2 A surveying instrument from the Fayum, dating from
Ptolemaic times 68
4.1 The medieval and Renaissance tradition of Euclid’s Elements 127
5.1 Roman abacus (replica) 147
5.2 Gearwork from the Antykythera object 1525.3 Sundial found in Pompeii 154
5.4 Reconstruction of a groma from Pompeii, c. first
century
AD 155
5.5 Centuriation from Africa Proconsularis 156
5.6 Fragment 7 of Cadaster Orange A, c. AD 77 156
Tables
1.1 Milesian or Ionian notation 11
1.2 Acrophonic or Attic notation 11

viiABBREVIATIONS
CAG Commentaria Aristotelis Graeca
CMG Corpus Medicorum Graecorum
CIL Corpus Inscriptionum Latinarum
DK H. Diels and W . Kranz (eds), Die Fragmente der
Vorsokratiker , 6th edn, Zurich 1952
DSB Dictionary of Scientific Biography , New York 1969–
IG Inscriptiones GraecaeILS H. Dessau, Inscriptiones Latinae Selectae , Berlin 1892–
1916
LS H.G. Liddell and R. Scott, A Greek-English Lexicon ,
9th edn, rev. H.S. Jones, Oxford 1968
O. Bodl. J.G. Tait, Greek Ostraka in the Bodleian Library and
Other Collections , 1, London 1930
OCD The Oxford Classical Dictionary , Oxford 1996
P . Amh. Amherst Papyri , B.P . Grenfell and A.S. Hunt (eds), 2
vols, London 1900–1
P . Berol. Berlin Papyri
P . British Museum Greek Papyri in the British Museum , F .G. Kenyon and
H.I. Bell (eds), London 1893–1917
P . Cairo B.P . Grenfell and A.S. Hunt (eds), Greek Papyri,
Catalogue général des Antiquités égyptiennes du Musée
du Caire , vol. 10, Oxford 1903
P . Cairo Isidor. The Archive of Aurelius Isidorus , A.E.R. Boak and H.C.
Youtie (eds), Ann Arbor 1960
P . Cairo Zen. C.C. Edgar, Zenon Papyri, Catalogue général des
Antiquités égyptiennes du Musée du Caire , 79, 4 vols,
Cairo 1925–31
P . Col. Zen. W .L. Westermann and L. Sayre Hasenoehrl (eds),
Zenon Papyri. Business Papers of the Third Century B.C.
Dealing with Palestine and Egypt , 2 vols, New York
1934–40

viiiP . Ent. Publications de la Société royale égyptienne de
Papyrologie, T extes et Documents , ed. O. Guéraud, 1,
Cairo 1931–32
P . Fayum B.P . Grenfell, A.S. Hunt, D.G. Hogarth, Fayum Towns
and their Papyri , London 1900
P . Flind. Petr. The Flinders Petrie Papyri , J.P . Mahaffy and J.G. Smily
(eds), Dublin 1891–1905
P . Freib. Mitteilungen aus der Freiburger Papyrussammlung , in
Sitzungsberichte der Heidelberger Akademie derWissenschaften, Phil.-hist. Klasse , 1914–16
P . Herc. Papyri Herculanenses , see M. Capasso, Manuale di
papirologia ercolanese 1991
P . Lips. L. Mitteis, Griechische Urkunden der Papyrussammlung
zu Leipzig , 1, 1906
P . Lond. Greek Papyri in the British Museum , F .G. Kenyon and
H.I. Bell (eds), London 1893–1917
P . Mich. Michigan Papyri
P . Oxy. Oxyrhynchus Papyri , B.P . Grenfell and A.S. Hunt (eds),
London 1898–
P . Rev. Laws B.P . Grenfell, Revenue Laws of Ptolemy Philadelphus ,
Oxford 1896
P . T ebt. T ebtunis Papyri , B.P . Grenfell, A.S. Hunt, J.G. Smyly,
E.J. Goodspeed and C.C. Edgar (eds), 3 vols,
London/New York 1902–38
P . Vindob. Papyrus Vindobonensis
SelPap Select Papyri , Engl. tr. A.S. Hunt and C.C. Edgar, 5
vols, Cambridge, MA/London 1932–34ABBREVIATIONS

ixLIST OF PLACES MENTIONED
1 Agrigentum I8
2 Aosta (Augusta
Taurinorum) D6
3 Bologna D7
4 Canosa (Canusium) F95 Capua F9
6 Carthage I7
7 Cosa F7
8 Cumae G9
9 Enns B9
10 Hippo I6Màlaga off the map
towards I1
11 Marseille (Massilia) E512 Marzabotto E7
13 Metapontum G10
14 Milan D715 Naples G9
16 Orange (Arausius) D4
17 Ostia F8
Osuna (Urso) off the
map towards I118 Paestum G9
19 Pompeii G9
20 Rimini E821 Rome F8
22 Syracuse I9
23 Tarentum G1024 Thurii G10
25 Verona C7
26 Worms A7LIST OF PLACES
MENTIONED
Western Mediterranean

xLIST OF PLACES MENTIONED
1 Alexandria G6
2 Antinoopolis H63 Aphrodisias D5
4 Arsinoe G7
5 Ascalona G8
6 Berenice I9
7 Chersonesos on the
Black Sea A5
8 Coptus I7
9 Cyrene G210 Doura Europus F10
11 Elephantine Island I712 Fayum, region of H6
13 Gadara F8
14 Gerasa F8
15 Hermopolis H6
16 Karanis H617 Krokodilopolis H6
18 Laodicea D5
19 Olbia on the Black Sea A520 Oxyrhyncus H6
21 Palmyra F922 Perga E6
23 Philadelphia H6
24 Ptolemaïs G2
25 Sidon F8
26 Soli E727 Syene I7
28 Thebes in Egypt I7Eastern Mediterranean25

xiLIST OF PLACES MENTIONED
1 Abydus C7
2 Argos G4
3 Arta, Gulf of D2
4 Athens F5
5 Byzantium A9
6 Chaeronea F47 Chalcis F5
8 Chios E7
9 Clazomene E8
10 Cnidus G9
11 Colophon E8
Constantinople, see
Byzantium
12 Cyzicus B713 Delos G6Greece and the Aegean
14 Delphi F415 Didymus F8
16 Elis F3
17 Ephesus F8
18 Epidaurus G4
19 Halieis G520 Heraclea E4
21 Heraclea on
Latmus F8
22 Iasus F8
23 Lamia E424 Magnesia F8
25 Megara F5
26 Mende D527 Messene G428 Miletus G8
29 Naupactus F3
30 Nicaea B9
31 Pergamum D8
32 Phleious F433 Pitane E8
34 Salamis F5
35 Samos F8
36 Smyrna E8
37 Sparta H438 Thasos C5
39 Thebes F5
40 Thera H741 Thessalonika C4
42 T ralles F9

xiiLIST OF PLACES MENTIONED
ACKNOWLEDGEMENTS
The publishers and author would like to thank all copyright holders who
have given permission to reproduce material in their possession in Ancient
Mathematics . While the publishers and author have made every effort to
contact all copyright holders of material printed, they have not alwayssucceeded, and would be grateful to hear from any they have been unable
to contact with a view to rectifying omissions in forthcoming editions.

1INTRODUCTION
INTRODUCTION
Years ago, I was in a taxi somewhere in the British Isles. Being asked, by way
of small talk, what I did for a living, I said I was a historian of ancient
mathematics. When my driver stopped the car, he got a small notebook outof the glove compartment. As it turned out, he liked to keep a record of the
weirdest jobs his customers did, and I (he assured me) could qualify for the
top ten.
That was an instructive taxi ride. If they ask me now, I say I am a teacher
or, better still, a tourist. Besides, I had to admit that the driver had a point.
The history of ancient mathematics is not the most mundane of subjects.T raditionally, it has concentrated on advanced or high-level Greek mathe-
matics – something that combines the wide appeal of a complex subject-
matter with the charm of a pretty abstruse language. Again traditionally,the history of ancient mathematics has tended to focus on internal issues
(textual analysis, links between various texts and authors, heuristics) rather
than trying to relate mathematical practices to their historical contexts.
Which, apart from being questionable from a historiographical point of
view, is not very likely to make converts from neighbouring fields. In sum,to put it in brutal terms, ancient mathematics can be perceived as mostly
incomprehensible and largely irrelevant; a double challenge which this
volume will strive to address.
So, I will cast the net wider than usual, and give an account not just of
the advanced, high-brow practices, but also of ‘lower’ and more basic levels
of mathematics, such as counting or measuring. I think such a choice willpay off in at least two senses: we will achieve a better-balanced picture, with
a full spectrum of activities rather than an isolated upper end; and, since
counting and measuring affect a greater section of the population thansquaring the parabola or trisecting the angle, looking at them will give us
more insight into everyday, everyperson, ‘popular’
1 views of mathematics. I
will avoid unnecessary technicalities. The reader will find sections (denotedby rules above and below) with samples of mathematical texts: they are

2INTRODUCTION
meant to offer a taste of what the more advanced end of the spectrum was
all about. Non-mathematically-inclined readers can just ignore them, while
mathematically-inclined readers who, their appetites whetted, are hungryfor more, can refer to the bibliography. Finally, I will endeavour to relate
mathematical practices to their times and places, and to other contemporary
cultural activities. This will occasionally lead me outside strictly scientificcontexts, so we are going to look for mathematics in some strange places,
including history and theology.
My choices have come at a cost: given the restrictions of space, I had to
leave something out, and the axe has often fallen on ‘advanced’ mathematics.
Thus, some readers may be scandalized to find that I devoted nearly as
much space to Iamblichus as I did to Apollonius of Perga. Then again, theycan learn more about Apollonius’ mathematics from several other books,
which is more than can be said in the case of Iamblichus. This volume aims
to fill a few gaps – it is to be seen as complementary, not as substitutive, ofprevious treatments. I also had to omit pre-Hellenistic Egyptian and Mesopo-
tamian mathematics, both for reasons of space, and of personal ignorance
(I cannot read the primary sources in the original languages).
2 Moreover,
there is a risk that much of what I discuss in the book will not be recognized
as mathematics by some of the readers. I trust that I have used actors’
categories throughout: if a land-surveyor says that mathematics is part ofhis job, that is enough for me to take him into account, whether he did use
mathematics at all or not. The claims are of as much historical value to me
as the actual practice, especially since actual practice is almost impossible toreconstruct, and claims tend to be all we have. Finally, I am very aware that
a lot of what I say is tentative and vague. The usual excuse ancient historians
give for the fuzziness of their work is that the evidence does not allow us tobe certain about anything. It is a very good excuse, and, together with the
space restrictions and the impossibility of being a specialist for the whole
millennium covered by this book, I adopt it as my official excuse whole-heartedly. My aim has been to open paths and stimulate interest, rather than
provide fully thought-out arguments for each topic I discuss or question I
raise.
The volume is organized chronologically – an odd-numbered chapter
dealing with the ‘facts’ is followed by a companion, even-numbered chapterwhich problematizes them. Facts comes with quotes because, even though
my aim in the odd-numbered chapters has been to provide as exhaustive a
survey of the material as possible, selectiveness has crept in. Ideally, odd-numbered chapters should give the reader a full view of what we know
about the mathematics of that period, what sources are available, a sketch
of what is in the sources. The sections are divided according to the type ofevidence, with particularly significant authors (Plato, Euclid, Ptolemy)

3INTRODUCTION
getting individual sections. Each first section of an odd-numbered chapter
is about what I have improperly called ‘material’ evidence: archaeological
data, but also epigraphical, papyrological and, in chapter 7, legal sources.They are ‘material’ only in the sense that I do not have a better term to
distinguish them from literary ones. The even-numbered chapters deal each
with two questions arising from the evidence collected in the previous com-panion chapter. Thus, chapter 4 will explore the issue of the authentic text
of Euclid, who has a whole section devoted to him in chapter 3. Of course,
the questions I ask are only a fraction of the questions that could be askedof the evidence in each case, and they reflect my interests in, for instance,
the historiography of mathematics, the relation between mathematics and
politics, the self-image of mathematicians. The reader will notice that Itend not to answer my own questions in any final way, but then that is
probably a good thing. Hopefully, and this especially if the book is used for
teaching, people will ask their own questions, and give their own answers.
A final disclaimer. My own research deals especially with the mathematics
of later antiquity (the
AD years), with so-called applied mathematics, and
with mathematics in non-mathematical sources. Consequently, I feel evenmore responsible for what I wrote about those topics than I do for the rest
of the book. I have of course relied extensively on the publications, com-
ments, encouragement and criticisms of others, who unfortunately cannotbe blamed for any of my mistakes. In fact, I would like to thank a few of
those brave and generous people: Richard Ashcroft, Domenico Bertoloni
Meli, Paul Cartledge, Sarah Clackson, Silvia De Renzi, John Fauvel, MarinaFrasca Spada, Roger French, Peter Garnsey, Campbell Grey, Victoria
Jennings, Geoffrey Lloyd, Reviel Netz, Robin Osborne, Richard Stoneman,
Bernard Vitrac, Andrew Warwick. And, of course, God bless that taxi driver,wherever he may be now.
Notes
1 I put the term in quotes because, given the state of the evidence, many times it means
views of mathematics held by wealthy, educated, socially respected people who would not
have identified themselves as mathematicians.
2 For Egyptian and Mesopotamian mathematics, see Ritter (1995a) and (1995b). For other
accounts of Greek and Roman mathematics, see Heath (1921), Smith (1951), van derWaerden (1954), Knorr (1986), Dilke (1987), Fowler (1999), Netz (1999a).

4EARLY GREEK MATHEMATICS: THE EVIDENCE
1
EARLY GREEK
MATHEMATICS:
THE EVIDENCE
There is nothing in all the state that is exempt from audit,
investigation, and examination.1
Our story officially begins in the Greek-speaking world around the late
sixth–fifth century BC, during a period also known as the Classical Age of
Greece. Most of what we know about mathematics from this period comes
from Athens. The most powerful and among the richest Greek states at the
time, Athens was also the cultural centre for art, rhetoric, philosophy.
Consequently, it attracted people, including mathematicians, from all over
the Mediterranean. Moreover, Athens was a democracy, in that its citizen
body (comprising only free native adult males), over a period of time andthrough various upheavals, including periods of tyranny, had gained right
to political representation. There was a general public assembly; many public
offices were in principle open to any citizen; the law was administered to asignificant extent by jury courts, filled by lot from the assembly. The citizen
body also manned the army and navy.
Athenian public life, conducted in spaces such as the theatre, the
marketplace, the courts, was particularly lively. The Athenians, both rich
and poor, landowners and cobblers, come across in our literary sources as
particularly fond of debate and discussion about the most various topics:politics, justice, beauty. It has been argued that such a context affected intel-
lectual life in general, and mathematics in particular:
in the law-courts and assemblies many Greek citizens gained exten-
sive first-hand experience in the actual practice of argument and
persuasion, in the evaluation of evidence, and in the applicationof the notions of justification and accountability.
2
Whereas the mathematics we find earlier in Egypt or Mesopotamia consis-
ted of specific exercises with verification of the result but no justification of

5EARLY GREEK MATHEMATICS: THE EVIDENCE
the method employed, Greek mathematics introduced the quest for general
propositions which could be proved in such a way as to be objectively
persuasive. In other words, where the Egyptians had been able to calculatethe volume of a certain cylinder and verify that the result was correct, or at
least suitable for their, usually practical, purposes, the Greeks found the
general formula for the volume of any cylinder and proved why that formulawas right. The public life and political circumstances of Athens and other
contemporary Greek states have been seen as a fundamental factor in creating
this difference.
Although the context for the emergence of early Greek mathematics has
been well described, its details remain fairly obscure, because no strictly
mathematical text has survived from the fifth or fourth century
BC. I say
‘strictly’ mathematical because we do have a good number of texts which
have a lot to say about mathematics, written by people like Plato or Aristotle,
who can be more properly called philosophers. They were less interested inproviding an accurate depiction of contemporary mathematicians and
mathematics than they were in making philosophical points – what they
say cannot be taken as ‘neutral’ information. Also, both Plato and Aristotlehad very strong views about mathematics and its value as a form of
knowledge, and we do not know to what extent those views were shared by
other people, even from their same social and economic background. Otherevidence about our topic is found in historians such as Herodotus, play-
wrights such as Aristophanes, orators such as Lysias, who were not particularly
interested or versed in mathematics, but used it as an example, or for simpleoperations such as counting ships in a naval battle. Herodotus, Aristophanes,
Lysias can be taken as representative of the educated common person, who
was probably numerate but not an expert in mathematics, and, althoughlike Plato and Aristotle they had their own agenda, major mathematical or
philosophical tenets do not seem to have figured on it.
Hundreds of public documents which have survived from this period
are also evidence for the use of mathematics in official contexts such as
compiling inventories, recording tribute or accounting for the expense of
building a temple. Finally, we have some archaeological material, for instance
traces of geometrical town planning or objects which can be identified as
counting-boards or abaci (from the Greek abax, ‘tablet’).
I have already remarked on the problems linked to the nature of the
evidence in the introduction, but let me repeat myself here. The sources I
have collected in this chapter are fragmentary, scattered over time and place,or so concentrated in one place (Athens) as to make any generalization
dangerous. They are biased, unclear and silent on many points (mathematics
in education, for instance) on which one would love to know more. In fact,chapter 2 will be partly devoted to an analysis of later testimonies about

6EARLY GREEK MATHEMATICS: THE EVIDENCE
early Greek mathematics, and we will see that most of what we know (or we
think we know, or we thought we knew) comes in fact from authors who in
some cases write many centuries after the event. Readers will see for them-selves how the work of the historian approaches the fluky clues-gathering
and educated-guessing of the detective.
Material evidence
To begin with, a little exception to our division of evidence by type. I quote
the earliest Greek story on the origins of geometry. According to Herodotus,
the Egyptian king Sesostris
divided the country among all the Egyptians by giving each an
equal square parcel of land, and made this his source of revenue,
appointing the payment of a yearly tax. And any man who wasrobbed by the river of a part of his land would come to Sesostris
and declare what had befallen him; then the king would send men
to look into it and measure the space by which the land wasdiminished, so that thereafter it should pay in proportion to the
tax originally imposed. From this, to my thinking, the Greeks learnt
the art of measuring land ( geometria ).
3
In the use Sesostris made of it, geometry was both part of the procedure
of exacting taxes and a guarantee to his people that the assessment of theamount of tax to be exacted was fair. This story encapsulates the strict connec-
tions between mathematics and practical tasks, between mathematics and
the political sphere, and between mathematics and persuasion; connectionswhich we find again in early Greek instances of geometry in the literal sense.
From as early as the eleventh century
BC, a combination of land hunger,
social unrest and sense of opportunity prompted several Greek communitiesto branch out in the Mediterranean, founding new cities or colonies, among
them Miletus in T urkey, Cyrene in North Africa, Massilia (Marseille) in
France. The colonists aspired not just to a change in location, but to a new
life, ideally with improved status and economic circumstances. The land
newly ‘acquired’ – the presence of indigenous people was variously negotiated– was often to be equally divided between the settlers, ignoring any distinc-
tions there may have been among them in the mother community. For
instance, we learn from one of the foundation accounts of Cyrene (a colonyof Thera) that
[the Therans] are to sail on equal and similar terms, by family, one
son to be chosen † those in their prime and free men from the rest

7EARLY GREEK MATHEMATICS: THE EVIDENCE
of the Therans † to sail. If the settlers establish a settlement, any of
their relatives who later sails to Libya is to receive citizenship and
rights and a share of undistributed land.4
Archaeological surveys of the sites of several Greek colonies (Metapontum,
Chersonenos in the Black Sea, Massilia, perhaps Pharos in Croatia) haverevealed land-division grids which follow regular geometrical patterns (see
Figure 1.1).
Figure 1.1 Plan of Metapontum and surrounding area
(reproduced with permission from Atti dell’Accademia Nazionale dei Lincei,
Notizie degli scavi di antichitá, Supplement to 1975,
8th series, 29 (1980), fig. 10 p. 32)

8EARLY GREEK MATHEMATICS: THE EVIDENCE
It is very difficult to date this kind of evidence, but at least in some cases
the grids seem to be contemporary with the foundation of the city. This
suggests an orderly land distribution, where equality of share was guaranteedby the equality of the geometrical figures (usually rectangles), in its turn
dependent on the accuracy of the relevant measurements.
5
Better attested still is urban geometrical land division. Many ancient
cities, mostly from the fifth and fourth century BC, although there are earlier
examples, exhibit an orthogonal plan – Greek foundations, such as Rhodes,
Agrigentum, Paestum; Etruscan ones such as Marzabotto and Capua, and,in the fourth and especially third century
BC, Roman colonies such as Alba
Fucentia (near Rome) and Cosa. All these cities have straight streets cutting
each other at right angles, and urban blocks shaped like rectangles or squares,sometimes all of the same size. Sometimes two or more streets seem to have
been designated as ‘main’ streets, in that they meet in the middle of the city
area or are recognizably larger than the ‘side’ streets.
As well as having concrete advantages (e.g. facility of orientation), ortho-
gonal city planning was imbued with social and political meanings: in a
geometrically laid-out city, unlike a casually-conglomerated one, everythinghad a proper place. Spatial order suggested orderings of other kinds. Mathe-
matics thus served both as a tool for solving practical tasks, and as a carrier
of meanings, a combination confirmed by Aristotle’s discussion on howbest to organise and arrange a city. He mentioned Hippodamus of Miletus,
the same who invented the art of planning cities, and who also
laid out the Piraeus […] The city of Hippodamus was composed
of 10,000 citizens divided into three parts – one of artisans, one of
husbandmen, and a third of armed defenders of the state. He alsodivided the land into three parts, one sacred, one public, the third
private […] The arrangement of private houses is considered to be
more agreeable and generally more convenient if the streets areregularly laid out after the modern fashion which Hippodamus
introduced.
6
Apart from the port of Athens (the Piraeus), later ancient sources credit
Hippodamus with having planned Rhodes and Thurii in Southern Italy(founded in 443
BC). His provisions for an ideal city went from geometrical
land division to social and economic tripartitions, all probably informed by
a general notion of order and regularity.
Hippodamus may be taken as an early specimen of ‘applied’ Greek
mathematician, together with others whose memory has come down to us
in connection with their buildings or other achievements. The names ofarchitects like Kallikrates or Philo have been preserved in inscriptions which

9EARLY GREEK MATHEMATICS: THE EVIDENCE
link them to the Parthenon and to the Arsenal in Athens, respectively.7 We
can get an idea of the possible mathematical content of the activities of
those men by looking at some features of ancient Greek architecture. It hasbeen argued that Greek temples, from as early as 540
BC (date of the temple
of Apollo at Corinth) were built so that their width was the mean propor-
tional between their length and height. Many fifth-century works, such asthe Parthenon in Athens ( c. 447–420
BC), incorporated refinements ‘to
correct optical illusions which would make a truly regular temple look
irregular’.8 For instance, the axes of the columns incline inwards; the platform
on which the temple stands is not perfectly horizontal but curved; the corner
columns are slightly thicker than the others. It is thought that these
refinements were obtained through a combination of mathematical rules ofproportion and rule of thumb. Also, the dimensions of each element of a
building could be determined proportionally on the basis of a pre-established
standard or ‘building block’, through arithmetical manipulations.
9 Plato
testifies that architecture enjoyed a ‘superior level of craftsmanship over
other disciplines’ because of ‘its frequent use of measures and instruments
which give it high accuracy’.10
Another stunning example of the accuracy that could be achieved by
Greek builders is linked to the name of Eupalinus of Megara.11 The so-
called Eupalinus tunnel, built c. 550–530 BC, is part of a water-supply system
for the city of Samos, and runs for some 1036 m under a mountain.
Excavations have revealed that the tunnel was made by two teams of workers,
who started to dig at the two opposite sides of the mountain and managedto meet in the middle (see Figure 1.2).
When trying to understand how this could be achieved, archaeologists
have focussed on three main problems: where to start the tunnel, how tohave the two ends start on the same level, so as to meet on the same plane,
and how to have the two ends meet in the middle. Most scholars agree that
mathematics was used to solve the second and third problem. In order tostart the two ends on the same level, the builders may have used ‘a lot of
very basic arithmetic (i.e. additions) to measure the height up one side of
the mountain and down the other’, together with sighting instruments and
stakes. Many different solutions to the third problem have been proposed,
some simpler, some more complex, but all basically involving a system oftriangulations, effected with the help of instruments such as the gnomon or
the chorobates , and drawing on geometrical notions about the properties of
similar triangles. We can imagine that Eupalinus knew enough mathematicsto plan the tunnel and instruct his workforce adequately.
But there is more; series of numbers have been found on the walls of the
tunnel, underground. The most complete sequence appears on both stretchesof the conduit; starting from near the entrances, the numbers, situated at

10EARLY GREEK MATHEMATICS: THE EVIDENCE
regular intervals, go like this: 10 20 30 40 50 60 70 80 90 100 10 20 30 …
200 (the south-end sequence stops at 200) 10 20 … 300 (here the north-end sequence stops). It looks as if the builders were assessing the distance
and keeping tabs on their work by means of simple counting.
12
The numbers on the walls of the Eupalinus tunnel are one of the earliest
example of the so-called Milesian or Ionian notation, where numbers are
represented by letters of the alphabet as shown in Table 1.1.
Commonly used in mathematical texts, including papyri, the Milesian
notation gained ground from about the middle of the third century BC and
eventually replaced an earlier system, the so-called acrophonic or Atticnotation, which is attested from the seventh century and seems to have
disappeared by the end of the first century
BC.13 As can be seen in Table 1.2
(and this is what acrophonic means), the signs usually derive from the initialof each number’s name in Greek: e.g. 10= D for deka, 1,000= X for chilioi .
The acrophonic notation included signs not only for numbers, but also
for amounts of money, notably for the obol, one of the main currency unitsin Attica, and some of its subdivisions, for the drachma (worth six obols)Figure 1.2 The Eupalinus tunnel
(reproduced with permission from Rihll and T ucker (1995), p. 406, fig. 18.2)

11EARLY GREEK MATHEMATICS: THE EVIDENCE
and for the talent (worth 6,000 drachmae), and occasionally for areas of
land and grain measures. This fact is a strong reminder of the contexts
where numbers were used: trade and commerce, transactions involving landand produce. The acrophonic system, unlike the Milesian, is not attested in
literary texts: we find it above all in inscriptions, as we have mentioned, and
on counting-boards.
The so-called Salamis table (Figure 1.3) is an example of what a Greek
abacus may have looked like.
14
Similar slabs of stone inscribed with signs for numbers and/or for money
have been found in Thyrium and in the Aegean islands of Naxos and
Amorgos.15 We have no direct information as to how calculations were carried
out on these counting-boards, but reconstructions are possible on the basisof medieval European or modern Chinese equivalents. Pebbles, beans or
purpose-made tokens would have been used as counters, the lines would
have served as markers for tens or hundreds, and the signs would haveindicated the value of counters laid on them.
16
We have an illustration of a somewhat different counting-board from a
fourth-century BC vase (Figure 1.4).
The scene depicted has been interpreted as the payment of tribute to the
Persian king Darius, and the person with a beard is probably a tax collector.
It has been suggested that he is sitting at a normal table with coins scatteredon it, but the presence of letters would rather indicate that this was a a sort
of desktop abacus, with a frame around it and legs to make it into a smallTable 1.1 Milesian or Ionian notation
a = 1 ia = 11 ja = 21 s = 300
jb = 22
b = 2 ib = 12 k = 30 t = 400
c = 3 ic = 13 l = 40 u = 500
d = 4 id = 14 m = 50 v = 600
e= 5 ie = 15 n = 60 w = 700
y = 6 iy = 16 o = 70 x = 800
f= 7 if = 17 p = 80 > = 900
g = 8 ig = 18 < = 90´a = 1,000
h = 9 ih = 19 q = 100´b = 2,000
i = 10 j = 20 r = 200´i or L = 10,000
T able 1.2 Acrophonic or Attic notation
I = 1 DDD = 30 1 = 50 5 = 1 drachma 6 = 5 talents
II = 2 G = 100 2 = 500 I = 1 obol 7 = 10 talents
0 = 5 V = 1,000 3 = 5,000 C = half obol 8 = 100 talents
D = 10 L = 10,000 4 = 50,000 T = 1 talent 9 = 1,000 talents

12EARLY GREEK MATHEMATICS: THE EVIDENCE
Figure 1.3 The so-called Salamis abacus
(marble, 1.5 m × 0.75 m, now in the Epigraphical Museum at Athens,
reproduced with permission from Pritchett (1965))

13EARLY GREEK MATHEMATICS: THE EVIDENCE
table. The man is holding a wax tablet with the abbreviated words ‘A hundred
talents’, probably the record of a transaction.17
The second main source for the acrophonic notation, i.e. inscriptions
containing numbers, is much more abundant, with examples from all over
the Greek-speaking world, although the great majority comes from Attica.Many of these documents are inventories of temple property or building
records; we also have contribution lists; judicial sentences that imply payment
of a fine; records of sale of property; acknowledgements of loans or of therepayments of loans, where one of the parties involved is often the state or
a temple. Given their sheer quantity, I will focus on three examples: a tribute
list, a public account from Athens and a building record from Epidaurus.
The tribute list (425
BC) concerns a re-assessment of the sums that the
Greek cities politically subject to Athens had to pay each year.18 The decree
established the procedure for the periodical assessment of tribute – onethousand jurors (drawn by lot) would join with the council at a certain
time of the year, and work intensively until the quotas to be paid had been
expressed. It also assigned suitable punishment to the magistrates who failedto carry out their duty and contemplated the possibility that the subject
cities may complain or be unable to pay. The inscription ends with a list of
all the cities which were to pay tribute, with the amount due by each. Anexcerpt (imagine this arranged in a tall column):
T ribute from the islands/Paros 30 talents/Naxos 15 talents/Andros
15 talents/[…] Thera 5 talents/Ceos 10 talents/[…] Pholegandros
2,000 drachmae/Belbina 300 drachmae/[…] Myrina on Lemnos
4 talents/Imbros 1 talent/Total of the tribute from the islands 163talents 413 drachmae/T ribute from the Ionian region/[…]Figure 1.4 The so-called Darius vase, found at Canosa
(in Apulia) in Southern Italy, 1.3 m high, now in the
Archaeological Museum at Naples

14EARLY GREEK MATHEMATICS: THE EVIDENCE
Even though the text of the tribute quotas is fragmentary, it is possible
to observe that the cities are grouped by region, and that the total amount
of tribute for each regional group is summed up at the end of each subsectionof the list. Notice that the inscription has no mention of a census of the
resources of each tributary community: the Athenian jurors and councillors
‘shall not assess a smaller amount of any city than it was paying before,unless because of impoverishment of the country there is a manifest lack of
ability to pay more’ (21–2). There is an explicit indication that the assess-
ment has to be made ‘in due proportion’ (18).
19
The second document from Athens (426/5 to 423/2 BC) records the
loans given out by the temple treasuries of Athena and other gods:20
The accountants calculated the following dues in the four yearsfrom Panathenaea to Panathenaea; the treasurers handed over the
following, Androcles of Phlyeus in charge together with thefinancial secretaries … and in charge together with the commanders
Hippocrates of Cholargus and in charge together with Cecropis
being the prytany for the second time, the council had been infunction for four days, Megakleides first was secretary, under the
archon Euthunos, 20 talents; the interest on these produced 5695
talents and one drachma. The second donation under the secondprytany of the Cecropis, the remainder were, seven days into the
prytany, 50 talents, interest to this 2 talents and 1970 drachmae.
[…]
The list is punctuated by sum totals of moneys due, usually at the end
of each year and at the end of each four-year period (in correspondencewith the festival of the Great Panathenaea). The sums are sometimes
written out both in full and in numerical signs, and include the
calculation of the interest that accrues to them. The text also mentionsstate accountants ( logistai ), who seem to have had two main tasks: to
supervise the accounts of public financial bodies, as exemplified by this
inscription, and to supervise the accounts that each public official had
to render at the end of his term of service. In the latter case, they were
drawn by lot from the assembly of all the citizens and formed boards ofthirty; in the former, they were drawn from the more restricted council
of five hundred and formed boards of ten.
21 It is possible that in either
case the actual task of producing the accounts was left to secretaries,while the accountants went through the final result to make sure that
the records were correct. The logistai were able to impose fines or bring
charges of peculation, and were very much in the public eye: on at leastone occasion, a person who had held several public posts had his time

15EARLY GREEK MATHEMATICS: THE EVIDENCE
as accountant singled out, because of the opportunities the job provided
for bribery and illegal profits.22
The third inscription comes from Epidaurus, where extensive building
activities were carried out throughout the fourth and third centuries BC.
Since at least the end of the sixth century, the city had been one of the main
sites for the cult of Asclepius, a healer god, to whose sanctuary people wouldtravel for a cure. The oldest set of inscriptions ( c. 370
BC) is a comprehensive
account of building expenses, who did what bit of the work, and how much
they were paid for it.23 For instance,
To the plasterer Antiphilos, for models, 60 drachmae. Journey
money to Ariston, 10 dr. To Isodamos, for nails for the fencing,7 dr. 3 ob. To Aristonos for bricks, 1 dr. To the herald for Thebes,
2 ob. To Aristaios for pitching the workshop doors, 2 dr. 5 ob. […]
For a lock and key, to Isodamos, 15 dr.
24
A later Epidaurus account contains, like the Athenian documents, recur-
rent recapitulations, with the totals of the sums spent up to a certain point.
Inscriptions involving numbers are important evidence of the role of
mathematics in the public life of several Greek states. The use of inscriptions
as historical sources, however, is not unproblematic. For instance, theirabundance in fifth- and fourth-century Athens was seen as linked to the
presence of a democratic government, to the open character and account-
ability of its administration, to the high level of literacy of its population,and to the relation between all these factors. More recently, it has been
shown that no simple equation can be made between public written docu-
ments and form of government – we have inscriptions from non-democraticstates, including those in other parts of Greece – and that Athens remained
a largely oral culture, where only very few people could read and write.
Thus, revised interpretations of the significance of inscriptions in the classicalage try to recapture the complexity of their meaning, their ‘utilitarian’
(recording and publishing data, providing information, allowing public
scrutiny) as well as their symbolic functions: commemorative, celebratory,
religious, intimidatory, even magical. Indeed, ‘there were certainly cases in
fifth-century Athens when the symbolism of the record was of as muchimportance as the ritual recorded’.
25
With particular reference to the inscriptions we have discussed, then, we
can speculate that, if indeed at least some accountants were chosen by lot,your average Athenian citizen must have been sufficiently numerate to super-
vise, if not produce, accounts. Moreover, and perhaps more significantly,
these documents testify that mathematics had a significant public presence.To quote Robin Osborne again:

16EARLY GREEK MATHEMATICS: THE EVIDENCE
The accounts […] preserved, and made visible to all, rituals of the
sort which the Council so frequently oversaw, rituals of counting
and handing over in which what was seen to be done was far moreimportant than the intrinsic significance of the act.
26
Although it would have often been impossible actually to check all the
figures on an inscription, they were out there in the public domain. I have
emphasized that in several cases lists of figures were punctuated by subtotals
– a way for the maker of the document to keep count, but also to makereading easier for a potential public. If anyone at all went through these lists
to check that everything added up, expressing subtotals would have been an
indication that the administrators of the state shared the process of account-ing with the rest of the citizen body. We can also infer that people, even
those who were not fluently numerate, were familiar with both the utilitarian
and symbolic functions of mathematics, and that they associated it with anumber of public areas: the administration of the state, its financial wheelings
and dealings, foreign policy, war, religion, owing, distributing and receiving.
Historians, playwrights and lawyers
Herodotus is the earliest literary source we will consider. Of course, mathe-
matics existed in Greece, and in Greek literature, before him. For instance,
Homer compared a thick battle engagement with two men who, measuring-
rod in hand, fight about the boundary of their common field, and ‘in anarrow space contend each for his equal share’, while Theognis associated
truthfulness and justice with measuring instruments – a man sent as envoy
must ‘take care to be more true than scale or rule or lathe’.
27 But Herodotus,
as we have seen with his account of the Egyptian origins of geometry, is the
first to report on the history of mathematics. His work also contains a good
deal of counting: he counted the days in a man’s life, the generations ofEgyptians, the tribute due to the Persian king Darius, the length of Darius’
journey to the West (which he undertook with an aim to conquering Greece),
the troops and various armaments of Darius’ successor Xerxes, and the
soldiers in the Greek army.
28 All these figures have at least two purposes:
one, they reinforce some impressions Herodotus wants to convey (that theEgyptians are a really ancient people, that the Persians are much more
numerous, richer and more powerful than the Greeks, who nevertheless
manage to win against the odds); two, they add to Herodotus’ constructionof himself as an accurate reporter of facts. From the point of view of the
present volume, they are again evidence that mathematics was seen as a
means of rendering an account of things that were of common interest, andaffected the body politic. Accurate figures, where the accuracy was guaranteed

17EARLY GREEK MATHEMATICS: THE EVIDENCE
by carrying out the operation in full detail, appeared more trustworthy
than just qualifying the tribute of ‘huge’ or the troops of ‘countless’.
A rather different type of story about the origins of mathematics comes
from the first playwright we will look at, Aeschylus or whoever is the author
of Prometheus Bound . The Titan, chained to a rock at the edge of the world,
his liver perpetually eaten by an eagle, explains the reason for his punishmentthus:
PROMETHEUS : For the suffering race of humankind [Zeus] cared nothing,
he planned to wipe out the whole species […] I saved humanity from
going down smashed to bits into the cave of death. For this I’m wrenched
by torture: painful to suffer, pitiable to see. […] Humans used to foreseetheir own deaths. I ended that. […] What’s more, I gave them fire.
CHORUS : Flare-eyed fire!? Now! In the hands of these things that live and
die!?
PROMETHEUS : Yes, and from it they’ll learn many arts ( technai ) […] hear
what wretched lives people used to lead, how babyish they were – until
I gave them intelligence, I made them masters of their own thought.[…] they knew nothing of making brick-knitted houses the sun warms,
nor how to work in wood. They swarmed like bitty ants in dugouts in
sunless caves. They hadn’t any sure signs of winter, nor spring flowering,nor late summer when the crops come in. All their work was work
without thought, until I taught them to see what had been hard to see:
where and when the stars rise and set. What’s more, I gave them number-ing, chief of all the stratagems. And the painstaking, putting together
of letters: to be their memory of everything, to be their Muses’ mother,
their handmaid! […] In a word: listen! All the arts are from Prometheus.
29
Mathematics, like the other arts an offspring of Prometheus, is, like him,
associated with craft, resourcefulness and even trickery, the ability to prevailin a conflict against a stronger adversary. At the same time, it epitomises the
passage of humankind from a feral state to civilization, it is both a sign and
a cause of its newly-acquired intelligence and self-awareness. The ‘chief of
all the stratagems’ is depicted in action in this passage from a comedy staged
in 422
BC:
And not with pebbles precisely ranged, but roughly thus on your
fingers count the tribute paid by the subject states, and just considerits whole amount; and then, in addition to this, compute the many
taxes and one-per-cents, the fees and the fines, and the silver mines,
the markets and harbours and sales and rents. If you take the totalresult of the lot, ’twill reach two thousand talents or near. And

18EARLY GREEK MATHEMATICS: THE EVIDENCE
next put down the Justices’ pay, and reckon the sums they receive
a year: six thousand Justices, count them through, there dwell no
more in the land as yet, one hundred and fifty talents a year Ithink you will find is all they get.
30
The author, Aristophanes, was one of the most successful playwrights of
his day. The passage above is like a comically distorted version of the inscrip-
tions we have seen in the previous section: the character in the play even
refers to a tribute list. It sends up a situation where your average citizen readthrough the inscription, rehearsed the calculations in his head or on his
fingers, and had the enlightening and very democratic experience of finding
out where the money was going. Here, to the people’s jury courts, the targetof satire in this particular play. In general, Aristophanes’ works reflect contem-
porary concerns, ranging from bitter commentaries on the long war between
Athens and Sparta, to irreverent portraits of magistrates, philosophers, poetsand mathematicians. In the Birds , performed in 414
BC, a new city has to
be founded from scratch. The main character, Peisthetaerus, is visited by
various people who offer their services.
Enter
METON : I come amongst you –
PEISTHETAERUS : Some new misery this! Come to do what? What’s your
scheme’s form and outline? What’s your design? What buskin’s on your
foot?
METON : I come to land-survey this Air of yours, and mete it out by acres.
PEISTHETAERUS : Heaven and earth! Whoever are you?
METON : Whoever am I? I’m Meton, known throughout Hellas and Colonus.
PEISTHETAERUS : Aye, and what are these?
METON : They’re rods for Air-surveying. I’ll just explain. The Air’s, in outline,
like one vast extinguisher; so then, observe, applying here my flexiblerod, and fixing my compass there, – you understand?
PEISTHETAERUS : I don’t.
METON : With the straight rod I measure out, that so the circle may be
squared; and in the centre a market-place; and streets be leading to it
straight to the very centre; just as from a star, though circular, straightrays flash out in all directions.
PEISTHETAERUS : Why, the man’s a Thales!31
Meton combined what are the now separate skills of an engineer, an
astronomer and (judging from his role in the play) a land-surveyor, town-
planner or architect. We know from later sources that he was responsiblefor a reform of the Athenian calendar, that he set up an instrument for the

19EARLY GREEK MATHEMATICS: THE EVIDENCE
observation of solstices on the Pnyx (the hill in Athens where the general
assembly met), and that he built a fountain and/or a time-keeping device
and/or a water-operated time-keeping device in the Agora, on a hill calledColonus Agoraios (hence the reference in the passage). Meton must have
been famous enough to be recognized by the general public attending the
theatre; the same goes for Thales, a sixth-century
BC mathematician and
philosopher from Miletus with whom he is ironically equated. The audience
must have also been able to understand the comic allusion to the problem
of squaring the circle, which was considered impossible to solve, or absurd.
Stretching our interpretation, we could suggest that Meton was made
into a figure of fun because he engaged in a type of mathematics which not
everybody recognized as relevant or important, or which was simply toosophisticated for everybody to understand. In another play where
Aristophanes lampoons the new intellectual education made popular by
the sophists, the seminal question is asked:
STREPSIADES : And what’s this?
STUDENT : Geometry.
STREPSIADES : So what’s that useful for?
STUDENT : For measuring land.
STREPSIADES : You mean land for cleruchs?
STUDENT : No, land generally.
STREPSIADES : A charming notion! It’s a useful and democratic device.32
A cleruch was a settler-soldier who received land as a reward for his service.
A pun is lost in translation: the expression ‘land generally’ also means ‘land
for everybody’, as if a land distribution bonanza was intended, but all thestudent is saying is that the geometry they do at his school concerns itself
with general and abstract questions: squaring the circle perhaps, solving the
economic problems of the multitude definitely not.
A contemporary of Aristophanes and together with Herodotus the greatest
historian of the classical period, Thucydides also makes interesting use of
numbers. He describes how, in a situation of extreme emergency during the
Peloponnesian War ( c. 431
BC), Pericles, the Athenian leader, tried to persuade
his fellow citizens of a certain course of action. Part of his argument consistedof a long list of revenues, which once again reads like one of the financial
inscriptions:
And he bade them be of good courage, as on an average six
hundred talents of tribute were coming in yearly from the allies
to the city, not counting the other sources of revenue, and therewere at this time still on hand in the Acropolis six thousand

20EARLY GREEK MATHEMATICS: THE EVIDENCE
talents of coined silver (the maximum amount had been nine
thousand seven hundred talents, from which expenditures had
been made […]).
Next, Pericles quantified defence structures (‘the length of the Phalerian
wall was thirty-five stadia to the circuit-wall of the city, and the portion ofthe circuit-wall itself which was guarded was forty-three stadia’) and human
resources (‘the cavalry […] numbered twelve hundred, including mounted
archers, the bow-men sixteen hundred, and the triremes that were sea-worthythree hundred.’).
33 Those items would not have normally been on an account,
but including them as quantifiable resources adds to the sense that collective
counting, reckoning one’s strengths and taking decisions consequentlyamounts to acquiring self-awareness as a community. Another episode from
the same war involved the people of Plataea, who found themselves allied
to Athens but closely threatened by Spartan troops, whose envoy,
endeavouring to reassure them […], said: ‘You need only consign
the city and your houses to us, […] pointing out to us the boundariesof your land and telling us the number of your trees and whatever
else can be numbered; […] as soon as the war is over we will give
back to you whatever we have received; until then we will hold itall in trust’.
34
T rust whose only support seems to have been a list of whatever could be
numbered – once again, something that would have looked like an inventory
inscription.
Mathematics appeared in yet another public context: the legal courts. In
Athens a great number of cases were tried in front of a jury drawn by lot
from the entire citizen body or those of them who volunteered. The number
of people in the jury varied from case to case, with the final verdict reachedby majority of votes. Tokens – solid ones if the juror found for the defendant,
perforated tokens otherwise – were collected by attendants into vessels,
emptied out onto an abacus and counted. The abacus seems to have been
of a special type, with holes on it to host the tokens; in particular it had as
many holes as there were jurors, to make sure that everybody’s vote wasaccounted for. One of our main sources for this procedure, Aristotle, specifies
that this was so that the tokens would be ‘set out visibly and be easy to
count, and that the perforated and the whole ones may be clearly seen bythe litigants’.
35 T ransparency of procedure was signified by a shared
calculation, by doing mathematics together.
The legal courts were of course an arena for the performance of speeches,
many of which are extant. No professional lawyers were supposed to exist

21EARLY GREEK MATHEMATICS: THE EVIDENCE
(being paid to give a defence or accusation speech was illegal), but speeches
were often written by third parties with outstanding rhetorical skills. Since
the audience was, like that for Aristophanes’ plays, a general one, theirlanguage and allusions needed to be sophisticated but comprehensible,
persuasive but also somewhat pandering to common tastes.
A great many court cases were about financial matters, so that the
relevant speeches contain accounts, inventories, and other evidence
involving numbers.
36 An example is Lysias’ On the Property of Aristophanes ,
delivered around 388–87 BC. Some relatives of the defendant had been
condemned to death and their property confiscated. The amount
confiscated was found to be much less than expected, and the defendant’s
father was accused of having hidden property. When he died, the chargewas inherited, as it were, by the defendant, his only son. He had to make
the case that the property in question was actually much less than the
state had anticipated – a case, basically, about quantifiable value. He (orrather Lysias, who wrote the speech for him) started by establishing the
good character of his father, who, far from witholding property from the
state, was a generous benefactor: ‘Yet, gentlemen, my father in all his lifespent more on the state than on himself and his family, – twice the amount
that we have now, as he often reckoned in my presence’.
37 A ‘public’ account
whose only witness was the defendant – but, if that sounds unconvincing,an itemized account is also produced:
Now, Aristophanes had acquired a house with land for more than
five talents, had produced dramas on his own account and on his
father’s at a cost of five thousand drachmae, and had spent eighty
minae on equipping warships; on account of the two, no less thanforty minae have been contributed to special levies; for the Sicilian
expedition he spent a hundred minae, and for commissioning the
warships […] he supplied thirty thousand drachmae to pay thelight infantry and purchase their arms. The total of all these sums
amounts to little short of fifteen talents.
38
Many of the expenses are in fact for the benefit of the community, from
subsiding the theatre to helping out in the Peloponnesian War. The jurorsare taken through the calculations, so that they can not only appraise the
actual amount of Aristophanes’ father’s generosity, but also appreciate his
civic commitment.
In some cases, the audience could be faced with competing calculations.
In the speech Against Diogeiton , also written by Lysias around 400
BC, a
guardian was accused by his wards of having robbed them of their fortune.According to the plaintiff, Diogeiton had been given

22EARLY GREEK MATHEMATICS: THE EVIDENCE
five talents of silver in deposit; […] seven talents and forty minae †
and two thousand drachmae invested in the Chersonese. [The
father of the wards] charged him, in case anything should happento himself, to dower his wife and his daughter with a talent each
[…]; he also bequeathed to his wife twenty minae and thirty staters
of Cyzicus. […] He was killed at Ephesus […] [Diogeiton] gave[the dead man’s wife] in marriage with a dowry of five thousand
drachmae, – a thousand less than her husband had given her. Seven
years later the elder of the boys was certified to be of age; […]Diogeiton summoned them, and said that their father had left
them twenty minae of silver and thirty staters.
39
Diogeiton’s deceit was first exposed by the widow, who claimed that he
had received five talents in deposit, and
convicted him further of having recovered seven talents and four
thousand drachmae of bottomry loans, and she produced the
record of these […]. She also proved that he had recovered ahundred minae which had been lent at interest on land mortgages,
besides two thousand drachmae and some furniture of great
value.
40
More figures follow:
Gentlemen of the jury, I ask that due attention be given to this
calculation […] [Diogeiton] has had the face to […] make out a
sum of seven talents of silver and seven thousand drachmae asreceipts and expenses on account of two boys and their sister during
eight years. So gross is his impudence that, not knowing under
what headings to enter the sums spent, he reckoned for the viandsof the two young boys and their sister five obols a day; for shoes,
laundry and hairdressing he kept no monthly or yearly account,
but he shows it inclusively, for the whole period, as more than a
talent of silver. For the father’s tomb, though he did not spend
twenty-five minae of the five thousand drachmae shown, he chargeshalf this sum to himself, and has entered half against them. Then
for the Dionysia […] he showed sixteen drachmae as the price of a
lamb […] for the other festivals ad sacrifices he charged to theiraccount an expenditure of more than four thousand drachmae;
and he added a multitude of things which he counted in to make
up his total, as though he had been named in the will as guardianof the children merely in order that he might show them figures

23EARLY GREEK MATHEMATICS: THE EVIDENCE
instead of the money […] I will now base my reckoning against
him on the sum which he did eventually confess to holding.41
Diogeiton did produce an account – only, such a blatantly inflated
one that it was possible to unmask him by going through the numbers
verifying at each step their implausibility. Accurate accounting is identifiedby the speaker with good behaviour – bad accounting, in all its details
concerning crucial moments of Athenian civic life (the festivals, burying
one’s father, coming of age, dowries), is associated not only with greedand lack of family feeling, but also with transgression of religious and
social rules and, in other cases, with corruption and/or undemocratic
political leanings. Aeschines, in Against Ctesiphon (330
BC), stressed the
importance of the Athenian law according to which every magistrate had
to give account: ‘In this city, so ancient and so great, no man is free from
the audit who had held any public trust’.42 Being prepared and willing to
give an account showed not just financial honesty, but also political
transparency, the recognition that one’s actions could be checked,
scrutinized, and discussed by the public.
The identification of jurors and accountants was again invoked by
Demosthenes, in a speech delivered against Aeschines as part of the same
case against Ctesiphon.
I shall prove without difficulty that [Aeschines] has no right to ask
you to reverse that opinion – not by using counters, for politicalmeasures are not to be added up in that fashion [he refers to a
logismos , a calculation], but by reminding you briefly of the several
transactions, and appealing to you who hear me as both the wit-nesses and the auditors ( logistai ) of my account.
43
Peppered with puns, the plea Demosthenes was making in his defence was
for the jurors to go through the steps of his argument the way they would
go through the steps of a calculation, except that his speech did not involve
any counters or abacus. Compare Aeschines’ response:
But if such a statement as I have just made, falling suddenly on your
ears, is too incredible to some of you, permit me to suggest how
you ought to listen to the rest of my argument: When we take our
seats to audit the accounts of expenditures which extend back along time, it doubtless sometimes happens that we come from
home with a false impression; nevertheless, when the accounts
have been balanced, no man is so stubborn as to refuse, before heleaves the room, to assent to that conclusion, whatever it may be,

24EARLY GREEK MATHEMATICS: THE EVIDENCE
which the figures themselves establish. I ask you to give a similar
hearing now.44
The jurors were called upon to be not just witnesses, but also calculators
– in principle, a logistes could tell with absolute certainty whether the account
presented to him worked out or not, whether everything added up. It wasthis kind of persuasiveness, mathematical persuasiveness, that both speakers
wished to claim for their arguments. In chapter 2, I will further discuss some
of the implications of the evidence collected in this section. For now, let meobserve that accounts – collective, public counting – were a pervasive practice
in classical Athens. We find them not only in inscriptions, but also in various
genres of literature. Along with its practical functions, public counting wasassociated with political accountability, and in fourth-century legal speeches
it seems increasingly to symbolize the role itself of the Athenian citizen.
Plato
It is very difficult to put Plato’s philosophy in a nutshell, both because it is
extremely complex, and because in different works he sometimes says rather
different things on the same question. He wrote in the form of dialogues,
with his teacher Socrates usually cast as the main speaker and mouthpieceof his views. Each and every part of Plato’s thought has been the object of
much debate ever since antiquity, although perhaps none as much as the so-
called theory of forms. Generalizing and simplifying, this is the belief thatthere are two levels of reality, corresponding to two levels of knowledge of
that reality: the sensible world which we live in, and of which we have
knowledge primarily through the senses, and a world of changeless entities,called forms, of which we have true and certain knowledge through our
intellect. Things in the world are imperfect reproductions of the forms,
which constitute ‘thing in themselves’, i.e. not this or that horse, but the
horse, not this or that act of justice, but justice itself.
Most of Plato’s work is devoted to the discussion of particular forms
(Justice, Love, Pleasure, the Good), to the relation between forms and things
in the world, and to how true knowledge can be attained. Mathematics was
of special interest to him, because, while it dealt with sensible things andwas employed in fields such as architecture or the military, it also concerned
itself with the general, the abstract, the unchanging. In a calculation, for
instance, one operates not just with three oxen or five fingers, but with‘three’ and ‘five’, which have their own characteristics (e.g. they are both
odd numbers) quite independently of the objects they are assigned to. Again,
one measures a specific triangular field, but determines how the area of ageneral triangle is to be found. Plato claimed that, by practising mathematics,

25EARLY GREEK MATHEMATICS: THE EVIDENCE
the mind got used to turning from sensible particulars towards abstract
concepts, and was thus in a better position to gain knowledge of the forms.
Moreover, the results of mathematics were convincing – everybody wasprepared to believe that three and five made eight, whereas opinions about,
say, the good state were bound to differ. Mathematics thus provided an
example of persuasive discourse. There is also a third aspect: in one of hisdialogues, the Timaeus , Plato described how the universe and all the things
in it, including humans, came to be, and how the physical elements were
shaped following a geometrical pattern (more details below). He thus positedmathematics at the very foundations of the world around us.
Now, there are two basic ways in which Plato can be used as a source for
the history of early Greek mathematics: as a philosopher and as a historian.In the first capacity, he engaged with questions such as what kind of objects
numbers and geometrical figures are, what is their relation to forms on the
one hand and things in the world on the other, what is the value ofmathematical knowledge. Plato addressed all these issues, but did not answer
them unequivocally. According to some interpretations, he considered
mathematical entities to be forms; then again, he may have seen them asintermediate between the realm of forms and that of sensibles; some evidence
can also be adduced that he thought of all forms as mathematical entities,
or that mathematical entities were the only true forms there were. His viewof mathematical knowledge, on the other hand, seems more straightforward:
he gave it a crucial role in the training of a good philosopher, because of the
above-mentioned habit it imparts, of turning the mind to the general andthe abstract. At the same time, mathematics for him was not perfect, because
it still relied on undemonstrable principles (e.g. geometers assume ‘the odd
and the even, the various figures, the three kind of angles […] as if they wereknown’), and it still concerned itself with objects (e.g. triangles drawn on a
whiteboard). If one visualized knowledge as a line divided into segments
(an analogy Plato adopts in the Republic ) with total ignorance at one end
and perfect knowledge at the other, disciplines such as mathematics would
fall immediately short of perfect knowledge, in a segment labelled dianoia ,
argumentative reasoning.
45
In his capacity as a historian, Plato provided information about mathe-
maticians contemporary to, or earlier than him, and referred to issues thatwere being discussed, had been solved or constituted common knowledge.
Sometimes he reported whole mathematical passages to make a philosophical
point, thus giving us insight into what his readers would have been expectedto know or understand. In this book, I will only deal with this second aspect
of Plato as a source – I will not discuss his philosophy of mathematics at
any greater length than I have done above. That said, I am very aware thatit is not possible sharply to separate Plato the philosopher from Plato the

26EARLY GREEK MATHEMATICS: THE EVIDENCE
historian of mathematics. We just have to be alert to the fact that any
historical information we may glean is embedded within a much wider and
more complicated context.
The picture Plato presents of the mathematics of his day is both diverse
and vast. First of all, he talks about the mathematicians’ method, how they
did what they did. He characterizes their way of proceeding as hypothetical:they used claims that were not proved but only assumed to be valid, and on
which other claims were then based. Thus, proving a mathematical statement
consisted of working back from that statement to other known statementsof which the first was a consequence, until one found an accepted hypothesis
which needed no further proof and could then serve as the starting-point
(reversing the logical process that led to it). By ‘hypothesis’ (literally some-thing that is put under) Plato seems to have meant an assumption or starting-
point in quite a general sense, including both definitions and fully-fledged
propositions:
For example, if [geometers] are asked whether a specific area can
be inscribed in the form of a triangle within a given circle, one ofthem might say: ‘I do not yet know whether that area has that
property, but I think I have, as it were, a hypothesis that is of use
for the problem, namely this: If that area is such that when onehas applied it as a rectangle to the given straight line in the circle it
is deficient by a figure similar to the very figure which is applied,
then I think one alternative results, whereas another results if itimpossible for this to happen. So, by using this hypothesis, I am
willing to tell you what results with regard to inscribing it in the
circle – that is, whether it is impossible or not. […]’
46
Plato distinguishes several branches of mathematics according to their
main object of enquiry: arithmetic and logistics (the science of calculation),which both studied numbers, albeit in different ways; geometry and stereo-
metry, which dealt with geometrical objects in two and three dimensions,
respectively; harmonics and astronomy. As for the content of the various
mathematical disciplines, we are told that arithmetic and logistics study the
‘odd and even’, a phrase Plato seems to use as a synonym for ‘numbers’, andhe also mentions square and solid numbers and the harmonic and
arithmetical mean, plus intervals of various kinds.
47 As for geometry, Plato
often deals with the issue of incommensurable or irrational magnitudes,which he also calls ‘unaccountable’ ( alogos ) or ‘inexpressible’ ( arhetos ). The
classical example of an incommensurable line is the diagonal of a square,
which cannot be measured by the same unit as its side. Plato observed thatincommensurables may be such when taken individually but not such when

27EARLY GREEK MATHEMATICS: THE EVIDENCE
taken together,48 and seemed to put great store by a grasp of their complexi-
ties. In the Laws we are told that if
all we Greeks believe [lines, surfaces and volumes] to be commen-
surable when fundamentally they are incommensurable, one had
better address these people as follows (blushing the while on theirbehalf): ‘Now then, most esteemed among the Greeks, isn’t this one
of those subjects […] it was disgraceful not to understand […]?’.
49
It could be that at the time Plato wrote this, his last dialogue, a knowledge
of incommensurables had indeed yet to become common, but it could also
be that he overemphasized the ignorance of his fellow Greeks as a way ofcontrasting the superficial opinions held by the many and the truth, which
is fully understood only by a few. The same issues underlie a famous passage
in the Meno , where a slave, asked the right questions by the right person
(Socrates), successfully tackles a geometrical problem involving, again,
incommensurables. Anybody can gain, or, as Plato argues, retrieve correct
mathematical knowledge, but it is important not to trivialize even apparentlysimple operations such as doubling a square, and to recognize the need for
appropriate guidance.
[Socrates wants to illustrate to Meno his idea that knowledge is
recollection, by taking an uneducated slave and showing that, with
appropriate questioning, even the slave can exhibit mathematical
knowledge (see Diagram 1.1)] Socrates . Tell me now, boy, do you
know that a square is a figure like this? Boy. Yes. S. A square thenDiagram 1.1

28EARLY GREEK MATHEMATICS: THE EVIDENCE
is a figure that has all these sides equal, and they are four? B. Yes
indeed. S. Does it not also have these lines through the middle
equal? B. Yes. S. […] If […] this line was two feet, and this two feet,
how many feet would the whole be? Look at it this way: if therewere two feet here, but only one foot here, would the figure not be
once two feet? B. Yes. S. Since then there are two feet here, would
it not become twice two feet? B. It would. […] S. How many feet
then is twice two feet? Calculate and tell me. B. Four, Socrates. S.
Now there could be another figure double this one, which has all
the sides equal like this one. B. Yes. S. How many feet will it be? B.
Eight. S. Come on then, try to tell me how long each of its sides
will be. The side of this is two feet; what about each side of the
double one? B. It is clear, Socrates, that it is the double. S. […] Tell
me, are you saying that the double figure is based on the doubleline? […] B. I do. S. Now the line becomes double if we add another
the same from here? B. Yes indeed. S. Say, the eight-foot figure
will be based on this, if four sides the same are generated? B. Yes.
S. Let us draw from this four equal sides then. Would this not be
what you say is the eight-foot figure? B. Yes. S. Now, within this
there are four, each of which is equal to the four-foot one? B. Yes.
S. How big is it then? Is it not the quadruple of this? B. And how
not? S. Is this, the double, then the one which is quadruple? B.
No, by Zeus. S. But how many times larger is it? B. Four times. S.
Therefore, boy, the figure based on the double line will be notdouble but quadruple? B. You speak the truth. S. Now, four times
four is sixteen, is it not? B. Yes. S. The eight-foot figure, on how
long a line should it be based? […] The four-foot figure is on thishalf line here, is it not? B. Yes. S. Very well. Is the eight-foot figure
not double this and half that? Will it not be on a line greater than
this and lesser than that? Is that not so? B. I think that it is so. S
.
Good, you answer what you think. And tell me, was one not twofeet, and the other four feet? B. Yes. S. Therefore the side of the
eight-foot figure must be greater than this one of the two-feet
figure, and lesser than that of the four-feet figure? B. It must be. S.
T ry to express then how long you say this is. B. Three feet. S. Now
if it is three feet, let us add the half of this, and it will be three feet;
for these are two, and this is one. And from here in the same waythese are two and this is one, and so the figure that you say is
generated? B. Yes. S. Now if there are three feet here and three feet

29EARLY GREEK MATHEMATICS: THE EVIDENCE
here, the whole figure be three times three feet? B. It is evident. S.
How much is three times three feet? B. Nine feet. S. And the double
had to be how many feet? B. Eight. S. So the eight-foot figure
cannot be based on the three-foot line? B. Certainly not. S. But
on how long a line? T ry to tell us with precision, and if you do not
want to calculate with numbers, then show me on how long a
line. B. By Zeus, Socrates, I do not understand. S. […] Tell me, is
this not for us a four-foot figure? You understand? B. Yes. S. We
add to it this other figure which is equal to it? B. Yes. S. And we
add this third figure equal to either of them? B. Yes. S. Now could
we fill in the corner like this? B. Certainly. S. Would these four
equal figures not originate? B. Yes. S. Well then, how many times
is the whole larger than this? B. Four times. S. But it should have
come out double – do you not remember? B. Certainly. S. Now,
does not this line from corner to corner cut each of these figures in
two? B. Yes. S. Now these are four equal sides surrounding this
figure. B. They are. S. Look now: how large is this figure? B. I
don’t get you. S. Does not each internal line cut half of each of
these four figures? B. Yes. S. How many areas this big are there in
this? B. Four. S. How many in this? B. T wo. S. What is the four to
the two? B. Double. S. How many feet in this? B. Eight feet. S. On
what line? B. This one. S. On the line stretching from corner to
corner of the four-foot figure? B. Yes. S. The sophists call this
diagonal; so that if diagonal is its name, do you say, o Meno’s boy,that the double figure would be based on the diagonal? B. Most
certainly, Socrates.
50
Plato’s Theaetetus , set in 399 BC, introduces us to two experts on the
topic of incommensurables: Theodorus of Cyrene51 and his pupil Theaetetus,
who died in 369 BC. We are told that Theodorus of Cyrene had studied
with the sophist Protagoras, but had ‘very soon inclined away from abstract
discussion to geometry’,52 and that he had explored some particular cases of
squares with area 3, with area 5, and so on, observing that their sides werenot commensurable with the unit. Theaetetus, at the fictional time of the
dialogue still a whiz kid, is credited with broader enquiries. He found a
general principle that allowed him to identify which lines were incommen-surable in length but not when squared. For instance, as in the case
Theodorus himself was studying, two squares whose sides are (in modern
symbolism) √3 and √5 will be commensurable in area (3 and 5) but not in

30EARLY GREEK MATHEMATICS: THE EVIDENCE
length, i.e. as far as their sides are concerned. This much seems clear, but
debate has raged over the precise details of the procedures used by Theodorus
and Theaetetus, their respective contributions, and the actual state of theissue at the time (had incommensurables just been discovered, or had they
only just been brought to Plato’s notice?).
53
Another topic about which Plato provides precious information is that
of the five regular bodies, also called Platonic bodies. In the Timaeus , which
is a story of the origin of the universe, he introduces a sort of creator figure,
the demiurge (in Greek, that means ‘craftsman’ or ‘artisan’). The demiurgemoulds the cosmos using the forms as models, and gives air, earth, fire and
water (the four traditional elements of early Greek natural philosophy) a
geometrical structure. The basic constituents of matter are two types ofright-angled triangle: isosceles and scalene; everything else can be put together
using those two. Thus, fire is made from minuscule pyramids, made from
equilateral triangles, in their turn made from six scalene right-angled triangles;air is made from octahedra, water from icosahedra, earth from cubes.
Dodecahedra are used to ‘decorate’ the universe.
54 Why those five solids
and not others? Well, the universe in the Timaeus is spherical, and pyramid,
cube, octahedron, icosahedron and dodecahedron are the only five regular
solids which can be inscribed in a sphere. A proof of this statement is not
found until Euclid. On the other hand, the geometry contained in theTimaeus is put in very vague terms, and possibly deliberately so, since the
dialogue claims to be a ‘likely story’, not an accurate report. Plato may have
chosen not to include a more rigorous mathematical justification of his
grouping of the five bodies. In any case, it seems that geometrical research
by the time he was writing the Timaeus had identified the solids involved,
and probably noticed their common properties.
55
Apart from having both caught Plato’s attention, incommensurables and
regular bodies have another thing in common: both have been linked by
later sources to the Pythagorean school. Some sources even accused Plato ofhaving plagiarized Pythagorean writings to produce the Timaeus . Plato indeed
does mention the Pythagoreans, but primarily about their theories on the
soul and perhaps (in a passage where no names are named) for their research
into music. Indeed, everything we know about their mathematical discoveries
and interests comes from later, often much later, centuries and is generallythought to be unreliable – which is why the reader will not find much about
Pythagoras and the Pythagoreans in this chapter. Most scholars will agree
that there was a Pythagorean school of philosophy from the sixth untilprobably the fourth century
BC, that they were involved in politics and that
they had certain beliefs about life and the universe, including perhaps the
tenet that ‘everything is number’, or that number holds the key to under-standing reality. But most scholars today also think, for instance, that

31EARLY GREEK MATHEMATICS: THE EVIDENCE
Pythagoras never discovered the theorem that bears his name.56 I will return
to this topic in chapter 2, and more briefly in the next section.
In sum, what information we can get from Plato is fascinating and rich,
and fraught with problems. Beyond issues of detailed reconstructions and
precise attributions, which may never be solved with any certainty, he
definitely is testimony to the vitality of mathematics at the time, and to theinterest that mathematical issues aroused in the educated public at large.
Plato will return in the next chapter – let us now turn to another illustrious
member of that educated public at large.
Aristotle
Aristotle was Plato’s most famous pupil, and one of his most stringent critics:
among other things, he disagreed with his views about knowledge, the way
the universe worked and whether forms really did exist. For him,mathematical knowledge was not of things that existed in the real world –
it provided certainty, surely, but about entities which were abstractions from
physical objects rather than objects with a separate existence (a belief thiswhich he attributed to the Platonists). Since, grossly put, knowledge for
Aristotle was ultimately knowledge of physical reality, the validity of mathe-
matics as a form of knowledge, while granted for its strict domain of applica-tion, i.e. mathematical objects, was jeopardized by the fact that its relation
with physical reality was at best problematic. At the same time, however,
Aristotle considered mathematics an important model for scientific discourse,and devoted large sections of his work to a discussion of the nature of
mathematical objects. He explored the logical structure of mathematics,
the way it started from undemonstrated principles, on which consensushad been gained, and then proceeded rigorously to conclusions that held
generally rather than particularly, thus becoming unassailable by objections
or criticisms.
As we have done with Plato, we can artificially distinguish the information
we can obtain from Aristotle as a philosopher, and Aristotle as a historian.
Again, I will focus on the latter rather than the former, and again, I am
aware that no such distinction is really possible. There seems to be a
consensus, however, that Plato was more of a mathematician than Aristotle,that he got more involved and participated more directly in the subject.
Consequently, what Aristotle says about mathematics, even in his capacity
as a philosopher, would seem to reflect historical circumstances to a greaterdegree than Plato. But we will be cautious anyway.
Like Plato, Aristotle was very interested in the mathematicians’ method,
and it is on mathematical procedures that some of his philosophicaldiscussions concentrate. He provides the earliest extant account of what

32EARLY GREEK MATHEMATICS: THE EVIDENCE
criteria should be followed to obtain demonstrative validity, and several of
his works are devoted to an analysis of demonstration, argumentation, forms
of discourse – not just how knowledge is to be gained, but how it is to beorganized, expressed and defended against objections. Although nowhere
did Aristotle explicitly state that the mathematics of his day had reached
the ideal described in his works, or that it was his discipline of choice, manyof the examples deployed in the Posterior Analytics , which dealt with scientific
demonstration particularly, were taken from mathematics. For instance, he
stated that all scientific argumentations must have indemonstrable starting-points or principles: subject-matters such as unit or magnitude, which are
assumed to exist; statements such that ‘if equals are taken from equals, the
remainders are equal’, which are assumed to be true; definitions of ‘odd’ or‘even’ or ‘irrational’ or ‘verging’, which are assumed to be understood in the
same way by all the participants.
57 An element of necessity is also required:
one has to be aware that what is in a such-and-such way could not beotherwise, for instance the diagonal cannot be commensurable. In fact, the
proof that the diagonal cannot be commensurable (of which more below) is
used by Aristotle to exemplify the so-called privative demonstration, alsoknown as reductio ad absurdum or proof per impossibile : if the consequences
of a certain statement are absurd, the statement itself is false. Finally,
universality, another characteristic a scientific demonstration should possess,is discussed in these terms:
Even if you prove of each triangle either by one or by different
demonstrations that each has two right angles – separately of the
equilateral and the scalene and the isosceles – you do not yet know
of the triangle that <it has> two right angles […] nor <do you knowit> of triangle universally, not even if there is no other triangle apart
from these. […] So when do you not know universally, and when do
you know simpliciter ? […] Clearly whenever after abstraction it
belongs primarily – e.g. two right angles will belong to bronze
isosceles triangle, but also when being bronze and being isosceles
have been abstracted. But not when figure or limit have been.
58
Aristotle’s analysis of demonstration stemmed from his, and his contem-
poraries’, interest and enquiries into rhetoric and persuasive discourse. He
realized that the method of mathematics was distinctive with respect to
other forms of argumentation: ‘it is evidently equally foolish to acceptprobable reasoning from a mathematician and to demand from a rhetorician
demonstrative proofs’.
59 Mathematics was thus rather generally associated
not only with a certain subject-matter (numbers, geometrical figures), butalso with a certain style:

33EARLY GREEK MATHEMATICS: THE EVIDENCE
Some people do not listen to a speaker unless he speaks mathematic-
ally, others unless he gives instances, while others expect him to
cite a poet as witness. And some want to have everything doneaccurately, while others are annoyed by accuracy […] Therefore
one must be already trained to know how to take each sort of
argument, since it is absurd to seek at the same time knowledgeand the way of attaining knowledge, and neither is easy to get.
The minute accuracy of mathematics is not to be demanded in all
cases, but only in the case of things which have no matter.
60
The last sentence is a reminder of mathematics’ limitations: it may be a
powerful tool for persuasion, but its range of applicability, or the suitabilityof its method, are restricted.
Like Plato, Aristotle often refers to incommensurables, and, like Plato,
he uses them to say things about knowledge in general. In a famous passagefrom the Metaphysics , he has philosophy originate from curiosity:
Everybody begins […] by wondering that things are as they are, as
one does about self-moving puppets, or about the solstices or the
incommensurability of the diagonal; for it seems wonderful to all
who have not yet seen the cause, that there is something whichcannot be measured even by the smallest thing. But we must end
in the contrary and, according to the proverb, the better state, as is
the case in these instances too when one learns the cause; for thereis nothing which would surprise a geometer so much as if the
diagonal turned out to be measurable.
61
It would then seem that, by Aristotle’s time, mathematicians had learnt thecause of incommensurability in the square – indeed, elsewhere Aristotle
says that they could prove ‘that the diagonal of a square is incommensurablewith its side by showing that, if it assumed to be commensurable, odd
numbers will be equal to even’. In other words, a reductio ad absurdum
proof was available.
62 Results which may have already been known to Plato,
e.g. that the sum of the angles of a triangle is equal to two right angles, or
that the angle in a semicircle is a right angle, are used by Aristotle as ‘clearstock examples’.
63 He also hinted that proportion theory was by then a
well-established field, and specified that some theorems which had previously
been proved separately for numbers, lines, solids, and times, had now beenproved universally.
64 Not only was Aristotle able to view mathematical
developments over time, he also made the double equation, already under-
lying the Theaetetus , between ‘before’ and ‘after’ and ‘specific’ and ‘universal’,

34EARLY GREEK MATHEMATICS: THE EVIDENCE
as if the natural course of mathematics was in the direction of greater and
greater generality.
On the other hand, Aristotle is testimony that not all was proceeding
smoothly in the mathematical field – at least not in his opinion. One problem
area he identifies is the theory of parallel lines: the people investigating
them ‘unconsciously assume things which it is not possible to demonstrateif parallels do not exist’.
65 In other words, their reasoning is circular. Another
dodgy field is the squaring of the circle, already encountered as a subject of
ridicule in Aristophanes. Aristotle knew of several attempts to solve theproblem, none of them in his view successful. He mentions, rather briefly,
Bryson, Antiphon (both sophists – Antiphon was also known as a poet) and
Hippocrates of Chios: about Antiphon, he says that it is not the business ofthe geometer to refute his solution because it was not based on geometrical
principles. Bryson’s quadrature of the circle relied on universal principles,
rather than principles proper only to geometry. As for the quadrature bymeans of lunules or segments, which interpreters traditionally attribute to
Hippocrates, although Aristotle’s text is fairly ambiguous on the matter,
exactly what was wrong with it has been the object of much debate. Itseems clear that, unlike Antiphon’s and Bryson’s attempts, it had to be taken
as a serious geometrical effort.
66 The obscurity of Aristotle’s mathematical
references has tantalized interpreters since later ancient times and, as weshall see in chapter 2, many people tried to reconstruct early Greek solutions
to the quadrature of the circle with varying degrees of implausibility.
We may ask the crucial question again: to what extent does Aristotle
reflect the actual mathematical practice of his day? The reader will probably
know the answer by now: we simply cannot tell for sure. His direct involve-
ment with mathematics may have been negligible and his admiration forthe subject remarkable, yet, most of the times Aristotle names mathemati-
cians it is in order to criticize them, either because of fallacious arguments,
or on even more exquisitely philosophical grounds. Thus, he attacks thePlatonists for believing that mathematical objects have an existence separate
from the things in the world. Eudoxus of Cnidus, associated with Plato’s
school and celebrated by other sources for his many achievements in geo-
metry, is acknowledged by him ‘only’ as an astronomer and a philosopher.
67
As for the ‘so-called Pythagoreans’, Aristotle is prepared to acknowledge
that they were the first to advance the study of mathematics
and having been brought up in it they thought its principles were
the principles of all things. […] in numbers they seemed to see
many resemblances to the things that exist and come into being
[…] such and such a modification of numbers being justice, anotherbeing soul and reason, another being opportunity – and similarly

35EARLY GREEK MATHEMATICS: THE EVIDENCE
almost all other things being numerically expressible; since, again,
they saw that the attributes and the ratios of the musical scales
were expressible in numbers; since, then, all other things seemedin their whole nature to be modelled after numbers, and numbers
seemed to be the elements of all things, and the whole heaven to
be a musical scale and a number. And all the properties of numbersand scales which they could show to agree with the attributes and
parts and the whole arrangement of the heavens, they collected
and fitted into their scheme; and if there was a gap anywhere, theyreadily made additions so as to make their whole theory coherent.
E.g. as the number 10 is thought to be perfect and to comprise
the whole nature of numbers, they say that the bodies which movethrough the heavens are ten.
68
This is just the prelude, however, to a systematic criticism of a whole
host of mistaken beliefs, interspersed with information about the so-called
Pythagoreans’ actual mathematical research: their investigations into music
and the connection between accords and numerical ratios, their interestinto the properties and definitions of odd, even, square and rectangular
numbers. Both with the Pythagoreans and with the followers of Plato, ideas
about mathematics are only a part of what Aristotle criticizes: he is clearlynot interested in faithfully reproducing his adversaries’ theories, and he
obviously does not care strictly to separate what is of interest for us (mathe-
matics) from other topics.
We can certainly use his testimony to conclude that the Pythagoreans,
whoever they were, took mathematics to be very significant, and attributed
moral, political and cosmological meanings to numbers, and that interest inmathematics was rife in Plato’s school in his last days and after his death. Yet,
any more detailed reconstruction contains, in my view, an excessive element
of speculation. Here I prefer to keep to the vagueness of our early sources,and leave the (relatively) brash clarity of our later ones to the next chapter.
Notes
1 Aeschines, Against Ctesiphon 22. Here and henceforth I have used Loeb translations,
unless otherwise indicated.
2 Lloyd (1990), 8. See also Vernant (1965), chapter 6; Lloyd (1972), (1979), (1987b);
Netz (1999a).
3 Herodotus, Histories II 109. Debate has raged over the actual links between Egyptian
and Greek mathematics, and on the dependence of the latter on the former, see e.g. Kahn
(1991), Bernal (1992).
4 This document is an inscription from the fourth century BC, found in Cyrene, and allegedly
reproducing a seventh-century BC original. The translation, discussion and references are
in Osborne (1996), 10–15.

36EARLY GREEK MATHEMATICS: THE EVIDENCE
5 See e.g. Carter (1990); Stancic and Slapsak (1999). For the political significance of rural
and urban land division, see Castagnoli (1956); Asheri (1966) and (1975); Boyd and
Jameson (1981).
6 Aristotle, Politics 1267b, 1330b, tr. B. Jowett, Princeton University Press 1984.
7 See the evidence collected in Svenson-Evers (1996); Philo’s work described in IG 22.1668
(347–46 BC).
8 Coulton (1977), 109.
9 Coulton (1977), 64.
10 Plato, Philebus 56b, tr. D. Frede, Hackett 1997. The ‘instruments’ in question are specified
a few lines later: straight-edge and compass, mason’s rule, plumbline and carpenter’s square.
11 Mentioned by Herodotus, Histories III 60.
12 Comprehensive evidence on the tunnel in Kienast (1995); extensive discussion of the
three problems in Rihll and T ucker (1995), quotation at 410; see also Burns (1971).
13 Tod (1911–12), 128. In fact, it would be more correct to refer to the acrophonic system
as a group of notations, because we have several versions of it from all over the Mediter-
ranean basin, with some signs differing rather widely from place to place . Attica is the
region around Athens.
14IG II2 2777 and see Smith (1951), II 162 ff. There has been debate as to whether the
Salamis object is a gaming table or a counting-board; both Heath (1921), I 46 ff. and
Pritchett (1965) incline for the first interpretation. There is literary and archaeological
evidence, however, that abaci were extensively used and it seems likely, as suggested even
by Heath (1921), I 50, that they may have at least looked like the Salamis table.
15IG IX 488; IG XII 99; IG XII 282, respectively. The abacus from Thyrium has been
interpreted as a fragment of the accounts of the state by its editors, but Tod (1911–12),
112, with whom I agree, thinks it ‘far more probable that the stone was a counting-board’. See also Leonardos (1925–26); Lang (1968); IG II
2 2778, 2779, 2781. Number
2780, from Eleusis, is catalogued as an abacus, but I think it may be a list of some sort,
because the same string of numbers is repeated three times, and this is neither typical of
other abaci, nor understandable on the basis of the slab itself being an abacus.
16 On counters and early Greek mathematics see Netz (forthcoming).17 Several other vases and sherds from the same period are inscribed with numbers to indicate
their capacity, price or weight. Sherds may have been used as an alternative to abaci when
no complicated counting was needed, cf. Lang (1956).
18 The complete inscription in IG I
3 71; tr. B.D. Meritt and A. West, Ann Arbor 1934;
commentary in Meritt et al. (1939), A 9 and in Meiggs and Lewis (1989).
19 An expression also used by the contemporary historian Thucydides to describe requisitions
of bakers in times of war, which had to be, again, proportionate to the size of their mills:
The Peloponnesian War , VI 22.
20IG I3 369; commentary in Meritt (1932), 128 ff., my translation.
21 Our (somewhat muddled) information comes from Aristotle, Constitution of Athens 48.3;
54.2 and from inscriptions like the one above, see Rhodes (1972) and (1981).
22 Aeschines, Against Timarchus , 107.
23 A complete study of the temple and of the inscriptions in Burford (1969), whose translation
I quote. The earliest set of inscriptions is: IG IV2 102, 104 and 743; SEG XI 417a; SEG
XV 208–9. The latest inscription, also known as the Tholos accounts, is IG IV2 103.
24IG IV 1484 B II 250–4.
25 Osborne (1994), 13. See also Thomas (1992).
26 Osborne (1994), 15.27 Homer, Iliad 12.421–3; Theognis 805 ff., cf. 543, tr. M.L. West, Oxford 1993.
28 Herodotus, Histories I 32; II 142; III 89–95; V 52–4; VII 184–7; IX 28–30, respectively.

37EARLY GREEK MATHEMATICS: THE EVIDENCE
29 Aeschylus, Prometheus Bound 343–378, tr. J. Scully and C.J. Herington, Oxford 1975,
with modifications.
30 Aristophanes, Wasps 656–63. The reckoning continues until verse 718.
31 Aristophanes, Birds 992–1009. For more information on Meton, see Bowen and
Goldstein (1988).
32 Aristophanes, Clouds 202–5; tr. A.H. Sommerstein, Aris and Phillips 1982.
33 Thucydides, The Peloponnesian War II 13.3–9.
34 Thucydides, The Peloponnesian War II 72.3.
35 Aristotle, Constitution of Athens 69. The abacus on which trial votes were counted is also
mentioned by Aristophanes, Wasps 332–3.
36 Several of our extant speeches were written for trials involving bankers or their relatives.
This is just one indication that banking activities were widespread at the time, and implieda whole host of arithmetical operations, presumably conducted on the abacus, perhaps
by specialized slaves: calculation of interest, exchange between different currencies, division
of profits between partners with different shares, see Bogaert (1976); Cohen (1992).
37 Lysias, On the Property of Aristophanes, Against the Treasury 9.
38 Lysias, ibid. 42–3. Similar accounts (going through the calculation in order to defend or
accuse) are given in e.g. Lysias, Defence Against a Charge of T aking Bribe 1–5 ( c. 403–402
BC); Isaeus, On the Estate of Hagnias 40–6 (between c. 396 and c. 378 BC); Demosthenes,
Against Aphobus I 9–11, 34–9, 47 (364 BC); Against Leptines 77, 80 ( c. 355 BC); For
Phormio 36–41 (mid-fourth century BC).
39 Lysias, Against Diogeiton 4–9.
40 Lysias, ibid. 13–15.
41 Lysias, ibid. 19–28.
42 Aeschines, Against Ctesiphon 9–27; quotation at 17. Cf. also e.g. Lysias, Against Nicomachus
5 (c. 399 BC, he refused to show his accounts for four years), 19–20 (he entered sacrifices
to an inflated excess amounting to six talents); On the Property of Aristophanes 50–1
(Diotimus is no longer suspected of embezzlement when ready to show his accounts).
See Tolbert Roberts (1982).
43 Demosthenes, On the Crown 229.
44 Aeschines, Against Ctesiphon 59.
45 Plato, Republic 509d–511e. The translation of the passage is pretty controversial, for
instance, the last phrase of our quotation (510c6) has also been rendered ‘since they were
known’. A discussion of the divided-line passage with further references in Mueller (1992).
46 Plato, Meno 87a f., see also Republic 510c.
47 For references on odd and even see Knorr (1975), 106n101. For square and solid numbers,
Plato, Timaeus 31c–32b. For means and intervals, ibid. 35b–36d.
48 Plato, Greater Hippias 303b. Cf. also Republic 534d (irrational lines are compared to
political rulers).
49 Plato, Laws 819e–820c, tr. T.J. Saunders, Hackett 1997, with modifications (note the
shift from ‘we Greeks’ to ‘these people’). Cf. also Parmenides 140c.
50 Plato, Meno 82b–85b, my translation. Note that the word for ‘line’ and ‘side’ is the same
(gramme ).
51 Theodorus is also mentioned by Xenophon, Memorabilia 4.2.10.
52 Plato, Theaetetus 165a, tr. M.J. Levett, rev. M.F . Burnyeat, Hackett 1997.
53 Some of the most recent contributions are: Szabó (1969); Knorr (1975); Fowler (1999).
54 Plato, Timaeus 54a–57c.
55 See e.g. Sachs (1917); Waterhouse (1972).56 See Burkert (1972); Huffmann (1993); Zhmud (1997).
57 See especially Aristotle, Posterior Analytics 76a ff.

38EARLY GREEK MATHEMATICS: THE EVIDENCE
58 Aristotle, Posterior Analytics 74a–b, tr. J. Barnes, Princeton 1984.
59 Aristotle, Nicomachean Ethics 1094b, tr. W.D. Ross, rev. by J.O. Urmson, Princeton 1984;
see also Posterior Analytics 79a.
60 Aristotle, Metaphysics 995a.
61 Aristotle, Metaphysics 983a, tr. W.D. Ross, Clarendon Press 1928, with modifications.
An almost comprehensive collection of mathematical passages in Aristotle is Heath (1949).
For the passages where Aristotle cites incommensurables, see the references in Fowler
(1999), 290–1.
62 Aristotle, Prior Analytics 41a, 50a.
63 Cf. Mendell (1984) for references.
64 Aristotle, Posterior Analytics 74a–b.
65 Aristotle, Prior Analytics 65a.
66 Evidence collected in Heath (1949). For an extensive discussion of the evidence about
Hippocrates of Chios, see Lloyd (1987a).
67 See Napolitano Valditara (1988).
68 Aristotle, Metaphysics 985b–986a.

39EARLY GREEK MATHEMATICS: THE QUESTIONS
2
EARLY GREEK
MATHEMATICS:
THE QUESTIONS
Early Greek mathematics was not one but many; there were various levels
of practice, from calculations on the abacus to indirect proofs concerning
incommensurable lines, and varying attitudes, from laughing off attemptsto square the circle to using attempts to square the circle as examples in a
second-order discussion about the nature of demonstration. In sum, different
forms of mathematics were used for different purposes by different groupsof people. Perhaps one common feature is clearly distinguishable: mathe-
matics was a public activity, it was played out in front of an audience, and it
fulfilled functions that were significant at a communal level, be they countingrevenues, measuring out land or exploring the limits of persuasive speech.
The first question addressed in this chapter is what I call the problem of
political mathematics. I take ‘political’ in the literal Greek sense, as somethingthat has to do with the polis, the city/community/state. When reading fifth-
and fourth-century
BC philosophical sources, I have always been struck by
the frequency with which mathematical images or examples are used tomake points which are not related to mathematics at all – often, points
about politics. Moreover, Plato has some very interesting statements on the
question of who mathematics should be for, and which mathematics oughtto be done by whom: he established parallel hierarchies between forms of
mathematics and categories of people. Once again, these were deeply political
statements. So, having warned the reader in the introduction that I will askquestions rather than answering them, the first section will expand and
muse on the theme, what were the political functions of early Greek
mathematics?
The second section will tackle a historiographical issue: how later ancient
sources depict early Greek mathematics, and what can be done with them.
It will be, I am afraid, an exercise in scepticism.

40EARLY GREEK MATHEMATICS: THE QUESTIONS
The problem of political mathematics
In chapter 1, we observed that not only was public counting associated
with political accountability, it became a symbol of, or a way of talking
about, political participation and the role of the citizen. Moreover, accounts
were but one of several mathematical activities that took place in a publiccontext: there were also commercial arithmetic, practised by traders and
bankers, the geometry of land division, and in general the mathematics
associated with the technai , for instance architecture. Land division and
commercial arithmetic can be connoted as ‘democratic’ mathematics: the
former was a guarantee of equal distribution, whereas the latter was identified
with moneyed economical exchange, as opposed to non-moneyed, non-quantified, status-dependent transactions, which had traditionally been
dominated by aristocratic value systems.
1 Aristotle, himself a supporter of
oligarchy rather than democracy, even put forth what we could call a mathe-matizing theory of monetary exchange, where the value of a thing can be, in
principle, completely reduced to a number, and transactions to arithmetical
operations.
[Aristotle on money as the measure of all things] All things that
are exchanged must be somehow commensurable. It is for this
end that money has been introduced, and it becomes in a sense an
intermediate; for it measures all things, and therefore the excessand the defect – how many shoes are equal to a house or to a givenamount of food. The number of shoes exchanged for a house must
therefore correspond to the ratio of builder to shoemaker. For if
this be not so, there will be no exchange and no intercourse. Andthis proportion will not be effected unless the goods are somehow
equal. All goods must therefore be measured by some one thing
[…] Money, then, acting as a measure, makes goods commensurateand equates them; for neither would there have been association if
there were not exchange, nor exchange if there were not equality,
nor equality if there were not commensurability. Now in truth itis impossible that things differing so much should become com-mensurate, but with reference to demand they may become so
sufficiently. There must, then, be a unit, and that fixed by agreement
(for which reason it is called money); for it is this that makes allthings commensurate, since all things are measured by money. Let
A be a house, B ten minae, C a bed. A is half of B, if the house is

41EARLY GREEK MATHEMATICS: THE QUESTIONS
worth five minae or equal to them; the bed, C, is a tenth of B; it is
evident, then, how many beds are equal to a house, that is, five.
That exchange took place thus because there was money is evident.2
Democratic mathematics, however, was but one version of the possible
political uses of mathematics, which in its various forms was used to articulate
conflicting positions about the polis, man, knowledge, and their interaction.
Take the case of accounts. Ancient authors themselves were quick to
point out that no simple equation could be made between accounts and
honesty or willingness to have one’s actions scrutinized, or between accountsand democracy. Legal rhetoric exposed examples of bad accountancy while
at the same time relying on accountancy as an image of clarity and objective
good judgement. Both Aristotle and Plato retained audits and auditors intheir oligarchic ideal states.
3 Aristotle pointed out that one of the conciliatory
methods tyrants may adopt in order to secure their power was rendering
accounts of receipts and expenditure, thus providing an illusion but not thesubstance of democracy,
4 and Plato depicted a real-life mathematical expert
as the embodiment of the dangers of knowledge inappropriately used. The
sophist Hippias of Elis is presented in the eponymous dialogues as a sort oftravelling salesman of general culture, who, as well as acting as envoy for his
city, instructed (for a fee) the youths of various parts of Greece in the art of
persuasive discourse, in grammar and history, and in astronomy, geometryand arithmetic. It is as a skilled calculator, who hangs out in the agora , ‘next
to the tables of the bankers’, that Socrates addresses him here:
SOCRATES […] If someone were to ask you what three times seven hundred
is, could you lie the best, always consistently say falsehoods about these
things, if you wished to lie and never tell the truth? […] So we should alsomaintain this, Hippias, that there is such a person as a liar about calculation
and number. […] Who would this person be? Mustn’t he have the power to
lie, as you just agreed, if he is going to be a liar? […] And were you not just
now shown to have the most power to lie about calculations? […] Do you,
therefore, have the most power to tell the truth about calculations? […]Then the same person has the most power both to say falsehoods and to
tell the truth about calculations. And this person is the one who is good
with regard to these things, the arithmetician?
HIPPIAS Yes.5
Mathematics, the transparent, accountable knowledge par excellence , the
knowledge which you should be able easily to control by running a check

42EARLY GREEK MATHEMATICS: THE QUESTIONS
on it, in this short passage is blown apart and revealed as the site of contra-
dictions: truth and falsity are almost indistinguishable – the arithmetician,
(by extension, and forgive me for speculating) the accountant, the personwho embodies democratic control over the workings of the polis, is shown
to be the potential master of deceit. Further scepticism had been voiced,
according to Plato, Aristotle and later reports, by Protagoras, also a sophist.He observed that geometrical objects are not really as the geometers say they
are (for instance, a material circle tangent to a material straight line will touch
it in more than one point), and in general that in mathematics ‘the facts arenot knowable, the words not acceptable’.
6
Not all critics of mathematics were that philosophically sophisticated.
Some people simply could not see the point of speculating on the quadratureof the circle and similar things. Aristophanes and his audience were at home
with counting or with geometry as land-division, but found Meton and the
Socratic student of the Clouds a bit of a joke. According to Xenophon,
Socrates himself
said that the study of geometry should be pursued until the student
was competent to measure a parcel of land accurately in case he
wanted to take over, convey or divide it, or to compute the yield
[…] He was against carrying the study of geometry so far as toinclude the more complicated figures, on the ground that he could
not see the use of them.
7
Plato’s contemporary Diogenes the Cynic apparently would wonder that
‘the mathematicians should gaze at the sun and the moon, but overlook
matters close at hand’, and thought that ‘we should neglect music, geometry,astronomy, and the like studies, as useless and unnecessary’.
8 Isocrates, while
allowing that astronomy and geometry could be practised as gymnastics for
the mind, and to keep young men occupied and out of harm’s way, reportednonetheless that ‘most men see in such studies nothing but empty talk and
hair-splitting, since none of these things is useful either in private or in
public life’. He also warned against the dangers of an excessive use of mathe-
matics, which, if pursued too intensely, would have impeded the harmonious
mental development of the youth, and in any case was too accurate to be ofreal use in everyday practical applications – too much accuracy was not
always necessary.
9 For more views on the role and dangers of mathematics
in education, let us turn back to Plato.
Since in Plato’s view knowledge and wisdom were the best entitlements
to political power, his ideal state was to be ruled by philosophers. They
would be brought up from early childhood following an educational curricu-lum: gymnastics, reading and writing, military training to begin with; later

43EARLY GREEK MATHEMATICS: THE QUESTIONS
on, between the ages of around twenty-two and thirty-two years old, mathe-
matics: geometry, stereometry, astronomy, harmonics, but, first of all, arith-
metic. Not only did this latter have practical applications in war, for whichthe rulers had to be prepared, it also helped the mind overcome the pitfalls
of sensible knowledge. Both the practical and the more philosophical uses
of arithmetic were appreciated by Plato; he drew, however, a crucialdistinction:
it would be appropriate […] to legislate this subject for those who
are going to share in the highest offices in the city and to persuade
them to turn to calculation and take it up, not as laymen do, but
staying with it until they reach the study of the natures of thenumbers by means of understanding itself, not like tradesmen and
retailers, for the sake of buying and selling, but for the sake of war
and for ease in turning the soul around, away from becoming andtowards truth and being.
Analogous claims are made for geometry, whose practitioners use a
language which is
very absurd, if very inevitable. […] They talk as if they were actually
doing something and as if the point of all their theorems was to
have some actual effect: they come up with words like squaring
and applying and adding and so on, whereas in fact the sole purposeof the subject is knowledge.
10
Mathematics has then a double character. There are two kinds of arith-
metic, that of the ‘many’, the money-oriented traders and merchants, and
that of the people who philosophize. The first kind concerns itself with
things which are given a number (two oxen, two armies), the second withnumbers considered independently of things. A parallel distinction is intro-
duced between geometry for a concrete purpose on the one hand and
philosophical geometry on the other. The distinction is not neutral: ‘the arts
which are stirred by the impulse of the true philosophers are immeasurably
superior in accuracy and truth about measures and numbers’.
11
Plato thus establishes a boundary between ‘good’ and ‘bad’, or, more
accurately, ‘better’ and ‘worse’ mathematics. The two operate in a continuum,
in that they both perform the same operations, or generally speaking talkabout the same things. Some people use numbers to count money, some
others to reflect about Forms. What in his view irremediably separates them,
and gives them different value, is the use they make of their subject-matter,
the purpose they have.

44EARLY GREEK MATHEMATICS: THE QUESTIONS
Plato’s reflections on the ethics of knowledge involved not only
mathematics, but also rhetoric, medicine, and, more generally, the technai ,
which were object of wide debate between the fifth and the fourth centuries
BC.12 Opinions about their nature and status veered between on the one
hand equating techne and science ( episteme ), or claiming that anybody who
wanted a reputation in philosophy had to learn as many of the technai as
possible, and, on the other hand, denying that some arts could exist at all,
or definitely subordinating techne to science, for reasons that included the
former’s variability or its lack of proof. One of the big issues for Plato waswhether moral knowledge and politics were technai : was there such a thing
as an expert in morals, the way there were experts in medicine, gymnastics,
horse-rearing? Could happiness and justice be taught and learnt, the wayone did with building a house or making a statue? And could one reach a
criterion that would enable him or her always to take the best decision?
What would seem to be our salvation in life? Would it be the art
of measurement or the power of appearance? While the power of
appearance often makes us wander all over the place confused andregretting our actions and choices, both great and small, the art of
measurement, in contrast, would make the appearances lose their
power by showing us the truth, would give us peace of mind firmlyrooted in the truth and would save our life. […] What if our salva-
tion in life depended on our choices of odd and even, when the
greater and the lesser had to be counted correctly […] What thenwould save our life? Surely nothing other than knowledge, specific-
ally some kind of measurement, since that is the art of the greater
and the lesser? In fact, nothing other than arithmetic, since it’s aquestion of the odd and even? Would most people agree with us
or not?
13
Despite this passage, nowhere in his works does Plato even attempt to
quantify actions, or goods, or pleasures. The attractiveness of a mathematical
model for ethics lies not in its actual feasibility, but in the fact that it evokes
accuracy and incontrovertibility. There never is any question that four is
greater than three, for instance:
If you and I were to disagree about number, for instance, which of
two numbers were the greater, would the disagreement about thesematters make us enemies and make us angry with each other, or
should we not quickly settle it by resorting to arithmetic? Of course
we should. Then, too, if we were to disagree about the relative sizeof things, we should quickly put an end to the disagreement by

45EARLY GREEK MATHEMATICS: THE QUESTIONS
measuring? Yes. And we should, I suppose, come to terms about
relative weights by weighing? Of course?14
In other words, if Plato was after a superior techne to use as a model for,
or somehow transfer to, the ethical and political field, mathematics was a
very strong candidate. In the event, he came to the conclusion in his middleand later works that the reduction of political and moral knowledge to a
techne , especially as promoted by the sophists, had a number of undesirable
consequences. First of all, it was not clear whether anybody could be an
expert in politics (in fact, the Republic puts forth that only the philosopher-
rulers could); also, it was necessary to distinguish true moral and political
knowledge from pseudo-political technai such as, above all, rhetoric. Further,
an expert in an art was better at lying about it than someone who was
simply ignorant of it, and being a technical expert did not amount to knowing
good from evil. Even in politics, one may have learnt how to persuade peopleof a course of action, but not how to determine the best course of action.
One may have known how to gain knowledge, but not how to put it to its
best use. To employ an analogy:
no part of actual hunting […] covers more than the province of
chasing and overcoming; and when they have overcome the creaturethey are chasing, they are unable to use it: the huntsmen and fisher-
men hand it over to the cooks, and so it is too with the geometers,
astronomers, and calculators – for these also are hunters in theirway […] – and so, not knowing how to use their prey, but only
how to hunt, I take it they hand over their discoveries to the dialecti-
cians to use properly.
15
Activities that produced or gained knowledge were to be subordinated tomore discerning activities, best able to use that knowledge. Accordingly,mathematics was subordinated to philosophy or dialectic.
In sum, Plato constructed a dichotomy between a mathematics that did
not let itself be guided by philosophy, and a mathematics which handed
over its direction to philosophers. The latter stayed true to the nature of
mathematics itself, at least as Plato saw it, and grew and prospered. Theformer instead pursued goals which were not necessarily good (such as
material enrichment) and thus remained at best fundamentally blind, stunted
and misguided, at worst, it led to abuse and wrongdoing. Clearly, his reflec-tions about knowledge and its uses are inseparable from a consideration of
the people involved. As there are two kinds of arithmetic, there are two kinds
of arithmeticians: tradesmen, builders, the common and morally undisci-plined layman on one side, true philosophers, philosopher-rulers, wise legis-

46EARLY GREEK MATHEMATICS: THE QUESTIONS
lators on the other. If some ways of doing mathematics had indeed come to
symbolize democratic activities or values, Plato’s reminder that mathematics
is only good when supervised by an ethically and philosophically informedelite is sending a clear political message, which is perhaps nowhere as clear
as in his last dialogue.
The Laws is a description of Plato’s second-best ideal state, conceived as
a colony. Its political leadership, as in the Republic , is both restricted to a
few individuals, and strictly associated with knowledge, including mathe-
matical knowledge of a certain kind. The number of citizens for the newstate is mathematically regulated: 5,040, not a person more, not a person
less. The advantage of the number 5,040 is that it has the largest number of
consecutive divisors, making it possible to employ it everywhere:
this is the mathematical framework which will yield you your
phratries and demes and villages, as well as the military companiesand platoons, and also the coinage-system, dry and liquid measures,
and weights. The law must regulate all these details so that the
proper proportions and correspondences are observed. […] [Thelegislator] will assume it is a general rule that numerical division
and variation can be usefully applied to everything – to arithmetical
variations and to the geometrical variations of surfaces and solids,and also to those of sounds, and of motions […] The legislator
should take all this into account and instruct all his citizens to hold
fast, so far as they can, to this system. For in relation to householdadministration, to politics and to all the arts ( technai ), no single
branch of educational learning has so great a power as the study of
numbers. […] These subjects [number and calculation] will provefair and fitting, provided that you can remove pettiness and greed
[…] otherwise you will find that you have unwittingly produced a
rascal […] instead of a sage: examples of this we can see today inthe effect produced on the Egyptians and Phoenicians and many
other nations by the petty character of their approach to wealth
and life in general.
16
It is precisely because numbers are so powerful that their use has to be
regulated: although everybody has to get a basic smattering of mathematics,
the subject is to be studied in depth by only a chosen few, who would know
how to use it in the right way.17 Indeed, not all mathematical notions are
beneficial: for instance, simple division of goods in equal shares, which would
amount to equality ‘according to measures, weights and numbers’, is a rash
idea, and not a realistic possibility. Political participation depends on another,much preferable, type of equality: the one according to nature, i.e. according

47EARLY GREEK MATHEMATICS: THE QUESTIONS
to what everybody deserves. Given that, admittedly, ‘natural’ equality is very
difficult to assess for humans who do not have ‘the wisdom and judgement of
Zeus’, this means that some people are more equal than others.
A similar type of mathematical politics, or political mathematics, is found
in Aristotle. Like Plato, he established a correspondence between epistemic
and social hierarchy. In the Metaphysics he put theoretical knowledge,
including mathematics, at the top of a value scale whose next steps down
were the techne of the master-worker, the experience of the manual worker
(‘we think the manual workers are like certain lifeless things which act indeed,but act without knowing what they do’) and finally sense-perception pure
and simple, like animals have. For Aristotle, the pursuit of knowledge
presupposed leisure, freedom from daily cares and independence, all of themprerogatives of the privileged classes. He emphasized the analogy between
knowledge and power:
of the sciences […] that which is desirable on its own account and
for the sake of knowing […] is more of the nature of wisdom than
that which is desirable on account of its results, and the superiorscience is more of the nature of wisdom than the ancillary; for the
wise man must not be ordered but must order, and he must not
obey another, but the less wise must obey him.
18
Aristotle’s version of the origin of mathematics is also quite revealing:
as more arts were invented, and some were directed to the necessities
of life, others to its recreation, the inventors of the latter were
always regarded as wiser than the inventors of the former, becausetheir branches of knowledge did not aim at utility. Hence when all
such inventions were already established, the sciences which do
not aim at giving pleasure or at the necessities of life werediscovered, and first in the places where men first began to have
leisure. This is why the mathematical arts were founded in Egypt;
for there the priestly caste was allowed to be at leisure.
19
While the geographical attribution, so to speak, is at odds with Plato’s pictureof the greedy Orientals, the value distribution is the same: the original and
true nature of mathematics is detached from any material, common, concrete
uses; the wiser sort of mathematician is not someone who has to work for a
living. The leisured man is also in a better position to cultivate philosophy,
and understand what is better for the state. In fact, two more passages in
Aristotle blithely associate the right sort of mathematics and the right sortof politics. The first reprises Plato’s double notion of equality:

48EARLY GREEK MATHEMATICS: THE QUESTIONS
Justice involves at least four terms, namely, two persons for whom
it is just and two shares which are just. And there will be the same
equality between the shares as between the persons, since the ratiobetween the shares will be equal to the ratio between the persons;
for if the persons are not equal, they will not have equal shares […]
Justice is therefore a sort of proportion; for proportion is not aproperty of numerical quantity only, but of quantity in general,
proportion being equality of ratios, and involving four terms at
least […] The principle of distributive justice, therefore, is theconjunction of the first term of a proportion with the third and of
the second with the fourth; and the just in this sense is a mean
between two extremes that are disproportionate, since the propor-tionate is a mean, and the just is the proportionate. This kind of
proportion is termed by mathematicians geometrical proportion
[…] But the just in private transactions […] is not the equalaccording to geometrical but according to arithmetical proportion.
For it makes no difference whether a good man has defrauded a
bad man or a bad man a good one […] the law looks only at thenature of the damage, treating the parties as equal […] Hence the
unjust being here the unequal, the judge endeavours to equalize it
[…] if we represent the matter by a line divided into two unequalparts, he takes away from the greater segment that portion by which
it exceeds one-half of the whole line, and adds it to the lesser
segment. When the whole has been divided into two halves, peoplethen say that they ‘have their own’, having got what is equal. This
is indeed the origin of the word dikaion (just): it means dicha (in
half).
20
There is a subtle distinction here between criminal justice (‘the just in
private transactions’) and economic, or distributive, justice. The first canafford to apply full equality, or arithmetical proportion. Even supporters of
oligarchic forms of government were prepared to subscribe to the principle
that everybody is the same in front of the law: even Plato’s ideal state has
auditors. When it comes to the big divide between rich and poor, however,
equality ‘by merit’ in the form of geometrical proportion rears its head. Letus look at a second Aristotelian passage:
party strife is everywhere due to inequality, where classes that are
unequal do not receive a share of power in proportion […] for
generally the motive for factious strife is the desire for equality.
But equality is of two kinds, numerical equality and equalityaccording to worth […] the proper course is to employ numerical

49EARLY GREEK MATHEMATICS: THE QUESTIONS
equality in some things and equality according to worth in others
[…] what is thought to be the extreme form of democracy and of
popular government comes about as a result of the principle ofjustice that is admitted to be democratic, and this is for all to have
equality according to number. […] But the question follows, how
will they have equality? Are the property-assessments of fivehundred citizens to be divided among a thousand and the thousand
to have equal power to the five hundred?
21
In a sense, the supporters of extreme democracy are making a mathemati-
cal mistake: their claims are disproportionate, as anybody can verify, because
it is rather absurd to try and divide five hundred by a thousand, or to equalone thousand to five hundred. It is as if political mistakes, and even revolu-
tions, are caused by the wrong application of mathematical concepts, or by
the application of the wrong mathematical concepts.
In conclusion, mathematics in classical Athens had a number of practical
uses, and was very visible. The city kept accounts of its financial operations,
officers were obliged to give accounts, any citizen could be required to dosums for the benefit of the polis. The public counting involved was a ritual
that celebrated the transparency of, and involvement of the Athenian citizens
in, the running of the state; it was constructed as a deeply democratic type of
mathematics. As such, we find it in another crucial arena, i.e. the law courts,
where verdicts were reached by simple counting of votes. Also, as Plato
reminds us, we find a lot of counting in public in yet another crucial arenaof democracy: the moneyed economy of the marketplace. Yet, traders prover-
bially cheat under your own eyes, accounts can be fudged, calculations can
trick you, mathematics in the wrong hands goes the wrong way, is misapplied,misunderstood and becomes bad. In some texts, a dark side of mathematics
emerges, which corresponds to the pitfalls and shortcomings of democracy.
I think it is evident that mathematics was a ‘polyvalent symbol within a
complex symbolic system’ and that there was a struggle over who controlled
its signification.
22 What made the stakes particularly high was the fact that
early Greek mathematics was, through accounts, commercial transactions,
architecture, in the simple and banal forms of measuring and counting,
part of the experience of many people, not just of the highly educated; itwas a techne , which means it could be learnt and taught by virtually anybody.
Moreover, already at the time, mathematics was viewed as an objective,
certain, persuasive, form of knowledge. It was, to use anachronistic terms, ascience. A lot of things can be used and are used as signifiers in all periods
to talk about, say, the social and political order – what makes the use of
science a particularly strong signifier is that it claims to be objectiveknowledge, it is used as a signifier because it projects that aura of objectivity

50EARLY GREEK MATHEMATICS: THE QUESTIONS
onto political and social discourse. This is at least part of the reason why
Plato, for instance, upheld an ideal of mathematics as a philosophical,
detached, elite, pursuit – in order to reappropriate it. Controlling thesignification of science, in other words, was, and continues to be, a
particularly powerful way to control other areas of discourse.
The problem of later early Greek mathematics
My introductory disclaimer, that, given the limitations of space, many things
had to be left out of this book, applies particularly to this section. In an
attempt to introduce the novice to the raw business of squeezing reliable
evidence out of unlikely informers, I will focus on seven ancient sources(Archimedes, Philodemus, Plutarch, Diogenes Laertius, Proclus, Simplicius
and Eutocius), rather than discussing modern reconstructions of early Greek
mathematics, especially since they inevitably use those same ancient sourcesanyway. Each of the chosen seven had his own reasons for citing pieces of
mathematics from the past, but we cannot always reconstruct their agenda
in full detail; so, while those reasons obviously affect their reliability, thishappens to an extent we can only (educatedly) guess.
Archimedes tells us precious little: that some of the ancient geometers
devoted many efforts to squaring the circle, without really succeeding becausethey supported their proofs with inadmissible lemmas. What they did
manage to prove was that circles are to each other like the squares on their
diameters; that spheres are to each other like the cubes on their diameters;and that pyramids and cones are one third of prisms and cylinders
(respectively) with the same base and the same height. All these results relied
on the lemma that two unequal surfaces can be continuously subtractedone from the other, until their difference is smaller than any given area.
Now, on another occasion, Archimedes explicitly attributes to Eudoxus the
proposition about cones and cylinders, adding that ‘one should give nolittle credit to Democritus, as the first to formulate the statement about
that figure, without proof’. We can safely assume, then, that Eudoxus was
the discoverer of the lemma and of the other two results, too.
23 This
attribution is very important because the lemma in question is extensively
employed in Euclid’s Elements – one can infer that the results contained in
the Elements which depend on the lemma may have also been discovered by
Eudoxus. How far one goes with attributions by inference depends of course
on how original one thinks Euclid was, or how advanced one thinksmathematics was in the fourth century
BC.
Little precious remarks can go a long way, and produce a sort of chain
reaction effect, in that they can corroborate the testimony of other authorsand reflect positively on their reliability in general. For instance, if we take

51EARLY GREEK MATHEMATICS: THE QUESTIONS
Archimedes to be reliable, then information about Democritus can be used
to substantiate a passage in Plutarch. The fact that Archimedes seems to
concord with Plutarch on one point makes Plutarch rather more reliable ingeneral, perhaps even when he is reporting things for which we have no
external corroboration.
In fact, Archimedes is considered a very trustworthy source. Not only
was he relatively close in time to Democritus and Eudoxus: he was a
mathematician, so we can assume that he understood the material and had
access to a wide range of treatises and results, either in person or throughhis extensive and well-documented contacts with mathematicians in
Alexandria. Not only was Archimedes a mathematician, he was a very good,
arguably the best ancient mathematician, so again that adds to his reliabilitybecause we tend to make an (anachronistic) equation between scientific
expertise and professional ethics. We believe that Archimedes was what we
could call intellectually honest, that he apportioned praise and blame whereit was due, so that when he said Democritus had done something, that is
something Democritus had indeed done. Nobody I know (not even me)
doubts Archimedes’ testimony on Democritus and Eudoxus. Nevertheless,I would invite reflection on one point: Archimedes himself is writing history
of mathematics. We are trusting him as a historian, because we trust him as
a mathematician. He identifies a development in the research on a particularset of problems, with Democritus formulating the statement but not
providing the proof, other ancient geometers coming up with proofs which
were incorrect because they relied on inadmissible principles, and finallyEudoxus discovering the right lemma and thus proving a number of results.
Archimedes himself builds on Eudoxus’ achievement and takes a version of
his lemma as a starting-point. It is a cumulative vision of mathematics,which also corresponds to a certain vision of the heuristic process: the
statement is arrived at first, then a proof is sought, using the right auxiliary
propositions. We know that this was more or less Archimedes’ own heuristicprocess in his own actual practice, because he tells us as much. The historical
development of mathematical research thus mirrors, or is made to mirror,
the individual experience of the mathematician. History reflects on a larger
scale what happens within a lifetime of work.
We next find Eudoxus as one of the main characters in Philodemus’
history of Plato’s Academy, in its turn based on Dicearchus:
At that time there was also a remarkable advancement in the
mathematical studies, with Plato as architect and propounder of
problems, and the mathematicians then researching them with
zeal. Thus metrology and the problems about definitions reachedan acme for the first time, while Eudoxus and his circle renewed

52EARLY GREEK MATHEMATICS: THE QUESTIONS
completely the original results of Hippocrates. Geometry also
advanced greatly. Then originated analysis and the propositions
about diorismoi , and overall geometry was taken greatly forwards.
Nor were optics or mechanics left behind.24
Apart from the reference to diorismoi , the testimony is rather vague: Plato’s
Academy fostered mathematical research, Eudoxus and other people were
associated with it and they had wide interests. This much is corroborated
by other sources. That said, we also have to remember that Philodemus waswriting a celebratory history of his philosophical school of choice, that we
have no indication that he was himself a mathematician, and that some
three centuries separate him from Eudoxus’ time. The image of the Academyas a fertile greenhouse of mathematical talent is a recurrent one in Platonist
traditions, as is the figure of Plato presiding over the mathematicians, telling
them what to research, and keeping their eyes steadily on the real prize. Theparallel here between Plato and an architect, someone who directs the work
of very skilled technitai , while being himself in possession of a superior kind
of knowledge, the only one who can see where it is all going, fits in very wellwith Plato’s views as sketched in the previous section. Continuity is stressed
by the reference to Hippocrates of Chios, but the emphasis is primarily on
renewal, originality and climaxing ‘for the first time’. Philodemus, and/orDicearchus, seem keen to characterize Platonist mathematics as different
from analogous, but not equally philosophically-informed, practices.
Both Eudoxus and Democritus figure in Plutarch, who reports a mathe-
matical puzzle attributed to the latter:
if a cone should be cut by a plane parallel to its base, what one
must suppose the surfaces of the segments prove to be, equal or
unequal? – for, if unequal, they will make the cone uneven by
giving it many step-like notches and asperities; and, if they areequal, the segments will be equal, and the cone, being composed
of circles that are equal and not unequal, will manifestly have got
the properties of the cylinder – which is the height of absurdity.
25
While, for Archimedes, Democritus realized the equivalence of a certain
cone and a certain cylinder, Plutarch reports a more detailed and polemical
discussion of the difficulties involved in studying the cone. No easy connec-
tion can be made between the two – it could be that Democritus exploredgeometrical paradoxes in an attempt to prove what he had simply formulated,
or that he happened upon the formulation in the context of his reflections
on geometrical paradoxes. Can we trust Plutarch as to the details? Perhapsyes: from the rest of his work it is clear that he was quite knowledgeable

53EARLY GREEK MATHEMATICS: THE QUESTIONS
about mathematics, and that he had access to many early sources. The
testimony is not out of character as far as Democritus, one of the founding
fathers of atomism, is concerned, but it also seems to be contained in anintermediate source, rather than being quoted from Democritus’ works
themselves – they were probably lost by the time Plutarch was writing. If
we move on to Eudoxus, we find ourselves again on rather shaky ground.Take this passage:
Plato himself reproached Eudoxus and Archytas and Menaechmus
for setting out to move the problem of doubling the cube into the
realm of instruments and mechanical constructions, as if they were
trying to find two mean proportionals not by the use of reasonbut in whatever way would work.
26
It was indeed to Plato, in Plutarch’s narration, that a delegation of people
from Delos turned for a solution to the duplication of the cube – once
again, Plato appears in the role of ringmaster of mathematical practice.27
Plutarch was well acquainted, if not with Philodemus/Dicearchus, with other
works in the same tradition which depicted the relation between Plato and
the mathematicians around him in pretty much the same light. Now, it
seems from other sources that Eudoxus, Archytas and Menaechmus did infact provide solutions to the duplication of the cube, so for some aspects
Plutarch appears to be trustworthy. Nevertheless, a certain image of Plato
was crucial to Plutarch’s view of mathematics as a whole, so it was importantfor him to emphasize the significance of Platonic input and overall super-
vision. The fact that our survey now counts two reports, both agreeing on
Plato’s ‘architectural’ function, does not necessarily increase their veracityquotient, because the earlier one, or a version of the earlier one, may have
been read by the later author, who shared similar interest and philosophical
loyalties. In fact, there is a notable tendency for stories like these toaccumulate more and more details as they are passed on: Eudoxus and his
circle have now become Eudoxus, Archytas, Menaechmus and Helicon of
Cyzicus. The past gets suspiciously clearer the further away we get from it.
Diogenes Laertius’ main work is a sort of biographical dictionary of
great philosophers, from Thales to Epicurus. He gathered as many pieces ofinformation as he could, sometimes in contrast with one another, always
indicating his sources (many of which are otherwise totally unknown to
us), occasionally assessing their credibility. Of Democritus, for instance,Diogenes confirms that he was ‘versed in every department of philosophy’,
including mathematics, which he had learnt while travelling in Egypt.
Eudoxus, ‘an astronomer, a geometer, a physician and a legislator’ is includedamong the famous Pythagoreans; he also discovered the properties of curves

54EARLY GREEK MATHEMATICS: THE QUESTIONS
and learnt geometry from Archytas. Democritus and Eudoxus were not
unique in their choice of topics for investigation: Thales, whom earlier
sources tend to depict as a rather unspecifically wise man, interested inastronomy, capable of engineering feats, according to Diogenes also learnt
geometry in Egypt, where he measured the height of the pyramids by means
of their shadow (probably using the properties of similar triangles). He ‘wasthe first to inscribe a right-angled triangle in a circle, whereupon he sacrificed
an ox. Others tell this tale of Pythagoras’. In the entry on Pythagoras,
however, the sacrifice is motivated by the discovery that in a right-angledtriangle the square on the hypotenuse is equal to the squares on the sides
containing the right angle, i.e. what is still called Pythagoras’ theorem.
Eudoxus’ alleged teacher, the Pythagorean Archytas of Tarentum, as well asbeing general of his city seven times in a row, applied mathematical principles
to mechanics and, conversely, ‘employed mechanical motion in a geometrical
construction, namely, when he tried, by means of a section of a half-cylinder,to find two mean proportionals in order to duplicate the cube’.
28
The wealth of details should not blind us to the fact that there is a general
plan, an ideally complete map of philosophy, from its remote origins, in acontinuous stream, all the way to more recent times. Diogenes systematized
his information in at least two ways: he classified people on the basis of the
school they belonged to, in some cases forcibly enlisting them to a school(how else would Eudoxus become a Pythagorean?), and he created teacher-
pupil links between them. The detectable presence of a general plan need
not make one entirely suspicious of individual elements – Diogenes after alladmits to uncertainty about Pythagoras’ discoveries, and his information
on Archytas may confirm what we know from Plutarch. Yet, one cannot
help being struck by the sheer abundance of information about very earlyfigures like Thales, who was already a sort of semi-mythical character by
Aristophanes’ time.
Thales figures again in Proclus’ potted history of mathematics:
Thales, who had travelled to Egypt, was the first to introduce
[geometry] into Greece. He made many discoveries himself and
taught the principles for many others to his successors, attacking
some problems in a general way and others more empirically. Nextafter him Mamercus […] is remembered as having applied himself
to the study of geometry […] Following upon these men, Pythagoras
transformed mathematical philosophy into a scheme of liberaleducation, surveying its principles from the highest downwards
and investigating its theorems in an immaterial and intellectual
manner. He it was who discovered the doctrine of proportionalsand the structure of the cosmic figures. After him Anaxagoras of

55EARLY GREEK MATHEMATICS: THE QUESTIONS
Clazomenae applied himself to many questions in geometry, and
so did Oenopides of Chios […] Following them Hippocrates of
Chios, who invented the method of squaring lunules, andTheodorus of Cyrene became eminent in geometry. For
Hippocrates wrote a book of elements, the first of whom we have
any record who did so. Plato, who appeared after them, greatlyadvanced mathematics in general and geometry in particular […]
At this time also lived Leodamas of Thasos, Archytas of Tarentum,
and Theaetetus of Athens, by whom the theorems were increasedin number and brought into a more scientific arrangement. Younger
than Leodamas were Neoclides and his pupil Leon, who […] was
able to compile a book of elements more carefully designed […]He also discovered diorismoi […] Eudoxus of Cnidus, a little later
than Leon and a member of Plato’s group, was the first to increase
the number of the so-called general theorems; to the three meansalready known he added three more and multiplied the number
of propositions concerning the ‘section’ which had their origin in
Plato, employing the method of analysis for their solution. Amyclasof Heracleia, one of Plato’s followers, Menaechmus, a student of
Eudoxus […] and his brother Dinostratus made the whole of
geometry still more perfect. Theudius of Magnesia […] producedan admirable arrangement of the elements and made many partial
theorems more general. There was also Athenaeus of Cyzicus, who
[…] became eminent in other branches of mathematics and mostof all in geometry. These men lived together in the Academy,
making their enquiries in common. Hermotimus of Colophon
pursued further the investigations already begun by Eudoxus andTheaetetus, discovered many propositions in the Elements , and
wrote some things about locus-theorems. Philippus of Mende […]
also carried on his investigations according to Plato’s instructions[…] Not long after these men came Euclid.
29
Some of the information in this passage is well-substantiated by contem-
porary sources: Aristotle may be taken to confirm that Hippocrates investiga-
ted the quadrature of lunules, Plato tells us that Theodorus and Theaetetusmade contributions to geometry. Some other statements match the sources
we have surveyed in this section: Thales made a trip to Egypt as in Diogenes
Laertius, Eudoxus employed the method of analysis as in Philodemus, aconnection is made between Eudoxus and Menaechmus as in Plutarch. As
we have pointed out before, however, this indicates a common source and a
partly shared agenda, more than constituting mutual corroboration. Severalof the mathematicians above are hardly mentioned by sources significantly

56EARLY GREEK MATHEMATICS: THE QUESTIONS
earlier than Proclus, and when they are, it is for their enquiries into
astronomy and natural philosophy rather than for any mathematical
contributions they might have made. Above all, many of the names onProclus’ list are not known outside it: Mamercus, Neoclides, Leon, Amyclas,
Theudius, Athenaeus, Hermotimus. There is at present no way of
establishing what these people did, who they were or (if one wants to bereally suspicious) whether they existed at all.
Proclus’ picture worryingly looks as if he had fleshed out a by then well-
established canon of names and lines of transmission, with the aim ofconstructing a progressive development towards greater generalization and
theoretical sophistication. His mathematicians seem to flow more or less
seamlessly into each other’s lives, inheriting problems from the previousgeneration; the role of Plato, the Academy and their particular interests is
seen as crucial; the production of perfectly-arranged ‘elements’ is presented
ever since Hippocrates as a major concern, culminating with Euclid. Onceagain, there is no cogent reason to think that individual details are wrong –
taking the picture wholesale as historically accurate, however, is a different
matter.
In fact, Proclus’ list is an interesting case, because of its possible association
with Eudemus. A pupil of Aristotle’s, Eudemus wrote a history of geometry,
now lost, from the unique vantage-point of a contemporary witness of theactivity between the fourth and third century
BC. Now, in other parts of the
same commentary to Euclid, Proclus alleges Eudemus as his source. Although
he is generally seen as too late, too philosophically involved and not enoughof a mathematician to be fool-proof, this has had the power to give him
some credibility. Not a few modern scholars seem to have clung to his report,
one of the most complete and detailed pieces of evidence we have about acrucial period of Greek mathematics, as the only rock in a sea of vague,
undetailed evidence. Yet, one should sceptically point out that, first, it is
fair to expect Proclus to be more interested in putting his own points acrossthan in quoting Eudemus word for word; second, that we have no guarantee
that Eudemus’ report was itself historically accurate and immune from philo-
sophical biases; third, that it is unlikely that Eudemus’ text would survive
in its original form until Proclus’ time.
Simplicius, next on our list, records another little miracle of his own,
also taken from Eudemus’ book. You may remember that Aristotle often
used mathematical examples. We are not the first readers to find his allusions
rather cryptic – a late ancient audience also needed to be filled in on thedetails. Simplicius was a sixth-century
AD writer whose main extant work
consists of commentaries to Aristotle’s books, including explanations of
passages which had by then become unclear. He explains Aristotle’s referenceto ‘squaring by means of sections’ thus:

57EARLY GREEK MATHEMATICS: THE QUESTIONS
The squaring by means of sections is the squaring by means of
lunules, discovered by Hippocrates of Chios; for the lunule is a
section of circle. Eudemus, however, in his history of geometrysays that Hippocrates demonstrated the quadrature of the lunule
not on the side of a square but in general […] For every lunule has
an outer circumference equal to a semicircle or greater or less, andif Hippocrates squared the lune having an outer circumference
equal to a semicircle and greater and less, it would seem that the
quadrature was proved in general. I shall set out what was said byEudemus word for word, adding a few things from Euclid’s Elements
for the sake of clarity since I am reminded of the summary style of
Eudemus, who set out the proofs concisely, according to the ancientcustom. He says this in the second book of the history of geometry.
And the quadratures of lunules, which seemed to be part of the
complex propositions because of its kinship with the circle, werefirst written about by Hippocrates, and seemed to be properly
carried out […] He made a starting-point, and posited as first thing
useful for these, that similar segments of circles have the sameratio to each other as the squares on their bases. And this he proved
by showing that the squares on the diameters have the same ratio
as the circles. […]
30
Simplicius’ text retains at least partly the existence of various layers of trans-mission; he shows awareness of the difference between what he himself issaying, what Eudemus was saying and what Hippocrates may have been
saying in his turn, but that is not his only concern, because he also wants
his readers to understand him, if necessary by modifying the text for thesake of clarity. Can we take Simplicius to be a ‘faithful’ testimonial of
Hippocrates’ mathematical practice? Then again, how can we be sure that
he really had Eudemus’ text, and that Eudemus really had Hippocrates’text?
To conclude with another little sixth-century
AD miracle, Eutocius. In
the second book of his Sphere and Cylinder , Archimedes takes the solution
to the problem of the two mean proportionals as granted. Some nine
centuries later, Eutocius of Ascalona, who was writing a commentary onArchimedes’ text, decided to expand on that point by reporting several
solutions to the problem. His anthology includes a duplication of the cube
attributed to Plato, which is carried out by means of a moving ruler (thuscontradicting Plutarch’s account), a solution by Menaechmus, one by
Archimedes’ contemporary Diocles, which is not attested elsewhere but has
been corroborated by the recovery of Diocles’ book in an Arabic version,and a long passage by (allegedly) Eratosthenes:

58EARLY GREEK MATHEMATICS: THE QUESTIONS
It was researched by the geometers in what way to double the
given solid, it retaining the same shape, and this problem was
called the duplication of the cube; for, having posited a cube, theysought to double it. Nobody having solved it for a long time,
Hippocrates of Chios first came up with the idea that, if two mean
proportionals taken in continued proportion were discoveredbetween two straight lines, of which the greater was double the
lesser, the cube would be doubled, so that the puzzle was by him
turned into another puzzle as great as the first. After a time, theysay, certain Delians, trying to double a certain altar in accordance
with an oracle, were stuck with the same puzzle, and were sent to
ask the geometers who were with Plato in the Academy if theycould discover what they sought. Having applied themselves
diligently and seeking to find two mean proportionals between
two given straight lines, Archytas of Tarentum is said to havediscovered them by means of the semicylinders, and Eudoxus by
means of the so-called curved lines; but it so happened that while
all these were written in a demonstrative fashion, it was not possibleto make them handy and put them to use, except to a certain
small extent Menaechmus, and even that with difficulty.
31
Again a micro-history of mathematics, this time focussed on the duplica-
tion of the cube, and contradicting some elements of Plutarch’s story, because,
far from being too compromised with practice, Archytas’ and Eudoxus’solutions are not considered practical enough. In fact, at least in the former
case, Eutocius’ readers could judge for themselves, because he reports
Archytas’ solution in full, and on the authority of Eudemus (see Diagram2.1):
Let the two given straight lines be AD, C; it is necessary to find
two mean proportionals between AD, C. Let the circle ABDF be
described around the greater straight line AD, and let AB be fitted
in equal to C and let it be prolonged until it meets the tangent to
the circle from D in the point P, let BEF be drawn parallel to
PDO , and let a right semicylinder be imagined on the semicircle
ABD , and on AD a right semicircle posited in the parallelogram
of the semicylinder. This semicircle rotating as if from D to B, the
end A of the diameter remaining at rest, it will cut the cylindrical
surface in its rotation and will describe a certain curve in it. Again,
if AD remains at rest and the triangle APD goes around with a
motion opposite to the semicircle, it will produce a conic surfacewith the straight line AP, which moving around will come across

59EARLY GREEK MATHEMATICS: THE QUESTIONS
the cylindrical line in a certain point; at the same time B will
describe a semicircle on the surface of the cone. Let the moving
semicircle have a position according to the point of junction of
the curves, such as the position D´JA, and the triangle moved in
the opposite direction a position DKA ; let the point of the said
junction be J, and let the semicircle described through B be BLF ,
and let BF be the section common to it and to the circle BDFA ,
and let there be drawn from J a perpendicular to the plane of the
semicircle BDA ; it will fall on the circumference of the circle
because the cylinder is right. Let it fall, and let it be KI, and let the
line joining I and A come across BF at the point H; let AK come
across the semicircle BLF at the point L, and let JD, LI, LH be
joined. Now, since each of the semicircles D´JA, BLF is perpen-
dicular to the underlying plane, therefore their common section
LH is also perpendicular to the plane of the circle; so that LH is
also perpendicular to BF. Therefore the rectangle from BH, HF,Diagram 2.1

60EARLY GREEK MATHEMATICS: THE QUESTIONS
which is the same as the rectangle from AH, HI, is equal to the
square on LH; therefore the triangle ALI is similar to each of
LIH, LAH , and the angle ILA is right. The angle D´JA is also
right. Therefore J´D, LI are parallel, and because of the similarity
of the triangles there will be the proportion: D´A is to AJ, as JA
is AI, as AI is to AL. Therefore the four straight lines DA, AK,
AI, AL are in continuous proportion. And AL is equal to C,
since it is equal to AB; therefore two mean proportionals, AJ, AI,
have been found between the two given straight lines AD, C.32
Strange as it may sound, Hippocrates’ squaring of the circle, as in
Simplicius, and Archytas’ duplication of the cube, as in Eutocius, are theearliest ‘extant’ Greek mathematics we have, and they are both credited to
Eudemus. Are they to be trusted, and the felicity of such transmissional
accidents celebrated? Or should we postulate that ‘Eudemus’ really was alate compilation, perhaps deliberately couched in archaicizing language,
perhaps even a forgery?
As I said, I have no ready answers. I will just issue a further warning. All
the sources we surveyed, including Archimedes, were writing their own little
histories of early Greek mathematics, and all for different reasons. One
should bear that in mind, and not view them as mere reservoirs of informa-tion. That we should gather more minutely detailed information about fifth-
century
BC geometry from sixth-century AD authors than we are able to do
from their own contemporaries, is only one of the succulent paradoxes ofhistory, and we may never be able completely to unravel it.
Notes
1 von Reden (1995) and (1997); Morris (1996); Kurke (1999).
2 Aristotle, Nicomachean Ethics 1133a–b, tr. W.D. Ross revised by J.O. Urmson, Princeton
1984, with modifications.
3 Plato, Laws 945b ff.; Aristotle, Politics 1318b; 1322b.
4 Aristotle, Politics 1314b.
5 Plato, Lesser Hippias 367a–c, tr. N.D. Smith, Hackett 1992. Cf. also Greater Hippias
281a, 285b–c; Protagoras 318d–e and Xenophon, Memorabilia 4.4.7.
6 See Untersteiner (1961), chapter 2 and P. H e r c . 1676 in Gaiser (1988).
7 Xenophon, Memorabilia 4.7.2–4.
8 In Diogenes Laertius, Lives of the Philosophers 6.28, 73.
9 Isocrates, Panathenaicus 26–9 (written c.342 BC); Antidosis 261–9 ( c.354 BC); Helen 5
(c.370 BC); Against the Sophists 8 (c.390 BC). Cf. Aristotle, Metaphysics 995a.
10 Plato, Republic 522c–527a. Is Plato here responding to Aristophanes’ lampoon of Meton,
also a learned man involved in the foundation of an ideal city? For similar themes see also
Philebus 56d–57a and [Plato], Epinomis 990a–991e.
11 Plato, Philebus 57d.

61EARLY GREEK MATHEMATICS: THE QUESTIONS
12 See Cambiano (1991), to which my discussion is heavily indebted; Lloyd (1963); Roochnik
(1996).
13 Plato, Protagoras 356d–357a, tr. S. Lombardo and K. Bell, Hackett 1992. See also Philebus
19a–b; Statesman 283d–285a. A discussion of this in Nussbaum (1986), 89 ff.
14 Plato, Euthyphro 7b–c.
15 Plato, Euthydemus 290c. Cf. also Sophocles, Antigone 332–63 (probably written c. 450–
40 BC).
16 Plato, Laws 746d–747c, Loeb translation with modifications. See Harvey (1965);
Cartledge (1996).
17 Plato, Laws 817e–818a.
18 Aristotle, Metaphysics 981b–982a; Posterior Analytics 76a, 87a.
19 Aristotle, Metaphysics 981b18–26.
20 Aristotle, Nicomachean Ethics 1131a–1132a.
21 Aristotle, Politics 1301b–1302a; 1318a.
22 Adapting what Kurke (1999), 23 says about money.
23 Archimedes, Quadrature of the Parabola 262.13 ff.; Sphere and Cylinder I 4.9 ff.; Method
430.1 ff., respectively.
24 Philodemus, The Academy col.Y 2–18 in Gaiser (1988), my translation.
25 Plutarch, Against the Stoics on Common Conceptions 1079e–f and cf. T able-T alk 718e–f.
26 Plutarch, ibid. 718e–f, Loeb translation with modifications.
27 Plutarch, On the E in Delphi 386e and On the Genius of Socrates 579a–d (mentioning
Eudoxus and Helicon of Cyzicus).
28 Diogenes Laertius, Lives of the philosophers 9.34 ff. for Democritus; 8.86 ff. for Eudoxus;
1.22 ff. for Thales; 8.1 ff. (esp. 11 ff.) for Pythagoras; 8.79 ff. for Archytas.
29 Proclus, Commentary on the First Book of Euclid’s Elements 65–68, tr. by G.R. Morrow,
Princeton University Press 1970, reproduced with permission. A different version of the
same list in [Hero], Definitions 136.1. On Proclus’ list see Vitrac (1996).
30 Simplicius, Commentary on Aristotle’s Physics 55.26 ff., my translation.
31 Eutocius, Commentary on Archimedes’ Sphere and Cylinder II 88.4–90.13, my translation.
32 Eutocius, ibid. 84.12–88.2, my translation.

62HELLENISTIC MATHEMATICS: THE EVIDENCE
3
HELLENISTIC
MATHEMATICS:
THE EVIDENCE
Some of the numbers to which I have given a name […]
surpass not only the number of grains of sandthat could fill the Earth […] but eventhe number of grains of sand that could fill the universe itself.
1
At the battle of Chaeronea in 338 BC, Philip II of Macedonia defeated a
coalition of Greek states, and established control over the peninsula. He
was succeeded in 336 BC by his son Alexander, soon to be known as Alexander
the Great, who embarked on a military campaign and, in the space of a few
years (he died in 323), brought down the already shaky Persian Empire and
appropriated its immense former domains, stretching as far as NorthernIndia, and including Asia Minor, Syria and Egypt. What in Alexander’s
intentions, had he lived long enough, would probably have been a unified
mega-empire was eventually divided up among his successors, most of whomhad been officers in his army. The members of the new ruling elite (the
monarchs, their families, their associates) were prevalently Greek or Greek-
speaking and promoted the settlement of Greeks on their newly-acquired
territories – hence the term Hellenistic kingdoms for their states, and of
Hellenistic age for the period in which they flourished, roughly third tosecond century
BC, when Rome gradually established predominance in the
Mediterranean basin.
A few words on the nature of the evidence. What we have seen happening
with Athens in the fifth and fourth centuries BC more or less happens in
this period with Egypt. Many mathematicians or philosophers from other
parts of the world went to, or worked in, or corresponded with, peopleworking in Alexandria, the capital of the new Ptolemaic state. A lot of our
material evidence consists of texts written on papyrus, whose very survival
is due to dry and hot climatic conditions, such as are rarely found outsideEgypt. The most remarkable thing of all about the evidence for Hellenistic
mathematics, however, is that (at last!) we have whole real mathematical
treatises instead of scrappy, indirect information. So, the sections being

63HELLENISTIC MATHEMATICS: THE EVIDENCE
once again organized by type, we will look at the material evidence first;
then at texts by non-mathematicians, arbitrarily divided into rest of the
world and philosophers. To follow, the mathematicians themselves, divided(again, arbitrarily) into little people and Big Guys (Euclid, Archimedes,
and Apollonius). The sections on the Big Guys are in their turn sub-divided
into a description of contents – the topics they dealt with – and of procedures.
Material evidence
Until the fourth century BC, the main forms of attack against a fortified city
seem to have been to starve its inhabitants, bribe someone in order to have
them open the gates, or build a giant wooden horse and hope the enemieshad not heard the story before. Siege devices such as beams with chains to
ram down the gates are mentioned by Thucydides and were probably
extensively employed.
2 Around 399 BC, the story changed with the invention
of a new type of weapon: the catapult.
Early catapults probably looked like oversized bows, but their initially
limited range and power quickly improved. There are no archaeologicalremains of war engines earlier than the first century
AD, but we do have
indirect evidence of their presence from changes in fortification construction.
Following not upon the discovery of the catapult immediately, but ratherwith its established and widespread use in the early to mid-third century
BC, thicker walls were built, different wall designs adopted, the better to
resist the impact of projectiles, and larger and differently-shaped towerserected. These elements are in evidence at sites such as Heraclea on Latmus
(c. 300
BC, with a second phase of construction around the mid-third century
BC), Ephesus ( c. 290 BC), Fort Euryalus at Syracuse, whose latest phases
date from the end of the third century BC, Iasus (late third century BC), and
Doura Europus in Syria ( c. 300 BC). See Figure 3.1.
Some Hellenistic fortifications, for instance Doura Europus, Ephesus or
Fort Euryalus, exhibit geometrical patterns or structural details strikingly
similar to those described in the earliest extant treatise on military
architecture, the second-century BC Poliorketika (siege-craft) by Philo of
Byzantium.3 He described a zig-zag-shaped wall whose outline was a regular
combination of triangles, and towers of various types, from square andquadrangular to hexagonal to semicircular. The building blocks for those
latter had to be shaped with particular accuracy, by measuring the external
circumference and preparing models in wood. The text was complementedby figures, now lost. Philo also dealt with the storing of provisions in the
eventuality of a long siege: he specified that granaries had to be built
symmetrically, with the height in proportion to the size, ‘determining theheight of the ceiling arcs on the basis of the foundations’.

64HELLENISTIC MATHEMATICS: THE EVIDENCE
Philo also wrote a Belopoiika (construction of propulsion instruments),
addressed to the same person as the Poliorketika . Evidently, the same person
could wear the hat both of the military architect and of the machine-maker.4
A further hat in the wardrobe of people like Philo, perhaps of his addresseeAriston, or of the Polyeidus of Thessaly, Dionysius of Alexandria, Charias
of Magnesia mentioned by other third-century sources, would have beenthat of town-planner and surveyor.
5 It was often the same people who were
responsible for devising the city layout and the fortifications around it.
While depending on the geographical situation, regular geometric patterns,square or rectangular, have been observed in many Hellenistic town plans
such as that of, again, Doura Europus, Kassopeia near the Gulf of Arta
(third century
BC), Goritzas and Demetrias in Thessaly (also third century
BC), and the Roman colonies of Cosa (273 BC) and Rimini (268 BC).6Figure 3.1 Doura Europus with city plan and fortifications in evidence
(adapted from Garlam (1974))

65HELLENISTIC MATHEMATICS: THE EVIDENCE
Roman colonists were often retired soldiers, so it is perhaps not surprising
that the plan of their town often resembled that of their military camps,
which, in the opinion of the historian Polybius, exemplified what could becalled a geometrical mind-set:
the manner in which [the Romans] form their camp is as follows.
[…] Fixing an ensign on the spot where they are about to pitch
[the general’s tent], they measure off round this ensign a square
plot of ground each side of which is one hundred feet distant, sothat the total area measures four plethra . […] They now measure a
hundred feet from the front of all these tents, and starting from
the line drawn at this distance parallel to the tents of the tribunesthey begin to encamp the legions […] Bisecting the above line,
they start from this spot and along a line drawn at right angles to
the first, they encamp the cavalry […] The whole camp thus formsa square, and the way in which the streets are laid out and its
general arrangement give it the appearance of a town. […] Given
the numbers of cavalry and infantry, whether 4000 or 5000, ineach legion, and given likewise the depth, length, and number of
the troops and companies, the dimensions of the passages and
open spaces and all other details, anyone who gives his mind to itcan calculate the area and total perimeter of the camp.
7
Archaeological research has established that, while nowhere as standardizedas Polybius described, early remains of stone-built Roman camps reveal
indeed regular geometrical patterns.
8 Polybius also drew out the implications
of geometrically-organized space in a comparison of the Roman camp withthe Greek one:
The Romans by thus studying convenience in this matter pursue,
it seems to me, a course opposite to that usual among the Greeks.
The Greeks in encamping think it of primary importance to adapt
the camp to the natural advantages of the ground, first because
they shirk the labour of entrenching, and next because they think
handmade defences are not equal in value to the fortificationswhich nature provides unaided on the spot. So that as regards the
plan of the camp as a whole they are obliged to adopt all kinds of
figures to suit the nature of the ground, and they often have toshift the parts of the army to unsuitable situations, the consequence
being that everyone is quite uncertain whereabouts in the camp
his own place or the place of his corps is. The Romans on theother hand prefer to submit to the fatigue of entrenching and

66HELLENISTIC MATHEMATICS: THE EVIDENCE
other defensive work for the sake of the convenience of having a
single type of camp which never varies and is familiar to all.9
The symbolic significance of imposing a geometrical pattern on a territory
was evident in another context. As at earlier times, the establishment of a
new city came with the apportionment of the territory around it, whichneeded to be surveyed and divided up. We have evidence of such divisions
in, for instance, Larissa (perhaps end of third century
BC) and Halieis (perhaps
second century BC) in Thessaly, and in some areas conquered by the Romans
such as Northern Italy and North Africa (second century BC).10 The most
common pattern was rectangular, while the square shape, so pervasive in
Polybius’ account, is more commonly associated with Roman foundationsand land-divisions. As well as facilitating allocation, which was often carried
out through drawing of lots, the fact that the pieces of land were geometrically
uniform and consequently of equal size provided at least an appearance ofjustice in the distribution. Geometrical patterns signified fairness.
We have further evidence about land-surveys from Egypt, where the
Ptolemies took over well-established pre-existing administrative structures.The king was the biggest landlord in the country – not only did he lease
portions of land to local farmers and new settlers, he also told them what to
grow, and established a monopoly on various products such as oil. Thus,no-one but government officials was allowed to sell oil, and the possession
of instruments such as presses was regarded as a criminal offence. In order
to maintain such a degree of control over properties, tenants and theiractivities, periodical surveys were necessary. A typical geometria (for this is
the term commonly used) would have looked like this:
To the west entering the north along the canals that have been
surveyed before, proceeding from the east, the seven-aroura
cleruchic holding of Pathebis son of Teephraios, one of Chomenis’soldiers: the remainder
6
1/2 1/41 1/41 1/8 1/10 1/6 1 <the area is> 1 1/4 1/10 1/6, (wheat).
To the west proceeding from the south, the seven-aroura cleruchic
holding of Besis son of Kollouthes, one of Chomenis’ soldiers:6
1/2, crown 1/2 1/4 1/8 1/10 1/6 1/30 1/2, <total> 7 1/4 1/8 1/10 1/6 1/30 1/2,
at 41/2 each
6 1/2 1/8 1/10 1/6 1/30 1/21 1/8 1/10 1/6 6 1/8 1/10 1/6 1/30 1/2
<the area is> 8, excess 1/2 1/30 1/2, black cumin, self cultivated. To
the north coming from the east along the seven-aroura cleruchicholding that has been surveyed before,
1/4 1/10 1/6 schoinoi revenuesame
1 1/4 1/10 1/6

67HELLENISTIC MATHEMATICS: THE EVIDENCE
from those 3 1/6.
6 1/2 1/30 1/2 1/4 1/8 1/10 1/6 1/30 1/26 1/2 1/4,
<the area is> 3 1/4 1/10 1/6, of which lentils 1 black cumin 2 1/4 1/10
1/6. To the north of the canal 1/10 1/6. To the north proceeding from
the west, of Chales son of Pasitos, crown 2 at 1 each
1/2 1/4 same 1/2 1/4 1/8
<the area is> 1 [1/2] 1/4 1/10 1/6 1/30 1/2, black cumin. From the east
revenue from those 91/2 1/4 1/8
611/2 1/4 1/851/2 1/8
<the area is> 111/2 1/4 1/8 1/10 1/6 1/30 1/2 1/60 1/4. To the east proceeding
from the south
1/8 1/10 1/6 [..] 1/8 nothing
<the area is> 1/8 1/30 1/2, <total> 91/2 1/4 1/8 1/10 1/6 1/30 1/2.11
A few words of explanation: although the document is in Greek, the numbers
are expressed as sums of series of parts, as was common in ancient Egyptian
mathematics. Apart from telling what piece of land was cultivated by whom,and often what was being grown, the survey measured the land, either by
simply providing a figure for the area, or by reporting the lengths of the
sides of a plot of land, written down with a horizontal line to separatethem. If the allotment was quadrangular (the most common case, but there
is also a triangular holding, where the writer notes ‘nothing’ in correspond-
ence of one of the sides), the area was then obtained by adding oppositesides, halving the resulting sums and multiplying them by each other. Tax
was calculated on the basis of the area. It has been observed that the method
of obtaining surfaces on the basis of the sides, while lacking accuracy,achieved the far more desirable result (in the government’s eyes at least) of
always overestimating the size of the allotment, thus forcing the tenant to
pay more tax.
12
Surveying instruments were employed for these operations, and remains
of some of them have been found. The instrument in Figure 3.2 wouldhave helped with the drawing of straight lines and right angles, and therefore
with the laying in place of an orthogonal grid.
We also have evidence (again from Egypt) about the geometers involved:
a letter sent c. 252
BC by a Polykrates to his father Kleon, whom we know to
have been an engineer employed in the public administration, mentions
that he was ‘proceeding in the study of geometry’.14 We do not know for
sure where Polykrates was writing from: perhaps Alexandria, perhaps1/2 1/30 1/2
2 1/4
2 1/4
1 [.]

68HELLENISTIC MATHEMATICS: THE EVIDENCE
Krokodilopolis. Datable to the same period (251 BC) is this letter, from the
Arsinoite district:
Antipater to Pythocles greeting. I append for you a copy of the
letter written to me by Phanias. As soon as you receive my letter,
inspect and survey ( geometreson ) all the holdings under your super-
intendence, as Phanias has ordered, and after making a list asaccurate as possible according to crops send me the survey
(geometrian ) to submit to Phanias. Do the work scrupulously in
the manner of one prepared to sign the royal oath. Goodbye. […]Phanias to Antipater greeting. […] As sowing has begun in your
Figure 3.2 A surveying instrument (the plumblines are a modern reconstruction)
from the Fayum, dating from Ptolemaic times13

69HELLENISTIC MATHEMATICS: THE EVIDENCE
district, take at once some expert surveyor and inspect all the
holdings under your superintendence and after surveying them
make a list according to crops, as accurate as possible, of the landsown in each holding, continuing until you have inspected all.
Do the work scrupulously in the manner of one prepared to submit
the survey to me with a sworn declaration. Take care that youpresent it to us.
15
The emphasis on scrupulousness and expertise was not purely rhetorical.
Disputes and confusions often arose because the standards of measure varied
from place to place, or because equal size did not amount to equivalent
quality of land, or because of miscalculations, or because not all officerswere as honest as their sworn declarations would have required.
16 A great
part of the surviving papyri from this period consists of accounts, receipts
of payment and lists (of tax-payers, of tenants). They often state that accountsmust be produced regularly and checked carefully. For instance:
Eukles to Anosis greeting. I learn that you have deposited in the
record office the accounting of the pottery without bringing into
it even the breakage which has occurred through the donkey drivers,
and, in the matter of the pay still due in the case of the potters,that you have entered 8 drachmas per hundred pots instead of the
6 drachmas pay which had been given them; […] and that, up to
the present, you have not handed in to the record office theaccounting of the hogs ready for slaughter; and that, on the whole,
you have begun to act like a scoundrel […] I have written regarding
these matters to Lykophron and to Apollonios that, if they discoverany discrepancies from the accounts handed in by you, they write
to me immediately in order that I, being present in person, may
have my case judged in relation to you. For it is right that you,who are entering excess charges and are not reporting the totals
correctly throughout the accounts, should pay the discrepancies,
not I.
17
We know that Eukles was epistates (administrator) of Philadelphia, and
therefore quite an influential personage. A much humbler position in society
was occupied by the author of this letter, who made similar claims:
Pemnas to Zenon greeting. About the money owed on the pigs in
previous time together with the rent of 371/2, Herakleides has made
a deal with Thoteus, and, without me, they have balanced theaccounts and have not allowed me to look anything through until

70HELLENISTIC MATHEMATICS: THE EVIDENCE
now, and then without a justification they have dared to give me
the account. And about these things I have complained to Jason
many times that it is not right that they make deals together. AndHerakleides also has all the documents about the pigs. I have
written to you about these things so that you know. Good-bye.
18
Many documents indicate that the correctness of accounts and, to some
extent, of land-surveys, could be guaranteed by collective checking – official
instructions for administrators stated that the calculations had to beexamined by more than one government officer, to make sure that
everything was in order. In other words, doing mathematics together was
seen as a guarantee of smooth and honest functioning of the state machine.Of course, this view was not confined to the Hellenistic period (see chapter
1), or to Egypt: for instance, we have evidence of collective surveying
from Heraclea in Southern Italy. T wo bronze tablets, datable to the earlythird century
BC, record a survey of holdings of the temples of Dionysius
and Athena. A group of boundary-men ( oristai ), elected by the city
assembly, ‘measured together’ the lands, expressing sizes and setting upboundary stones. The first table was signed by the citizens involved, as
well as by Aristodamus, son of Symmachus, as secretary, and by Chaireas
son of Damon, of Neapolis, as geometer.
19
The prevailing impression from the Egyptian documents, however, is
that, beyond the rhetoric of fairness that mathematics implemented,
checking the sums was not open to anyone – these texts were not publicinscriptions, and gaining access to them could be difficult, and perhaps
status- and ethnicity-related, as is illustrated by the case of the Egyptian
swineherd Pemnas.
20 Numeracy and literacy obviously went along with status
and, since getting an education chiefly meant getting a Hellenic education,
with cultural identity. A few documents from this period which contain
simple mathematical exercises, sometimes together with grammatical drillssuch as declensions of verbs or lists of names, can be situated within the
context of basic education. For instance, a third-century
BC schoolbook
from the Fayum starts with syllables, follows with the numbers up to 25,
names of (Greek) gods, quotations from (Greek) literature, including
Euripides and Homer, then a table of squares and, finally, a table ofsubdivisions of the drachma.
21
Perhaps also linked to an educational context is a third-century BC papyrus
from Hermopolis in demotic Egyptian, containing arithmetical andgeometrical problems: measurement of land of various shapes, amounts of
cloth, size of masts, square roots. The two examples on the facing page are,
respectively, an application of the theorem of Pythagoras and a version ofthe quadrature of the circle.
22

71HELLENISTIC MATHEMATICS: THE EVIDENCE
[Problem 34] A plot of land that <amounts to> 60 square cubits,
[that is rec]tangular, the diagonal (being) 13 cubits. Now how
many cubits does it make [to a side]? You shall [reckon 13, 13times: result 169]. You shall reckon 60, 2 times: result 120. You
shall [add] it to 1[69]: result 28[9]. Cause that it reduce to its
square root: result 17. You shall take the excess of 169 against120: result 49. Cause that it reduce to its square root: result 7.
Subtract it from 1[7]: remainder 10. You shall take to it
1/2: result
5. It is the width. Subtract 5 from 17: remainder 12. It is theheight. You shall say: ‘Now the plot of land is 12 cubits by 5 cubits’.<To> cause that you know it. Viz. You shall reckon 12, 12 times:
result 144. You shall reckon 5, 5 times: result 25. Result 169.
Cause that it reduce to its square root: result 1[3]. It is its diagonalof plot.
[Problem 37] A plot of land [that is round that amounts to 675
square cubits, the diameter being 30,] … a piece that is square[within] it, that has four corners up to the circumference of the
plot of land. Now the piece … within it makes how many [square]
cubits? You shall reckon 30, 30 times: result 900. You shall take toit [half]: result 450. Cause that it reduce to its square root: result
21
1/5 1/60. They are the [meas]urements of the piece. To cause that
you know it. … the plot of land in which is the piece (is) [6]75square cubits. Its plan. Viz. (see Diagram 3.1a)
Now [the piece] that is squared [… reckons] 21
1/5 1/60 divine-
cubits [2]11/5 1/60 times: result 450 square cubits. Four segments,
[what are] their measurements? Viz. Subtract 211/5 1/60 from the
Diagram 3.1a

72HELLENISTIC MATHEMATICS: THE EVIDENCE
diameter of the circumference, which (is) 30 cubits: remainder
82/3 1/10 1/60. Its quantity at 2 amounts to 41/3 1/20 1/120 at 1. It is the
middle height of the segment. Here is its plan. (See Diagram 3.1b.)
You shall add 41/3 1/20 1/120 to 211/5 1/60: result 251/2 1/10 1/20 1/120.
Take [their] half: result 122/3 1/10 1/30 1/240. You shall reckon 122/3
1/10 1/30 1/240, 41/3 1/20 1/120 times: result 561/4. You shall reckon
561/4, 4 times: [result 225]. Result: [6]7[5] cubits again.
The problems in the demotic papyrus are solved not generally, but for
specific cases, and, rather than a deductive proof, they contain a verification,
or check step, introduced by the expression ‘to cause that you know it’. A
teaching context is suggested by direct appeals to the reader and by statementssuch as this:
When another [add-fraction-to-them] (problem) is stated to you,
it will be successful according to the model. If you take the excess
of the small (number) against the large (number), you shall put it
opposite 1 until it completes.
23
A group of third or mid-second century BC ostraka (pottery fragments)
from the island of Elephantine in Southern Egypt have also been found,written over with mathematical texts. Their contents have been identified
with Euclid’s Elements 13.16: the construction of a regular icosahedron
inscribed within a sphere, i.e. quite a complex procedure. The Elephantineostraka raise a number of questions. While their contents denote a high
level of education, both the humble material and the location (a remote
outpost in the heart of ‘Egyptian’ Egypt) seem to jar with that conclusion.
Besides, was the person who produced them acquainted with Euclid’s work,
or with a different account of the same subject-matter?Diagram 3.1b

73HELLENISTIC MATHEMATICS: THE EVIDENCE
The earliest extant papyrological remains of Euclid date from the late
second century BC. They formed part of a treatise on geometry by the
Epicurean philosopher Demetrius of Laconia. He reported, for polemicalpurposes, a definition of the circle and statements on the bisection of the
angle which correspond to Elements , definition 1.15 and propositions 1.3,
1.9 and 1.10.
24 Again, was Demetrius actually referring to Euclid’s text, or
to something similar that has not come down to us?
Let us defer a discussion of those questions to chapter 4, and move on
with our examination of the evidence.
Non-mathematical authors: the rest of the world
For a section about the world (minus philosophers and mathematicians),
this is a pretty short survey. The problem is that, unlike the Demosthenes,
Aristophanes and Lysias of the previous chapter, not many writers of theHellenistic period seem to mention mathematics at all. Could that be a
reflection of the fact that mathematics played a much more ‘public’ public
role in the Athens of the fifth and fourth century than in the communitiesof the third and second century, where the majority of Hellenistic literary
texts was produced?
One exception, as we have started to see, is Polybius. In line with his
geometrized vision of Roman troops, he stated that a knowledge of mathe-
matics was a positive asset for military leaders. In order to be a good general,
one needed a combination of experience, observation and methodical knowl-edge. In particular, a military leader had to be acquainted with astronomy
and geometry. Astronomy was necessary to know the variations of day and
night, in order to calculate movements of the troops, days of march and soon. Geometry, on the other hand, helped calculate the right length for siege
ladders – if one did not know the height of the enemy walls, Polybius
reminded them, an easy method (probably via gnomon and similar triangles).was available to those who wanted to exercise themselves in the mathematical
studies. Geometry, especially proportion theory, also came in handy when
one wanted to change the size of the camp while retaining shape and
arrangement.
25
That not everybody was as knowledgeable in mathematics as they should
have been emerged on occasions. Here Polybius criticizes the historian
Callisthenes for his report of a battle:
It is difficult to understand how they posted all these troops in
front of the phalanx, considering that the river ran close past the
camp, especially in view of their numbers, for, as Callistheneshimself says, there were thirty thousand cavalry and thirty thousand

74HELLENISTIC MATHEMATICS: THE EVIDENCE
mercenaries, and it is easy to calculate how much space was required
to hold them. For to be really useful cavalry should not be drawn
up more than eight deep, and between each troop there must be aspace equal in length to the front of a troop so that there may be
no difficulty in wheeling and facing round. Thus a stade will hold
eight hundred horse, ten stades eight thousand, and four stadesthree thousand two hundred, so that eleven thousand two hundred
horse would fill a space of fourteen stades. If the whole force of
thirty thousand were drawn up the cavalry alone would very nearlysuffice to form three such bodies, one placed close behind the
other. Where, then, were the mercenaries posted, unless indeed
they were drawn up behind the cavalry? This he tells us was not so[…] For such mistakes we can admit no excuse. For when the
actual facts show a thing to be impossible we are instantly convinced
that it is so. Thus when a writer gives definitely, as in this case, thedistance from man to man, the total area of the ground, and the
number of men, he is perfectly inexcusable in making false
statements.
26
Using weak demonstrative formulae such as phaneron oti and delon oti (it is
clear that, it is evident that), which were common in mathematical discourse,Polybius proved Callisthenes wrong on the basis of the very figures quoted
in his account. In fact, because Callisthenes had given a seemingly accurate,
mathematical, report, it was all the easier for Polybius to check it throughand show that it was incorrect.
27
Using mathematics in historical reports, even if only to expose someone
else’s ignorance or misuse of mathematics, was a rhetorical device whichaccrued accuracy and even an air of objectivity to the report itself. More
mathematical mistakes in the following passage:
Most people judge of the size of cities simply from their circum-
ference. So that when one says that Megalopolis is fifty stades in
contour and Sparta forty-eight, but that Sparta is twice as large as
Megalopolis, what is said seems unbelievable to them. And when in
order to puzzle them even more, one tells them that a city or campwith a circumference of forty stades may be twice as large as one the
circumference of which is one hundred stades, what is said seems to
them absolutely astounding. The reason of this is that we haveforgotten the lessons in geometry we learnt as children.
28
Clearly, Polybius expected mathematical knowledge to be part of the culturalbaggage of a political leader – not too much, but enough to be aware for

75HELLENISTIC MATHEMATICS: THE EVIDENCE
instance of issues of isoperimetrism, which, it is implied with rhetorical
exaggeration, were part of elementary geometry. In fact, an in-depth treat-
ment of isoperimetrism is contained in Euclid’s Elements , book 13, to which,
as we have seen, one could relate the material in the Elephantine ostraka .
The references in Polybius suggest that his audience was sensitive to
certain aspects or functions of mathematics. The same can be said to explainthe numerical imagery in Theocritus, a poet who benefited from the
patronage of the Ptolemies:
From Zeus let us begin, and with Zeus in our poems, Muses, let
us make end, for of immortals he is best; but of men let Ptolemy
be named, first, last, and in the midst, for of men he is mostexcellent. […] Of what am I to make mention first, for beyond
myriads to tell are the blessings wherewith heaven has honoured
the best of kings? […] Infinite myriads and myriads of tribes ofmen with the aid of rain from heaven bring their crops to ripeness,
but none is so prolific as are the plains of Egypt […] Three hundreds
of cities are built therein, and three thousands and three times tenthousand therewith, and twice three and three times nine beside;
and of all Lord Ptolemy is king.
29
Theocritus uses numbers in two different ways: very large figures, whichare ‘accurate’ in the sense that they express a definite quantity, and numbers
so large that they are beyond counting. Both are meant to convey thegreatness of Ptolemy Philadelphus and of his domain: large, very large, so
large that numbers cannot suffice to describe it. These rhetorical uses of
numbers are not unique to Hellenistic poetry, but they acquire a particularresonance when set against the various acts of official measuring and
numbering that took place in Ptolemaic Egypt. At least in intention, accounts
and surveys were meant to quantify the domain of the king, say exactlyhow big, how great, how rich he and his land were. More than that: as we
will see in the section on little people on p. 85, geography, at the hands of
Ptolemaic employees such as Eratosthenes, became a geometrizing,
mathematizing effort. The Earth itself was measured, using Alexandria as
reference point. And in chapter 4, we will also see how Archimedes wentbeyond the indefinite infiniteness of myriads, stretching to extreme limits
the capacity of numbers to count things. He estimated the number of grains
of sand that could be contained in the entire universe, and expressed theirnumber, dedicating the work to his king, on whom the importance of
counting and measuring would certainly not have been lost.

76HELLENISTIC MATHEMATICS: THE EVIDENCE
Non-mathematical authors: the philosophers
Many philosophers between the third and the second century BC were con-
cerned with the problem of knowledge, and with mathematics in particular.
Getting to know their views is made difficult by two major factors. First,
we rarely have first-hand evidence about them – most of what we knowcomes from later reports. Second, in the wake of Plato and Aristotle, both
founders of schools, the Hellenistic period saw the emergence of several
philosophical movements, groups and sects: the Epicureans, the Stoics, theSceptics, the Cynics, plus Academics (Platonists) and Peripatetics
(Aristotelians). Although some people can definitely be assigned to one
school rather than another (for instance, Epicurus was an Epicurean), greatunclarity reigns as to how and whether those labels really corresponded to
institutions with unified curricula, to what extent certain theories were
distinctive of one school rather than another, and especially, how theintellectual history of those schools can be traced chronologically. Our
sources all too often tend to attribute theories to for example ‘the Stoics’ in
general, glossing over developments over time or diversity of opinions amongmembers of the same school. Moreover, in the case of some individuals it is
not clear that they subscribed to any philosophical creed exclusively.
A tradition which is even more difficult to profile than the others is
Pythagoreanism, which, as we mentioned in chapter 1, had been around
since the sixth century
BC. Interest in it not only continued, but seems to
have positively increased in the Hellenistic period, when many of the textsthrough which we know the story, or legend, of earlier Pythagoreans were
written. In other words, Hellenistic Pythagoreanism to a great extent ‘created’
earlier Pythagoreanism in the form of books or views, including specificmathematical discoveries, usually in the field of proportion theory, attributed
to Archytas, Philolaus and Pythagoras himself. The forgeries or imitations
produced in this period can be identified with some reliability because theyare only quoted by authors later than the fourth century
BC, and also because
they often use Aristotelian and/or Platonic philosophical terminology, or
anachronisms of a similar kind. Several of these texts deal with mathematicalissues, especially with the relation between numbers and the world, and
with the ethical or philosophical significance of arithmetic qualities such as
even or odd, or of numbers: five or ten and so on. Numbers are often madeto carry meanings beyond simple quantity – for instance, according to some
of these texts god is an irrational number.
30
The Peripatetics and Academics also inherited a tradition where mathe-
matics played a fundamental role. Plato’s immediate successors had been
particularly interested in the possibility that Forms were numbers, and that
numbers held the key to universal knowledge.31 Later Academics concerned

77HELLENISTIC MATHEMATICS: THE EVIDENCE
themselves with explaining the many mathematical passages of the Timaeus
or the Republic . Names that have been transmitted include Crantor (end of
fourth/beginning of third century BC), Theodorus and Clearchus, known
also as a Peripatetic, all from Soli. Of the people generally associated with
the Aristotelian tradition, Dicearchus was interested in geographical
measurements; Aristoxenus, as we will see, wrote about harmonics; the nameof Heraclides of Pontus is linked to astronomical theories; Eudemus of
Rhodes, as we have seen, wrote histories of geometry, but also of arithmetic,
astronomy and music.
32 The Aristotelian corpus preserves a work On
Indivisible Lines , aimed at showing that it is ‘neither necessary, nor believable
that there are indivisible lines’33 – a line basically shared by Stoics like
Chrysippus.34
Indivisibles conflicted with the assumption, common among geometers,
that magnitudes could be divided indefinitely. Their existence in nature,
on the other hand, was one of the main tenets of Epicurus, who put forthan atomic theory of matter inspired by Democritus. Indeed, as mentioned
in the section on material evidence on p. 73, the Epicurean Demetrius set
out to attack Euclid’s Elements , or at least some of the contents of the first
book. That, however, does not amount to evidence that the Epicureans
spurned mathematics entirely: Demetrius seems to have been knowledgeable
in the very subject he set out to criticize; two early third-century Epicureans,Polyaenus and Pythocles, allegedly studied mathematics; the works of
mathematicians like Apollonius and Hypsicles mention people (Philonides,
and Basilides and Protarchus, respectively), who can be identified withcontemporary Epicurean philosophers by the same name.
35
More general reflections about the nature of demonstration on the part
of Epicureans and Stoics are reported in Proclus:
Up to this point we have been dealing with the principles [defi-
nitions, postulates and common notions in Euclid’s Elements book
1], and it is against them that most critics of geometry have raised
objections, endeavoring to show that these parts are not firmly
established. Of those in this group whose arguments have become
notorious some, such as the Sceptics, would do away with all
knowledge […] whereas others, like the Epicureans, propose onlyto discredit the principles of geometry. Another group of critics,
however, admit the principles but deny that the propositions
coming after the principles can be demonstrated unless they grantsomething that is not contained in the principles. This method of
controversy was followed by Zeno of Sidon, who belonged to the
school of Epicurus and against whom Posidonius has written awhole book.
36

78HELLENISTIC MATHEMATICS: THE EVIDENCE
Both Zeno and Posidonius of Rhodes (this latter usually labelled a Stoic)
lived at the end of the second century BC, thus providing a chronological
framework to Proclus’ picture. His testimony about the Sceptics, on theother hand, is typical of the vagueness that often surrounds both individual
thinkers and what they thought. Diogenes Laertius offers a glimpse of what
the father himself of Scepticism, Pyrrho ( c. 365–275
BC), thought about
mathematical certainty. He had distinguished ten modes in which seemingly
undisputable questions turned out not to be so:
The seventh mode has reference to distances, positions, places and
the occupants of the places. In this mode things which are thought
to be large appear small, square things round; flat things appear tohave projections, straight things to be curved.
37
Later Sceptics added five modes to Pyrrho’s ten, focussing more explicitlyon demonstration and casting doubt on argumentative structures typical of
mathematics:
If you think, they add, that there are some things which need no
demonstration, yours must be a rare intellect, not to see that you
must first have demonstration of the very fact that the things yourefer to carry convinction in themselves.
38
Although the Stoics appear to be not as critical of mathematics, they did
not assign it a prominent place in their epistemology. In line with the belief
that knowledge is ultimately sense-based and should concern itself with the
real world, they designated medicine, divination, dialectic and virtue as thefour primary sciences. Mathematics, however, played a role in the Stoics’
physical theory, which included astronomy and causation, because it shared
its subject-matter with those latter.
39 Again, according to Diogenes Laertius,
points of contact existed between Stoic physical theory and Euclidean-style
preoccupation with the definition of geometrical objects:
Body is defined by Apollonius in his Physics as that which is
extended in three dimensions, length, breadth, and depth. This isalso called solid body. But surface is the extremity of a solid body,
or that which has length and breadth only without depth. […] A
line is the extremity of a surface or length without breadth, or thatwhich has length alone. A point is the extremity of a line, the
smallest possible mark.
40
A parallel, not in contents but in argumentative structure, may also be
drawn between mathematics and Stoic logic. As part of their enquiry into

79HELLENISTIC MATHEMATICS: THE EVIDENCE
the rationality of nature and of discourse, the Stoics analysed language,
defining and distinguishing not only its components (propositions, both
simple and complex), but also the way those components were combinedto form arguments. They studied and classified types of valid argument,
including the modus ponens and a form of modus tollens very similar to the
one used in many indirect mathematical proofs.
41 In other words, as well as
identifying features of demonstrative discourse that were shared by mathe-
matical demonstrative discourse, the Stoics seemed to be doing for common
language what Euclid was doing in the first book of his Elements : establishing
basic assumptions, notions and constructions, organizing knowledge into a
new, rigorous structure.
Little people
If indeed he lived between 360 and 290 BC, the earliest mathematician
whose work has come down to us directly is Autolycus of Pitane. We have
two treatises by him, on the moving sphere and on the risings and settings
of the stars, in which heavenly bodies are reduced to their geometrical shapeand the conclusions about them are drawn on the basis of the mathematical
properties of those geometrical shapes. Many of the proofs are indirect, by
reductio ad absurdum ; occasional appeal is made to things assumed to be
valid earlier in the treatise. All these features characterize Autolycus’ account
as mathematical astronomy, and are paralleled by other works from the
Hellenistic period. Euclid’s Phenomena , for instance, also treated astronomy
in a geometrical style. Basic notions (horizon, meridian, etc.) are defined at
the outset, and, although astronomical observations, including the use of a
dioptra, are mentioned, the argumentative structure is very similar to thatof the Elements , with its typical enunciation, setting-out, definition of goal
and proof. Along similar lines is Euclid’s Optics , which starts off with
assumptions whereby the phenomenon of vision is geometrized, andcontinues in deductive mode, results being proved on the basis of previously
established propositions.
In late antiquity, Autolycus’ On the Moving Sphere , Euclid’s Phenomena
and perhaps also Euclid’s Optics came to be part of the same collection as
three more Hellenistic treatises: Aristarchus of Samos’ On the Sizes and
Distances of the Sun and Moon and Theodosius of T ripoli’s Sphaerics and On
Days and Nights .
42 Aristarchus (early third century BC), well-known to us
for his heliocentric theory of the universe, started his book with hypotheses,i.e. undemonstrated statements, and on the basis of them proved several
propositions along deductive lines. It is to be noted that, although he
considered heavenly bodies from the point of view of their geometricalproperties (sizes and distances), physical characteristics such as movement,
illumination and perceptibility were not eliminated from his account.

80HELLENISTIC MATHEMATICS: THE EVIDENCE
[From Aristarchus’ work – an example of approximation via upper
and lower boundary (see Diagram 3.2)] The diameter of the moon
is less than two 45ths, but greater than one 30th, of the distance
that separates the centre of the moon from our eye. For let our eyebe at A, and let B be the centre of the moon when the cone
surrounding both the sun and the moon has the vertex at our eye.
I say that what said in the enunciation takes place. For let AB be
joined, and let the plane through AB be prolonged; it will produce
as section a circle in the sphere and lines in the cone. Let it then
produce the circle CED in the sphere, and the lines AD, AC in
the cone, and let BC be joined, and be prolonged to E. It is evident
then from what has been proved before that the angle BAC is the
45th part of half a right angle; and in the same way BC is less than
CA by one 45th. Therefore BC is much less than one 45th part of
BA. And CE is double BC; therefore CE is less than two 45th of
AB. And CE is the diameter of the moon, while BA is the distance
that separates the centre of the moon from our eye; therefore thediameter of the moon is less than two 45ths of the distance that
separates the centre of the moon from our eye. I say then that CEDiagram 3.2

81HELLENISTIC MATHEMATICS: THE EVIDENCE
is also greater than one 30th part of BA. For let DE and DC be
joined, and with centre A and radius AC, let a circle CDF be
described, and let DF equal to AC be fitted in the circle CDF. And
since the right <angle> EDC is equal to the right <angle> BCA,
but the <angle> BAC is equal to the <angle> HCB , therefore the
remaining <angle> DEC is equal to the remaining <angle> HBC ;
therefore the equiangular triangle CDE is equal to the triangle
ABC . Therefore it is as BA to AC, so EC to CD; and alternately as
AB to CE, so AC to CD, that is, DF to CD. But since again the
angle DAC is the 45th part of a right <angle>, therefore the
circumference CD is the 180th part of the circle; and the
circumference DF is the sixth part of the whole circle; so that the
circumference CD is the 30th part of the circumference DF. And
the circumference CD, which is less than the circumference DF,
has to the circumference DF itself a ratio less than that of the line
CD to the line FD; therefore the line CD is greater than the 30th
of DF. FD is then equal to AC; therefore DC is greater than the
30th of CA, so that CE is also greater than the 30th of BA. And it
has been proved to be also less than two 45ths.43
Theodosius (late second/early first century BC) explicitly relied on the
tradition of mathematical astronomy before him: he mentioned Euclid
(Elements and Phenomena ), as well as our old acquaintance Meton. Book 1
of the Sphaerics again starts off with basic definitions, such as that of ‘sphere’,
and continues in a deductive mode towards more and more complexpropositions involving, for instance, arcs and tangents to spherical segments.
Some of the theorems in book 3 of the Sphaerics are divided into sub-cases
each with a different diagram, a practice which, as we will see, is oftenfound in Apollonius.
44
Mathematical astronomy was not the only form of knowledge of the
skies: immensely popular since its composition and throughout the Graeco-Roman period was an astronomical poem, the Phenomena by Aratus of
Soli, written c. 276–74
BC at the invitation of King Antigonus II of
Macedonia (Aratus later moved to the court of Antiochus I Seleucid). ThePhenomena provided a wealth of information about constellations, the stars
that constitute them, and on how to interpret heavenly signs to forecast the
weather. Practical applications to agriculture, navigation and calendar-making were all mentioned, and positive statements were made about the
fact that sign-reading was more than random guessing – in fact, when sign

82HELLENISTIC MATHEMATICS: THE EVIDENCE
confirmed sign, the knowledge derived from them could for Aratus be
considered certain.45 Aratus offers a fascinating example of the relation and
contrast between mathematical and non-mathematical accounts of basicallythe same phenomena, especially if we consider that he apparently sourced
his knowledge from the almost certainly mathematical treatment of
astronomy by Eudoxus of Cnidus. The same issues recur in the only survivingwork by Hipparchus (of Nicaea, but he worked mostly in Rhodes around
the second half of the second century
BC): the Commentary on the Phenomena
of Aratus and Eudoxus . He is credited, among other things, with the discovery
of the precession of the equinoxes. The target of criticism in the Commentary ,
along with the two authors of the title, is an Attalus who had also written a
commentary on Aratus. Hipparchus’ book does not contain much in theway of mathematics, but it does draw a distinction between what Aratus
was doing (poetry, cannot be expected to get things right) and the enterprise
Hipparchus himself and Eudoxus are engaged in. A revealing passagecontrasts the belief, apparently held by Aratus, that circles like the equator
have an actual width, with what ‘all the mathematicians think’, i.e. that
they have none.
46
A similar contrast between mathematized and non-mathematized knowl-
edge is found, with a rather different outcome, in Aristoxenus, probably a
pupil of Aristotle. Although parts of his main work, the Elements of
Harmonics , are organized as enunciation-like statements followed by proofs,
some of them indirect, he made a point of distantiating himself from mathe-
matical treatments of music:
We try to give these matters demonstrations which conform to
the appearances, not in the manner of our predecessors, some ofwhom used arguments quite extraneous to the subject, dismissing
perception as inaccurate and inventing theoretical explanations,
and saying that it is in ratios of numbers and relative speeds thatthe high and the low come about. Their accounts are altogether
extraneous, and totally in conflict with the appearances. […] While
it is usual in dealing with geometrical diagrams to say ‘Let this be
a straight line’, we must not be satisfied with similar remarks in
relation to intervals. The geometer makes no use of the faculty ofperception: he does not train his eyesight to assess the straight or
the circular or anything else of that kind either well or badly: it is
rather the carpenter, the wood-turner, and some of the other craftsthat concern themselves with this. But for the student of music
accuracy of perception stands just about first in order of impor-
tance, since if he perceives badly it is impossible for him to give agood account of the things which he does not perceive at all.
47

83HELLENISTIC MATHEMATICS: THE EVIDENCE
Aristoxenus’ target could be exemplified by the Section of a Canon (perhaps
by Euclid), whose introduction justified the application of mathematics to
the physical phenomenon of musical sound on the basis of the fact thatnotes are composed of parts and parts have numerical ratios to each other.
48
The mathematization of harmonics thus postulated at the beginning iscarried out in full in the body of the treatise, with occasional references to(perhaps) the Elements themselves. That the question of mathematics’ ability
to provide adequate knowledge of nature was widely discussed is also
evidenced by Diocles:
Sometimes people who try to discredit the mathematical scientists
and say that they construct their subject on a weak foundationscoff <at this>: for some of them <the mathematicians> assert that
the radii of the sphere are known and that each one is greater than
the one <next to it> by more than 30 million stades, while othersassert <that it is greater> by more than 50 million stades.
49
Diocles’ only extant treatise, On Burning Mirrors (c. 200 BC) itself presents
an interesting interplay of geometrical proofs and recourse to ‘the real world’.
In his attempt to provide a mathematical description of a paraboloid mirror
which can concentrate the sun rays onto a target, setting it on fire, he indica-tes, for instance, that intensity of burning has to be assessed empirically (31
ff.) and employs material aids in mathematical demonstrations, such as the
problem reported in the exmple below, or his solution to the duplication ofthe cube, which relies on a curved ruler (195). A later tradition credited
Archimedes with the use of burning mirrors as weapons, but Diocles suggests
rather that they make accurate time-keeping devices, as shadow-less gnomonsthat burn a trace instead of casting a shadow (16–17). Or they can be employed
in temples to light the fire on an altar in a spectacular way (36–7).
[From Diocles’ book on burning mirrors – Do not try this at home
(see Diagram 3.3)] How do we shape the curvature of the burning-
mirror when we want the point at which the burning occurs to beat a given distance from the centre of the surface of the mirror?
We draw with a ruler on a given board a line equal to the distance
we want: that is line AB. We make AJ twice BA and erect BE
perpendicular and equal to AB; we join EJ. We make AF equal
to BE, and join EF: then ABEF is a square, and also EF is equal to
FK. We mark on BA two points, G, D, and make EZ equal to BG
and HE equal to BD. We join ZG, HD, and produce them on
both sides: let them meet EK in M, L. Then if, with A as centre

84HELLENISTIC MATHEMATICS: THE EVIDENCE
and GM as radius, we draw a circle, it cuts GM: let it cut in Q.
Then we continue to draw it in the same way until it cuts it in Y.
Again, if, with centre A and radius DL, we draw a circle, it cuts
DL: let it cut in N. Then we continue to draw it about centre A
until it cuts it again in P. Then we draw AX as an extension in a
straight line of KA and make it equal to it. Then points K, N, Q,
B, Y, P, X lie on a parabola. For we produce AB to R, letting BR
equal AB; let us draw RS perpendicular to AB and equal to KA,
and join SK, and draw from points L, M, Q, N to line RS
perpendiculars LW, MC, NO, QJ. Then when KE is produced in a
straight line it passes through R. So WL is equal to LD and MC is
equal to MG, because KER is a diagonal of square AS. But LD is
equal to NA and MG is equal to QA, and LW is equal to LD also
and MC is equal to MG. So AN is equal to LW is equal to NO and
AQ is equal to MC is equal to QJ. And AK is equal to KS and AR
is bisected at B. Since that is so, points B, Q, N, K lie on a parabola,
as we shall prove subsequently. Similarly points Y, P, X also. So if
we mark numerous points on AB, and draw through them lines
parallel to AK, and mark on the lines points corresponding to the
other points, and bend along the resultant points a ruler made of
horn, fastening it so that it cannot move, then draw a line along it
and cut the board along that line, then shape the curvature of thefigure we wish to make to fit that template, the burning from that
surface will occur at point A, as was proved in the first proposition.
50Diagram 3.3

85HELLENISTIC MATHEMATICS: THE EVIDENCE
A further variation on the question of mathematics and the real world is
found in the work of Eratosthenes of Cyrene ( c. 285–194 BC), who was
chief librarian at Alexandria. He had apparently studied philosophy at Athensand wrote a Platonicus , which, like all of his works, is only known through
fragments. In one of them, a passage from the Timaeus lends Eratosthenes
the opportunity to articulate the difference between diastema (distance)
and logos (ratio). There can be a ratio between things which are equal, but
distance is only between two things distinct from each other; ratio is not
the same both ways (2:1 is not the same as 1:2), whereas distance is. Thus,
ratio is a certain kind of relation of two magnitudes to each other
[…] ratio is the principle of proportion […] for every proportion<comes from> ratios, but principle of ratio <is> sameness.
51
Eratosthenes also discussed proportion in relation to geometrical figures,
and Sextus Empiricus confirms his interest in basic definitions of
mathematical concepts by reporting that he analyzed the nature of point
and line. Apart from the Platonicus , Eratosthenes wrote a book On Means ,
which may have contained the original version of a solution to the
duplication of the cube now extant in Eutocius and a method, called ‘the
sieve’, to find prime numbers, and a Geography . Tackling the subject
mathematically, he mapped the surface of the then-known world by means
of intersecting lines, while regions were approximated to geometrical figures
by having, for instance, Sicily shaped as a triangle and India as a rhomboid.He also measured the obliquity of the ecliptic and the circumference of the
Earth, which he put at 252,000 stades. We do not know for sure which unit
of measure was used, and his procedure is described in detail only by thelate fourth-century
AD Cleomedes. According to the latter, Eratosthenes
knew that in the city of Syene at noon on the summer solstice the gnomon
cast no shadow; he thus assumed that Syene was on the T ropic of Cancer.He also knew that Syene and Alexandria were on the same meridian; finally,
the distance between the two cities had been measured by the royal surveyors.
Eratosthenes thus measured the shadow cast by a gnomon at noon on the
summer solstice at Alexandria and, by means of the properties of similar
triangles, estimated that the shadow amounted to
1/25 of the hemisphere,
and thus to 1/50 of the whole circle.53
After music, astronomy, geography, we find mathematics applied to a
rather different set of phenomena in Biton’s treatise on the construction ofwar instruments, written between the mid-third and the early second century
BC and addressed to a king Attalus (one of the Pergamum monarchs). While
Biton portrays himself as knowledgeable about the use of the dioptra, andas the author of a book on optics, all the machines in the treatise are explicitly

86HELLENISTIC MATHEMATICS: THE EVIDENCE
attributed by him to other people, and the place where they were produced
is also specified. Thus, Charon of Magnesia built a certain machine at
Rhodes, Isidorus of Abydus another machine at Thessalonika, Posidoniusthe Macedonian devised the siege tower for Alexander the Great, the siege
ladder is due to Damis of Colophon. Zopyrus of Tarentum is credited with
two types of belly-bow (a type of small catapult), one made at Myletus, oneat Cumae. The resulting picture is one of geographical variety, both in the
places of origin of these people, and in where they end up working. A
community of engineers/machine-makers is evoked, whose memberstravelled often, changing employer or patron, sharing and transmitting
knowledge about materials, proportions in the construction of the machines,
anecdotes about their opportune use in actual situations.
Biton describes the shape of the devices and the dimensions of their
components, provides suggestions as to their overall structure, and
complements his account with illustrations (now lost). His specifications,or lists of dimensions, are in the form of sets of measurements to which the
various pieces of a machine have to correspond as accurately as possible.
The use of some geometrical instruments is required in making each pieceto the given specification with the required accuracy. Biton concludes the
account thus:
Whatever engines we considered most appropriate for you, we
have now described. We are convinced that you will be able to
discover similar forms through these. Do not be worried by thethought that, because we have used fixed measurements, it will be
necessary for you to use the same measurements, too. If you wish
to construct larger or smaller instruments, do so; only try topreserve the proportion ( analogia ).
54
Philo of Byzantium (third century BC) opened his book on war engines
(the Belopoiika ) by pointing out that previous treatments were in disagree-
ment both about the proportion ( analogia ) of the various components of
the machines and about the guiding element ( stoicheion ) of the construction,
i.e. the standard on the basis of which the dimensions of each element were
established. Therefore, he decided to ignore the old authors and to draw onthose newer methods
that can achieve what is required in the facts. I understand you do
not ignore that the art ( techne ) has something very obscure and
incomprehensible to the many; at any rate, many who have under-
taken the construction of instruments of equal size and who haveused the same arrangement and similar wood and the same metal

87HELLENISTIC MATHEMATICS: THE EVIDENCE
without even changing its weight, have made some with long range
and powerful impact and others which fall short of these. Asked
why this happened, they were not able to say the cause. Thus theremark made by Polykleitus the sculptor is proper for what I am
going to say; for he said that the good is generated little by little
through many numbers. Likewise, in this art, since the things thatare done are completed through many numbers, those who make
a small discrepancy in individual parts produce a large mistake
when adding up at the end.
55
One of the good things generated through many numbers was the
realization that, if one wanted to modify the dimensions of a catapultproportionately, they had to use the hole holding the torsion spring as
guiding element. Philo describes this discovery as a process happening over
time, through repeated experience, trial-and-error and accumulatedknowledge. The culmination of this technical development was the rigorous,
mathematical, formulation of the duplication of the cube, which was
necessary to change the size of the torsion spring hole. Philo’s solution tothe problem comes at the end of a passage where he offers lists of dimensions
for the construction of the machines, and draws up a mathematical table
where, with opportune approximations, weight of projectile and diameterof the torsion spring hole are correlated. The table of weights and diameters
is obtained as follows (see Diagram 3.4):
Reduce to units [i.e. drachmae] the weight of the stone for which
the machine must be put together. Make the diameter of the hole
of as many dactyls as there are units in the cubic root of the numberobtained, adding the tenth part of the root found. If the weight
does not have an expressible side, take the nearest one; but, if it is
above, try to diminish the tenth part in proportion and, if below,to increase the tenth part. […] It is possible, also, from one number,
[…] to put together the remaining diameters instrumentally by
doubling the cube […] Let there be a straight line given of the
diameter, of which it is necessary to find the double power, for
example A. I put B, which is double that, perpendicular to it, and
from the extremity of B I drew another <line> C perpendicularly,
indefinitely. And I drew a line J from the angle H, and cut it in
half; let the dividing point be at J. Using K as centre, and JH as
radius, I described a semicircle, cutting also through the angle F,
and taking a straight ruler I carefully connected <them>, cutting
both lines and keeping one part of the ruler on the angle F. Let the
ruler then be at F. I moved the ruler around at the same time

88HELLENISTIC MATHEMATICS: THE EVIDENCE
keeping one part of it touching the angle, and moved around until
it happened to me that the part of the ruler from the point of
contact C, to the point of contact of the circumference G, resulted
equal to the <line> from the point of contact D, to the angle F.
And DE will be the double power of EF, and HC of ED, and HF
of HC.56
Philo’s is, along with Diocles’, the earliest first-hand extant solution to the
problem of the duplication of the cube. It, too, employs mechanical aids, in
this case a moving ruler, and adds to the conclusion that mathematics was
widely practised, benefited from exchanges and accumulation of results overtime, and was applied to a number of socially and politically prominent
fields.
Euclid
For such a famous mathematician, Euclid remains a very shadowy figure.
We do not know with certainty when or where he lived, and even the author-
ship of his major work, the Elements , has come under question. The most
educated guess we can make is that Euclid was active in Alexandria, possiblyenjoying some form of royal patronage, around the very beginning of the
third century
BC. His surviving works, apart from the Elements , are the
Data , the Phenomena , the Optics , the Section of a Canon , the Divisions of
Figures ; titles of lost books include Conics and Loci with Respect to a Surface .57
In what follows I will concentrate on some features of the Elements and of
the Data .Diagram 3.4

89HELLENISTIC MATHEMATICS: THE EVIDENCE
Contents
The contents of the Elements can be briefly summarized thus: book 1 and 2
are on plane rectilineal geometry, book 3 on the circle, book 4 on regular
polygons, book 5 on proportion theory, book 6 on plane geometry with an
use of proportions (e.g. similar polygons), books 7 to 9 on number theory,book 10 on irrational lines, books 11 and 12 on solid geometry, and, finally,
book 13 on regular polyhedra and their relation with the sphere.
58
Book 1 sets off with a number of basic starting points: definitions, which
include that of a point, a line, a circle, the three kinds of angle; postulates,
which mostly establish the possibility of elementary constructions, such as
let it be required to have drawn a straight line from any point to
any point […] and a circle to be drawn with any centre and radius
and all right angles to be equal to each other […]59
common notions to the effect that, for instance, ‘things which are equal tothe same <thing> are also equal to one another’ and that ‘the whole is greaterthan the part’. Many of the theorems in book 1 were already familiar before
Euclid, as we know because they are mentioned by for instance Plato and
Aristotle. For example, that in any isosceles triangle the angles at the baseare equal to each other (1.5), that the internal angles of a triangle are equal
to two right angles (1.32) or that in a right-angled triangle the square on
the side subtending the right angle is equal to the squares on the sidescontaining the right angle, or theorem of Pythagoras (1.47). Some other
theorems, such as those on the equality of two triangles, seem new to the
Elements , even though that may be due to lack of evidence. In general,
Euclid inscribes both already known and not previously known results into
a systematic demonstrative framework, where the results themselves, the
assumptions they use, the possibility of the geometrical constructions theyrequire, and even the geometrical objects they deal with, are all firmly
grounded and justified.
Like book 1, books 2, 3 and 4 present a combination of results familiar
from elsewhere, results whose simplicity suggests that they were perhaps
commonly assumed but not proved in full, and what may have been new
results. They are all organized within a deductive structure, with startingpoints (in the form of definitions) and propositions proved on the basis
either of the starting points or of previous propositions. The last item in
book 2 is the seminal problem of building a square equal to a given rectilinearfigure, or squaring a rectangle, which is equivalent to finding a mean
proportional between two straight lines.

90HELLENISTIC MATHEMATICS: THE EVIDENCE
[Elements 2.14 (see Diagram 3.5)] To put together a square equal
to a given rectilinear figure. Let the given rectilinear figure be A; it
is necessary then to put together a square equal to A. Let a right-
angled parallelogram BD be put together equal to the rectilinear
figure A; if then BE really is equal to ED, what has been prescribed
would be produced. The square BD equal to the rectilinear figure
A has in fact been put together. If instead this is not so, one of the
BE, ED is greater. Let the BE be greater, and let it be prolonged
until F, and let EF be posited equal to ED, and let BF be divided
in two in the point G, and with G as centre and one of the GB,
GF as radius let the semicircle BHF be drawn, and let DE be
prolonged until H, and let GH be conjoined. Since then the line
BF is divided in equal parts by G, but in unequal parts by E,
therefore the rectangle formed by BE, EF plus the square on EG
is equal to the square on GF. For GF is equal to GH. Therefore
the <rectangle formed> by BE, EF plus the <square> on GE is
equal to the <square> on GH. But the <square> on GH is equal to
the squares on HE, EG; therefore the <rectangle formed> by BE,
EF plus the <square> on HE is equal to the <squares> on HE,
EG. Let the common square on HE be subtracted; the remainder
therefore, the rectangle formed by BE, EF is equal to the square
on EH. But the <rectangle formed> by BE, EF is BD, for EF isDiagram 3.5

91HELLENISTIC MATHEMATICS: THE EVIDENCE
equal to ED; therefore the parallelogram BD is equal to the square
on HE. But BD is equal to the rectilinear figure A. Therefore also
the rectilinear figure A is equal to the square built on EH. Therefore
a square has been put together, the <one> built on EH equal to
the given rectilinear figure A; which it was necessary to do.
Among the definitions that open book 5, one finds this passage:
ratio is a sort of relation of two homogeneous magnitudes according
to size. It is said that <those> magnitudes have a ratio one with
the other, which are capable, multiplied, to exceed one another. Itis said that magnitudes are in the same ratio the first with the
second and the third with the fourth, if multiplying the same times
the first with the third, and multiplying the same times the secondwith the fourth, in any multiplication whatsoever, the former will
either alike exceed, or alike be equal or alike fall short of the latter,
respectively taken in the corresponding order. Let magnitudes thathave the same ratio be called in proportion.
60
The starting points of book 5 are stated in a sequence: notice how theconcept of ratio is first vaguely explained, then substantiated by a working
definition, while the concept of magnitudes in the same ratio is first pain-
stakingly laid out, and then followed by a slimmer synonym. Again, althoughratios and proportions were used before Euclid, this is the first extant clear
(-ish) statement of what they are.
Book 6 contains, among other things, a definition of similarity for recti-
linear figures, a solution to the problem of how to cut a segment into extreme
and mean ratio (6.30) and a solution to the problem of finding a mean
proportional (see Diagram 3.6):
To find a mean proportional to two given lines. Let the two given
lines be AB, BC; it is necessary then to find a mean proportional
to AB, BC. Let them be posited on a line, and let a semicircle
ADC be drawn on AC, and let BD be drawn from the point B
perpendicular to the line AC, and let AD, DC be joined. Since <it
is> an angle in a semicircle, the angle ADC is right. And since DB
is drawn in a right-angled triangle, <namely> ADC , from the right
angle to the base perpendicularly, DB therefore dividing the base
is the mean proportional between AB, BC. Therefore DB has been
found <to be> mean proportional between the given lines AB,
BC; which it was necessary to do (6.13).

92HELLENISTIC MATHEMATICS: THE EVIDENCE
The next three books are usually called ‘arithmetical’ because they do for
basic number theory what book 1 does for geometry. They are in a strictsequence, book 7 beginning with definitions of concepts that are employed
throughout books 8 and 9. Euclid says that ‘unit’ is the thing ‘according to
which it is said that each of the beings <is> one’ (7. definition 1), while a‘number’ is ‘a multitude put together from units’ (7. definition 2). The unit
is therefore not a number. Other definitions include that of odd and even,
of prime number, of multiplication, of plane, square and solid numbers,and (again) of proportionality, this time specifically for numbers: ‘numbers
are in proportion, if the first of the second and the third of the fourth, are
the same times either multiple of, or the same part of, or the same parts of’(7. definition 20).
61
Results established in the arithmetic books include how to find the
greatest common measure of two or three numbers (7.2 and 7.3) and theleast common number which two or three numbers measure (7.34 and
7.36), as well as propositions on prime numbers and on numbers in
proportion, including means and continuous proportion. In book 8 it isstated that between two square numbers and between two similar plane
numbers there is one mean proportional (8.11 and 8.18), while between
two cube numbers and between two similar solid numbers there are two
mean proportionals (8.12 and 8.19). Proposition 8.12 amounts to a proof
that the problem of the duplication of the cube can be reduced to that offinding two mean proportionals between two given lines. Some of the
theorems are very general – the one that follows concerns numbers as a
whole (see Diagram 3.7):
The multitude of prime numbers is more than all the multitude
of proposed prime numbers. Let the proposed prime numbers beA, B, C; I say that prime numbers are more than A, B, C. Let theDiagram 3.6

93HELLENISTIC MATHEMATICS: THE EVIDENCE
least dividend of A, B, C have been taken and let it be DE, and let
the unit DF be added to DE. EF then will either be prime or not.
Let it first be prime; therefore the found prime numbers A, B, C,
EF are more than A, B, C. But let EF be not prime; therefore it
will be divided by some prime number. Let it be divided by F; I
say that H is not the same as A, B, C. If indeed it is possible, let it
be <so>. Then A, B, C divide DE and therefore H divides DE.
And it will divide EF as well. And the number G divides the
remaining unit DF; which is absurd. Therefore G will not be the
same as A, B, C. And it was assumed to be prime. Therefore the
prime numbers A, B, C, G that have been found are more than
the multitude A, B, C that had been assumed; which it was neces-
sary to prove (9.20).
Book 10, the longest of the Elements , again exhibits a combination of
old and new. Its main focus is incommensurability – of magnitudes in
general, but more especially of lines, which can be incommensurable inlength and in square, as we know from earlier sources including Plato’s
Theaetetus . In fact, book 10 matches the project of thorough classification
of incommensurables attributed to Theodorus and/or Theaetetus, as well
as extending to magnitudes in general results that had been established for
numbers in the arithmetical books, and exploring their further ramifications.
For instance, a series of propositions (10.17 onwards) discusses the
application of parallelograms to straight lines, a topic already found in books
2 and 6, with the difference that here the notion of incommensurability isbrought on board. Thus, cases are distinguished according to whether the
segments resulting from the application are incommensurable with each
other or with the original line. We also have a method to find the so-calledPythagorean triplets (see Diagram 3.8):Diagram 3.7

94HELLENISTIC MATHEMATICS: THE EVIDENCE
To find two square numbers, such that their sum will also be a
square. Let two numbers be posited, AB, BC, let them be either
odd or even. And since, if even is subtracted from even, and if odd
from odd, the remainder is even, therefore the remainder AC is
even. Let AC be divided in two in the point D. Let AB, BC be
either similarly plane or square, and these are similarly plane.
Therefore the product of AB and B C plus the square on CD is
equal to the square on BD. And the product of AB, BC will also
be square, since it has been proved that if two similarly plane
<numbers> multiplied by each other produce something, the resultis a square. Therefore the two numbers, the product of AB and
BC and the square on CD, will be found to be square, and those
numbers added to each other make the square on BD. And it is
evident that again the square on BD and the square on CD will be
found to be square numbers, so that the excess of them is the
square on AB, BC, if AB, BC are similarly plane. If instead they
are not similarly plane, the square on BD and the square on DC
will be found to be square, then the excess of the product of AB,
BC is not a square; which it was necessary to prove (10. first lemma
after 10.28).
Several types of incommensurable lines are introduced and given specific
names in the course of the book: medial at 10.21, binomial at 10.36,
apotome at 10.73 and a whole group of first, second, up to sixth binomial
between propositions 47 and 48, and first, second, up to sixth apotomebetween propositions 84 and 85.
The following books 11, 12 and 13, on solid geometry, establish, among
other things, the relation between pyramid and prism with the same base
and same height (12.7) and between cone and cylinder with the same base
and same height (12.10). The reader may remember from the section onthe problem of later early Greek mathematics p. 50, that those results were
by Archimedes attributed to Eudoxus – once again, it would seem that
Euclid weaves previous material into the texture of his account.
The main purpose of book 13, the final book of the Elements , is the
investigation of regular solids with respect to the sphere. Since the faces of
regular solids are regular polygons, much space is devoted to those latter,for instance to the relation between the side of a polygon and the radius ofDiagram 3.8

95HELLENISTIC MATHEMATICS: THE EVIDENCE
the circle within which that polygon is inscribed. Materials from previous
books are relevant here, but Euclid also found that in some cases the relation
between side and radius, as defined above, or the relation between the sideof a polyhedron and the radius of the sphere circumscribing it, could be
expressed in terms of incommensurable lines such as had been classified in
book 10. He thus brings to fruition several of the topics scattered throughoutthe work: for instance, results from books 5 and 6 are combined with
elements from book 10 to produce 13.6: ‘If a rational line is divided into
extreme and mean ratio, each of the segments is an irrational, the so-calledapotome’ or 13.8: ‘If lines subtend two successive angles of an equilateral
and equiangular pentagon, they cut each other in extreme and mean ratio,
and of these the greater segments are equal to the side of the pentagon’ (seethe example below).
[Elements 13.8 (see Diagram 3.9)] If lines subtend two successive
angles of an equilateral and equiangular pentagon, they cut each
other in extreme and mean ratio, and of these the greater segmentsare equal to the side of the pentagon. Let the lines AC, BE
subtending the two successive angles on A, B of an equilateral and
Diagram 3.9

96HELLENISTIC MATHEMATICS: THE EVIDENCE
equiangular pentagon ABCDE , cut each other in the point H. I
say that each of these is cut in extreme and mean ratio in the point
H, and of these the greater segments are equal to the side of the
pentagon. Let the circle ABCDE be circumscribed to the pentagon
ABCDE . And since the two lines EA, AB are equal to AB, BC
and they surround equal angles, therefore the base BE is equal to
AC, and the triangle ABE is equal to the triangle ABC , and the
remaining angles will be equal respectively to the remaining angles,
<those> by which the equal sides are subtended. Therefore theangle BAC is equal to ABE ; therefore the <angle> AHE is double
BAH . The <angle> EAC is double BAC , because the arc EDC is
also double the arc CB. Therefore the angle HAE is equal to AHE ,
so is the line HE to EA, that is, it is equal to AB. And since the
line BA is equal to the line AE, the angle ABE is also equal to
AEB . But it has been proved that the <angle> ABE is equal to
BAH ; and therefore the <angle> BEA is equal to BAH . And the
angle ABE is common to the two triangles ABE and ABH .
Therefore the remaining angle BAE is equal to the remaining
angle AHB . Therefore the triangle ABE has the same angles as
the triangle ABH ; therefore in proportion as EB is to BA, so AB
to BH
. But BA is equal to EH. Therefore as BE to EH, so EH to
HB; but BE is greater than EH; therefore EH is also greater than
HB. Therefore BE is divided in extreme and mean ratio in the
point H, and the greater segment HE is equal to the side of the
pentagon. Similarly we prove that AC is divided into extreme and
mean ratio at H, and the greater segment of that, CH, is equal to
the side of the pentagon; which it was necessary to prove.
The last five propositions of the Elements are devoted to the construction
of the five regular polyhedra and to their comparison with one another: theicosahedron is the greatest, then the dodecahedron, the octahedron, the
cube and the pyramid. An adjunct to prop. 13.18 is as follows:
I say then that apart from the five mentioned figures no other
figure is put together surrounded by equilateral and equiangular
<figures> equal to each other. For a solid angle is not put togetherfrom two triangles or in general two planes. From three triangles
the <solid angle> of the pyramid, from four that of the octahedron,
but five than of the icosahedron; from six equilateral and

97HELLENISTIC MATHEMATICS: THE EVIDENCE
equiangular triangles put together in one point a solid angle will
not <derive>. Since the angle of an equilateral triangle is two thirds
of a right <angle> the <angles of the> six <triangles> will be equalto four right <angles>; which is impossible; for any solid angle is
surrounded by less than four right angles. Through this, also a
solid angle will not be put together by more than six plane angles.The angle of the cube is surrounded by three squares; it is
impossible <that it be surrounded> by four; for again they will be
four right angles. The <angle> of the dodecahedron by threeequilateral and equiangular pentagons; it is impossible by four;
since the angle of the equilateral pentagon is a right <angle> and a
fifth, the four angles will be greater than four right angles; whichis impossible. A solid angle will not be surrounded by other
polygons because of this absurdity. Therefore except for the said
five figures, no other solid figure will be put together fromsurrounding equilateral and equiangular <figures>; which it was
necessary to prove.
In some aspects as basic a work as the Elements , the Data also starts off
with definitions, which specify what it means that a point, an angle, a
rectilinear figure, a circle, a ratio are given in magnitude, position or form.For instance, ‘a rectilinear figure is said to be given in form when its angles
are given one by one and the ratios of the sides with each other are given’.
Areas, lines and angles are given (in magnitude) when ‘to them we can findequals’.
62 The definitions are extensively deployed in the proofs, as is clear
from the example below.
[Data 39 (see Diagram 3.10)] If each of the sides of a triangle is
given in magnitude, the triangle is given in form. So, let each of
the sides of a triangle ABC be given in magnitude. I say that the
triangle is given in form. So, let a straight line DL be given in
position, then prolonged towards the point D, and without limit
towards the remaining point, and let DE be posited equal to AB.
But AB is given; therefore DE is also given; but it is given also in
position; and D is given; therefore E is also given; and EF is equal
to BC; and BC is given; therefore EC is also given; but it is given
in position as well; and E is given; therefore F is also given; and
FG is equal to AC. And AC is given; therefore FG is also given.
But it is also given in position. And F is given; therefore G is also
given. And with centre the point E and radius ED let the circle

98HELLENISTIC MATHEMATICS: THE EVIDENCE
DJH be drawn; therefore DJH is given in position. Again, with
centre the point F and radius FG let the circle GJ K be drawn;
therefore GJ K is given in position; and the circle DHJ is given in
position; therefore the point J is also given. Let then each of the
E, F be given; therefore each of the JE, EF, FJ is also given in
position and in magnitude; therefore the triangle JEF is given in
form. And it is equal and also similar to ABC ; therefore the triangle
ABC is given in form.
‘Being given’ means that the mathematical object is determined and
identifiable on the basis of its size, position or form. In terms of a diagram,
it means that the element is fully identified; when it comes to the solution
of a geometrical problem, it means that we can take that element as knownand move on in our demonstration. The propositions in the Data mostly
aim at establishing correlations between givens, so that the reader can
recognize how, under certain conditions, the being-given of certain elementsentails the being-given of certain other elements. The book seems aimed at
a more advanced public than the Elements ; this is confirmed by its survival
in later antiquity as part of a group of texts, the so-called ‘treasure of analysis’,which were used to acquire and hone problem-solving skills.
Procedure
We have already hinted that propositions in Euclid are proved on the basis
either of the starting points or of previous propositions. While this isDiagram 3.10

99HELLENISTIC MATHEMATICS: THE EVIDENCE
generally true, several ripples corrugate the smooth logically-ordered surface
of the Elements . For one, the fifth postulate of book 1:
if a straight line meeting two straight lines produces the internal
angles on the same side less than two right angles, [let it be required
that] the two straight lines having been produced indefinitely meeteach other on that side on which are the two <angles> less than
two right angles
was criticized since antiquity for being not a postulate, but rather a statement
requiring proof. Moreover, terms are introduced which have not been
previously defined; things are established that are not subsequently usedbut seem too marginal to have been established just for their own sake;
apparently inexplicable detours are not infrequent and there are, as in the
case of proportionality, duplicate accounts whose mutual relation is notclear. Some demonstrative procedures are introduced, used a couple of times
and then abandoned. For instance, the idea of ‘fitting’ geometrical figures
onto one another is only deployed at 1.4 and 1.8, both about equality oftriangles (see Diagram 3.11):
If two triangles have the two sides equal to the two sides respectively
and have the angle surrounded by the equal lines equal to the
angle, they also have the base equal to the base, and the triangle
will be equal to the triangle, and the remaining angles will beequal to the remaining angles respectively <those> by which the
equal sides are subtended. Let the two triangles be ABC , DEF ,
having the two sides AB, AC equal to the two sides DE, DF
respectively, AB to DE and AC to DF and the angle BAC equal
to the angle EDF . I say that also the base BC is equal to the base
EF, and the triangle ABC will be equal to the triangle DEF , and
the remaining angles will be equal to the remaining angles
respectively <those> by which the equal sides are subtended, ABC
Diagram 3.11

100HELLENISTIC MATHEMATICS: THE EVIDENCE
to DEF , and ACB to DEF. Indeed the triangle ABC having been
fitted on the triangle DEF and the point A being put on the point
D and the line AB on the line DE, the point B will also fit on E
because AB and DE are equal. AB having been fitted on DE, the
line AC also <fits> on DF because the angles BAC and EDF are
equal. Therefore the point C will also fit on the point F because
again AC and DF are equal. But also B will fit to E; therefore the
base BC will fit on the base EF. If then B having been fitted on E,
and C on F, the base BC did not fit on EF, two straight lines would
surround an area, which is impossible. Therefore the base BC will
fit on EF and will be equal to it. Therefore also the whole triangle
ABC will fit on the whole triangle DEF and will be equal to it, and
the remaining angles will be on the remaining angles and will be
equal to them, ABC to DEF and ACB to DFE . If therefore two
triangles have the two sides equal to the two sides respectively andhave the angle surrounded by the equal lines equal to the angle,
they also have the base equal to the base, and the triangle will be
equal to the triangle, and the remaining angles will be equal to theremaining angles respectively <those> by which the equal sides are
subtended, which it was necessary to prove (1.4).
Some modern interpreters see the ‘fitting’ method as a relic from the
pre-Euclidean past, because its reliance on a certain ‘physicality’ of
geometrical objects would indicate a lower level of abstraction, and therefore
an earlier stage of development. I hardly need to point out that the
assumption that there is a correlation between level of abstraction, stage of
development and relative chronology is founded exclusively on one’ssubjective views of mathematics and of intellectual history. Because reliable
detailed information about early Greek mathematics is so scarce, we are in
no position safely to discriminate between earlier or later procedures. Also,greater or lesser development or level of advancement tend to be measured
against the yardstick of Archimedes and Apollonius. This may overestimate
the degree to which advanced authors affected mathematical practice across
the spectrum, and oversimplifies the variety of mathematical traditions which
persisted and in fact flourished even after the gathering of consensus arounda certain mathematical model embodied by Euclid, Archimedes and
Apollonius (our Big Guys). In other words, the introduction and successful
wide application of axiomatico-deductive structure, indirect proof and soon did not cause the total extinction of alternative methods, so that the use
of a procedure rather than another is no univocal indication of a later or
earlier date, or of a supposedly corresponding more or less advanced stageof development.

101HELLENISTIC MATHEMATICS: THE EVIDENCE
Many of the propositions in book 1 of the Elements are in the form of
problems: to construct an equilateral triangle on a given finite straight line
(1.1), to bisect a given rectilinear angle (1.10) or more complex tasks, suchas ‘to apply to a given straight line in a given rectilinear angle a parallelogram
equal to a given triangle’ (1.44). Problems of this kind are normally solved
by means of straight ruler and compass, i.e. nothing more than circles orarcs of circle are required for the construction. Take for instance 1.12 (see
Diagram 3.12):
To draw a perpendicular straight line to a given infinite straight
line from a given point which is not on the same line. Let then thegiven infinite straight line <be> AB and the given point which is
not on the same line C. It is necessary to draw a perpendicular
straight line to the given infinite straight line AB from the given
point C, which is not on the same line. Let then be taken on the
other side of the line AB a point whatever D, and with centre C
and radius CD let the circle EFG be drawn, and let the line EG
be divided into two at H, and let the lines CG, CH, CE be joined.
I say that CH has been drawn perpendicular to the given infinite
straight line AB from the given point C, which is not on the same
line. Since GH is equal to HE; HC is common; the two GH, HC
are equal to the two EH, HC respectively; and the base CG is equal
to the base CE; therefore the angle CHG is equal to EHC . And
they are one next to the other. If then a line standing onto a line
makes the angles one next to the other equal, each of the equalangles is right, and the line that has been erected is called perpen-Diagram 3.12

102HELLENISTIC MATHEMATICS: THE EVIDENCE
dicular to the line on which it has been erected. Therefore, a perpen-
dicular CH has been drawn to the given infinite straight line AB
from the given point C which is not on the same line; as it was
necessary to do.
The proofs I have quoted so far are direct, but several of the propositions
in book 1 are proved indirectly, by showing that the negation of what they
affirm is self-contradictory. For instance (see Diagram 3.13),
If a line which intersects two lines produces the alternate angles
equal to each other, the lines will be parallel to each other. Let
indeed a line EF intersecting two lines AB, CD produce the
alternate angles AEF , EFD equal to each other. I say that AB is
parallel to CD. If indeed it was not the case, AB, CD having been
prolonged will meet either on the parts B, D or on A, C. Let them
have been prolonged and let them have met on the parts B, D in
G. But the external angle of the triangle GEF , AEF, is equal to
the internal and opposite <angle> EFG ; which is impossible;
therefore AB, CD prolonged will not meet on the parts B, C.
Similarly it will be proved that they do not meet on the parts A,
C. But the <lines> which do not meet on any part are parallel;
therefore AB is parallel to CD. If a line which intersects two lines
produces the alternate angles equal to each other, the lines will be
parallel to each other, which it was necessary to prove (1.27).
A particular type of indirect method, first found in Euclid, is commonly
known as the ‘method of exhaustion’, although no ancient source designatesit by any name in particular, and we are not absolutely sure as to what
extent it was a standardized method. Used especially in order to determine
the relation between circular and rectilinear objects, the method ofexhaustion worked by proving that all possible alternatives to the result oneDiagram 3.13

103HELLENISTIC MATHEMATICS: THE EVIDENCE
was trying to establish were absurd. It relied on manipulations of ratios
between figures, and on a lemma, which various geometers stated in various
versions. The one in the Elements is as follows:
T wo unequal magnitudes having been posited, if from the greater
more than the half is subtracted and more than the half from theremainder, and this is done continuously, a magnitude will be left
which will be less than the posited lesser magnitude (10.1).
63
This lemma, which is used by Euclid to prove, among other things, thatany pyramid is a third of the prism with the same base and the same height
(12.7), corresponds with a lemma reported by Archimedes and traceable toEudoxus. Once again, it would seem that Euclid was revisiting old ground,
not just in terms of contents but also of procedures. A famous example of
method of exhaustion is Elements 12.2 (see Diagram 3.14):
Circles are to each other as the squares on their diameters. Let
ABCD , EFGH be circles, their diameters BD, FH. I say that the
circle ABCD is to the circle EFGH as the square on BD is to the
square on FH. If indeed it is not <the case that>, as the circle
ABCD to the EFGH , so the square on BD to that on FH, it will
be as the <square> on BD to that on FH, so the circle ABC D
either to an area less than the circle EFGH or to <an area> greater.
Let it be first to the lesser, the <area> R. And let the square EFGH
be inscribed in the circle EFGH ; the inscribed square is greater
than half the circle EFGH , because if tangents to the circle are
drawn through the points E, F, G, H, the square EFGH is half
the square circumscribed to the circle, and the circle is less thanthe circumscribed square. So the inscribed square EFGH is greaterDiagram 3.14

104HELLENISTIC MATHEMATICS: THE EVIDENCE
than half the circle EFGH . Let EF, FG, GH, HE be divided into
two in the points J, K, L, M, and conjoin EJ, JF, FK, KG,
GL, LH, HM, ME; therefore each of the triangles EJF , FKG ,
GLH , HME is greater than the half of its segment of circle, because
if tangents to the circle are drawn through the points J, K, L, N,
and the parallelograms on the lines EF, FG, GH, HE are
completed, each of the triangles EJF , FKG , GLH , HME will be
half of its parallelogram, but its section <of the circle> is less than
the parallelogram. So each of the triangles EJF , FKG , GLH ,
HME is greater than the half of its section of circle. Dividing then
the remaining arcs in two and conjoining the lines and doing this
continuously some parts of circle are left over which will be lessthan the excess by which the circle EFGH exceeds the area R. For
it is proved in the first theorem of the tenth book that two unequal
magnitudes having been posited, if more than the half is subtractedfrom the greater and more than the half from the remainder and
this is done continuously, a magnitude is left which is less than
the posited lesser magnitude. Let it be left then, and let the sectionsof the circle EFGH on EJ, JF, FK, KG, GL,
LH, HM, ME be
less than the excess by which the circle EFGH exceeds the area R.
Therefore the leftover polygon EJFKGLHM is greater than the
area R. Let the polygon ANBOCPDQ similar to the polygon
EJFKGLHM be inscribed in the circle ABCD . Therefore as the
square on BD is to the square on FH, so the polygon ANBOCPDQ
is to the polygon EJFKGLHM . But also as the square on BD is
to that on FH, so the circle ABCD is to the area R. And therefore
as the circle ABCD is to the area R, so the polygon ANBOCPDQ
is to the polygon EJFKGLHM . Conversely therefore as the circle
ABCD is to the polygon inscribed in it, so the area R is to the
polygon EJFKGLHM . But the circle ABCD is greater than the
polygon in it. Therefore the area R as well <is greater> than the
polygon EJFKGLHM . But it is less; which is impossible. There-
fore it is not <the case that> as the square on BD is to the square
on FH, so the circle ABCD to some area less than the circle EFGH .
Similarly we prove that it is not <the case that> as the square onFH to the square on BD, so the circle EFGH to some area less
than the ABCD . I say then that it is not <the case that> as the
square on BD is to the <square> on FH, so the circle ABCD is to
some area greater than the circle EFGH . If indeed it is possible,
let it be to a greater area R. And again therefore as the square on
FH is to that on BD, so the area
R to the circle ABCD , but as the
area R to the circle ABCD , so the circle EFGH to some area less

105HELLENISTIC MATHEMATICS: THE EVIDENCE
than the circle ABCD . And therefore as the <square> on FH to that
on BD, so the circle EFGH to some area less than the circle ABCD ;
which it has been proved to be impossible. Therefore it is not <thecase that> as the square on BD is to that on FH, so the circle
ABCD to some area greater than the circle EFGH . It has been
proved that it is not to a lesser <area>. Therefore it is as the squareon BD to that on FH, so the circle ABCD to the circle EFGH .
Circles are to each other as the squares on their diameters; which
it was necessary to prove.
In the fragment attributed to Hippocrates of Chios and reported on
p. 57 we read:
He made a starting-point, and posited as first thing useful for
these, that similar segments of circles have the same ratio to eachother as the squares on their bases. And this he proved by showing
that the squares on the diameters have the same ratio as the circles.
Let us ignore for a moment the problems related to Simplicius’ testimony,
and let us say that Hippocrates in the early fourth century
BC already knew
the result on the ratio of circles. What he may not have known, however,was how to prove his result rigorously, in a way that was immune from the
criticisms of self-contradiction or circularity or inappropriateness variously
levelled by Aristotle. Eudoxus then formulated a lemma which allowed himto talk rigorously about what happens when processes of subtraction,
addition or multiplication are carried out indefinitely. The combination of
Hippocrates’ result and Eudoxus’ lemma is seen in Euclid’s Elements , whether
he reproduced it from an extant source or put the two things together himself.
The old and the less old are yet again combined to powerful effect.
Archimedes
Archimedes was one of the many victims of the many wars fought in theHellenistic period. He was killed in 212
BC when his city Syracuse was taken
by the Romans after two years of siege, as part of a larger conflict betweenRome and Carthage. Syracuse, as we mentioned, had excellent fortifications
in which Archimedes may have been involved. He also played a major role
in the devising of military machines. Polybius says that his engines werecapable of throwing projectiles both at long and close range, and describes
an iron hand attached to a chain which would clutch a ship, lift it up in the
air and then drop it into the sea with disastrous consequences.
64 There are
also reports of machines Archimedes built in peace time – a weight-lifting

106HELLENISTIC MATHEMATICS: THE EVIDENCE
instrument with which he launched an enormous ship, and a sphere that
imitated the heavens.65
Apart from his adventurous life and wondrous inventions, Archimedes
is famous as the author of several mathematical works: Sphere and Cylinder ,
in two parts (= SC); Quadrature of the Parabola (QP); Measurement of the
Circle (MC); Spirals (SP); Equilibrium of Planes in two parts ( EP); Sand-
Reckoner (AR); Conoids and Spheroids (CS); Floating Bodies in two parts
(FB); Method to Eratosthenes ; The Cattle Problem . Various shorter works of
more uncertain attribution are also extant ( Stomachion ; Lemmas; On Circles
T angent to Each Other ), as are the titles of several lost works, one or more of
them on mechanics. Nearly each work is introduced by a letter, where
Archimedes explains and summarizes the contents, and occasionally providesa history of the issues at hand. His addressees are Dositheus, Eratosthenes
and, for the Sand-Reckoner , Gelon, who succeeded Hiero as king of Syracuse.
Archimedes mentions that a former correspondent, Conon, hasrecently died, and that he had sent arithmetical studies to a Zeuxippus.
Like Eratosthenes, both Dositheus and Conon were probably based at
Alexandria, and many historians think that Archimedes himself spent sometime there.
66
Contents
Archimedes’ name is linked to some discoveries in plane and solid geometry,
particularly the determination of the area and volume of curvilinear figures.Most famously, he established the equivalence between a circle and a recti-
linear figure, thus finally squaring the circle (see Diagram 3.15):
Diagram 3.15

107HELLENISTIC MATHEMATICS: THE EVIDENCE
Any circle is equal to a right-angled triangle, whose radius is equal
to one of the <sides> around the right angle, while the perimeter<is equal> to the base. Let the circle ABCD have to the triangle E
<the relation> as assumed; I say that it is equal. If possible, let the
circle be greater, and let the square AC be inscribed, and let the
arcs be divided in half, and let the segments already be less than
the excess by which the circle exceeds the triangle; therefore the
rectilinear figure is even greater than the triangle. Let a centre M
be taken and a perpendicular MN; therefore MN <is> less than the
side of the triangle. Then the perimeter of the rectilinear figure is
also less than the remainder, since <it is also less> than the perimeterof the circle. Therefore the rectilinear figure is less than the triangle
E; which is absurd. Let then the circle, if possible, <be> less than
the triangle E, and let the square be circumscribed, and the arcs
be divided in half, and tangents throught the points drawn.
Therefore the <angle> OAQ is right. Therefore OQ is greater than
LQ; for QL is equal to QA; and the triangle QOP therefore is
greater than half of the figure OFAL . Let the segments similar to
PFA then have been left less than the excess by which E exceeds
the circle ABCD ; therefore the circumscribed rectilinear figure <is>
even less than E; which is absurd; it is in fact greater, since MA is
equal to the perpendicular of the triangle, and the perimeter is
greater than the base of the triangle. Therefore the circle is equalto the triangle E.
67
While the theorem expresses the area of a circle in terms of such-and-
such a triangle, another proposition from the same book provided a
numerical expression of the same problem, by stating that ‘Thecircumference of any circle is three times the diameter plus an amount
which is less than the seventh part of the diameter, and more than ten
seventy-one parts <of the diameter>’ ( MC 3). Archimedes also determined
the area and volume of a sphere, and proved several results about spherical
segments.
68 His interest in the relationship between rectilinear and curvilinear
figures may have been behind his enquiries into geometrical objects whichare not in Euclid’s Elements . For instance, the conics, produced by cutting a
cone with a plane (especially the parabola, which Archimedes calls ‘section
of a right-angled cone’) or conoids and spheroids, obtained by rotatingconics, or the spiral, which originates from a segment rotating around one
of its extremities with a point moving along it at the same time.
69 As with
more traditional geometrical objects, his efforts were directed at determiningthe area or volume of these figures.

108HELLENISTIC MATHEMATICS: THE EVIDENCE
Archimedes also tackled topics which belong to what we would call
mathematical physics: centres of gravity of plane figures, the problem of
determining conditions of equilibrium and the problem of determiningwhether a body will sink or float when immersed in a liquid. He stated and
proved the so-called law of the lever, according to which magnitudes, whether
commensurable ( EP 1.6) or incommensurable ( EP 1.7), are in equilibrium
when their distance from the fulcrum of a balance from which they are
suspended is inversely proportional to their weight. Unfortunately, none of
these works comes with a prefatory letter, and none of them provides adefinition of centre of gravity. Archimedes does refer, however, to other
mechanical books, which may have introduced basic notions.
Procedure
Like Euclid, Archimedes employed both direct and indirect proofs. He often
used the method of exhaustion, for instance to prove the theorem about the
area of the circle reported above. Another procedure worth noting is
exemplified as follows (see Diagram 3.16):
To cut the given sphere with a plane, so that the surfaces of the
segments have to each other the same ratio as a given one. Let it
be done, and let the greatest circle of the sphere be ADBE, a
diameter of that be AB, and let a perpendicular plane have been
dropped towards AB, and let the plane produce a section DE in
the circle ADBE, and let AD, BD be conjoined. Since now the
ratio of the surface of the segment DAE to the surface of theDiagram 3.16

109HELLENISTIC MATHEMATICS: THE EVIDENCE
segment DBE is <given>, but the surface of the segment DAE is
equal to the circle in which the radius is equal to DB, as the said
circles to each other, so the <square> on AD to that on DB, that is
AC to CB, therefore the ratio of AC to CB has been given; so that
the point C is given. And DE is perpendicular to AB; therefore the
plane across DE is also given in position. Let it be put together
like this; let there be a sphere ABDE, in which <there is> a greatest
circle, and a diameter AB, and the given ratio that of Z to H, and
let AB be cut in C, so that it is as AC to BC, so Z to H, and let the
sphere be cut across the plane C perpendicularly to the line AB,
and let DE be the common section, and let AD, DB be conjoined,
and let two circles H, K, be posited, and H having the radius equal
to AD, while K has the radius equal to DB; therefore the circle H
is equal to the surface of the segment DAE, while the <circle> K
<is equal> to the segment DBE; this indeed has been already proved
in the first book. And since the <angle> ADB is right and CD is
perpendicular, it is as AC to CB, that is Z to H, <so> the <square>
on AD to that on DB, that is the <square> on the radius of the
circle H to that of the radius of the circle K, that is the circle H to
the circle K, that is the surface of the segment DAE to the surface
of the segment DBE of the sphere ( SC 2.3).
In the solution to this problem, Archimedes first assumes that the problem
is already solved (‘Let it be done’), then works from this assumption back-wards, so to speak, until every element of the problem is accounted for and
given. This part is called analysis, a ‘breaking down’ the construction into
its elements. The second part of the process, the synthesis or ‘putting
together’, consists of a normal proof, where every step is basically repeated,
in the confidence that it can be justified. Analysis-and-synthesis procedures,
which had been described by Aristotle and were not confined to mathematics,allow a glimpse into the way ancient geometers attained their results before
organizing them in a demonstrative structure – what is called their heuristics
(process of discovery). Most of the proofs one finds are in the form of
synthesis; Archimedes is pretty unusual in appending analyses as well.
Further unparalleled glimpses into the heuristics of Greek mathematics
are afforded by the Method – in the introductory letter, Archimedes explains
to Eratosthenes that he wants to make public the way in which many of his
results had occurred to him, so that people can benefit from it and discovereven more theorems. The procedure combines geometry and mechanics,
and can be applied when one needs to establish the equivalence between a
certain object (say, a sphere) and another object whose characteristics arebetter known (say, a cone). The two objects are imagined to be at the two

110HELLENISTIC MATHEMATICS: THE EVIDENCE
ends of a balance – if the conditions for their equilibrium can be established,
then so can the conditions for their equivalence . This is achieved using on
the one hand the results about centres of gravity and equilibrium that
Archimedes had formulated in EP, and, on the other hand, by considering
each figure to be made up by an infinite number of lines, for each of which
the equilibrium conditions are then proved to apply. In other words, mecha-
nics can be applied to geometry because a connection can be made betweengeometrical being, extension, which determines area or volume, and
mechanical or physical being, which has to do with equilibrium and centre
of gravity.
An example of a result proved ‘normally’: [ QP 24 (see Diagram
3.17)] Any segment surrounded by a line and a section of right-
angled cone [a parabola] is four thirds of the triangle having thesame base and equal height. Let ADBEC be a segment surrounded
by a line and a section of right-angled cone, let ABC be a triangle
having the same base and equal height, let the area K be four thirds
of the triangle ABC. It is to be proved that <the area K> is equal to
the segment ADBEC. If in fact it is not equal, it is either greater or
lesser. First, if possible, let the segment ADBEC be greater than
the area K. I have inscribed the triangles ADB, BEC, as it is said, I
have inscribed also in the remaining segment all the triangles having
the same base as the segments and the same height, and always in
the next segments let two triangles be inscribed having the samebase as the segments and the same height; the segments which areleft over will be less than the excess by which the segmentDiagram 3.17

111HELLENISTIC MATHEMATICS: THE EVIDENCE
ADBECexceeds the area K. So that the inscribed polygon will be
greater than K, which is impossible. Since then there are posited
areas one next to the other in a quadruplicate ratio, first the triangle
ABC is four times the triangles ADB, BEC, since these same
<triangles> are four times those inscribed in the said segmentsand always like that, it is clear that all the areas together are less
than a third of the greatest, while K is four thirds of the greatest
area. Therefore the segment ADBEC is not greater than the area
K. Let it then, if possible, be lesser. Let the triangle ABC be posited
equal to Z, H <being> a fourth of Z, and H similar to H, and let it
be set always one next to the other, until the last comes out less
than the excess by which the area K exceeds the segment, and let I
be less; then the areas Z, H, H, I and the third of I are four thirds
of Z. K is also the four thirds of Z; therefore K is equal to Z, H, H,
I and a third part of I. Since now the area K exceeds Z, H, H, I by
less than I, but the segment by more than I, it is clear that even
more the areas Z, H, H
, I are greater than the segment; which is
impossible; it has been proved in fact that if the area taken next toit in quadruple ratio is as large, the greatest is equal to the triangleinscribed in the segment, the areas all together will be lesser than
the segment. Therefore the segment ADBEC is not lesser than the
area K. It has been proved that it is not greater either; therefore it
is equal to K. The area K is then four thirds of the triangle ABC;
and therefore the segment ADBEC is four thirds of the triangle
ABC.
The same result established through a combination of geometry,
mechanics and infinitesimals: [ Method 1 (see Diagram 3.18)] Let
the segment ABCbe surrounded by the line AC and by the section
of right-angled cone ABC, and let AC be divided in half at D, and
let DBE be drawn <parallel to> the diameter, and let AB, BC be
conjoined. I say that the segment ABC is four-thirds of the triangle
ABC. Let from the points A, C be drawn AZ <parallel> to DBE,
and CZ tangent to the section, and let < CB> be prolonged <to K,
and let KH be posited equal to CK>. Let the balance CH be

112HELLENISTIC MATHEMATICS: THE EVIDENCE
imagined and its centre K and a parallel whatever to ED, MN. Since
now CBA is a parabola, and CZ is a tangent and CD is an ordinate,
EB is equal to BD; this in fact has been proved in the elements;
because of this, and because ZA, MN are parallel to ED; MN is
also equal to NN, and ZK to KA. And since it is as CA to AN, so
MN to NO, on the other hand as CA to AN, so CK to KN, and CK
is equal to KH, therefore <it is> as HK to KN, so MN to NO. And
since the point N is centre of gravity of the line MN, because MN
is equal to NN, if therefore we set TH equal to NO and H as its
centre of gravity, so that TH is equal to HH, THH is in equilibrium
with MN, this latter remaining <in place>, because HN is divided
inversely with respect to the weights TH, MN, and as HK to KN,
so MN to HT; so that the centre of gravity of both is K. Similarly
also if one draws all the parallels to ED in the triangle ZAC, they
are in equilibrium, remaining <in place>, with the <lines> that they
cut off from the section <of right-angled cone>, transported to the<point H, so that> the centre of gravity <of both is> K. And since
the triangle GZA is made up from the <lines> in the triangle CZA,
the segment ABC is made up of the <lines> taken in the section
<of right-angled cone> similarly to NO, therefore the triangle ZACDiagram 3.18

113HELLENISTIC MATHEMATICS: THE EVIDENCE
will be in equilibrium remaining <in place> with the segment of
section <of right-angled cone> having been posited around the
centre of gravity H in the point K, so that the centre of gravity of
both is K. Let CK be divided in X, so that CK will be three times
KX; therefore the point C will be the centre of gravity of the triangle
AZC; it has been proved in fact in the Equilibria . Since then the
triangle ZAC remaining <in place> is in equilibrium with the
segment BAC staying in K around the centre of gravity H, and the
centre of gravity of the triangle ZAC is X, therefore it is as the
triangle AZG to the segment ABG posited around the centre H, so
HK to XK. But HK is three times KX; therefore the triangle AZC
will also be three times the segment ABC. The triangle ZAC is four
times the triangle ABC because ZK is equal to KA and AD to the
DC; therefore the segment ABC is four thirds of the triangle ABC.
Archimedes made it quite clear that his mechanical method did not
amount to a rigorous proof, either because he was wary of using mechanics
to yield geometrical results, or because he had a problem with infinitesimals,
or because of both. He seemed aware of a distinction between the contextof discovery and that of justification, and of the need to package a discovery
in a way that was acceptable. Discovering a result could turn into nothing
if the supporting proof did not work, or could not be formulated.Archimedes also intimated that a result may be felt to be correct but it
could still take a long time for a valid proof to be found.
‘Acceptable’, ‘valid’ are all terms that imply the existence of an accepting
or rejecting public on the one hand, and of an agreed-upon set of rules on
the other hand. Both seem to have been present at Archimedes’ time – his
addressees, both the ones he names and the many unnamed ones, constituteda sort of community, where people knew each other, exchanged work, shared
interests, a language and, at least to some extent, some criteria of what
constituted ‘good’ and ‘bad’ mathematics.
Apollonius
Apollonius, according to Eutocius, was born at Perga in Pamphylia during
the reign of Ptolemy III (246–21 BC). He lived between Alexandria and
Pergamum, and mentioned a stay at Ephesus. The only surviving work byhim is the Conics , originally in eight books, of which four are extant in
Greek and three in Arabic only. There were other works, of which the titles
and occasional excerpts are preserved: on rules for multiplication and/or a

114HELLENISTIC MATHEMATICS: THE EVIDENCE
system to express large numbers, on cutting off a ratio, an area and a deter-
minate section, on tangencies, plane loci, neuseis , on a comparison of the
dodecahedron with the icosahedron. Apollonius is also credited with studieson the spiral, on astronomy and on optics. Like Archimedes, he prefaced
his work with letters: the first two parts of the Conics are addressed to
Eudemus of Pergamum, the last four to an Attalus, who might be kingAttalus I of Pergamum. Apollonius also mentioned the geometers Naucratis,
Philonides, Conon of Samos and Nicoteles of Cyrene, as well as Euclid.
70
Contents
Apollonius himself summarized the contents of the Conics in his introduction
to the first book:
At the time when I was with you [Eudemus] in Pergamum, I saw
that you were eager to get a copy of the Conics which I had worked
out. So I send you the first book, which I have corrected, and the
remaining ones will be sent off when I am satisfied with them. ForI think that you do not forget that you heard from me how I
undertook the composition of this matter at the request of the
geometer Naucrates, at the time when he was relaxing with usafter he arrived in Alexandria; and how, having elaborated it in
eight books, I immediately gave copies to him in a hurry, without
revising them, since he was on the point of sailing away: instead Iput down everything as it occurred to me, with the intention of
coming back to it in the end. Hence, having now got the oppor-
tunity, I am publishing each part as it gets its revision. Now sinceit so happened that some others of those who came into contact
with me got copies of the first and second books before they were
corrected, do not be surprised if you come across these in a differentversion. Of the eight books, the first four constitute an elementary
introduction. The first contains the methods of generating the
three sections and the opposite branches [of the hyperbola], and
their basic properties, developed more fully and more generally
than in the works of the other [writers on conics]; the secondcontains the properties of the diameters and axes of the sections,
the asymptotes, and other matters which have typical and essential
applications in diorismoi […] The third contains many surprising
theorems useful for the synthesis of solid loci and for diorisms; of
these the greater part and the most beautiful are new. It was the
discovery of these that made me aware that Euclid has not workedout the whole of the locus for three and four lines, but only a

115HELLENISTIC MATHEMATICS: THE EVIDENCE
fortuitous part of it, and that not very successfully; for it was not
possible to complete the synthesis without additional discoveries.
The fourth deals with how many times the conic sections mayintersect each other and the circumference of a circle, and other
matters in addition; neither of these two problems have been
written about by our predecessors, namely in how many points aconic section or circumference of a circle <can intersect [opposite
branches] and in how many points opposite branches> can intersect
<opposite branches>. The remaining [books] are more particular:one deals somewhat fully with minima and maxima , another with
equal and similar conic sections, another with theorems concerning
diorismoi , another with determinate conic problems.
71
Conics were not a new topic, but it is evident that, as well as distancing
himself from some of his predecessors by declaring his account fuller, clearerand more general than theirs, Apollonius was keen to add new things and
redefine concepts. For instance, he re-christened as parabola, hyperbola and
ellipsis what were already known as sections of a right-angled, obtuse-angledand acute-angled cone, respectively. He showed how all three curves could
be produced by cutting the same cone, rather than three different cones,
and redefined them on the basis of the relation between some of theirelements. For instance (see Diagram 3.19),
If a cone is cut by a plane through the axis, and it is cut also by
another plane cutting the base of the cone along a line
perpendicular to the base of the triangle through the axis, and the
diameter of the section prolonged meets one side of the trianglethrough the axis outside the vertex of the cone, any <line> which
is drawn parallel from the section to the common section of the
cutting plane and the base of the cone until the diameter of thesection, is equal in square to some area applied to some line, to
which <line> the <line> which is the prolonging of the diameter
of the section, and which subtends the angle outside the triangle,
has a ratio which <is> the square on the <line> drawn from the
vertex of the cone parallel to the diameter of the section until thebase of the triangle to the <rectangle> formed by the bases of the
section, as produced by a <line> drawn, having as latitude the
<line> cut off by this from the diameter to the vertex of the section,exceeding by a shape similar and similarly posited the <rectangle>
formed by the <line> subtending the external angle of the triangle
and by the parameter; let then this section be called hyperbola[…] (1.12).
73

116HELLENISTIC MATHEMATICS: THE EVIDENCE
Also, for each type of conic, Apollonius extended the properties defined
for one particular diameter, i.e. the axis, to all diameters, and the properties
defined for ordinates perpendicular to the diameter to all ordinates (1.50–
51). This allowed him to solve some problems of construction – for instance,how to find a certain conic given certain data at the outset. Above all, he
can be said to have streamlined the treatment of conics; by defining them
on the basis of a common origin and by generalizing their properties, heconstituted them as fully-determined, rigorously-described, geometrical
objects.
Procedure
Apollonius used both direct and indirect methods; a proof of the conditions
of equality for two parabolas remarkably echoes Euclid’s superposition or
‘fitting’ procedure (see Diagram 3.20):Diagram 3.19

117HELLENISTIC MATHEMATICS: THE EVIDENCE
Parabolas in which the parameters of the perpendiculars to the
axes are equal are themselves equal, and if parabolas are equal,
their parameters are equal. Let there be two parabolas, with axesAD, ZH, and equal parameters, AE, ZM. Then I say that these
sections are equal. When we apply axis AD to axis ZH, then the
section will coincide with the section so as to fit on it. For if itdoes not fit on it, let there be a part of section AB which does not
fit on section ZH. We mark point B on the part of it which does
not coincide with ZH, and draw from it [to the axis] perpendicular
BK, and complete rectangle KE. We make ZK equal to AK, and
draw from point K a perpendicular to the axis [meeting the section
in H], KH, and complete rectangle KM. Then lines KA, AE are
equal to lines KZ, ZM, each to its correspondent; therefore the
rectangle from KA, AE is equal to the rectangle from KM, ZM.
And the square on KB is equal to the rectangle EK, as is proven in
proposition 11 of the first book. And similarly too the square on
KH is equal to the rectangle KM; therefore KB is equal to KH. So
when the axis [of one section] is applied to the axis [of the other],
line AK will coincide with line ZK, and line KB will coincide with
line KH, and point
B will coincide with point H. But it was
supposed not to fall on section ZH: that is absurd. So it is
impossible for the section not to be equal to the section.
Furthermore, we make the section equal to the section, and makeAK equal to line ZK, and draw the perpendiculars [to the axis]
from points K, K, and complete rectangles EK, MK: then section
AB will coincide with section ZH, and therefore axis AK will
coincide with axis ZK. For if it does not coincide with it, parabolaDiagram 3.20

118HELLENISTIC MATHEMATICS: THE EVIDENCE
ZH has two axes, which is impossible. So let it coincide with it.
Then point K will coincide with point K, because AK is equal to
ZK. And point B will coincide with point H. Therefore BK is
equal to KH; therefore the rectangle EK is equal to the rectangle
KM. And AK is equal to ZK, therefore AE is equal to ZM.74
In fact, Apollonius explicitly mentions Euclid in the introduction to the
first book of the Conics , as quoted before, and claims to have improved on
him. There are also interesting parallels between lost works by Apollonius,and subjects which Archimedes deals with: both studied the spiral, both
produced notation systems to express very large numbers, both calculated
the ratio between diameter and circumference of a circle. Eutocius evenreports that Apollonius had been accused of plagiarizing Archimedes in his
treatment of conics.
75 Be that as it may, Apollonius’ work clearly presupposes
an accumulation of mathematical results: his style is deductive in the mouldof Euclid and Archimedes, but also very conscious of the need to systematize
and order his material, and of the necessity carefully to distinguish between
similar results proved for cognate, but non-identical objects. See for instance1.26, where the case of a parabola and a hyperbola are considered separately
(see Diagram 3.21):
If in a parabola or a hyperbola a line is drawn to the diameter of
the section, it meets the section at only one point. Let there be aparabola first, with diameter ABC, and parameter AD, and let EZ
be drawn parallel to AB. I say that EZ prolonged will meet the
section. Let a point E be taken on EZ, and from E let EH be
drawn parallel ordinately, and let the <rectangle> formed by DACbeDiagram 3.21

119HELLENISTIC MATHEMATICS: THE EVIDENCE
greater than the <square> on HE, and from C let CH be erected
ordinately; therefore the <square> on HC is equal to the <rectangle>
formed by DAC. And the <rectangle> formed by DAC is greater
than the <square> on EH; therefore the <square> on HC will be
greater than that on EH as well; and therefore HC will be greater
than EH as well. And they are parallel; therefore EZ prolonged
cuts HC; so that it will also meet the section. Let it meet <the
section> in K. I say that it will meet it in only one point K. If in
fact possible, let it meet <the section> in K as well. Since then a
line cuts a parabola in two points, prolonged it will meet the dia-
meter of the section, which is absurd; for it is assumed to be parallel.
Therefore EZ prolonged meets the section in only one point. Let
the section be a hyperbola, with transverse side AB and parameter
AD, and let DB be conjoined and prolonged. Having constructed
these things let CM be drawn from C parallel to AD. Since the
<rectangle> formed by MCA is greater than that formed by DAC,
and the <square> on CH is equal to the <rectangle> formed by
MCA, and the <rectangle> formed by DAC is greater than the
<square> on HE, therefore the <square> on CH will be greater than
that on EH as well. So that CH is greater than EH as well, and the
same things will happen as before.76
The proof combines direct and indirect methods, is divided into sub-
cases, it takes advantage of Apollonius’ new definitions, and it builds onprevious results. Sub-cases consider variations in the characteristics of the
geometrical figures under examination, and are already found in the Elements ,
but they are particularly frequent in the Conics , where sometimes we have
more than one diagram for the same proposition (e.g. 4.56, 4.57).
Apollonius provides ample evidence that the objects of geometrical
inquiry were becoming increasingly complex: they no longer had simple,univocal properties and a limited field of variation ranging e.g. from acute
to right-angled to obtuse. Conics allowed much more room for manoeuvre;
the task of providing some grasp of their manifold configurations could be
articulated as a diorismos . The proposition below is an example: Apollonius
set the question and then considered how the solution varied with thevariation of elements in Diagram 3.22.
If there is a hyperbola, and the transverse diameter of the figure
constructed on its axis is not less than its parameter, then the para-
meter of the figure constructed on the axis is less than the parameter
of [any of] the figures constructed on the other diameters of thesection, and the parameter of [any of] the figures constructed on

120HELLENISTIC MATHEMATICS: THE EVIDENCE
diameters closer to the axis is less than the parameter of the figures
constructed on [diameters] farther from the axis. Let there be a
hyperbola with axis AC and center H, and with two of its diameters
KB, TT. Then I say that the parameter of the figure of the section
constructed on AC is less than the parameter of the figure of the
section constructed on KB, and that the parameter of the figure of
the section constructed on KB is less than the parameter of the
figure of the section constructed on TT. First, we make axis AC
equal to the parameter of the figure constructed on it. […] Further-more, we make axis AC greater than the parameter of the figure
of the section constructed on it […] Furthermore, we make line
AC less than the parameter of the figure constructed on it, but
not less than half the parameter of the figure constructed on it:Diagram 3.22

121HELLENISTIC MATHEMATICS: THE EVIDENCE
then I say that, again, the parameter of the figure constructed on
AC is less than the parameter of the figure constructed on KB,
and that the parameter of the figure constructed on KB is less than
the parameter of the figure constructed on TT. […] Furthermore,
we make AC less than half the parameter of the figure of the section
constructed on it: then I say that there are two diameters, [one]on either side of this axis, such that the parameter of the figure
constructed on each of them is twice that [diameter]; and that
[parameter] is less than the parameter of the figure constructed onany other of the diameters on that side [of the axis]; and the
parameter of figures constructed on the diameters closer to those
two diameters is less than the parameter of a figure constructed ona [diameter] farther [from them] […].
77
In sum, Apollonius’ use of subcases, his interest in generalizations and
in determination of the conditions of validity or solubility of a proposition,
his (explicit) references to Euclid and (implicit?) to Archimedes, as well as
the complexity itself of his topics, suggest the presence of cumulative mathe-matical knowledge, without which some operations (organizing, redefining,
being ‘clearer’, adding new things) would lack significance. His works hint
that a mathematical tradition was in the making, and that written culture(see the awareness he shows of editing, correcting, of having different copies
of his work circulating) played a giant role in the formation of that tradition.
Notes
1 Archimedes, Sand-Reckoner (Mugler 134–5), my translation.
2 Thucydides, The Peloponnesian War 1.102.2; 2.18.1; 2.58.1; 2.75 ff.
3 On Philo and fortifications, see Winter (1971), esp. 118 f.; Garlan (1974); Lawrence
(1979); McNicoll and Milner (1997). Some of the elements of Fort Euryalus are so
sophisticated that more than one scholar is inclined to suggest Archimedes as the engineer
behind them, see Lawrence (1946); Winter (1963). Cf. also Ober (1992).
4 Philo, Poliorketika 1.3–7; 1.39; 1.64; 1.87; 2.19 (quotation). The Poliorketika and the
Belopoiika were part of a larger work, entitled Mechanical Syntaxis , see Ferrari (1984).
5 References in Garlan (1974), 207 ff.
6 Castagnoli (1956); Owens (1991). We have references to surveying and town planning
in Callimachus, Aetia 1.24 (a ten-foot pole used both as a goad for oxen and a measure
for land), 2.fragment 43 (foundation rites for a city) and Plautus, Poenulus 46–9 (the
person explaining the plot of the play compares himself to a surveyor determining the
boundaries of a territory).
7 Polybius, Histories 6.27–32, Loeb translation with modification.
8 Pamment Salvatore (1996) – mid-second century BC evidence from various sites in Spain.
9 Polybius, Histories 6.42, Loeb translation with modifications.
10 Larissa in Salviat and Vatin (1974); Halieis and Chersonesos (fourth century BC) in Boyd
and Jameson (1981). For a Hellenistic dating of the Chersonesos land-division, seeDufková and Pecírka (1970); Wasowicz (1972). For Roman cases, see e.g. Dilke (1971),

122HELLENISTIC MATHEMATICS: THE EVIDENCE
Gabba (1984), Salmon (1985), Moatti (1993). For redistributions, see e.g. Austin (1981),
documents 180 ( c. 275 BC) from Asia Minor, 235 (259/8 BC) and 240 (257 BC) from
Egypt, 271 (240 BC) from Lycia.
11P . T ebt . 87 (late second century BC, probably from Ber enikis Thesmophorou in the Egyptian
Fayum), 46–62, partial translation in Thompson/Crawford (1971), 13, with modifica-
tions. For land-surveys in Egypt see also Déléage (1933), Cuvigny (1985), who indicates
that some of the papyri had diagrams or maps, 88. A survey like this was not the only
type available: we have at least one example of what was called simply a ‘determination of
boundaries’ ( periorismos ): see document 185 in Austin (1981), which is an inscription
from Didyma in Asia Minor, dating 254/3 BC. The main difference of this latter to a
geometria was that no size was indicated.
12 A similar procedure in P . Mich . 3245 (probably second century BC, perhaps from the
Arsinoite nome), in Bruins et al. (1988).
13 Reproduced from Lyon (1927).
14P . Flind. Petr . 2.11.2, cf. Lewis (1986), 42.
15P . Freib . 7, in Hunt (1934), number 412, Loeb translation with modifications.
16 See e.g. P . Cairo Zen . 59132 (256 BC), P . Cairo Zen . 59188 (255 BC); P . Lond . 2027 (not
dated, but also from Zenon’s archive); P . Col. Zen . 2.87.1–22 (244 BC), in Westermann
and Sayre Hasenoehrl (1934); P. E n t . 66 (218 BC), mentioned in Lewis (1986), 65 f.; P.
T ebt. 24 (117 BC), in Grenfell (1902).
17P . Col. Zen . 88 (243 BC), translation in Westermann and Sayre Hasenoehrl (1934). Cf.
also e.g. P . T ebt . 24 (see note above); P . Col. Zen . 54 (250 BC); P . Cairo Zen . 59355 (243
BC), and see Lewis (1986), 44, 53.
18P . Cairo Zen . 59330, my translation. In 59331, which is more fragmentary, Pemnas
manifests to Zenon the suspicion that Herakleides ‘has entered to his debit a larger quantitythan he really owes’. Both documents are dated 30 June, 248
BC, and are in Edgar (1925),
52–4 (the quotation is from his introduction to 59331).
19 The documents in Uguzzoni and Ghinatti (1968).
20 Official instructions in P . Rev. Laws (259/8 BC). For Hellenistic examples of public accounts,
cf. Burford (1969); Austin (1981), document 97, from Olbia on the Black Sea (latethird–early second century
BC), and document 194, from the temple of Apollo at Delos
(third century BC); Rhodes and Lewis (1997).
21P . Cairo 65445, in Guéraud and Jouguet (1938). Cf. also P . Berol . 21296, second century
BC, which has a table of parts, in Ioannidou (1996), 202. Fowler (1988) and (1995) list
other examples (multiplication and tables of parts, three from the second century BC, one
from either the second or the first century BC); see also Morgan (1998).
22P . Cairo J.E. 89127–30, 89137–43 translation and numbering in Parker (1972).
23P . Cairo J.E. 89127–30, 89137–43, problem 6, translation and numbering as above. The
reference is to problems about series of parts where one has to find the next element in
the series.
24 The ostraka in Mau and Müller (1987); the papyri in Angeli and Dorandi (1987) and
Dorandi (1994). See also Fowler (1999), 209 ff.
25 Polybius, Histories 9.14 ff.
26 Polybius, Histories 12.17–22.
27 According to Strabo, Geography 2.4.1–3, Polybius expressed similar criticisms about the
mathematical data reported by some geographers.
28 Polybius, Histories 9.26a, Loeb translation with modifications.
29 Theocritus, Idylls 17.1–85, tr. A.S.F . Gow, Cambridge 1952, with modifications.
30 The main collection is Thesleff (1965), with introductory material in Thesleff (1961).
Also important Burkert (1972).
31 See e.g. Napolitano Valditara (1988).

123HELLENISTIC MATHEMATICS: THE EVIDENCE
32 The Academics in Dörrie (1987), ch. 5-8; the Peripatetics in Wehrli (1959).
33 [Aristotle], On Indivisible Lines 969b, my translation. The date of the treatise is uncertain,
but there is a reference to apotome at 968b which might provide a date post quem since
the earliest extant definition of apotome is in Euclid’s Elements book 10. Unfortunately,
we do not know Euclid’s date with accuracy.
34 As reported by Plutarch, On Common Conceptions Against the Stoics 1079d–f.
35 Apollonius, Conics preface to book 2; Hypsicles, Book 14 of the Elements preface. See
Sedley (1976); Mueller (1982), Appendix; Angeli and Dorandi (1987); Mansfeld (1998),
36.
36 Proclus, Commentary on the First Book of Euclid’s Elements 199, translation as above .
37 Diogenes Laertius, Lives of the Philosophers 9.85.
38 Diogenes Laertius, ibid. 9.90–91.
39 Diogenes Laertius, ibid. 7.132–133.
40 Diogenes Laertius, ibid. 7.135. Apollonius is believed to have lived in the second century
BC.
41 Diogenes Laertius, ibid. 7.81.
42 Mansfeld (1998), 14 ff.43 Aristarchus, On the Sizes and Distances of the Sun and Moon 11, my translation.
44 References to Euclid’s Elements in the Sphaerics too numerous to list; the Phenomena at
Days and Nights 2.10, 126.33; Meton at ibid. 2.18, 152.1; the theorems with subcases at
Sphaerics 3.9, 3.10.
45 Aratus, Phenomena 1140 ff.
46 Hipparchus, Commentary on Aratus and Eudoxus’ Phenomena 90.20–4.
47 Aristoxenus, Elements of harmonics 2.32–3, tr. A. Barker, Cambridge 1989.
48 [Euclid], Section of a Canon 148–9.
49 Diocles, On Burning Mirrors 4, tr. G.J. Toomer, Springer 1976, with modifications,
reproduced by kind permission of Spinger-Verlag, Inc. The text is only extant in an Arabic
translation.
50 Diocles, On Burning Mirrors 97–111, translation as above.
51 In Theon of Smyrna, Account of Mathematics Useful to Reading Plato 81.17 ff., my trans-
lation. Eratosthenes’ definition of ratio is similar to that in Euclid’s Elements book 5.
52 Sextus Empiricus, Against the Geometers 28.
53 Cleomedes, On Circular Motion 1.10; cf. Dicks (1971), 390.
54 Biton, Construction of War Instruments 67–8, translation Marsden (1971), with modifica-
tions.
55 Philo, Construction of Catapults 49.12–50.9, translation Marsden (1971), with modifi-
cations.
56 Philo, ibid. 51.15–52.17, translation as above.
57 See Heath (1926); Murdoch (1971); Caveing (1990).
58 The summary is adapted from Mueller (1981), viii–ix.
59 Euclid, Elements 1. Postulates 1–3, translations of Euclid are mine unless stated otherwise.
60 Euclid, Elements 5. definitions 4–6.
61 Multiplication, which here takes place between numbers, had already been assumed in
the definitions of ratios between magnitudes at the beginning of book 5. This and the
double definition of proportionality have led some scholars to conclude that for Euclid
numbers are not magnitudes, see Mueller (1981), 121 ff., 144 ff.
62 Euclid, Data definitions 1 and 2.
63 Alternative versions concern addition or multiplication, rather than subtraction, of magni-
tudes.
64 Polybius, Histories 8.3 ff.; among other sources are Livy, From the Founding of the City
24.34, 25.31; Cicero, On the Greatest Good and Bad 5.19.50, Tusculan Disputations 1.25.63,

124HELLENISTIC MATHEMATICS: THE EVIDENCE
5.23.64, 5.32.64 ff., Against Verres 2.58.131, On the Commonwealth 1.21.14; Diodorus,
Historical Library 26.18–19.
65 References in Dijksterhuis (1956).
66 Dositheus in SC, CS, SP, and QP; he is also mentioned (if it is the same person) in
Diocles, On Burning Mirrors 6. Eratosthenes in Method and Cattle Problem . Conon (of
Samos, if again it is the same person) is also mentioned in Apollonius, Conics 1.Preface
and Diocles, On Burning Mirrors 3. Zeuxippus in AR Introduction.
67 Archimedes, MC 1. T ranslations of Archimedes are mine unless stated otherwise.
68 Archimedes, SC 1.33, 1.34 and e.g. 1.35, 1.37, 1.38, 2.2 (volume of a spherical segment).
69 The conics were probably known to Euclid, who is credited with a treatise on them, and
perhaps already to fourth-century BC geometers like Menaechmus. Conoids and spheroids
seem to have been Archimedes’ own invention. As for the spiral, Pappus explains that itwas proposed by Conon of Samos and proved by Archimedes ( Mathematical Collection
234.1–4).
70 For biographical data, see Huxley (1963); Toomer (1970).
71 Apollonius, Conics 1.Preface, tr. G.J. Toomer (1990), xiv–xv, with modifications, repro-
duced by kind permission of Springer-Verlag, Inc.
72 Apollonius, Conics 1.11, 1.12 and 1.13, respectively.
73 My translation. The diagram here reproduced is from T oomer (1990) 667, by kind
permission of Springer-Verlag Inc.
74 Apollonius, Conics 6.1, translation as above.
75 Eutocius, Commentary on Apollonius’ Conics 1.5 ff.
76 My translation. The expression ‘lines drawn ordinately’ corresponds to what we call
‘ordinates’, i.e. a line bisected by the diameter of the conic, while an ‘abscissa’ is the
segment of diameter cut off by an ordinate.
77 Apollonius, Conics 7.33–5, translation as above. The diagram is also reproduced from
Toomer (1990), 822, by kind permission of Springer-Verlag Inc.

125HELLENISTIC MATHEMATICS: THE QUESTIONS
4
HELLENISTIC
MATHEMATICS:
THE QUESTIONS
The previous chapter depicts mathematics as a collective enterprise:
Ptolemaic officers were told to check the accounts in groups; the surveyors
at Heraclea measured the land together; the central element of a catapultwas identified thanks to accumulated experience. The prefaces to several
mathematical texts recount of networks of people, who sometimes
communicated by letter, sometimes physically met and talked to each other,maybe pored over a diagram together. Some other times, they communed
across the boundaries of time, in a dialogue with past works – take
Archimedes and Eudoxus, or Apollonius and Euclid. One of the twoquestions raised in this chapter will be about communities of
mathematicians. I will try better to describe them, and to relate them to the
wider context of Hellenistic culture. But before I do that, I will tackle anotherquestion.
If from the group picture sketched above we change focus to a close-up
of one individual – Euclid – we get a very blurry image. As we have said, weknow nothing about his life – he is, simply, the author of the Elements . Or
is he? Archimedes’ testimony on Eudoxus implies that substantial parts of
book 10 and possibly of book 12 may not be by Euclid. Still, if the Elements
contain pre-Euclidean material, at least we can source it for precious
information about early Greek mathematics. Or can we? The following
three lines by the fourth-century
AD mathematician Theon of Alexandria
have, in Maurice Caveing’s words, dominated the history of the text of the
Elements :
But that within equal circles sectors are to each other as the angles
on which they insist, we have proved in our edition of the elements
at the end of book six.1
Now, the great majority of manuscripts on which we depend for our
text of Euclid unselfconsciously contain the proposition described above,

126HELLENISTIC MATHEMATICS: THE QUESTIONS
as if it just belonged to the text. Had Theon not mentioned it, we may
never have known that the corollary of Elements 6.33 is not Euclid’s at all.
How do we know that the same is not true for other parts of the text? Howdo we go about searching for the real Euclid? This is what we shall try to
find out next.
The problem of the real Euclid
To get an idea of the story behind our text of the Elements , take a look at
Figure 4.1. And if that looks complicated, you can imagine a similar, if
simpler one, attached at the top of the archetypal G, to signify the pre-
Euclidean sources incorporated into the Elements . Given the rather desolate
scenario of the evidence about early Greek mathematics, it comes as no
surprise that historians have clung to Euclid with near-obsessive interest.
The quest for the real Euclid has thus acquired a double purpose: to accessthe original work for its own sake, and because an ‘unadulterated’ version
would show more transparently traces of the past.
Because the problem is one of transmission of texts, it is useful to turn
to the actual manuscripts through which a work is known to us. Many of
them are arranged thus:
The notes made at the margins of the main text in order to explain or
complement it, are called scholia. They became widespread especially from
the third century
AD onwards, as papyrus scrolls were gradually replaced by
parchment or vellum codexes as the main writing medium. A codex hadlarge stackable pages and adding writing to an already extant body of text
was made easier by its format (you did not have to unroll it and it stayed
open by itself). It is thought that sometimes scholia got incorporated into the
main text due to copying accidents – many interpolations (passages contained
in a work which are not original to it) perhaps started their lives that way.
In parallel with the huge number of Euclidean manuscripts, an enormous
quantity of scholia to the Elements have come down to us, and they are only
partially edited. Insofar as it has been possible to date them, mostly onpaleographical grounds, the earliest group may have been produced around
the sixth century
AD, and it draws heavily on Proclus’ and almost certainly
Pappus’ commentaries to the Elements . Scholia often offer clues about the
original authorship of one piece or another of Euclid’s text. For instance,with the main body of writing in the
middle and smaller pieces of writing on the
side, sometimes even on both sidessuch as
this, which could go
on for an entire
paragraph or
this here

127HELLENISTIC MATHEMATICS: THE QUESTIONS
they attribute proposition 1.26 (on the conditions of equality for triangles)
to Thales, on the authority of Eudemus; 1.47 (the so-called theorem of
Pythagoras) to (unsurprisingly) Pythagoras; 10.9 (on incommensurables insquare and length) to Theaetetus. Their reliability as sources, however, should
be assessed in the same way as that of any other source: they are late, and,
like modern historians, they may be unduly stretching the vague informationfound in Plato or other earlier authors.
The identification of pre-Euclidean material more often relies on
conceptions of logical coherence or of how mathematics proceeds. On thisrather risky basis, it has been shown that parts of the Elements do not fit
with the rest, and consequently must be pieces from other, presumably
previous, works. One of the most famous examples of such reconstructions
is the last part of book 9, which was by Oskar Becker ascribed to the
Pythagoreans. The subject of the book is arithmetic, in particular primenumbers and divisibility into factors, but after theorem 9.20 the text switches
to a different topic, and the next fourteen items (9.21 to 9.34) deal with
odd and even numbers (e.g. 9.26: if from an odd number an odd numberbe subtracted, the remainder will be even). Only the two final propositions
seem to return to the main path, both in content and procedure. Now, odd
and even are topics that we know to have been studied by the Pythagoreans,and Becker was able to recast the simple demonstrations of propositionsFigure 4.1 The medieval and Renaissance tradition of Euclid’s Elements
(adapted from Murdoch (1971))

128HELLENISTIC MATHEMATICS: THE QUESTIONS
9.21 to 9.34 in terms of so-called pebble arithmetic, a technique which had
also been linked to the Pythagoreans and which consists in arranging pebbles
in strings, squares or gnomons. Thus, an even number is represented by adouble string where all the pebbles are in pairs. Prop. 9.21, that the sum of
even numbers is even, can for instance be shown to be correct via this method,
because any string of even numbers/pebbles remains double when any evennumber of pebbles is added to it.
2 Becker concluded that this part of the
Elements was a relic from a former mathematical age, framed uneasily within
the greater sophistication of later arithmetical theory.
Another interesting case of ‘missing link’ is the so-called proposition
10.117, as follows (see Diagram 4.1):
Let it be proposed by us to prove that the diagonal of square figures
is incommensurable in length with the side. Let ABCD be a square,
its diagonal AC; I say that CA is incommensurable with the length
AB. If in fact possible, let it be commensurable; I say that it will
happen that the same number is even and also odd. Since it is
evident that the <square> on AC is double that on AB. And since
CA is commensurable with AB, therefore CA has a ratio to AB as
of a number to a number. Let it have the ratio of EZ to H, and let
EZ, H be the least of those having the same ratio as them; therefore
EZ is not a unit. If in fact EZ was a unit, it would have the <same>
ratio to H as AC has to AB, and AC is greater than AB, therefore
EZ would also be greater than the number H. Which is absurd.
Therefore EZ is not a unit; therefore it is a number. And since it is
as CA to AB, so EZ to H, and therefore as the <square> on CA to
that on AB, so that on EZ to that on H. The <square> on CA is
double that on AB; therefore the <square> on EZ is also double
that on H; therefore the <square> on EZ is even; so that EZ is also
even. If in fact it was odd, the square on it would also be oddbecause if a certain number of odd numbers are added to eachDiagram 4.1

129HELLENISTIC MATHEMATICS: THE QUESTIONS
other and their quantity is odd, the whole is odd; therefore EZ is
even. Let it be divided in half along H. And since EZ, H are the
least of those having the same ratio, they are prime with respect toeach other. And EZ is even; therefore H is odd. If in fact it was
even, two would measure the EZ, H; for a whole even has a half
part; <and they> are prime with respect to each other; which isimpossible. Therefore H is not even; therefore it is odd. And since
EZ is double EH, therefore the <square> on EZ is four times that
on EH. The <square> on EZ is double that on H; therefore the
<square> on H is double that on EH; therefore the <square> on H
is even. Therefore H, through what has been said, is even; but it is
also odd; which is impossible. Therefore CA is not commensurable
with AB in length; which is what it was necessary to prove.
3
This is a proof of the incommensurability of the side and diagonal of the
square, which uses reductio ad absurdum to show that, if diagonal and side
were commensurable, the same number would be both odd and even. But
Elements book 10 contains another, more general, proof to the same effect,
that ‘squares which do not have to one another the ratio of a square number
to a square number have their sides incommensurable in length’ (10.9,
attributed by a scholion to Theaetetus). Proposition 10.117, on the otherhand, matches the description of a proof mentioned by Aristotle:
In the latter [ sc. arguments which are brought to a conclusion per
impossibile ], even if no preliminary agreement has been made, men
still accept the reasoning, because the falsity is patent, e.g. the
falsity of what follows from the assumption that the diagonal iscommensurate, viz. that then odd numbers are equal to evens.
4
According to one line of interpretation, if 10.9 was discovered by
Theaetetus, then the proof by Aristotle, which is less general, must have
been discovered earlier, but by whom? The best candidates are the
Pythagoreans, who seem to have been interested both in incommensurability
and in odd and even. On this reading, 10.117 would be another relic from
a former mathematical era, tacked on more sophisticated material. Adifferent, and now widely accepted, interpretation was given by Wilbur
Knorr, who observed that the third-century
AD author Alexander of
Aphrodisias, commenting on that very same passage from the Prior Analytics ,
quoted material from the Elements but did not appear to know 10.117.
Knorr argued that 10.117 was produced at some point between Alexander
of Aphrodisias and Theon, whose version of the Elements already contained
it. The ‘pre-Euclidean’ proof was ‘a result of the continuing activity of later
Aristotelian commentators’.

130HELLENISTIC MATHEMATICS: THE QUESTIONS
These two cases illustrate the difficulties facing the historian who wants to
use the Elements as a source for earlier material. The crucial question is Euclid’s
attitude to past mathematics: did he incorporate entire treatises into his textwith the seams still showing, or did he drastically intervene, changing them
beyond recognition? And how can we really tell what kind of editor Euclid
was, if first, we do not know his sources – they are all lost to us, and, second,even the Euclid we do have is not the real Euclid anyway? I retreat to a safe
and cosy agnostic position with respect to the retrievability of pre-Euclidean
mathematics from the body of the Elements . When Plato, Aristotle or Archi-
medes testify that a result was already known, or the general features of a
procedure already established, then we have some ground, but hardly a lot of
detailed ground, to run on. Suspending judgement with respect to pre-Euclid-ean material, however, still leaves the other horn of the problem unattacked.
Leaving aside the (virtual) tree at the top of G in Figure 4.1, we still have the
many ramifications underneath it, and first of all the Theonine question.
In 1808 the French scho lar François Peyrard observed that an Euclidean
manuscript in the Vatican library in Rome (known as Vat. gr . 190, or P ,
tenth century) showed no trace of the infamous corollary to proposition6.33 and was in other respects different from the majority of other
manuscripts of the Elements . He proceeded to conjecture that P was pre-
Theonine, unscathed by Theon’s intervention, and therefore closer to Euclid’soriginal. J.L. Heiberg, editor of what is still widely considered the standard
edition of the Greek text of the Elements , accepted Peyrard’s conclusions.
While he used manuscripts from the so-called Theonine family, he held P‘to represent the authentic Euclidean text’.
5
But not even P could be said to contain the real Euclid. In an article
entitled ‘The Wrong Text of Euclid’, Knorr questioned the primacy of P .He revived a previous hypothesis by M. Klamroth, a contemporary of
Heiberg, to the effect that the Arabic tradition of the Elements , which also
lacked the corollary to 6.33, may have derived from a better text, or texts,
of Euclid than P . Indeed, the Arabic Elements differ from Heiberg’s on several
counts, such as shorter proofs or a number of missing corollaries. Heiberg
had entertained but rejected, on the basis of several factors, including incom-
pleteness, the idea that the Arabic tradition may be truer to the real Euclid.
Yet, as Knorr pointed out, ‘incompleteness’ is a subjective factor. But thereis more. Already Heiberg had occasionally chosen the lesson of Theonine
manuscripts over that of P wherever he suspected scribal error. One
manuscript from the Theonine family, called b (from Bologna, eleventhcentury) is unique in that it agrees with the rest of the group except for the
last part of book 11 and the whole of book 12. Knorr showed that the
divergent parts of b aligned with the Arabic tradition, and could then beused further to support his thesis. He concluded his article thus:

131HELLENISTIC MATHEMATICS: THE QUESTIONS
We have never had a ‘genuine’ text of Euclid, and we never will have
one […] But we can do much better than has been done so far.6
One of the ways to go would be to get a clear picture of the extent of
Theon’s intervention on the text. Comparison with what he did in his com-
mentary on Ptolemy could be useful to identify operations such as correctionor improvement where he believed Euclid was mistaken or confused, or
additions where he thought that the text was difficult to understand. We
have to keep in mind that the Elements were seen first and foremost as a
reservoir of results, not necessarily as a work that should be preserved in its
linguistic or methodological integrity. Although the notion of preserving
ancient texts did exist – Hellenistic editions of Homer, say, were producedwhich respected archaic language – it seems to have been applied only to
literary works. No philological concern seems to have prevented later mathe-
maticians
7 from correcting, simplifying, updating previous material. In fact,
Theon was not the first. In the centuries between the early third BC and late
fourth AD, other people like him had read, used and reworked the Elements .
We know, to begin with, of at least two commentaries on Euclid prior toTheon: Hero’s and Pappus’, and Proclus’ commentary mentions many unspe-
cified others. It is possible that these commentators edited parts of the
Elements , or that they made copies, perhaps for personal use, with additions
or modifications which later came to be incorporated into the text.
Unique access to pre-Theonine Euclidean material is given by three
papyri: one, Hellenistic, mentioned in the section on material evidence inchapter 3, contains the definition of a circle (1. definition 15) and proposi-
tions 1.9 and 1.10; a second- or third-century
AD papyrus from the Fayum
has the equivalent of 1.39 and 1.41 and a third- or fourth-century AD papyrus
fragment from Oxyrhynchus has the equivalent of 2.5. The following
example will allow the reader to compare the versions in the papyri with
those in P .
[P. H e r c . 1061 – 1. definition 15 (see Diagram 4.2)] … circle is a
plane figure surrounded by one line, all the lines falling on it fromone point of those situated within the circle are equal; for in this
way the geometers define it […] [1.9] let, they say, A be the given
angle surrounded by the side AB and by the <side> AC and a point
of the side AB <is> D, it is necessary to subtract a length AD
sufficient … since in fact AD is equal to AE and AZ is common,
therefore the two AD and AZ are equal to the two AZ and AE and
the base DZ to EZ; for the triangle has been put together equilateral

132HELLENISTIC MATHEMATICS: THE QUESTIONS
… [1.10] on the given line MN, they say, let it be put together the
equilateral triangle … divide the given line into two … for if it is
<possible?> to divide into two the given line … if MQ and QN are
equal and the given line MN is divided into two, the same will be
also … NM is equal to NN the angle O to P and MN will be
divided into two, so that …
[Vat. gr . 190 – 1. definition 15 (see Diagram 4.2)] circle is a plane
figure surrounded by one line, which is called circumference, all
the lines falling on it, on the circumference of the circle, from onepoint of those situated within the figure are equal to each other.
[1.9] […] for let a point whatever D be taken on AB, and let AE
equal to AD be subtracted from AC […] since in fact AD is equal
to AE and AZ is common, the two DA, AZ are equal to the two
EA, AZ respectively; and the base DZ is equal to EZ; therefore the
angle DAZ is equal to EAZ […] [1.10] […] Let an equilateral triangle
ABC be put together on it […] I say that the line AB will be divided
into two along the point D. Since in fact AC is equal to CB, CD is
common, the two AC, CD are equal to the two BC, CD respectively;
and the angle ACD is equal to BCD […] therefore the given
delimited line AB will be divided into two along D […]Diagram 4.2

133HELLENISTIC MATHEMATICS: THE QUESTIONS
[P . Fayum 9 – 1.39 (see Diagram 4.3)] are … on the same side …
parallels … on the same base … let AD be conjoined … is to BC …
parallel to BC … ABC … on the same base … parallels … is equal to
BDC … the greater to the lesser … AE is parallel to BC … we have
proved that it is not other … therefore AD is parallel to BC … [1.41]
if a parallelogram as a triangle … the same and in the same … the
parallelogram will be … parallelogram … the base … double … let
it be conjoined … to the triangle EBC … of BC and … to BC AE
but … parallelogram … and of EBC […]
[Vat. gr. 190 – 1.39] Equal triangles which are on the same base
and on the same side are also in the same parallels. […] For let AD
be conjoined; I say that AD is parallel to BC. […] and the triangle
ABC is equal to the triangle EBC; for it is on the same base […]
and in the same parallels. But ABC is equal to BDC; and DBC is
equal to EBC the greater to the lesser […] neither therefore is AE
parallel to BC […] [1.41] If a parallelogram has the same base as a
triangle [etc.]Diagram 4.3

134HELLENISTIC MATHEMATICS: THE QUESTIONS
[P. O x y. 29 – 2.4 (see Diagram 4.4)] to the surrounded rectangle
… [2.5] If a straight line is divided into equal and unequal
<segments>, the whole rectangle formed by the unequal segmentsplus the square on the sections in between is equal to the square
on the half.
[Vat. gr. 190 – 2.4] to the rectangle; which it was necessary to
prove. Corollary. From this it is evident that in square areas theparallelograms formed around the diagonal are squares. If a straight
line is divided into equal and unequal <segments>, the whole
rectangle formed by the unequal segments plus the square on thesections in between is equal to the square on the half.
The Euclidean papyri offer a good sample of ways in which a mathe-
matical text could undergo modifications. The first account was probablypolemical, and distanced the author from the ‘geometers’ through use of
‘they say’ to mark the two propositions, which in their turn differ quite
significantly from P . Yet, the correspondence as far as the definition of circlegoes is remarkable. The Fayum papyrus has preserved only the inner core of
columns of text, and P seems to fill it in rather nicely – but the papyrus goes
from what we now call 1.39 straight to 1.41. That means that the Elements
known to the person who wrote the Fayum papyrus did not contain our
present 1.40. As for the third papyrus, along with good parallelism in theDiagram 4.4

135HELLENISTIC MATHEMATICS: THE QUESTIONS
enunciation of 2.5 and the end of 2.4, there is a major element of divergence
in that the papyrus does not have the corollary (which Heiberg himself
considered an interpolation).
In sum, the quest for the real Euclid seems fraught with serious, if not
completely insurmountable, difficulties. There is in it a lesson for the
historian: collecting, reworking, explaining, systematizing previous textsbecame one of the main features of ancient mathematical practices at least
since Euclid himself, thus basically since our very first ‘big’ work. At least in
part, this is the result of the growing presence of the written medium, soimpressively represented by the hundreds and hundreds of scrolls which are
said to have been stacked on the shelves of the library in Alexandria. We
need better to understand those operations of collecting and reworking, ifwe are to understand ancient mathematics itself. Moreover, the relation
between the Elements and the other surviving works by Euclid, whose
editorial history is marginally less complicated, could fruitfully be explored.Finally, it bears reflection that the people who wrote the Fayum and
Oxyrhynchus papyri are representative of a large, unnamed public for mathe-
matics, who may have contributed no new result to it, but may have nonethe-less shaped its practice by the mere act of reproducing and transmitting
texts.
The problem of the birth of a mathematical community
A rather large, if not entirely unnamed, public for mathematics emerges also
from the works of Archimedes and Apollonius, but also of lesser figures like
Philo or Diocles. At the very beginning of his account on burning mirrors,
the latter mentions a number of other people who were interested in thesame topic as him: Zenodorus, Pythion from Thasus who suggested a certain
problem to Conon, and Dositheus, who gave a practical solution to it.
8 A
similar scenario is evoked by Hypsicles (mid-second century BC), who
addressed the so-called book 14 of the Elements to a Protarchus and
mentioned a Basilides of Tyre. Basilides shared a passion for mathematics
with Hypsicles’ own father and, together with him, had written corrections
to one of Apollonius’ texts. Hypsicles in his turn had been so intrigued by a
proof of Apollonius’ that he decided further to examine the issue, and tosend Protarchus the results, so that they could be assessed by those who ‘are
proficient, thanks to their experience, in all studies and in particular in
geometry’.
9
In a recent study, Reviel Netz has examined the language and argumenta-
tive structure of a cross-section of mathematical texts, primarily Euclid,
Archimedes and Apollonius. He has established that they shared a methodof proof based on deduction and on the use of the lettered diagram. He has

136HELLENISTIC MATHEMATICS: THE QUESTIONS
also convincingly demonstrated that the lexicon used by these texts was
small, often organized formulaically, and that they relied on a small system
of background knowledge, a set of mathematical results which were takenfor granted. Acquaintance with this toolbox on the part of the reader was
also taken for granted, and amounted to the expertise required for a full
comprehension of the text.
10 The practices described by Netz (agreement
on procedures, compartmentalized language, notion of expertise) pinpoint
a group, among whose members we can count the usual suspects – Euclid,
Archimedes and Apollonius – and their immediate audience, i.e. the peoplementioned in their works. So, the internal mathematical structure itself of
at least a selection of our texts points to the existence of a community of
mathematicians who, even across time, shared a specialized language, discur-sive conventions, criteria of validity and rigour.
How did this community originate? A partial answer can be found in
the cultural policies of the Hellenistic kings; moreover, we could talk aboutmultiple births, of mathematical communities and of similar groups in other
fields. The most famous cultural space of the Hellenistic period was the
Museum of Alexandria, founded and financed by Ptolemy I. Although manydetails of the functioning of this institution remain obscure, we know that
it was home to a diverse group of scholars, mostly Greek-speaking but not
limited to people of Greek descent; that there was a massive library; andthat there was some degree of communal life, perhaps with shared meals
and in-house accommodation. People associated with the Museum included
poets, grammarians, historians, philosophers, doctors and, crucially for ushere, natural philosophers, geographers, machine-builders, astronomers,
geometers. Other cities such as Pergamum and Rhodes were also active
cultural centres, complete with libraries and sundry scholars.
Given the obvious expense entailed by patronage on such a large scale, a
natural question could be, what was in it for the Ptolemies (and the Attalids,
and the Seleucids)? In some cases, promoting science had immediate practicalbenefits. It is rather obvious why Hellenistic monarchs would want to support
military engineers like Biton or Philo. Geography and astronomy could be
used for map-making and time-keeping. But there was more than simply
utility in sight for the royal patrons of the arts and sciences: rulers could
now present and represent themselves as well-educated in the Greek way,and could flaunt knowledge as yet another jewel in their crown. In this
sense, the scientists and scholars of the Museum constituted yet another
collection, along with that of books and of rare and exotic animals. Indeed,they were described accordingly by a contemporary: ‘many are kept to graze
in populous Egypt, well-fed bookworms, who quarrel without end in the
Muses’ bird-cage’.
11

137HELLENISTIC MATHEMATICS: THE QUESTIONS
Another second-hand testimony also conveys significant aspects of the
Ptolemaic rule:
[…] the Grand Procession […] was led through the city stadium.
First of all marched the sectional procession of the Morning Star,
because the Grand Procession began at the time when thataforementioned star appeared. […] After them marched the poet
Philikos, who was the priest of Dionysius, and all the guild of the
artists of Dionysius […] After them a four-wheeled cart was ledalong by sixty men … twelve feet wide, on which there was a seated
statue of Nysa twelve feet tall, wearing a yellow chiton woven with
golden thread […] This statue stood up mechanically withoutanyone laying a hand on it, and it sat back down again after pouring
a libation of milk from a gold phiale. […] The figure was crowned
with golden ivy leaves and with grapes made of very precious jewels.[…] Next, another four-wheeled cart, thirty feet long by twenty-
four feet wide, was pulled by three hundred men, on which there
was set up a wine-press thirty-six feet long by twenty-two and ahalf feet wide, full of ripe grapes. […] Next there came a four-
wheeled cart, thirty-seven and a half feet long by twenty-one feet
wide, which was pulled by six hundred men. On it was an askosmade of leopard skins which held three thousand measures. As
the wine was released little by little, it also flowed over the whole
street. […] At the very end, the infantry and cavalry forces marchedin procession, all of them fully armed in a marvellous fashion. […]
Besides the armour worn by all these troops, there were also many
other panoplies kept in reserve, whose number is not easy to record,but Kallixeinos gave the full count.
12
Kallixeinos’ detailed recounting of this public celebration provides us witha full picture of power itself: the gold, the abundance of food and drink, the
connection with the gods, the more explicit embodiment of might marching
at the end, and, crucially for us, the display of knowledge. The timing itself
of the procession (which included both morning and evening star floats)
seems to have been based on astronomical calculations. A poet walked along.The statue of Nysa (the mythical wetnurse of Dionysius), gleaming with
gold, was an automaton , a self-moving object, whose functioning was to be
later explained, mathematically, by Hero. The automaton represented an
important element of the self-image the Ptolemies wished to convey to the
public (we know that delegates from many other states had been invited to
the festivities), namely the capacity to create wonder and amazement. The

138HELLENISTIC MATHEMATICS: THE QUESTIONS
fact that common observers were not able to explain the apparently impos-
sible things they were experiencing – an enormous statue moving ‘without
anyone laying a hand on it’ – reflected back positively on those who knewwhat was going on, or indeed could produce such phenomena: the machine-
maker, but above all the patron. Even straightforwardly useful research such
as geography compounded effectiveness with wondrousness; Philo was keenthat his catapults should look as terrifying as they would prove when actually
used; Diocles was very well aware of the potentialities of burning mirrors as
spectacle.
13
Having to operate within such an environment shaped cultural practices
in several ways. First, thrown together by the service of a common patron,
people were exposed to other forms of knowledge, and cross-pollinationoccurred frequently. According to recent reconstructions, for instance,
Hellenistic medical treatises used mechanical terms that we otherwise only
find in Philo of Byzantium. The heroes of Apollonius of Rhodes’ Argonautica
travel through a mythical world whose features seem to have been updated
following the latest geographical treatises.
14 Euclid wrote on optics,
astronomy and harmonics as well as mathematics. Eratosthenes wasinterested both in philosophy and in mathematical novelties; he was chief
librarian at Alexandria as well as measuring the circumference of the Earth.
Archimedes was a machine-maker, and Livy reports that he was well knownfor his observations of the skies.
15 Philo and Diocles offered solutions to the
problem of the two mean proportionals within the context of machine-
building, but Archimedes took the same problem for granted (it evidentlywas part of the geometers’ toolbox) in his Sphere and Cylinder .
16 Second,
reference to the past and collection and accumulation of cultural tokens,
especially books, were crucial factors in the construction of the self-imageof Hellenistic kings, and they were, or they became, crucial factors in the
construction of the self-image of groups of intellectuals as well. The Hellenis-
tic period, to an extent not seen before, saw the emerging of philosophical,medical and philological or grammatical traditions or schools.
This has been amply documented, so I need not rehearse well-known
evidence here. Let me just point out a few aspects shared by some of these
schools or traditions: above all, the ‘canonizations’ of some texts or authors.
Already Lycurgus, public treasurer at Athens between 337 and 325/4
BC,
had elevated the triad of Aeschylus, Sophocles and Euripides to the status
of official classics, promoting the establishment of an authentic text of their
works. Indeed, the Hellenistic age was a great period for fakes, imitationsand works written in the style of a chosen authorial model: just think of the
mass of Pythagorean texts we mentioned in the section on philosophers in
chapter 3, or of faux-Platonic or Aristotelian works like the Epinomis and
On Indivisible Lines . This not only indicates that there was a market for

139HELLENISTIC MATHEMATICS: THE QUESTIONS
books, but also that some names had become prestigious enough to inspire
imitation or downright forgery. Thus, more manuscripts of the same work
were collected, in an attempt to establish a canonical or ‘best’ version, whichinvolved selecting between different manuscripts of the same work, or
different readings of the same phrase, or different explanations of the same
passage. The choice had to do with claiming to oneself the authority todecide what was the real Sophocles, or what Plato really meant, or what
Euclid missed out. In the case of Homer, Alexandrine scholars preserved
versions that they did not consider authentic, but even so they were stilladvocating to themselves the expertise to discriminate between genuine
Homer and later imitations.
17
It was in connection with this process of canonization that grammar was
born. Codification of texts implied attention to the language employed and
formulation of criteria to distinguish between authentic and not authentic.
Now that Greek was being learnt by more and more non-native Greekspeakers, some criterion had to be articulated systematically to teach correct
from incorrect. Grammar thus was a codification of the language, whereby
some speakers, taking into account common usage on the one hand and theusage of the canonical texts on the other hand, decided what was good
Greek and what was not quite the right Greek. Grammarians defined their
pursuit as a techne , in the same league as medicine and mathematics. Parallels
can indeed be run: both the grammar and the mathematics of this period,
as exemplified chiefly by Euclid’s Elements , operate on already existing
material and order it according to criteria of validity. Although people mayspeak and write correctly even if they ignore grammar, grammar is an
explicitation of concepts already in use that by its very act establishes those
concepts on firmer foundations. Thus, notions such as point, line, triangle,had been unproblematically used long before Euclid posited them as
rigorously defined starting points. The creation of grammar also created
the grammatical expert, a person who can claim a more rigorous knowledgethan the person in the street, through the firm grasp he has acquired of the
canonical texts, combined with observation and participation in common
usage and practices. This description fits with what Netz has identified as
the features of the mathematical expert implied by the works of Euclid,
Archimedes and Apollonius.
Unfortunately, not much survives of the grammatical work done in
Alexandria between the third and the first century
BC: the earliest treatise
we have is the Grammatical art by Dionysius of Thracia (late second century
BC). Its authorship has sometimes been questioned, but enough scholars
deem it authentic for me to include it here. The work is organized systematic-
ally: it opens with a definition of grammar as methodic knowledge basedon the poets and on prevalent usage, and then proceeds to define, define,

140HELLENISTIC MATHEMATICS: THE QUESTIONS
define each and every element of speech, from vowels to consonants, to
modes to tenses to conjunctions, usually specifying how many parts are
subsumed under each element. If we take these definitions as first principlesor starting points, the foundation work here can again be compared to that
in the first book of Euclid’s Elements . Everybody in a sense knows what a
line or a circle are, and everybody in a sense knows what a vowel or consonantare – Euclid’s and Dionysius’ action in defining them are parallel.
On the other hand, the accumulation of cultural resources, both human
and papyraceous, seems to have gone hand in hand with a desire todistinguish and define oneself and one’s specific pursuit. There was a need
for self-identification and distinction, which might have derived from an
environment which was both multicultural and slanted towards oneparticular culture, or could be related to competition for patronage. The
hostility, as well as the exchanges, between different schools or sects within
the philosophical and medical fields are very well documented. Also,remember the philosophical attacks on mathematics, or the dismissive
remarks about mathematicians reported by Aristoxenus and Diocles, or, on
the other hand, Hipparchus’ not-so-high opinion of Aratus. Or, and this isone of the richest texts available to us for an exploration of the complexities
of Hellenistic mathematical practice, let us consider Archimedes’ Sand-
Reckoner . In it, arithmetical and astronomical interests are combined, and
Archimedes both put forth a new system of notation for very large numbers
and launched into astronomical enquiries. His claim is that, even if one
filled the whole universe with sand, it would still be possible to express thatmultitude of grains of sand with a number. Astronomy comes in when he
has to give an estimate for the size of the universe, in order to assess precisely
how many grains of sand it could contain. The final number would nothave been expressible in ordinary Greek (Milesian) notation, but is ‘a
thousand myriads of eighth numbers’ according to Archimedes’ new system
– 10
24 in modern figures.
The text is addressed to a king, probably a provider of patronage;18 it
presents the author both in relation to his colleagues and to previous authors,
astronomers and mathematicians past and present; it plays up the wondrous-
ness factor. The anecdotal evidence abounds in tall stories about Archimedes’
machines and their marvellous effects, but here we have the man himself, inhis own words. Imagine, he says to the king, being able to count not just
the sand on the local beach of Syracuse, nor even the sand of the whole of
Sicily, but the sand of the entire universe, if the universe were filled withsand. Archimedes reassures Gelon that the tenor of the demonstrations is
not too difficult, and that he will be able to follow them, and, after navigating
an assured course between astronomical observations and arithmeticalcalculations, concludes the treatise thus:

141HELLENISTIC MATHEMATICS: THE QUESTIONS
I understand, king Gelon, that these things will seem not believable
to the many who do not share in the studies, but to those who
instead have taken part [in them] and have reflected on the distancesand the sizes of the earth and the sun and the moon and the whole
universe, they will be persuasive because of the proof; which is
why I thought that it was not inappropriate for you to considerthese things.
19
King Gelon need not know, or notice, that the Sand-Reckoner is not struc-
tured like Archimedes’ other treatises, presumably reserved for his mathema-
tical peers: it has a narrative mode of argumentation, but no starting points
and no deductive chains. The king, on the other hand, may appreciate beingmade a honorary member of the restricted community of people who know,
as opposed to ‘the many who do not share in the studies’.
As we have already noticed, astronomy was a popular subject, especially
with patrons, as testified by Aratus and by poems such as Callimachus’
celebration of the lock of Berenice, a constellation named after the hair of a
Ptolemaic queen:
Having examined all the charted sky, and where the stars move …
Conon saw me also in the air, the lock of Berenice, which shededicated to all the gods.
20
Conon (the same as in Archimedes’ and Diocles’ works?) must have cleverlycombined science and ego-boosting by inscribing the Ptolemies in the
heavens; but it is interesting that Callimachus made the episode his own by
celebrating it in style. Aratus’ poetic appropriation of Eudoxus, as describedby a none-too-pleased Hipparchus, may have been along similar lines. Think
also of Eratosthenes’ little poem, as reported by Eutocius (and if it is
authentic), with which he celebrates his solution to the problem of the twomean proportionals, praises Ptolemy and criticizes a couple of mathematical
predecessors. More than direct competition, one could describe this as a
dialectic relationship between mathematicians/astronomers and poets. Was
Archimedes’ Sand-Reckoner directly answering Theocritus’ evocation of a
countless universe? Was he proclaiming the centrality of mathematics tothe understanding and managing of an enlarged world? And was he, the
best representative of a small and extremely sophisticated mathematical
community, also recognizing the necessity of patronage, and the inescap-ability of the real world?

142HELLENISTIC MATHEMATICS: THE QUESTIONS
Notes
1 Theon, Commentary on Ptolemy’s Syntaxis 1.10 492.7-8. Theon’s edition is mentioned in
the scholia to the Elements , 1.2; 4.4; references in Mansfeld (1998), 25. See Caveing
(1990), 45 ff., on which I rely extensively.
2 Becker (1933).
3 Euclid, Elements 10. appendix 27 (contained in most manuscripts, including P).
4 Aristotle, Prior Analytics 50a, quoted from Fowler (1999), 292, and see also Knorr (1975),
228 ff.
5 Which is not to say that P is the oldest extant manuscript: B, in Oxford, can be dated
with accuracy to AD 888. Cf. Knorr (1996), 212.
6 Knorr (1996), 261.
7 Or, as G.E.R. Lloyd points out to me, later medicine writers.
8 Diocles, On Burning Mirrors 4, 3 and 6, respectively.
9 Hypsicles, Book 14 of the Elements preface.
10 Netz (1999a).
11 The words are attributed to Timon of Phleisus (third century BC) by Athenaeus,
Deipnosophistae 22d, Loeb translation with modifications.
12 Kallixeinos of Rhodes ap. Athenaeus, ibid. 197c–203a, translation in Rice (1983). The
procession described is thought to have taken place ca. 280–75 BC.
13 Philo, Mechanica IV (Construction of Catapults) 61.29–62 ff.; 66.17 ff.
14 Cf. von Staden (1998), Hurst (1998), respectively.
15 Livy, From the Foundation of the City 24.34.2.
16 Archimedes, SC 2.1.
17 See Irigoin (1998) and, on grammar, Montanari (1993).
18 Archimedes is defined in a source sungenes , ‘related’, to the king of Syracuse – the term
has often been taken literally, but in fact could be used as an honorific title to denote a
royal familiar, cf. LS s.v.
19 Archimedes, AR (Mugler 156–7).
20 Callimachus, Aetia fragment 110.

143GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
5
GRAECO-ROMAN
MATHEMATICS: THE
EVIDENCE
That certain persons have studied, and have dared
to publish, the dimensions [of the universe] is mere madness[…] as if indeed the measure of anything could be taken by him that knows not the measure of himself.
1
By the end of the first century BC, Rome had turned from a republic into an
empire, and, in the course of the first two centuries AD, it secured control
over most of Europe, North Africa, Egypt and the Near East. The empire is
a constant presence in our evidence from this period, and it enters mathe-
matical discourse in several ways. Managing an army, collecting taxes, keepinga census on such a vast scale implied centralized administrative practices
(accounts, tax rolls, land surveys). Mathematics was also used to articulate
views about politics, society and morals. It would be impossible to describeour period in a few words: let us just say that the world had become even
larger than after Alexander’s expedition, exchanges of all types increased;
and the textual past kept accumulating in the form of books and libraries.
T urning to the evidence, apart from the usual survey of material sources,
there are individual sections on Vitruvius and Hero. For the rest, authors havebeen assigned to the two sections ‘Other Greeks’ and ‘Other Romans’ on
the basis of the language they worked in – geographically, they come from
all over the place and they all belonged to the same Empire. A more earnestexploration of the Greek/Roman divide will be taken up in chapter 6.
Material evidence
The great majority of papyri from this period comes, as usual, from Egypt.They include the earliest extant philosophical commentary, on Plato’sTheaetetus . The author, who mentions other commentaries, including one
to the Timaeus , has not been identified. Given the contents of the original
dialogue, this papyrus is quite rich in mathematical passages, explaining for

144GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
instance the construction of a square on a given line (reference is made to
Plato’s Meno ), the difference between square and rectangular numbers,
between incommensurability in square and in length, and between ‘wedge’,‘brick’ and ‘beam’ numbers. I append a sample in the following section
(and see Diagram 5.1).
Let ABC be a square with side of one foot, AB. It is clear that the
<square> on this side will be one square foot; for one by one is one.And let a line be prolonged from the line AB and let BD equal to
AB be cut on it, and let the square BCDE be drawn on BD; the
<square> on BD will be equal to that on AB; the whole AE is not
a square but a parallelogram. Again, let a line be prolonged fromthe line AD and let DZ equal to BD be cut on it, and let the square
DZEH be drawn on it. The square DZEH is equal to either of the
squares set out before, and the whole area AH is a parallelogram.
Again let a line be prolonged from the line AZ, and let ZH equal
to DZ be cut, and let AH be divided into half by the point D. And
with centre D and radius DA, let a semicircle AKH be drawn, and
let ZK be drawn perpendicular to HE, and let KD be joined. Since
the line AH is divided into equal parts by the point D and in
unequal parts by Z, the <rectangle> formed by AZ, ZH plus the
<square> on the <line> between the sections, DZ, is equal to the
<square> on
DH. But DK is equal to DH. Therefore the <square>
on DK is equal to the <rectangle> formed by AZ, ZH and theDiagram 5.1

145GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
<square> on DZ. But the <squares> on DZ, ZK are equal to that
on DK. Therefore the <squares> on DZ, ZK are equal to the
<rectangle> formed by AZ, ZH and the <square> on DZ. Let the
<square> on DZ which is common be subtracted; therefore the
<square> on ZK which is left is equal to the <rectangle> formed
by AZ, HZ which is left. The <rectangle> formed by AZ, ZH is the
<rectangle> formed by AZ, ZH. For ZH is equal to ZH; therefore
the <square> on ZK is equal to the parallelogram AH. The <square>
on ZK is then incommensurable to the parallelogram AH, which
surrounds inside itself three squares of one foot equal to each other.2
Whereas the proposition above follows very closely Euclid’s Elements
2.14, some other parts of the text (e.g. the ‘wedge’ and ‘brick’ numbers) do
not. If the commentary was meant for the general public (its tone is quitesimple, and interest in Plato was quite widespread), their mathematical
knowledge would have been compounded of more than one tradition, both
Euclidean and ‘other’.
Evidence for the diffusion of the Euclidean tradition is provided by three
papyri whose contents are traceable to the Elements . All seem to have been
written in good, if at times hasty, hands, i.e. they seem to have been producedby educated adults. In fact, two of them, which only have the enunciations
and diagrams without proofs, look as if they were written for personal perusal,
as if the author wanted to work through the demonstration for himself(herself?).
3 As for the ‘other’ tradition, it is represented by at least one papyrus
with problems such the following (see Diagram 5.2):
If another incomplete cone is given, which has the vertex 2, the
base 10, and the inclinations each 5, subtract the 2 of the vertex
from the 10 of the base = 8, of which 1/2 = 4; by itself = 16; and the
5 of the inclination by itself = 25. From these subtract the 16 = 9,
of which the root = 3. This is the height. And add the vertex to the
base, the 2 to the 10 = 12, of which 1/2 = 6. Describe a circle whose
diameter is 6. Multiply this by itself = 36, of which 1/4 = 9. Subtract
9 <from 36> = 27. And subtract the 2 of the vertex from the 10 =
8, of which 1/2 = 4. Describe a circle whose diameter is 4, the area
is 12, of which 1/3 = 4. Add to the 27 = 31. This multiplied by the
3 of the height = 93. The stone will be of as many feet.4
The reference to the truncated cone as a ‘stone’ ties in with several other
clues in this text: some of the geometrical objects are given names of archi-

146GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
tectural elements, and one of the problems, now completely illegible, is
illustrated by a cylindrical column, with a well-defined capital. The papyrus
was probably aimed at architects for training or reference, and provided
information that could be extended from the particular cases, each involvinga solid with such-and-such dimensions, to problems with similar features.
In fact, it often specifies that the procedure can be applied in analogous
circumstances: one of the problems does not even append a solution, leavingthe reader to fill it in, and there is a sequence of four propositions about the
volume of a cylinder with a full calculation in the first two and an abbreviated
version in the latter two, as if they were, basically, a drill.
We have numerous other papyri containing tables of division, addition
and multiplication; financial documents and metrological texts, where
instructions are given to convert one unit of measure (of land, of money)into another. One papyrus starts off with a table of parts, and follows with
a list of arithmetical problems concerning conversion between different
coinages and calculation of freight charges. The problems are mostly aboutspecific examples, but two of them refer to a ‘proof’ ( apodeixis ) – essentially
a brief counter-check:
The width of a field is [2
1/2] schoenia; what will be the length so
that it makes 200 arouras? As is necessary, reduce the [21/2] schoenia
to] 1/2, = 5; and the 20 arouras to 1/2 = 40 of which the 5th part =
8; the length will be of as many <units> as that; proof; multiply
the 21/2 schoenia of the width by the 8 of the length = the above-
said 20 arouras.5Diagram 5.2

147GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
Another papyrus has two problems about the distribution of money
into unequal parts. T wo long calculations are arranged in columns, so that
the procedure, as well as explained, is visualized, as if it was being carriedout on an abacus. We have at least three concrete examples of abacus from
this period, all quite small and in bronze (see Figure 5.1). The most recent
find is from a late first-century
AD grave in Aosta.
Like earlier Greek ones, these abaci had signs and columns both for
numbers and for sums of money. Compared to our extant Greek examples,
however, which are slabs of stone, the bronze Roman abaci seem to havebeen more easily transportable. The fact that the counters are grafted on,
rather than loose, must also have facilitated use, and suggests that, although
only a few actual items survive, abaci were widely used.
Facility of calculation was also the rationale behind a rather numerous
group of papyri: planetary tables. They were used mainly in connection with
astrological practice; how they were produced is not entirely clear, but most
scholars think that their authors drew on Mesopotamian observations and
data. Knowing how to use a planetary table required expertise, and some ofour papyri provide instructions for calculations requiring arithmetical skills.
7
Once the position of the planets at a certain point in time was determined,one could compile horoscopes. Most of the ones we have tend to be simple,listing what ‘house’ each planet is in and little else. The astrologers who made
them need not have known very much about astronomical observations or
celestial models: interpreting the chart was of course a different matter, andFigure 5.1 Roman abacus (replica)
(© copyright Science Museum/Science and Society Picture Library, London)

148GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
that was where for most people the real skill lay. But, as the public for astrology
was vast and diverse, the corresponding practice was just as varied. Along the
simple horoscopes we find more sophisticated examples:
The Egyptian men of old who had faithfully studied the heavenly
bodies and had learned the motions of the seven gods, compiledand arranged everything in perpetual tables and generously left
to us their knowledge of these things. From these I have accurately
calculated and arranged for each one (of the seven gods) accordingto degree and minute, aspect and phase, and, simply, not to waste
time in enumerating each item, whatever concerns its
investigation. For thus the way of astrological prediction is madestraight, unambiguous, that is, consistent. […] And Phosphoros,
the star of Venus, had completed in Pisces 16 degrees and four
minutes, which is the fifteenth part of a degree; in the sign ofJupiter; in its own exaltation; rising at dawn; at the Southern
Fish; like crystal; in the terms of Mercury; distant two lunar
diameters from the Star in the Connecting Cords. […] TitusPitenius computed it as is set forth. […] Computed in
Hermopolis, where the horizon has the ratio seven to five. The
time of pregnancy: 276 days. With good fortune.
8
This planetary picture is quite complex; the astrologer prizes accuracy andemphasizes that the calculations are his own, the result of personal input.By referring to the eternal planetary tables handed down from the distant
past, he inscribes his practice within a valuable tradition. In sum, astrology
catered for a diverse public, with diverse pockets. While the use of planetarytables was a constant, calculating skills, the ability to make one’s own astro-
nomical observations and the resulting greater accuracy made the difference
between a luxury horoscope and an ordinary one.
Our papyrological evidence includes of course a great many accounts,
receipts, and lists for tax purposes, whose format shows remarkable continuity
with earlier periods. The notion that accounts can be tampered with, and
that accountability goes hand in hand with political honesty, are also already
familiar.
9 Since the time of the Republic, Roman magistrates had had to
give accounts at the end of their term of office, and deposit a copy with the
central archive in Rome; the charters of newly-founded cities, or of cities
which had newly entered the Roman sphere of dominance, often mentionthe obligation to give accounts.
10 These practices are known from previous
chapters; the difference is in whom one is accountable to. An inscription
from Messene ( AD 35–44) puts accounts in the context of a Greek city
faced with Roman visitors:

149GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
[Aristocles], secretary [of the councillors], introduced the assess-
ment of the eight obol tax into the council, with an account of the
[revenues accruing from] it and how each has been spent on thepurposes ordained, showed that he had used it for nothing [other
than these purposes], and made his statement of the sums of money
still owing from the tax in the theatre before non-members of thecouncil and in the presence of Vibius the praetorian legate […] In
this connection also the councillors approved his care and integrity
and with Vibius the praetorian legate were eager to do him thehonour of a bronze statue, while Vibius the praetorian legate per-
sonally gave him in the presence of all the citizens the right to wear
a gold ring […] Whereas Aristocles the son of Callicrates […] gavehis attention to the clear daily writing up on the wall of all the
financial transactions of the city by those responsible for handling
any business of the city, setting a beneficial example to worthy menof integrity and justice in the conduct of office […] in entertaining
governors and numerous other Romans too he devotes the
expenditure of his own money to the advantage of the city. […]And on account of the merits inscribed above, Memmius the pro-
consul and Vibius the praetorian legate have each in recognition
of his conduct given him the right to wear the gold ring, as hasthe council also […] [Cresphon]tis <tribe>; one hundred and
twenty-two talents, thirty minae, a stater, eight obols, a half obol;
[Daiphonti]s <tribe>; one hundred twenty-two talents, fifty-sixminae, five staters, eight obols [etc.].
11
The writing on the wall and its corollary of transparency in the use of
resources may be seen as a throwback to the golden times of the Greek polis
(which Messene, founded in the fourth century BC, never experienced
anyway), yet everything happens under the eyes of Vibius the praetorianlegate, whose presence gives a whole new meaning to the Greek officers’
accountability.
Another figure familiar from earlier times is that of the public accountant,
in Latin numerarius or tabularius . He generally worked for upper financial
officers called rationales or a rationibus ; moreover, secretaries or scribae , whose
duties were less specific, were usually numerate as well as literate. To these
one should add the accountants and financial secretaries working in big
private households, who tended to be slaves or freedmen. In the case ofimperial slaves or freedmen, the boundary between ‘state’ and ‘private’
accountants becomes very blurred. Our abundant epigraphical evidence
indicates that, especially in the second half of the first century
AD, many of
the numerarii and tabularii were freedmen, when not slaves, while from the

150GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
mid-second century onwards we find a greater proportion of free-borns. A
connected group whose features are not entirely clear is that of the
calculatores , or ratiocinatores . They do not occur in inscriptions very
frequently: the examples I could find were a doctor (teacher) of calculation
in a provincial town, an Italian sevir Augustalis (member of a civic order
made up of rich freedmen), and a slave child-prodigy from Ostia, who diedat the age of thirteen and was fondly remembered by his teacher as a
wondrous calculator who had written commentaries on the art.
12 Finally,
accountants, secretaries and perhaps architects were included, among others,in a category of public clerks, the apparitores , which was only open to men
of free or freedman status. We know that several apparitores , including the
poet Horace, improved their status – he went from son of a freedman tomember of the equestrian order, so it would seem that belonging to the
category was a good avenue for social mobility.
13
With the obvious exception of loans from the state to private citizens and
taxes and duties,14 public accounts in the Roman Empire were not usually
displayed in the form of inscriptions. What we do find instead is for instance
this:
Gaius Appuleius Diocles, charioteer of the Red Stable, a Lusitanian
Spaniard by birth, aged 42 years, 7 months, 23 days. He drove hisfirst chariot in the White Stable, in the consulship of Acilius Aviola
and Corellius Pansa. […] He won his first victory in the Red Stable
in the consulship of Laenas Pontianus and Antonius Rufinus.Grand totals: He drove chariots for 24 years, ran 4,257 starts, and
won 1,462 victories, 110 in opening races. In single-entry races he
won 1,064 victories, winning 92 major purses, 32 of them(including 3 with six-horse teams) at 30,000 sesterces, 28 (including
2 with six-horse teams) at 40,000 sesterces, 29 (including 1 with a
seven-horse team) at 50,000 sesterces […] He won a total of35,863,120 sesterces.
15
A charioteer’s achievements are celebrated, and enumerated and added
up, the better to emphasize his extraordinary career. On the principle that
spending money for the community was virtuous and politically a goodinvestment, there are also inscriptions advertising the amounts lavished on
public works by private individuals. For instance, the long bilingual
document known as Res Gestae :
Below is a copy of the achievements of the deified Augustus, whereby
he brought the whole world under the rule of the Roman people,and of the sums which he expended on the Republic and people of

151GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
Rome. […] I held a lustrum after an interval of forty-two years. At
this lustrum , four million sixty-three thousand Roman citizens were
registered. I performed a second lustrum […] At this lustrum four
million two hundred and thirty-three thousand Roman citizens were
registered. […] To every man of the common people of Rome I paid
three hundred sesterces in accordance with my father’s will; and inmy own name I gave four hundred sesterces from the spoils of war
[…] again in my tenth consulship I gave every man a gratuity of
four hundred sesterces from my own patrimony […] These gratuitiesof mine reached never fewer than two hundred and fifty thousand
persons. […] On four occasions I assisted the treasury with my own
funds, paying over a hundred and fifty million sesterces to those incharge of the treasury. […] I provided the public spectacle of a naval
battle on the other side of the Tiber […] having excavated an area a
thousand and eight hundred feet long by a thousand and twohundred feet wide, where thirty beaked ships […] joined in battle.
[…] Of those who fought at that time under my standards, more
than 700 were senators. 83 of these have become consuls […] andapproximately 170 became priests. […] Italy possesses 28 colonies
established under my authority […] At the time of writing, I am in
my seventy-sixth year.
16
Close to the perceived end of his life, Augustus takes comprehensive stock,quantifying and counting not only sums of money he has spent, but alsopeople, territories of the empire, members of the élite and the extent of
their participation to the highest offices; down to his own age. A whole life
and a whole piece of history are set down in numbers.
Augustus had other ways of making mathematics work for him: a monu-
mental sun-dial was set up in Rome between 10 and 9
BC.17 Its gnomon was
an Egyptian obelisk whose shadow indicated hours, days and months andwhose pediment reminded the public of Augustus’ victory over Antony and
Cleopatra. The obelisk is still extant, albeit in reconstructed form and in a
different position, and archaeological excavations have unearthed bronze-
inlaid time lines, dating from Domitian’s period, which may have been a
restoration of the earlier sun-dial.
18 Pliny the Elder ( AD 23–79) declared the
working of Augustus’ massive clock a ‘thing worthy of being known’ and
named the mathematician Facundus Novius as its maker:
[Facundus] was said to have understood [its] principle from <the
shadow cast by?> a human head. The readings have been out of
line for about 30 years now, either because the course of the sunitself is out of tune and has been altered by some change in the

152GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
way the heavens work, or because the whole earth has shifted
slightly from its central position.19
The production of sundials was indeed an important branch of mathematics,discussed by authors such as Vitruvius, Hero and Ptolemy. Interestingly,
Pliny was prepared to believe that the whole universe was out of synchrather than doubt the accuracy of Facundus’ creation. We have another,
earlier, spectacular example of time-keeping object in the so-called
Antikythera device, retrieved from a shipwreck off the Greek island by thesame name (see Figure 5.2).
A small bronze mechanism, datable to c. 87
BC, it consists of some thirty
toothed wheels of different sizes, connected by pinions, encased in a box.Although its precise function is not entirely clear, it seems that various aspects
Figure 5.2 Gearwork from the Antykythera object (reproduced with permission of
the American Philosophical society from de Solla Price (1974), fig. 14 p. 24)

153GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
of mathematical knowledge went into its making: it appears to be based on
calendrical cycles, and its gear mechanism required accurate calibration of
the diameter and number of teeth of the various wheels, in ways later dis-cussed by Hero and Pappus. The main dial is inscribed with the names of
the signs of the Zodiac and those of the months in the Egyptian calendar
(widely used since Hellenistic times). T wo longer inscriptions, unfortunatelyfragmentary, are a parapegma and some user’s instructions. As in the case of
planetary tables, the person producing the mathematical instrument is not
necessarily the one using it, but mediation or (partial) transmission ofexpertise is provided in the form of instructions or explanations. The
Antikythera device was a precious, prestige object: the craftsmanship involved
is remarkable; also, the ship cargo included other luxury items such as statues,some of them in bronze, and amphorae. It is further testimony to the interest
members of the upper classes had in astronomy and time-keeping.
Along with these two extraordinary examples, ‘normal’ sundials were
common in the Graeco-Roman period, although it is often difficult to date
them with accuracy. They have been found in private houses and working
establishments, as well as in squares and markets, in cities such as Pompeii,Athens, Carthage, and Palmyra. Usually rather small, they were often made
of stone with a metallic gnomon and with the time-lines engraved and
painted in. Most of them only indicated hours, others days, months, solsticesand equinoxes; also, their shapes varied from hemispherical to conical to
cylindrical, and some dials were more complicated to make than others. We
find, in short, a situation similar to that of horoscopes: the product wasdifferentiated on the basis of its customers and their financial means. Besides,
differentiation was achieved through a greater or lesser import of mathema-
tical knowledge, as well as through decoration, scale and quality of materialemployed. To give an example, the sundial in Figure 5.3, found in Pompeii
in the so-called granario (a sort of corn exchange), shows ‘that the hour lines
were constructed using arbitrary parallel circles by makers who were notparticularly concerned with exact seasonal markings’.
20 On the other hand,
other sundials were more accurate and their making must have required
some knowledge of conic sections. Examples from Pompeii include some
found in wealthy private houses and one found in a shop whose owner was
called Verus:21 it is of small dimensions and encased in a box, hence trans-
portable, and it is made of ivory and carefully carved, so it must have been
expensive. Verus’ establishment has turned out other objects, mostly in
bronze (candelabra, vessels), graffiti that identify him as a faber and, above
all, writing implements, a ruler, compasses and the pieces of a surveying
instrument, which has been reconstructed as in Figure 5.4
The reliability of this reconstruction is confirmed by a similar find in
Bavaria, and by engravings on tombstones. Roman land-surveying has left

154GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
many archaeological remains; the most spectacular are large-scale grids, still
visible in many parts of France, Spain, the former Yugoslavia and especially
Italy and North Africa. Figure 5.5 is an example.
These patterns are recoverable by various methods: digging, aerial
photography, electronic remote sensing or field survey archaeology. On the
ground, the lines are walls or roads, or remains of ditches which produce
surface irregularities, in their turn detectable from the air or by instruments.Often, boundary stones are excavated in correspondence with these patterns;
their function was to indicate the ownership or lease of a plot of land or its
position with respect to some reference points. The laying-out of a grid likethe one pictured in Figure 5.5 usually started from two designated main
perpendicular axes (mostly roads or paths), generally orientated in the
directions of the four cardinal points, the decumanus east–west and the
cardo north–south. All the other lines were then laid in place parallel and
perpendicular and at regular distances from the main axes. The groma and
other sighting instruments were used to keep the lines straight; measuringrods, the extremities of which have also been recovered from Verus’ shop
and from Enns in Austria,
23 were used to ascertain distances; and sun-dials,
probably not unlike the one again found in Verus’ shop, helped lay thecardo and decumanus in the right directions. The whole operation was known
Figure 5.3 Sundial found in Pompeii (reproduced with permission from Gibbs
(1976), plate 4 p. 137, © copyright Yale University Press)

155GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
as ‘centuriation’ (the territory as ‘centuriated’) from centuria , a unit of
measure corresponding to two hundred iugera , equivalent to two-thirds of
an acre. Physical or man-made obstacles, such as rivers or irregular borders
or temples, often got in the way of the surveyor, but, on the whole, thecenturiated territory became in effect a geometrical landscape.
Many inscriptions document land-surveying activities. Especially for
emperors like Vespasian or Nerva, who both came to power after very turbu-lent periods, the division and apportionment of land had strong political
and financial motivations: political, because they were presented as the
restoration of order and justice; financial, because remeasurement andredistribution of land often meant the establishment or reassertion of tax
demands. The emperor often figured as the author of the survey, even when
this had been in fact carried out by mensores . Clearly, surveying the land
was a mark of power; it was up to Caesar to divide, parcel out and assign.
One of the most famous monuments of land restitution is the so-called
Orange cadaster (see Figure 5.6).Figure 5.4 Reconstruction of a groma from Pompeii, c. first century AD
(© copyright Science Museum/Science and Society Picture Library, London)

156GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
Figure 5.6 Fragment 7 of Cadaster Orange A, c. AD 77
(reproduced with permission of the Musée d’Orange, Vaucluse, France)
Figure 5.5 Centuriation from Africa Proconsularis
(from Paul MacKendrick, The North African Stones Speak , p. 31, fig. 2.2, ©
copyright 1980 by the University of North Carolina Press.
Used by permission of the publisher)

157GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
Now in a fragmentary state, it was a large-scale marble map of the region
around the city of Arausius (Orange) in France. As one can see from the
picture, space had been geometrized and divided into equal squares, andthen inscribed with some landscape features (a river) and letters to indicate
who occupied the land and, in some cases, how much tax they were supposed
to pay.
24 The Orange cadaster was accompanied by an inscription, dated AD
77, with which Vespasian announced that he was returning public land to
the state. What that in fact meant was that he was claming the right to tax
land that had until then been in private use. Although no surveyors arementioned, it was they, the ‘silent technicians’, who measured out the land
and produced, or led to the production of, the map. It was their mathematical
outlook that made both the act of measurement and the representation ofaccomplished measurement (the map) possible.
25
In other cases the surveyor is present in the picture, as an expert called
upon to give the benefit of his specialized skill. We have several examples ofthis in inscriptions about boundary disputes, which occurred very frequently
not only between private individuals (in which cases the state was unlikely
to intervene), but also between neighbouring communities:
Decreed by the proconsul Quintus Gellius Sentius Augurinus, read
out from the tablets on the kalends of March. Since the good andgreatest emperor T rajan Hadrian Augustus had written to me in
order that I, having employed surveyors and having investigated
the case of the boundary controversy between the people of Lamiaand those of Hypata, mark off the boundaries, and since I have
been to the place in question rather often and for successive days,
and have investigated in the presence of the representatives of eithercity, having employed the surveyor, Julius Victor, veteran of
Augustus, it is resolved that the beginning of the boundaries is
from the place where I found out that Siden was, which is belowthe precinct dedicated to Neptune, and from that place going down
one keeps a straight line until the Dercynna spring […].
26
Although the emphasis is squarely on Augurinus, in his full proconsular
judicial capacity (he inspects the sites concerned, listens to the partiesinvolved, walks along the boundaries), a mensor is mentioned and even
named. We know from other inscriptions of this kind that the expert
testimony of the land-surveyor was only one of the various types of evidenceused in a boundary dispute. Those were often sensitive cases: for instance,
Greek communities, which in the past had resolved these matters through
their own boundary-men and their own arbitrations, now were told whatto do by a Roman officer and by surveyors sent from Rome. It is not surpris-

158GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
ing that decisions were taken after listening to local witnesses, looking at
precedent adjudications and searching the terrain for old boundary markers.
The expertise of the land-surveyor in some cases fades into the background:the role of the governmental officer as arbiter is brought to the fore, and accu-
rate measuring becomes just one of the many guarantees of a fair decision.
The fact that many inscriptions about land-surveying give the point of
view of the person using the services of the surveyor does not mean of course
that the surveyor did not have a point of view. Had the inscription above
been set up by Julius Victor, we may have heard a different version, of howthe dispute was settled mostly thanks to his expert intervention. We have
one such different point of view in a document found in Algeria:
Both the most splendid city of Saldae and I, together with the people
of Saldae, beg you, o lord, to exhort the surveyor Nonius Datus,
veteran of the third Augustan legion, to come to Saldae, to completethe works. I set out on the journey, and was attacked by brigands;
naked and wounded my men and I managed to escape; I arrived at
Saldae; I met Clemens the provincial governor. He took me to themountain, where they were uncertain and weeping about the tunnel,
on the point of giving up the whole thing, because the tunnellers
had covered a distance greater than that from side to side of themountain. It turned out that the cavities diverged from the straight
line, to the point that the upper end of the tunnel was leaning to the
right southwards, and analogously the lower end of the tunnel wasleaning to its right northwards: so the two parts were diverging,
deviating from the straight line. But the straight line had been marked
off with stakes on the top of the mountain, from east to west. […]When I assigned the work, to make them understand how to do the
tunnelling, I set a competition between the team from the navy and
the team from the javelin division, and in this way they met in themiddle of the mountain. […] Having completed the work, and
released the water, the provincial governor Varius Clemens
inaugurated it. 5 modii of capacity […].
27
Nonius Datus, a retired military surveyor, describes his adventurous
trip in his own voice. Unlike the Eupalinus tunnel (see chapter 1), the
two arms of the underground water conduit in Saldae had not managed
to meet in the middle: their course was divergent, and the people werethrown into despair. Datus went (literally) to set things straight: his
intervention is compounded of managerial skills (motivating and
organizing the workforce) and surveying expertise. The method of markingthe path off with stakes required some simple geometry and accurate

159GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
measurements, and is described in at least one almost contemporary land-
surveying treatise.28
We have many other, shorter, inscriptions commemorating land-surveyors
themselves.29 Many of them, like Datus, were or had been in the military;
several were freedmen. I think it is indicative of a certain pride in the
profession that we have surveying instruments engraved on the tombstonesof at least two mensores , in both cases freedmen.
Although it has not been possible to reconstruct their training in any
detail, the combined evidence of inscriptions and technical treatises goes toshow that the surveyors had indeed a sense of belonging to something like
a professional group, with shared knowledge, work ethics and mathematical
reference points such as Euclid and, to a lesser extent, perhaps Archimedes.At the same time, again both epigraphical and literary evidence also show
that there were alternative, when not competing, views about surveying
and land division on the part of non-technical administrators and, as wewill see, of some members of the educated general public.
Vitruvius
A contemporary of both Julius Caesar and Octavian Augustus, Vitruviusworked as an architect and military engineer and may have been anapparitor .
30 His only work, the Architecture , deals with a variety of topics:
book 10, for instance, is entirely devoted to machines, including catapults,
and there is a long section on sun-dials. Mathematics is a ubiquitous presence:Vitruvius uses it to lay the groundplan of a building; to track down the
directions of the winds; to calculate the proportions of the various elements
of a temple starting from a module or standard element; to build soundingvessels that amplify voices in the theatre according to the principles of
harmonics; to construct an analemma, the scheme on which dials are based.
He provides lists of ready-made measurements for catapults, calculated onthe basis of the weight of their intended projectile, because those who are
not ‘familiar with the numbers and multiplications through geometrical
procedures’ may find themselves at a loss for time if they need that crucial
information during a siege.
31
At the very beginning of the Architecture , after dedicating it to the
emperor, mentioning his service to the emperor’s excellent father, his good
relations with the emperor’s sister, and praising the emperor’s own building
achievements, Vitruvius states that the architect
should be a man of letters, an expert draughtsman, know geometry
very well, he should have learnt many histories, have listened dili-gently to philosophers, known music, not be ignorant of medicine,

160GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
have learnt the decisions of jurists and be familiar with astrology
and the principles of the sky. The reasons why this should be so
are these. […] Geometry furnishes many resources to architecture.It teaches the use of rule and compass which greatly facilitate the
laying out of buildings on their sites and the arrangement of set-
squares, levels and lines. […] By arithmetic, the cost of building issummed up, the procedures of mensuration are explained, and
difficult questions of symmetry are solved with geometrical proce-
dures and methods. […] But those individuals on whom naturehas bestowed so much activeness, acumen, memory that they can
know geometry, astronomy, music and the other disciplines
thoroughly, go beyond the duties of architects and produce mathe-maticians. […] Such men, however, are rarely met. We can point
to Aristarchus of Samos, Philolaus and Archytas of Tarentum,
Apollonius of Perga, Eratosthenes of Cyrene, Archimedes andScopinas from Syracuse.
32
Vitruvius is intent on building an image for his profession which situates
it firmly within high culture. He makes it clear that architects are educated,
fully-rounded individuals who can be trusted with the wider civic and
political implications of their activities. Especially if he was an apparitor ,
the image of the architect he puts forth can be read as a paradigm of the
gentleman technician, ready, willing and able to serve the state and climb
the social ladder at the same time. That Vitruvius distinguishes architectsfrom mathematicians is due not only to a consideration of how many things
one can realistically be expected to do in a lifetime, but also to the emphasis
he chooses to put on the architect’s duties, his commitment to what is goodand useful for the state. Geometry and arithmetic are justified on the basis
of their use, not of their role in the advancement of knowledge.
Vitruvius’ praise of mathematics is reprised and expanded at the
beginning of book 9; once again, its value is derived from its actual benefits,
from the things mathematics does. Athletes traditionally win fame and
honour, yet, in Vitruvius’ view, their achievements do not really signify
anything for humankind. How much juster it would be, he comments, if
similar honours were bestowed on learned men! He mentions Pythagoras,Democritus, Plato and Aristotle as examples, and recounts their many
discoveries, ‘which have been useful for the going forth of human life’.
Probably on the basis of the Meno , Plato is reported as the author of the
duplication of the square, which is introduced as a practical problem:
‘Suppose there is a square area, or field with equal sides, and it is necessary
to double it’. Vitruvius stresses that Plato provided a solution by meansof lines, because the results produced through numbers, i.e. via simple

161GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
multiplications, would not be correct. Next, he introduces a contrast
between the mathematician and the craftsman:
Then again, Pythagoras showed how to find a set-square without
the constructions of a craftsman, and whereas the craftsmen who
make a set-square with great labour are hardly able to get it closeto the truth, the same thing corrected by procedures and methods
is by him explained with instructions. For if three rulers are taken,
of which one be 3 feet, another 4 feet and the third 5 feet, thoserulers combined with each other will touch one another at their
extremities making the shape of the triangle, and will form a
corrected square. Moreover, if single squares with equal sides bedescribed along the several rulers, when the side is of three, it will
have 9 feet of area, the one of 4, 16, the one which will be 5, 25.
[…] The same calculation, as it is useful in many things andmeasurements, so it applies to buildings in the construction of
staircases, for the adjustment of the steps.
33
Pythagoras and the artisan both want the same thing, but the former
obtains a correct solution without ‘great labour’. Vitruvius follows this up
with the story of how Archimedes discovered a method to tell whether anallegedly golden crown made by a craftsman for king Hiero was made entirely
of gold, or of an alloy of gold and baser metal. Finally, he praises Archytas
and Eratosthenes for their solutions to the duplication of the cube, which,in his report, originated from a request by Apollo of an altar double the
extant one. Although philosophy and poetry are also mentioned, the
examples at the core of Vitruvius’ argument are all mathematical, startingwith the duplication of the square and concluding with that of the cube.
Mathematics thus exhibits its utility for building, agriculture, the service of
the king or devotion to the gods. Mathematics also allows Vitruvius implicitlyto associate himself with famous figures from the past, and to contrast their
virtue with the incapacity or moral defects of others.
Hero of Alexandria
Hero lived, probably in Alexandria, around AD 62 (the date of an eclipse
he mentions). Of his works many have survived: Automata , Pneumatica ,
Belopoeiika , Cheiroballista , Mechanica (in an Arabic translation), Dioptra ,
Metria. There are also texts attributed to him whose authorship is debated:
Stereometria and Geometrica may contain Heronian material but were put
together at a later stage; the Definitions may be by Diophantus.
Mathematics plays a number of roles in Hero’s work: he deals with

162GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
straightforward geometrical and arithmetical questions; his mechanics is
heavily geometrized, and he has a number of pronouncements about
mathematics and mechanics as forms of knowledge. I will discuss thislatter aspect in chapter 6.
Let us start with the Definitions , which covers plane and solid figures,
proportion, equality and similarity, infinity and divisibility and what it meansto measure something. The introduction, addressed to a Dionysius, describes
it as a preparation to Euclid’s Elements , and indeed [Hero] explains and
expands where Euclid had only concisely defined. One example is thedefinition of line, which in the Elements is ‘length without breadth’:
Line is length without breadth and without depth or what first
takes existence in magnitude or what has one dimension and is
divisible as well; it originates when a point flows from up down-
wards according to the notion of continuum, and is surroundedand limited by points, itself being the limit of a surface. One can
say that a line is what divides the sunlight from the shadow or the
shadow from the lighted part and in a toga imagined as acontinuum <it divides> the purple line from the wool or the wool
from the purple. Already in customary language we have an idea
of the line as having only length, but neither breadth nor depth.We say then: a wall is according to hypothesis 100 cubits, without
considering the breadth or the thickness, or a road is 50 stades,
only the length, without also concerning ourselves with its breadth,so that the calculation of that as well is for us linear; it is in fact
also called linear measurement.
34
The Definitions tends explicitly to relate basic concepts to external reality
or to mathematical operations one may perform in the real world. Rather
than axiomatico-deductive, its structure is demonstrative, in that the topicsfollow an increasing order of complexity, and taxonomic schemes are
extensively used. While reporting as definitions things that in Euclid figure
as postulates, axioms or even theorems, it does not cover the same ground
as the Elements : it introduces many more geometrical objects (conics, to
name but one) and occasionally provides alternative definitions altogether(for instance, of parallel lines). Moreover, some philosophical issues are
addressed, such as the continuum principle mentioned above or the idea of
divisibility:
A part is a magnitude smaller than a greater magnitude, in the
case when the greater is measured up exactly into equals. One sayspart here neither in the same sense that the earth is part of the

163GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
universe, nor that the head <is a part> of a human being, but not
even in the sense that, having drawn a perpendicular to the diameter
of a circle from its extremities, we say that an angle taken outsidethe semicircle is a part of the perpendicular; for it is impossible for
a right angle to be measured up exactly by this angle, which is
called horn-shaped, since the horn-shaped <angle> is smaller thanany rectilinear angle. We will thus rather take the part among
magnitudes which are of the same kind and call the part among
magnitudes accordingly, that is we say that the angle of a third ofright angle is a part of the right angle. So we can leave aside the
common sophistical saying, that if the part is what measures up
exactly, then also what measures up exactly is part, but the solid ismeasured up exactly by lines one foot long, therefore the line one
foot long is a part of the solid, which is absurd. A line one foot
long measures not the solid, but the length of the solid and itsbreadth and its depth, which are of the same kind as the line itself.
35
On the whole, although the question of the authorship may ultimatelyremain unsolved, the Definitions constitutes interesting evidence for the
existence of accounts of basic mathematics other than Euclid.
As for Philo of Byzantium, mathematics is for Hero an important element
in the development of military technology: through experience and adjust-
ments, subtracting a little here, adding a little there, the engineers found
‘harmonious’ measurements which could be expressed in mathematicalterms.
36 Indeed, Hero’s machines are described as if they were geometrical
objects, through lettered diagrams. T o find the dimensions of the hole of a
stone-thrower, one has to proceed as follows:
Multiply by one hundred the weight in minas of the stone to be
discharged; take the cube root of the product; and of whateverunits you have found the root to be, add to what you have found
the tenth part, and make the diameter of the hole that number of
fingers. For instance let the stone be of eighty minas; one hundred
times these produces 80,000; the cubic root 20 and the tenth of
these 2 produces 22; of as many <fingers> will be the diameter ofthe hole. If the product does not have a cube root, it is necessary
to take the nearest and add the tenth part.
37
The same result can be obtained with the duplication of the cube, for
which Hero provides a solution via a moving ruler.38 We thus have an example
of a problem solved in two different ways, neither of which seems to beprivileged with respect to the other: a set of general instructions for calcula-

164GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
tion, followed by an arithmetical solution which takes approximation, and
thus a certain degree of inaccuracy, into account; and a geometrical ‘instru-
mental’ solution whose results are substantiated but not necessarily handyfor use. Along with the Belopoeiika , many of Hero’s treatises ( Mechanica ,
Automata , Dioptra , Pneumatica ) describe machines for several purposes:
lifting weights, enlarging or copying two- or three-dimensional figures,cutting thick pieces of wood, supporting a roof, putting up a puppet show,
putting out a fire with a water-pump, producing music, measuring distances
or the size of the moon. As with the catapults, Hero pays great attention tothe materials and the construction specifications of his machines, and
mathematizes them, reducing their components to geometrical objects. In
some cases, their working itself is geometrically explained: for instance, aknowledge of centres of gravity and points of suspension, which Hero derives
from Posidonius and Archimedes, can help to lift or support a weight. The
hidden mechanism of a puppet-theatre must be constructed in respect ofsolid geometry; the functioning of a pump is explained by means of an
indirect geometrical proof.
39 In all these cases, mathematics is put to use,
and utility is not divorced from what we would consider simpleentertainment. The Automata and Pneumatica describe self-moving statues
of Dionysius, or ‘bottomless’ cups that keep spouting wine: they may have
been built as part of a well-appointed banquet or a public celebration, whereputting up an impressive show was one of the duties of the host or organizer.
Entertainment was far from useless; it was an important way for the elite to
display their wealth and power.
The Dioptra and especially the Metrica are the most mathematical of
Hero’s works. The dioptra is of course another machine, and the beginning
of the book describes in detail how it is to be built, and duly proclaims itsinnumerable uses. After that, Hero puts it to work on a number of problems
related to land-surveying, astronomy and engineering. To quote just a few
examples, it can be used to measure the width of a river or the depth of aditch, to help dig underground water conduits or build a well, to measure a
piece of land or determine a boundary, or to divide pieces of land even
when they are inaccessible because, for instance, they are thickly wooded. A
mathematically constructed instrument, deploying geometrical principles,
becomes the means through which the practitioner measures, counts, divides,and grasps a vast deal of the world that surrounds him. The facing page
contains an example of dioptra-assisted land-division (see Diagram 5.3).

165GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
To divide the given area through the given point in the given parts.
Let the given point be for instance a water-spring, so that all the
parts one takes use the same water. Let the given area be surroundedby straight lines AB, BC, CD, DE, EZ, ZH, HH, HK, KK, LK; if
the lines surrounding the area are not straight, but some unordered
line, let points in succession be taken on it, so that the lines between
each successive <point> are straight. Let the given point be M, and
let it be required to divide the area into seven equal parts through
the point M. Let MN be drawn perpendicular to AB by means of
the dioptra, so that if we imagine MA, MB joined, it will be possible
to measure the triangle AMB . For the <rectangle> formed by AB,
MN is double the triangle ABM . It is also possible to measure, as
it has been written before, the whole area. If the triangle ABM is
equal to a seventh part of the whole area, the triangle ABM will be
one of the parts; if greater, it is necessary to subtract from it, having
drawn MN, and one produces the triangle AMN equal to the
seventh part of the whole area; if the triangle ABM is less than a
seventh, it is necessary to subtract from the triangle BCM the
triangle BMO , which, together with the triangle AMB , will be a
seventh part of the whole area […] In this way we calculate theremaining triangles as well, and we divide the area into the given
parts from the point M.
40Diagram 5.3

166GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
The Metrica deals, as the title suggests, with measurement and division
of plane and solid figures. Hero indicates from the start that his task is
essential to geometry itself, which means, as the ‘old account teaches us’,the measurement and division of land. The universal utility of these opera-
tions propelled research at the hands of authorities such as Eudoxus and
Archimedes, whose main discoveries (volume of the cylinder, area of thesphere) only confirm that geometry is quintessentially about measurement.
His own work, Hero says, will be a combination of past results and personal
contributions to the field.
41
The structure of the Metrica will become clear if we look at one topic:
the area of the triangle. The order of treatment is as follows: first the simplest
case – the area of a right-angled triangle is half the area of a rectangle withthe same sides. There follow the case of an isosceles triangle and of a scalene
one, distinguished into the two sub-cases of height falling inside or outside
the triangle. Hero then provides a general method, which allows one tocalculate the area of any triangle, given its sides. This can be quoted as a
good example of Hero’s procedure throughout the Metric. First of all he
demonstrates the method on a specific triangle, taking the reader through
all the calculations, including the extraction of an approximated square root,
which I have omitted here:
There is a general method to find the area of any triangle whatever,
given the three sides and without the height; for instance let the
sides of the triangle be of 7, 8, 9 units. Add the 7 and the 8 andthe 9; it makes 24. Of these take the half; it makes 12. Subtract
the 7 units; the remainder is 5. Again subtract from the 12 the 8
<units>; the remainder is 4. And further the 9; the remainder is 3.Multiply the 12 by the 5; they make 60. These by the 4; they
make 240; these by the 3; it makes 720; take the root of these and
it will be the area of the triangle.
After the arithmetical part, Hero has a geometrical one, presented as a
proof, and thus as a justification of what precedes it (see Diagram 5.4):
The geometric proof of that is as follows: Given the sides, to find
the area of a triangle. It is in fact possible to find the area of the
triangle when one draws the height and obtains its magnitude,
but it is required to obtain the area without the height. Let thegiven triangle be ABC and let each of AB, BC, CA be given; to find
the area. Let a circle DEZ be inscribed in the triangle, whose centre
is H, and let AH, BH, CH, DH, EH, ZH be conjoined. Therefore
the <rectangle> formed by BC EH is double the triangle BHC, the

167GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
<rectangle> formed by CA ZH is double the triangle ACH and the
<rectangle> formed by AB DH is double the triangle ABH .
Therefore the <rectangle> formed by the perimeter of the triangle
ABC and by EH, that is the radius of the circle DEZ, is double the
triangle ABC. Let CB be prolonged, and let BH be taken equal to
AD; therefore CBH is half the perimeter of the triangle ABC since
AD is equal to AZ, DB to BE, ZC to CE. Therefore the <rectangle>
formed by CHEH is equal to the triangle ABC. But the <rectangle>
formed by CH EH is the side of the <square> on CH multiplied by
the <square> on EH; therefore the area of the triangle ABC
multiplied by itself will be equal to the <square> on HC multiplied
by the <square> on EH. Let HK be drawn perpendicular to CH,
BL to CB, and join CK. Since each of the <angles> CHK, CBK is
right, therefore CHBK in the circle is a square; therefore the
<angles> CHB, CKB are equal to two right angles. Then also the
<angles> CHB, AHD are equal to two right angles, since the
<angles> around H are divided in half by AH, BH, CH andDiagram 5.4

168GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
the <angles> CHB, AHD are equal to AHC, DHB and all the
four right angles are equal; therefore the <angle> AHD is equal
to CKB. And the right angle ADH is also equal to the right
angle CBK; therefore the triangle AHD is similar to the triangle
CBK. Therefore as BC to BK, AD to DH, that is BH to EH,
and conversely, as CB to BH, BK to EH, that is BK to KE
since BK is parallel to EH, and composing as CH to BH, so
BE to EK; thus also as the <square> on CH to the <rectangle>
formed by CH HB, so the <rectangle> formed by BEC to that
formed by CEK, that is to the <square> on EH; in fact in the
right-angled <triangle> EH is drawn from the right angle to
the base perpendicularly; so that the <square> on CH
multiplied by that on EH, the side of which is the area of the
triangle ABC, will be equal to the <rectangle> formed by CHB
multiplied by that formed by CEB. And each of CH, HB, BE,
CE is given: for CH is half of the perimeter of the triangle
ABC; BH is the excess by which the half of the perimeter
exceeds CB; BE is the excess by which the half of the perimeter
exceeds AC; EC is the excess by which the half of the perimeter
exceeds AB, for this reason EC then is equal to CZ, BH to AZ,
because it is also equal to AD. Therefore the area of the triangle
ABC is also given.
The geometrical proof is followed in its turn by a ‘synthesis’, once again
a calculation carried out on a specific triangle, in this case one with sides 13,
14 and 15 units long.42 We thus have both handy measuring procedures, in
this case repeated twice, and the rationale behind them, the reason whythose procedures work. These two parts are considered as a whole: the geo-
metrical proof is the ‘analysis’ and the arithmetical calculation the ‘synthesis’.
Analysis and synthesis were distinct faces of the same coin. The fact thatmeasuring happens in the real world, on objects which are not perfect
geometrical entities, is never forgotten. Hero calculates the area of irregularly-
shaped everyday objects by covering their surface with papyrus or linen;
what is more, he measures the volume of irregular solid bodies by means of
a method analogous to that devised by Archimedes. The reader is oftenreminded that the bodies in question are in fact a bathtub, a shell (in the
sense of a decorative architectural element), a vault. In such a world no
perfect accuracy is possible, and often one has to do with the best approxima-tion, but the emphasis is less on the limitations than on the enormous
versatility and power of mathematics when put to use for the necessities of
life.

169GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
The other Romans
In parallel with the extensive Roman land-surveying activities, we have a
sprawling body of works in Latin, known as the Corpus Agrimensorum
Romanorum . The Corpus is a very mixed bag: it includes technical treatises;
laws relative to land administration; abstracts from Euclid’s Elements trans-
lated into Latin; lists of colonies. It is also a mixed bag of interpretative
problems: its contents are difficult to date; its Latin is hard to decipher, and
its manuscript versions are many and scattered around the world.
The earliest datable author in the Corpus is Julius Sextus Frontinus, a
general, four times a consul, an augur and, around AD 97, a supervisor of
the water supply for the city of Rome. He wrote three treatises: on land-surveying, on military stratagems, and on aqueducts. When Frontinus took
up the job as water supervisor, he had no specific competence in the field
and was ready to admit it – the treatise is the result of his desire to getacquainted with the task at hand. He found the water supply of Rome in a
state of chaos; many pipes of many different sizes conveyed water from the
public reservoirs to private establishments without authorization, a myriadunregulated rivulets escaping state control. The abuse took place with
impunity because Frontinus’ staff, the water-men, regularly profited from,
and in fact encouraged, fraud by private citizens. In order to recover thesituation, Frontinus employed mathematics extensively: he gave a streamlined
description of the nine aqueducts Rome had at the time, each of them
denoted by sets of measurements for length and capacity. Then he did thesums and, comparing input to output, saw that things did not add up: not
all the water that went in at the source came out once inside the city. In fact,
the extent itself of the abuse was detected because of his calculations.
It has been argued that the book on aqueducts served as a report of
Frontinus’ activity to the Senate.
43 If so, then the figures he abundantly
provides would have been equivalent to rendering accounts: they demon-strated that he had the situation under control, and allowed the senators, if
they so wished, to satisfy themselves that it was indeed so. The point is not
so much whether they are an accurate reflection of the water supply inRome c.
AD 97 – indeed, as it has been observed, water output cannot be
accurately measured on the basis of the diameter of the pipes, the way
Frontinus does. The point is that mathematizing water supply played amajor role in Frontinus’ rhetoric of good administration. This impression
is reinforced by the fact that he set out to standardize the pipes, choosing
one type, the quinaria , as the official size. Up to then, different pipes had
been used, making both repair and checking of irregularities extremely
difficult. Frontinus decided that it was a good idea to set a standard, officially
stamped, type of pipe, thus reducing the confusion of sizes to a common

170GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
unit of measure. In his words, ‘everything that is bounded by measure must
be certain, unchanged and equal to itself’.44 We find similar issues in his
treatise on land-surveying, where he describes the measuring art as follows:
It is impossible to express the truth of the places or of the size
without calculable lines, because the wavy and uneven edge of anypiece of land is enclosed by a boundary which, because of the
great quantity of unequal angles, can be contracted or expanded,
even when their number [of the angles] remains the same. Indeedpieces of land which are not finally demarcated have a fluctuating
space and an uncertain determination of iugera . But, in order that
for each border its type is established and the size of what is enclosedwithin is determined, we will divide the piece of land, to the extent
allowed by the position of the place, with straight lines. […] We
also calculate the area enclosed within the lines using the methodof the right angles. […] Having assigned boundaries to its space,
we restore the place’s own truth. […] For any smallest part of the
land which is to be in the power of the measurer must be boundwith the method of the right angles.
45
The mathematical act of measurement allows an accurate division, and
makes the business of apportioning and administering land easier. At the
same time, mathematics has connotations of stability and certainty. In fact,
geometrization is elevated by Frontinus to the status of a restoration of theessence itself of the land – the truth of the place consists in its being trans-
formed into a geometrical object. The land-surveyor, like the water supply
supervisor, is an agent of order over and against chaos and confusion atboth a material and a moral level.
The Corpus also includes Categories of Fields by Hyginus Gromaticus
(second century
AD), a general treatise aimed at the professional which
includes a history of land-surveying, an explanation of the surveyor’s legal
competences, as distinct from the magistrate’s, and instructions on how to
decipher inscribed boundary stones. Hyginus points out very often that
surveyors make mistakes because of their lack of experience or knowledge.
One of the hardest tasks is laying out the decumanus and cardo properly.
Many people apparently were unable to find the true cardinal points; a
good surveyor, however, should know some astronomy, and Hyginus, after
citing Vergil, Lucan, and Archimedes, who ‘wrote how much sand the worldcould contain if it was filled up’, provides an orientation procedure that
uses a gnomon and a simple geometrical construction.
46
Like Vitruvius’ architect, the land-surveyor in the Regulation must have
an extensive field of competence, to include law, mathematics and astronomy.

171GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
It is the possession of this knowledge and of the related skills that distinguish
the surveyor from other state officers, and the good surveyor from the bad
one, all the more strongly in a sensitive field where mistakes could havepolitical consequences.
Even keener on the necessity for the surveyor to have mathematical
knowledge is Marcus Junius Nipsus (second century
AD), author of Measure-
ment of a River , Replacing a Boundary and Measurement of an Area . This
latter work contains definitions of measures and angles, followed by a number
of problems, mostly about triangles, including ‘given an odd number, forma right-angled triangle’, ‘given an even number, form a right-angled triangle’,
and
to measure the area of all triangles via one method, say, right-
angled, acute-angled and obtuse-angled. We would find it out in
this way. I unite into one the three numbers of any of the threetriangles. That is, the right-angled one, whose numbers are given,
the cathetus 6 feet, the base 8 feet, the hypotenuse 10 feet, I unite
these three numbers into one, and they make 24. Of this I alwaystake the half. It makes 12. This I set aside, and from this number,
that is, from 12, I subtract the <other> individual numbers. I sub-
tract 6: I put the rest under 12. Analogously I subtract the base, 8feet, from 12: I put the rest under 6. I subtract the hypotenuse, 10
feet, from 12, the rest is two, I put it under four. Then I multiply
6 by 4. It makes 24. This I multiply by 2, and it makes 48. This Imultiply by 12. It makes 576. Of this I take the root, and it makes
24. It will be the area. And the area of the other triangles will be
calculated in the same way.
47
The other mathematical problems in Nipsus’ treatise are in the same
vein: they read like a set of instructions, provide no proof and have as objectsspecific triangles, whose sides have a definite measure. A connection with
Hero of Alexandria has been suggested, and is not unlikely; a similarity
with the procedural style of other mathematicians from this period is evident.
A further version of mathematics for the surveyor is found in Balbus’
book. He starts by telling his addressee Celsus of how he had been requiredto assist the emperor T rajan on a military expedition:
After we took the first step on hostile territory, immediately, Celsus,
the earthworks of our Caesar began to demand of me the calcula-
tion of measurements. When a pre-arranged length of marching
had been completed, two parallel straight lines had to be producedat which a huge defensive structure of palisaded earthworks would

172GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
rise up, for the protection of communications. By your invention,
when a part of the earthworks was cut back to the line of sight, the
use of the surveying instrument extended these lines. In regard toa survey of bridges, we were able to state the width of rivers from
the bank close at hand, even if the enemy wished to harass us.
Then it was that calculation, worthy of a god’s respect, showed ushow to know the heights of mountains that had to be captured. It
was calculation that I began to revere more fervently, as if she were
worshipped in all the temples, after experiencing and taking partin these great enterprises.
48
Celsus was evidently himself a mensor – Balbus refers to ‘our profession’,
and his tone is rather deferential, as if Celsus was the more senior or more
expert of the two. The utility of surveying is displayed right from the start:
Balbus comes on the scene as the busy servant of the state and of ‘our mostholy emperor’, his expertise contributing to Rome’s military supremacy. He
takes the opportunity to promote the profession itself, because one of the
inventions he uses is by Celsus, and it is their relationship, their sharedknowledge, that enables him to deploy it. He also states the importance of
calculation, indeed, his religious respect for it – in his view, it should become
the object of universal worship, or, in other words, what surveyors do shouldgain everybody’s recognition and respect.
Balbus’ treatise was meant to be the first in a series, and it consists entirely
of definitions; it presents a taxonomy of basic geometrical concepts, startingwith a definition of measure and a list of units of measurement, and following
with point and line, down to three-dimensional figures. While some of the
definitions correspond to those in Euclid’s Elements (‘Point ( signum ) is that
of which there is no part’, ‘Line is length without breadth, and the limits of
a line are points’),
49 Balbus’ account is geared to his chosen public, i.e. other
surveyors. Thus we find:
The kinds of lines are three, straight, circular and curved. A straight
line is that which lies equally with respect to its straight points;
circular, <that> whose path will be different from the arrangement
of its points. A curved line is multishaped, like fields or ridges orrivers; the border of unsurveyed ( arcifiniorum ) lands is delimited
by such lines, and similarly many things, which by nature are shaped
by an irregular line.
50
Balbus often explains a concept with examples (fields, boundaries,
elements of the landscape) or vocabulary ( arcifinius ) drawn from the
surveyor’s experience. Also, he often refers to the Greek equivalents of the

173GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
terms he is defining, transliterating them into Latin. In sum, his catalogue
of definitions seems to have been compiled in order to enable experienced
surveyors better to ground their knowledge. What was previously simplythe line typical of an unsurveyed piece of land can now be subsumed into
the geometrical domain as one of the three main lines, the ‘curved’ one.
That some key terms are introduced in their Greek version may be meantto facilitate the understanding of Greek geometry in the original. In other
words, Balbus’ treatise is not about teaching the surveyor what to do in the
field. His intended reader probably knew that already, and in any case, as ithas been pointed out, one does not need to know the mathematical definition
of right angle to be able to draw one. The treatise was rather about enhancing
the surveyor’s knowledge by enabling him to explain, order and classifyconcepts that he was already using. Another passage from the beginning of
this text is quite revealing:
It would seem shameful to me if, having been asked how many
kinds of angle are there, I answered ‘many’: therefore I have
examined the types, qualities, characters, modes and numbers ofthe things that are relevant to our profession, as much as my occupa-
tions allowed.
51
Many surveyors practised their job in contexts where their authority
could be disputed, or at least set against other sources of authority; knowing
one’s right angles may have boosted the claims of any exponent of ‘ourprofession’. In sum, we could say that the surveyor envisaged by Balbus,
and typified by himself and Celsus, is again a gentleman technician, who
does his duty when he is required to, but can also appreciate returning tohis studium and his otium , like any well-born, well-educated Roman of the
time. His expertise is demonstrated not just by the fact that his devices
work, but also by his being well schooled in the foundations of his art,including to some extent the Greek tradition of geometry.
We find similar issues in fields other than land-surveying: for instance,
in the Astronomy , addressed by the author, (another) Hyginus (perhaps
second century
AD) to a Marcus Fabius. Hyginus praises Fabius’ learning
and discernment, and contrasts the education and judgement of the periti
(experts) with those of the non-experts. The account he gives of astronomy
is quite thorough, starting with basic definitions of sphere, pole and so on,
and ending with a full star-catalogue, which takes up information fromprevious astronomers, including Eratosthenes, and expands the mythological
stories behind many of the constellations’ names. The star-catalogue comes
complete with a calculation of how many stars there are in each constellation.For instance:

174GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
The Snake has two stars at the top of the head, under the head
four all in the same place; two towards the left hand of Ophiuchus
[a neighbouring constellation], but that which is closer to its bodyis the brighter; and on the back of the Snake towards the junction
of the body five, and in the first coil of the tail four, in the second
coil towards the head it has six stars. So all together it is twenty-three stars.
52
Astronomical expertise implies that one is able to recognize the stars, even
those which are not very bright, assign them to the right constellation, and
express their precise number. Apart from Hyginus, we have extensive literary
evidence of the astronomical interests of many illustrious Romans: Cicerotranslated Aratus into Latin, and Germanicus wrote a commentary on the
same text. Manilius chose Augustus as dedicatee of his Astronomy , which, as
well as detailed arithmetical and geometrical procedures, contains an explicitparallel between the hierarchy of the stars and that of human society.
53
Less lofty matters are treated by the jurist Lucius Volusius Maecianus in
a short book he dedicated to Marcus Aurelius, with the aim to inform himon the divisions of the basic Roman currency unit – the as. After a brief
introduction where he asserts the necessity for the emperor to acquaint
himself with the topic because of its utility for inheritances ‘and many otherthings’, Maecianus goes into the details of each subdivision of the main
money denomination, what part it is of the larger unit and what written
sign denotes it (presumably in order to recognize it in accounts or on anabacus). Things are made more complicated by the fact that an as, for
example, can be divided in two different ways: into equal parts, like half,
thirds, and so on (up to twelfth parts, or unciae ) or into unequal parts, like
five-twelfths, seven-twelfths and so on (up to eleven-twelfths). Maecianus’
premise is that the emperor is not completely at home in the complicated
world of money, and that some dexterity is required fully to commanddivision into parts, whose nature, as he says more than once, is infinite.
The skills in question, one would think, were normally possessed by money
lenders, secretaries and/or their slaves. Indeed, Maecianus says that he has
got some of his information from ‘reckoners’ ( ratiocinatores ).
54
The expectation that a powerful political figure ought to know his twelfth
parts of an as may sound far-fetched. On the other hand, Quintilian ( c. AD
35–90) insists on the necessity for the good orator to have rather extensive
mathematical knowledge, both for its persuasive style of argumentation andfor its contents, and checking accounts had been mentioned by Cato the
Elder (234–149
BC) as one of the duties of landowners, to be done together
and often by overseer and master.55 Arithmetic was used in agricultural
treatises to calculate the workforce necessary to run a certain establishment,

175GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
or the time required to produce a certain yield. Here is a passage by Columella
(mid-first century AD):
This calculation shows us that one yoke of oxen can meet the
requirements of one hundred and twenty-five modii of wheat and
the same of legumes, so that the autumn sowing may total twohundred and fifty modii , and even after that seventy-five modii of
three-months crops may still be sown. The proof of this is as follows:
Seeds that are sown at the fourth ploughing require, for twenty-five iugera , one hundred and fifteen days’ labour of the ploughmen;
for such a plot of ground, however hard, is broken in fifty days, re-
ploughed in twenty-five, ploughed a third time and then sown inforty days. […] Forty-five days also are allowed for rainy weather
and holidays, on which no ploughing is done; likewise thirty days
after the sowing is finished, in which there is a period of rest. Thusthe total amounts to eight months and ten days.
56
Columella provides extensive information about land measurement too:
he starts with a detailed description of units of measure, the iugerum and its
many subdivisions, in a passage which reads rather like Maecianus’ account
of money. He then considers how to obtain the area of pieces of land ofdifferent shapes: square, rectangular, wedge-like, triangular, circular, semi-
circular, curved but less than a semicircle, hexagonal, in increasing order of
complexity. Finally, he explains how to calculate the number of trees a fieldof a given size can contain, given the distance between the plants. And yet,
he says, land-surveying has been included in his account only out of friend-
ship towards the addressee, Silvinus – Columella had already turned downa similar request made by another friend:
I replied that this was the duty not of a farmer but of a surveyor,
especially as even architects, who must necessarily be acquainted with
the methods of measurement, do not deign to reckon the dimen-
sions of buildings which they have themselves planned, but think
that there is a function which befits their profession and another
function which belongs to those who measure structures after theyhave been built and reckon up the cost of the finished work by
applying a method of calculation. […] But […] I will comply with
your wish, [Silvinus,] on condition that you harbour no doubt thatthis is really the business of geometers rather than of countrymen.
57
Notice the dialectic between knowing how to do something and actuallyhaving to do that thing because that is your job – the architect knows how

176GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
to measure the dimensions of buildings, but does not have to carry out that
operation himself; the landowner does not have to survey his land, but it is
good for him to know how surveying works. There is a definite idea thatknowledge of these matters is valuable and entails a form of power, even
when it is disassociated from actual application.
The ability to check accounts and land-surveys could be required of rich
Roman citizens not just to keep their own house in order. Here is an official
report of what Pliny the Younger set out to do on his arrival to Bythinia as
imperial legate:
[…] I did not reach Bythinia until 17 September. […] I am now
examining the finances of the town of Prusa, expenditure, revenues,and sums owing, and finding the inspection increasingly necessary
the more I look into their accounts […] I am writing this letter, sir,
immediately after my arrival here.
I entered my province, sir, on 17 September, and found there
the spirit of obedience and loyalty which is your just tribute from
mankind. Will you consider, sir, whether you think it necessary tosend out a land surveyor? Substantial sums of money could, I think,
be recovered from contractors of public works if we had dependable
surveys made. I am convinced of this by the accounts of Prusa,which I am handling at the moment.
58
Pliny’s letters to T rajan are punctuated by requests for architects to sort
out buildings which were falling apart, engineers for the same purpose,
and, as in this case, mensores . All these mathematical experts appear to have
been significant components of the administration of the provinces.
The Elder Pliny, Pliny the Younger’s uncle, wrote a monumental Natural
History addressed to the emperor, ‘his’ Vespasian. A self-declared collector
of information, Pliny the Elder said that he had concentrated 2,000 worksinto the 36 books of his History . Indeed, the whole of the first book is a
summary of contents, divided into topics covered and sources used, those
latter in their turn distinguished between Roman and other. The summary
is organized as an account, a quantified report of the extent of Pliny’s
knowledge: at the end of the summary of each book, we are given the totalof facts, or famous rivers, or towns and peoples, or types of plants, that have
been discussed. Yet, for other aspects quantification and calculation should
not be pushed too far:
Posidonius holds that mists and winds and clouds reach to a height
of not less than forty stades from the earth […] The majority ofwriters, however, have stated that the clouds rise to a height of

177GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
ninety. These <figures> are really unascertained and impossible to
disentangle, but it is proper to put them forward because they
have been put forward already, although they are matters in whichthe method of geometrical gathering, which is never fallacious, is
the only one that it is possible not to reject, if anybody likes to
pursue these things further, not in order to establish a measure(for to want that is a sign of an almost insane whiling away one’s
time) but only a conjectural extimate. […] And when they have
dared to predict the distances of the sun from the earth they dothe same with the sky […] with the consequence that they have at
their finger’s ends the measure of the world itself. […] This
calculation is a most shameful business, because [the multiplication]produces <a number> which is even beyond reckoning.
59
While inveighing against the heaven-measurers, Pliny praises Hipparchus
for his catalogue of stars, and has absolutely no problem in reporting
Eratosthenes’ and Hipparchus’ measurements of the Earth. Indeed, the
History , whose introduction affects a casual attitude to knowledge and
learning, is interspersed with passionate defences of their importance, and
with lamentations on the ignorance of the time, when everybody is seeking
profit rather than wisdom or learning.60 Pliny also follows the practice,
exemplified by many ancient historians, of using accurate figures to bolster
some points. For example, in his section on metals he launches into a long
discussion of the evils brought about by gold, especially its role in the corrup-tion of old Roman customs, and adds:
It follows that there was only 2,000 pounds [in weight] at most
when Rome was taken [by the Gauls], in the year 364, although
the census showed there were already 152,573 free citizens. From
the same city 307 years later the gold that Caius Marius the youngerhad conveyed […] amounted to 14,000 in weight […] Sulla had
likewise […] carried in procession 15,000 in weight of gold.
61
The numbers emphasize the parallel growth of Rome’s power and of the
menacing mass of metal. Pliny continues with cautionary tales of the excessesdue to misuse of gold, in their turn punctuated by figures and accurate
expenditures – he calculates just how out of proportion the exploits of people
like Caligula or Nero were. He even comes to the conclusion that the veryexistence of numbers past a hundred thousand is a by-result of usury and of
the introduction of coined money.
62 Similar mathematical moralizing we
find in Seneca ( c. 4 BC–AD 65). He denied that mathematics was a part of
philosophy; in fact, its role consisted merely in measuring and counting the

178GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
natural phenomena studied by philosophy. He also commented that the
only study worth pursuing is that of wisdom, and that poetry, music,
mathematics, astronomy are of only limited value:
A geometer teaches me to measure my estate; but I should rather
be taught to measure how much is enough for a man to own. Heteaches me to do sums and put my fingers to the service of greed,
but I should prefer him to teach me that those calculations have
no importance […] What good is there for me in knowing how todivide a piece of land into shares, if I know not how to share it
with my brother? What good is there in adding together carefully
the feet in a iugerum and including even something that has escaped
the measuring rod, if I get upset by an arrogant neighbour who
encroaches on my land? […] O noble art! You can measure curved
things, you reduce any given shape to a square, you enunciate thedistances of the stars, there is nothing which falls outside your
measure: if you are so good at your art, measure a man’s soul, say
how big or how small it is. You know what a straight line is; whatgood is that to you if you do not know what a straight life is?
63
The reader can compare the pronouncements of some of the land-
surveyors with these statements, which appear to deny that geometry or
arithmetic can have any wider or higher significance. In conclusion, we can
say that, contrary to the belief that there is no such thing as Romanmathematics, the Romans did a lot of counting and measuring, and they
did a lot of thinking about counting and measuring. Some of them were
experts whose knowledge included mathematics and whose service to thestate depended on their mathematical knowledge; some of them used
mathematics, sometimes critically, to articulate views about the relation of
man to nature and the limits of human knowledge.
The other Greeks
Strabo, like Vitruvius a witness of the transition from Roman republic toempire, also opens his book by saying that the study of his subject, geography,requires extensive knowledge: philosophy, natural history, and especially
astronomy and mathematics. He states that one need not know those last
two disciplines thoroughly, but should be acquainted at least with somebasics: for instance, ‘what a straight line is, or a curve, or a circle, [or] the
difference between a spherical and a plane surface’.
64 Mathematics also helps
Strabo score points against other geography writers:

179GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
Eratosthenes is so simple that, although he is a mathematician, he
will not even hold fast to the solid opinion of Archimedes, who in
his Floating Bodies says that the surface of every liquid body at rest
and in equilibrium is spherical, the sphere having the same centre
as the earth. All those who have studied mathematics at all accept
in fact this opinion. […] And as testimonies for [his] ignorantopinion [Eratosthenes] produces architects, even though the mathe-
maticians have declared architecture a part of mathematics.
65
Eratosthenes’ crucial mistake was choosing the wrong supporter for his
opinion: not Archimedes, evidently the chief authority on the topic, but
architects. Appealing to the right source was by itself a sign of inclusionamong those ‘who have studied mathematics’, because a shared tradition of
results and canonical texts was part of what constituted them as a group.
Strabo uses mathematics to cast doubt on other geographers elsewhere: atone point he sums up the distances they provide and demonstrates that the
world thus obtained would be impossibly large, or shows that their data are
incompatible with the relative positions and even climates of certaincountries. Yet, he is keen to remark that criticisms based on mathematics
are only fair when the theory to be criticized itself uses mathematical accuracy
to gain credibility.
66
Space in Strabo is often geometrized and enclosed within lines parallel
or perpendicular to each other, to form a map. Countries are often assimilated
to geometrical shapes. On several occasions, however, he makes it clear thatgeography is not a completely mathematical discipline. Knowledge can be
gleaned from the experience of sailors, in preference or in alternative to
geometrical reasonings. Or again, unless a country is well-defined by riversor mountains, ‘in lieu of a geometrical definition, a simple and roughly
outlined definition is sufficient’.
67 On the other hand, Strabo launches in a
full-length justification of the importance of mathematics in order even todefine who the audience for his book is:
The sailor on the open sea, or the man who travels through a level
country, is guided by certain popular notions, and these notions
impel not only the uneducated man but the man of affairs as wellto act in the self-same way, because he is unfamiliar with the
heavenly bodies and ignorant of the varying aspects of things with
reference to them. For he sees the sun rise, pass the meridian, andset, but how it comes about he does not consider; for, indeed,
such knowledge is not useful to him with reference to the task
before him, any more than it is useful for him to know whether or

180GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
not his body stands parallel to that of his neighbour. But perhaps
he does consider these matters, and yet holds opinions opposed to
the principles of mathematics – just as the natives of any givenplace do; for a man’s place occasions such blunders. But the geo-
grapher does not write for the native of any particular place, nor
yet does he write for the man of affairs of the kind who has paidno attention to the mathematical sciences properly so-called; nor,
to be sure, does he write for the harvest-hand or the ditch-digger,
but for the man who can be persuaded that the earth as a whole issuch as the mathematicians represent it to be, and also all that
relates to such an hypothesis. And the geographer urges upon his
students that they first master those principles and then considerthe subsequent problems; for, he declares, he will speak only of
the results which follow from those principles; and hence his
students will the more unerringly make the application of histeachings if they listen as mathematicians; but he refuses to teach
geography to persons not thus qualified.
68
In this passage, exposure to mathematical knowledge and the ability to
be persuaded by its argumentations institute a hierarchy: harvest-hands,
ditch-diggers, the uneducated at the bottom; the man of affairs perhaps ina sort of limbo – he only cares about what is useful and does not pay enough
attention, but may be redeemable; finally, the students who can listen as
mathematicians at the top, as Strabo’s audience of choice. In my view, such
pronouncements, especially since we do not find them actually applied in
the Geography , whose position on mathematics is much more ambiguous,
are an example of how claims about mathematics are also about somethingelse. The remarks about not wanting to write for the natives of any given
place or for the man of affairs signal Strabo’s ambition that his account be
universal rather than particular; disengaged from the mercantile outlookof, for instance, the tales of commercial travellers; made more authoritative
by being made exclusive: one has to be qualified in order fully to understand
it.
69 Mathematics was associated with all of these characteristics, so that
invoking it, even without actually deploying it, helped Strabo articulate
some of the features of his chosen way of doing geography.
Later than Strabo and thus a witness of the consolidation of Roman
rule, Philo of Alexandria (late first century BC/early first century AD) was a
Greek-speaking Jew, and most of his works revolve around interpreting theBible, to which task he applied philosophy. The story of creation, for instance,
is a Platonizing account of order emerging from chaos and, since ‘order
involves number’, it is punctuated by numerical symbolism. Philo drawsout the properties and wider significance of all the numbers involved: six

181GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
days of activity for God, the heavens arranged on the fourth day, seven days
to complete the creation. Thus, six is the first perfect number, being both
the sum and the product of one, two and three; four is the base and sourceof ten (1+2+3+4=10) and of musical concords; seven is so rich of meanings
that Philo himself does not know where to start.
70 As for the role of mathe-
matics itself, Philo assigns geometry, along with grammar, as a maid tophilosophy, since it needs philosophy to express definitions for its subject-
matter. The study of geometry is deemed useful to learn about equality,
symmetry, proportion and consequently justice;
71 indeed, equality and
proportion are manifested everywhere in the world and are a sign of divine
presence. In a sense, then, God is the best mathematician of all: not only
does his creation exhibit symmetry, but he is the only one who can knowthe world, mathematically speaking, in a perfect way. Absolute accuracy in
measuring or counting are only possible to God: pace the land-surveyors,
He is the only one who can carry out a perfect division.
72
Reading meanings in numbers, or numerology, was also a prominent
feature in the Arithmetical Theology , now lost, by Nicomachus of Gerasa.
He has variously been called a neo-Pythagorean or a neo-Platonist mathe-matician, and indeed he does mention Pythagoras and Plato often and
favourably. Those categorizations, however, have had the unfortunate effect
of pushing Nicomachus to the margins of modern histories of ancient mathe-matics, which tend to dismiss material associated with the neo-Pythagoreans
as wacky mysticism. This is unfortunate, because Nicomachus was probably
one of the most popular mathematicians of antiquity. The type ofmathematics he did is widely represented in philosophical and literary works,
he is often mentioned, and his success is attested by later commentaries and
by an alleged translation into Latin, by Apuleius ( c.
AD 125–70), of his
Introduction to Arithmetic .73
Apart from the Introduction to Arithmetic , we have an Introduction to
Harmonics (an Introduction to Geometry and a Life of Pythagoras are lost).
The Arithmetic discusses topics such as odd and even numbers, their various
sub-species, including perfect numbers, division into factors, prime numbers,
multiples and parts, ratios, triangular, square and polygonal numbers, and
finally proportions and means. The argumentative style is discursive, with
examples employing specific numbers, but no proof in the axiomatico-deductive style. Persuasion seems to be brought about by mere showing,
and by enabling the reader to see for himself
74 that what is being said is
verified in actual instances. Nicomachus also provides tables, for instance ofmultiples, for quick reference. An example will clarify his way of proceeding:
We shall now investigate how we may have a method of discerning
whether numbers are prime and incomposite, or secondary and

182GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
composite […] Suppose there be given us two odd numbers and
some one sets the problem and directs us to determine whether
they are prime and incomposite relatively to each other orsecondary and composite, and if they are secondary and composite,
what number is their common measure. We must compare the
given numbers and subtract the smaller from the larger as manytimes as possible; then after this subtraction, subtract in turn from
the other as many times as possible; for this changing about and
subtraction from one and the other in turn will necessarily endeither in unity or in some one and the same number, which will
necessarily be odd. Now when the subtractions terminate in unity
they show that the numbers are prime and incomposite relativelyto each other; and when they end in some other number, odd in
quantity and twice produced, then say that they are secondary
and composite relatively to each other, and that their commonmeasure is that very number which twice appears. For example, if
the given numbers were 23 and 45, subtract 23 from 45, and 22
will be the remainder; subtracting this from 22 as many times aspossible you will end with unity. Hence they are prime and
incomposite to one another, and unity, which is the remainder, is
their common measure. But if one should propose other numbers,21 and 49, I subtract the smaller from the larger and 28 is the
remainder. Then again I subtract the same 21 from this, for it can
be done, and the remainder is 7. This I subtract in turn from 21and 14 remains; from which I subtract 7 again, for it is possible,
and 7 will remain. But it is not possible to subtract 7 from 7;
hence the termination of the process with a repeated 7 has beenbrought about, and you may declare the original numbers 21 and
49 secondary and composite relatively to each other, and 7 their
common measure in addition to the universal unit.
75
If we compare the passage with its equivalent in Euclid’s Elements (7.1),
many differences will emerge: Euclid’s proposition is enunciated in a way
which requires proof, i.e. in the form of a theorem, and in fact it provides a
proof, whereas Nicomachus just adds ‘necessarily’ a couple of times andsimply states what is going to be the case. On the other hand, Euclid deals
with two non-specified numbers, while Nicomachus appends two examples
with specific numbers, whose function is both persuasive and pedagogic,because they enable the reader to go through the operation. You (my reader)
will have started to notice that the presence of specific examples, exercises
for the reader, as it were, with or without a corresponding general proof, isan usual feature in the authors of this period.

183GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
The first thing the Introduction to Arithmetic does is also what many other
mathematical treatises did: justify its existence. Unlike Hero, who played the
card of usefulness and social relevance of mathematics, Nicomachus presentsit as an altogether nobler and more intimate enterprise, aimed at personal
happiness, which, he concedes, is ‘accomplished by philosophy alone and by
nothing else’.
76 Philosophy, on the other hand, means love of wisdom, and
wisdom looks for the truth in things, that is, it looks for what does not flow
and change, but stays always the same: magnitudes and multitudes, or,
formulated differently, size and quantity. In other words, real happiness canonly be achieved through the study of mathematics, and awareness of this
ultimate aim informs the whole book. As a premise to the demonstration
that some numerical relations originate from the relation of equality, which isprimary and seminal, Nicomachus states that he wants to present
very clearly and indisputably […] the fact that that which is fair
and limited, and which subjects itself to knowledge, is naturally
prior to the unlimited, incomprehensible, and ugly […] it is
reasonable that the rational part of the soul will be the agent whichputs in order the irrational part, and passion and appetite […] will
be regulated by the reasoning faculty as though by a kind of equality
and sameness. And from this equalizing process there will properlyresult for us the so-called ethical virtues, sobriety, courage,
gentleness, self-control, fortitude, and the like.
77
Acquiring wisdom amounts to recognizing that the universe and everythingwithin it are well-ordered and good. The many interweaving relationships
between numbers are a way, an excellent and straighforward way, for thehuman intellect to get to this essential truth.
Similarly enough, Ptolemy (mid- to late second century
AD) opened what
was to become his most famous work with the following praise ofmathematics:
[…] Aristotle divides theoretical philosophy […] into three primary
categories, physics, mathematics and theology. […] the first two
divisions of theoretical philosophy should rather be called guess-work than knowledge, theology because of its completely invisible
and ungraspable nature, physics because of the unstable and unclear
nature of matter; hence there is no hope that philosophers willever be agreed about them; […] only mathematics can provide
sure and unshakeable knowledge to those who practise it, provided
one approaches it rigorously. […] With regard to virtuous conductin practical actions and character, this science, above all things,

184GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
could make men see clearly; from the constancy, order, symmetry
and calm which are associated with the divine, it makes its followers
lovers of this divine beauty.78
The divine beauty in question is that of the universe, whose regular motionsPtolemy takes upon himself to describe geometrically by means of uniformcircular orbits. Understanding how the universe works holds the key to
grasping its order and harmony, and thus constitutes a philosophical
enterprise. As a deeply moral, indeed religious task, astronomy thus requiresvirtuous behaviour on the part of its practitioners. In concluding the Syntaxis ,
he reminds his addressee, Syrus, that his aim has been ‘scientific usefulness’,
not ‘ostentation’.
79 Part of this attitude translated into respect for the
astronomers of the past (he quotes Archimedes, Apollonius, Eratosthenes
and especially Hipparchus), and appreciation of what valid contributions
they have made. Ptolemy even shows willingness to stand corrected bysomeone else’s methods if they provide more accurate results than his.
80
As well as accuracy, Ptolemy aimed at rigour and certainty, which he
thought were attainable via a combination of geometrical and arithmeticalproofs. Overall, the Syntaxis , parts of which are mathematically very complex,
together with the Planetary Hypotheses , which gives further measurements
of the distances of the planets, can be said to provide a total mathematizationof the heavens. Ptolemy also turned his attention to the phenomena of
vision and hearing, again mathematized in the Optics and Harmonics , and
to the terrestrial world in the Geography , where each major city of the time
is listed with its coordinates, and maps are provided which aim geometrically
to represent the whole inhabited earth.
In terms of methodology, apart from deductive-style proofs, Ptolemy
sometimes accompanied the general proof of a proposition with calculations
on actual numbers, in order to show that the results matched.
81 He devoted
large parts of the Syntaxis to the construction of tables (which circulated
separately as Handy tables ), so as to simplify calculations for time-keeping
or chart-making. Like maps, tables were a kind of instrument – Ptolemy
also explains how to build a quadrant, an equinoctial ring and an astrolabe.82
Crucially, general proofs, arithmetical operations, tables, and observation,aided or not by instruments, were meant to work together. As an example,I report Ptolemy’s instructions on how to find the position of the sun at any
given time:
we take the time from epoch to the given moment (reckoned with
respect to the local time at Alexandria), and enter with it into the
table of mean motion. We add up the degrees corresponding to thevarious arguments, add to this the elongation, 265 and 15 parts,

185GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
subtract complete revolutions from the total, and count the result
from Gemini 5 and 30 parts rearwards through the signs. The point
we come to will be the mean position of the sun. Next we enter withthe same number, that is the distance from apogee to the sun’s mean
position, into the table of anomaly, and take the corresponding
amount in the third column. If the argument falls in the firstcolumn, that is if it is less than 180 parts, we subtract the [equation]
from the mean position; but if the argument falls in the second
column, i.e. is greater than 180 parts, we add it to the mean posi-tion. Thus we obtain the true or apparent [position of the] sun.
83
Or again, the theorems which justify the way in which the values of a tableof chords have been obtained, can in their turn be used to test and correct
the values in the said table, if in the future they turn out to be incorrect
because of mistakes in scribal transmission.
84
Ptolemy could be seen as an ideal counter-point to Sextus Empiricus.
Sextus was a sceptic: his surviving works are an outline of Pyrrhonian philo-
sophy and a long attack on any form of knowledge possible, from ethics,rhetoric and grammar to the mathematical sciences. T raditionally, mathe-
matics had been seen as producing certainty: who would doubt that two
plus two is four? Well, Sextus would: as far as arithmetic was concerned, forinstance, he aimed to undermine its foundations in order to pull down the
whole edifice. In Against the Arithmeticians , he relentlessly criticized basic
notions such as number, unit or addition and subtraction. He came to theconclusion that number is nothing.
85 A similar strategy was deployed against
the geometers: although they consider themselves safe because they use
hypotheses, Sextus says, the very use of hypotheses can be criticized in severalways. The definitions themselves of point (‘a sign without dimensions’) and
line (‘a flux of the point’ or ‘length without breadth’) are ill-founded – how
can a point, which is incorporeal and with no dimension, generate a line,which has dimension?
86 And even assuming for the sake of argument that
basic definitions are valid, geometrical operations such as bisecting an angle
are shown to be impossible.87
Who are the mathematicians targeted by Sextus? He mentions Eratos-
thenes, especially his definition of line as ‘flowing’ from a point, and, mostfrequently, Pythagoras and the Pythagoreans. With a few exceptions (e.g.
the definition of point), the mathematical notions contained in Sextus look
less like what we find in Euclid than in Nicomachus. Sextus also affordsinsights into the public aspect of mathematics, the roles it played in the
community. His doubts that arithmetical or geometrical certainties are not
as incontrovertible as the mathematicians would have them are not takento their fullest consequences:

186GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
Life judges everything on the basis of standards which are the
measures of number. Surely if we abolish number, the cubit will
be destroyed, which consists of two half-cubits and six palms andtwenty-four fingers, and the bushel will be destroyed and the talent
and the remaining standards; for all these, as composed of a
multitude <of things> are at once forms of number. Hence all theother things too are held together by number, loans, testimonies,
votes, contracts, times, periods. And in general it is infeasible to
find anything about life that does not participate in number.
88
A different strand of Graeco-Roman philosophy is represented by
Alcinous (second century AD), who wrote an introduction to Platonism
where mathematics figured as part of theoretical philosophy. He followed
Plato’s Republic in believing that mathematics, as well as being useful for
practical purposes, sharpens the intellect, hones and elevates the soul andgives accuracy. While reiterating that the mathematical sciences are subordi-
nate to philosophy, Alcinous also says that they are a ‘prelude’ to contempla-
tion, compares them to gymnastics and even deems music, arithmetic,astronomy and geometry ‘initiation rites’ and ‘preliminary purifications’ of
our spirit, before greater studies are begun.
89 A fuller idea of what mathe-
matics Alcinous may have had in mind is gleaned from another, probablycontemporary, author for whom Plato was also a major reference point:
Theon of Smyrna.
In his Account of Mathematics Useful to Reading Plato , Theon has a detailed
image of the mathematical disciplines (five in his case, he adds stereometry)
as the stages of an initiation rite which culminates with philosophy. While
following the Republic and the Epinomis quite closely, he quotes from
numerous other authors, including the Pythagoreans, Eratosthenes and
(twice, briefly) Archimedes. One of the images that recurs most often in his
only partially preserved treatise is that of a correspondence between variouslevels of reality: man, the cosmos, the city, the physical elements. Conse-
quently, some mathematical notions, such as the tetrad (a group of four, a
four-some), are ubiquitous: the four elements are a tetrad, as are the geometric
solids that correspond to them (here Theon follows Plato’s Timaeus ); the
‘common things’ (man, house, city quarter, city) are also four. Basic arithme-tical terms are defined in a way similar to Nicomachus; methods are provided
to find various types of numbers (for instance, square or perfect number)
and to generate proportions or means; a general statement is often accompa-nied by a specific example by way of demonstration and both ‘numerical’
and ‘geometrical’ procedures are occasionally given to find the same result.
90
The text as we have it ends with a section on astronomy, including a detailed
discussion of eccentrics and epicycles.

187GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
And, to conclude, Galen. He was one of the most famous doctors of his
times, physician to the emperor Marcus Aurelius and author of a huge
number of books. Galen believed that a good doctor should also be aphilosopher,
91 and often paired medicine and philosophy when talking about
supreme knowledge in general. Both dealt with the two inseparable aspects
of a human being: body and soul; also, medicine was a particularly complexform of knowledge, which a philosophical mind would understand better
than a non-philosophical one. Formulating a diagnosis or predicting the
outcome of an illness implied reading from signs, making causal connections,distinguishing bad inferences from good ones; communicating one’s response
convincingly and often in competition with other doctors’ responses
demanded that one’s medical skills be supported by rhetorical capacities, bythe ability to demonstrate that what one was stating was right. The well-
rounded education provided by philosophy thus served an important purpose
in the life of a doctor. The problem with philosophy, however, was thatnobody seemed to agree on anything, and different schools bickered endless-
ly; the medical world was equally divided into sects, whose exponents debated
about causes of diseases, appropriate cures and just about anything else.Looking for some certainty and (like Ptolemy and like many philosophers
of this period) for a criterion to distinguish truth from falsity, Galen came
to admire the rigour of mathematical proofs, and the consensus they engen-dered among geometers, arithmeticians and astronomers. He was impressed
by the fact that nobody for instance would doubt the results contained in
Euclid’s Elements or in his Phenomena .
92 In fact, mathematics, although hard
to follow for the general public, seemed to achieve universal persuasion
among not only its own practitioners, but also philosophers and rhetoricians.
On top of its compelling form of argumentation, and the positive
consequences this had in the establishment of shared belief, mathematics
deserved recognition because of its concrete workings in the world. Galen
never lost sight of the fact that the people engaged in mathematical practices(calculators, geometers, architects, astronomers, musicians, gnomon-makers)
produced something: predictions of eclipses, buildings, instruments like
sundials and waterclocks. He brought out the full implications of this in
the following passage:
Imagine that a city is being built, and its prospective inhabitants
wish to know, not roughly but with precision, on an everyday
basis, how much time had passed, and how much is left beforesunset. According to the method of analysis, this problem must
be referred to the primary criterion, if one is to solve it in the
manner that we learnt in our study of the theory of gnomons;then, one must go down the same path in the opposite direction

188GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
in order to do the synthesis […] When we have in this way found
the path which is to be followed in all cases, and once we have
realized that this kind of measurement of periods of time withinthe day must be carried out by means of geometric lines, we must
then find the materials which will receive the imprint of such lines
and of this gnomon. […] If you have no desire to find out thismethod, my friend – what can one say? You have obviously failed
to recognize your own conceit, and the fact that one who is ignorant
of these problems will never discover anything in the whole courseof a year, indeed, in the whole course of a life. For they were not
discovered in the lifetime of a single man. Geometrical theory was
there previously, and was first used to discover those theoremswhich are known as ‘elements’; once they were discovered, the
men who came later added to these theorems that most wonderful
science to which I have attributed the name ‘analytical’, and gavethemselves and anyone else who was interested a most thorough
training in it. And they have yet to produce a more wonderful
product of their ingenuity than those of the sundial and the water-clock.
93
For Galen, mathematical truth is demonstrated both by its products and
by its proofs, and its validity is guaranteed by the role it has in the community,
by shared assent and collective persuasion. Assent in mathematical proofs is
generated by the experience itself of going through the demonstration or oflearning a certain method to solve geometrical problems – one can see that
it works. Analogously with the embodied mathematics of sundials, water-
clocks, predictions of eclipses or architectural calculations: one can see thatthey work, they too are proofs of the incontrovertible truths of mathematics,
and a proof which is often out there in the street, under everybody’s eyes.
94
Thus, measuring, counting, the most basic mathematical activities are identi-
fied as fundamental elements of humankind’s shared knowledge of the world.
Notes
1 Pliny Sr., Natural History 2.3–4, Loeb translation with modifications.
2 Anonymous, On Plato’s Theaetetus 29.42–31.28 ( P . Berol. 9782, probably from Hermopolis
Magna, dated to the second century AD), my translation.
3P. O x y . 1.29 (redated in Fowler (1999), 211 to the late first/early second century AD) has
enunciation and diagram of El. 2.5, in a quickly written but competent hand on bad
papyrus; P . Fayum 9, latter half of second century AD, containing El. 1.39 and 41, looks
like a professional scribe’s copy but is very different from the text of Euclid we have (see
chapter 4); P. B e r o l . 17469, second century AD, has figures and enunciations of El. 1.8, 9
and 10, the figures drawn with a ruler, the text carefully written. All this in Fowler (1999),
209 ff.

189GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
4P . Vindob. 19996, in Gerstinger and Vogel (1932), exercise 25, my translation. Cf. also
Fowler (1999), 253 ff.
5P . Mich . 3.145.3.6.5–8 (second century AD), tr. J.G. Winter, Ann Arbor 1936, with
modifications.
6P . Mich . 3.144 (early second century AD).
7 See Jones (1997a), (1997b) and (1998).
8P . Lond . 130 ( AD 81), translation in Neugebauer and van Hoesen (1959), 23–4.
9 See e.g. Cicero, Against Verres 2.2.69.170 ff., Letters to Atticus 5.21 and cf. Fallu (1973);
P . Amh. 77 ( AD 139), in SelPap ; P . T ebt. 315 (second century AD), in SelPap ; Fronto,
Letters to the Emperor 5.
10 Cf. the charter of Tarentum, CIL 12.590 (bronze tablet, between 88 and 62 BC); Urso
(Osuna) in Spain, 44 BC, CIL 12.594; and Málaga, AD 81–4, all in Lewis and Reinhold
(1990), 1.414–5, 1.424, and 2.236, respectively.
11IG 5.1 numbers 1432 and 1433, translation in Levick (1985), number 70, with
modifications.
12CIL 13.6247 (near Worms in Germany, not dated, Lupulius Lupercus); CIL 5.3384 (P .
Caecilius Epaphroditus, Verona, not dated); CIL 14.472 (probably AD 144, the child was
a house-born slave), respectively. Cf. Martial, Epigrams 10.62, who mentions a calculator
with numerous pupils.
13 Purcell (1983).
14 E.g. the second-century Italian alimenta inscriptions and the tariff list from Coptus ( AD
90) in Lewis and Reinhold (1990), 255–9 and 66, respectively, or the bilingual tax and
duties inscription (in Greek and Aramaic) from Palmyra ( AD 137) in Smallwood (1966),
number 458. On the alimenta cf. Andreau (1999), 119 with more bibliography.
15CIL 6.10048 (Rome, AD 146), translation in Lewis and Reinhold (1990), 146–7.
16The achievements of the divine Augustus , translation in Chisholm and Ferguson (1981),
number A1, with modifications. See also Lewis and Reinhold (1990), 170, 260, 264–5;
Nicolet (1988); Andreau (1999), 49.
17 Cf. Buchner (1982) and contra Schütz (1990).
18 Domitian’s interest in mathematics and its applications is attested by a treatise On specific
gravities (only extant in Arabic translation) addressed to him. The author, Menelaus, also
wrote a Sphaerics (again only extant in Arabic translation) cited by Ptolemy in his Syntaxis .
In the Gravities , Menelaus mentions an earlier treatise on centres of gravity addressed to
the emperor Germanicus, who in his turn wrote a (still extant) commentary on Aratus’
Phenomena .
19 Pliny Sr., Natural History 36.72 f., Loeb translation with modifications.
20 Gibbs (1976), 17.
21 See Della Corte (1954). I thank Henry Hurst for his help on this point.22 See Dilke (1971), 66 ff.; Panerai (1984), 116.
23 Dilke (1971), 73.
24 Fragments of similar maps, albeit for different purposes, have been found in Rome and
Spain: see Chouquer and Favory (1992).
25 See also the examples mentioned in Hinrichs (1974); Smallwood (1966), numbers 434,
441, 446, 454, 465; Levick (1985), numbers 50, 51, 57; Sherk (1988), numbers 87, 91,
96; O. Bodl . 2.1847 in Fowler (1999), 231 ff.
26ILS 5947a, from Lamia, in Smallwood (1966) number 447, my translation.
27CIL 8.2728= ILS 5795 (
AD 157, from Tazoult), my translation. The top of the stele is
adorned with three figures of women, labelled ‘Patience’, ‘Virtue’ and ‘Hope’. I take itthat the beginning of the inscription (which is fragmentary) is a letter copied for infor-
mation by Datus.

190GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
28 Frontinus, On land-surveying 18–19.
29 Cf. Panerai (1984).
30 Vitruvius, On Architecture 1 Preface 2. An identification with Mamurra, praefectus fabrum
(officer of engineers) under Julius Caesar, has been proposed, see Purcell (1983), 156,who also argues for his being an apparitor .
31 Vitruvius, On Architecture 1.2.2; 1.6.6 ff.; 4.3.3; 5.5.1; 9.7.1 ff.; 10.11.1–2, respectively.
32 Vitruvius, ibid. 1.1.3–4, 17, Loeb translation with modifications.
33 Vitruvius, ibid. 9 Introduction, quotation at 6–8, Loeb translation with modifications.
34 Hero, Definitions 2, my translation.
35 Hero, ibid. 120, my translation.
36 Hero, Belopoiika (Construction of Catapults) 112.
37 Hero, ibid. 114, translation in Marsden (1971) with modifications.
38 The duplication of the cube at Construction of Catapults 115–18; the same solution in
Mechanics 1.11.
39 Hero, Mechanics 1.24 ff.; Automata 1.6-7; Pneumatica 1.1, respectively. Cf. also the
explanation of the five simple powers by means of properties of the circle, Mechanics 2.7.
40 Hero, Dioptra 26, my translation.
41 Hero, Metrica 1. Preface.
42 The whole proposition at Hero, Metrica 1.8, my translation. The same proposition also
in Dioptra 30.
43 See DeLaine (1996).
44 Frontinus, On the Aqueducts of the City of Rome 34.
45 Frontinus, On Sand-surveying 15–16; text in Hinrichs (1992), my translation. See Cuomo
(2000b).
46 Similar observations on the necessity of geometry and astronomy for town-planners in
Strabo, Geography 1.1.13, 2.5.1.
47 M. Junius Nipsus, Measurement of an Area 300.11–301.5, my translation. For an argument
that the Roman land-surveyors knew Hero’s work, see Guillaumin (1992).
48 Balbus, Explanation and Account of All Measures 92.7 ff., translation in Sherk (1988),
number 113, with modifications. Dilke (1971), 42, thinks that the emperor in questionis Domitian.
49 Balbus, Explanation and Account of all Measures 97.15, 98.15, respectively.
50 Balbus, ibid. 99.3–10, my translation.
51 Balbus, ibid. 93.11–15, my translation. See also Athenaeus Mechanicus, On Machines
15.2–4, who comments that being able to explain clearly how a machine works earnsmore praise even than building it.
52 Hyginus, On Astronomy 3.87, my translation.
53 Manilius, Astronomica 5.734 ff.
54 L. Volusius Meacianus, Distribution 65.
55 Quintilian,
Handbook of Oratory 1.10.3, 1.10.34–49; Cato, On Agriculture 2.2, 2.5, 5.4.
Cf. also Cicero, On the Orator 1.128, 1.158.
56 Columella, On Agriculture 2.12.7–9. Similar passages in Cato, On Agriculture 10 ff. and
Varro, On Agriculture 1.18.
57 Columella, On Agriculture 5.1.2–4, Loeb translation with modifications.
58 Pliny the Younger, Letters 10.17a–17b ( c. AD 110). Pliny complains about his official
duties (‘I sit on the bench, sign petitions, produce accounts, and write innumerable –
quite unliterary – letters.’) at ibid. 1.10.9.
59 Pliny the Elder, Natural History 2.85–8, Loeb translation with modifications. On similar
issues see also Lucian, Icaromenippus 6, with a probable reference to Archimedes’ Sand-
Reckoner .

191GRAECO-ROMAN MATHEMATICS: THE EVIDENCE
60 Pliny the Elder, Natural History 2.95; 2.247; 14.4.
61 Pliny the Elder, ibid. 33.15-16, Loeb translation with modifications.
62 Pliny the Elder, ibid. 33.133.
63 Seneca, Letters to Lucilius 88.10–13, translation based on Loeb and on C.D.N. Costa,
Aris and Phillips 1988.
64 Strabo, Geography 1.1.21.
65 Strabo, ibid. 1.3.11, Loeb translation with modifications.
66 Strabo, ibid. 2.1.23 ff. See also ibid. 2.4.3.
67 Strabo, ibid. 2.1.11; 2.1.30, respectively. A country which cannot be described in accurate
geometrical terms is e.g. Italy, ibid. 5.1.2.
68 Strabo, ibid. 2.5.1.
69 See Clarke (1999).70 Philo of Alexandria, On the Creation 13–14, 48, 89–128. Cf. also e.g. On the Migration
of Abraham 198–205.
71 See especially Philo, On Mating with the Preliminary Studies e.g. 16, 75, 146.
72 Philo, Who is the heir of divine things and on the division into equals and opposites 141 ff.
73 The information is due to a late source, Cassiodorus, see OCD on Apuleius.
74 Or herself – Nicomachus’ Harmonics is addressed to a woman, further indication of the
wide appeal of his work.
75 Nicomachus, Introduction to Arithmetic 1.13.10–13, tr. M.L. D’Ooge, University of
Michigan Press 1926.
76 Nicomachus, ibid. 1.2.3, translation as above. At ibid. 1.3.7, Nicomachus cites the passage
in Plato’s Republic 527 ff. where the useful side of mathematics is subordinated to its role
in the elevation of the soul.
77 Nicomachus, Introduction to Arithmetic 1.23.4–5, translation as above.
78 Ptolemy, Syntaxis 1.1, tr. G.J. T oomer, Duckworth 1984, with modifications.
79 Ptolemy, ibid. 13.11.
80 On this latter point, see Ptolemy, ibid. 4.9.
81 E.g. Ptolemy, ibid. 3.3; instructions are given to find the position of the moon and of the
five planets both geometrically and arithmetically in For the Use of the T ables 6–7, 9–10,
respectively. A similar practice in Vettius Valens, Anthology e.g. 1.8, 1.9, 1.15 (second
century
AD).
82 Ptolemy, ibid. 5.1; 5.12; 8.3, respectively. Pappus, Theon and Proclus also describe the
construction of astronomical instruments.
83 Ptolemy, ibid. 3.8, translation as above.
84 Ptolemy, Syntaxis 1.10.
85 Sextus Empiricus, Against the Arithmeticians 34; cf. also Against the Physicists 2.309.
86 Sextus Empiricus, Against the Geometers 22 ff., 29 ff.
87 Sextus Empiricus, ibid. 109 ff.
88 Sextus Empiricus, Against the Logicians 105–6, Loeb translation with modifications.
89 Alcinous, Handbook of Platonism 161–2.
90 Cf. e.g. Theon of Smyrna, Account of Mathematics Useful to Reading Plato 116.23–118.3.
91 The title of one of his works was The Best Doctor is also a Philosopher – at 53, he claims
that, according to Hippocrates, astronomy and consequently mathematics are central to
the study of medicine.
92 Galen, On his own books 11.39-41; On the Sentences of Hippocrates and Plato 112; 482–6;
Logical Institution 12; 16; On Prognosis 72–4; 120
93 Galen, The Affections and Errors of the Soul 80–7, translation in Singer (1997), with
modifications.
94 Cf. Galen, On his own books 11.41.

192GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
6
GRAECO-ROMAN
MATHEMATICS:
THE QUESTIONS
This chapter is about two well-known passages by Cicero:
With the Greeks geometry was regarded with the utmost respect,
and consequently none were held in greater honour than mathe-maticians, but we Romans have delimited the size of this art to
the practical purposes of measuring and calculating.
1
and Plutarch:
But all this [the Roman engines at the siege of Syracuse, 212 BC]
proved to be of no account for Archimedes and for Archimedes’
machines. To these he had by no means devoted himself as work
worthy of serious effort, but most of them were trifles of geometryat play, since earlier the ambitious king Hiero had persuaded
Archimedes to turn some of his art from the things of the mind to
those of the body […]. For the art of making instruments, now socelebrated and admired, was first originated by Eudoxus and
Archytas, who embellished geometry with subtlety, and gave
problems which had not been solved by reason-like or geometricalproof the support of sense-like and instrumental arguments. […]
But Plato was incensed at this, and inveighed against them as
corruptors and destroyers of the excellence of geometry, whichhad turned from incorporeal things of the mind to sensible things,
and was making use, moreover, of bodies which required much
vulgar handicraft. For this reason mechanics, having been banished,was distinguished from geometry, and being overlooked for a long
time by philosophy, has become one of the military arts.
2
Even though the two authors were not contemporaries, their statements
are connected. First of all, they both set out divides, between practical and

193GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
non-practical mathematics, and between Greece and Rome. Second, both
Cicero and Plutarch have been immensely influential, and many modern
views of ancient mathematics are still tinged by their opinions. Manyhistorians have bought wholesale into the view that Greek mathematics
‘typically’ was pure and speculative, whereas the Romans, that ‘characteristic-
ally’ pragmatic people, only cared for concrete applications. The fact thatone is talking about mathematics, an objective science, has clouded another
important matter: the images of ‘Greek’ and ‘Roman’ knowledge, like all
historical images of knowledge, were a construction, not a neutral reflectionof reality. That they have been so successful does not indicate so much that
they were true, as that they tapped into persistent beliefs about mathematics,
knowledge and the world in general.
We will explore some of the issues arising from those two passages: the
first section deals with Cicero, Plutarch, the myth of Archimedes and the
Greek/Roman divide. The second section takes its start from the pure/applieddivide and then moves on to look at the distinctions mathematicians
themselves were making: what boundaries did they draw, how did they see
themselves? In a sense, it is a contrast between a view from above and a viewfrom below, of (in principle) the same thing: mathematical practices in the
Graeco-Roman period.
The problem of Greek versus Roman mathematics
Like many upper-class Romans of his time, Cicero had travelled to Greece
as a young man, attended the lectures of famous philosophers and rheto-
ricians and started to collect a library. He espoused the widely-held opinion
that mathematics was one of the constituents of a liberal education, and,again like many Romans at the time, had an interest in astronomy (he trans-
lated Aratus’ Phenomena and wrote a treatise on astrology). As a magistrate,
and a land-owner engaged in business, he was numerate enough to under-stand accounts and detect financial frauds through them. In sum, Cicero
was not a mathematician, but he was not completely ignorant of mathematics
either; in line with his philosophical beliefs and everyday demands, he
recognized mathematics as an important form of knowledge. I do not think
that he had a specifically mathematical agenda – I think that what he saidabout mathematics was also about something else.
3
Let us look at the context of the short passage we have quoted at the
beginning. It is part of the Tusculan Disputations , written c. 45 BC at a moment
of forced rest in Cicero’s previously hectic public life. The work consists of
five dialogues, inspired by the Platonic model, set in the peaceful atmosphere
of a country villa and concerned with big themes such as death, pain andvirtue. Each dialogue is preceded by an introduction; the passage on

194GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
mathematics comes from the first of those, where Cicero recalls his own
major role in making Greek philosophy accessible to the Romans.4 Admitting
that the masters of the world had something to learn from another peoplewas not unproblematic, and Cicero qualifies the Romans’ debt to the Greeks
in several ways: Romans are better at war and household management, they
are doing really well at oratory and, although they have nothing as ancientas Homer and Hesiod, their literature is now flourishing. In fact, if some
areas of Roman culture are underdeveloped it is for lack of appreciation,
not of talent – Cicero laments that not enough honours are paid to poetry,painting, music and (our passage) geometry.
At a simple level, then, Cicero is matter-of-factly comparing two cultures:
the balance is rather even, because, if the Greeks have started earlier, theRomans have learnt faster, and sometimes improved upon their teachers.
The statement about Greek and Roman mathematics need not mean more
than what it says; Cicero’s picture of the situation is in fact pretty accurate,in that there is absolutely no evidence to contradict it on the Roman side. I
would be hard put to adduce a Latin equivalent of Euclid, Archimedes or
Apollonius. And yet, I do not think that is the main point – it is not thequestion that interests me here. Cicero’s sentence should not be taken as a
simple description of a state of things any more than any other general
statement about Greeks and Romans – in fact, no more than what Cicerosays in the same passage about philosophy and music, only to contradict it
later. After declaring that music is virtually an unknown art among the
Romans, Cicero claims that, thanks to Pythagorean influences, the earlyRomans practised singing on a large scale. Philosophy is first said to have
been successfully introduced to the Romans from the Greeks, yet later Cicero
laments that it is far from being praised and appreciated for its proper value.
5
As historical testimony, Cicero’s picture, while superficially not entirely false,
fails to take into account the fact that practical activities were not limited to
the Romans: a lot of mathematics done in Greek was about measuring andcounting. He ignores the views measurers and calculators, Roman or
otherwise, had of themselves and their activities – as we shall see in the next
section, they did not agree with his. In other words, the point of Cicero’s
picture is not that it tells the story as it was, because it does not, or not to
the extent that historians would have it do. Its huge significance lies ratherin what it can tell us about the perception, and uses of, mathematics at
Cicero’s time. A passage by Plutarch will help me set the scene for discussion.
In the Life of Cicero (paired with the Life of Demosthenes ), Plutarch
describes Cicero’s early years and his educational trip to Greece:
In Rhodes [Cicero’s] teacher in oratory was Apollonius […] It is
said that Apollonius, who did not understand Latin, asked Cicero

195GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
to declaim in Greek. Cicero was very glad to do so […] So he
made his declamation and, when it was over […] finally Apollonius
said: ‘Certainly, Cicero, I congratulate you and I am amazed atyou. It is Greece and her fate that I am sorry for. The only glories
that were left to us were our culture and our eloquence. Now I see
that these too are going to be taken over in your person by Rome’.
6
The equation between acquisition of Greek culture and acquisition ofpower was arguably as valid for Cicero as it was for Plutarch, albeit indifferent ways, given the lapse of time between the two. Mastering the
Greek language and literature, including philosophy and scientific treatises,
was an important way for second- and first-century
BC Romans to affirm
their entitlement to master the Greek world itself. This is stated explicitly
in a letter to Cicero’s brother Quintus, who in 60 BC had been confirmed
as proconsul of the main Greek province for the third year running. Ciceroreminds Quintus that rulers, according to Plato’s doctrine, should be
philosophers – there should be no hegemony without education.
7 And in
the Tusculan Disputations themselves, he calls for a ‘birthday’ of philosophy
in Latin, for his fellow Romans to ‘wrest’ philosophy away from the failing
grasp of Greece. An important element in the justification of the Romans’
rule over the Greeks was that they had assimilated and espoused Greekvalues, at a time when the Greeks themselves were in decline. But there is
more. Greek culture was not used just so that the Romans could construct
their entitlement to rule over the Greeks and a fortiori over the barbarians.
There was a battle to be fought at home too: at the same time that Rome
was expanding into the Mediterranean and negotiating its encounters and
co-existence with the Greek-speaking world, enormous changes were takingplace within Rome and Italy itself.
8 Acquiring an empire had created a lot
of new wealth; almost uninterrupted civil wars throughout the first century
BC had caused a quick turnover in the small group of those who were rich
and powerful enough to be either targets or emissaries of political
persecution. As a result, the senatorial and equestrian orders saw a number
of sudden departures and new arrivals. Cicero himself was a ‘new man’,
the first generation of his family to enter the Senate. For him (or indeed
for his brother) a good education, being at the centre of a cultural circle,possessing a well-stocked library, knowing Greek philosophy to the point
actually to introduce it to other, less sophisticated Romans, were all
fundamental ways of signalling status and political clout. Take the twobrothers as represented in the letter to Quintus: they are philosophers,
who have learnt the Greek lesson, and that is precisely why they are now
entitled to rule over Greece, and why they are better qualified to do so
than other Romans.

196GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
A Greek education enabled Romans to justify their power not just to the
Greeks but also to other Romans. Cicero’s divide was not just about Greek
and Roman, and not just about pure and applied mathematics, but alsoabout class and internal social conflict. After all, how does he describe
publicani , tax officers who must have done a lot of calculating? They are
powerful and have become indispensable, but they are also universally hatedand a headache for a provincial governor who has to reconcile the local
populations to their presence. Again, Cicero’s works contain many stories
about accounts that have been cleverly doctored – another example ofmathematical ability that can turn to the service of corruption. On occasions,
secretaries ( scribae ) are described as ‘entrusted with the public accounts and
the reputations of our magistrates’, and presented as dangerous upstarts,
examples of social mobility gone wrong.
9 Again, how does Cicero depict
land-surveyors? We have some insight into that question thanks to three
speeches given in 63 BC, first to the Senate, then to the popular assembly,
about the necessity to oppose an agrarian law put forth by the tribune of
the plebs Publius Servilius Rullus. In the latter’s proposal, a commission of
ten people was to be set up and equipped with technical staff, with the taskof administering all public land: selling it, distributing it, founding new
colonies if they thought it necessary. Cicero did not like the proposal for
several reasons: he claimed that it did not make financial sense, and wouldin fact impoverish the state; his arch-enemy Julius Caesar was rumoured to
be behind it; and above all it would relinquish enormous power into the
hands of people whose social credentials were not immaculate.
The second speech significantly opened with a long apology of the social
status of Cicero himself. He was, as we have said, a homo novus , but he
emphasized that he had earned his status and reputation through goodservice, thus he would continue his good service to the people by warning
them against the dangers of the agrarian law. He insisted that the excessive
power trusted to the agrarian officers would subvert the whole structure ofthe state:
[Rullus] gives the decemvirs [the agrarian officers] an authority
which is nominally that of the praetors but is in reality that of a
king. […] he provides them with apparitors, clerks, secretaries,heralds, and architects […] he draws money for their expenses from
the treasury and supplies them with more from the allies; two
hundred surveyors from the equestrian order, and twenty attendantsfor each are appointed as the servants and henchmen of their power.
[…] Just observe what immense power is conferred upon them;
you will recognize that it is […] the intolerable insolence of kings.[…] They are allowed to establish fresh colonies, to restore old

197GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
ones, to fill all Italy with their own; they have absolute authority
to visit all the provinces, to confiscate the lands of free people, to
sell kingdoms […] [they can] delegate their power to a quaestor,send a surveyor and ratify whatever the surveyor has reported to
one man by whom he has been sent.
10
Surveyors – also characterized by Cicero as ‘a picked band of young
surveyors’ and Rullus’ ‘handsome surveyors’11 – are here a hefty crowd of
henchmen, ready to divide out and sell the whole world to the highestbidder. That Rullus’ surveyors should belong to the equestrian order is
evidence of the growing importance of that group, and of its association
with publicani and apparitores – they were threatening to push against
boundaries that Cicero and people like him may have wanted to keep in
place.
12 ‘Surveyor’ is elsewhere in Cicero almost a term of abuse, to the
point of making me think that it need not necessarily imply that the targetsof criticism were actual practitioners. In his speeches to the Senate against
Mark Antony, Cicero calls Antony’s brother Lucius, among other things, ‘a
heap of crime and iniquity’ and, sarcastically, ‘that most just surveyor’(decempedator , literally ‘wielder of a ten-foot rod’, pun involuntary?). Other
associates of Antony include Nucula and Lento, ‘the parcellers of Italy’, and
Decidius Saxa, a tribune of the plebs from Celtiberia and a ‘cunning andexpert land-surveyor’, who started in the army, measuring out the camp,
and now is hoping to measure out the city of Rome herself.
13 In all cases,
surveyors are socially mobile individuals: children or grand-children ofequestrians, army veterans, natives of semi-barbaric provinces come good
through political intrigue.
On the other hand, Archimedes, the quintessential Greek mathematician,
is for Cicero an example of virtuous life. He plays a big role in the Tusculan
Disputations : he is likened to Plato’s demiurge, because he built a sophisticated
astronomical sphere in imitation of the heavens, and appears again in thefifth dialogue, where Cicero’s contention is that only the practice of virtue
can make one really happy. For instance, Dionysius of Syracuse, a man who
had wealth and power but no virtue, was deeply unhappy.
With the life of such a man, and I can imagine nothing more
horrible, wretched and abominable, I shall not indeed compare
the life of Plato or Archytas, men of learning and true sages: I shall
call up from the dust on which he drew his figures an obscure,insignificant person belonging to the same city […] Archimedes.
When I was quaestor I tracked out his grave, which was unknown
to the Syracusans (as they totally denied its existence), and foundit enclosed all round and covered with brambles and thickets; for

198GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
I remembered certain […] lines inscribed, as I had heard, upon his
tomb, which stated that a sphere along with a cylinder had been
set up on top of his grave. Accordingly, after taking a good look allround […] I noticed a small column rising a little above the bushes,
on which there was the figure of a sphere and a cylinder. And so I
at once said to the Syracusans (I had their leading men with me)that I believed it was the very thing of which I was in search.
Slaves were sent in with sickles who cleared the ground of obstacles,
and when a passage to the place was opened we approached thepedestal fronting us; the epigram was traceable with about half
the lines legible, as the latter portion was worn away. So you see, one
of the most famous cities of Greece, once indeed a great school oflearning as well, would have been ignorant of the tomb of its one
most ingenious citizen, had not a man of Arpinum pointed it out.
14
The contrast between Greeks and Romans is inscribed within widercontrasts: the virtuous and content life of the wise man and the evil and
miserable existence of the non-philosophized one; the people who ought toexert power because they have wisdom and those who in this bad, bad
world end up sitting on a throne while wallowing in ignorance. What side
Cicero was on needs no explaining – remember that he was writing whileon exile from a political arena where he had not received his due deserts
and had been unjustly persecuted. His pilgrimage to the tomb of Archimedes
is both the cathartic act of expiation of a Roman towards the Greek geniusunjustly extinguished by an earlier Roman,
15 and an identification with the
deceased, a sort of looking through the thickets for a representation of
himself, the wise man neglected by his own country fellows.
The Tusculan Disputations is not the only work by Cicero where we find
Archimedes, and Cicero is not the only author who writes about him. The
Syracusan mathematician had become a symbol of indomitable intellectand amazing ingenuity since at least Polybius, who describes his machines
at length, and Livy. The details varied, but the story of him being killed
more or less by mistake, while distracted by a mathematical demonstration,
is common to various versions. Plutarch, to whom we now turn, was writing
in an already long tradition, and some of the features of his account fit withearlier reports. Yet, in no other source do we find the remarks I have quoted
at the beginning about what kind of mathematics Archimedes preferred, or
what he really thought about his machines. The natural question would thenbe, is Plutarch’s account accurate – is that what Archimedes really thought?
Let us say, first of all, that Plutarch probably had no way of knowing
Archimedes’ inner thoughts, any more than we do, unless he had access tosome work (now lost) where Archimedes voiced them. It is unlikely that he

199GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
did, and it would have been I think even more unlikely not to mention this
Archimedean source, had he had it. Thus, Plutarch’s report may or may not
be true, but there is no sure way of finding out. Historians have of coursedrawn on Archimedes’ extant works and on the role mechanics plays in
them to side one way or the other, but the evidence is inconclusive. So, as in
the case of Cicero, I prefer to shift the question from, is Plutarch’s reporttrue, to, what does it tell us about the way mathematics was perceived and
used at his time?
In general, mathematics for Plutarch brought pleasure to those who
practised it, but was also, in Plato’s mould, a serious business, and there was
a right way and a wrong way of going about it. For instance, while he did
engage in numerological speculations, on several occasions he poked fun atpeople who attributed meanings to numbers inappropriately.
16 One of the
T able-T alks , which purport to be reports of conversations held over dinner,
has as subject ‘What Plato meant by saying that God is always doing geo-metry’. One of the dinner guests likens the mathematical sciences to ‘smooth
and undistorted mirrors’ where the objects of intellectual knowledge, basically
Plato’s Forms, can be seen reflected, and in this connection repeats a storywe have heard before:
Plato himself reproached Eudoxus and Archytas and Menaechmus
for setting out to remove the problem of doubling the cube into
the realm of instruments and mechanical constructions, as if they
were trying to find two mean proportionals not by the use of reasonbut in whatever way would work. In this way, he thought, the
good of geometry was dissipated and destroyed, since it slipped
back towards sensible things instead of soaring upward and layinghold of the eternal and immaterial images in the presence of which
god is always god.
17
This passage is followed by a description of how the Platonic god elimi-
nates arithmetical proportion, which corresponds to a desire for political
and economical equality, and sustains geometric proportion, a different
kind of equality where everybody is rewarded according to their
(opportunely assessed) worth. The political meaning of the right sort ofmathematics is clearly expressed, and linked to the reproach that hits the
‘practical’ mathematicians Eudoxus, Archytas and Menaechmus. Plutarch’s
discourse is not just about mathematics, of the cosmically and religiouslyproper sort – mathematics is used to say something about the pursuit of
philosophy in the context of a particular social and political order. If one
proceeds with the text, one finds good geometry as the guarantor of cosmicorder itself:

200GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
Matter is always struggling to break out into unboundedness, and
seeking to avoid being subjected to geometry; but reason seizes
upon it and encloses it in lines and marshals it in the patterns anddistinctions which are the source and origin of all that comes to
be.
18
Recent studies of Greek literary culture between the first and the second
century AD have argued that the flourishing of interest in grammar, lexico-
graphy and rhetoric, with its emphasis on a revival of atticizing Greek, was‘building defences around educated Greek’.
19 It can be argued along similar
lines that Plutarch was building defences around what he saw as the ‘right’
mathematics and science, and deploying heroical examples from the past(Archimedes but also Plato) to that purpose. It will be useful to quote some
more from the Life of Marcellus :
And yet Archimedes possessed so great a mind and so deep a soul
and such a wealth of theories that, although he had gained from
those [from the machines] name and reputation not human, butof some superhuman being, he did not want to leave behind
something written about them, but considering the business of
mechanics and all the arts that as a whole touch upon utility low-born and fit for vulgar craftsmen, he directed his ambition only to
all the things in which the beautiful and the extraordinary are not
mixed with the necessary.
20
The siege of Syracuse marked an important step in the Roman
subjugation of the Greek world. As a result of the victory, Archimedes waskilled, the city sacked, and its art treasures transported to Rome. Marcellus,
however, is depicted as an unwilling conqueror: he weeps over Syracuse
about to be pillaged, he tries to save Archimedes’ life, he is, as Plutarchpoints out, the living example that even Romans could harbour gentleness
and humanity. Marcellus is in a sense converted by Archimedes’ martyrdom,
and acknowledges the value of Greek culture first by mourning his death
and then by showing good taste in his choice of Syracusan pieces of art as
triumphal spoils. ‘Before this time’, Plutarch comments, ‘Rome neither hadnor knew about such elegant and exquisite productions, nor was there any
love there for such graceful and subtle art’.
21 In fact, if Archimedes is made
to embody the conflict between disinterested knowledge and direpragmatism, after his death Marcellus takes up a similar role, and becomes
the promoter of exquisite taste and love of beauty against the criticism of
his compatriots:

201GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
They blamed Marcellus […] because, when the population [of
Rome] was accustomed only to war or agriculture, and was
inexperienced in luxury and recreation […] he made them leisurelyand loquacious about arts and artists, so that they spent a great
part of the day in discussions. Marcellus however spoke of this
with pride even to the Greeks, declaring that he had taught theignorant Romans to admire and honour the wonderful and
beautiful productions of Greece.
22
To conclude: both in the case of Cicero and Plutarch, the divides they
constructed and deployed were much more complicated than simple Greek/
Roman or pure/applied dichotomies. Those divides had a political signi-ficance, not just in a cross-national, but also in a cross-social-strata sense.
For instance, they upheld leisure, detachment, ‘playing’ with knowledge –
all luxuries that not everybody could afford. Plutarch’s emphasis on cosmicorder is unequivocal, as are his general attitude to technical knowledge, and
his remarks about the low and vulgar artisans. In short, mathematics gave
upper-class Romans and Greeks a way of articulating their positions aboutthe state and the individual, power and knowledge. But what did the mathe-
maticians themselves, whether Greek or Roman, think they were doing?
The problem of pure versus applied mathematics
This section is a survey, in chronological order; I will draw some conclusions
at the end.
Let us start with Geminus (first century AD). We know from the extent
of the citations in the scientific and philosophical literature well into thefifth century
AD, that his work was very influential, although the Introduction
to the Phenomena is the only thing that has survived. The Introduction deals
with astronomy in a rather discursive style, thus following a well-establishedgenre of which Aratus, with his poem, was one of the most popular exponents.
Geminus, however, emphasizes that mathematicians and physicists are
different from poets and also from philosophers, and that astronomy and
meteorology, which Aratus had put together, are in fact quite distinct. Objects
of knowledge (the heavenly phenomena) and purposes of knowing (weatherforecast, astrological predictions) may be shared, but primacy is claimed for
mathematical and physical explanations. For instance, the grammarian Crates
is criticized because, basing himself on Homer, he had affirmed that theOcean was situated between the tropics. What is more, he had had the
effrontery to claim that this view was supported by mathematics. Instead,
Geminus thinks that Crates’ theory is contrary both to physical and to

202GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
mathematical accounts, and remarks that it had not been adopted by any of
the ancient mathematicians. Thus, he presents himself as a more reliable
interpreter of the past mathematical tradition than Crates; in fact, Geminuseven dares occasionally to criticize the ancients. His rebuttal can be read as
a delimitation of the relative spheres of competence of on the one hand
poets, grammarians and (in a later passage) philosophers, and on the otherhand of mathematicians like himself.
23 Geminus is just as cutting in his
description of what he calls the meteorology of the inexperts: their calendars
are based on repeated experience, not on well-defined principles; they lacktechne and their conclusions are not necessary.
24 On the contrary, Geminus’
mathematical version of astronomy enjoys greater certainty and reliability;
its predictions of an eclipse or of the rising of a star can be depended upon.Moreover, it has great ethical significance, as established in the introduction,
where the regularity and uniformity of the heavens’ motions is compared to
the irregularities and continuous changes of human existence.
25
Geminus exemplifies features which we will find, in various modes and
combinations, in the rest of this section: he draws a line between mathematics
and other disciplines (further lines are sometimes drawn within the mathema-tical domain between different modes of practice); he adds authority to his
statements by appeal to a tradition of which he is a representative; he empha-
sizes the ethical, social or political significance of his form of knowledge.
Vitruvius engages in similar boundary-drawing. As well as proclaiming
what the good architect is, Vitruvius clarifies what he is not: for a start, he
knows a good deal of mathematics, but is not a mathematician; he knowsabout building and making things, but is not a craftsman. Those two bounda-
ries are far from well-defined, since the passages where he takes his distance
from mathematicians and craftsmen could be juxtaposed to other parts of thetext, where we find a slightly different story. At the beginning of the sixth
book, Vitruvius tells the anecdote, widely circulated in this period, of the
philosopher (various names in various versions) who was shipwrecked andcast on a strange shore, where he saw geometrical diagrams written in the
sand. He exulted at the sight, because they were the signs of human civilization.
The point of the tale is that mathematics is a possession for ever, ‘immune
from the stormy injustice of fortune, the changes of politics and the ravages
of war’.
26 In fact, learning a craft also fits that description and Vitruvius reveals
that he was trained in the architectural art (although he underlines that his
education was both literary and technical). The system of apprenticeship
common in the crafts has, he says, the advantage that future practitioners arevetted not only on the basis of their skill, but also of their reliability and
loyalty. The good craftsman, skillful, loyal and immune from the storms of
fortune, is then contrasted with the bad architect, who lacks both experienceand training and gives a bad name to the profession as a whole.
27

203GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
In other words, the relation between architects and mathematicians on
the one hand, and architects and craftsmen on the other hand, is defined
and re-defined instrumentally. When Vitruvius targets excessive specializationor unreliability, he emphasizes differences. When his aim is to mark out
good architecture against bad, or assert the quasi-philosophical value of
architectural knowledge, he flashes his mathematical credentials andreminisces about his technical roots. These issues are explicit in another
passage, where Vitruvius seems to arrange the practice of the arts, and conse-
quently their practitioners, in a sort of spectrum, ranked on the basis offortune and social standing, from the poor and obscure to the wealthy and
officially recognized. Crucially, one’s position in this spectrum is not
determined by mere skill or industry – out of a list of painters and sculptors,some became famous, while others remained obscure. Yet, those latter lacked
neither industry nor devotion to the art nor skill: but their reputation was
hindered, either by scanty possessions, or poor fortune, or the victory ofrivals in competitions.
28
The real value of craftmanship is not appreciated by the general public,
because they can only judge what they see, and what they see is unfortunatelynot ‘manifest and transparent’ – the non-experts are not able to recognize
real expertise. If the validity of the arts basically cannot speak for itself,
social graces and adulation are what influences people’s judgement. In otherwords, what should be a matter of greater art or skill, thus ultimately of
greater (technical) knowledge, becomes a matter of wealth or social status.
One needs to fight back, and this is why Vitruvius published his book: todisplay ‘the excellence of our knowledge’ and make a stand for technicians
whose well-rounded education, honesty, significance for the community
and roots in a long-standing tradition ought to counteract the respectablefamily, good connections, and vacuous charms of the bad architects.
Another mathematician in a pugnacious mood, Hero of Alexandria,
opened his Belopoeiika as follows:
The largest and most essential part of philosophical study is the
one about tranquillity ( ataraxia ), about which many researches
have been made and still are being made by those who pursue
learning; and I think research about tranquillity will never reachan end through reasonings. But mechanics has surpassed teaching
through reasonings on this score and taught all human beings how
to live a tranquil life by means of one of its branches, and thesmallest – I mean, of course, the one concerning the so-called
construction of artillery. By means of it, when in a state of peace,
they will never be troubled by reason of resurgences of adversariesand enemies, nor, when war is upon them, will they ever be

204GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
troubled by reason of the philosophy which it provides through
its engines.29
Everybody yearns for a tranquil, undisturbed life, and pursues happiness. Aperson, however, cannot enjoy true ataraxia without political and social
security, and tranquillity must be reached both at an individual and at acommunal level. The existence of this common ground, shared by everybody,
makes Hero’s comparison between forms of knowledge possible; if they aim
at the same thing, they can be assessed one against the other on the basis oftheir results. In other words, he creates a context for competition between
philosophy and mechanics where none need exist, because he is keen to
show the wider relevance of his form of knowledge for society. He employsa string of oppositions: philosophy uses reasonings, which are mere words,
mechanics uses machines, which you can see and touch, and so can the
enemy; the largest part of philosophy is pitted against the smallest branchof mechanics. The rest of his account, as we have seen in chapter 5, employs
mathematics to confer certainty on the workings of catapults, and is
organized as a progressive history, where the achievements of engineersaccumulate over time producing better and stronger machines. On the whole,
mechanics is depicted as a science which can bring about peace and happiness
for the individual and for the state, produces tangible results, justifies themon reliable mathematical bases and is practised by people who belong to a
glorious tradition.
Reliance on previous authors and claims to general utility are also found
in the Pneumatics , which deals with the properties and effects of vacuum
and of moving air ( pneuma , as distinct from still or elemental air). In the
introduction, Hero again compares philosophers and mechanicians.Pneumatics, he says, has been studied by both, ‘the former proving its
power discoursively, the latter from the action of sensible bodies’.
30 The
contrast is the same as in the Belopoeiika , empty words on one side, concrete
and tangible effects on the other. Hero repeats it when he later opposes
the fabricated arguments of those who assert things on the nature of
vacuum but do not offer any sensible proof, and those who can sustain
what they say by appeal to sensible phenomena.31 Yet another piece of
evidence from Hero is the introduction to the third book of the Metrica ,
about division of geometrical figures. After reminding the reader, as he
had already done at the beginning of the first book, of the Egyptian origins
of geometry, he continues:
To assign an equal region to those who are equal and a greater
<region> to those who deserve it according to proportion isconsidered completely useful and necessary. The whole earth is in

205GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
fact divided by nature herself according to merit; in fact on her
live a great people who have been assigned a great region, some on
the other hand have a small region, being small in relation to theformer ones; no less the individual cities are also divided according
to merit; to those who govern and to the others who are able to
rule, a greater part according to proportion, to those on the otherhand who are not able to do that, small places and villages are left,
and to those with smaller personalities farms and things like that;
but those things somehow take on an extremely gross and uselessproportion; if on the other hand you want to divide the regions
according to the given ratio, so that, as it were, not even a grain of
millet of the proportion exceeds or falls short of the given ratio,one needs geometry alone; in it <are> equal fitting, justice to the
proportion, the uncontested proof about these things, which none
of the other arts or sciences promises.
32
The themes of reliability and usefulness of mathematics recur; its signifi-
cance is stated with respect to the order itself of things in the world. Heroestablishes a correlation between nature, the characteristics of peoples, the
size of the regions they live in and their political role. He then inserts mathe-
matics in this scenario, by identifying division of areas as a primary concern.As in the introduction to the Belopoeiika , he uses contrasting terms: great
peoples and great regions, small peoples and small regions, those who rule
and those ‘with smaller personalities’ who do not even have the potential torule,
33 cities on the one side and villages and farms on the other. Reading
between the lines, one could see this as a disillusioned depiction of a world
dominated by inequality; in particular, dominated by a great people whohad acquired enormous quantities of land. Like many non-Romans faced
with the inevitability of Roman rule, Hero invokes nature and merit as
justifications of the status quo . Yet, his images of mathematics and mathe-
maticians as the guarantors of tranquillity and justice over uncertainty and
disproportion can be read as a powerful statement that guarantees were
needed, and tranquillity and justice strongly demanded.
Whereas Hero tends to compare mathematicians and mechanicians with
‘outsiders’, i.e. philosophers, in astrological texts different types of boundariesare drawn. As we have seen, both material and textual sources point to the
existence of different levels and different traditions of astrological practice.
A first distinction between astrologers is introduced by the degree of accuracyof their calculations and predictions – our authors tend to equate accuracy
with the possession of mathematical skills.
34 A particularly well-articulated
statement of these issues is in Ptolemy. He distinguishes astronomy fromastrology thus:

206GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
Of the means of prediction through astronomy, o Syrus, two are
the most important and valid. One, which is the first both in
order and in power, is that whereby we apprehend the aspects ofthe movements of sun, moon and stars in relation to each other
and to the earth […]; the second is that in which by means of the
natural character of these aspects themselves we investigate thechanges they bring about in that which they surround. The first
of these, which has its own theory, valuable in itself although it
does not attain the result given by its combination with the second,has been expounded to you as best we could in its own treatise in
a demonstrative way. We shall now give an account of the second
and less self-sufficient <means of prediction> in a properlyphilosophical way, so that one whose aim is the truth might never
compare its perceptions with the sureness of the first, unvarying
one.
35
In order to demonstrate that astrology is feasible at all, he points out
that celestial phenomena do affect what happens on earth: the sun andmoon obviously, but also constellations or planets, whose appearance in the
sky was made to correspond, ever since Hesiod’s time, to changes in the
weather. Astrology for Ptolemy is the reading of signs in the heavens fortheir consequences in the terrestrial region. Yet, due to the imperfect nature
of the physical elements down here, which introduce unpredictable changes
and disruptions, astrology can only be conjectural. Still, in principle, it shouldget more credence than other, generally accepted, types of sign-reading,
such as farmers and herdsmen observing the heavens for agricultural
purposes, or sailors detecting the onset of a storm. Astrologers are morereliable than all those people, because of the amount of knowledge they
bring into their reading of signs. Ptolemy calls this knowledge ‘theory’,
because it is based on a correct understanding of the times, places and periodicmotions of the planets. A hierarchy of semiotic knowledge is thus constituted,
with ‘dumb animals’ at the bottom, then farmers, herdsmen and sailors
whose job entails regular observations, and finally people who have been
thoroughly trained in mathematical knowledge of the heavenly phenomena.
They can refer to the same things as the ‘ignorant men’ above, but withmore accuracy and certainty. The skilled astrologer is still not infallible, but
he is less likely to err in his predictions. A second distinction is drawn within
the field of astrology itself. Bad practitioners, Ptolemy says, bring theprofession into disrepute, either because they are not properly trained (‘and
they are many, as one would expect in a great and manyfold science’
36) or
because they give the name of astrology to divinatory practices that havegot nothing to do with it.

207GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
In sum, Ptolemy upholds the value of astrology, which had come under
many attacks,37 by emphasizing its links with mathematical knowledge (of
the kind expounded in the Syntaxis and condensed in the Handy T ables )
and its value for the pursuit of a happy and tranquil life.38 At the same time,
he restricts the field of those who can practice it to thoroughly-trained people
who are not only interested in their own gain.
Slightly later than Ptolemy, Galen, though not a mathematician in a
strict sense, argues, in ways analogous to what we have encountered so far,
that mathematics gives certainty, agreement and concord, while philosophydoes not, and that the consensus thus generated coagulates a community
around canonical texts (especially Euclid’s works) and accepted procedures.
Galen’s sympathy partly stemmed from the fact that mathematics, likemedicine, was a techne . In his Exhortation to the Arts , medicine, together
with a whole host of mathematical disciplines, from geometry and arithmetic
to architecture and gnomon-building, is part of the band of Hermes, the‘lord of the word’. The god’s followers are well-behaved, keep their place
and are honoured not on the basis of political reputation, noble family and
wealth, but of a good life and of excellence in their profession. They arepitted against the followers of the goddess Fortune, whose mob includes
the great tyrants of the past (Cyrus, Dionysius) as well as demagogues,
betrayers of friends, murderers and robbers.
The ethical and political significance of mathematical professions is again
clearly expressed by some of the authors in the Corpus Agrimensorum . We
have already drawn attention to Frontinus and Balbus. Another example isSiculus Flaccus with his Categories of Fields . The treatise seems addressed to
other surveyors, in order to make them aware of the various elements
involved in working on the terrain, not only the administrative differencesbetween types of land, but also the huge number of methods people in
different regions have of marking boundaries. Flaccus frequently insists that
the surveyor must pay attention to, and, whenever possible, respect, thecustoms of the place; he suggests that his expertise consists not only of
technical abilities, but also of diplomatic skills and of experience in the
ways of the world, gained from having travelled widely in the service of the
Empire.
39 The knowledge of the surveyor encapsulates Roman history itself:
sometimes the names given to a type of land bring the imprint of pastevents. For instance, Flaccus derives the etymology of territorium from the
‘terror’ felt by the populations when they abandoned their land, after being
defeated.
40 Some other times the reader is alerted to the difficulty of
disentangling successive layers of land management: for instance, the
Gracchan and Sullan surveyors in the second and early first century BC
respectively set boundary markers which sometimes were still in place in
the second century AD and needed to be recognized. Or again, the vicissitudes

208GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
of history, including the civil wars of the first century BC, could cause
confusion in the central archive, supposed to contain maps of all the
territories of the empire. The quick succession of events between JuliusCaesar and the ascent of Octavian Augustus had left its trace on the records
of land ownership, and only through calculation of the total areas involved
was Flaccus able to set the record straight.
41 The business of surveying taps
into history in an even more seminal way: the reason why fields are classified
into various categories, about which the surveyor has to be knowledgeable,
is that some peoples have fought the Romans; others, having experiencedRoman military might, have settled for peace; others still have traditionally
been allies of the Romans, to the point of fighting alongside them.
42 The
land these different peoples occupy will have different statuses, in fact acquirea different nature in the surveyor’s eyes.
Thus, the natures of the fields are many and diverse, and they are
unequal either because of the circumstances of war, or because of
the interests of the Roman people or, as they say, because of
injustice.
Flaccus also observes that ‘wars are the reason why fields were divided’.
43
What Hero had put down to nature Flaccus unflinchingly attributes to war,
personal interest and injustice; what the two have in common, though, is
the recognition that, one way or the other, mathematicians, land surveyors,
the exponents of their knowledge, had a fundamental role to play in society.
Let us start to draw some conclusions: we seem to have two opposing
fronts. On the one hand, Seneca, Pliny the Elder, Cicero, Plutarch are
questioning or downright denying the significance of certain mathematicalpractices. On the other hand, the authors we have grouped in the second
section are saying that certain mathematical practices have great moral and
political significance. In some cases, they even argue specifically that theirversion of mathematical knowledge is better than other forms of knowledge
at attaining moral and political goals, that it gives concrete, reliable results
in fields of common interest. For instance, philosophy, according to some
the knowledge proper of political rulers, is made the butt of criticism not
only for its supposed inability to do anything for anybody in the real world,but also because its exponents kept squabbling with each other.
44 In sum,
the point of contention was not just, which mathematics is better – it was a
debate about knowledge and its role in society where mathematics was onlya part of the picture, and it was a debate about the role within society of the
people whose knowledge identified them as philosophers, geometers, or
architects. Some of the more vocal mathematicians (Vitruvius, the land-surveyors) are known to have belonged to strata of Roman society which

209GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
were socially on the up, or aspired to be so. Yet, this is not a straightforward,
simplistic, conflict of upper-class snobs vs. middle-class social climbers.
The positions in the social spectrum of our protagonists are infinitely morecomplicated than that, as are their individual variations on the theme
‘knowledge is power – but what knowledge’? An interesting case, for instance,
is Frontinus, a homo novus like Cicero, and probably an example of upward
mobility through the army: as a member of the elite, a senator and a consul,
surely he was not expected to write a treatise on land-surveying. Let us
focus, however, on his preoccupation with wanting to match the expertiseof his subordinates:
There is nothing as dishonourable for a decent man as to conduct
an office entrusted to him on the basis of the prescriptions of his
assistants, which it is necessary to do, every time that the ignorance
of the person in charge has recourse to the experience of thosewho, even though they are parts necessary to the task, should still
be like some sort of hand and instrument of the agent.
45
That must have been in fact a real, if rarely recognized, problem: the
knowledge that the elite looked down upon (and we are talking land-
surveying, accounting, machine-building, the nitty-gritty of administrationin general) was also the knowledge needed by the elite to run the state and
thus maintain their elite status. On a different level, it was the knowledge
needed to manage an estate or a workshop; management was usually dele-gated to slaves or freedmen, but they could not be trusted completely. Cicero,
Columella, and others remind the reader and themselves more than once of
the fine line they had to tread between leaving accounts, lists, decisions,money administration in the hands of their expert subordinates, and still
retaining some control over them. It is the paradox embodied by Columella
not wanting to talk about land measurement because it is below his dignitybut still talking about it because one needs after all to know one’s own land;
by Pliny the Younger who would like to write literature and is forced to
check accounts, but at the same time checking the accounts is what signifies
his power with respect to, for instance, the population of Bythinia; it is the
paradox of Cicero’s scribae in whose hands and within whose public accounts
the reputation of a magistrate lay. It is behind, I think, Galen’s words:
[People] prize horses trained for war, and dogs for hunting, more
than any other kind; and they usually have their household staff
trained in some skill ( techne ), often at considerable expense. None
the less they neglect their own education. But is it not disgracefulthat the slave should be worth as much as 10,000 drachmas, while

210GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
the master is not worth one? One drachma? No one would take
such a fellow even as a gift.46
Vitruvius, Hero, Balbus, Flaccus realized what they were worth and the
wider implications of their knowledge. Consequently, they claimed recogni-
tion for it and for themselves. In the process, they often took care to comeacross as a group – the land-surveyors had a well-defined juridical identity
– and/or as a tradition, stretching back sometimes to the remote past. For
instance, Vitruvius saw mechanics and architecture as having cruciallycontributed to human civilization itself.
47 Interestingly, Archimedes,
appropriated by some authors as the hero of good, philosophical, abstract
mathematics is to some extent reclaimed by the mathematicians andmechanicians. Vitruvius and Hyginus mention his achievement and its
importance for their work; Hero inscribes him squarely within the tradition
of measuring and dividing, which is how he sees geometry. In a sense, they
were all deploying a man from the past to talk about their present – they
were all writing their own histories of mathematics for contemporary use.
Notes
1 Cicero, Tusculan Disputations 1.2.5, Loeb translation with modifications.
2 Plutarch, Life of Marcellus 305.4-6, Loeb translation with modifications. The life parallel
to Marcellus’ is Pelopidas’.
3 I adapt one of the main arguments in Edwards (1993).
4 An achievement recognized by e.g. Beard and Crawford (1999).
5 Cicero, Tusculan Disputations 4.3–5 and 5.5–6, respectively.
6 Plutarch, Life of Cicero 863a, tr. R. Warner, Penguin 1972. Cf. Swain (1990b).
7 Cicero, Letters to his Brother Quintus 1.27 ff. Cf. Ferrary (1988).
8 See e.g. Hopkins (1978), Wallace-Hadrill (1998).
9 See e.g. Cicero, Against Piso 61; Letters to his Brother Quintus 1.32 ff.; Against Verres
3.184. Cf. also Badian (1972); Andreau (1999), and, for scribae as apparitores , Purcell
(1983).
10 Cicero, On the Agrarian Law 2.32–34, Loeb translation with modifications.
11 Cicero, ibid. 2.45, 53.
12 See Nicolet (1970), Badian (1972), Alföldi (1975).
13 Cicero, Philippics 11.12, 13.37, 14.10 (delivered in 43 BC).
14 Cicero, Tusculan Disputations 5.64–6. The passage with Archimedes as the demiurge at
ibid. 1.63. It is worth noting that the Syracusans’ neglect of Archimedes contradicts
Cicero’s initial statement that the Greeks held geometry in great honour. We find thesame issues (Archimedes’ globe, Archimedes and Dionysius) in On the Commonwealth
1.21–2 and 1.29, respectively.
15 See Gigon (1973).
16 Plutarch, T able-Talk 738d–739a.
17 Plutarch, ibid. 718e–f, Loeb translation with modifications.
18 Plutarch, ibid. 719e.
19 Cf. Swain (1997).
20 Plutarch, Life of Marcellus 307.3–4, Loeb translation with modifications.

211GRAECO-ROMAN MATHEMATICS: THE QUESTIONS
21 Plutarch, ibid. 310.1.
22 Plutarch, ibid. 310.6–7, Loeb translation with modifications.
23 Geminus, Introduction to the Phenomena 16.22–23, 17.32, 7.18, 7.22.
24 Geminus, ibid. 17.1–25.
25 Geminus, ibid. 1.19–21.
26 Vitruvius, On Architecture 6 Preface 2. Cf. Cicero, On the Commonwealth 1.29; Galen,
An Exhortation to Study the Arts 8.
27 Vitruvius, On Architecture 6 Preface 6–7.
28 Vitruvius, ibid. 3 Preface 3, Loeb translation with modifications.
29 Hero, Construction of Catapults 71–73.11, translation in Marsden (1971) with modifi-
cations. See also Athenaeus Mechanicus, On Machines 4-5, where the many words of the
philosophers are contrasted with the really useful things produced by the mechanicians.
30 Hero, Pneumatics 1 Preface.
31 Hero, ibid. 1 Preface.
32 Hero, Metrica 3 Preface, my translation.
33 The term I have translated with ‘having small personalities’ is mikropsuchos and is not
used very much outside Aristotle’s work. Guillaumin (1997) argues for Platonist,Aristotelian and Stoic echoes in this passage.
34 See Vettius Valens, Anthology e.g. 4.11, 5.6, 9.9.
35 Ptolemy, T etrabiblos 1.1, Loeb translation with modifications.
36 Ptolemy, ibid. 1.2.6, Loeb translation with modifications.
37 Cf. Cicero, On Divination ; Sextus Empiricus, Against the Astrologers .
38 This latter especially at Ptolemy, T etrabiblos 1.10 ff.
39 Siculus Flaccus, Categories of Fields e.g. 59, 71, 135, 145. As for other instances of
diplomatic skills, Hyginus (thought to be not the same as Hyginus ‘Gromaticus’, also inthe Corpus ) recounts the episode of a surveyor assigned by T rajan to the distribution of
land to veterans, who, as well as inscribing the size of the allotments on bronze tablets,
noted down their borderline and position, so as to avoid fights and arguments among
them: Hyginus, Categories of Fields 84.
40 Siculus Flaccus, Categories of Fields 26.
41 Siculus Flaccus, ibid. 272–6.
42 Siculus Flaccus, ibid. 5–7.
43 Siculus Flaccus, ibid. 33 and 210 respectively, my translation.
44 This according not just to mathematicians, but also to people like Galen and Lucian, in
e.g. Philosophies for Sale , who in fact would consider themselves philosophers.
45 Frontinus, On Aqueducts of the City of Rome 2.
46 Galen, An Exhortation to Study the Arts 6, translation as above.
47 Vitruvius, On Architecture
10.1.4

212LATE ANCIENT MATHEMATICS: THE EVIDENCE
7
LATE ANCIENT
MATHEMATICS:
THE EVIDENCE
We cannot say that the number six is perfect because God
perfected his work in six days, but that God perfectedhis work in six days because the number six is perfect.In fact, even if his works did not exist, that number would still beperfect.
1
The period I will be examining in the present chapter (third to sixth century
AD) begins with an Empire still in place and ends after its dissolution in the
West. By the fourth century, Christianity had become the religion of the
emperor (with the brief exception of Julian), and the Church emerged as amore and more powerful institution, with buildings, books, schools,
canonical authors of its own. The interaction between new and old religion,
between the various Christian sects, and between church and state are someof the key issues in the history of this period. One of the most dramatic
ways in which a mathematician could be affected by religious changes is
exemplified by the death of Hypatia of Alexandria, who taught philosophyand mathematics. She was lynched by a Christian mob in
AD 415, for reasons
that probably had to do with her being an educated, politically visible, pagan
woman.2 Other, more subtle, traces of Christianity in the field of mathe-
matics will be explored in the next chapter. Here we devote individual sections
to Diophantus, Pappus, and Eutocius, and general ones to ‘Philosophers’
(mainly Iamblichus and Proclus) and ‘Rest of the world’. To begin with, asusual, I will review the material evidence, which will include (new to this
chapter) legal sources.
Material evidence
As for earlier periods, the great majority of our papyrological sources come
from Egypt. A particularly interesting cluster of third-century AD papyri, at

213LATE ANCIENT MATHEMATICS: THE EVIDENCE
present scattered around at least five different collections, has been identified
as belonging to the archive of a rich landowner called Appianus, whose
estates were in the Arsinoite nome. Many of the documents are accounts ofa rather complex type. In fact, it has been said that ‘the system of accounting
on the Appianus estate […] is the most sophisticated presently known from
the Graeco-Roman world’.
3 Each administrator on each sub-division of the
estate drew up his own little accounts, for day-to-day running of the farm,
payment of the workforce and so on. The information relative to the produc-
tion of crops, the sale of produce, the use of animals, the general expenditureon the staff (including the administrators themselves), was then summarized
from these smaller accounts into larger monthly ones, in their turn pasted
together (literally, as pieces of papyrus roll) into one big yearly account foreach particular sub-division of the estate. Entries were arranged by sector,
with cash expenses and gains extrapolated from all the different sectors.
It has been argued that the function of these accounts was not only to
verify the honesty of the administrators, or to keep a record of things, but
also to plan the management of the estate. Appianus and his own staff of
secretaries could at little more than a glance get a clear idea of where profitsand losses were being made, what parts were not productive, and even
compare different sub-divisions of the estates for productivity. Accounts of
this kind gave the owner the opportunity to take better economic decisionsbecause the information was purposefully selected and arranged. It has also
been observed that the accounting systems across different sub-divisions of
the estate or across different estates look very similar, although variationsmay have been deliberately introduced by at least one administrator,
Heroninos.
4 This suggests that the training undergone by the various
administrators must have also been similar.
In sum, it seems that third-century Egyptian administrators were highly
numerate, that their skills could be an instrument of social advancement
(Heroninos came up through the administrative ranks, his son Heronasfollowed in his footsteps), and that they may have undergone a streamlined
system of accountancy training. We have quite abundant evidence – papyri,
wax tablets, wooden tablets bound together to make a booklet, potsherds –
that financial mathematics was extensively taught at an elementary level. In
terms of quantity (and it could of course be a mere accident of transmission)we have in fact more documents of this kind from late antiquity than from
any other post-pharaonic period, and they are in Latin and Coptic as well as
in Greek. Most of them contain division, multiplication and currency-conversion tables, very often together with literary material ranging from
paraphrases of Homer to psalms. Sometimes the tables contain verifications
that the result, usually provided without explanation, is indeed correct. Insome cases we are given the name of the author of the document: thus, a

214LATE ANCIENT MATHEMATICS: THE EVIDENCE
wooden schoolbook probably from the sixth century AD is inscribed with
the words ‘Papnouthion <son> of Iboïs’.5 One booklet contains both addition
tables and a fragment of what looks like a land-survey;6 a sixth-century
wooden board preserves division tables on one side and nine problems
(written in three different hands) on the other. The problems, specific rather
than general and set in question-and-answer form, with the solutions writtenbetween the lines just above the questions, deal with everyday situations
such as calculating interest on loans, or payments on leases of land.
7
A man once owning one island, a man came … I say that I want tosow the island. I offer you in respect of half the island … artabae of
lachana per arura; in respect of the rest, for the third part 4 [or 1]artabae of lachana per arura; and in respect of the remaining sixth
part 3 artabae of lachana. He came at the time of the harvest …
artabae of lachana. Find the number of arurae he
150 sowed.8
The fourth century saw a number of administrative reforms, perhaps
initiated by the emperor Diocletian and definitely followed by his successors;they included a new way of assessing tax liability.
9 T raditionally, the amount
of tax, in cash or kind, was assessed on the basis of the size of the land, its
quality and its crops. There is evidence from Egypt that a tax on the person(a poll tax) was also levied. The system we find at work in the fourth century
combined these two types of taxation into a unified system, whereby the
total tax liability of a town or rural community could be expressed. Landwas measured in iugera , or other local standards which in some cases were
fixed by law; whereas people were counted by heads, capita . There is evidence
that, as well as the two separate units, iugera and capita , there was a general
unit, the iugera vel capita , which implied abstracting from what was actually
being measured (property or people) for the purpose of calculating tax. In
other words, the late ancient taxation system may have involved a sort ofmathematization of resources, but evidence is too scant to make any definite
claims. We can assume that, whatever the details of the system, if indeed there
was a unified, in some sense ‘abstract’, numerical, unit of measure for both
people and property, formulas for conversion between them must have been
devised, taught, learnt and applied by skilled people. Calculations andaccount-keeping must have been required, and demand for them may have
effectively increased from the end of the third century onwards.
The so-called archive of Aurelius Isidorus from Karanis (late third–early
fourth century
AD) offers more evidence for the use of mathematics in
everyday administrative practices (mostly accounts), as well as providing
insights into the fiscal system: for instance, it documents measurements ofestates apparently for tax purposes. In three of these cases, the surveys were

215LATE ANCIENT MATHEMATICS: THE EVIDENCE
carried out by the same geometrai , Aurelius Aphrodisius and Aurelius
Paulinus, in the presence of state officials.10 This suggests that fiscal reforms
may have boosted the need for the services of surveyors, as well as ofcalculators.
We have quite a lot of other evidence about land-surveying in late anti-
quity: for instance, an entry in the so-called Price Edict (
AD 301), with which
Diocletian tried to establish a maximum for prices and salaries, refers to
wages for a teacher of geometria ;11 a letter by Pope Gregory I about a boundary
dispute ( AD 597) recommends recourse to a surveyor, as does another letter
by Cassiodorus (which will be discussed in the section on the rest of the
world later in this chapter).12 Mensores are mentioned as part of governmental
staff by the Notitia dignitatum , a sort of catalogue of the late ancient adminis-
trative machine.13 Most of our information, however, can be gleaned from
the law codexes. Under the emperors Theodosius II ( AD 401–450) and
Justinian ( AD 483–565), jurists were employed to collect decrees and rescripts
from the time past, organizing them under topical headings for quick
reference. Thus, the Codex and Digesta of Justinian comprise laws that date
from the third century, with names like Ulpian, Paul and Modestinus, tothe sixth, with recent or current emperors and provincial governors.
Using legal evidence can be tricky for a variety of reasons: the range of
applicability of the law is not sure, both because some laws seem to belimited to a particular province (and we do not know whether similar laws
existed for other provinces), and because we cannot easily chart the relation
between what a law suggests and actual circumstances. For instance, a lawestablishing harsh punishment for thieves may indicate that there was an
increase in theft, that the current emperor wanted a zero tolerance policy
for propaganda reasons, or that he particularly hated thieves. Whether lawswere actually applied is difficult to tell: law enforcement must often have
been impossible. That said, legal sources contain extremely interesting
information about mathematics and mathematicians. In general, I assumethat laws promoting architects, for instance, signify that the social status of
architects was, or had potential to be, on the up. Thus, as far as land-surveying
is concerned, a law to the effect that ‘The chief of the land surveyors after
completing two years [of service] is assigned the lowest office of agens in
rebus ’ confirms the impression that surveyors were an important part of the
administrative body, within which they had a regulated career path.
14 We
know that, as in the past, land-surveyors could act as main arbitrators in
boundary disputes.15 On the other hand, an entry on ‘If a land-surveyor
declares the wrong size’ established various sanctions against land-surveyors
who did not do their job properly. The jurists (Ulpian and Paul) mentioned
architects and accountants ( tabularii ) as parallel cases where specialized
expertise was required but had to be legally controlled.16

216LATE ANCIENT MATHEMATICS: THE EVIDENCE
An important group of laws concerns the fiscal status of various
‘professionals’, including people engaged in mathematical practices. Citizens
were supposed to contribute to the upkeep of the Empire in cash, kind orin taking responsibility for the maintenance of roads, billeting of soldiers
and so on. Certain categories of people, however, were exempt from some
or all fiscal obligations, on various grounds that are not always specified butmay have depended on other services they paid to the general welfare of the
Empire, or on some special status they may have enjoyed. Thus, fiscal
privileges were granted to, on the one hand, soldiers and shippers of graincargoes, on the other hand, to the Christian clergy (in
AD 313, following
Constantine’s conversion). The choice of worthy categories was a matter
for continuous negotiation, through which we can not only assess the officialview of some activities, but also realize that definitions could be problematic
and involved issues of self-image and of the ethical import of one’s
knowledge. Take the case of philosophers: we have scattered evidence thatpeople did sometimes allege their being a philosopher as a reason not to pay
tax.
17 On the other hand, the laws often scoff at philosophers, who, far
from being exempt from paying, should scorn money and thus be morethan happy to contribute to the welfare of their fellow-citizens.
18
Various categories of mathematicians figure in the extant tax legislation:
apart from occasional occurrences of geometres where the term does not
seem to denote a land-surveyor, and must thus denote a ‘real’ geometer, we
have architects/mechanicians, astrologers/astronomers, and teachers of arith-
metic, geometry, astronomy and music. Those latter were affected by blanketlaws which aimed to promote education and thus established fiscal immunity
for them (teachers of liberal arts), for rhetoricians, occasionally grammarians,
and often also for doctors.
19 Every city with the exception of Rome was
apparently given a fixed quota of teachers to whom immunity could be
granted, and hopeful individuals may have had to undergo a public assess-
ment.20 As we have said, however, distinguishing between worthy and not-
so-worthy was a complicated process, well exemplified by the following
passage:
The governor of a province regularly settles the law on salaries,
but only for the teachers of the liberal studies. […] Rhetors will beincluded, grammarians, geometers. The claim of doctors is the
same as that of teachers, perhaps even better, since they take care
of men’s health, teachers of their pursuits. […] But one must notinclude people who make incantations or imprecations or […]
exorcisms. For these are not branches of medicine […] But are
philosophers also to be included among teachers? I should notthink so, not because the subject is not hallowed, but because they

217LATE ANCIENT MATHEMATICS: THE EVIDENCE
ought above all to claim to spurn mercenary activity. […] Although
also masters of an elementary school are not teachers, nonetheless,
the custom has arisen that cases involving them should be heard,also those involving archivists and shorthand writers and
accountants or ledgerkeepers. But the governor of a province must
not hear outside the regular system cases of workers or craftsmenin fields other than those involving writing or short-hand.
21
This law is an exercise in boundary-drawing: first to include liberal
studies, then to include doctors, but also to exclude dubious healing practices.
Philosophers are excluded on moral grounds, but primary teachers and other
borderline categories of literate and numerate people such as librarii, notarii
and calculatores are accommodated. At the same time, workers or artisans
whose tasks have nothing to do with written signs are kept out. Thus, the
criteria of inclusion seem to be literacy/numeracy, usefulness or recognitionof an actual state of things which would be difficult to change; the criteria
of exclusion may reflect social and moral concerns, some sort of notion of
what constitutes ‘liberal’ or ‘illiberal’ pursuits. Yet, such a distinction couldbe blurred: while the law quoted above is inclusive of ‘applied’ mathema-
ticians, another law excludes teachers of civil law and geometers from
privileges accorded to philosophers, doctors, rhetoricians and grammarians,and a regulation issued by Diocletian and Maximian states that a certain
exemption law includes teachers of liberal studies, but excludes calculators.
22
On the other hand, we know that accountants, called numerarii or tabularii ,
were very visible: practically every department of the administration, whether
civilian or military, had a couple of them on its staff.23 Another entry in the
codexes underlines that public accounting was a key job, not to be left intoservile hands:
We prescribe by general law that, if […] tabularii are needed, free
men be appointed and that apart from them nobody who is subject
to servitude be given access to this office. And if some master has
allowed his slave or tenant to deal with public acts – for we want
to punish collusion, not ignorance – the person himself, to the
degree to which it would benefit public utility, should pay thepenalty for the computations which were dealt with by his slave or
tenant, while the slave, having damaged the fiscus , should be taken
to the suitable flogging.
24
The case of astronomy/astrology also presents several layers. While we
know that the practice of astrology continued well into Christian times andbeyond, late ancient documents abound in scathing comments on, and harsh

218LATE ANCIENT MATHEMATICS: THE EVIDENCE
punishments for, astrologers.25 But they also introduce distinctions between
a good and a bad use of the knowledge of the skies: time-keeping and
calendar-making are allowed, if not encouraged.26 Constantine toned down
his own regulation, specifying that it was a bad thing to use magic arts
‘against the well-being of people’ but that there was nothing wrong in
forecasting the weather for agricultural purposes. On the whole, thedifference was made by the use one made of one’s knowledge – the import
of such knowledge being somewhat immaterial, what mattered were the
ethic or civic qualities of the mathematicus , whether he wanted to do evil or
to benefit ‘the divine duties and men’s work’.
27 Another explicit distinction
was made by Diocletian and Maximian: ‘It is to the public advantage to
learn and to practice the art of geometry. But the damnable astrological artis prohibited under any circumstances’.
28 The praise of geometry, which
here could mean land-surveying, suggests once again that this form of
knowledge had a rather high profile. A similar view of the relation betweenknowledge and its uses in society underlies a group of laws which establish
full fiscal immunity for architects, doctors, painters, carpenters and other
thirty-three technical categories, with the exhortation that, in the spare timefrom their activities, they teach their profession to other people, in particular
to their children.
29 One of these laws explicitly maintains that ‘one needs as
many architects as possible’ and exhorts the prefect of the African provincesto encourage towards that career any youths in their twenties who have had
a taste of the liberal studies; another describes the collective duties of
‘mechanicos et geometras et architectos’ as administering boundaries,measuring (it is not specified what) and looking after aqueducts.
In sum, late antiquity saw a continuation or even, arguably, an intensifica-
tion, due to the government’s fiscal policies and to its emphasis on efficientadministration, of mathematical practices such as accountancy, land-
surveying, and architecture. These forms of knowledge had a legal identity
and a public profile, which was ambiguously articulated, but employed apretty constant language of utility and public benefit.
Diophantus
We have two manuscript traditions, Greek and Arabic, for Diophantus’main work, the Arithmetic . Of the original thirteen, six books have been
preserved in Greek, seven in Arabic, but, while the first three books of both
traditions are the same, the remaining ones do not match. Scholars seem tobe of the opinion that the Arabic tradition has kept the original sequence,
and that the books numbered four to six in the Greek are extracts from the
original books eight to thirteen.
30 A rather recent article has attributed the
pseudo-Heronian Definitions to Diophantus, thus moving him to an earlier

219LATE ANCIENT MATHEMATICS: THE EVIDENCE
third-century or even a mid- to late first-century AD date, in this latter case
to have him a contemporary of Hero. The earlier third-century date would
identify the Dionysius to whom both Arithmetic and Definitions are addressed
as the leader of the Christian school at Alexandria.31
The debate surrounding the real text and the real date of Diophantus
mirrors the uncertainty about his place within Greek mathematics. He hasanachronistically been hailed, since at least early modern times, as the founder
of algebra, and consequently ‘ahead of his times’, whereas little study has
been devoted to situating him and his work within an ancient context(whether third or first century
AD). In fact, the arithmetic we find in
Diophantus looks rather unlike what we have encountered so far, if only for
its complexity and the dexterity of his solutions. An example will be usefulat this point – I have tried to reproduce the symbols used in the Greek
manuscript tradition, some of which (crucially, the minus sign and the sign
for an unknown quantity) are introduced by Diophantus in the preface tobook 1. Numbers are underlined when they are constant and not linked
with the unknown quantity:
To divide the prescribed number into two numbers twice, so that
one <number> from the first division has to one from the second
division a given ratio, while the remaining <number> from thesecond division has to the remaining from the first division a given
ratio. Let it be prescribed then to divide the 100 into two numbers
twice, so that the greater from the 1st division is 2ice the lesserfrom the 2nd division, while the greater from the 2nd division is 3
times the lesser from the 1st division. Let the lesser from the 2nd
division be posited, x, therefore the greater from the 1st divisionwill be 2x; the lesser from the 1st division therefore will be
100 –
2x; and since the greater from the 2nd division is triple of this, it
will be 300 – 6x. Besides, the sum of the 2nd division makes 100;
but the sum makes 300 – 5x; these are equal to 100, and it makes
the x <equal to> 40. To the initial matter. The greater from the 1st
division is posited 2x, it will be 80; while the lesser from the same
division 100 – 2x will be 20; while the greater from the 2nd division
300 – 6x will be 60; while the lesser from the 2nd division x will
be 40. And the proof is evident.32
A general problem is enunciated, which is then solved on specific numbers
– that is, rather than a general method to find all possible solutions to the
problem, one specific solution is provided. The move from general to
particular takes place as the enunciation of the problem is followed by aclause which puts forth specific quantities and determines the givens (in the

220LATE ANCIENT MATHEMATICS: THE EVIDENCE
case above, the given ratios). It is a move somewhat parallel to the setting-
out in Euclidean-style geometry, where a general enunciation is applied to
a specific diagram. After a number of manipulations that result in theunknown quantity being made equal to a known quantity, a clause indicates
a return of the argumentation ‘to the initial matter’, or, in other cases, a
moving on to the ‘positions’, that is, a return to the enunciation which isfinally filled in with solutions. In the proposition above, we also have a
concluding reference to an ‘evident’ proof. What is Diophantus referring
to? Perhaps he means that the specific solution ought to persuade the readerthat the problem in general is solvable (and indeed, a problem may be
considered solved if one solution is found), or also that it is left to the reader
to work out a more general procedure.
[An example of problem with four unknown quantities] To find
four numbers so that the sum of the four squares, if any of them is
added or subtracted, produces a square. Since the square on the
hypotenuse of all right-angled triangles, if one adds or subtractsthe two <squares> on the <sides> around the right angle, producesa square, let one search first of all four right-angled triangles which
have equal hypotenuses; this is the same as dividing a square into
two squares, and we have learnt to divide the given
/G6F into two /G6Fs
in infinite ways. Now then let us produce two right-angled triangles
from the least numbers, such as 3, 4, 5; 5, 12, 13. And multiply
either of the <triangles> posited by the hypotenuse of the other,and the 1st triangle will be 39, 52, 65; the <other> 25, 60, 65.
33
And they are rectangles which have the hypotenuses equal. Also,the 65 naturally is divided into squares in two ways, into the 16and the 49, but on the other hand also into the 64 and the
1. This
happens because the number 65 is the product of the 13 and the
5, either of which is divided into two squares. Now of the posited
<numbers>, the 49 and the 16, I take the roots; they are 7 and 4,and I form the right-angled triangle from two numbers, the 7 and
the 4, and it is 33, 56, 65. Similarly also the roots 8 and 1 of the
64 and the
1, and I again form from those a right-angled triangle
where the sides <are> 16, 63, 65. And four right-angled triangles
are produced which have the hypotenuses equal; returning to the
problem at the beginning, I posit the sum of the four, 65x, any ofthese four being 4x
2 times of the area, that is the 1st 4056×2, the
2nd 3000×2, the 3rd 3696×2, and the 4th 2016×2. And the four are

221LATE ANCIENT MATHEMATICS: THE EVIDENCE
12768×2 equal to 65x and it makes x equal to 65 of the 12768 part
[x = 65/12768]. To the positions. The 1st will be 17136600, the
2nd 12675000 of the same part, the 3rd 15615600 of the same
part, the 4th 8517600 of the same part; the part <will be>163021824. [x1=17136600/163021824; x2=12675000/
163021824, and so on]
34
This proposition is another example of Diophantus’ complex
manipulations: the reader is invited to go through it, play around with the
numbers, perhaps try alternative numbers and see if they fit the bill and canlead to a solution. The Arithmetic have no evident axiomatico-deductive
structure: we are presented with some operative notions at the beginning,
and then plunged into a sequence of problems which invite repetition orindividual experimentation. Persuasion is achieved through rehearsing, or
even reproducing, the mathematics in question. All solutions are specific,
rather than general; in fact, the knowledge embodied by the treatise andthat ideally should pass from the author to the reader is not so much a
wealth of results, of truths that can be memorized, as a set of problem-
solving skills. There is a rather strong sense then that the Arithmetic was an
advanced training ground – Diophantus himself says as much in the
introduction to the first book:
Knowing that you are anxious, my most esteemed Dionysius, to
learn how to solve problems in numbers, I have tried, beginning
from the foundations on which the subject is built, to set forth thenature and power in numbers. Perhaps the subject will appear to
you rather difficult, as it is not yet familiar, and the minds of
beginners are apt to be discouraged by mistakes; but it will be easyfor you to grasp, with your enthusiasm and my teaching; for keen-
ness backed by teaching is a swift road to knowledge. […] Now let
us tread the path to the propositions themselves, which contain a
great mass of material compressed into the several species. As they
are both numerous and very complex to express, they are onlyslowly grasped by those into whose hands they are put, and include
things hard to remember; for this reason I have tried to divide
them up according to their subject-matter, and especially to place,as is fitting, the elementary propositions at the beginning in order
that passage may be made from the simpler to the more complex.
For thus the way will be made easy for beginners and what theylearn will be fixed in their memory.
35

222LATE ANCIENT MATHEMATICS: THE EVIDENCE
And indeed Diophantus occasionally intervenes in the text with
instructions,36 or comments ‘this is easy’,37 or divides problems into
categories, so that the reader has a pointer to the solution: if two problemsbelong to the same category, they can be solved along similar lines; plus, if
one learns to recognize a certain category of problems, one will also know
how to go about solving them.
38 Or again, the reader is told what to do in
the case of equations with terms of the same species but different coefficients,
and how one should try and reduce everything to the equation of one, or at
most two, terms. Some of the manipulations most frequently used in thetext are clearly set out at the beginning of the first book, which also contains
basic notions, for instance of the different types of numbers or of the minus
sign. While defining this latter, Diophantus explains that in multiplicationa minus by a minus makes a plus and a minus by a plus makes a minus. The
focus throughout is on acquiring problem-solving arithmetical knowledge,
with the help of practice, repetition, a certain arrangement of the material,and several tricks of the trade. It is a fascinating combination: amazing
complexity and great abstraction on the one hand, as represented above all
by the introduction of what we can call negative quantities. On the otherhand, practical concerns, as represented for instance by proposition 5.30,
which is cast as an epigram:
A person has mixed eight-drachmae and five-drachmae cups, and
has been ordered to make a profit for people who sail together.
For the price of all of them, he has paid a square number which,added to a given number, gives you yet again a square whose root
is the total number of cups. Distinguish, O youth, and say how
many were the eight-drachmae and how many the five-drachmaecups.
39
The apparent ‘anachronicity’ of Diophantus’ work may be striking if we
set it against Euclid’s or, for that matter, Nicomachus’ arithmetic, but it
diminishes if we look at other, less obvious, texts for terms of comparison.
For instance, mathematical problems in the form of little everyday anecdotes
or in verse are known from other late ancient sources: for instance, the
booklet mentioned on p. 214, or book 2 of Pappus’ Mathematical Collection
(see the next section). There are also, of course, parallels with Egyptian
mathematics and with Hero’s Metrica , especially for its recourse to specific
solutions rather than general procedures. That Diophantus was familiarwith the format of deductive mathematics as well emerges from the much
shorter treatise On Polygonal Numbers , whose argumentative structure is
Euclidean in style. His work might then be characterized as a hybrid – half-way between Greek and Egyptian arithmetic, between theory and practice,

223LATE ANCIENT MATHEMATICS: THE EVIDENCE
between general and particular, or even between the Christian bishop to
whom it may have been addressed, and the pagan martyr Hypatia who
wrote a commentary on it. A hybrid, perhaps – but then no more so thanmany other ancient mathematical texts, late or otherwise.
Pappus
Pappus of Alexandria’s surviving works are a Mathematical Collection in
eight books, seven of which are extant; commentaries on Ptolemy’s Syntaxis ,
books 4 and 5, and on Euclid’s Elements , book 10 (this latter in an Arabic
translation); and a Geography (in an Armenian paraphrase). The Collection
may have been put together at a later stage, since the various books differ incharacter, are not all addressed to the same person and cover many different
subjects: arithmetic (book 2); means, including the two mean proportionals
(book 3); the five regular solids (books 3 and 5); mathematical paradoxes(book 4); linear curves (book 4); isoperimetrism (book 5); astronomy (book
6); advanced geometry (book 7); mechanics (book 8). Much of the material
comes from, or relates to, earlier texts. The relation between Pappus and hissources is very complex: he can endorse them, criticize them, or rework
them in several ways. We will take a closer look at his use of the past in chapter
8 – here for reasons of space we will limit ourselves to a description of someof the contents of the Collection .
Pappus preserves pieces of ancient mathematics that would be little known
otherwise: linear curves and ‘analytical’ geometry are just two examples.Take the first: Archimedes devoted a book to the spiral, and Eutocius talks
about the cissoid and the cochloid/conchoid in the context of doubling the
cube, but Pappus provides additional information about the properties andfurther uses of these curves, for instance, in the trisection of the angle.
Moreover, he describes the quadratrix (see Diagram 7.1):
A certain line was applied to the squaring of the circle by Deino-
stratus and Nicomedes and some other more recent ones; it takes
the name from its characteristic property; for it is called quadratrix
by these <people> and it has the following origin. Let a square
ABCD be put forth and around the centre A let a circumference
BED be drawn, and then let AB be moved in such a way that BC,
remaining always parallel to AD, follows the point B which is
carried along BA, and at the same time AB being moved uniformly
traverses the angle BAD, that is the point B <traverses> BED, and
BC passes through the straight line BA, that is the point B is brought
along BA. It clearly happens that the straight line AD coincides
with both AB and BC. This movement having been originated,

224LATE ANCIENT MATHEMATICS: THE EVIDENCE
the straight lines BC BA cut each other in transit along a certain
point which is always being carried along with them, by whichpoint a certain line is drawn in the space between the straight lines
BAD and the circumference BED concave with respect to those,
<a certain line> which is BZH , which seems to be useful to find
the square equal to a given circle. Its original characteristic property
is the following. If any <straight line> whatever is drawn to the
circumference, the circumference will be to ED as the straight line
BA to ZH; this in fact is evident from the origin of the line.
Pappus goes on to report that the description of the origin of the quadra-
trix was criticized by Sporus on the basis both of circularity and of indeter-
minacy. According to the first objection, the curve cannot be constructed
unless the ratio of the velocities of AB and BC is already given, or unless the
ratio of a radius to a quadrant is already given (Sporus seems to assume that
the two are equivalent). But the condition that the ratio of a radius to a
quadrant be given cannot be satisfied without circularity, because it amounts
to squaring the circle, and the quadratrix is itself being devised in order to
square the circle. Sporus’ second objection revolves around the fact that theexact position of the points of intersection which constitute the curve cannot
be determined:
The extremity of the curve which is used for the squaring of the
circle, that is the point by which the line AD is cut [ sc. the point
H], is not found. […] Even if the lines CB and BA, having been
drawn, come back to the same point, they will coincide with theDiagram 7.1

225LATE ANCIENT MATHEMATICS: THE EVIDENCE
line AD and will produce no section by intersecting each other.
The intersection in fact ends before the coincidence along the line
AD, the same intersection which instead should become the limit
of the curve, in which the two lines meet the line AD. Unless
someone says that the curve is imagined prolonged until it meets
AD – this does not follow from the principles set at the beginning,
but rather would assume the point H, having taken as a premise
the ratio of the line to the circumference.40
Pappus subsequently tries to get round the problem of indeterminacy
with an alternative proposition. The sections concerning the quadratrix are
part of a larger account, book 4, that includes mathematical paradoxes andalso the following classification:
We say that there are three kinds of problems in geometry, and
some of them are called planar, some solid, some linear. Those
which can be solved by means of straight line and circumference
are justly called planar, because the lines by means of which theseproblems are discovered have their origin in a plane. Those
problems whose solution is found applying one or more of the
conic sections are called solid, because for their construction it isnecessary to employ surfaces of solid figures (I mean it is necessary
to employ conic surfaces). A third kind of problems remains, the
so-called linear; other lines in fact, apart from the ones wementioned, are applied to the construction [of a problem]; their
origin is more variegated and rather constrained and they are
generated from more disorderly surfaces and from interwovenmotions. Those are the lines found in the so-called loci with respect
to a surface, and others more variegated than these […]. They have
many wondrous properties. […] lines of this kind are spirals,quadratrices, cochloids, cissoids. It seems to the geometers that it
is no small mistake when [the construction of] a planar problem
is discovered by someone by means of conics or of linear [curves],
and in general when it is solved by means of a kind not its own.
41
Although curves like the quadratrix were problematic objects, not easily
definable, they were at the same time a fundamental tool for problem-solving.
As in Diophantus, teaching or learning about problem-solving implies somekind of ordering of the material – for instance, the definition of common
procedures or of the tools employed, which subsumes functional, yet
variegated and unorganized, curves into a systematized and comprehensivewhole. Moreover, the classification reported above is prescriptive, because,

226LATE ANCIENT MATHEMATICS: THE EVIDENCE
right at the end, it formulates a rule about problems and the procedures
appropriate to their solution. It is thus an attempt at ordering in a strong
sense. Operations of this kind required Pappus to have a whole wealth ofmaterial at his disposal, so that he could take stock of the different solutions
that had already been given to problems of each kind and put them into
categories which, he then claimed, had a value for future problem-solving.
The Collection abounds in ‘meta-mathematical’ passages where Pappus
steps outside first-order discourse and classifies, defines or systematizes. For
instance, he opens book 7 with a definition of analysis and synthesis, and inthe course of the text also provides short definitions of problems, theorems,
porisms, loci, neuseis . It is a way for him not only to clarify things for the
reader, but also to set reference points. Book 7, sometime called the ‘treasureof analysis’, complements a group of canonical texts including Euclid’s Data
and Porisms , Apollonius’ Conics, Plane loci, Section of a Ratio and Section of
an Area and Eratosthenes’ On Means . Book 6 is a similar companion text,
this time to a set of astronomical works, ranging from Autolycus’ Moving
Sphere to Theodosius’ Sphaerics . In both cases, the mathematical results
from the past are annotated in various ways. The wealth of materials fromthe past in book 7 is subject to various orderings: there is a sense of an
historical order, because the canon or treasure of analysis at hand is presented
as the outcome of a process that has taken place over time, through thecontributions of several people. Or again, Pappus orders each of the texts in
book 7 as exemplified here by his description of Apollonius’ Section of an
Area:
The first book of the section of an area has 7 modes, 24 cases, and
7 diorisms, of which 4 are maxima, three minima. And there is amaximum in the second case of the first mode, as well as in the
first case of the 2nd mode and in the 2nd of the 4th and in the
third of the 6th mode. The one in the third case of the third modeis a minimum, as is that in the 4th of the 4th mode, and in the
first in the sixth mode. The second book of the section of an area
contains 13 modes, 60 cases, and for diorisms those of the first,
because it reduces to it. The first book has 48 theorems, the second
76.
42
A list is compiled of how many cases, modes and theorems a book
contains, and sometimes it is specified what the principal propositions are.All this amounts to a systematization of the canonical texts; knowledge of
them has to be not only comprehensive, but has to follow the right order,
use the right reference points. Of the several things at work in book 7, oneis definitely the desire to enable, or indeed enpower, the reader to solve

227LATE ANCIENT MATHEMATICS: THE EVIDENCE
problems in advanced geometry; it is implied that this derives not just from
what you know but from how you know it, from the contents and from a
certain approach to them. I have mentioned above that the mathematicalresults from the past are annotated by Pappus in various ways. I report an
example, again from book 7, relative to Apollonius’ Section of a Ratio (see
Diagram 7.2):
[Problem for the second of the section of a ratio, useful for the
summation of the 14th mode]. Given two straight lines AB, BC,
and producing line AD, to find a point D that makes the ratio BD
to DA the same as that of CD to the excess by which ABC together
exceeds the line that is equal in square to four times the rectangle
formed by ABC. The combination cannot be made in any other
way, unless DE, AC together are equal to the excess EA and all DA
to all AB and also that EA, AB, BC have to one another the ratio of
a square number to a square number, and that CB is twice DE. Let
it be done, and let the excess be AE, for we have found this in the
previous. Then as BD is to DA, so CD is to AE. And alternating
and separating area to area, therefore the rectangle formed by BC,
EA is equal to that formed by CDE. But that formed by BC, EA is
given. Therefore that formed by CDE is also given. And it lies
along CE
, which is given, exceeding by a square. Therefore D is
given. It is synthesized as follows […]43
Unfortunately, in this as in the majority of cases, we do not have the
original text against which to assess the extent of Pappus’ intervention. Let
us note, however, his preoccupation that all the elements of a problem should
be fully determined, and his attention to different cases and modes, with
related different conditions of possibility and impossibility of a solution to
the problem. Taken together with Pappus’ emphasis on counting anditemizing the contents of a book, and on definition and classification, it
would seem that he aimed to map the territory of mathematical knowledge
as extensively as possible. That Pappus’ interests were wide-ranging isevidenced not just by the sheer number of sources that seem to have been
available to him, but also by the variety of topics he chose to discuss. Book
2, for instance, which as we have it is incomplete, deals with rules for themultiplication of multiples of ten. It consists almost exclusively of specificDiagram 7.2

228LATE ANCIENT MATHEMATICS: THE EVIDENCE
examples rather than general propositions, and ends with an epigram whose
letters are transformed into the equivalent numbers (remember that he was
using the Milesian notation) and then multiplied by each other. Thefollowing is an example:
Let a multitude of numbers be the multitude A, of which
<numbers> each is less than a hundred and divisible by ten, and
another multitude of numbers, the multitude B, of which
<numbers> each is less than a thousand and divisible by a hundred,and let it be required to tell the product of the As and Bs without
multiplying them. Let then the <numbers> H, the units 1, 2, 3
and 4, be basic numbers of the <numbers> A and let the <numbers>
H, the units 2, 3, 4 and 5, be basic numbers of the <numbers> B
and assuming that the product of the basic numbers, of 2, 3, 4, 2,
3, 4, 5 that is, E, is of 2880 units, let the multitude of <numbers>
A added to twice the multitude of <numbers> B be first divided
by four, with the result Z, for it divides them <exactly>. And
Apollonius proves that the product of all the As and Bs is as many
myriads as there are units in E with the same name as the number
Z, that is 2880 triple myriads. In fact one myriad with the same
name as Z, that is triple, multiplied by E, that is 2880, makes the
number of the products of the <numbers> A and B. Indeed, the
product of the numbers of the As and the Bs is as many myriads as
there are units in E, with the same name as the number Z. But let
the multitude of <numbers> A, added to double the multitude of
<numbers> B, divided by four, first have the remainder of one.
And Apollonius gathers that the product of the numbers A and B
is as many myriads with the same name as Z, multiplied by ten
times E; but if the aforesaid multitude divided by four has the
remainder of two, the product of the numbers of the As and Bs is
as many myriads with the same name as Z, multiplied by a hundred
times the number E; if the remainder is three, the product of those
numbers is equal to as many myriads with the same name as Z
multiplied by a thousand times the number E.
44
Apart from the references to Apollonius (from a work now lost), the
only justification of the validity of Pappus’ multiplication method is that,
once it is carried out over and over again on several sets of specific numbers,one can see that the method works. Thus, the arithmetic of book 2 does not
seem to belong to an Euclidean or a Nicomachean tradition, if indeed those
labels correspond to anything definite, and it is tempting to see a parallel ofits many exercises and of its final piece of poetic mathematics in the work

229LATE ANCIENT MATHEMATICS: THE EVIDENCE
of Diophantus. The ‘applied’, ‘concrete’ mathematics of counting and
calculating is also a possible scenario. Another part of Pappus’ work with a
probably close relation to practice is book 8, devoted to mechanics, a congerieof disciplines (more than a single discipline) which is praised extensively in
the introduction. Pappus declares that mechanics has been highly esteemed
by philosophers and by anybody interested in learning; emphasizes thatArchimedes, far from looking down on it as some have said, owed his enor-
mous reputation to his mechanical feats, and concludes by saying:
Geometry is in no way injured, but is capable of giving content to
many arts by being associated with them, indeed it, being as it
were mother of the arts, is not injured by dealing with constructionof instruments and architecture; indeed it is not injured by being
associated with land-division, gnomonics, mechanics and sceno-
graphy, on the contrary, it seems to favour these arts and also to beappropriately honoured and adorned by them.
45
These statements about the complementarity and mutual benefit of
mechanics and geometry are representative of Pappus’ approach throughout
book 8. Although he does not come across as an actual machine-builder
(unlike, say, Hero), he is very favourable to the use of instruments ingeometry. In fact, his own version of the duplication of the cube employs a
moving ruler, and he claims that such procedures are often much more
convenient than methods involving conics. He occasionally discussesmathematical problems useful for architecture – an example from book 8
follows (see Diagram 7.3).
In mechanics the so-called instrumental problems are separated
from the domain of geometry, for instance […] that proposed by
the architects about the cylinder damaged along both bases. For
they ask, given part of the surface of a right cylinder, of which nopart of the circumference of the bases is preserved whole, to find
the thickness of the cylinder, that is the diameter of the circle from
which the cylinder has originated. It is found with the followingprocedure. Let two points A B be taken on the given surface and
with these centres with one opening <of the compass> let first the
point C be marked on the surface, and again with these centres A
B and the opening greater than the first let D be marked, and with
another opening E, and with another Z, and with another H. The

230LATE ANCIENT MATHEMATICS: THE EVIDENCE
5 points C D E Z H will then be in one plane because the <line>
joining each of them as vertex of an isosceles triangle to the half-
point of the straight line joining A B as common basis of the
triangles is perpendicular to AB and the 5 lines have originated in
one plane, and it is clear that <so have> the points C D E Z H . We
set these in a plane as follows; from the three straight lines joining
the triangle C D E in the plane let HKL be put together, and from
the three joining D E Z <let> KLM <be put together>, from the
three joining the points E Z H let the triangle LMN be put
together; therefore the three triangles HKL KLM LMN will be
taken instead of the triangles CDE DEZ EZH . If then we draw an
ellipse around the points H K L M N , its lesser axis will be the
diameter of the circle which produced the cylinder.16
In sum, Pappus’ work is testimony to the variety of sources, interests,
even audiences of late ancient mathematics. Book 3 was sent to a woman,
Pandrosion, who taught mathematics, and it mentions the philosopherHierius, who was interested in geometry. It also criticizes a contemporary
for an incorrect solution to the duplication of the cube. Book 5 again
criticizes certain philosophical approaches to mathematics – the tone of theDiagram 7.3

231LATE ANCIENT MATHEMATICS: THE EVIDENCE
argumentation indicates that the addressee, Megethion, may have been a
non-specialist.47 Books 7 and 8, written for Hermodorus, are instead more
advanced. Books 2 and 8 suggest links with mathematical practices such ascalculation and architecture; the commentary on Ptolemy includes a rather
detailed description of how to make an astronomical instrument. In sum,
while Pappus’ works looked back to the past, they were also firmly rootedin the present.
Eutocius
Eutocius of Ascalona (sixth century AD) wrote commentaries on some of
Archimedes’ works and on Apollonius’ Conics , and probably taught
philosophy. He may have been connected to the group of architects who
built the cathedral of Hagia Sophia in Constantinople, because he mentions
one of them, Anthemius of T ralles.48 The commentary on Archimedes’ Sphere
and Cylinder is addressed to an Ammonius, who is described as an expert of
philosophy in general and mathematics in particular, while Eutocius
represents himself as young and relatively unexperienced. He also statesthat this, the first commentary he writes on Archimedes, is the first com-
mentary on Archimedes to be written at all. This may be due to objective
difficulty: Eutocius says at the beginning of the commentary on SC 1 that
Archimedes’ mathematical style is rather obscure and requires a lot of
explanations and filling in of demonstrative steps. The Measurement of the
Circle , on the other hand, to which he devotes himself next, is much clearer
because it deals with one issue; also, squaring the circle, Eutocius says, has
been of great interest to the philosophers of old and, according to a biographer
of Archimedes, is necessary for the needs of life. Its few propositions, unlikethose of SC, do not skip any steps and can be understood by the average
reader.
Eutocius’ declared intention is carried out in the body of his work: he
tends not to comment on whole propositions, but to pick up a detail that
needs clarifying. A representative example is given below.
[On SC 1.13: some phrases are quoted and then explained (see
Diagram 7.4)] Let us imagine then a <polygon> circumscribed and
inscribed in the circle B, and circumscribed to the circle A and similar
to that circumscribed to the circle B. How a polygon similar to
another inscribed is inscribed in a given circle, is clear, and is stated
also by Pappus in the commentary to the Elements; on the other
hand, <on the topic of how> to circumscribe to a given circle a

232LATE ANCIENT MATHEMATICS: THE EVIDENCE
polygon similar to that circumscribed to another circle we do not
have anything analogous that has been said; which is to be saidnow. Let us inscribe then in the circle A a <polygon> similar to
that inscribed in the circle B, and to the same <circle> A <a
polygon> similar to that inscribed, as in the theorem 3; and it willbe similar also to that circumscribed to B. And since the rectilinear
figures circumscribed to the circles A, B are similar, they have the
same ratio as the power of their radii. This has been proved for the
inscribed <figures> in the Elements, but not for the circumscribedones; it will be proved thus. Let us imagine the inscribed and
circumscribed rectilinear figures separately and from the centres
of the circles let KE, KM, KH, KN be drawn; it is evident then
that KE, KH are radii of the circles around the circumscribed
polygons and that to each other they are in power as the
circumscribed polygons. And since the <angles> KEM , KHN are
half of the angles in the polygons, while the polygons are similar,
it is clear that they are also equal. But also the <angles> M, N are
right; therefore the triangles KEM , KHN have equal angles, and it
will be as KE to KH, so KM to KN; so also the <squares> on them.
But as the <square> on KE to that on HK, so the circumscribed
<polygons> to each other; and therefore as the <square> on KM
to that on KN, so the circumscribed <polygons> to each other.
49
Eutocius will also explore the validity of a statement by considering every
possible variation in the diagram, or assess the appropriateness of the
terminology, or add demonstrations that confirm that Archimedes’ propo-Diagram 7.4

233LATE ANCIENT MATHEMATICS: THE EVIDENCE
sitions remain valid in any case. He will supply a synthesis when Archimedes
only has an analysis; or insert extra demonstrative steps when Archimedes
has left them implicit; or, in the numerous cases where, in an exhaustionprocedure, Archimedes considers circumscribed and inscribed figures at the
same time, he will consider them separately. On occasions, the clarification
of a detail turns into a long excursus: at SC 2.1, Archimedes had assumed
that two mean proportionals, necessary for a certain construction, could be
found, but had not explained how. Eutocius supplies twelve different
solutions ascribed to earlier authors ranging from Plato (a likely misattribu-tion) to Pappus. Some of the solutions are quite similar to each other (a fact
underlined by Eutocius himself), so the rationale behind his providing a
full anthology may have been, at least in part, to show that he had access toa wealth of ancient sources.
In the commentary to the ‘much clearer’ MC, Eutocius concentrates on
the numerical proposition 3, supplying all the calculations of whichArchimedes only provides the end result, and often indicating (they are all
approximated values) how much they fall short or in excess of the more
exact figure. In his work on Archimedes’ Equilibria of Planes , he adds a
proposition whereby the geometrical objects in question are given a number,
i.e. measured; he can then take a certain element in the diagram as the unit
and work out a numerical relationship between that unit and the otherelements in the proposition.
50
In the commentary on Apollonius, who himself had a marked interest
in generalizations and in the study of sub-cases, Eutocius again takes thereader through variations on the elements of a diagram, in fact produces
sub-cases from a manipulation of the diagram, and confirms the validity of
the proposition in each case. He occasionally quantifies the total number ofpossible cases.
51 Or again, he invites the reader to reflect on the overarching
demonstrative structure – for instance, he remarks on how the first ten
propositions of Apollonius’ Conics work as a sequence.52
As well as the contents, Eutocius is keen to explain what the intent of a
mathematical treatise was, and that some interpreters misunderstood the
aims of earlier authors, and thus did not read their works correctly:
The numbers mentioned by [Archimedes] have been sufficiently
explained; one must know on the other hand that Apollonios of
Perga in the Okytokion proved the same by means of different
numbers and with a greater approximation. If this seems to bemore accurate, on the other hand it is not useful for Archimedes’
purpose; for we have said that his purpose was in that particular
book to find the approximation for the uses in life. Thus neitherSporus of Nicaea will be found right blaming Archimedes for not

234LATE ANCIENT MATHEMATICS: THE EVIDENCE
having found with accuracy to what straight line is equal the
circumference of the circle […] [and] saying that his teacher Philo
from Gadares had led to more accurate numbers than those foundby Archimedes […]; all these people, one after the other, seem to
ignore his purpose.
53
Eutocius establishes a very close connection between himself and
Archimedes: on top of explaining the details of his proofs, and on top of
deciding which of his texts were obscure and which were not, he claims tounderstand Archimedes’ real intentions about the uses of his work. If indeed
this was the first commentary on Archimedes to be written, Eutocius emerges
as the only authority; and definitely so, by contrast with others who haveallegedly misunderstood the Syracusan mathematician.
There is yet another aspect to Eutocius’ approach: we know (he tells us)
that the very same works of Apollonius which he was clarifying were beingedited by him. He had several versions of the Conics , and compared and
collated them to obtain one text. Not only was Eutocius’ intervention on
the past a primary feature of his mathematics, the past itself as he had it(and probably as we have it now) was to some extent the result of Eutocius’
intervention.
The philosophers
Iamblichus of Chalcis (mid- to late second century AD) was one of the most
influential philosophers of his time. He is said to have inspired the policies
of the emperor Julian; a letter mistakenly attributed to the same emperor
declares Iamblichus ‘the saviour, so to speak, of the whole Hellenic world’.54
Several of his books survive, including a paraphrase (usually called a commen-
tary) of Nicomachus’ Introduction to Arithmetic ,55 a Life of Pythagoras , an
Exhortation to Philosophy , a treatise on General Mathematical Knowledge (the
four were originally part of a vast work On Pythagoreanism ), and an account
of the Egyptian Mysteries . We also have fragments of his commentaries to
several works by Plato and Aristotle, and an Arithmetic theology has been
attributed to him, but is probably spurious and loosely based on Nicomachus’
lost work by the same title.
Iamblichus’ most important reference points seem to have been ‘the
divine’ Plato, to some extent Aristotle, whose psychology he was influenced
by,56 and above all Pythagoras. His philosophy, generally classified as neo-
Platonism or neo-Pythagoreanism, is far too complex to be summarized
here: let us just say that Iamblichus espoused a general Platonist cosmology,
as described in the Timaeus . He believed that there is an order in the world,
and that there is a correspondence between human souls, the divinity and

235LATE ANCIENT MATHEMATICS: THE EVIDENCE
the cosmos. Specifically, souls have fallen into the bodies but aspire to go
back to where they belong (a sort of heaven) and in order to do so they must
live philosophically, rid themselves of material things and perform theurgicrites, through which they can experience a sort of ecstasy. This did not
exclude participation in political life: in fact, as in Plato’s Republic , the wise
man’s love for other human beings leads him to counsel and guide them.
The role of mathematics in all this was enormous. Not only did
Iamblichus take up some Platonic views on the question (mathematics as
dianoia or as gymnastics that trains the soul to turn from material to abstract
things): he developed them to the point of positing the One and the Many
(plethos , literally ‘multitude’, ‘quantity’) as the very principles of the universe
and of any knowledge we may have of it. As in the Timaeus, the demiurge
produced the world mathematically, therefore time, space, the motions of
the heavenly bodies, the generation and corruption of plants and animals,
the musical concords, all are regulated according to mathematical propor-tions. The soul itself is a number for Iamblichus, but not in an arithmetical
sense, because it embraces geometry, music and sphaerics (here equivalent
to astronomy) as well; the theurgic rites which are indispensable for thesoul’s ascent to the divinity seem to have included mathematical rituals.
Mathematics was thus in a sense even more fundamental than philosophy
itself: this is neatly encapsulated by various statements of Iamblichus’ to theeffect that Plato took second place to Pythagoras or to the Pythagoreans,
and his doctrines to their pervasive notion that everything is number.
If mathematics was so important, it was also important to state clearly
what mathematics one was talking about, and what it consisted of. This is
where the books on Nicomachus and on Common Mathematical Knowledge ,
plus some lost works on mathematics in ethics and politics, come in. As faras strictly mathematical content goes, Iamblichus concentrates on definitions
of odd, even, odd-even, even-odd, polygonal, cubic, perfect and other types
of numbers, their properties and classification. Arithmetical notions areoften visualized via geometrical figures such as a line, an angle or a gnomon,
or more pictorial images. He describes a square number as a race-course,
formed of successive numbers from the unit to the root of the square, which
is a ‘turning-point’.
57 We do not have much about geometry per se , and
Iamblichus’ definition of point and line as flowing from a point fit in withhis Nicomachean model. Again like Nicomachus’, Iamblichus’ treatises do
not have an axiomatico-deductive structure, and he uses specific examples
rather than general proofs. In fact, he shows some disrespect for traditionalmathematical authorities:
Here then it is a blatant mistake by Euclid not to distinguish even-
odd from odd-even, nor to recognize that one of them on the one

236LATE ANCIENT MATHEMATICS: THE EVIDENCE
hand is opposed to the even in an even way, the other on the other
hand is the mixture of both […] For he says thus: an even-odd number
is that measured by an even number odd-times. Odd-even is thesame; for it is measured by an odd number even-times, as for instance
6; for if on the other hand we say twice three, it is even-odd, if
instead three times two, it is odd-even; it is totally simple. But inthe third of the Arithmetic books he confuses the three in one, being
subject evidently to the appearance of the name. For he says: if an
even number has the half odd, then it is odd even-times and evenodd-times, evidently saying the same as those before. Next he adds:
if an even <number> has neither the half even nor is produced by
multiplication from the unit, then it is at the same time even even-times and odd even-times and even odd-times. And Euclid thus;
for us on the other hand let rather what is commonly formed and
shaped from both […] be called the third type.
58
This passage is not the only occasion on which Iamblichus criticizes
Euclid, and is remarkable for several reasons. First, it is testimony to thefact that mathematics, like other cultural practices, was multi-stranded,
multi-layered and conflictual. Even well-respected authors like Euclid could
be attacked (and no mincing of words) for alleged incompetence. Second,it is evidence of the fact that there was something to get competitive about:
mathematics was so important within Iamblichus’ conception of the world
that it was crucial to get it right. The points of contention (odd-even, even-odd) may appear banal to us, but for Iamblichus they must have had such a
weight as to render Euclid’s mistakes extremely significant. Third, by the
time Iamblichus was writing, Euclid represented a (prestigious) mathematicalpast. Iamblichus was keen to create continuities between his philosophy
and that of Pythagoras and Plato. In mathematical terms, that meant often
attributing one discovery or another to the Pythagoreans, or enlisting to thePythagorean field people who did not belong to it, such as Eudoxus, or
composing micro-histories of particular fields, such as means theory, where
Pythagorean import was presented as crucial. That the promotion of one’s
view of mathematics should take place through a rewriting or appropriation
of the mathematical past is one of the prominent features of this period,and we will discuss it in chapter 8.
The reasons behind Iamblichus’ mathematical choices, such as that of
Nicomachean arithmetic (basically assimilated to Pythagorean arithmetic)over a Euclidean one, are explained as follows:
Pythagorean mathematics is not like the mathematics pursued by
the many. For the latter is largely technical and does not have a

237LATE ANCIENT MATHEMATICS: THE EVIDENCE
single goal, or aim at the beautiful and the good, but Pythagorean
mathematics is preeminently theoretical; it leads its theorems
towards one end, adapting all its assertions to the beautiful andthe good, and using them to conduce to being.
59
If one believes that mathematics has to do with the larger order of theuniverse, that it is much more than mere crunching theorems or attaining
formulas for areas or volumes, Iamblichus’ choice makes perfect sense. Neo-
Pythagorean or neo-Platonist mathematics has often been dismissed as ratherincoherent and unoriginal, but passages like the one quoted above show
that in fact these philosophies valued mathematics very highly, because of
the possibility through it to acquire knowledge of the universe and ultimatelyof your own self. That said, it has to be kept in mind that neo-Pythagoreans
and neo-Platonists were not sharply differentiated, and that they did not
present a unified front. In fact, their positions on mathematics, and particu-larly on what version of mathematics should be adopted, were rather varied.
Proclus, for instance, although (or so the story goes) it was revealed to him
in a dream that he was the reincarnation of Nicomachus, chose Euclid as theauthority to comment upon. He also studied Ptolemy, on whose Harmonics
the earlier neo-Platonist Porphyry had written a commentary. A fellow-
student of Proclus, Domninus of Larissa, wrote a mathematical treatise whichhas been seen as a return to Euclid against current Nicomachean trends.
60
Let us turn to Proclus. As well as the Commentary on the First Book of
Euclid’s Elements , his rather prolific production includes The Elements of
Theology , Physics, A Sketch of the Astronomical Positions , and a number of
other commentaries, mainly to Plato’s works. Proclus believed that
mathematics was intrinsic to the fabric of things, and that its languageenabled one to articulate the knowledge of reality: the relation between
whole and parts could be expressed in terms of divisibility; the bonds between
disparate parts of the same universe could be seen as proportions;
61 the
generation of beings from the divine Monad, the One which is the principle
of everything, was a sort of multiplication, and so on. Mathematical objects
themselves are intermediate between the intelligibles and the sensibles and
consequently reflect the human condition – in fact, the soul itself is inter-
mediate between indivisible principles and divided corporeal ones, betweeneternal existence and temporal activity.
62 Mathematics is then the knowledge
most appropriate to human nature. Proclus also talks of a ‘general’ or ‘whole’
mathematics, which should deal with the whole spectrum of its applications,from lower forms such as mechanics, optics and catoptrics to the knowledge
of divine beings.
63
The declared aim of Proclus’ commentary to the Elements is to show
that Euclid is at bottom a Platonist. It opens with a long discussion of the

238LATE ANCIENT MATHEMATICS: THE EVIDENCE
nature and use of mathematics, and of the necessity for the good geometer
to make the right distinctions and categorizations. When dealing with
Euclid’s basic definitions and first principles especially, Proclus’ mode ofoperation is not different from his philosophical style of commentary: he
expands the meaning of the text in a number of possible directions,
comparing and contrasting the views of previous interpreters, eventually toshow (in the majority of the cases) that every utterance of the author works
from various angles, from the cosmological to the ontological. This is an
indication that all levels of existence, the microcosm and the macrocosm,are connected and part of the same reality. Thus, explaining the definition
of point amounts to showing what that definition means ontologically,
cosmologically, epistemologically – it amounts to showing that it means
something in each one of those domains, that it is much more than just a
geometrical notion, and that by understanding it in full we can increase our
knowledge in a number of unsuspected directions. The fact that everythingin Proclus means a lot of things at the same time has elegantly been termed
semantic superabundance.
64 An example of this, about the distinction
between equilateral, scalene and isosceles triangle, is the following:
From these classifications you can understand that the species of
triangle are seven in all, neither more nor less. […] You can alsounderstand from the differences found in their sides the analogy
they bear to the orders of being. The equilateral triangle, always
controlled by equality and simplicity, is akin to the divine souls,for equality is the measure of unequal things, as the divine is the
measure of all secondary things. The isosceles is akin to the higher
powers that direct material nature, the greater part of which isregulated by measure, whereas the lowest members are neighbours
to inequality and to the indeterminateness of matter; for two sides
of the isosceles are equal, and only the base is unequal to the others.The scalene is akin to the divided forms of life that are lame in
every limb and come limping to birth filled with matter.
65
The relation between mathematical objects and their wider meanings is
indicated by terms like ‘analogy’, ‘symbol’, ‘akin’ or ‘likeness’. Proclus’ styleof commentary is slightly different when he is working on the propositions
themselves of the first book of the Elements (rather than the preliminary
material): once again he informs the reader of other interpretations andassesses them, but he also glosses the text by explaining anything that in
Euclid is left unsaid. For instance, what principles or previous propositions
the proof appeals to; what other propositions the statement at hand is usedfor; in the case of indirect proofs, what principles the proof contradicts. He

239LATE ANCIENT MATHEMATICS: THE EVIDENCE
sometimes underlines the relevance of a problem or theorem for other fields,
such as land-surveying or astronomy,66 or the rationale of Euclid’s vocabulary,
structure and phrasing. An example is given in the following section.
[Commenting on Elements 1.14, ‘If with any straight line, and at
a point on it, two straight lines adjacent to one another and not
lying on the same side make the adjacent angles equal to two rightangles, the two straight lines will be in a straight line with oneanother’, (see Diagram 7.5).] That it is possible for two adjacent
lines drawn at the same point on a straight line and lying on the
same side of it to make angles on the straight line equal to tworight angles we can demonstrate thus, after Porphyry. Let AB be a
straight line. Take any chance point on it, say C, and let CD be
drawn at right angles to AB, and let angle DCB be bisected by CE.
Let a perpendicular EB be dropped from E, let it be extended, and
let BF be equal to EB and CF be joined. Then since EB is equal to
BF, BC is common, and these sides contain equal angles (for they
are right angles), base EC is equal to base CF, and all corresponding
parts are equal. Angle ECB is therefore equal to angle FCB. But
angle ECB is half of a right angle, for a right angle was bisected by

240LATE ANCIENT MATHEMATICS: THE EVIDENCE
EC; hence FCB is half of a right angle. Angle DCF is therefore one
and one-half of a right angle. But DCE is half of a right angle;
therefore on line CDand at point C on it there are two adjacent
straight lines CE and CF lying on the same side of it and making
with it angles equal to two right angles, CE making an angle equal
to half of a right angle, and CF an angle equal to one and one-half
of a right angle. Thus to prevent our drawing the impossibleconclusion that CE and CF, which make angles with DC equal to
two right angles, lie on a straight line with one another, our
geometer had added the phrase ‘not lying on the same side’. Hencethe lines that make with a line angles equal to two right anglesmust lie on opposite sides of the line, though starting at the same
point, one extending to this and the other to that side of the straight
line.
67
Furthermore, Proclus is interested in classifying and arranging his
material, for which purpose he deploys a whole battery of labels and
definitions: he distinguishes between theorem and problem, returning to
the issue over and over again; he divides a mathematical proposition intoparts; he has names for various kinds of converse propositions, for
interventions on the text in the form of sub-cases or objections; he defines
locus theorems and among them distinguishes plane from solid ones.
68 As
in Iamblichus, establishing the proper order is fundamental, it is what
distinguishes the good geometer from, for instance, some past authors who,
in his view, had muddled categories or misunderstood the place of someprinciples or statements. Alternative proofs are sometimes rejected because
they violate the order of the Elements as Proclus had them, in that they
present things that are not simple enough for that part of the account, ornot appropriate for the introductory tenor of the book. For him, the
structured progression of the proofs, the architecture of demonstration, is a
likeness of the progress of the soul towards knowledge and self-elevation:the order of geometry has to be kept because it corresponds to an epistemic
and cosmological order.
69
In fact, Proclus’ Elements of Theology are arranged according to an
Euclidean model, with enunciation followed by a demonstration, often
conducted via reductio ad absurdum or complete with clauses like ‘therefore’
(ara) and ‘it is clear that’ ( phaneron oti ). The propositions or theorems are
sometimes deductively connected, in that some proofs directly appeal to
ones which have been previously established.

241LATE ANCIENT MATHEMATICS: THE EVIDENCE
We should not take mathematics, however, to be the supreme form of
knowledge for Proclus, because he knew that its application to reality was
problematic. He took issue with the lack of accuracy of some physicalarguments; in his discussion of astronomy, he listed the problems tackled
by Ptolemy and others, critically assessed their solutions and rejected some
of their tenets, including the precession of equinoxes, eccentrics, and epicycles(those latter had the additional fault of not being mentioned by Plato).
70
In sum, both our representative philosophers, Iamblichus and Proclus,
assigned mathematics a tremendous role in their cosmology, epistemologyand ethics. They turned to previous mathematical authors in order to
comment and reflect on them, and in order to inscribe them in their own
framework, or reject their contribution. While some of their positions orchoices may be quite different from what we find in authors like Pappus,
the way they approached the past, their emphasis on classification, right
order, teaching, suggest once again that sharp boundaries between philoso-phy, mathematics and between different mathematical or philosophical
traditions have to be problematized.
The rest of the world
Apart from our legal sources, we have evidence for ‘applied’ mathematics in
some literary sources. Yet another author called Hyginus wrote a short and
now fragmentary manual on the division of military camps. He provided
various size specifications, depending on the terrain on which the camp wasto be built and on the number of people to be accommodated. Those latter
were measured in military units such as the cohors . He also specified what
parts of the army should be located in what part of the camp, and mentionedthe intervention of army surveyors, who lay out the camp with the help of
a groma .
71 The work may have been addressed to the emperor of the day;
Hyginus declares himself a trainee in the subject, but that does not preventhim from suggesting a few innovations in the way the right specifications
for a certain number of legions are to be calculated.
72
A fourth-century source on land-surveying is Agennius Urbicus, who
again distinguishes his form of knowledge from other ones, by remarking
that, while the Stoics assert that the world is one, if one really wants toknow what the world is like, and how big, one needs geometrical knowl-
edge.
73 He describes the methodology of his form of knowledge as follows:
Thus, of all the honourable arts, which are carried out either
naturally or proceed in imitation of nature, geometry takes the
skill of reasoning as its field. It is hard at the beginning and difficultof access, delightful in its order, full of beauty, unsurpassable in its

242LATE ANCIENT MATHEMATICS: THE EVIDENCE
effect. For with its clear processes of reasoning it illuminates the
field of rational thinking, so that it may be understood that geo-
metry belongs to the arts or that the arts are from geometry.74
More than contributing to land-surveying lore, Urbicus is enhancing
the status of geometria , which is presented as beautiful but also effective, an
inquiry into reason but also one of the arts. These themes are well known to
us from the land-surveyors of earlier periods, as is his insistence on the
ethical significance of land-surveying:
In making a judgement the land surveyor must behave like a good
and just man, must not be moved by any ambition or meanness,must preserve the reputation both by his art and by his conduct.
[…] [F]or some err because of inexperience, some because of
impudence: indeed this whole business of judging requires anextraordinary man and an extraordinary practitioner.
75
Apart from his continuation of traditional themes, Urbicus wrote com-
mentaries on Frontinus’ land-surveying works, operating on the text in
such a way that it is almost impossible to distinguish commentary from
original. From some remarks of his, it would also seem that he was involvedin teaching. In sum, like many other late ancient mathematicians, his
practice is a combination of old and new: he appropriates traditional values
and traditional texts in order to show the contemporary relevance of hisform of knowledge. Further appreciation of land-surveying comes from
Cassiodorus. In a letter ( c.
AD 507–11) written on behalf of King
Theodoric, he invites two spectabiles viri between whom a boundary
dispute had arisen to entrust their case to the capable hands of a land-
surveyor, who will solve it ‘by means of geometrical forms and land-
surveying knowledge’ rather than with weapons. Perhaps in an attempt todrive home its authoritativeness, Cassiodorus appends a micro-history of
land-surveying: how it originated with the Chaldeans, how it was taken
up by the Egyptians and eventually by Augustus, who carried out an
extensive programme of land-division; he mentions the ‘metrical author’
Hero as the person who ‘made it into a written doctrine’. Land-surveyingenjoys a great reputation indeed when compared to other branches of
knowledge: arithmetic, Cassiodorus says, is taught to empty schoolrooms;
geometry ‘insofar as it discusses heavenly things’ is only known to scholars;astronomy and music are learnt just for their own sake, but the agrimensor
‘shows what he says, and proves what he has learnt’.
76 The emphasis on
actual performance is paired with a reconstruction of the discipline’s past,of its tradition.

243LATE ANCIENT MATHEMATICS: THE EVIDENCE
And indeed, all the mathematicians we find in this period look back to
previous work. In his two books, On the Section of a Cylinder and On the
Section of a Cone , both addressed to a Cyrus, Serenus of Antinoopolis looks
at Euclid and above all Apollonius as main reference points. His declared
aim is to rectify some erroneous opinions the public has about Apollonius
and to research further the topics that Apollonius has already dealt with.There is some element of ordering of the earlier text (he mentions propo-
sitions from the Conics by number) and an emphasis on generality: for
instance, he wants to examine the sections of an oblique, not just a right,cylinder. Analogously, he contrasts the Euclidean definition of cone, which
only describes a right cone (Euclid is however not named explicitly) with
Apollonius’, which includes oblique cones as well.
77 Again, Theon of
Alexandria, as well as ‘editing’ Euclid’s Elements (see the section on the real
Euclid in chapter 4 p. 125), wrote two commentaries on Ptolemy’s Handy
T ables and one on the Syntaxis , in collaboration with his daughter Hypatia.
In the commentary on the Syntaxis , he remarked that his readers were not
very competent in mathematics, and this has often been taken to apply to
late ancient readers in general. Yet, apart from the fact that those are ratherstock remarks, typical of introductions,
78 we know that late ancient readers
of Ptolemy, for instance, had a whole variety of interests and mathematical
abilities. They included ‘proper’ mathematicians like Pappus who too wrotea commentary on the Syntaxis ; Platonist philosophers such as Porphyry,
who commented on the Harmonics and Proclus, who however had various
problems with Ptolemaic astronomy; and of course astrologers like Paul ofAlexandria and Hephaestio of Thebes, who mention Ptolemy as an example
of accuracy in calculations.
79 On the other hand, the very fact that Theon
wrote two different commentaries to the same work (the Handy T ables )
should alert us to his desire to cater for a diverse public. The ‘big’ commentary
in five books, now lost, must have been for a more advanced level of reader-
ship, while the simpler version, as he himself says, was aimed at those of hisstudents who had difficulty following multiplications and divisions, and
knew little geometry. The main interest of this not-terribly-mathematical
public was probably in the use of Ptolemy’s tables for astrological forecast.
80
Yet another likely late reader of Ptolemy was Synesius of Cyrene, bishop
of Ptolemais. He studied with Hypatia in Alexandria, and correspondedwith her afterwards. The astronomical training he must have received is
evident from a letter to Paeonius, an officer he had met at the imperial court
in Constantinople. The letter, usually entitled On the Gift , was accompanied
by an astrolabe, made by skilled craftsmen to Synesius’ own specifications.
The object, he said, included all that he had learnt from Hypatia. After
praising Paeonius for combining political power and philosophy, Synesiusextolled astronomy as the stepping stone to theology, its demonstrations

244LATE ANCIENT MATHEMATICS: THE EVIDENCE
being certain because they rely on geometry and arithmetic. Hipparchus,
Ptolemy and their successors had contributed to the knowledge of astrolabe-
making, but, according to Synesius, even more was known at the present day:
Every allowance must be made for these men [Hipparchus,
Ptolemy, etc.] if they worked on mere hypotheses, because theimportant questions were still incomplete, and geometry was still
at the nursing stage. But we, in return for the splendid mass of
knowledge that we owe to their achievements, without labour onour own part, should be grateful to these happy men who forestalled
us. At the same time we esteem it an ambition by no means
unworthy of philosophy, to attempt to bring in now certainadornments, to make a work of art, and to elaborate further. For
even as cities when first founded look only to the necessities of
life, to whit how they may be preserved, and how they may continuein existence, but as they advance are no longer content with what
is needful, and rather expend money on the beauty of the porticoes
and gymnasia, and the splendour of the forum; so in the case ofknowledge, the beginning is engaged with the necessary; only the
development with the excellent.
81
Hipparchus’ and Ptolemy’s times ‘the nursing stages’ of geometry? The latefourth century
AD an age of splendour and escape from bare necessities
(incidentally, the very conditions which in Aristotle and Proclus favour theflourishing of mathematics)? Yet another example of how late ancient
reconstructions of the past may confound modern expectations.
Moving into the early sixth century, Marinus, a pupil of Proclus, following
perhaps his teacher’s lead, wrote a commentary on Euclid’s Data , which,
along with his biography of Proclus, is his only surviving work. In fact, the
commentary as we have it, is only an introduction, and deals principallywith definition and classification of terms, a feature which we have found
in several other late ancient mathematicians. An anonymous negative remark
about Euclid is reported,
82 to the effect that in the Data , unlike the Elements ,
the order of exposition from general to particular is not respected. On a
couple of occasions Marinus engages in mathematical history, indicatingfor instance that there was a ‘before’ and ‘after’ Archimedes: some things in
mathematics were known to be knowable only after he made them so.
83
To conclude, a few words on the Palatine Anthology , a late ancient
collection of short poems on the most various subjects, and by several
different authors. As well as epitaphs, riddles and religious poetry, there are
many short mathematical verse compositions, some of them attributed tothe fourth-century
AD grammarian Metrodorus. They are all in the form of

245LATE ANCIENT MATHEMATICS: THE EVIDENCE
riddles, the fictional situations being historical, mythological or just everyday.
Many are problems of distribution into different parts, couched in terms of
dividing golden apples among goddesses, or inheritances among ordinarymortals; there are also calculations of the length of a journey, the duration
of one’s life (in the form of a horoscope or an epitaph) or of the time it takes
for a spout to fill up a fountain tank. A mythological example:
Heracles the mighty was questioning Augeas, seeking to learn the
number of his herds, and Augeas replied: ‘About the streams ofAlpheius, my friend, are the half of them; the eighth part pasture
around the hill of Cronos, the twelfth part far away by the precinct
of Taraxippus; the twentieth part feed in holy Elis, and I left thethirtieth part in Arcadia; but here you see the remaining fifty
herds’.
84
Another example, this time from real life:
O woman, how hast thou forgotten Poverty? But she pressed hard
on thee, goading thee ever by force to labour. Thou didst use to
spin a mina’s weight of wool in a day, but thy eldest daughter spun
a mina and one-third of thread, while thy younger daughtercontributed a half-mina’s weight. Now thou providest them all
with supper, weighing out one mina only of wool.
85
Intriguingly, we find ‘advanced’ examples of poetic mathematics in
association with illustrious mathematicians like Archimedes (author of the
Cattle Problem ) and Diophantus:
This tomb holds Diophantus. Ah, how great a marvel! the tomb
tells scientifically the measure of his life. God granted him to be aboy for the sixth part of his life, and adding a twelfth part to this,
he clothed his cheeks with down; He lit him the light of wedlock
after a seventh part, and five years after his marriage He granted
him a son. Alas! late-born wretched child; after attaining the
measure of half his father’s life, chill Fate took him. After consolinghis grief by this science of numbers for four years he ended his life.
86
The rest of the manuscript, as we have said, contains various materials,
including riddles which are not mathematical – something similar to what
we find, on a smaller scale, in the so-called educational texts, with division
tables on one side and paraphrases of Homer on the other. One has toconclude that these mathematical games could be read and solved (or not,

246LATE ANCIENT MATHEMATICS: THE EVIDENCE
as the case may be – some of them have not been solved to this day) by the
general educated public, and used both for diversion and advanced
instruction, in a way analogous to gnomic poetry.87 Indeed, the Diophantus
puzzle, like gnomic poetry, has a moral message, about the caducity of life
and the tragedy of death and loss – a further example of how use of the
mathematical tradition could work in many different ways.
Notes
1 Augustine, The Genesis Interpreted Literally 4.14, my translation.
2 Hypatia was not a unique example: book 3 of Pappus’ Mathematical Collection is addressed
to a woman, Pandrosion, whose pupils are also mentioned, and several examples of women
philosophers are mentioned in Eunapius, Lives of the Philosophers .
3 Rathbone (1991), 331.
4Ibid. (1991), 348.
5 Boyaval (1973); in general see Fowler (1999).6 Brashear (1985).
7 Crawford (1953).
8 T ranslation in Crawford (1953), with modifications; the text is very difficult to reconstruct.
The 150 between the lines is the solution.
9 There is debate as to the effective origin of the fiscal reform, see Carrié (1994).
10P . Cairo Isidor. 3, 4 and 5 (all
AD 299).
11 The meaning of geometria here (land-surveying or geometry?) is problematic, see Cuomo
(2000a).
12 Both letters in Dilke (1971).
13Notitia dignitatum Orientis 7.66, 11.12.
14Codex of Justinian =CJ 12.27.1 ( AD 405). The duties of the agentes in rebus included making
reports on the provinces, and serving as couriers between the court and the provinces.
They gained a reputation as secret police (see OCD s.v. ).
15Codex of Theodosius =CT 2.26 (from AD 330 to 392); CJ 12.27.1 ( AD 405); Digesta 10.1
(including Gaius and Julian from the second century AD, Paul, Ulpian, Modestinus and
Papinianus from the third).
16Digesta 11.6.7 (Ulpian).
17 Pliny Jr., Letters 10.58; Dio Cassius, History 78.7.3; Aelius Aristides, Sacred Discourse 73;
Philostratus, Lives of the Sophists 490 and 622–3; P . Lips . 47 ( c. AD 372).
18Digesta 27.1.6 (Modestinus, quoting Antoninus Pius, Paul, Ulpian, Commodus, Severus
and Antoninus), 50.13.1 (Ulpian); CJ 10.42.6 (between AD 286 and 305), 10.52.8 ( AD
369).
19 E.g. CT 13.3 (from AD 321 to 428); Digesta 50.5.10.2 (Paul); CJ 10.52 (from second
century AD to AD 414).
20Digesta 27.1.6.
21Digesta 50.13.1, translation University of Pennsylvania Press 1985, with modifications.
22 Ulpian, De excusatione 149 and CJ 10.52.4 (between AD 286 and 305), respectively.
23CJ 10.12.49.2; 10.12.49.4; 10.72.13; 12.7.1; 12.28.3; 12.29.3.1; 12.49.2; 12.49.4;
12.49.6; 12.49.10; 12.49.12; 12.59.3; 12.60.6. See also the diagram at the end of Seeck’s
edition of the Notitia Dignitatum .
24CT 8.2.5 ( AD 401), my translation.
25CT 9.16 (from AD 319 to 409); CJ 9.18 (between second century AD and 389); 1.4.10
(AD 409). Cf. also MacMullen (1966); Straub (1970); Grodzynski (1974).

247LATE ANCIENT MATHEMATICS: THE EVIDENCE
26 For portable dials from around the fourth century see de Solla Price (1969), Evans (1999).
As already explained in chapter 5, the construction of sundials required some mathematical
knowledge in order to produce plane projections of time lines. For calendars, see Salzman
(1990); Maas (1992), ch.4.
27CJ 9.18.4 ( AD 321).
28CJ 9.18.2 (between AD 286 and 305), my translation.
29CT 13.4 (from AD 334 to 374); CJ 10.64 ( AD 337 and 344); see also Digesta 50.6.7
(Tarruntenus Paternus, late second century AD).
30 See the entry on Diophantus in the OCD (by G.J. Toomer).
31 The later dating is based on Michael Psellus (eleventh century), according to whom
Anatolius, bishop of Laodicea from AD 270, had dedicated a treatise on Egyptian arithmetic
to Diophantus, see entry on Diophantus in DSB (by K. Vogel). For the earlier dating, see
Knorr (1993). I am not sure as to the attribution of the Definitions , and take Diophantus’
date to be the traditional mid- to late third century AD.
32 Diophantus, Arithmetica 1.12, my translation.
33 Take two numbers, a and b. Let us posit x=a2 + b2; y=a2 – b2; z=2ab. It will be x2=y2 + z2.
One says that the triangle with sides x, y and z is formed by the numbers a and b.
34 Diophantus, ibid. 3.19, my translation.
35 Diophantus, ibid. Preface, translation in Thomas (1967), with modifications.
36 Diophantus, Arithmetica 3.10, 3.15.
37 Diophantus, ibid. 5.1.
38 E.g. Diophantus, ibid. 2.11, 2.13, 3.12, 3.13, all examples of ‘double equation’.
39 Diophantus, ibid. 5.30, my translation.
40 I.e. that the point H is given depends on the ratio of line and circumference being given.
Pappus of Alexandria, Mathematical Collection 4.250.33–252.25, 254.10–22, my
translation. Heath (1921), I 229–30 thought that ‘both Sporus’s objections are valid’;
van der Waerden (1954), 192, disagreed.
41 Pappus, Mathematical Collection 4.270.3–31, my translation.
42 Pappus, ibid. 7.642.6–18, tr. A. Jones, Springer 1986, with modifications.
43 Pappus, ibid. 7.700.14–702.1, translation as above.
44 Pappus, ibid. 2.6.6–8.11, my translation.
45 Pappus, ibid. 8.1026.21–1028.3, my translation.
46 Pappus, ibid. 8.1072.31–1076.11, my translation.
47 Cf. Cuomo (2000a).
48 Eutocius, Commentary on Apollonius’ Conics Introduction.
49 Eutocius, Commentary on Archimedes’ Sphere and Cylinder 13, my translation. See Decorps-
Foulquier (1998).
50 Eutocius, Commentary on Archimedes’ Equilibria of planes 2.2 (176–8 Mugler).
51 Eutocius, Commentary on Apollonius’ Conics e.g. 1.262.28–264.2, 264.22, 264.25–26,
266.24.
52 Eutocius, ibid. 1.214.6–216.12, cf. also 1.285.1–288.6.
53 Eutocius, Commentary on Archimedes’ Measurement of the Circle 3 (162–3 Mugler), my
translation.
54 [Julian], Letters 78.419a.
55 At least three more commentaries on Nicomachus’ Arithmetic have survived from late
antiquity: one by Asclepius of T ralles, one by Johannes Philoponus, and an anonymous
one.
56 Shaw (1995).57 Iamblichus, On Nicomachus’ Arithmetical Introduction 75.22 ff. A detailed description in
Heath (1921), I 113 ff.
58 Iamblichus, Ibid. 23.18–24.17. Cf. also 20.10–14; 25.24; 30.28; 74.23 ff.

248LATE ANCIENT MATHEMATICS: THE EVIDENCE
59 Iamblichus, On Common Mathematical Knowledge 91.3–11, translation in Mueller
(1987a).
60 Proclus’ commentary to the first book of Euclid’s Elements , which is all we have, was
perhaps part of a commentary on the whole of the Elements , cf. Proclus, Commentary on
the First Book of Euclid’s Elements 398.18-19, where he may be referring to a commentary
on the second book; 432, where he seems to leave the task to others. For Domninus of
Larissa, see Tannery (1884) and (1885).
61 Proclus, Commentary on Plato’s Timaeus 3.18.20 ff.
62 Proclus, Elements of Theology 190–1; Commentary on Plato’s Timaeus 127.26 ff.;
Commentary on Euclid’s Elements 4. In general see Charles-Saget (1982).
63 Proclus, Commentary on Euclid’s Elements 19–20, 44.
64 Charles-Saget (1982).65 Proclus, Commentary on Euclid’s Elements 168, tr. G.R. Morrow, Princeton University
Press 1970, reproduced with permission.
66 Proclus, ibid. 236–7, 268–70, 403. Cf. Serenus, Section of a Cylinder 117, who claims
that a certain problem, which properly belongs to optics, cannot be solved without
geometry.
67 Proclus, Commentary on Euclid’s Elements 297-298, translation as above.
68 See e.g. Proclus, ibid. 252 ff., 394–5, and cf. Netz (1999b).
69 Proclus, ibid. e.g. 377.
70 Proclus, Commentary on Plato’s Republic 16 227.23–235.3; cf. also his Sketch of Astronomy
and Physics .
71 Hyginus, On the Division of Military Camps 12. He mentions compasses at 55. The
dating of this work has been very controversial: it could be second, late second or third
century
AD.
72 Hyginus, ibid. 45–7. About mathematics and the military in late antiquity, Vegetius (late
fourth/early fifth century AD) implies that numeracy, in the form of arithmetical skills,
was required, or was considered desirable, for soldiers and potential recruits, see On Military
Things 2.19, 3.15.
73 Agennius Urbicus, Controversies about Fields 22.7–8.
74 Agennius Urbicus, ibid. 25.15–27, my translation.
75 Agennius Urbicus, ibid. 50.9–15, my translation.
76 Cassiodorus, Letters 3.52. Cf. also 7.5.
77 Serenus, On the Section of a Cylinder 3–5 (fourth century AD).
78 Mansfeld (1998).79 Paul of Alexandria, Elements of Astrology e.g. 33.12, 79.10; Hephaestio of Thebes, Astrology
e.g. 32.10 (the ‘divine’ Ptolemy), 88.20, 126.1. Both are late fourth century
AD.
80 See Segonds (1981), 15.81 Synesius, Letters 160 ( To Paeonius, on the Gift ), tr. A. Fitzgerald, Oxford 1926, with
modifications.
82 Marinus, Commentary on Euclid’s Data 252.14–18.
83 Marinus, ibid. 244.1–8 (that the spiral is ‘feasible’), 248.3–8.
84Palatine Anthology 14.4. The solution, according to the Loeb editor, is 240 (120+30+20+
12+8+50).
85Palatine Anthology 14.134. The solution, as above, is: ‘The mother in a day 6/17, the
daughters respectively 8/17 and 3/17’.
86Palatine Anthology 14.126. The solution, as above, is: ‘He was a boy for 14 years, a youth
for 7, at 33 he married, at 38 he had a son born to him who died at the age of 42. Thefather survived him for 4 years, dying at the age of 84’.
87 See Morgan (1998).

249LATE ANCIENT MATHEMATICS: THE QUESTIONS
8
LATE ANCIENT
MATHEMATICS:
THE QUESTIONS
Some years ago, there were people who still believed or, more worryingly,
put in print that late antiquity was a period of decline and decadence formathematics. A rather good time could be had collecting quotes to the
effect that after Apollonius darkness had covered the earth, that those were
degenerate times, that Pappus’ Collection was the requiem of Greek
mathematics. Historiographical attitudes have changed, but years of neglect
and misunderstanding have had the result that, at the time of writing, very
little study has been done on the mathematics of late antiquity, includingmajor topics such as the later reception of Ptolemy or Nicomachus, or
mathematics in the Aristotelian commentators. This chapter, like the one
preceding it, will then be more a sampler of what late ancient mathematicshas to offer than a finished product. The first section is devoted to
mathematics in Christian authors, and to the related issue of numerology –
semantic superabundance as found in simple counting. Of coursenumerology did not start – indeed it did not end – in late antiquity, but it
is in this period that some of its most sublime manifestations are to be
found. In the second section, I will discuss the relation between late ancientmathematics and its past – the reader will have noticed that the majority of
the works we have surveyed refer to, or depend on, earlier works in various
ways. To phrase it differently, late ancient mathematical texts tend to be‘deuteronomic’
1 – they tend to be about other, ‘primary’ texts. Why is that,
and how can this relationship be better characterized? Let us start with high
and holy matters.
The problem of divine mathematics
Christianity required some mathematical practices: daily prayer in monastic
communities was strictly regulated, it had to begin at a certain time and last
a certain number of hours. Also, and this applied to all the faithful, Eastercame on a different day every year, and that day had to be determined with

250LATE ANCIENT MATHEMATICS: THE QUESTIONS
accuracy. Therefore, Christian authors and authorities were interested in
accurate time-keeping, by means of astronomical observations and
instruments.2 Another area where mathematics and the Christian faith inter-
sected was in understanding God and his creation. As we have seen in chapter
5, Philo of Alexandria had analyzed the meaning of various numbers in the
Bible: the seven days of creation, the forty days of the flood, the twelvetribes of Israel. Inspired by Philo and/or by the Platonism which partly
inspired Philo himself, Christian interpreters often carried out the same
hermeneutic operation, comforted in this by Scriptural phrases such as‘Everything You [God] ordered with measure, number and weight’ ( Book of
Wisdom 11.21). If the world since its creation followed a mathematical
order, then numerology was an important instrument of exegesis. Moreover,mathematics provided a type of reasoning that was certain and undisputable,
while at the same time it did not crucially rely on the senses – it was abstract.
It could work as a model for the knowledge of spiritual entities, such asGod or the soul. Finally, mathematics was an integral part of the non-
Christian educational systems or philosophies to which nearly all educated
Christians had been exposed. Thus, to the extent to which mathematicsrepresented, say, Platonist philosophy, or more generally a liberal education,
and to the extent to which a liberal education and Platonist philosophy
represented the old pagan order, talking about mathematics was part ofwider discourses about the relation between new faith and old values,
between Hellenic institutions and the new Jewish-born, in principle
cosmopolitan, religion and its own institutions. I will explore these issuesthrough two examples: Clement of Alexandria and Augustine of Hippo.
Clement of Alexandria is a bit early for our purposes, because he died in
AD 216, but he offers some interesting insights into the dialectic between
traditional and new Christian values. One of his works, for instance, is
devoted to arguing that the evangelic pronunciation against wealthy people
(it would be easier for a camel to go through a needle’s eye than for a richman to enter heaven) is actually not to be taken literally, and that the wealthy
need not worry too much about their salvation, as long as they redistribute
some of their possessions.
3 An example of how the new beliefs could be
accommodated to extant and definitely unchanged social and economic
hierarchies. Mathematics comes onto Clement’s horizon for several reasons.First, as part of the pagan educational curriculum and thus as a Greek form
of knowledge. In this guise, the potential dangers of some branches of mathe-
matics are made amply clear: in the course of interpreting the prophet Enoch,Clement even concludes that humans were taught astronomy by the evil
angels.
4 On a less dramatic note, astronomy and geometry (and dialectic)
are seen as futile – they do not teach the real truth; or they (in this casegeometry and music, and grammar and rhetoric) are represented as the

251LATE ANCIENT MATHEMATICS: THE QUESTIONS
handmaids of philosophy, which itself, while limited, at least is a quest for
truth.5 But Clement cuts the ‘Greekness’ of traditional knowledge down to
size. He underlines that Greek philosophy is heavily indebted to non-Greekone, and that all technai , philosophy, geometry, astrology and the division
of the year into months were invented by non-Greeks.
6 On the other hand,
he appreciates the value of some philosophical authors, especially Plato andPythagoras, and also of some mathematics. Moses, he tells us, studied
arithmetic and geometry, and mathematics (following a common Platonist
tenet) is useful to train the soul towards non-material, higher, realities.Moreover, mathematics represents order, in particular the harmony and
regularity of God’s creation: in line with what is stated in the Bible, He is
the measure, weight and number of the universe, the one who has countedthe depth of the oceans and the number of hairs on everyone’s head. Clement
states explicitly that God possesses mathematical knowledge; he draws a
parallel between faith and science, which both start from undemonstratedfirst principles, and engages in quite a lot of numerology. The size of the ark
of the covenant in the tabernacle in Jerusalem is discussed from four different
mathematical points of view: arithmetical, geometrical, astronomical andmusical.
7 Here is part of the arithmetical explanation:
‘And the days of the men will be’ it says ‘120 days’ […] the hundred-
and-twenty is a triangular number and has been formed from the
equality of the 64, of which the composition part by part gives
birth to squares, 1 3 5 7 9 11 13 15, on the other hand from theunequality of the 56, seven of the even numbers starting from
two, which give birth to the rectangles, 2 4 6 8 10 12 14. According
to yet another interpretation the number hundred 20 has beenformed from four <numbers>, one, the triangular <number>
fifteen, another, the square <number> 25, the third the pentagonal
<number> 35, the fourth the hexagonal <number> 45.
8
For the definition of triangular, pentagonal and hexagonal numbers, as wellas for an explanation of the ‘birth’ of squares and rectangles, the reader has
to be referred to Nicomachus or Iamblichus – it is material that does not
appear prominently in Euclid’s Elements , where we only find definitions of
square and rectangular numbers.
Yet more is found in Augustine. Some mathematical notions seem to
have puzzled him for their philosophical consequences: for instance, hewondered whether number, measure and weight pre-existed creation and,
if yes, in what sense, and addressed the question of whether the soul could
be talked about as if it was a number or a geometrical object.
9 Augustine’s
interest in mathematics dated from his young days: as a child, he had had

252LATE ANCIENT MATHEMATICS: THE QUESTIONS
rhythmically to chant mathematical tables; later, he had received a good
liberal education, which included music and arithmetic, and got interested
in astrology when attracted to Manichaeism. Even in the troubled phase ofhis conversion, when he was ready to jettison his pagan education entirely,
he could not rid his memory of mathematical notions, because they were
abstract, neither Greek nor Latin, nor related to any material object inparticular.
10 Augustine ended up a Christian, but, instead of getting rid of
mathematics, he put it to Christian use. T rue, he often inveighed against
curiosity, and insisted that knowledge without knowledge of God is worthnothing. That included mathematical knowledge:
A man who knows that he owns a tree and gives thanks to you
[God] for the use of it, even though he does not know exactly how
many cubits high it is or what is the width of its spread, is better
than the man who measures it and counts all its branches but doesnot own it, nor knows and loves its Creator. In an analogous way
the believer has the whole world of wealth and possesses all things
as if he had nothing by virtue of his attachment to you whom allthings serve; yet he may know nothing about the circuits of the
Great Bear. It is stupid to doubt that he is better than the person
who measures the heaven and counts the stars and weighs theelements, but neglects you who have disposed everything by
measure and number and weight.
11
But Augustine also wrote a detailed treatise on music, with the last sections
devoted to a distinction between different types of numbers and their relative
order from the point of view of spiritual excellence.12 Through numbers,
for instance the numbers contained in music and rhythm, human beings
could get closer to God, because if numbers were beautiful, ordered and
balanced, they would induce admiration and love in all those who sawthem. That love would then extend to the creator of things in which numbers
were contained, and also creator of numbers themselves. Augustine also
carefully re-calculated the number of generations before Christ, in order to
refute criticisms according to which their number was different in different
gospels,
13 and used numbers to elucidate theological concepts.14 For instance,
immutability: the simple propositions ‘T wo and four is six’ and ‘Four is the
sum of two and two: this sum is not two: therefore, two is not four’ were
used as example of unchanging reasoning.15 Or again, in the Soliloquies ,
which are an early work, Augustine engaged in dialogue with his own reason,
about (among other things) the relation between the knowledge of God
and other, more familiar, forms of knowledge such as sense knowledge ormathematical knowledge. The importance of this latter lay in the fact that

253LATE ANCIENT MATHEMATICS: THE QUESTIONS
it provided evidence for the existence of unassailable certainties about
abstract, non-material objects. Augustine often compared the type of
certainty offered by mathematical truths with the certainty that should beinspired by faith, and the existence of mathematics gave him ammunition
to reject sceptic attacks on the possibility itself of knowing things like the
soul. For instance, in On the Quantity of the Soul , he contended that under-
standing the soul can start from a process similar to the understanding of
mathematical objects. It was however clear to Augustine that knowledge of
God and mathematical knowledge were not the same thing (a notion enter-tained and then dismissed in the Soliloquies ), because their objects and the
respective value of those latter were different. Nonetheless, ‘the dissimilarity
rests on a difference of objects and not of understanding’.
16 Mathematics,
for all its limitations, could indeed provide a step-ladder to divinity.
A further facet of mathematics for Augustine, as for Philo and Clement,
was numerology. In one dazzling passage of On the trinity , he finds meanings
in, among other things, the six days of creation (six is the first perfect
number), the years a certain woman who was miraculously cured suffered
from her illness, the years a certain fig tree was left alone before beingpunished for being unfruitful, the length of the year according to both the
moon and the sun calendars, and other episodes in the life of Christ. He
informs us that we live in precisely the sixth age of man, which has beeninaugurated by the birth of Christ, and determines the exact day of Christ’s
conception. He concludes defiantly:
And now a word about the reasons for putting these numbers in
the Sacred Scriptures. Someone else may discover other reasons,
and either those which I have given are to be preferred to them, orboth are equally probable, or theirs may be even more probable
than mine, but let no one be so foolish or so absurd as to contend
that they have been put in the Scriptures for no purpose at all, andthat there are no mystical reasons why these numbers have been
mentioned there. But those which I have given have been handed
down by the Fathers with the approval of the Church, or I have
gathered them from the testimony of the divine Scriptures, or
from the nature of numbers and analogies. No sensible personwill decide against reason, no Christian against the Scriptures, no
peaceful man against the Church.
17
Augustine’s firm belief in the mystical rationale for the numbers in the
Bible should be seen in connection with his notion, mentioned above, that
numbers and order in the universe may draw people closer to God. Numer-ology had a strong ethical component, which was not limited to intensely

254LATE ANCIENT MATHEMATICS: THE QUESTIONS
Christian contexts. For instance, the author of the panegyric for Constantius
(AD 297–8) first deploys mathematics for rhetoric purposes – he declares it
impossible to do justice to Constantius’ and his family’s achievements, sohe will limit himself to counting them, except that even counting proves
impossible because of the sheer quantity of great things done by the emperor.
Then he moves on to numerological speculations, which provide him witha device to connect cosmological and human/political order. Thus, the
tetrarchy (two emperors, each with an appointed successor) reflects the
number of the seasons, of the elements, of the divisions of the Earth, evenof the ‘lamps’ in the sky.
18 Numerous other instances can be found in neo-
Platonic/neo-Pythagorean authors:
The number five is highly expressive of justice, and justice compre-
hends all the other virtues. […] if we suppose that the row of
numbers is some weighing instrument, and the mean number 5 isthe hole of the balance, then all the parts towards the seven, starting
with the six, will sink down because of their quantity, and those
towards the one, starting with the four, will rise up because oftheir fewness, and the ones which have the advantage will altogether
be triple the total of the ones over which they have the advantage,
but 5 itself, as the hole in the beam, partakes of neither, but italone has equality and sameness. […] thanks to the fact that five is
a point of distinction and reciprocal separation, if the disadvantaged
one which is closest to the balance on that side is subtracted fromthe one which is furthest from the balance on the excessive side
and added to the one which is furthest from the balance on the
other side – if, to effect equalization, 4 is subtracted from 9 andadded to 1; and from 8, 3 is subtracted, which will be the addition
to 2; and from 7, 2 is subtracted, which is added to 3; and from 6,
1 is subtracted, which is the addition to 4 to effect equalization;then all of them equally, both the ones which have been punished,
as excessive, and the ones which have been set right, as wronged,
will be assimilated to the mean of justice. For all of them will be 5
each; and 5 alone remains unsubtracted and unadded, so that it is
neither more nor less, but it alone encompasses by nature what isfitting and appropriate.
19
The presence of numerology across religious divides, with holy men as
different as Clement and [Iamblichus] attributing fundamentally similar
meanings to simple arithmetical properties, or to simple numbers, raises a
few questions about the religious divides themselves. Numerology presup-posed a belief in the link between mathematics and the divine, whether

255LATE ANCIENT MATHEMATICS: THE QUESTIONS
located in the Christian God or in the neo-Platonic One/Monad. The fact
that, despite the evident parallels, the Christian God and the NeoPlatonic
One were not the same thing opened a potential ground for conflict. Ifnumbers were the key to a higher, deeper and more stable form of knowledge,
who was most qualified to read them? There are instances of direct com-
petition between pagan and Christian holy men, and it has been suggestedthat the numerologist par excellence , Pythagoras, as depicted by above all
Iamblichus, was ‘the pagan response and counterpart to Christ’.
20 Keeping
in mind Clement’s dismissal of the Greekness of philosophy and of severalmathematical disciplines, let us read a micro-history of mathematics by the
rampantly anti-Christian emperor Julian, apparently an admirer of
Iamblichus:
The theory of the heavenly bodies was perfected among the
Hellenes, after the first observations had been made among thebarbarians in Babylon. And the study of geometry took its rise in
the measurement of the land in Egypt, and from this grew to its
present importance. Arithmetic began with the Phoenicianmerchants, and among the Hellenes in course of time acquired
the aspect of a regular science.
21
The work containing this passage is devoted to showing that the Jewish
people, the people Jesus belonged to, never invented anything or did anything
remarkable. In sum, there are elements to suggest that mathematics mayhave been one of the many battling grounds for late ancient discussions
over the divine signification of history and of the universe in general. On
both fronts, numbers were transmitted through sacred ancestral texts – thePythagorean symbola in the Arithmetical Theology , the Scriptures in Clement
and Augustine – which awaited interpretation. Augustine’s appeal to a trinity
of authorities (reason, written tradition and the Church) in order to defendthe importance of numerology may then be seen as addressed not exclusively
to fellow Christians who did not subscribe to that particular way of doing
mathematics, or simply had different opinions about the meaning of parti-
cular numbers. It may also have served as a strong reminder that decoding
of numbers, both those found in the Bible and those inscribed in the bookof creation, was a Christian thing – mathematics was God’s own activity.
The hapless person who counts the stars but neglects God, the Manichean
astrologers of Augustine’s early days and the pagan believers in mathematicaltheurgy were all examples of bad mathematics: bad not because of its
contents, nor even because of its methods, but because of the meanings
that one ascribed, or failed to ascribe, to it, because of the use one made ofit.

256LATE ANCIENT MATHEMATICS: THE QUESTIONS
The problem of ancient histories of ancient mathematics
As we discussed in the section on the problem of the birth of a mathe-
matical community in chapter 4 p. 135, the past was already an inevitable
presence in the second century BC; not just for mathematics, but also for
philosophy, medicine, grammar. As knowledge accumulated in the formespecially of books, scrolls at first, codexes later, collected in the libraries of
rich Romans or later of bishops and abbots, the presence of the past and its
inevitability grew even larger. It is thus not surprising that late antiquitysaw a proliferation of ‘deuteronomic’ texts, not only in mathematics but in
practically any other form of knowledge in which the written medium played
an important role. I will first try better to characterize how late ancientmathematicians behaved towards previous sources – this has been tackled
in chapter 7, so here I will only say some more with reference to the same
authors: Pappus, Proclus, Eutocius. Second, I will look at explicit statementsthese authors made about the past, at the histories they wrote. Third, I will
address the question of why the past was used so extensively.
We have already pointed out that authors writing about other authors
put a lot of emphasis on clarity and intelligibility, often for the benefit of
students who had different levels of ability, or of an addressee who is
described as excellently gifted but not a mathematical expert. Thus, lateancient deuteronomic texts supply missing steps in the demonstrations, re-
define or footnote the terminology, explore the limits of possibility or
determinacy of a solution through variations in the diagram. The presenceof a tradition also leads to the examination of sub-cases and generalizations:
Pappus extended the theorem of Pythagoras to any triangle; Proclus, the
first problem of the first book of Euclid’s Elements (‘On a given finite straight
line to construct an equilateral triangle’) to isosceles and scalene triangles.
22
The point of, for instance, sub-cases is to ensure that the problem is fullydetermined and that no further manipulation of its elements will produceconfigurations for which the solution is not valid. Sub-cases also provide
exercise for the student because of their variety; and allow the reader to
grasp the full implications of the proposition, its complete ramifications inthe scheme of things mathematical. Acquiring power in problem-solving
through an extensive command of the material is a recurrent theme in the
mathematics of this period.
Late ancient mathematicians classified, defined and systematized quite
a lot; they attached great importance to the right order of exposition;
occasionally they formulated rules and prescriptions. A case in hand isPappus’ treatment of the duplication of the cube in book 3 of the
Mathematical Collection : first he criticizes a solution that had been sent to
him, then launches into a classification of problems and of the proceduresappropriate to solve them,
23 then appends three previous solutions to the

257LATE ANCIENT MATHEMATICS: THE QUESTIONS
problem (Eratosthenes’, Nicomedes’ and Hero’s), all of which are in line
with his prescriptions, and finally provides his own solution. Since Eutocius’
anthology of solutions to the duplication of the cube includes those quotedby Pappus, a comparison can be made between the two reports, and it can
be argued that Pappus modified the earlier sources to have them fit within
his classificatory scheme and with his own solution.
But the ways in which late ancient mathematicians reworked their sources
could be even more pervasive. Faced with several, somewhat diverging,
manuscripts of the Conics , Eutocius thought that they might be the different
editions that Apollonius mentioned in the preface to the first book, and set
out to collate them into one text, which would correspond to Apollonius’
original intentions.
24 In other words, Eutocius produced the ‘original’ text
he was commenting on to an extent which can be assessed only partly.
Some insight can be gained by analyzing his guiding criteria, for instance
his notion of clarity,25 a highly subjective notion which he emphasized over
and over again, and corresponded to an ability to distinguish clear from
obscure, and to understand this latter to the point of explaining it to others.
At one point, Eutocius justifies his selection of a proof as Apollonius’ proofover another from an alternative version of the Conics with the remark that
he ‘produces clear light for the readers’.
26
Eutocius’ interventions reflect a situation common to other late ancient
authors, where they tried to collect as many manuscripts of the same texts
as possible. They were aware of problems of scribal transmission both for
the text and the diagrams, and assessed the antiquity and genuineness of amanuscript on the basis of, in the case of Archimedes, the use of Doric
dialect (but note that this could and was forged in the production of Pytha-
gorean writings); the use of old-fashioned terminology (principally the oldterms for conic sections, but that could have been forged as well); the filling
of significant gaps (again a good sign of forgery). Eutocius refused to include
into his account a solution to the duplication of the cube attributed toEudoxus, no less, because the manuscript he had claimed that curved lines
would be used, and then they were not, and a discontinuous proportion
was used as if it was continuous. Yet, at the same time, he unquestioningly
accepted a solution to the same problem by Plato, of which there is no
record elsewhere.
Eutocius’ luck at times does seem remarkable: in the course of comment-
ing on SC 2.4, he mentioned that Archimedes at some point promised to
provide a certain result but never did, at least not in any manuscript thatEutocius had seen. The lacuna, he said, had been there for a long time:
already Diocles and a Dionysodorus whose date is uncertain had allegedly
tried to fill the gap in Archimedes’ account. But now, Eutocius had comeacross a very old book, in very bad condition, full of errors, which happened

258LATE ANCIENT MATHEMATICS: THE QUESTIONS
to contain just the right material, complete with traces of Doric dialect and
old-fashioned terminology. The contents of the old book are not reported
as they stand, however: we are provided with the results of Eutocius’ verycareful examination, cleaning of the text’s errors and rephrasing of it in
more usual and clearer language.
27 Clearly, the commentator is the medium
through whom the readers can appreciate the greatness of Archimedes andbe guided through his difficulties: more than that, he retrieves the past on
their behalf.
28
So, the point is not just interpreting the texts and ordering their contents
in a newly-arranged mathematical universe ruled by new classifications –
deuteronomic practice is also about the correct interpretation of the author’s
intention, the possibility itself to make the past understandable and meaning-ful for the present. A case taken up both by Pappus and Eutocius is
Apollonius’ criticism of Euclid in the Conics . Pappus interpreted it in these
terms:
The locus on three and four lines that [Apollonius] says, in [his
account of] the third [book], was not completed by Euclid, neitherhe nor anyone else would have been capable of; no, he could not
have added the slightest thing to what was written by Euclid, at
any rate using only the conics that had been proved up to Euclid’stime, as he himself confesses when he says that it is impossible to
complete it without what he was forced to establish first. But […]
Euclid, out of respect for Aristaeus as meritorious for the conicshe had published already, did not anticipate him, […] for he was
the fairest of men, and kindly to everyone who was the slightest
bit able to augment knowledge as one should, and he was not atall belligerent, and though exacting, not boastful, the way this
man [Apollonius] was, – […] [Apollonius] was able to add the
missing part to the locus because he had Euclid’s writings on thelocus already before him in his mind, and had studied for a long
time in Alexandria under the people who had been taught by
Euclid.
29
Instead, in Eutocius’ view, Apollonius is simply referring to another book
by Euclid, now lost, so that the real meaning of his sneer is in a sense for
ever irretrievable.30 The aspect of transmission and the second element I
want to explore in this section, history-making, are strictly related again inPappus’ book 7:
The so-called domain of analysis, Hermodorus my son, is, taken
as a whole, a special resource that was prepared after the production

259LATE ANCIENT MATHEMATICS: THE QUESTIONS
of the common elements for those who want to acquire a power
in geometry that is capable of solving problems set to them; and it
is useful for this alone. It was written by three men: Euclid theelementarist, Apollonius of Perga, and Aristaeus the Elder.
31
The domain of analysis as a body of work has been produced through
successive accretions over time; the transmission of a text embodies the
history of that text and of the materials it contains. Pappus’ history, however,
is of course not neutral, as already evidenced by his unlikely reconstructionof the relationship between Euclid and Aristaeus: the three men he singles
out here, for instance, are not the only authors of texts cited in book 7, and
there is no evidence that they planned to produce a body of work. Neo-Platonist histories of mathematics tend not to be neutral either. Iamblichus
rewrote traditional accounts of discoveries in a Pythagorean light, as in this
passage, reported second-hand through Simplicius, where Sextus the Pytha-gorean supplants Archimedes as the discoverer of the area of the circle:
Aristotle did not know this [the formulation for the area of the
circle] at all, it seems; but it was discovered by the Pythagoreans,
says Iamblichus, as it is clear from the proofs of Sextus the
Pythagorean, who at the beginning learnt the method of the proofsaccording to the tradition. And afterwards, he says, Archimedes
by means of † [the names of other mathematicians follow:
Nicomedes, Apollonius, Carpus], as Iamblichus reports. And it ismost extraordinary that the most learned Porphyry omits this – in
fact he seems to say that there is a proof, according to which there
is a figure of square that approximates the circle, as for the otherfigures, but does not transmit in any way that it was discovered.
32
Again, when discussing the discovery of different types of means,
Iamblichus’ main reference points are Pythagoras and the Pythagoreans
(Archytas, Ippasus of Metapontus, Philolaus), and anyone else is either cut
down to size or forcibly enlisted. Thus, Plato is mentioned, but Iamblichus
often specifies that things Plato said had been said before him by Pythagoras
or the Pythagoreans, and Eudoxus figures as ‘the Pythagorean’.33 When
Pappus deals with the very same topic, his sources turn out to be Nicomachus
‘the Pythagorean’ and Plato, but also Ptolemy and Eratosthenes.34 And, like
most authors before or after him, Pappus reports that the squaring of thecircle was first discovered by Archimedes.
It is clear already from these few examples that using the past in the sense
of writing a history of mathematics was not only not neutral, but could conflictwith someone else’s non-neutral use of that very same past. In other words,

260LATE ANCIENT MATHEMATICS: THE QUESTIONS
the mathematical past could be contested ground, with alternative
reconstructions being proposed. A further interesting example of this is Proclus’
micro-history of mathematics, discussed in chapter 2, p. 54. The aspect Iwant to emphasize here is that, while its value as information about the fifth
and fourth centuries
BC may be not very much, it is of course precious evidence
about Proclus’ own time and approach. Whatever his source or sources, Procluscreated a more or less seamless chain that leads towards greater and greater
generalization and rigour. While geometry and arithmetic may have had their
origin in necessity, and while Thales, importing mathematics to Greece, mayhave retained a measure of empiricism, Pythagoras started the process of
‘scientification’ of mathematics. Everybody else who is mentioned joins in a
path whose ultimate end is never put in doubt: systematization into elements,increased rigour, increased generality. Mathematicians across centuries are
presented as aiming at a common end, embodied by Euclid, who
belonged to the persuasion of Plato and was at home in this
philosophy; and this is why he thought the goal of the Elements as
a whole to be the construction of the so-called Platonic figures.
35
Proclus created a mathematical tradition, culminating in Euclid, and
related it indissolubly to the philosophical tradition he saw himself as partof – Platonism. But again, Proclus’ story is only one version of ancient
history of ancient mathematics: Pappus had other heroes, with Archimedes
and Ptolemy playing a role that in Proclus is basically non-existent. Wecould even go as far as saying that Augustine had his own mathematical
tradition, with the Sacred Scriptures as key texts to a twin understanding of
the mathematical texture of reality and divinity. Those different stories, oralternative traditions, (I repeat) were in potential or actual conflict:
Iamblichus writing Archimedes and Euclid out of the history of mathematics
on the one hand, Pappus writing them in and attacking philosophers fortheir incompetence in mathematics (in book 5 of the Collection ) on the
other, Proclus revealing the real Platonic motives of Euclid’s Elements ,
Eutocius revealing the real practical motives of Archimedes’ Measurement of
the Circle .
With reference to the third point raised in this section, then, one possible
motive for the use of the past was that it made a good polemical weapon.
One’s authority could be enhanced strategically against competing claims
by creating continuities between oneself and the figures of the tradition.
36
Late antiquity was a period of radical change, with traditional points of
reference – religious authority, number of emperors, state boundaries, capital
of the Empire, organization of the administrative machine – shifting, andnew structures emerging. The past at least could be presented as something

261LATE ANCIENT MATHEMATICS: THE QUESTIONS
fully mapped and properly ordered, a stabilizing presence, and establishing
links with it was a way to promote oneself and obtain legitimation.
Deuteronomic or revivalist practices were so widespread and pervasive
that they may appear to be the one defining feature of late ancient culture.
Recourse to the past, however, was a strategy used already at earlier times,
nor was it the only strategy available. For instance, as far as late ancientmathematicians were concerned, utility, benefit for the community, both
variously defined, and ethical significance of their knowledge were also values
with which many of them wished to be associated, and which were officiallypromoted. Along with the rhetoric of ‘past is good’, we also find a notion
that ‘new is better’: we have seen what Synesius thought about the astronomy
of his own time, and that Iamblichus criticized Euclid; Apollonius for hispart was criticized by Pappus, Proclus and Eutocius; even the great
Archimedes was reprimanded by Pappus because he had not followed his
prescriptions about problem-solving (which, incidentally, he had no reasonto be aware of). Occasionally, Pappus indicated that there had been progress
in mathematics with respect to the past: for instance, discovery of better
solutions, clarification of some issues, additions to the body of knowledge.On one occasion, he warned the reader that ‘we do not have to trust the
opinion of the men who <first> discovered <this>’.
37
In sum, late ancient mathematicians used the past as it suited them and
their present concerns: their accounts were geared to different audiences,
and mixed old and new both in their deployment of the sources and in
their appreciation of what constituted good and bad mathematical practice.They make a fitting conclusion for this book, in that they were among the
first to hand down histories of mathematics. Their histories are a warning
that no history, not even that of mathematics, can ever be neutral.
I have set forth my agenda in the introduction: here I just want to repeat
that I have chosen to present a vast range of heterogeneous material, because
of my belief that mathematical practices, in antiquity as at probably anyother time, are extremely complex, multilayered and occasionally baffling
things. Distinguishing between what is and is not ‘real’ mathematics is a
choice one has to make, and I have chosen to be inclusive of everyday
mathematics, of non-mathematical views of mathematics and of ‘applied’
mathematics. If that has meant cutting down on some topics, I hope I havestill managed to convey some of the brilliance and rigour of the theorems,
discoveries and insights of ancient mathematicians. More than anything,
people in the nearly thousand years we have surveyed could get incrediblypassionate about mathematical matters, be they the properties of incommen-
surable lines or the appropriateness of measuring the heavens – my wish is
that some of that passion has rubbed over onto you, the reader.

262LATE ANCIENT MATHEMATICS: THE QUESTIONS
Notes
1 The term in this sense is introduced in Netz (1998).
2 McCluskey (1990).
3 Clement of Alexandria, The Rich Man’s Salvation .
4 Clement of Alexandria, Eclogues on the Prophets 53.4 (commenting on 1 Enoch 8.3).
5 Clement of Alexandria, Stromata 6.11.93 and 1.5.29, respectively.
6 Clement of Alexandria, ibid. 1.15.66–1.16.80, cf. also 1.21.101.
7 Moses: Stromata 1.23.153; God as measure of the universe: Exhortation to the Greeks
6.60; God having mathematical knowledge, interpretation of the ark of the covenant:Stromata 6.11; faith and science: Stromata 2.4.14.
8 Clement of Alexandria, Stromata 6.11.84–85, my translation. The quote is from Genesis
6.3.
9 Augustine, The Genesis Interpreted Literally 4.7, On the Quantity of the Soul 3 ff.
10 Augustine, Confessions 10.19.
11 Augustine, Confessions 4.7, tr. H. Chadwick, Oxford 1991.
12 Augustine, On Music esp. book 6.
13 Augustine, On the Agreement of the Evangelists 2.4.
14 Augustine, On Free Will 2.20–24; On the Immortality of the Soul 1.5.
15 Augustine, On the Immortality of the Soul 2.2, tr. G. Watson, Aris and Phillips 1990. Cf.
also similar examples at e.g. On Free Will 2.8, 2.12; On the Immortality of the Soul 1, 6
(examples with geometry); Against the Academics 3.11; Confessions 6.6; Letters 14.4.
16 Augustine, Soliloquies 1.11, translation as above. Cf. also ibid. 2.33.
17 Augustine, On the Trinity 4.10, tr. S. McKenna, Washington 1963. Cf. also On the Trinity
4.7, The City of God 11.30, The Genesis Interpreted Literally 4.7.
18Panegyric 8 1.3–5 and 4.1–2, respectively.
19 [Iamblichus], Theologia Arithmetica 35.6 ff., tr. R. Waterfield, Phanes Press 1988, with
modifications. Cf. also Anatolius, On the Decad , a very similar work written by a Christian.
20 O’Meara (1989), 214.
21 Julian, Against the Galileans 178a–b.
22 Pappus, Mathematical Collection 4.176.9–178.13; Proclus, Commentary on the First Book
of Euclid’s Elements e.g. 218–19, 228 ff., 323–6. See Netz (1998), Cuomo (2000a).
23 The same classification is reported in book 4 and quoted in the previous chapter.
24 Eutocius, Commentary on Apollonius’ Conics 1.176.17–22, 230.13–16, 246.15–17,
250.16–22.
25 On this see Decorps-Foulquier (1998).26 Eutocius, Commentary on Apollonius’ Conics 2.296.6–7, cf. also 4.354.5–7.
27 Eutocius, Commentary on Archimedes’ Sphere and Cylinder 2.4 (88 ff. Mugler).
28 Eutocius, ibid. 2.4 (100 Mugler).
29 Pappus, Collection 7.674.20–682.23, tr. A. Jones, Springer 1986.
30 Eutocius, Commentary on Apollonius’ Conics 1.186.1–10.
31 Pappus, Collection 7.634.3–10, translation as above with modifications.
32 Simplicius, On Aristotle’s Categories VII 192.15–25 (the relative passage in Aristotle is
7b15), my translation.
33 E.g. Iamblichus, On Nicomachus 105.2–11.
34 Pappus, Collection 68.17 ff.
35 Proclus, Commentary on Euclid 68, tr. G.R. Morrow, Princeton University Press 1970,
reproduced with permission. Cf. also ibid. 82.
36 Parallels of this in fields other than mathematics are a huge number, see for further
references Maas (1992), Cuomo (2000a).
37 Pappus, Collection 4.254.23–24, my translation.

263GLOSSARY
GLOSSARY
Application : the operation that consists in constructing a parallelogram or
a rectangle under certain conditions and which has as side a given
straight line.
Arithmetic mean , see Mean, arithmetic .
Binomial : consisting of two terms; an expression which contains the sum
or difference of two terms ( OED ).
Chorobates : an instrument for finding the level of water, a ground-level ( LS).
Delian problem , see Duplication of the cube .
Delos, problem of , see Duplication of the cube .
Diorismos (pl. diorismoi ): statement of the limits of possibility of a problem;
particular enunciation of a problem ( LS).
Duplication of the cube : to find a cube double a given cube. Found to be
equivalent to the problem of finding two mean proportionals betweentwo given lines, i.e. given the two lines A and B, one is required to find
X and Y such that A:X=X:Y=Y:B. Also called Delian problem or problem
of Delos because of a story that the god Apollo asked for a new altar forhis temple at Delos double the size of the extant cubic one.
Equestrian order : originally members of the Roman cavalry with a certain
social eminence, from which officers and the staffs of governors andcommanders were drawn. Under the Empire, they constituted the
second aristocratic order, ranking below the senators, provided the
officer corps of the Roman army and held a wide range of posts in thecivil administration. In general (although precise criteria for membership
of the order remain disputed) all Roman citizens of free birth who
possessed the minimum census qualification of 400,000 sestercesautomatically qualified as members of the order ( OCD s.v. ).
Extreme and mean ratio : a line segment is divided into extreme and mean
ratio when the shorter segment resulting from the division is to thelarger segment resulting from the division as the larger segment is to
the whole original segment.

264GLOSSARY
Geometric mean , see Mean, geometric .
Gnomon : pointer of a sundial; carpenter’s square; number added to a figurate
number to obtain the next number of the same figure ( LS); in Euclid
(El. 2. def. 2), any of the parallelograms about the diagonal of a parallelo-
grammic area with the two complements.
Harmonic mean , see Mean, harmonic .
Interval : the difference of pitch between two sounds ( OED ).
Incommensurable : having no common measure with another quantity
(OED ).
Isoperimetrism : set of mathematical problems that deals with two or more
figures having equal perimeter ( OED ).
Locus (pl. loci): the figure composed of all the points which satisfy certain
conditions, or are generated by a point, line, or surface moving in accord-
ance with certain conditions ( OED ).
Mean, arithmetic : of three terms, the first exceeds the second by the same
amount as the second exceeds the third. I.e. B is the arithmetic mean
between A and C if A – B = B – C.
Mean, geometric or proportional : of three terms, the first is to the second
as the second is to the third. I.e. B is the geometric mean between A
and C if A : B = B : C.
Mean, harmonic : of three terms, by whatever part of itself the first term
exceeds the second, the middle term exceeds the third by the same part
of the third. I.e. B will be the harmonic mean between A and C if C :
A = (B – A) : (C – B).
Medial : pertaining to or designating a mathematical mean, or a line or area
which is a mean proportional ( OED ).
Maximum (pl. maxima ): the greatest value which a variable may have; a
point at which a continuously varying quantity ceases to increase and
begins to decrease ( OED ).
Minimum (pl. minima ): the least value which a variable may have; a point
at which a continuously varying quantity ceases to decrease and begins
to increase ( OED ).
Modus ponens : an argument employing the rule that the consequent q
may be inferred from the conditional statement if p then q and the
statement p (OED ).
Modus tollens : an argument employing the rule that the negation of the
antedecent p (i.e. not-p ) may be inferred from the conditional statement
if p then q and the consequence not-q (OED ).
Neusis (pl. neuseis ): lit. inclination; a line segment of a given length, whose
terminal points have to lie on given straight lines or curves and whose
extension is to pass through a given point.
Nome : administrative subdivision of the territory in Egypt.

265GLOSSARY
Parapegma : astronomical and meteorological calendar, inscribed on stone,
the days of the months being inserted on movable pegs at the side of
the text ( LS).
Parameter of a conic : also called latus rectum , it is the straight line which is
in the parabola the constant height of a rectangle which has as base the
abscisse of a point of the conic, and whose area is equal to the square ofthe ordinate of the same point. In the hyperbola it is the straight line
which is in the parabola the constant height of a rectangle which has as
base the abscisse of a point of the conic, and whose area, plus anothergiven area, is equal to the square of the ordinate of the same point. In
the ellipse, it is the straight line which is in the parabola the constant
height of a rectangle which has as base the abscisse of a point of theconic, and whose area, minus another given area, is equal to the square
of the ordinate of the same point.
Plane number : a number resulting from the multiplication of two other
numbers.
Porism : ‘A porism occupies a place between a theorem and a problem; it
deals with something already existing, as a theorem does, but has tofind it (e.g. the centre of a circle), and, as a certain operation is therefore
necessary, it partakes to that extent of the nature of a problem, which
requires us to construct or produce something not previously existing.’(Heath (1921), I 434).
Precession of the equinoxes : the earlier occurrence of the equinoxes in each
successive sidereal year (due to precession of the Earth’s axis) ( OED ).
Prime number : a number whose only integral factors are the number itself
and the unity.
Problem of Delos , see Duplication of the cube .
Problem of the two mean proportionals , see Duplication of the cube .
Quadrature of the circle : the problem of expressing the ratio between circum-
ference and diameter of a circle, so as to reduce its area to a knownrectlinear area.
Rectangular number : a number which is the product of the multiplication
of two different numbers.
Senatorial order : originally a body of wealthy men of aristocratic birth, most
of them ex-magistrates, which supervised the magistrates’ work andcould invalidate laws. Under the Empire, an official property qualifica-
tion of one million sesterces was introduced, as well as a strong hereditary
element ( OCD s.v. ).
Setting-out : the stage in a geometrical proof where the general enunciation
is repeated with reference to a particular diagram.
Similar numbers : two numbers are similar if a mean proportional exists
between them.

266GLOSSARY
Similar polygons : they are similar when they contain the same angles and
have the same shape ( OED ).
Solid number : a number which is the product of the multiplication of three
other numbers.
Square number : a number which is the product of the multiplication of a
number by itself.
Squaring the circle : see Quadrature of the circle .
Techne (pl. technai ): art, craft, knowledge, activity which can be taught and
learnt, and which produces some kind of artefact or result. Examplesof technai include medicine, rhetoric, hunting, architecture, poetry,
cooking, mathematics, mechanics.

267BIBLIOGRAPHY
BIBLIOGRAPHY
Primary sources
Aeschines, Against Ctesiphon, Engl. tr. C. Darwin Adams, London/Cambridge, MA:
Heinemann and Harvard University Press 1958.
—— Against Timarchus , Engl. tr. C. Darwin Adams, London/Cambridge, MA:
Heinemann and Harvard University Press 1958.
Aeschylus, Prometheus Bound , Engl. tr. J. Scully and C.J. Herington, Oxford: Oxford
University Press 1975.
Agennius Urbicus, Controversies about Fields , ed. K. Thulin, in Corpus agrimensorum
romanorum , vol.1 fasc.1, Leipzig: T eubner 1913.
Alcinous, The Handbook of Platonism , Engl. tr. J. Dillon, Oxford: Clarendon Press
1993.
Anatolius, On the Decad , French tr. P . Tannery, in Mémoires scientifiques 3, Paris/
T oulouse: Gauthier-Villars and Privat 1915 (first publ. 1900), 12–31.
Palatine Anthology , Engl. tr. W.R. Paton, London/Cambridge, MA: Heinemann
and Harvard University Press 1918, 5 vols.
Apollonius, Conics , ed. J.L. Heiberg, Leipzig: Teubner 1891–3.
Aratus, Phaenomena , Engl. tr. G.R. Mair, London/Cambridge, MA: Heinemann
and Harvard University Press 1955.
Archimedes, Cattle Problem , ed. C. Mugler, Paris: Les Belles Lettres 1970–2.
—— Conoids and Spheroids , ed. C. Mugler, Paris: Les Belles Lettres 1970–2.
—— Equilibria of Planes I and II , ed. C. Mugler, Paris: Les Belles Lettres 1970–2.
—— Measurement of the Circle , ed. C. Mugler, Paris: Les Belles Lettres 1970–2.
—— Method , ed. C. Mugler, Paris: Les Belles Lettres 1970–2.
—— Quadrature of the Parabola , ed. C. Mugler, Paris: Les Belles Lettres 1970–2.
—— Sand-Reckoner , ed. C. Mugler, Paris: Les Belles Lettres 1970–2.
—— Sphere and Cylinder I and II , ed. C. Mugler, Paris: Les Belles Lettres 1970–2.
—— Spirals , ed. C. Mugler, Paris: Les Belles Lettres 1970–2.
Aristarchus of Samos, On the Sizes and Distances of the Sun and Moon , Engl. tr. T .L.
Heath in id., Aristarchus of Samos. The Ancient Copernicus , Oxford: Clarendon
Press 1913.
Aelius Aristides, Sacred Discourse , Engl. tr. C.A. Behr, Leiden: E.J. Brill 1981.
Aristophanes, Birds , ed. N. Dunbar, Oxford: Clarendon Press 1995.

268BIBLIOGRAPHY
—— Clouds , Engl. tr. A.H. Sommerstein, Warminster: Aris and Phillips 1982.
—— Wasps , Engl. tr. B. Bickley Rogers, Cambridge, MA/London: Harvard
University Press and Heinemann 1978.
Aristotle, Constitution of Athens , Engl. tr. H. Rackham, Cambridge, MA/London:
Harvard University Press and Heinemann 1952.
—— Metaphysics , Engl. tr. H. T redennick, London/Cambridge, MA: Heinemann
and Harvard UP 1935, 2 vols.
—— Nicomachean Ethics , Engl. tr. W.D. Ross, rev. by J.O. Urmson, Princeton:
Princeton University Press 1984.
—— Politics , Engl. tr. H. Rackham, Cambridge, MA/London: Harvard University
Press and Heinemann 1944.
—— Politics , Engl. tr. B. Jowett, Princeton: Princeton University Press 1984.
—— Posterior Analytics , Engl. tr. J. Barnes, Princeton: Princeton University Press
1984.
—— Prior Analytics , Engl. tr. H. T redennick, Cambridge, MA: Harvard University
Press 1938.
[Aristotle], On Indivisible Lines , Engl. tr. W.S. Hett, London/Cambridge, MA:
Heinemann and Harvard University Press 1955.
Aristoxenus, Elements of Harmonics , Engl. tr. A. Barker, Cambridge 1989.
Athenaeus, Deipnosophistae , Engl. tr. C. Burton Gulick, London/New York: Heine-
mann and Putnam 1928.
Athenaeus Mechanicus, On Machines , in R. Schneider, ‘Griechische Poliorketiker
III’, in Abhandlungen der königlichen Gesellschaft der Wissenschaften zu Göttingen ,
Philol.-hist. Kl. N.F . 12, heft 5 (1912).
Augustine, Against the Academics , ed. J.-P . Migne, Paris 1865, vol.1.
—— The City of God , Engl. tr. G.E. Mc Cracken et al., London/Cambridge, MA:
Heinemann and Harvard University Press 1957–72.
—— Confessions , Engl. tr. H. Chadwick, Oxford: Oxford University Press 1991.
—— On the Quantity of the Soul , ed. J.-P . Migne, Paris 1865, vol.1.
—— The Genesis Interpreted Literally , ed. and French tr. P . Agaësse and A. Solignac,
Desclée de Brouwer 1970–2, 2 vols.
—— On the Immortality of the Soul , Engl. tr. G. Watson, Warminster: Aris and
Phillips 1990.
—— On Free Will , ed. J.-P . Migne, Paris 1865, vol.1.
—— On Music , ed. J.-P . Migne, Paris 1865, vol.1.
—— On the Trinity , Engl. tr. S. McKenna, Washington: Catholic University of
America Press 1963.
—— Letters , ed. J.-P . Migne, 1861, vol.2.
—— On the Agreement of the Evangelists , ed. J.-P . Migne, Paris 1865, vol.3.
—— Soliloquia , Engl. tr. G. Watson, Warminster: Aris and Phillips 1990.
Autolycus of Pitane, La Sphère en Movement. Levers et Couchers héliaques. T estimonia ,
ed. and French tr. G. Aujac, Paris: Les Belles Lettres 1979.
Balbus, Explanation and Account of all Measures , in Gromatici Veteres , eds F . Blume,
K. Lachmann, A. Rudorff, Berlin: Reimer 1848–52, vol.1.
Biton, Construction of War Instruments , Engl. tr. E.W. Marsden, in Greek and Roman
Artillery: T echnical Treatises , Oxford: Clarendon Press 1971.

269BIBLIOGRAPHY
Callimachus, Aetia , ed. and Engl. tr. C.A. T rypanis, London/Cambridge, MA:
Heinemann and Harvard University Press 1968.
Cassiodorus, Letters , ed. Th. Mommsen, Berlin: Weidmann 1894 (Monumenta
Germaniae Historica 12).
M. Porcius Cato, On Agriculture , Engl. tr. W.D. Hooper and H. Boyd Ash, London/
Cambridge, MA: Heinemann and Harvard University Press 1934.
Cicero, Against L. Calpurnius Piso , Engl. tr. N.H. Watts, Cambridge, MA/London:
Harvard University Press and Heinemann 1931.
—— Against Verres , Engl. tr. L.H. Greenwood, Cambridge, MA: Harvard University
Press 1935.
—— Letters to Atticus , Engl. tr. E.O. Winstedt, London/Cambridge, MA:
Heinemann and Harvard University Press 1912–18.
—— Letters to his Brother Quintus , Engl. tr. W. Glynn Williams, Cambridge, MA/
London: Harvard University Press and Heinemann 1972.
—— On the Commonwealth , Engl. tr. C.W. Keyes, London/Cambridge, MA:
Heinemann and Harvard University Press 1948.
—— On Divination , Engl. tr. by W.A. Falconer, London/Cambridge, MA:
Heinemann and Harvard University Press 1923.
—— On the Greatest Good and Bad , Engl. tr. H. Rackham, Cambridge, MA/
London: Harvard University Press and Heinemann 1971.
—— On the Agrarian Law , Engl. tr. J.H. Freese, Cambridge, MA/ London: Harvard
University Press and Heinemann 1967.
—— On the Orator , Engl. tr. H. Rackham, London/Cambridge, MA: Heinemann
and Harvard University Press 1967, 2 vols.
—— Philippicae , Engl. tr. W.C.A. Ker, London/Cambridge, MA: Heinemann and
Harvard University Press 1963.
—— Tusculan Disputations , Engl. tr. J.E. King, London/Cambridge, MA:
Heinemann and Harvard University Press 1950.
Clement of Alexandria, Eclogues on the prophets , ed. O. Stählin, Berlin: Akademie
1970.
—— Exhortation to the Greeks , Engl. tr. G.W. Butterworth, Cambridge, MA/
London: Harvard University Press and Heinemann 1919.
—— Stromata , ed. O. Stählin, Berlin: Akademie 1985, 2 vols.
—— Stromata 5 , ed. A. Le Boulluec, Paris: Éditions du Cerf 1981.
—— The Rich Man’s Salvation , Engl. tr. G.W. Butterworth, Cambridge, MA/
London: Harvard University Press and Heinemann 1919.
Cleomedes, On the Circular Motion of Heavenly Bodies , French tr. R. Goulet as
Théorie élémentaire , Paris: Vrin 1980.
Codex of Justinian , ed. E. Herrmann, Leipzig: Baumgartner 18755.
Codex of Theodosius , eds Th. Mommsen and P .M. Meyer, Berlin: Weidmann 1954,
3 vols.
L. Junius Moderatus Columella, On Agriculture , Engl. tr. H. Boyd Ash, London/
Cambridge, MA: Heinemann and Harvard University Press 1941.
Demosthenes, Against Aphobus I , Engl. tr. A.T. Murray, London/Cambridge, MA:
Heinemann and Harvard University Press 1958.

270BIBLIOGRAPHY
—— Against Leptines , Engl. tr. J.H. Vince, London/Cambridge, MA: Heinemann
and Harvard University Press 1954.
—— For Phormio , Engl. tr. A.T. Murray, London/Cambridge, MA: Heinemann
and Harvard University Press 1958.
—— On the Crown , Engl. tr. C.A. Vince and J.H. Vince, Cambridge, MA/London:
Harvard University Press and Heinemann 1971.
Digesta of Justinian , eds Th. Mommsen and P . Krueger; Engl. tr. ed. by A. Watson,
Philadelphia: University of Pennsylvania Press 1985, 4 vols.
Dio Cassius, History , Engl. tr. E. Cary, Cambridge, MA/ London: Harvard University
Press and Heinemann 1969, 9 vols.
Diocles, On Burning Mirrors , ed. and Engl. tr. G.J. Toomer, Berlin/Heidelberg/
New York: Springer 1976.
Diodorus, Historical library , Engl. tr. C.H. Oldfather, London/Cambridge, MA:
Heinemann and Harvard University Press 1933.
Diogenes Laertius, Lives of the Philosophers , Engl. tr. R.D. Hicks, London/
Cambridge, MA: Heinemann and Harvard University Press 1959.
Diophantus, Arithmetica , ed. P . Tannery, Leipzig: T eubner 1893.
—— On Polygonal Numbers , ed. P . Tannery, Leipzig: T eubner 1893.
—— Les arithmétiques IV-VII , ed. and French tr. R. Rashed, Paris: Les Belles Lettres
1984, 2 vols.
Euclid, Data , ed. H. Menge, Leipzig: T eubner 1896.
—— Elements , ed. J.L. Heiberg, revised by E.S. Stamatis, with Scholia , Leipzig:
T eubner 1969–73, 5 vols.
—— Optics , ed. J.L. Heiberg, Leipzig: Teubner 1895.
—— Phenomena , Engl. tr. J.L. Berggren and R.S.D. Thomas, New York/London:
Garland 1996.
[Euclid], Sectio Canonis , Engl. tr. A. Barker, Cambridge 1989.
Eunapius, Lives of the Philosphers and Sophists , Engl. tr. W. Cave Wright, London/
Cambridge, MA: Heinemann and Harvard University Press 1968.
Eutocius, Commentary on Archimedes’ Equilibria of Planes , ed. and French tr. C.
Mugler, Paris: Les Belles Lettres 1972 (in Archimedes, Works , vol.4).
—— Commentary on Archimedes’ Measurement of the Circle, ed. and French tr. C.
Mugler, Paris: Les Belles Lettres 1972 (in Archimedes, Works , vol.4).
—— Commentary on Archimedes’ Sphere and Cylinder, ed. and French tr. C. Mugler,
Paris: Les Belles Lettres 1972 (in Archimedes, Works , vol.4).
—— Commentary on Apollonius’ Conics , ed. J.L. Heiberg, Leipzig: Teubner 1893
(in Apollonius, Works , vol. II).
Siculus Flaccus, Fields Regulation , ed. and French tr. M. Clavel-Lévêque, D. Conso,
F . Favory, J.-Y. Guillaumin and P . Robin, Napoli: Jovene 1993.
Frontinus, On the Aqueducts of the City of Rome , Engl. tr. C.E. Bennett, London/
New York: Heinemann and Putnam 1925.
—— On Land Surveying , in Corpus Agrimensorum Romanorum , ed. K. Thulin,
Leipzig: Teubner 1913.
Fronto, Letters to the Emperor , Engl. tr. C.R. Haines, London/Cambridge, MA:
Heinemann and Harvard University Press 1919–20.

271BIBLIOGRAPHY
Galen, An Exhortation to Study the Arts , Engl. tr. P .N. Singer, Selected Works , Oxford:
Oxford University Press 1997.
—— On his Own Books , ed. K.G. Kühn, Leipzig 1830 (repr. Hildesheim: Olms
1965), vol.19, Engl. tr. P .N. Singer, Selected Works , Oxford: Oxford University
Press 1997.
—— On the Sentences of Hippocrates and Plato , Engl. tr. P . de Lacy, Berlin: Akademie
1981–4, 3 vols. ( CMG 5.4.1.2).
—— On Prognosis , Engl. tr. V. Nutton, Berlin: Akademie 1979 ( CMG 5.8.1).
—— Logical Institution , Engl. tr. J. Spangler Kieffer, Baltimore: The Johns Hopkins
University Press 1964.
—— The Affections and Errors of the Soul , Engl. tr. P .N. Singer, Selected Works ,
Oxford: Oxford University Press 1997.
—— The Best Doctor is also a Philosopher , Engl. tr. P .N. Singer, Selected Works ,
Oxford: Oxford University Press 1997.
Geminus, Introduction to the Phenomena , ed. and French tr. G. Aujac, Paris: Les
Belles Lettres 1975.
Hephaestio Thebanus, Astrology , ed. D. Pingree, Leipzig: Teubner 1973.
Hero of Alexandria, Automata , ed. W. Schmidt, Leipzig: T eubner 1899, vol.1.
—— Belopoiika , Engl. tr. E.W. Marsden, in Greek and Roman Artillery: T echnical
Treatises , Oxford: Clarendon Press 1971.
—— Dioptra , ed. H. Schöne, Leipzig: Teubner 1903, vol.3.
—— Mechanica , eds L. Nix and W. Schmidt, Leipzig: T eubner 1900.
—— Metrica , ed. H. Schöne, Leipzig: T eubner 1903.
—— Pneumatica , ed. W. Schmidt, Leipzig: T eubner 1899, vol.1.
[Hero], Definitions , ed. J.L. Heiberg, Leipzig: T eubner 1912.
Herodotus, Histories , Engl. tr. A.D. Godley, Cambridge, MA/London: Heinemann
and Harvard University Press 1926, 4 vols.
Hipparchus, Commentary to the Phenomena of Eudoxus and Aratus , ed. and German
tr. K. Manitius, Leipzig: T eubner 1894.
Homer, Iliad, Engl. tr. A.T. Murray, London/Cambridge, MA: Heinemann and
Harvard University Press 1924, 2 vols.
Hyginus Astronomicus, On Astronomy , ed. and French tr. A. Le Bœuffle, Paris: Les
Belles Lettres 1983.
Hyginus, Categories of Fields , in Gromatici Veteres , eds F . Blume, K. Lachmann and
A. Rudorff, Berlin: Reimer 1848–52.
Hyginus 3, On the Division of Military Camps , ed. A. Grillone, Leipzig: T eubner
1977.
Hyginus Gromaticus, On Boundary Regulation , in Gromatici Veteres , eds F . Blume,
K. Lachmann and A. Rudorff, Berlin: Reimer 1848–52, vol.1.
Hypsicles, Book 14 of the Elements , in Euclid, Elements , ed. J.L. Heiberg, Leipzig:
T eubner 1888, vol.5.
Iamblichus, De Mysteriis Aegyptorum , ed. and French tr. É. des Places, Paris: Les
Belles Lettres 1966.
—— Exhortation to Philosophy , Engl. tr. Th. Moore Johnson, Grand Rapids: Phanes
Press 1988.

272BIBLIOGRAPHY
—— Life of Pythagoras (On the Pythagorean Life ), Engl. tr. G. Clark, Liverpool:
Liverpool University Press 1989.
—— On Common Mathematical Knowledge , ed. N. Festa, Leipzig: Teubner 1891.
—— On Nicomachus’ Introduction to Arithmetic , ed. E. Pistelli, Leipzig: T eubner
1894.
[Iamblichus], Theologia Arithmetica , Engl. tr. R. Waterfield, Grand Rapids: Phanes
Press 1988.
Isaeus, On the Estate of Hagnias , Engl. trans. E.S. Forster, London/New York: Heine-
mann and Putnam 1927.
Isocrates, Against the Sophists, Engl. tr. G. Norlin, London/New York: Heinemann
and Putnam 1929.
—— Antidosis , Engl. tr. G. Norlin, London/New York: Heinemann and Putnam
1929.
—— Helen , Engl. tr. L. van Hook, London/Cambridge, MA: Heinemann and
Harvard University Press 1945.
—— Panathenaicus , Engl. tr. G. Norlin, London/New York: Heinemann and
Putnam 1929.
Julian, Against the Galileans , Engl. tr. W. Cave Wright, London/New York: Heine-
mann and Macmillan 1913.
[Julian], Letters , Engl. tr. W. Cave Wright, London/New York: Heinemann and
Macmillan 1913.
Livy, From the Foundation of the City , Engl. tr. F . Gardner Moore, London/
Cambridge, MA: Heinemann and Harvard University Press 1951.
Lucian, Icaromenippus , Engl. tr. A.M. Harmon, London/New York: Heinemann
and Macmillan 1915.
—— Philosophies for Sale , Engl. tr. A.M. Harmon, London/New York: Heinemann
and Macmillan 1915.
Lysias, Against Diogeiton , Engl. tr. W.R.M. Lamb, London/New York: Heinemann
and Putnam 1930.
—— Against Nicomachus , Engl. tr. W.R.M. Lamb, London/New York: Heinemann
and Putnam 1930.
—— Defence Against a Charge of T aking a Bribe , Engl. tr. W.R.M. Lamb, London/
New York: Heinemann and Putnam 1930.
—— On the Property of Aristophanes: Against the Treasury , Engl. tr. W.R.M. Lamb,
London/New York: Heinemann and Putnam 1930.
L. Volusius Maecianus, Distribution , ed. F . Hultsch, in Metrologicorum Scriptorum
Reliquiae II , Leipzig: T eubner 1866.
Manilius, Astronomica , Engl. tr. G.P . Goold, Cambridge, MA/London: Harvard
University Press 1992.
Marinus, Commentary on Euclid’s Data , in Euclid, Opera , vol.VI, ed. J.L. Heiberg
and H. Menge, Leipzig: Teubner 1896.
Martial (Marcus Valerius Martialis), Epigrammata , Engl. tr. D.R. Shackleton Bailey,
London/Cambridge, MA: Harvard University Press 1993, 3 vols.
Nicomachus, Introduction to Arithmetic , Engl. tr. M.L. D’Ooge, in R. Maynard
Hutchins (ed.), Great Books of the Western World, Chicago: Benton 1952 (the
translation first publ. 1926), vol.11.

273BIBLIOGRAPHY
Marcus Junius Nipsus, Measurement of a River, Replacing a Boundary, Measurement
of an Area , in Gromatici Veteres , eds F . Blume, K. Lachmann and A. Rudorff,
Berlin: Reimer 1848–52, vol.1.
Notitia dignitatum , ed. O. Seeck, 1876, reprinted Frankfurt a. M.: Minerva 1962.
Panegyric 8 , in C.E.V. Nixon and B. Saylor Rodgers, In Praise of Later Roman
Emperors. The Panegyrici Latini , Engl. tr. of the Latin texts R.A.B. Mynors,
Berkeley/Los Angeles/Oxford: University of California Press 1994.
Pappus of Alexandria, Mathematical Collection , ed. and Latin tr. F . Hultsch, 3 vols,
Berlin: Weidmann 1876–8.
—— Book VII of the Collection , ed. and Engl. tr. A. Jones, New York/Berlin/Heidel-
berg/Tokyo: Springer 1986, 2 vols.
Paulus Alexandrinus, Elements of Astrology , ed. E. Boer, Leipzig: Teubner 1958.
Philo of Alexandria, On Mating with the Preliminary Studies , Engl. tr. F .H. Colson
and G.H. Whitaker, London/New York: Heinemann and Putnam 1932.
—— On the Creation of the World According to Moses , Engl. tr. F .H. Colson and
G.H. Whitaker, London/New York: Heinemann and Putnam 1929.
—— On the Migration of Abraham , Engl. tr. F .H. Colson and G.H. Whitaker,
London/New York: Heinemann and Putnam 1932.
—— Who is the Heir of Divine Things and on the Division into Equals and Opposites ,
Engl. tr. F .H. Colson and G.H. Whitaker, London/New York: Heinemann andPutnam 1932.
Philo of Byzantium, Book 8 of the Mechanical Syntaxis (Poliorketika ), ed. and French
tr. by Y. Garlan, in Garlan (1974), 279–404.
—— Mechanica IV (Belopoiika ), Engl. tr. E.W. Marsden, in Greek and Roman
Artillery: T echnical Treatises , Oxford: Clarendon Press 1971.
Philostratus, Lives of the Sophists , with Engl. tr. W. Cave Wright, London/New
York: Heinemann and Putnam 1922.
Plato, Euthydemus , Engl. tr. W.R.M. Lamb, Cambridge, MA/London: Harvard
University Press and Heinemann 1924.
—— Euthyphro , Engl. tr. H.N. Fowler, London/New York: Heinemann and Putnam
1914.
—— Greater Hippias , Engl. trans. H.N. Fowler, London/New York: Heinemann
and Putnam 1926.
—— Laws , Engl. tr. T.J. Saunders, Indianapolis/Cambridge: Hackett 1997.
—— Lesser Hippias , Engl. tr. N.D. Smith, Indianapolis/Cambridge: Hackett 1997.
—— Meno , Engl. tr. G.M.A. Grube, Indianapolis/Cambridge: Hackett 1997.
—— Parmenides , Engl. tr. H.N. Fowler, London/Cambridge, MA: Heinemann
and Harvard University Press 1926.
—— Philebus , Engl. tr. H.N. Fowler, London/Cambridge, MA: Heinemann and
Harvard University Press 1925.
—— Philebus , Engl. tr. D. Frede, Indianapolis/Cambridge: Hackett 1997.
—— Protagoras, Engl. tr. W.R.M. Lamb, Cambridge, MA/London: Harvard
University Press and Heinemann 1924.
—— Protagoras , Engl. tr. S. Lombardo and K. Bell, Indianapolis/Cambridge:
Hackett 1997.

274BIBLIOGRAPHY
—— Republic , Engl. tr. P . Shorey, London/Cambridge, MA: Heinemann and
Harvard University Press 1946, 2 vols.
—— Republic , Engl. tr. Robin Waterfield, Oxford: Oxford University Press 1993.
—— Statesman , Engl. tr. C.J. Rowe, Indianapolis/Cambridge: Hackett 1997.
—— Theaetetus , Engl. tr. M.J. Levett and rev. M.F . Burnyeat, Indianapolis/
Cambridge: Hackett 1997.
—— Timaeus , Engl. tr. R.G. Bury, London/Cambridge, MA: Heinemann and
Harvard University Press 1966.
[Plato], Epinomis , Engl. tr. A.E. Taylor, in The Collected Dialogues of Plato , eds E.
Hamilton and H. Cairns, Princeton: Princeton University Press 1963.
Plautus, Poenulus , Engl. tr. P . Nixon, London/New York: Heinemann and Putnam
1932.
Pliny Jr., Letters , Engl. tr. W. Melmoth and W.M.L. Hutchinson, London/
Cambridge, MA: Heinemann and Harvard University Press 1947, 2 vols.
Pliny Sr, Natural History , Engl. tr. by D.E. Eichholz, London/Cambridge, MA:
Heinemann and Harvard University Press 1962 (10 vols).
Plutarch, Against the Stoics on Common Conceptions , Engl. tr. H. Cherniss, Cam-
bridge, MA/London: Harvard University Press and Heinemann 1976.
—— Life of Cicero , Engl. tr. R. Warner in Fall of the Roman Republic. Six Lives by
Plutarch , London: Penguin 1972.
—— Life of Marcellus , Engl. tr. B. Perrin, London/Cambridge, MA: Heinemann
and Harvard University Press 1968.
—— On the E in Delphi , Engl. tr. F .C. Babbitt, London/Cambridge, MA:
Heinemann and Harvard University Press 1969.
—— On the Genius of Socrates , Engl. tr. P .H. De Lacy and B. Einarson, London/
Cambridge, MA: Heinemann and Harvard University Press 1968.
—— T able-Talk , Engl. tr. E.L. Minar, London/Cambridge, MA: Heinemann and
Harvard University Press 1961.
Polybius, Histories , Engl. tr. W.R. Paton, London/Cambridge, MA: Heinemann
and Harvard University Press 1960.
Proclus, Commentary on the First Book of Euclid’s Elements , Engl. tr. G.R. Morrow,
Princeton: Princeton University Press 1970.
—— Commentary on Plato’s Republic , French tr. A.J. Festugière, Paris: Vrin 1970.
—— A Sketch of the Astronomical Positions , ed. and German tr. K. Manitius, Leipzig:
T eubner 1909.
—— Physics , ed. and German tr. A. Ritzenfeld, Leipzig: T eubner 1912.
—— The Elements of Theology , ed. and Engl. tr. E.R. Dodds, Oxford: Clarendon
Press 1963.
—— Commentary on Plato’s Timaeus , ed. E. Diehl, Leipzig: T eubner 1903–6, 3
vols.
Ptolemy, Astrology (T etrabiblos ), Engl. tr. F .E. Robbins, London/Cambridge, MA:
Heinemann and Harvard University Press 1940.
—— For the Use of the T ables , in Opera astronomica minora, ed. J.L. Heiberg, Leipzig:
T eubner 1907.
—— Geography, Engl. tr. E.L. Stevenson, New York: Dover 1991 (first publ. 1932).

275BIBLIOGRAPHY
—— Harmonics , ed. I. Düring, Göteborg 1930, repr. New York/London: Garland
1980.
—— On the Criterion, Engl. tr. by the Liverpool/Manchester Seminar on Ancient
Philosophy, in P . Huby and G. Neals (eds), The Criterion of Truth , Liverpool:
Liverpool University Press 1989, 179–230.
—— Optica, French tr. from Latin tr. of Arabic tr. A. Lejeune, Leiden: Brill 1989.
—— Planetary Hypotheses , in Opera astronomica minora, ed. J.L. Heiberg, Leipzig:
T eubner 1907.
—— Syntaxis , Engl. tr. G.J. Toomer, London: Duckworth 1984.
Quintilian (Marcus Fabius Quintilianus), Handbook of Oratory , Engl. tr. H.E. Butler,
London/Cambridge, MA: Heinemann and Harvard University Press 1963, 4vols.
Seneca, Letters to Lucilius , Engl. tr. C.D.N. Costa, Warminster: Aris and Phillips
1988.
Select Papyri , Engl. tr. A.S. Hunt and C.C. Edgar, Cambridge, MA/London: Harvard
University Press and Heinemann 1932–4, 5 vols.
Serenus of Antinopolis, On the Section of a Cylinder, On the Section of a Cone , ed.
J.L. Heiberg, Leipzig: T eubner 1896.
Sextus Empiricus, Against the Arithmeticians , Engl. tr. R.G. Bury, London/
Cambridge, MA: Heinemann and Harvard University Press 1936.
—— Against the Astrologers , Engl. tr. R.G. Bury, London/Cambridge, MA:
Heinemann and Harvard University Press 1936.
—— Against the Geometers , Engl. tr. R.G. Bury, London/Cambridge, MA: Heine-
mann and Harvard University Press 1936.
—— Against the Logicians , Engl. tr. R.G. Bury, London/Cambridge, MA: Heine-
mann and Harvard University Press 1935.
—— Against the physicists , Engl. tr. R.G. Bury, London/Cambridge, MA: Heine-
mann and Harvard University Press 1936.
Simplicius, Commentary on Aristotle’s Categories , ed. K. Kalbfleisch, CAG 8, Berlin:
Reimer 1907.
—— Commentary on Aristotle’s Physics , ed. H. Diels, CAG 9–10, Berlin: Reimer
1882, 1895.
Sophocles, Antigone , Engl. tr. H. Lloyd-Jones, Cambridge, MA/London: Harvard
University Press 1994.
Stobaeus Johannes, Anthology , eds C. Wachsmuth and O. Hense, Berlin: Weidmann
1884–1923, 6 vols.
Strabo, Geography , Engl. tr. H.L. Jones, London/New York: Heinemann and Putnam
1917.
Synesius of Cyrene, Letters, Engl. tr. A. Fitzgerald, London/Oxford: Milford and
Oxford University Press 1926.
Theocritus, Idylls , Engl. tr. A.S.F . Gow, Cambridge: Cambridge University Press
1952.
Theodosius T ripolites, On Stations. On Days and Nights , ed. R. Fecht, in Abhand-
lungen der Gesellschaft der Wissenschaften zu Göttingen , phil.-hist. Kl., n.s. 19
no. 4, 1927.

276BIBLIOGRAPHY
—— Sphaerica , ed. J.L. Heiberg, in Abhandlungen der Gesellschaft der Wissenschaften
zu Göttingen , phil.-hist. Kl., n.s. 19 no. 3, 1927.
Theognis, Works in Greek Lyric Poetry , Engl. tr. M.L. West, Oxford: Clarendon
Press 1993.
Theon of Alexandria, Commentary on Ptolemy’s Syntaxis , ed. A. Rome, Cittá del
Vaticano: Biblioteca Apostolica Vaticana 1936 (Studi e Testi 72).
—— Greater Commentary on Ptolemy’s Handy T ables , eds J. Mogenet and A. Tyhon,
Citta’ del Vaticano 1985–91.
—— Lesser Commentary on Ptolemy’s Handy Tables , ed. A. Tyhon, Citta’ del Vaticano
1978.
Theon of Smyrna, Account of Mathematics Useful to Reading Plato , ed. E. Hiller,
Leipzig: T eubner 1878.
Thucydides, The Peloponnesian War , Engl. tr. C. Forster Smith, London/Cambridge,
MA: Heinemann and Harvard University Press 1928.
Ulpian, De excusatione in Iurisprudentiae anteiustinianae reliquias , eds P .E. Huschke,
E. Seckel and B. Kuebler, Leipzig: T eubner 1911, 2 vols.
M. Terentius Varro, On Agriculture , Engl. tr. W.D. Hooper and H. Boyd Ash,
London/Cambridge. MA: Heinemann and Harvard University Press 1935.
Vegetius, On military things , Engl. tr. L.F . Stelten, New York/Bern/Frankfurt a.M./
Paris: Lang 1990.
Vettius Valens, Anthology , ed. D. Pingree, Leipzig: Teubner 1986.
Vitruvius, On Architecture , Engl. tr. F . Granger, London/Cambridge, MA: Heine-
mann and Harvard University Press 1962.
Xenophon, Memorabilia , Engl. tr. E.C. Marchant, London/New York: Heinemann
and Putnam 1923.
Secondary sources
Alföldi G. (1975), Römische Sozialgeschichte , Engl. tr. by D. Braund and F . Pollock
as The Social History of Rome , London/Sydney: Croom Helm 1985.
Andreau J. (1999), Banking and Business in the Roman World , Engl. tr. J. Lloyd,
Cambridge: Cambridge University Press.
Angeli A. and Dorandi T. (1987), ‘Il pensiero matematico di Demetrio Lacone’,
Cronache ercolanesi 17, 89–103.
Asheri D. (1966), ‘Distribuzioni di terre nell’antica Grecia’, Memorie dell’Accademia
di scienze di Torino, classe di scienze morali, storiche e filologiche , serie 4, 10,
1–127.
—— (1975), ‘Osservazioni sulle origini dell’urbanistica ippodamea’, Rivista storica
italiana 87, 5–16.
Austin M.M. (ed.) (1981), The Hellenistic World from Alexander to the Roman
Conquest. A Selection of Ancient Sources in Translation , Cambridge: Cambridge
University Press.
Badian E. (1972), Publicans and Sinners. Private Enterprise in the Service of the
Roman Republic , Ithaca/London: Cornell University Press.
Beard M. and Crawford M. (1999), Rome in the Late Republic. Problems and Inter-
pretations , London: Duckworth (2nd edn).

277BIBLIOGRAPHY
Becker O. (1933), ‘Eudoxos-Studien I. Eine voreudoxische Proportionenlehre und
ihre Spuren bei Aristoteles und Euklid’, Quellen und Studien zur Geschichte der
Mathematik, Astronomie und Physik Abt. B 2, 369–87.
—— (ed.) (1965), Zur Geschichte der griechischen Mathematik , Darmstadt: Wissen-
schaftliche Buchgesellschaft.
Bernal M. (1992), ‘Animadversions on the Origins of Western Science’, Isis 83,
596–607.
Bogaert R. (ed.) (1976), Epigraphica. III: T exts on Bankers, Banking and Credit in
the Greek World , Leiden: Brill.
Bowen A.C. and Goldstein B.R. (1988), ‘Meton of Athens and astronomy in the
late fifth century BC’, in E. Leichty, M. de J. Ellis and P . Gerardi (eds), A Scientific
Humanist: Studies in Memory of Abraham Sachs , Philadelphia, 39–81.
Boyaval B. (1973), ‘Tablettes mathématiques du Musée du Louvre’, Revue Archéo-
logique , 243–60.
Boyd T.D. and Jameson M.H. (1981), ‘Urban and land division in ancient Greece’,
Hesperia 50, 327–42.
Brashear W. (1985), ‘Holz- und Wachstafeln der Sammlung Kiseleff’, Enchoria 13,
13–23.
Bruins E.M., Liesker W.H.M. and Sijpesteijn P .J. (1988), ‘A Ptolemaic Papyrus from
the Michigan Collection’, Zeitschrift für Papyrologie und Epigraphik 74, 23–8.
Buchner E. (1982), Die Sonnenuhr des Augustus , Mainz a.R.: Philipp von Zabern.
Burford A. (1969), The Greek T emple Builders at Epidauros. A Social and Economic
Study of Building in the Asklepian Sanctuary, During the Fourth and Early ThirdCenturies B.C ., Liverpool: Liverpool University Press.
Burkert W. (1972), Weisheit und Wissenschaft: Studien zu Pythagoras, Philolaos und
Platon , Nürnberg: H. Carl 1962, revised Engl. tr. E.L. Minar Jr., Lore and
Science in Ancient Pythagoreanism , Cambridge, MA: Harvard University Press.
Burns A. (1971), ‘The T unnel of Eupalinus and the T unnel Problem of Hero of
Alexandria’, Isis 62, 172–85.
Cambiano G. (1991), Platone e le tecniche , Roma/Bari: Laterza (2nd edn).
Carrié J.-M. (1994), ‘Diocletien et la fiscalité’, Antiquité tardive 2, 33–64.
Carter J. Coleman (1990), ‘Dividing the chora: Metaponto in the sixth century
BC’, American Journal of Archaeology 94, 320.
Cartledge P . (1996), ‘Comparatively equal’, in J. Ober and C. Hedrick (eds), Demo-
kratia. A Conversation on Democracies, Ancient and Modern, Princeton: Princeton
University Press, 175–85.
Castagnoli F . (1956), Ippodamo di Mileto e l’urbanistica a pianta ortogonale , Roma:
De Luca.
Caveing M. (1990), ‘Introduction générale’, in B. Vitrac (ed.), Les éléments de Euclide ,
Paris: PUF , 13–148.
Charles-Saget A. (1982), L ’architecture du divin. Mathématique et philosophie chez
Plotin et Proclus , Paris: Les Belles Lettres.
Chisholm K. and Ferguson J. (eds) (1981), Rome. The Augustan Age. A Sourcebook ,
Oxford: Oxford University Press.
Chouquer G. and Favory F . (1992), Les arpenteurs romains. Théorie et pratique ,
Paris: Errance.

278BIBLIOGRAPHY
Clagett M. (1956), Greek Science in Antiquity , New York: Abelard-Schuman.
Clarke K. (1999), Between Geography and History. Hellenistic Constructions of the
Roman World , Oxford: Clarendon Press.
Cohen E.E. (1992), Athenian Economy and Society. A Banking Perspective , Princeton:
Princeton University Press.
Coulton J.J. (1977), Greek Architects at Work. Problems of Structure and Design ,
London: P . Elek.
Crawford D.S. (1953), ‘A mathematical tablet’, Aegyptus 33, 222–40.
Cuomo S. (2000a), Pappus of Alexandria and the Mathematics of Late Antiquity ,
Cambridge: Cambridge University Press.
—— (2000b), ‘Divide and rule: Frontinus and Roman land-surveying’, Studies in
History and Philosophy of Science 31, 189–202.
Cuvigny H. (1985), L ’arpentage par espéces dans l’Égypte ptolémaïque d’après les
papyrus grecs , Bruxelles: Fondation Égyptologique Reine Élisabeth.
Decorps-Foulquier M. (1998), ‘Eutocius d’Ascalon éditeur du traité des Coniques
d’Apollonios de Pergé et l’exigence de ‘clarté’: un exemple des pratiquesexégétiques et critiques des héritiers de la science alexandrine’, in G. Argoudand J.-Y. Guillaumin (eds), Sciences exactes et sciences appliquées à Alexandrie ,
Saint-Étienne: Publications de l’Université de Saint-Étienne, 87–101.
DeLaine J. (1996), ‘“De aquis suis”?: The “Commentarius” of Frontinus’, in C.
Nicolet (ed.), Les littératures techniques dans l’antiquité romaine. Statut, public et
destination, tradition , Genève: Vandœuvres, 117–45.
Déléage A. (1933), ‘Les cadastres antiques jusqu’à Dioclétien’, Études de Papyrologie
2, 73–228.
Della Corte M. (1954), Case ed Abitanti di Pompei , Roma: Bretschneider.
de Solla Price D.J. (1969), ‘Portable sundials in antiquity, including an account of
a new example from Aphrodisias’, Centaurus 14, 242–66.
—— (1974), Gears from the Greeks. The Antikythera Mechanism – A Calendar
Computer from c. 80 B.C. , Philadelphia: The American Philosophical Society
(T ransactions of the American Philosophical Society N.S. vol.64, part 7).
Dicks D.R. (1971), ‘Eratosthenes’, Dictionary of Scientific Biography , New York:
Scribner, vol.4, 388–93 .
Dijksterhuis E.J. (1956), Archimedes , Engl. tr. C. Dikshoorn, Princeton: Princeton
University Press (reprinted 1987).
Dilke O.A.W. (1971), The Roman Land Surveyors. An Introduction to the Agrimen-
sores, Newton Abbot: David and Charles.
—— (1987), Mathematics and Measurement , British Museum Publications.
Dörrie H. (1987), Die geschichtlichen Wurzeln des Platonismus , Stuttgart: Frommann
Holzboog.
Dorandi T. (1994), ‘La tradizione papiracea degli “Elementi” di Euclide’, in W .
Bülow-Jacobsen (ed.), Proceedings of the XX International Congress of Papyrologists ,
Copenhagen: Museum T usculanum Press, 306–11.
Dufková M. and Pecírka J. (1970), ‘Excavations of farms and farmhouses in the
chora of Chersonesos in the Crimea’, Eirene 8, 123–74.
Edgar C.C. (ed.) (1925), Zenon Papyri (Catalogue général des antiquités égyptiennes
du musée du Caire ), Cairo: Institut Français d’archeologie orientale 1925–31,
4 vols.

279BIBLIOGRAPHY
Edwards C. (1993), The Politics of Immorality in Ancient Rome , Cambridge:
Cambridge University Press.
Evans J. (1999), ‘The material culture of Greek astronomy’, Journal for the History
of Astronomy 30, 237–307.
Fallu E. (1973), ‘Les rationes du proconsul Cicéron. Un exemple de style adminis-
tratif et d’interprétation historique dans la correspondance de Cicéron’, in H.T emporini (ed.), Aufstieg und Niedergang der Römischen Welt , Berlin/New York:
De Gruyter, vol.1.3, 209–38.
Fellmann R. (1983), ‘Römische Rechentafeln aus Bronze’, Antike Welt 14, 36–40.
Ferrari G.A. (1984), ‘Meccanica allargata’, Atti del convegno‚ La scienza ellenistica‚
Pavia 1982 , Napoli: Bibliopolis 1984, 227–96.
Ferrary J.-L. (1988), Philhellénisme et impérialisme. Aspects idéologiques de la conquête
romaine du monde hellénistique, de la seconde guerre de Macédoine à la guerrecontre Mithridate , Rome: École Française de Rome.
Fowler D.H. (1988), ‘A catalogue of tables’, Zeitschrift für Papyrologie und Epigraphik
75, 273–80.
—— (1995), ‘Further arithmetical tables’, Zeitschrift für Papyrologie und Epigraphik
105, 225–8.
—— (1999), The Mathematics of Plato’s Academy. A New Reconstruction , Oxford:
Clarendon Press (2nd edn).
Gabba E. (1984), ‘Per un’interpretazione storica della centuriazione romana’ in
Misurare la terra: centuriazione e coloni nel mondo romano , Modena: Panini,
20–7.
Gaiser K. (1988), Philodems Academica. Die Berichte über Platon und die alte
Akademie in zwei herkulanensischen Papyri , Stuttgart: Cannstatt.
Garlan Y. (1974), Recherches de poliorcétique grecque , Athens: École Française
d’Athènes.
Gerstinger H. and Vogel K. (1932), ‘Eine stereometrische Aufgabensammlung im
Papyrus Graecus Vindobonensis 19996’, Mitteilungen aus der Papyrussammlung
der Nationalbibliothek in Wien (Papyrus Erzherzog Rainer) , N.S. 1 Wien, 11–
76.
Gibbs S.L. (1976), Greek and Roman Sundials , New Haven/London: Yale University
Press.
Gigon O. (1973), ‘Posidoniana-Ciceroniana-Lactantiana’, in Romanitas et Christia-
nitas. Studia I. H. Waszink , Amsterdam/London: North Holland, 145–80.
Grenfell B.P ., Hunt A.S. and Gilbart Smyly J. (eds) (1902), The T ebtunis Papyri ,
London: Frowde.
Grodzynski D. (1974), ‘Par la bouche de l’empereur’, in J.-P . Vernant et al., Divina-
tion et rationalité , Paris: Seuil, 267–94.
Guéraud O. and Jouguet P . (1938), Un livre d’écolier du iiie siècle avant J.-C., Cairo:
Institut Français d’archéologie orientale.
Guillaumin J.-Y. (1992), ‘La signification des termes contemplatio et observatio chez
Balbus et l’influence héronienne sur le traité’, in J.-Y. Guillaumin (ed.), Mathéma-
tiques dans l’antiquité , Saint-Étienne: Publications de l’Université de Saint-
Étienne, 205–14.

280BIBLIOGRAPHY
—— (1997), ‘L’éloge de la géometrie dans la préface du livre 3 des Metrica d’Héron
d’Alexandrie’, Revue des Études anciennes 99, 91–9.
Harvey F .D. (1965), ‘T wo kinds of equality’, Classica et Mediaevalia 26, 101–46.
Heath T .L. (1921), A History of Greek Mathematics , Oxford: Clarendon Press, 2
vols.
—— (1926), The Thirteen Books of Euclid’s Elements , New York: Dover 1956 (first
publ. 1926).
—— (1949), Mathematics in Aristotle , Oxford: Clarendon Press.
Hinrichs F .T. (1974), Die Geschichte der Gromatischen Institutionen , Wiesbaden:
Steiner.
—— (1992), ‘Die “agri per extremitatem mensura comprehensi”. Diskussion eines
Frontinstextes und der Geschichte seines Verständnisses’, in O. Behrends andL. Capogrossi Colognesi (eds), Die römische Feldmeßkunst. Interdisziplinäre
Beiträge zu ihrer Bedeutung für die Zivilisationsgeschichte Roms , Göttingen:
Vandenhoeck and Ruprecht, 348–74.
Hopkins K. (1978), Conquerors and Slaves. Sociological Studies in Roman History ,
Cambridge: Cambridge University Press.
Huffman C.A. (1993), Philolaus of Croton , Cambridge: Cambridge University Press.
Hurst A. (1998), ‘Géographes et poètes: le cas d’Apollonios de Rhodes’, in G.
Argoud and J.-Y. Guillaumin (eds), Sciences exactes et sciences appliquées à
Alexandrie , Saint-Étienne: Publications de l’Université de Saint-Étienne, 279–
88.
Huxley G. (1963), ‘Friends and Contemporaries of Apollonius of Perga’, Greek,
Roman and Byzantine Studies 4, 100–5.
Ioannidou G. (ed.) (1996), Catalogue of Greek and Latin Literary Papyri in Berlin ,
Mainz a. R.: von Zabern.
Irigoin J. (1998), ‘Les éditions de poètes à Alexandrie’, in G. Argoud and J.-Y.
Guillaumin (eds), Sciences exactes et sciences appliquées à Alexandrie , Saint-Étienne:
Publications de l’Université de Saint-Étienne, 405–13.
Jones A. (1997a), ‘Studies in the astronomy of the Roman period, I. The standard
lunar scheme’, Centaurus 39, 1–36.
—— (1997b), ‘Studies in the astronomy of the Roman period, II. Tables for solar
longitude’, Centaurus 39, 211–29.
—— (1998), ‘Studies in the astronomy of the Roman period, III. Planetary epoch
tables’, Centaurus 40, 1–41.
Kahn C.H. (1991), ‘Some remarks on the origins of Greek science and philosophy’,
in A.C. Bowen (ed.), Science and Philosophy in Classical Greece , New York/
London: Garland, 1–10.
Kienast H.J. (1995), Die Wasserleitung des Eupalinos auf Samos , Bonn: Habelt (Samos
19).
Knorr W.R. (1975), The Evolution of the Euclidean Elements , Dordrecht/Boston:
Reidel.
—— (1986), The Ancient Tradition of Geometric Problems , Boston/Basel/Berlin:
Birkhäuser.
—— (1993), ‘Arithmêtikê stoicheîosis: On Diophantus and Hero of Alexandria’,
Historia Mathematica 20, 180–92.

281BIBLIOGRAPHY
—— (1996), ‘The Wrong Text of Euclid: On Heiberg’s T ext and its Alternatives’,
Centaurus 38, 208–76.
Kurke L. (1999), Bodies, Games, Coins, and Gold. The Politics of Meaning in Archaic
Greece , Princeton: Princeton University Press.
Lang M. (1956), ‘Numerical notation on Greek vases’, Hesperia 25, 1–24.
—— (1968), ‘Abaci from the Athenian agora’, Hesperia 37, 241–3.
Lawrence A.W. (1946), ‘Archimedes and the design of Euryalus fort’, Journal of
Hellenic Studies 66, 99–107.
—— (1979), Greek Aims in Fortification , Oxford: Clarendon Press.
Leonardos B. (1925–6), ‘ Aluiaqeίot έpicqauaί ’, Aqvaiolocijή έugleqiy , 9–45.
Levick B. (1985), The Government of the Roman Empire. A Sourcebook , London/
Sidney: Croom Helm.
Lewis N. (1986), Greeks in Ptolemaic Egypt. Case Studies in the Social History of the
Hellenistic World , Oxford: Clarendon Press.
Lewis N. and Reinhold M. (1990), Roman Civilization. Selected Readings , New
York: Columbia University Press, 2 vols, (3rd edn).
Lloyd G.E.R. (1963), ‘Who is attacked in On ancient medicine ?’, Phronesis 8, 108–
26, reprinted in Methods and Problems in Greek Science. Selected Papers ,
Cambridge: Cambridge University Press 1991, 49–69.
—— (1972), ‘The social background of early Greek philosophy and science’, in D.
Daiches and A. Thorlby (eds), Literature and Western Civilization, I: The Classical
World , London: Aldus 1972, 381–95, reprinted in Methods and Problems in
Greek Science. Selected Papers , Cambridge: Cambridge University Press 1991,
121–40.
—— (1979), Magic, Reason, and Experience , Cambridge: Cambridge University
Press.
—— (1987a), ‘The Alleged Fallacy of Hippocrates of Chios’, Apeiron 20, 103–28.
—— (1987b), The Revolutions of Wisdom. Studies in the Claims and Practice of
Ancient Greek Science , Berkeley/Los Angeles/London: University of California
Press.
—— (1990), Demystifying Mentalities , Cambridge: Cambridge University Press.
Lyons H. (1927), ‘Ancient surveying instruments’, The Geographical Journal 69,
132–43.
Maas M. (1992), John Lydus and the Roman Past. Antiquarianism and Politics in the
Age of Justinian , London/New York: Routledge.
MacMullen R. (1966), Enemies of the Roman Order. Treason, Unrest and Alienation
in the Empire , London/New York: Routledge 1992 (first publ. Harvard: Harvard
University Press).
—— (1971), ‘Social History in Astrology’, Ancient Society 2, reprinted in Changes
in the Roman Empire: Essays in the Ordinary , Princeton/London: Princeton
University Press 1991, 218–24.
Mansfeld J. (1998), Prolegomena Mathematica. From Apollonius of Perga to the Late
Neoplatonists With an Appendix on Pappus and the History of Platonism, Leiden/
Boston/Köln: Brill.
Mau J. and Müller W. (1962), ‘Mathematische Ostraka aus der Berliner Sammlung’,
Archiv für Papyrusforschung 17, 1–10.

282BIBLIOGRAPHY
McCluskey S.C. (1990), ‘Gregory of T ours, monastic timekeeping, and early Chris-
tian attitudes to astronomy’, Isis 81, 9–22.
McNicoll A.W. and Milner N.P . (1997), Hellenistic Fortifications from the Aegean to
the Euphrates , Oxford: Clarendon Press.
Meiggs R. and Lewis D. (eds) (1989), A Selection of Greek Historical Inscriptions to
the End of the Fifth Century B.C ., Oxford: Clarendon Press.
Mendell H. (1984), ‘T wo geometrical examples from Aristotle’s Metaphysics ’,
Classical Quarterly 34, 359–72.
Meritt B.D. (1932), Athenian Financial Documents of the Fifth Century , Ann Arbor:
University of Michigan Press.
Meritt B.D. and Brown West A. (1934), The Athenian Assessment of 425 B.C., Ann
Arbor: University of Michigan Press.
Meritt B.D., Wade-Gery H.T. and McGregor M.F . (1939), The Athenian Tribute
Lists, Cambridge, MA: Harvard University Press, vol.1.
Moatti C. (1993), Archives et partage de la terre dans le monde romain ( IIe siècle
avant – 1er siècle après J.-C.), Roma: École Française.
Montanari F . (1993), ‘L’erudizione, la filologia e la grammatica’, in G. Cambiano,
L. Canfora and D. Lanza (eds), Lo spazio letterario della Grecia antica. Vol.1: La
produzione e la circolazione del testo. Tomo 2: l’Ellenismo , Roma: Salerno, 235–
81.
Morgan T. (1998), Literate Education in the Hellenistic and Roman Worlds ,
Cambridge: Cambridge University Press.
Morris I. (1996), ‘The strong principle of equality and the archaic origins of Greek
democracy’, in J. Ober and C. Hedrick (eds), Demokratia: A Conversation on
Democracies, Ancient and Modern , Princeton, 19–48.
Mueller I. (1981), Philosophy of Mathematics and Deductive Structure in Euclid’s
Elements , Cambridge, MA/London: MIT Press.
—— (1982), ‘Geometry and Scepticism’, in J. Barnes, J. Brunschwig, M. Burnyeat
and M. Schofield (eds), Science and Speculation. Studies in Hellenistic Theory
and Practice , Cambridge/Paris: Cambridge University Press and Maison des
Sciences de l’Homme, 69–95.
—— (1987a), ‘Mathematics and Philosophy in Proclus’ Commentary on Book I
of Euclid’s Elements’, in J. Pépin and H.D. Saffrey (eds), Proclus. Lecteur et
Interprète des Anciens , Paris: CNRS, 305–18.
—— (1987b), ‘Iamblichus and Proclus’ Euclid Commentary’, Hermes 115, 334–
48.
—— (1992), ‘Mathematical method and philosophical truth’, in R. Kraut (ed.),
The Cambridge Companion to Plato , Cambridge: Cambridge University Press,
170–99.
Murdoch J. (1971), ‘Euclid: T ransmission of the Elements’, Dictionary of Scientific
Biography , New York: Scribner, vol. 4, 437–59.
Napolitano Valditara L.M. (1988), Le idee, i numeri, l’ordine: la dottrina della
‘mathesis universalis’ dall’Accademia antica al Neoplatonismo , Napoli: Bibliopolis.
Netz R. (1998), ‘Deuteronomic texts: late antiquity and the history of mathematics’,
Revue d’Histoire des mathématiques 4, 261–88.

283BIBLIOGRAPHY
—— (1999a), The Shaping of Deduction in Greek Mathematics. A Study in Cognitive
History , Cambridge: Cambridge University Press.
—— (1999b), ‘Proclus’ division of the mathematical proposition into parts: how
and why was it formulated?’, Classical Quarterly 49, 282–303.
—— (forthcoming), ‘Counter-culture and early Greek numeracy’.Neugebauer O. and Van Hoesen H.B. (1959), Greek Horoscopes , Philadelphia:
American Philosophical Society.
Nicolet C. (1970), ‘Les finitores ex equestri loco de la loi Servilia’, Latomus 29, 72–
103.
—— (1988), L ’inventaire du monde: géographie et politique aux origines de l’empire
romain , Fayard, Engl. tr. by H. Leclerc, Space, Geography, and Politics in the
Early Roman Empire , Ann Arbor: University of Michigan Press 1991.
Nussbaum M. (1986), The Fragility of Goodness. Luck and Ethics in Greek Tragedy
and Philosophy , Cambridge: Cambridge University Press.
Ober J. (1992), ‘Towards a typology of Greek artillery towers: the first and the
second generations ( c. 375–275
BC)’, in S. van de Maele and J.M. Fossey (eds),
Fortificationes antiquae , Amsterdam: Gieben, 147–69.
O’Meara D. (1989), Pythagoras Revived: Mathematics and Philosophy in Late
Antiquity , Oxford: Clarendon Press.
Osborne R. (1994), ‘Introduction. Ritual, finance, politics: an account of Athenian
democracy’, in R. Osborne and S. Hornblower (eds), Ritual, Finance, Politics:
Athenian Democratic Accounts Presented to David Lewis , Oxford: Clarendon Press,
1–21.
—— (1996), Greece in the Making 1200 –479 BC, London/New York: Routledge.
Owens E.J. (1991), The City in the Greek and Roman World , London: Routledge.
Pamment Salvatore J. (1996), Roman Republican Castrametation. A Reappraisal of
Historical and Archaeological Sources , Oxford: BAR.
Panerai M.C. (1984), ‘Gli strumenti: un agrimensore a Pompei’, in Misurare la
terra: centuriazione e coloni nel mondo romano , Modena: Panini, 115–19.
Parker R.A. (1972), Demotic Mathematical Papyri , Providence, RI/London: Brown
University Press and Lund Humphreys.
Pritchett W.K. (1965), ‘Gaming tables and IG I2 324’, Hesperia 34, 131–47.
Purcell N. (1983), ‘The apparitores : a study in social mobility’, Papers of the British
School at Rome 51, 125–73.
Rathbone D. (1991), Economic Rationalism and Rural Society in Third-Century
A.D. Egypt. The Heroninos Archive and the Appianus Estate , Cambridge:
Cambridge University Press.
Rhodes P .J. (1972), The Athenian Boule , Oxford: Clarendon Press.
—— (1981), A Commentary on the Aristotelian Athenaion Politeia , Oxford:
Clarendon Press.
Rhodes P .J. and Lewis D.M. (1997), The Decrees of the Greek States , Oxford:
Clarendon Press.
Rice E.E. (1983), The Grand Procession of Ptolemy Philadelphus , Oxford: Oxford
University Press.
Rihll T .E. and T ucker J.V. (1995), ‘Greek engineering. The case of Eupalinus’ tunnel’,
in A. Powell (ed.), The Greek World , London: Routledge, 403–31.

284BIBLIOGRAPHY
Ritter J. (1989a), ‘Babylon – 1800’, in M. Serres (ed.), A History of Scientific Thought ,
Oxford/Cambridge, MA: Blackwell 1995, 17–43, Engl. tr. of Eléments d’histoire
des sciences , Paris: Bordas.
—— (1989b), ‘Measure for measure: mathematics in Egypt and Mesopotamia’, in
M. Serres (ed.), A History of Scientific Thought , Oxford/Cambridge, MA:
Blackwell 1995, 44–72, Engl. tr. of Eléments d’histoire des sciences , Paris: Bordas.
Roochnik D. (1996), Of Art and Wisdom. Plato’s Understanding of T echne , University
Park, PA: Pennsylvania State University Press.
Sachs E. (1917), Die fünf Platonische Körper , Berlin: Weidmann.
Salmon E.T. (1985), ‘La fondazione delle colonie latine’, in Misurare la terra:
centuriazione e coloni nel mondo romano. Cittá, agricoltura, commercio: materialida Roma e dal suburbio , Modena: Panini, 13–19.
Salviat F . and Vatin Cl. (1974), ‘Le cadastre de Larissa’, Bulletin de correspondance
hellénique 98, 247–62.
Salzman M.R. (1990), On Roman Time. The Codex-Calendar of 354 and the Rhythms
of Urban Life in Late Antiquity , Berkeley/Los Angeles/Oxford: University of
California Press.
Schütz M. (1990), ‘Zur Sonnenuhr des Augustus auf dem Marsfeld’, Gymnasium
97, 432–57.
Sedley D. (1976), ‘Epicurus and the Mathematicians of Cyzicus’, Cronache ercolanesi
6, 23–54.
Segonds A. (1981), ‘Introduction’, in Jean Philopon, Traité de l’Astrolabe , Paris:
Astrolabica.
Shaw G. (1995), Theurgy and the Soul. The Neoplatonism of Iamblichus , University
Park, PA: Pennsylvania State University Press.
Sherk R.K. (1988), The Roman Empire: Augustus to Hadrian , Cambridge: Cambridge
University Press.
Smallwood E.M. (1966), Documents Illustrating the Principates of Nerva, Trajan
and Hadrian , Cambridge: Cambridge University Press.
Smith D.E. (1951), History of Mathematics , New York: Dover 1951–3 (first edn
1923–5), 2 vols.
Stancic Z. and Slapsak B. (1999), ‘The Greek field system at Pharos: a metric
analysis’, Revue des études anciennes 101, 115–24.
Straub J. (1970), ‘Severus Alexander und die Mathematici’, in J. Straub (ed.), Bonner
Historia-Augusta-Colloquium 1968/1969, Bonn: Habelt, 247–72.
Svenson-Evers H. (1996), Die griechischen Architekten archaischer und klassischer
Zeit, Frankfurt a.M.: Peter Lang.
Swain S. (1990b), ‘Plutarch’s lives of Cicero, Cato and Brutus’, Hermes 118, 192–
203.
—— (1997), ‘Plutarch, Plato, Athens, and Rome’, in J. Barnes and M. Griffin
(eds), Philosophia Togata II. Plato and Aristotle at Rome , Oxford: Clarendon
Press, 165–87.
Szabó Á. (1969), Anfänge griechischen Mathematik, Engl. tr. A.M. Ungar as The
Beginnings of Greek Mathematics , Dordrecht: Reidel 1978.
Tannery P . (1884), ‘Domninos de Larissa’, in Mémoires scientifiques II, Toulouse/
Paris: Privat and Gauthier-Villars 1912, 105–17.

285BIBLIOGRAPHY
—— (1885), ‘Notes critiques sur Domninos’, Revue de Philologie 9, 129–37.
Thesleff H. (1961), An Introduction to the Pythagorean Writings of the Hellenistic
Period , Åbo: Åbo Akademi.
—— (ed.) (1965), The Pythagorean T exts of the Hellenistic Period , Åbo: Åbo Akademi.
Thomas I. (ed.) (1967), Greek Mathematical Works , 2 vols, London/Cambridge,
MA: Heinemann and Harvard University Press.
Thomas R. (1992), Literacy and Orality in Ancient Greece , Cambridge: Cambridge
University Press.
Thompson D.J. (writing as Crawford) (1971), Kerkeosiris. An Egyptian Village in
the Ptolemaic Period, Cambridge: Cambridge University Press.
Tod M.N. (1911–12), ‘The Greek numeral notation’, The Annual of the British
School at Athens 18, 98–132.
Tolbert Roberts J. (1982), Accountability in Athenian Government , Madison/London:
University of Wisconsin Press.
Toomer G.J. (1970), ‘Apollonius of Perga’, Dictionary of Scientific Biography , New
York: Scribner, vol.1, 179–93.
—— (ed.) (1990), Apollonius. Conics – Books V to VII , New York/Berlin/etc.:
Springer, 2 vols.
Uguzzoni A. and Ghinatti F . (1968), Le tavole greche di Eraclea , Roma: Bretschneider.
Untersteiner M. (1961), Sofisti. T estimonianze e Frammenti , eds and Italian tr. M.
Untersteiner and A. Battegazzore, Firenze: La Nuova Italia 1961–2, 4 vols.
van der Waerden B.L. (1954), Science Awakening , Engl. tr. by A. Dresden,
Groningen: P . Noordhoff.
Vernant J.-P . (1965), Mythe et pensée chez les Grecs. Etudes de psychologie historique ,
Paris: Maspero, Engl. tr. as Myth and Thought among the Greeks , London:
Routledge 1983.
Vitrac B. (1996), ‘Mythes (et réalités?) dans l’histoire des mathématiques grecques
anciennes’, in C. Goldstein, J. Gray and J. Ritter (eds), L ’ Europe mathématique.
Histoires, mythes, identités , Paris: Éditions de la Maison des Sciences de l’Homme,
33–51.
von Reden S. (1995), Exchange in Ancient Greece , London: Duckworth.
—— (1997), ‘Money, law and exchange: coinage in the Greek polis’, Journal of
Hellenic Studies 117, 154–76.
von Staden H. (1998), ‘Andréas de Caryste et Philon de Byzance: médecine et
mécanique à Alexandrie’, in G. Argoud and J.-Y. Guillaumin (eds), Sciences
exactes et sciences appliquées à Alexandrie , Saint-Étienne: Publications de l’Univer-
sité de Saint-Étienne, 147–72.
Wallace-Hadrill A. (1998), ‘To be Roman, go Greek. Thoughts on Hellenization
at Rome’, in M. Austin, J. Harries and C. Smith (eds), Modus Operandi. Essays
in Honour of Geoffrey Rickman , London: Institute of Classical Studies, 79–91.
Wasowicz A. (1972), ‘T races de lotissements anciens en Crimée’, Mélanges de l’école
Française de Rome 84, 199–229.
Waterhouse W.C. (1972), ‘The Discovery of the Regular Solids’, Archive for History
of Exact Sciences 9, 212–21.
Wehrli F . (ed.) (1959), Die Schule des Aristoteles , Basel: Schwabe 1944–59.

286BIBLIOGRAPHY
Westermann W.L. and Sayre Hasenoehrl L. (eds) (1934), Zenon Papyri. Business
Papers of the Third Century B.C. Dealing with Palestine and Egypt , New York:
Columbia University Press 1934–40, 2 vols.
Winter F .E. (1963), ‘The chronology of the Euryalos fortress at Syracuse’, American
Journal of Archaeology 67, 363–87.
—— (1971), Greek Fortifications , London: Routledge and Kegan Paul.
Zhmud L. (1997), Wissenschaft, Philosophie und Religion im frühen Pythagoreismus ,
Berlin: Akademie.

287INDEX
INDEX
abacus, 11–13, 17, 20, 23, 128, 147,
174
accountability, see accounts
accounts, 4, 14–16, 20–4, 40–2, 45,
49, 69–70, 75, 125, 143, 148–50,
169, 174, 176, 193–4, 196, 209,
213–5, 217–18
accuracy, 8–9, 16–17, 23, 33, 42–4,
63, 67–9, 74–5, 82–3, 86, 148,
152–3, 158, 162–4, 168–70, 177,
179, 181, 184, 186, 205–6, 224,
228, 233–4, 241, 243, 249–50,252–3
Aeschines, 4, 23–4
Aeschylus, 4, 17, 138
Alcinous, 186
Alexander of Aphrodisias, 129Alexandria, 51, 62, 64, 67, 75, 85,
88, 106, 113–14, 125, 135–41,
161, 180, 184, 212, 219, 223,243, 250, 258
analysis, 52, 55, 98, 108–9, 168,
187–8, 223, 226, 233, 258–9
Antykythera device, 152–3
Apollonius of Perga, 77, 81, 100,
113–21, 125, 135–6, 139, 160,
184, 226–8, 231, 233–4, 243,
249, 257–9, 261
Apollonius of Rhodes, 138
Arabic tradition, 57, 113, 130–1,
161, 189n18, 218, 223
Aratus, 81–2, 140–1, 174, 193, 201
Archimedes, 50–1, 57, 60, 62, 83,
94, 100, 103, 105–13, 118, 121,
125, 135–6, 138–41, 159–61,
164, 166, 168, 170, 179, 184,186, 192, 194, 197–201, 210,
223, 229, 231–4, 244–5, 257–61
architecture, 8–9, 15, 18, 24, 40, 49,
63–4, 146, 150, 159–61, 168,175–6, 179, 187, 196, 202–3,
207–9, 215–16, 218, 229–31
Archytas, 53–60, 76, 160–1, 192,
197, 199, 259
Aristarchus of Samos, 79–81, 160
Aristophanes, 5, 17–19, 21, 34, 54,
60n10
Aristotle, 5, 8, 20, 31–5, 40–2, 47–9,
55–6, 76–7, 82, 89, 105, 109,
129–30, 138, 160, 183, 211n33,
234, 244, 249, 259
Aristoxenus, 77, 82–3, 140
arithmetic, 9, 17, 24–6, 30, 33–5,
41–8, 62, 67, 70, 72, 75–7, 82,
85, 87, 89, 92–4, 106, 114, 118,
127–9, 140, 144–7, 160, 162,164–8, 171, 174, 178, 180–7,
199, 207, 212, 218–23, 228–9,
233–6, 242, 244–5, 250–5, 260
astrology, see astronomy
astronomy, 17–19, 26, 30, 34–5,
41–3, 45, 53–4, 56, 73, 77–82,
85, 106, 114, 136–8, 140–1,
147–8, 152–3, 160–1, 164, 170,
173–4, 177–9, 181, 184, 186–8,
193, 197, 201–2, 205–7, 216–18,223, 226, 231, 234–5, 237, 239,
241–3, 245, 250–5, 261
Athenaeus Mechanicus, 190n51,
211n29
Athens, 4–5, 9, 13–15, 18–20, 49,
85, 138, 153

288INDEX
Augustine, 212, 250–3, 255, 260
Autolycus of Pitane, 79, 226
Balbus, 171–3, 207, 210
Biton, 85–6, 136
calculators, see accounts
Cato the Elder, 174–5
Cicero, 174, 192–8, 201, 208–9Clement of Alexandria, 250–1,
253–5
clocks, 19, 83, 151–4, 159, 187–8,
250
colonies, 6–8, 46, 64–5, 151, 169,
196
Columella, 175–6, 209
commentary, 56–7, 82, 126, 129,
131, 143, 145, 150, 174, 181,
223, 231–4, 237–44, 249,
257–8
conics, 58–60, 83–4, 88, 106–7,
110–21, 153, 162, 225–6, 229,230–1, 233–4, 243, 257–8
curves, 53, 58–60, 106–7, 114, 118,
172–3, 175, 178, 223–5, 257
Cynics, 42, 76
Cyrene, 6–7, 29, 85, 114, 243
Democritus, 50–4, 77, 160
demonstration, 4–5, 23, 25–6, 30,
32–4, 39, 44, 50–2, 57, 72, 74,
77–9, 82–3, 89, 98–105, 108–13,
116–21, 127, 129–30, 135–6,
140–1, 145–6, 161–2, 164–8,
174, 180–2, 184–8, 192, 204–5,207, 219–20, 222, 231–5, 238,
240–1, 256–7, 259
Demosthenes, 23, 194Diocles, 57, 83–4, 88, 135, 138,
140–1, 257
Diogenes Laertius, 50, 53–5, 78
Diophantus, 161, 218–23, 225, 229,
245–6
Domninus of Larissa, 237
duplication of the cube, 53, 57–60,
83, 85, 87–8, 92, 138, 141, 161,
163, 199, 223, 229–30, 233,
256–7
duplication of the square, 27–9,
160–1economy, 10–15, 40–1, 43, 46, 49,
146–7, 174, 177, 180, 213, 255
education, 19, 25, 33, 42–6, 49, 54,
70, 72, 74, 82, 139, 146, 150,159, 173, 180–2, 187–8, 193–6,
202–3, 206–7, 209–10, 212–22,
225, 235, 241–3, 245, 250–2, 259
Egypt, 4, 6, 16, 46–7, 53–5, 62,
66–73, 75, 136, 143–8, 151, 153,204, 212–15, 222, 242, 255
Epicurus, 53, 73, 76–7
Eratosthenes, 57–8, 75, 85, 106, 109,
138, 141, 160–1, 173, 177–9,
184, 186, 226, 257, 259
Euclid, 30, 50, 55–7, 72–3, 75,
77–9, 81, 83, 88–105, 107, 114,
118–9, 121, 125–36, 138–40, 145,159, 162–3, 169, 172, 182, 185,
187, 207, 220, 222–3, 226, 228,
231–2, 235–40, 243–4, 251, 256,
258–61
Eudemus of Rhodes, 56–60, 77, 127Eudoxus of Cnidus, 34, 50–5, 58, 82,
94, 103, 105, 125, 141, 166, 192,
199, 236, 257, 259
Eupalinus tunnel, 9–10, 158
Eutocius, 50, 57–60, 85, 113, 118,
141, 223, 231–4, 257–8, 260–1
expertise, 41–5, 51–2, 69, 98, 136,
139, 147–8, 153, 157–9, 162,169–71, 173–4, 176, 178, 187,
202–3, 205–9, 213–5, 240, 243,
256, 260
Flaccus, 207–8, 210
Frontinus, 169–70, 209, 242
Galen, 187–8, 207, 209–10
Geminus, 201–2
geography, 18, 64, 75, 77, 85–6,
122n11, 136, 138, 157, 176–80,
184, 207–8, 223, 254
geometry, 6–9, 19, 25–30, 32–4,
40–6, 48, 50–60, 63–7, 70, 73–5,
77–9, 82–3, 85, 89, 94, 98–9, 103,
106–7, 109–14, 116, 119, 131,
134–6, 138, 145, 155, 157–68,
170, 172–5, 177–9, 181, 184–8,192, 194, 199–200, 202, 204–5,
207–10, 216, 220, 223, 225, 227,

289INDEX
229–30, 233, 235, 238, 240–4,
250–1, 255, 259–60
grammar, 41, 70, 136, 138–40, 181,
185, 200–2, 216–17, 244, 250,256
Hero of Alexandria, 131, 137, 152–3,
161–8, 171, 183, 203–5, 208, 210,
218–19, 222, 229, 242, 257
Herodotus, 5–6, 16–17
heuristics, 51, 109–12
Hipparchus, 82, 140–1, 177, 184, 244Hippocrates of Chios, 34, 51–2,
55–60, 105
Homer, 16, 70, 131, 139, 194, 201,
213, 245
Hypatia of Alexandria, 212, 223, 243,
246n2
Hypsicles, 77, 135
Iamblichus, 234–7, 240–1, 251,
254–5, 259–61
incommensurability, 26–7, 29–30,
32–3, 76, 89, 93–5, 108, 127–9,
144–5
instruments, 9, 16, 18, 53, 67–8,
86–7, 152–5, 159, 164, 172, 184,
187, 191n82, 192, 197, 199, 203,
225, 229, 231, 243, 250, 254
justice, 8, 14, 16, 34, 40, 46–9, 66, 70,
155, 181, 199, 202, 205, 208, 254
land surveying, 6–8, 11, 16, 18–20, 40,
42, 64, 66–71, 75, 85, 146, 153–9,164, 166, 169–73, 175–6, 178,
181, 196–7, 204–5, 207–10,
214–18, 229, 239, 241–2, 255
law courts, 4, 18, 20–4, 49
literacy, see numeracy
logic, 26, 31–3, 78–9, 127, 187
Lysias, 5, 21–3
Maecianus, 174–5
Marinus, 244
means, 9, 26, 48, 53–5, 57–60, 85,
89–92, 138, 141, 181, 186, 199,
223, 226, 233, 236, 259
mechanics, see technology
medicine, 44, 78, 136, 138–40, 159,187–8, 207, 216–18, 256
Mesopotamia, 4, 11, 16, 147, 242, 255
method of ‘exhaustion’, 102–5,
106–8, 233
Meton, 18–9, 42, 60n10, 81
Miletus, 6, 8, 10–11, 19, 86, 140,
228
music, 26, 30, 35, 42–3, 46, 77,
82–3, 138, 159–60, 164, 178,181, 184, 186–7, 194, 216, 235,
237, 242–3, 250–2
NeoPlatonism, see Plato
Nicomachus of Gerasa, 181–3,
185–6, 222, 234–5, 237, 249,
251, 259
Nipsus, 171numeracy, 5, 15–16, 70, 114, 126,
135–6, 140, 143, 149, 193, 213,
217, 256
numerical notation, 10–13, 118, 140,
219, 228
origins of mathematics, 6, 17–18, 47,
204, 242, 255, 260
Palatine Anthology, 244–6
Pappus, 124n69, 126, 131, 153, 222–
31, 233, 241, 243, 246n2, 249,
256–61
parallels postulate, 34, 99, 162
Pergamum, 85, 113–14, 136
Philodemus, 50–3, 55
Philo of Alexandria, 180–1, 250, 253
Philo of Byzantium, 63–4, 86–8,
135–6, 138, 163
philosophy, 19, 24–5, 30, 33–4,
42–7, 52–6, 73, 76–9, 85, 136,138, 140, 143–4, 159, 162,
177–8, 180–1, 183–4, 186–7,
192–5, 198–9, 201–10, 212,
216–17, 229–31, 234–41,
243–4, 250–1, 255–6, 260
Phoenicians, 46, 255
Plato, 5, 9, 24–31, 33–5, 41–7,
49–53, 55–8, 76–7, 85, 89, 93,
127, 130, 138–9, 143–5, 160,
180–1, 186, 192–3, 195, 197,199, 233–7, 241, 243, 250–1,
254–5, 257, 259–60

290INDEX
Pliny the Elder, 143, 151–2, 176–7,
208
Pliny the Younger, 176, 209
Plutarch, 50–5, 57–8, 192–5,
198–201, 208
politics, 4–8, 11, 13–17, 19–21, 23–5,
30, 35, 40–50, 62, 66, 70, 74–5,
81, 85, 88, 106, 114, 136, 138,
140–1, 143, 148–51, 155, 157,159–61, 169, 171–2, 174, 176–8,
187, 189n18, 192, 195–9, 201–2,
204–5, 207–9, 212, 214–16,234–5, 241–3, 254–5, 260
Polybius, 65–6, 73–5, 105, 198
Pompeii, 153–5
Porphyry, 237, 239, 243, 259
Proclus, 50, 54–6, 77–8, 126, 131,
237–41, 243–4, 256, 260–1
proportion, 9, 14, 33, 40, 46, 48–9,
53–4, 57–60, 63, 73, 76, 85–7,
89–92, 95–6, 99, 108, 138, 141,
159, 162, 181, 186, 199, 204–5,223, 233, 237, 257
Ptolemaic kings, 62, 66, 75, 113, 125,
136–7, 141
Ptolemy, 131, 152, 183–5, 187,
205–7, 223, 231, 237, 241,
243–4, 249, 259–60
Pythagoreans, 30–1, 34–5, 53–5,
70–1, 76, 89, 93–4, 127–9, 138,160–1, 181, 185–6, 194, 234–7,
251, 254–6, 259–60
quadrature of the circle, 19, 34,
42–3, 50, 55–7, 60, 70–2, 106–7,223–4, 231, 259
ratio, 40, 48, 57, 81, 85, 91, 95–7, 105,
108–11, 114–15, 118, 128–9, 148,
205, 219, 224–7, 232
reductio ad absurdum, 32–3, 79, 82,
100, 102, 116, 119, 129, 164, 238,
240
religion, 9, 14–15, 23, 58, 70, 76, 83,
137, 155, 159, 161, 172, 180–1,
184, 199, 212, 215–16, 218–19,
223, 231, 237–8, 244, 249–55
Rhodes, 8, 77–8, 82, 86, 136, 138, 194Rome, 62, 65, 105, 143, 148, 150–1,
169, 177, 195–7, 200–1, 216Scepticism, 76–8, 185–6, 253
Seneca, 177–8, 208
Serenus, 243
Sextus Empiricus, 85, 185–6Simplicius, 50, 56–7, 60, 105, 259
Sparta, 18, 20, 74
Sporus, 224, 233
Stoicism, 76–9, 241
Strabo, 122n27, 178–80Synesius, 243–4
Syracuse, 63, 105–6, 140, 160, 192,
197, 200
tables, 70, 86–7, 146–8, 159, 164,
181, 184–5, 207, 213–14, 241,
243, 245, 252
tax, 6, 11, 13–14, 16–19, 21, 67,
69, 148–50, 155, 157, 196,
214–18
techne , see technology
technology, 6, 8–10, 17–18, 30, 33,
40, 44–7, 49, 52–4, 63–4, 67–8,82, 85–8, 105–6, 108–13,
136–40, 150, 157, 160–4,
169–73, 176, 178, 192, 196,198–205, 207–10, 216–18, 223,
229, 237, 241–4, 251
Thales, 18–19, 53–5, 127, 260
Theaetetus, 29–30, 55, 93, 127, 129
Theodorus of Cyrene, 29–30, 55,
93
Theodosius of T ripoli, 79, 81, 226
Theon of Alexandria, 125–6,
129–31, 243
Theon of Smyrna, 186Thucydides, 19, 63
trisection of the angle, 223
Vitruvius, 152, 159–61, 170, 202–3,
208, 210
war, 4, 16–21, 24, 42–3, 46, 62–5,
73–4, 85–8, 105, 125, 136–8,143, 151, 158–9, 163–4, 169,
171–2, 192, 194–5, 197, 200–4,
208–9, 217, 241
water-supply, 9, 158–9, 164–5,
169–70, 218
women, 22, 137, 141, 191n74, 207,
212, 230, 243, 245, 246n2