Poetry and mathematics might seem to be worlds apart. Nevertheless, a number
of Greek and Roman poets incorporated counting and calculation within their
verses. Setting the work of authors such as Callimachus, Catullus and
Archimedes in dialogue with the less well-known isopsephic epigrams of
Leonides of Alexandria and the anonymous arithmetical poems preserved in
the Palatine Anthology, this book reveals the various roles that number played
in ancient poetry. Focusing especially on counting and arithmetic, Max Leventhal
demonstrates how the discussion, rejection or enacting of these two operations
was bound up with wider conceptions of the nature of poetry. Practices of
composing, reading, interpreting and critiquing poetry emerge in these texts as
having a numerical component. The result is an illuminating new way of
approaching Greek and Latin poetry and one that reaches across modern
disciplinary divisions.
max leventhal is Bye-Fellow and College Lecturer in Classics at Downing
College, Cambridge. He was previously the Thole Research Fellow at Trinity
Hall, Cambridge and a British Academy Postdoctoral Fellow in the Faculty of
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cambridge classical studies
General editors
j. p. t. clackson, w. m. beard, g. betegh, r. l. hunter,
m. j. millett,
s. p. oakley, r. g. osborne, c. vout, t. j. g. whitmarsh
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Downing College, Cambridge
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4 my family and friends, with love;
2 Alex, the 1 that counts the most.
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Acknowledgements page viii
List of Abbreviations and Editions xi
Introduction: Numbers Up 1
1.1 Poetic Figures 1
1.2 Poetry by Numbers 7
1.3 Poetic Numeracy and Greek Mathematics 17
Part I Counting and Criticism 21
1 Callimachus and His Legacy 23
1.1 Counting in Callimachus Reply to the Telchines 25
1.2 Erinna and the Epigrammatists 41
1.3 Roman Reckonings 52
2 Leonides of Alexandrias Isopsephic Epigrams 73
2.1 Callimachus Compressed 79
2.2 Cups and Sources 96
2.3 Pebbles in the Stream 102
Part II Arithmetic and Aesthetics 113
3 Archimedes Cattle Problem 121
3.1 Archimedes Art 127
3.2 Cattle and Catalogues 139
3.3 Calculating Cattle and Cultural Competition 151
4 The Arithmetical Poems in AP 14 162
4.1 An Archaeology of Arithmetical Poetry 165
4.2 The Cultural Capital of Calculation 178
4.3 Arithmetic Anthologised 189
4.4 Arithmetical Poetry beyond Late Antiquity 197
Conclusion: Summing Up Poetry 209
Bibliography 213
Index 227
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This book has had a considerably long gestation and several
origins. I rst began thinking about the intersection of poetry
and number as a Masters student. The result was a dissertation
on Archimedes Cattle Problem, and it would eventually become
an article in Ramus (2015), which is substantially reworked and
expanded here as Chapter 3. I then pursued a doctorate on
a different topic. Still, Leonides of Alexandria and his isopsephic
epigrams constituted one distraction from the main task of my
thesis; in fact, some of the material of Chapter 2 had originally
been submitted to the Cambridge Classical Journal as an article.
I thank the anonymous reviewers of that piece for their challenging
but supportive feedback. A failed Oxford interview in 2008 bears
special mention, too, since it was there that I rst encountered
Catullus cc. 5 and 7 and I have been ruminating on those poems
ever since. The beginnings of a postdoctoral position afforded me
the time to write up these ideas while, once again, I should have
been working on other things. Thus, this book is quite a different
beast to ones typical rst book in academia. It is the product of
a lengthy obsession and not the direct result of my doctoral studies.
Yet I hope and believe that the passage of so much time has
imbued it with a certain intellectual maturity of the kind which
I did not yet possess as a fresh postgraduate and has streamlined
its arguments so as to avoid the bloatedness that is a residual
characteristic of many a thesis turned monograph.
Although this is not the book of the thesis, I nevertheless owe
a debt of gratitude to several supervisors and mentors, for the
profound inuence they have had on me as a scholar and as
a person. Liba Taub supervised my Masters thesis and has been
a continual source of support. She frequently saw the value of my
thinking before I did and encouraged me to pursue many lines of
enquiry that are now to be found in this book. In that sense, it could
Published online by Cambridge University Press
not have been written without her. Renaud Gagné supervised my
doctoral thesis, and I am indebted to his inexhaustible generosity,
which has been truly formative. Richard Hunter, my second super-
visor, has read over more drafts of my various musings than I can
recall; he has vastly improved them all with his knack for asking
just the right question to make one reect on both matters of detail
and the larger picture. Tim Whitmarsh served as the internal
examiner of my thesis along with Annette Harder, who served as
the external examiner: their feedback was as constructive as it was
honest. I have been fortunate also to have Tim as my postdoctoral
mentor; his kindness, support and advice on all manner of subjects
have been a real inspiration.
It gives me great pleasure also to acknowledge those colleagues
and friends whose efforts have improved this nal product. As
well as Richard Hunter and the anonymous reviewer at CCS, drafts
of this entire book were read by Jan Kwapisz, Arthur Harris and
Sam Oliver, who all brought their own expertise to bear in enhan-
cing its clarity and avoiding any infelicities. Likewise, individual
chapters were read over and sharpened by Simon Goldhill, Talitha
Kearey, Thomas Nelson, Floris Overduin and Ivana Petrovic. The
Introduction in particular beneted from being exposed to the
exacting eyes of the Postdoctoral Work in Progress Seminar
(PWiPS) in Cambridge. I would also like to thank Francesco
Grillo, Richard Hunter and Jan Kwapisz, who all shared their
unpublished work with me.
I was lucky enough to receive funding as a postgraduate student
from Corpus Christi College, Cambridge, the Faculty of Classics,
University of Cambridge and the Jebb Fund. I was even more
fortunate subsequently to be awarded a Research Fellowship at
Trinity Hall, Cambridge and a British Academy Postdoctoral
Fellowship at the Faculty of Classics. These institutions have
provided me with the time, space and (relative) job security with
which to pursue several projects, including this present book.
Cambridge has been a truly wonderful place to grow intellec-
tually, and this is down to its enriching and lively community. If
sometimes I have felt the weight of academia as a solitary endeav-
our pulling me under, it has been the friendship of Talitha Kearey,
Thomas Nelson and, above all, of Hannah Price that has buoyed
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me up. I owe them so much. Of course, there is more to life than
books (but not much more), and I want also to thank those friends
outside academia who have kept me sane and have brought me
great joy beyond the job: Sam Isaacs, Dan Keane, Raj Mistry, Joe
Smallman, Sam Oliver and Hannah Price (again!).
The most personal debts, though, I cannot ever hope to repay.
For their love, I am forever grateful to my parents, Sue, Michael,
Ross and Kirsteen, to my siblings Rebecca, William and Robbie,
and to my grandparents, Peter, Mary and Pam. But the deepest
thanks go to my partner, Alex, whose incalculable love, support
and immeasurable patience when it comes to academic tribula-
tions and more besides have kept me going now for a full decade
(but whos counting?).
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Below is a list of abbreviation conventions and editions that I
employ in this book. For all else I follow the abbreviations of
names and works of Greek and Latin authors as given in the
Oxford Classical Dictionary.
AB Austin, C. and Bastianini, G. (2002) Posidippi
Pellaei quae supersunt omnia. Milan.
AP Anthologia Palatina (Palatine Anthology).
Broggiato Broggiato, M. (2001) Cratete di Mallo. I
Frammenti: edizione, introduzione e nota.La
Cougny Cougny, E. (1890) Epigrammatum anthologia
Palatina cum Planudeis et appendice nova epi-
grammatum veterum ex libris et marmoribus
ductorum, 3 vols. Paris.
FGE Page, D. L. (1981) Further Greek Epigrams .
GP Gow, A. S. F. and Page, D. L. (1968) The Greek
Anthology: The Garland of Philip and Some
Contemporary Epigrams, 2 vols. Cambridge.
Harder Harder, M. A. (2012) Callimachus: Aetia, 2 vols.
HE Gow, A. S. F. and Page, D. L. (1965) The Greek
Anthology: Hellenistic Epigrams, 2 vols.
IEG West, M. L. (198992) Iambi et elegi Graeci ante
Alexandrum cantati, 2nd ed., 2 vols. Oxford.
IMEGR Bernand, É. (1969) Inscriptions métriques de
lÉgypte gréco-romaine: Recherches sur la
poésie épigrammatique des Grecs en Égypte.
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KA Kassel, R. and Austin, C. F. L. (1983) Poetae
comici Graeci. Berlin and New York.
Kerkhecker Kerkhecker, A. (1999) Callimachus: Iambi.
Lightfoot Lightfoot, J. (2009) A Hellenistic Collection.
Cambridge, MA.
Mugler Mugler, C. (1972) Archimède, 4 vols. Paris.
Pf. Pfeiffer, R. (194953) Callimachus, 2 vols.
SH Lloyd-Jones, H. and Parsons, P. (1983)
Supplementum Hellenisticum. Berlin and New
Sider Sider, D. (2020) Simonides: Epigrams and Elegies.
List of Abbreviations and Editions
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I.1 Poetic Figures
This book explores Graeco-Roman poetrys engagement with and
use of numbers. What I mean by this can best be explained by
turning to Homers self-presentation in the Iliad, where the matter
of enumeration intersects with the question of poetic expression.
σπετε νν μοι, Μοσαι λμπια δματ᾿ἔχουσαι
μεςγρ θεαίἐστε πρεστ τε στ τε πντα,
μεςδ κλος οον κοομεν οδ τι δμεν
ο τινες γεμνες Δανανκα κορανοι σαν.
πληθνδ᾿ οκ ν γ μυθσομαι οδ᾿ὀνομνω,
οδ᾿ ε
μοι δκα μνγλσσαι, δκα δ στματ᾿ εεν,
φων δ᾿ἄρρηκτος, χλκεον δ μοι τορ νεη,
ε μὴὈλυμπιδες Μοσαι, Διςαγιχοιο
θυγατρες, μνησααθ᾿ὅσοι πὸἼλιον λθον·
ρχοςα νην ρωνῆάς τε προπσας.
(Homer Iliad 2.48493)
Tell me now, you Muses who have dwellings on Olympus for you are goddesses
and are present and know all things, but we hear only a rumour and know nothing
who were the leaders and lords of the Danaans. But the multitude I could not tell or
name, not even if ten tongues were mine and ten mouths and a voice unwearying,
and the heart within me were of bronze, unless the Muses of Olympus, daughters of
Zeus who bears the aegis, were to call to my mind all those who came beneath Ilion.
Now I shall tell the leaders of the ships and all the ships.
The passage addresses the presence in poetry of numerical as well
as heroic gures. Faced with the prospect of describing the entirety
of the gathered Achaean troops in the ninth year of the war, Homer
turns to address the Muses again.
While it precedes the Catalogue
The Greek text of Homer follows Allen (1920), with translations adapted from Murray
and Wyatt (1999) for the Iliad and from Murray and Dimock (1995) for the Odyssey.
Scholarship on the Invocation and Catalogue is vast. One traditional concern has been the
historical period and geographical politics it encapsulates, see Allen (1921); Burr (1944);
Hope Simpson and Lazenby (1970); Visser (1997). In terms of the make-up of the Iliad,it
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of Ships and the detailed counting up of the troops, it also follows
on from Homers series of similes variously describing the gath-
ering. In the run of similes, the hosts armour shines like a re
ravaging a forest, the troops appear like ocks of birds gathering in
a meadow, like all the leaves and owers in a meadow, and like
a swarm of ies round a milk pail (Il. 2.44573), and their organ-
isation is then likened to goatherds ordering their ocks ( 4747).
The Invocation thus functions as a hinge, mediating between
poetic modes: the similes poetics of likeness and the
Catalogues poetics of enumeration. Yet it is not frequently
observed that the passage is an extended reection on the tension
between poetic content (how many things you want to describe)
and the poetic resources required to recount it (how many verses it
will take). Prior to accounting for the ships at length in the
Catalogue, in other words, the poet is exploring and commenting
upon his enumerative abilities.
The role of the subsequent enumeration in the Catalogue
depends on the interpretation of this passage. On the one hand,
in contrast to the similes, which require no introduction or justi-
cation, the Catalogues poetics of enumeration need the support of
the Muses in order to be achieved. The ability to fully recall the
host lies solely with the Muses. On the other hand, the Muses
support in recounting the entire host is in fact a condition (note the
optative mood of μνησααθ), and the poet turns instead to recount-
ing only the leaders and the ships. The poet admits that the
problem is one of poetic capacity. The implication of his claim
that he could not tell or name the multitude, not even if ten
tongues were his and ten mouths (Il. 2.48890) is that a great
amount of content requires a concomitant extension of the poem,
which, in this case, even a division of labour by a multiplication of
mouths can do nothing to foreshorten. (Later Latin poets enact
their own numerical expansion from ten mouths to a hundred, but
has often been considered a later insertion, more appropriate to the gathering of the troops
at Aulis than to the troops on the Trojan plain in the ninth year of the war, see Allen
(1921) 16970; Wade-Gery (1952) 537; Jachmann (1958); Kullmann (1960) 63. For
a more literary evaluation of the dislocation see Sammons (2010) 1408. The following
is a necessarily brief account of the passage which glosses over certain interpretative
issues; see Chapter 3, Section 2 for more detail about the varying interpretations and for
my approach.
Numbers Up
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equally to no avail.)
His solution to the expected increase in
extension is remarkable. The Catalogue accounts for the number
of men per ship, the number of ships per leader and the number of
leaders. So, in lieu of counting up the number of warriors who
went to Troy (σοι πὸἼλιον λθον, 2.492), he allows for the
audience to reach a total instead based on the leaders and their
ships (ρχοςα νην ρωνῆάς τε προπσας, 2.493). Rather than
being an exhaustive count, the number of the host can be inferred
through what would now be called multiplication.
This strategy is used earlier in Iliad 2 by Agamemnon to calcu-
late the relative sizes of the Achaean and Trojan forces. He
imagines groups of ten Achaeans being served wine by one
Trojan and concludes that not all the Achaeans would be served
(Il. 2.11928): the Achaeans outnumber Trojans by more than ten
to one. Equally, one of the similes preceding the Invocation dis-
plays a similar thinking. The leaders are described as organising
their troops like goatherds, and among them stands Agamemnon at
a higher level above those leaders (47483). In both cases, indi-
vidual soldiers are organised into groups so as to make their
conceptualisation more manageable, and these groups are then
organised further: by Agamemnon when he compares the
Trojans with the Achaeans, and again by Agamemnon who rules
over the leaders who have already arranged their troops. The
organisation that enables the poet to encapsulate the host for the
audience is one which was understood both by gures within the
poem and by its audiences (to whom the simile is directed). Of
course, the use of multiplication is a traditional means of express-
ing quantity in Archaic epic.
What is so striking in Iliad 2 is that
the poet has harnessed these resources in order to explicitly reect
on his capacity as a poet and how certain types of calculation shape
the catalogue as a poetic form.
Enn. Ann. 46970 Skutsch; Hostius fr. 3 Courtney; Verg. G. 2.424 and Aen. 6.6357.
Ovid gives up the count and settles for many mouths (pluraque ... ora, Tr. 1.5.54).
Gowers (2005) 1713.
For example at Il. 8.5623 (1,000 res, 50 men by each); 9.85 (7 Greek leaders, each with
100 men); 9.383 (200 warriors coming out of each of the 100 gates of Thebes); and
16.16871 (50 ships for Achilles, 50 men at hole pins in each and 5 leaders).
I.1 Poetic Figures
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Later readings of the passage, moreover, focus on and respond
to Homers counting. In arguing that the scale of the Trojan War
was not as great as often assumed, Thucydides makes his own
count based on Homers Catalogue (Thuc. 1.10.12). He rst
surmises there to be 1,200 ships, which is not far from the 1,186
ships that is reckoned in modern scholarship on the basis of the
Catalogues count. He then recognises that only the Boeotian
contingent and Philoctetes contingent are given explicit numbers
of men per ship, at 120 and 50 men respectively, and conjectures
that this is the upper and lower limit of the men per ship (Thuc.
1.10.4). From this he states but does not calculate that if one
were to take the mean number of men per ship (85) the force would
still be small at 102,000 men (Thuc. 1.10.5). Setting to one side
whether this is in fact a small contingent by ancient standards, he
brings to bear his own numerical abilities in reading Homers
Catalogue and so elevates the numerical aspect as a key point of
Other readers, though, could come to different totals. The
mythological handbook attributed to Apollodorus of Athens (a
historian and geographer) concludes in the relevant chapter that
the total number of ships was 1,013 (νες μνονα πσαι ͵αιγʹ,
Apollod. Epit. 14). Similarly, in the Latin mythological handbook
attributed to Hyginus, the Fabulae, a count is made, although it is
marred by textual corruption. The chapter gives the reckoning of
the ships as 245 (summa naues CCXLV, Fab. 97.55) despite the
individual numbers given in Hyginus list adding up to 1,286
(which is a round 100 from the accepted 1,186). These counts of
ships and people also guided later readers approaching Homers
Catalogue. A scholium to the beginning of the Catalogue directly
invokes Thucydides language and his method of taking the mean
number of men per ship in explaining that Homer further, does
have something to say about the multitude [of the host] (καίτοι
λέγει κα περ το πλήθους,bT-scholia on Homer Iliad 2.488) and
that the reader is readily able to compute the total. The lack of
a nal sum in the poem, which in an original oral context may have
contributed to a purposeful overload of information for the audi-
ence, became a prompt to engage numerically with epic for later
readers encountering Homeric poetry on the page.
Numbers Up
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These readings also fed into new poetic compositions and their
reformulations of the poet and his work. To keep with the
Catalogue of Ships, a more overt mathematicisation is found in
the Contest of Homer and Hesiod and its reimagining of Homers
Invocation. When asked by Hesiod how many men sailed to Troy,
he replies with a calculation much shorter in length than the Iliadic
πεντήκοντ σαν πυρς σχάραι, νδὲἑκάστ
πεντήκοντ βελοί, περ δ κρέα πεντήκοντα·
τρςδ τριηκόσιοι περὶἓν κρέας σαν χαιοί.
(Contest of Homer and Hesiod 1435 Bassino)
(50 × 50 × 900 = 2,250,000)
There were fty hearths of re, in each were fty spits, and around each were
fty pieces of meat: three times three hundred Achaeans were around one piece of
The Homer of the Contest has progressed from the Catalogue that
counts to the more complex calculation that is multiplication. For
the audience(s) of the Iliad, it was necessary to estimate the
number of men in each ship and add together the troops under
each leader in order to reach a sum for the entire Achaean contin-
gent, in the manner that Thucydides had theorised. The Homer of
the Contest bypasses the need to display his counting, or indeed to
place his counting abilities under any scrutiny. He reaches
a number for the entire contingent in only a few lines where the
Iliadic Homer had professed his inability to account for the multi-
tude at all (πληθύν, Il. 2.488).
A further reworking of the Catalogue in Latin focuses on the
numerical abilities of the reader. The Ilias Latina, a Neronian-era
poem attributed to Baebius Italicus, compresses the key events of
the Iliad into 1,070 hexameter verses.
Its rewriting of the
Catalogue is prefaced by its own second invocation Vos mihi
nunc, Musae ...referte (recount to me now, Muses, 1612) and
it begins also with the Boeotian contingent: Boeoti decies quinas
egere carinas | et tumidos ualido pulsarunt remige uctus (the
I offer a more detailed analysis of this scene in the introduction to Part II.
For his name and date see Scaffai (1997) 1529. The Latin text also follows Scaffai,
while the translations are my own.
I.1 Poetic Figures
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