Below is the Greek text of Archimedes’ Cattle Problem (henceforth CP) and the anonymous prose introduction which provides the context of its composition, a translation and a delineation of the equations represented algebraically.Footnote ¹⁵

These twenty-two couplets capitalise on Homer’s depiction of the Cattle of the Sun in Odyssey 12 and its numerical aspect, where Circe explains to Odysseus that on Thrinakia, ‘there many cows and stout sheep of Helios graze, seven herds of cows and just as many fine flocks of sheep and fifty in each’ (Od. 12.127–30). The description in the CP of the related proportions of black, white, brown and dappled herds of cattle, which are then configured geometrically on Sicily, creates a strikingly colourful image. Just as striking is the author’s decision to respond to Homer’s scene with a poem that fills the verses almost exclusively with the ratios of cattle. Reading through the work it becomes clear that the mathematics is more complex than that of Homer’s Odyssey.

Since the work’s discovery, scholars have essayed solutions to Archimedes’ mathematical complexity.Footnote ¹⁸ It was only in 1965 that the smallest solution was able to be written out in full (a number whose digits filled forty-two sheets of paper).Footnote ¹⁹ What makes the problem particularly fiendish is the addition of the further parameters. The poem first outlines a series of ratios which in modern notation can be written as a series of simultaneous equations. The problem is interesting in that, since there are seven equations and eight unknowns (again this is a modern way of phrasing the problem), one cannot find a single solution, but instead infinitely many solutions.Footnote ²⁰ It is the subsequent stipulation that the white bulls and black bulls together form a square number and that the brown bulls and dappled bulls form a triangular number that makes the (infinitely many) solutions to the problem become truly astronomical in size. Unsurprisingly, attention has largely been paid to the mathematics, with historians of mathematics keen to highlight how the CP attests to an ancient awareness of complex arithmetic and of its limitations.Footnote ²¹ Approaches that have eschewed the mathematics inevitably do so only to discuss authenticity, a thorny riddle as unsolvable as the equations.Footnote ²²

The obsession with solving the mathematics and the question of authenticity has meant that the importance of the CP’s medium has been understudied and undervalued. Discussions of the text have failed to appreciate the CP as a poem and to understand the cultural and literary context which engendered it. Most, if not all, readers have been left bewildered by the mathematical demands of Archimedes’ prescribed proportions and configurations and read no deeper. Certainly, the confrontation of Homeric epic and mathematics is central to the work, yet its importance lies not in the complex calculations alone, but in how the mathematics is co-opted to manipulate a readership. It seems clear, given the time and effort modern scholars have put into solving Archimedes’ ratios, that his recipient, Eratosthenes, would have been unable to solve the arithmetical challenge.Footnote ²³ A more productive approach is to accept that the problem would have been arithmetically unsolvable and then to analyse Archimedes’ unique intersection of arithmetic and Homeric reception.

In that respect, it is important to observe that in other surviving treatises Archimedes shows himself to be a versatile and erudite author in his writing up of mathematics. In the Sand Reckoner, he engages with that most poetic trope, counting the number of the sands (e.g. Il. 2.800, 9.385; Pind. Ol. 2.98–100), and attempts to calculate the number of grains of sand that would be required to fill the universe. The treatise is dedicated to Gelon II, the ruler of Syracuse, and localised in relation to Sicily: Archimedes specifies that some people think the number of sands is infinite, the number ‘not only around Syracuse and the rest of Sicily, but in every region, both inhabited and uninhabited’ (2.134.1–6 Mugler). It stands apart from other, more typical mathematical texts in that it is not characterised by a pared-down, impersonal style focused on geometric proof, but ‘is ruled throughout by Archimedes speaking in his own voice, occasionally breaking his speech so as to give room for mathematical proof’.Footnote ²⁴ Similarly, in his Stomachion – which will be treated in more detail in the next chapter – he discusses the Greek game called στομάχιον (‘Belly-teaser’), in which a square cut into fourteen shapes can be rearranged into many other figures. From what survives of the text, his first aim was to compute the total number of different ways that the pieces could be combined to produce a square, the answer being 17,152. How the treatise then proceeded is unclear, but it is probable that it introduced further parameters which result in a solution for the number of different combinations being so large that it can only be approximated.Footnote ²⁵ In a not dissimilar vein to the Sand Reckoner, Archimedes takes an idea within Greek culture as a springboard for mathematical demonstration and as an opportunity for creating what Reviel Netz has called a ‘carnival of calculation’.Footnote ²⁶ In addition to this showmanship, there is the far more personal work of Archimedes’ Method, also addressed to Eratosthenes, which describes a mechanical method for calculating the volume of certain solids.Footnote ²⁷ He reminds Eratosthenes of geometrical problems he had sent him previously (4.82.1–8 Mugler) and praises his pedagogical commitment and mathematical enquiries (4.83.18–24 Mugler), before launching into an account of his discovery of the method which is strikingly biographical (4.84.10–25). My intention here is thus to situate the CP within the Archimedean corpus as equally sophisticated and literary, both capable of dazzling the reader with mathematical display and forged by his long-standing dialogue with Eratosthenes.

In what follows, then, I make three interrelated arguments. First, I show that the poem is a refined composition which resembles in form and content many other works produced in the Hellenistic era. In terms of the poet’s allusiveness, I suggest that the narrative of the Odyssey is not just a useful image with which to encode the mathematics, but that it is at the heart of the poem, and in particular, that epic’s concern with the location and name of Sicily. These aspects gain further significance when it is appreciated that the CP is sent between two scholar-poets in different Hellenistic kingdoms. In the following section, I show in detail that a further key intertext of the CP is the Catalogue of Ships in Iliad 2 and the surrounding scenes, including the Invocation to the Muses. Appreciating this intertext allows one to observe how Archimedes conceives of, and presents to the reader, the very project of providing calculations in verse. By appealing to this foundational context in which the Homeric poet deals with numbers and must call on the help of the Muses, he addresses the issue of mathematical knowledge and its limits. In Section 3, I combine the geopolitical reading of the CP proposed in the first section with the focus on poetic catalogues developed in the second section. I draw on a range of catalogic scenes from Archaic and Hellenistic poetry in order to demonstrate that an abiding association in these passages is enumeration as geographical possession: whoever is able to make a symbolic census – be it of cities, crops or livestock – has a claim to the control and ownership of the land. In offering the reader the opportunity to calculate the Cattle of the Sun, I argue, Archimedes makes a political point about the (im)possibility of possessing Sicily by means of arithmetic. This arithmetical poem, in short, advances a very particular aesthetic which not only characterises the competitive context of the challenge posed, but also probes precisely what it means to simultaneously compose poetry and produce arithmetic.

3.1 Archimedes’ Art

Archimedes was a great mathematician, but how good was his poetry? In this section I examine the literary aspects of the CP, its generic positioning and its allusions to earlier poetry. Whereas the focus has traditionally been on the complex enumeration encoded in the CP, here I provide a description of Archimedes the poet, a figure as erudite with words as he is sophisticated with mathematics. What will emerge, importantly, is not simply a scientific writer who draws on a Hellenistic education in order to ‘versify’ a series of equations, but a scientific writer able to handle a range of genres and generic expectations as well as to produce a poem full of intertexts and playful allusions to earlier works. Just like his correspondent Eratosthenes, Archimedes deserves to be ranked alongside the great Hellenistic poets as well as the greatest mathematicians.

To begin: the CP offers a number of different reading frameworks in its opening. The epistolary prose introduction frames the recipient as Eratosthenes and Archimedes as the sender. But is this Archimedes’ voice in the poem? The phrase ἐν ἐπιγράμμασιν εὑρών is ambiguous: it could mean he discovered the poem ‘among some epigrams’ or that he devised it ‘in elegiac couplets’.Footnote ²⁸ It is not inconceivable that he would have found the poem in a pre-existing collection, but given the complexity of the mathematics I think it is more likely that Archimedes himself composed the poem. In any case, it is an intentional communicative gesture to Eratosthenes on his part. If the poem were read without assuming the context of the prose introduction, a reader would probably consider themselves to be the addressee and the speaker to be the author of the poem. In characterising the relationship between the speaker and the addressee, one can also look towards the generic history of epigram. For public inscriptions and literary epigrams, the address to a παροδίτης (‘passer-by’, ‘traveller’), ὁδοιπόρος (‘wayfarer’, ‘traveller’) or ξένος/ξεῖνος (‘stranger’, ‘wanderer’) is a competitive manoeuvre intended to catch the reader’s eye, on busy public thoroughfares or on the scroll.Footnote ²⁹ φροντίδ’ ἐπιστήσας (2) could be taken not only as ‘set one’s mind to’ but also ‘halt one’s mind’, converting the traditional call to a passer-by to physically stop into a request for one to halt mentally. This aspect, as is often noted, is fruitfully exploited by epigrammatists of the Classical and Hellenistic period.Footnote ³⁰ As Michael Tueller has shown, depending on whether the epigram is sepulchral, dedicatory or amatory, the relationship between speaker and addressee differs.Footnote ³¹ Archimedes’ ξεῖνε hints towards the genre, though it is unclear into which subgenre the CP fits. In the present case, a subsequent, probably purposeful, ambiguity arises as to whether Eratosthenes is a ‘foreigner’ (ξεῖνος) or a ‘guest-friend’ (ξεῖνος).

The CP is also indebted to the language in the Odyssey where Circe addresses Odysseus.

An alert reader may infer a similar dynamic in the CP: Odysseus as the addressee and Circe the speaker. Indeed, Odysseus as a ξεῖνος is a key theme in the Odyssey, and its use in the epigram is a possible exegetical signpost.Footnote ³² Is this Odysseus quite literally (or textually?) in disguise? Without any clear indication to whom these Circean words are directed, the reader may well place themselves as the Odyssean addressee. If the reader has before them the prose introduction, they could also imagine that Archimedes has taken on the role of Circe and therefore that Eratosthenes has been made to play the role of Odysseus. In either case, the addressee’s characterisation as Odysseus presents them as the cunning, wily figure who is skilled in speech, according to Calypso (Od. 5.182–3) and Alcinous (Od. 11.367–8). The challenge as the poem proceeds is whether they can match up to that archetypal figure of intelligence and solve the mathematical puzzle.

Moreover, the opening line and address taken together point towards a further generic form:

In the initial hexameter line there is an invocation (ὦ ξεῖνε), a command (μέτρησον) and a topic (πληθύν) modified by an extended description (Ἠελίοιο βοῶν). It structurally recalls the opening lines of many hexameter poems, including the Iliad and the Odyssey.Footnote ³³

They too open with their subject, an invocation, a command and often a polysyllabic adjective. Epic invocations are employed to request information from the poet’s goddess or Muse: they, and not the poet, have true knowledge and information.Footnote ³⁴

The verse-initial use of πληθύν (‘multitude’) is uncommon in Homer (cf. Il. 9.641, 11.305, 11.405 and 15.295), and by far the most well-known usage is in the Invocation to the Muses in Iliad 2. This word, I argue in Section 2, provides a connection to the Invocation prior to the Catalogue of Ships and its positioning of the poet’s knowledge in relation to the Muses’. What can be said here is that the CP’s epic invocation is instead addressed to the reader and solver; will you be as successful as the omniscient Muses of epic in solving the problem? In one sense, the idea of knowing the number of the Cattle of the Sun parallels the knowledge of the Muses. Teiresias’ underworld explanation of Odysseus’ future stop on Thrinakia and encounter with the livestock represents the Sun in terms similar to the Muses. The description of the Sun who ‘looks over everything and hears everything’ (πάντ’ ἐφορᾷ καὶ πάντ’ ἐπακούει, Od. 11.109) is reminiscent of the Muses as described in the Invocation, who ‘are gods and are present and know everything’ (θεαί ἐστε πάρεστέ τε ἴστέ τε πάντα, Il. 2.485). The Cattle of the Sun are not a subject that the Muses dealt with directly, but the purview of the Sun allows for the possibility of their number being a matter of divine, superhuman and Muse-like knowledge nevertheless: an epic invocation in a Sicilian mode.

Equally, there is the influence of archaic elegy. Geoffrey Benson has argued that the stanzaic structure, the key terms of wisdom (σοφία) and measure or proportion (μέτρον) and the address to a ξεῖνε mean that ‘the main motifs imitate archaic elegy’.Footnote ³⁵ Wisdom and a sense of proportion appear in both archaic elegy and the CP, although the use of those terms in an emphatically mathematical context complicates the association; Archimedes enters into a dialogue with, but does not necessarily imitate, elegy. As Benson further notes, moreover, elegy continued to be composed in the Hellenistic period, and in particular it is the form used for some longer catalogue poems.Footnote ³⁶ So while the prose introduction suggests that the poem ought to be read as a long epigram, the CP’s metrical form together with its listing of ratios rather places it in the tradition of Hellenistic catalogues. Generically speaking, then, the CP positions itself at the intersection of a number of poetic forms; both epigram and elegy are in play, and the period attests amply to how both genres reinterpret and rework Homeric material. Echoing archaic elegy, for example, no doubt lent an air of intellectual superiority and didactic wisdom to the imagined speaker. The disjunction between a lengthy catalogue and the short, compact works of epigram will return more pointedly in the following section.

Besides generic dexterity, Archimedes shows himself to be in touch with contemporary literary scholarship, and the prose introduction suggests some sort of dialogue with those working in Alexandria specifically.

As the scholia to the Odyssey suggest, the number of the Cattle of the Sun was a subject of enquiry; ζητεῖν picks up the common term in the scholia for describing scholarly research.Footnote ³⁸ The scholia present the seven herds of fifty cows and seven of fifty sheep as representing the days and nights of the year, and the sun the whole year. According to one scholium, it is a claim made by anonymous thinkers: ἥλιον ἐνταῦθα τὸν χρόνον λέγουσιν εἶναι, βόας καὶ μῆλα τὰς ἡμέρας (‘they say that here the sun is time and the cows and sheep the days’, B-scholia on Odyssey 12.128). A further scholium specifies that Aristotle had considered the meaning of the Cattle of the Sun: Ἀριστοτέλης φυσικῶς τὰς κατὰ σελήνην ἡμέρας αὐτὸν λέγειν φησὶ τνʹ οὔσας (‘Aristotle explains in the manner of natural enquiry that he [Homer] says that the days under the moon are 350’, B-scholia on Odyssey 12.129). It thus appears that this passage created a ‘Homeric problem’ as early as the fourth century.Footnote ³⁹ Reincorporating scholarly cruces into new compositions is a hallmark of the early Hellenistic poets. Here Archimedes goes one step further and makes a Homeric zêtêma the subject of an entire poem.

Archimedes is no less scholarly in his vocabulary; his lexical choices suggest a keen awareness of Homeric language. As the reader proceeds through the poem, Archimedes plays with the idea of the reader as being Odysseus-like in their progress. After a gap of twenty-two lines, in which Archimedes elucidates the ratios of the herds of the Cattle of the Sun, he addresses the reader.

This signpost is not for the unlettered. It is an allusive reference underscoring the work’s scholarly nature and its ludic application of Homeric philology. The adjective describing the addressee, ἄϊδρις (‘unskilled’), occurs twice in Homer, once in the Iliad and once in the Odyssey. In the Iliad, Antenor describes Odysseus feigning foolishness while on an embassy to Troy as ἄϊδρις (Il. 3.219). In the Odyssey, after he has arrived on Aeaea and his crew have been transfigured into pigs, Hermes halts Odysseus and provides him with the protective moly before confronting Circe.

This is not the sly Odysseus of the Iliad, but of the Odyssey, constantly wandering and wondering to which land he has been blown, guided by the divine assistance of Athena.Footnote ⁴⁰ Similarly, the related noun ἀϊδρείη (‘ignorance’) is twice applied to Odysseus’ men who ‘with ignorance’ entered Circe’s palace: οἱ δ’ ἅμα πάντες ἀϊδρείῃσιν ἕποντο (‘they all at the same time entered with ignorance’, Od. 10.231 = 10.257). Superficially, this adjective seems to be a congratulatory compliment to the reader and hopeful solver. What might the attentive reader infer about Archimedes’ allusive description of them and another possible reference to Odysseus literally and textually disguised before them?

The Odyssean passage is emphatically geographical: Odysseus has no knowledge of where he is. How does this square with the CP? Broadly, the reader’s halted progress parallels Odysseus’ movement along the hilltops – δι’ ἄκριας ἔρχεαι – intercepted by Hermes.Footnote ⁴¹ A problem that arises, however, is the transposition from Aeaea in the Odyssey, to Thrinakia in the CP. A claim of oversight on Archimedes’ part is a possibility, but this does not really explain why such a specific textual allusion would lead to a readerly ‘dead end’. Rather, I suggest, for the reader recognising both their adopted Odyssean role and the incongruity of the Homeric intertext, they best Odysseus by orienting themselves in line with Homeric geography, textually and figuratively. Thus, Archimedes’ line could be reread as ‘you will not be called unskilled (as Odysseus was, geographically speaking)’. In geographic terms, the allusion asks the reader if they can locate Odysseus. For Eratosthenes, questions of Odyssean geography are highly contentious. Broadly speaking, Homeric scholars had two positions on Odysseus’ wanderings. Some located the wanderings within the Mediterranean, so Strabo records, such as himself and Callimachus (Strabo 1.2.37),Footnote ⁴² while others pinpointed them beyond the Pillars of Hercules, including Apollodorus of Athens and Eratosthenes (Strabo 7.3.6–7).Footnote ⁴³ Sicily was identified as an especially likely candidate for the mythical island, and by the Hellenistic period the association was common. This was no doubt bolstered by Thucydides’ folk etymology: Θρινακίη (Thrinakia), or as it was also known, Τρινακρία (Trinakria), a back-formation based on Sicily’s three capes, τρεῖς-ἄκρας (lit. ‘three points’, Thuc. 6.2.2).Footnote ⁴⁴ However, employing mythology to elucidate contemporary geography was found by some scholars to be methodologically dubious. Eratosthenes was a particularly vocal opponent. As a scientist and philosopher, as well as a literary critic and poet, he argued that although he was not against Homer’s poetry per se, Homer’s Odyssey had no place in the burgeoning discipline of geography.Footnote ⁴⁵

Yet prior to this proposed ‘geographical’ intertext, Archimedes had already signalled for the reader his intellectual allegiances.

Archimedes’ account of Sicily as Thrinakia signals no debate: the suggested geographical equivalence becomes fact. The association would pose no problem for the average reader, used to the mythical heritage of the island: cultural terra firma. For Eratosthenes, however, the equation of Sicily as Thrinakia is an impossibility. From the beginning, Eratosthenes’ acceptance of the mathematical challenge and the readerly journey would jar. The Odyssean allusion, then, advances Archimedes’ strategy. To decode Archimedes’ allusion, the reader must take on the Odyssean role, journeying through a text and a myth firmly located on Thrinakia, a Thrinakia that is in fact Sicily. The allusion sets the reader at the interstices of Homeric geography and Homeric philology. Yet Eratosthenes, whom one would expect to notice this allusion, interprets the Odyssey in a way which does not allow Archimedes’ (playful) geography and philology to intersect. The characterisation of the reader as οὐκ ἄϊδρις in a geographical sense gains piquancy when it is imagined to be aimed at Eratosthenes. Praise about knowing where one is, is a pointed compliment for Eratosthenes the revolutionary geographer. But the setting of Archimedes’ poem and the Odyssean allusion which would constitute this praising set such a compliment on the precipice of ridicule. Eratosthenes may know where he is in this poem through textual allusions, but as a geographer, does he really know Homeric geography? Archimedes displays a sophisticated literary strategy, not only testing the reader’s educated status, but offering a view of the literary challenge he sets up for Eratosthenes.

The final lines of the CP express a conditional tone, and again the possibility of a solution seems to be undercut by the literary references. Archimedes employs language reminiscent of Greek epinician poetry.

Proceeding as one who is κυδιόων νικηφόρος, the reader proudly carries off their victory. In the context of this intellectual contest, ἔρχεο is as much a sphinx-like ‘you may pass’ – having solved the problem – as it is a secondary epigrammatic command to go forth, having contemplated an inscription. The initial conditionality of the challenge – εἰ μετέχεις σοφίης (2) – is here resolved in a neat ring composition. Having completed these calculations, you have been judged wise; not only is it no longer a case of ‘if’, but the successful solver is ‘rich’ in a species of wisdom. The νικηφόρος so reminiscent of Pindaric epinician should also make one read an agonistic context in κεκριμένος – ‘having been judged in contest’ (cf. Pind. Isthm. 1.22; Nem. 3.67; Ol. 2.5, 13.14). This novelty should not be overlooked. The challenge exchanged between the two scholars, a battle of learning and culture, offers a noticeably different view of competing individuals and poleis in the Greek world. Success is not gained through sporting prowess, but in giving an account of mathematical dimensions and aspects of Homeric poetry.

Through his use of allusion Archimedes points to both the geographical and intellectual stakes of his problem: it is concerned with Sicily and with the parameters of human knowledge and the limits of the wise. Before exploring how these two issues are dealt with on the scale of the poem as a whole and its catalogic form, I want to consider further the integration, confrontation and elision of various epigrammatic forms. The dense, allusive reworking of Homeric material positions the CP within the genre of epigrammatic riddles. The differing levels of assumed knowledge on the part of the reader have something to say about the CP’s context of production and reception.

To what extent is this allusion to Odyssean geography in the CP to be noticed by an astute reader? An epigram by Philetas of Cos underscores how Hellenistic riddle epigrams engage with Homeric material in intricate ways, employing both philology and a broader cultural knowledge.

Peter Bing, rejecting variant views of the alder tree as a poet or a woman, suggested that it refers to a writing tablet.Footnote ⁴⁷ More recently though, Jan Kwapisz highlights how the noun κλήθρη refers to the alder tree out of which Odysseus constructs his raft on Calypso’s island.Footnote ⁴⁸ The noun is a Homeric dis legomenon, only appearing in the scene where Odysseus builds the raft (Od. 5.64, 239), and it is the key for decipherment. If the pronoun μέ refers to the alder, then the ‘alder-slayer’ who knows ‘the marshalling of words’ and ‘toils’ is Odysseus, traits formulaically ascribed to him. Much as in the CP, the character of Odysseus is revealed to us through a philological signpost. How convincing is this reading? Philetas’ epigram balances the reader’s broad cultural exposure to Odyssean material with a textual allusion. Retrospectively, the reader might congratulate themselves for having noticed the unique κλήθρην. It is possible that an ancient reader would have deciphered the epigram simply from the references to a man who is good with speech, has struggled, but nevertheless knows many ways.Footnote ⁴⁹ These are, after all, Odysseus’ characteristic traits. This is crucial when considering literary riddles. Within a riddle, the information supplied is never itself erroneous; rather, it is obscurely expressed. With Philetas, as with Archimedes, their language describing Odysseus employs both philological specificities and ingrained cultural formularity. Not only does Archimedes repeatedly address the reader as a ξεῖνος (‘stranger’, ‘guest’) – Odysseus being the archetypal ξεῖνος – but the very situation is uniquely Odyssean. The novelty of this type of riddling epigram, it seems to me, lies in the ability to observe the author at work covering up the identity of a figure in Greek culture, mentioning but not mentioning the great Homeric hero. For the astute reader, a philological allusion is a further sign of the poet’s skill in pointing to, but not explicating, the well-known subject.

The following riddle functions similarly, leaving its subject, a key Homeric figure, initially hidden from the reader.

The epigram’s features are not outwardly Homeric, nor are there any philological pointers; rather, a certain level of knowledge of Homer’s epics is required. To solve this riddle and identify the figure as Andromache, one must know that her first husband Hector was killed by Achilles, who became her father-in-law when she married Neoptolemus, who had killed her first father-in-law Priam, and that Andromache’s brother-in-law Paris killed her father-in-law Achilles, who had killed her father Eetion. The epigram presents a set of propositions concerning certain members of an unknown person’s family which are relatively straightforward. The repetitious language compounding the four interrelations, however, spawns complexity. With Philetas the identity of Odysseus is a textual matter; this Homeric epigram weaves a knot of interconnection around Andromache out of the broader cultural currency of epic. Archimedes operates in like fashion. There is a certain superficial simplicity in offering up the ratios of herds of cattle. When considered thoroughly, though, it becomes obvious that things are more complicated. Both epigrams underscore how difficult it can be to untangle the mass of culture that is the Homeric tradition. The denouement of the epigram on Andromache is successful because it offers the reader resolution; there are simple answers to knotty cultural interrelations.

In these riddles, the workings of cultural capital can be seen at play. Hellenistic literate education and knowledge of Homer in particular could create a shared identity uniting the educated Greek elite, but it is also the means through which individuals could gain intellectual distinction by demonstrating the extent and depth of their learning.Footnote ⁵⁰ The agonistic intellectualism of the Andromache epigram seems clear, for Philetas this is probable, and in the case of the CP, the epistolary header is highly suggestive. Clearly, a philological note demands deeper knowledge than heroic genealogies. Nonetheless, literary reference and popular knowledge are not mutually exclusive, and this is part of the craft of the riddle. In the CP, there is no enunciation of Odysseus. Yet his character and his narrative are never far from the reader’s mind. A reader of the CP, picking up the Odyssean cues, could congratulate themselves. Those who notice the philological intertext of ἄϊδρις will feel ‘intellectual’ and may additionally reflect whether Eratosthenes too noticed the intertext. Archimedes’ poem allows the reader to observe intellectual agonism ‘in action’, and the literary riddle is the ideal form through which to underscore this competitive interaction.

3.2 Cattle and Catalogues

Archimedes’ allusive art in the CP sets his poetic skills on a par with Hellenistic poets more traditionally viewed as scholarly and recondite. By redeploying key Homeric words, he alludes to the exclusive nature of being σοφός (‘wise’) and reconfigures Odyssean geography. This would have had a clear effect for a poem exchanged between himself and Eratosthenes, revolutionary geographer and curator of the largest Greek library ever seen. In addition to the allusive language, however, the catalogic form of the poem – its listing of the ratios of cattle – has a deep history in Homeric poetry. My interest in this section is the connection between the CP and Homer’s Invocation prior to the Catalogue of Ships. I argue that Archimedes frames the possibility of solving the ratios through a series of allusions to that passage and to Iliad 2 more broadly. My focus in particular will be on what this intertext implies about handling large numbers in verse and the possibility of the reader solving the ratios. Subsequently, I ask how this perspective is modified by the appeal to elegiac traditions that occur in the first pentameter. If the Iliadic Catalogue is signalled as an intertext in the opening hexameter, how does this picture change when it becomes clear that this is an elegiac catalogue of cattle? I ultimately want to argue that Archimedes actively strains generic forms that might be ascribed to the CP in order to highlight the limits of human knowledge. The series of allusions to Iliad 2 together with the programmatic opening couplet, in other words, explores the similarities between mathematical and poetic knowledge and the difficult compromises which arise when they interact.

Before turning to the first word of the CP, it is worth pointing out that Archimedes’ subject matter fits closely with the broader context of the Catalogue in Iliad 2. Immediately preceding the Invocation to the Muses for support in accounting for all the Achaeans at Troy, Homer describes the gathering host in a series of seven similes. They are likened first to a fire ravaging a forest (2.455–8), then to birds flocking on to a meadow (2.459–66), to the number of leaves in a meadow (2.467–8) and flies swarming round a milk pail (2.469–73). Following these four similes characterising the host, the poet turns to characterise their organisation.

The organisation of the troops is likened to goat-herding. The leaders who διεκόσμεον (‘ordered’) the troops recall Agamemnon’s notable numerical language earlier in the book, where he imagines both the Trojans and Achaeans being ‘counted up’ (ἀριθμηθήμεναι, Il. 2.125) and the Achaeans being ‘ordered into tens’ (ἐς δεκάδας διακοσμηθεῖμεν, Il. 2.127), in order to highlight that the Trojans are outnumbered. The counting of troops in this later scene is now a pastoral activity. Archimedes’ poem looks to a highly numerical passage regarding cattle in the Odyssey but, given its opening allusion to the Invocation prior to the Catalogue, also connects this with the herding imagery which immediately precedes the Invocation. In asking the reader to calculate the πληθύς (‘multitude’) of cattle, Archimedes realises the vehicle of the Homeric simile and transforms it into the actual subject of a calculation.

Now to the opening word itself: πληθύν. Primarily, it signifies a ‘multitude’. It also recalls Homer’s Invocation before the Catalogue. That passage’s popularity as a stand-alone section of the Iliad in Greek society, evidenced by papyri, affords the opportunity to take πληθύν seriously as a salient intertext and ask how this might affect a reading of the CP.Footnote ⁵¹ Here is the passage again.

With the prospect of (re)counting all the men at Troy the poet reaffirms his relationship to the Muses. The poet’s inability to deal with a large number of people contrasts with the Muses’ omniscience. This progression of thought raises interpretative issues. The poet’s lack of knowledge in comparison to the Muses and the inability to recall the entire πληθύς given his human limitations and mortal frame are traditional elements of catalogues.Footnote ⁵² The further conditional, however, could be interpreted as implying that the Muses can help the poet overcome those mortal deficiencies which he had outlined.Footnote ⁵³ I would follow Tilman Krischer and see this as being resolved by taking ὅσοι (Il. 2.492) to be an indirect interrogative and not a relative pronoun.Footnote ⁵⁴ The Muses, that is, can support the poet to recall the number of the πληθύς and select narratives, but nothing more: recalling the narratives of the entire πληθύς would demand a superhuman ability.Footnote ⁵⁵ His final resolution to speak about the leaders of the ships and the ships allows him to balance both demands.

How the passage in the Iliad might have been understood later in antiquity affects the sense that can be ascribed to the echo of πληθύς in the CP. On the broadest level, the opening use of πληθύς brings to mind the difficulty of dealing with large numbers that arose in Iliad 2 and raises the question whether the reader of the CP will be able to manage these large numbers too. In Iliad 2, the Invocation could be interpreted as signalling that the poet has divine support in giving the audience an account of the gathered troops, or it could be understood that his account based only on the leaders and the ships instead constituted the poet recounting the troops without divine aid. If Homer were understood not to have the support of the Muses in giving his enumerative catalogue, this may make more tangible the reader’s expectation that the catalogue of ratios is manageable and the πληθύς enumerable: Homer did this without the Muses, so might I. In my estimation, though, the condition of the Muses’ support in recalling how many went to Ilion (492) is what enables the poet to account for (to say nothing of naming) the πληθύς in the form of a catalogue. With Iliad 2 in mind, the Muses’ absence from Archimedes’ poem suggests that, just like the poet on his own, the reader will be unable to give the total number of Cattle of the Sun. This picks up a further aspect of the Catalogue and its calculations, namely that Homer never gives a final answer nor explicitly puts a number to the πληθύς of the troops. Even with the Muses’ help, the poet is only able to give a catalogue that counts the number of troops per ship and ships per leader, and fails to provide the numerical total. Since Archimedes’ ratios would have been irresolvable, his poem too remains a catalogue of numbers that does not yield a final numerical answer for the πληθύς.

Computing the ratios of the Cattle of the Sun thus becomes akin to attempting to count up all the heroes who went to Troy, but this connection extends well beyond the allusive opening word. Archimedes further draws from the deliberative scenes in Iliad 2 in order to characterise the potential solver of the problem. Consider again Archimedes’ apostrophe to the reader.

Verse 31 looks forward to the additional parameters which Archimedes will provide, but also continues to allude to Homer and to Odysseus. The Iliad and the Odyssey each contain a single occurrence of ἐναρίθμιος (‘numbered among’). Most pertinent is the Iliadic context where Odysseus seeks to persuade the Achaeans not to flee following Agamemnon’s test of the troops and false promise of return.Footnote ⁵⁶

Odysseus attempts to subtly talk over the other leaders among the Greeks, but he addresses those of the masses with harsher words. Here, being ἐναρίθμιος designates inclusion within a group, and a group that is marked out by its power and elite position within Homeric society. Odysseus’ denigration of the masses as not being ἐναρίθμιος within this group is offset by the Catalogue of Ships. If Odysseus uses the language of counting to define the lower social position of the average soldier, Homer nevertheless ensures that they are given some renown by being meticulously counted among those who went to Troy. The adjective’s Iliadic usage raises the possibility of the reader of the CP being counted among the wise in the same way that the leaders at Troy are promoted above the mere mass of soldiers.

Archimedes concludes his representation of the reader in the final lines of the CP and continues to draw on Iliad 2 in characterising the successful solver.

Important for my purposes, first, is that the participle κυδιόων (‘carrying oneself proudly’) is used to describe Agamemnon in his entry within the Catalogue.Footnote ⁵⁷

Even in a catalogue of heroes and their troops, Agamemnon nevertheless stands above them all in his pre-eminence. Identifying the figure of Agamemnon behind Archimedes’ representation of the reader in the final lines highlights the arithmetic progress being implied. The rhetorical movement in the CP from the solver as one who is ἐναρίθμιος to one who is like Agamemnon models Odysseus’ address to the soldiers. Following his denigration of the soldiers as not even ἐναρίθμιος in war or council he then calls for them to unite under Agamemnon – εἷς κοίρανος ἔστω | εἷς βασιλεύς (‘let there be one ruler, one king’, Il. 2.204–5). Characterising the solver now not simply as one of those who is counted among the generals of the troops as opposed to the mass of soldiers, but as the leader of the whole contingent, figures them as unique in their abilities. Agamemnon had already displayed his ability to make calculations regarding the troops earlier in the book (Il. 2.123–33), and he stands even above the other leaders ordering their troops in the simile before the Invocation and Catalogue (474–5, see above), both of which suggest his ability to handle and order numbers on a greater scale than the other leaders. At the end of the CP, Archimedes’ use of κυδιόων in a poem already recalling the Invocation and Catalogue raises the possibility that the reader will have full mastery over the number of cattle just as Agamemnon had control over the troops.

Equally, though, the conclusion can be read as hinting at the impossibility of the arithmetical task. The participle κυδιόων also appears in two almost identical similes comparing the heroes Paris and Hector to horses that have bolted the stable and enjoy their freedom glorying in their splendour (Iliad 6.506–11 = 15.263–8). With Paris, the image of a horse that delights too much in his appearance reflects Paris’ underlying nature, whereas Apollo, rousing Hector from his feeling of defeat, brings out in him the exulting confident defender of Troy. It is this onslaught, this final rallying against the Achaeans with Apollo’s aid, that leads to the death of Patroclus at Hector’s hands, and thus seals his fate at Achilles’ hands.Footnote ⁵⁸ Moreover, in the Homeric scholia both Paris and Hector are taken as paradigms of ‘boastfulness’ (ἀλαζονεία).Footnote ⁵⁹ To read echoes of either narrative is thus to hear a note of caution about believing in one’s own abilities.Footnote ⁶⁰ There may well be further irony, too. The adjective νικηφόρος plainly refers to the solver as a victor, but in Pindaric epinician poetry it can also be applied to horses (e.g. Ol. 2.5). Likewise, while the meaning of ὄμπνιος remains unclear, it seems that it was connected by a number of authors with nourishment, agricultural produce and grain.Footnote ⁶¹ The solver may well be ‘victorious’ and ‘well-fed’ or ‘nourished’, but like an overly proud horse; after all, Homer appeals to the Muses to account for horses, as well as for men, in his Catalogue (cf. Il. 2.760–2). Archimedes thus employs Homeric terms in order to create the expectation of a solution as well as to undercut it. Halfway through the CP, the reader is promised that they might become more than one of the masses and ἐναρίθμιος among the Greek leaders if they can solve the mathematics, and the conclusion elevates this to the possibility that they might be an Agamemnon having control over all the troops. Yet it is a decidedly ambiguous representation of the solver in the final lines. These allusions to Iliad 2 raise but do not confirm the possibility that the reader can compute the number of cattle in the same way that the poet counted the troops in his Catalogue after they had been herded by the leaders, in the imagery of Homer’s simile.

It is equally important to observe that the question of how easy it might be to grasp such a large amount is not only posed by the Iliadic intertexts. It also extends across the first couplet as a whole and particularly in the move from the opening hexameter to the following pentameter. In explaining the interrelation of the hexameter and pentameter, I consider to be instructive the one surviving fragment of the fifth-century Carian poet Pigres. This brother (Suda s.v. Πίγρης 1551) or son (Plut. Mor. 873f) of Artemisia, the ruler of Halicarnassus and ally of Xerxes, composed an Iliad in elegiacs, inserting after each of Homer’s hexameters a further pentameter. His modification to Il. 1.1 is as follows:

Pigres plugs Homer’s own concerns with the limits of mortal knowledge in the Invocation in Iliad 2 back into the opening invocation of the Iliad. He also reworks the proem into an elegiac couplet and introduces a notably elegiac theme. The term σοφία is common in Theognis’ articulation of wisdom in his sympotic elegies (563–6, 790, 876, 1074 IEG), and it is an attribute associated specifically with poets by Solon in his Elegy for the Muses: ἄλλος Ὀλυμπιάδων Μουσέων πάρα δῶρα διδαχθείς | ἱμερτῆς σοφίης μέτρον ἐπιστάμενος (‘Another, taught with gifts from the Olympian Muses, knowing the measure of lovely wisdom’, fr. 13.51–2 IEG). Similarly, μέτρον (‘measure’) is common in earlier elegiac poetry, denoting self-control in sobriety and desire.Footnote ⁶² In both Solon and Pigres, these terms sit in the pentameter, the line which differentiates the genre from epic hexameter. In Solon’s elegy, the pentameter negotiates the distinctiveness of elegy as a genre – with ἱμερτή suggesting a more erotic mode (cf. Theognis 1063–8 IEG) – and focalises the agency of the poet and his ability to know. Whereas Solon intimates the bounded nature of poetic knowledge per se through his use of μέτρον, Pigres’ pentameter emphasises how epic and elegiac poets differ in their claims to wisdom and authority. Rather than expanding the request for knowledge from the goddess across a series of lines, specifying the remit of the present song as was typical in early incipits, Pigres’ rewriting both curtails this request and emphasises the Muse’s supreme control over knowledge. His couplet does not position the elegiac poet as in control of sophia, but rather the Muse; it (re)asserts the authority of the Iliadic – and so, epic – Muse by means of an elegiac strategy. Moreover, despite Pigres doubling the length of the Iliad through pentameters, it is the Muse who retains ‘mastery’ over Homeric material. Archimedes likewise addresses the question of human and divine knowledge through the addition of the pentameter. There, he commands that the reader measure the multitude ‘if they have a share in wisdom’ (εἰ μετέχεις σοφίης, 2). Unlike Pigres, Archimedes does not make it immediately explicit who it is that possesses wisdom. He offers up to the reader the hope that they may gain wisdom but, given the irresolvable ratios, the CP demonstrates the exclusive and elusive nature of wisdom, something that Pigres’ elegiac addition had simply stated. That is, the pentameter supports the language and allusion of the hexameter in setting up another expectation for the hopeful solver that is destined to be unfulfilled.

The move from the hexameter to the pentameter hints at the potential impossibility of measuring the multitude in poetry in another manner, too. πληθύν in Iliad 2 signalled the opening of a hexameter catalogue. Similarly in the CP, the reader’s expectations are fulfilled when Archimedes provides his exposition of the ratios of the cattle, a catalogue of cattle responding to Homer’s imagery in Iliad 2. A catalogue in elegiac couplets, or epigram as the prose introduction has it,Footnote ⁶³ however, strains the concept of the generic form. Epigram is a traditionally compressed genre that would seem to be poles apart from the extended narratives of epic. A later Greek epigrammatist attempts to lay down the law when it comes to poetic length and its generic association, quipping in a single couplet that ‘a two-line epigram is very fine; but if you exceed three couplets, you are rhapsodising and are not saying an epigram’ (πάγκαλόν ἐστ’ ἐπίγραμμα τὸ δίστιχον· ἢν δὲ παρέλθῃς | τοὺς τρεῖς, ῥαψῳδεῖς κοὐκ ἐπίγραμμα λέγεις, AP 9.369).Footnote ⁶⁴ At a total of twenty-two couplets, the CP would rank as one of the longest extant epigrams. It could perhaps be compared to the equally ambitious Hellenistic inscription found at Salamacis on the history of Halicarnassus.Footnote ⁶⁵ By the same token, the blurred line between epigram and elegy that I noted in Section 1 reinforces the sense of strained generic forms; the recent advent of catalogue elegy represents a generic compromise between the concision of epigram and the expanse of epic.Footnote ⁶⁶ In an analogous vein, Archimedes combines a move into elegiacs with textual extension: his versified catalogue of the Cattle of the Sun is over ten times longer than Homer’s original (forty-four lines vs four lines). Yet, in Pigres’ case, doubling the length of the Iliad did not counteract the fact that the Muse is the one who possesses wisdom. The very meaning of his first inserted pentameter underscores this. The CP likewise offers the hope of wisdom in the pentameter but never in fact confers it upon the reader. In other words, length does not directly translate into more wisdom or knowledge contained within the poem. In that respect, too, the extension of the CP into a form of catalogue epigram or elegy simulates Homer’s own expansive catalogue of numbers and figures which for all its length does not in the end explicitly supply the total amount of the πληθύς for the audience. The opening couplet of the CP, then, introduces the challenge to the reader but also draws on language redolent of the quintessential epic catalogue, as well as of elegiac concerns about wisdom, precisely in order to suggest that such a feat might not be within the bounds of mortal knowledge.

On my reading, these Iliadic intertexts set up the expectation that calculating the number of cattle, and especially without the help of the Muses, will not be a success. This is subsequently supported with the pentameter’s turn to questions of wisdom and its attainability. Archimedes has set his sights on the question of human knowledge and its limits. This would have been a potent and political issue for Eratosthenes at the Library of Alexandria. In this respect, I want to tentatively suggest that the use of μέτρησον at the end of the opening hexameter is pointed. The verb μετρέω and its cognates are connected to measurement of all kinds from the earliest times, but it sees increasing use in the Hellenistic period in contexts which highlight not just a manipulation of, but a control over, Greek culture and its Homeric aspects. In the case of the Tabulae Iliacae, Michael Squire has demonstrated that the ability to circumscribe, condense and schematise Homeric narratives is constructed as a wondrous feat and an expression of mastery and wisdom (σοφία) by those who claim to have done so.Footnote ⁶⁷ Archimedes’ opening hexameter, flanked by πληθύν and μέτρησον, offers a similar possibility to the reader and to Eratosthenes, that they might succeed by employing the concrete tools of mathematics and have some grasp of one aspect of the Homeric tradition. There is an irony, moreover, in addressing the challenge to Eratosthenes in the library where Homer’s epics were most famously edited, ordered and commented upon in a way that sought mastery and control over the Homeric texts.Footnote ⁶⁸ How easily will this servant of the Muses calculate the πληθύς without the Muses’ explicit support? Even before the irresolvable ratios, Archimedes’ epic intertext and elegiac turn in the pentameter suggest that this is far from guaranteed. The opening couplet questions the possibility of measuring the multitude in poetry, a tension that additionally raises the possibility of, but also resists, the circumscribing of Homeric subject matter more widely.

3.3 Calculating Cattle and Cultural Competition

The CP represents itself as operating in line with Homer’s poetics of calculation in Iliad 2, but its metrical form also hints at the strain of composing a catalogue of calculations in verse. My argument in this section is that this tension that arises when one attempts to compress such a large amount of mathematical material into a poem has a specific cultural-political motivation. Here, I examine cataloguing and calculating in contemporary and earlier poetry. The calculations in these texts do not compare to the complexity of Archimedes’ ratios; they are for the most part displays of simple addition. The difference in the mathematical operation exhibited notwithstanding, I demonstrate that an abiding aspect of these passages is the enacting or performing of calculation as a form of geographical possession. This poetics of census-taking seems to have a particular aim in the context of the CP’s geographically focused claim to a Homeric Sicily. My proposal is that the very form of Archimedes’ calculating catalogue articulates a politics of space and identity in order to circumscribe the possibility of Sicily’s (metaphorical) possession.

I begin with perhaps the clearest contemporary instance of a poetics of census-taking. Theocritus, Archimedes’ older contemporary and fellow Syracusan, demonstrates the politics of a counting catalogue in his Encomium to Ptolemy (Idyll 17), where the fertility and productivity of Egypt are described.

As with Archimedes’ poem, the explanation of a number through calculation emphasises multiplicity, although of course Theocritus aims at nothing so complex. Both exhibit a similar means of connecting fertility and calculation. Just as the Nile’s fertile bubbling up (ἀναβλύζων, 80) is paralleled in the ensuing count of the many cities, so too do Archimedes’ cattle when ordered in a triangular formation ‘begin bubbling up from a single one’ (ἀμβολάδην ἐξ ἑνὸς ἀρχόμενοι, 38): numerical growth simulates natural and economic growth. Given that the passage concludes by stating that Ptolemy rules over this large number, however, its evocation of ‘the Egyptian and Ptolemaic passion for counting and census-making’ has the serious function of characterising political control through a control of numbers.Footnote ⁷⁰ The ability to express such a large number within just three lines further simulates this Ptolemaic control: the great number of cities in Egypt are still accountable to Ptolemy, and so their number is countable for a Ptolemaic poet.

A similar claim to land through enumeration can be seen in Lycophron’s Alexandra, a 1,474-line Hellenistic iambic poem which gives Cassandra’s final prophecy during the sack of Troy, which spans all the way from the time of the Trojan War through mythic history and down to the Hellenistic period itself. It describes Aeneas’ founding of Lavinium after fulfilling Helenus’ prophecy, in a narrative familiar from the Aeneid (3.390–2).

It is emphatically Aeneas’ enumeration here that leads to his founding of Lavinium and determines its number of towers. The Alexandra, although once considered to be early third-century, is most likely a product of the mid-second century.Footnote ⁷² This passage from the so-called Roman section is relatively early evidence for the development of Roman foundation myths, especially in a wider Greek context.Footnote ⁷³ While the prophecy on the enumeration of the sows is alluded to here first in poetry, as a myth it predates the Alexandra having been recorded by Fabius Pictor in the late third century (FGrH 809 F 2).Footnote ⁷⁴ In a less mythical – but no less fantastic – vein, the Alexander Romance (1.33.11) reports a numerical conundrum posed to Alexander in a dream by a god, who delineates a series of numbers (200-1-100-1-80-10-200) which reveals their nature when converted into letters (σ-α-ρ-α-π-ι-ς, Σάραπις, ‘Sarapis’).Footnote ⁷⁵ Certainly, this is a different form of mathematical challenge. Still, its appearance in the context of recognising the god so as to legitimate and support Alexander’s foundation of Alexandria highlights a further example of the intersection of counting and foundation. It is important to underscore in these examples that at the time Archimedes was composing the CP, scenes of enumeration were a productive means of staging (re)imaginations of political geography.

A further passage that has not been discussed in relation to the CP is Odysseus’ reunion with his father, Laertes. Having reunited with Penelope, Odysseus heads to the farm where his father lives and labours. Meeting him alone in the vineyard, he at first pretends to be someone else who had met Odysseus on his travels; only when Laertes breaks down in sorrow does Odysseus reveal himself to his father.Footnote ⁷⁶ In order to prove his identity, he offers the following tokens as evidence.

Odysseus gives two forms of evidence: the physical scar on his body and his mental recollection of the gifts that Laertes had promised to give him. Homer, through a variety of intermediaries, has already presented the scar and the narrative which accompanies it (cf. Od. 19.391, 393, 464, 507; 21.221; 23.74). The recounting of the trees, however, appears only here. The description of the trees and their count responds to the over-exposed sign of the scar; it represents not heroic deeds or the revealing and naming of the hero, but the naming of home (ὠνόμασας/ὀνόμηνας), its fixedness (τά … ἔμπεδα) and fecundity (the hapax διατρύγιος). As Odysseus reaches the heart of Ithaca at the end of the Odyssey, he reconnects with his roots and points to the one sign of belonging that he was unable to take with him but took account of nevertheless. For John Henderson, the enumeration is only one part of a wider rehearsal between father and son; Odysseus’ miming of ‘bodily commitment’, ‘his insistent deixis’ and his ‘remembered walk in the wake of his father across the very scene of utterance’ constitute a performance of sameness between father and son, a ‘monological evidentiality, a self-identical prestation’.Footnote ⁷⁷ I would emphasise in addition that within this recollection and rehearsal, the count of the trees figures Odysseus’ Ithacan inheritance at large: the variety of trees, their continual bearing of fruit throughout the seasons represents not just this plot of land, but also the fertility of Ithaca tout court. He has regained his son, his wife, his halls, and now he must recover the land. The passage from Lycophron’s Alexandra showed how the challenge of enumeration was employed to explain and legitimate claims over land following the Trojan war, and Odysseus’ enumeration at the end of the Odyssey is in a sense a prototype of the later Aeneas, although he is seeking to reclaim his Ithacan inheritance. As I have argued, the CP engages intricately with Odyssean geography; the tradition of claiming the land through counting also has an Odyssean lineage. Archimedes offers the possibility of another Odyssean ‘accounting’, and so the possibility of another claiming of land, only this time of a different island. He has taken one Odyssean claim to the possession of space and has transferred it to the equally Odyssean and equally numerical context of the Cattle of the Sun which had more significance for him as a Sicilian.

Odysseus’ and Aeneas’ travels and subsequent census-taking most likely arose in response to the Greek colonisation of the archaic period and to the need for myths to explain the foundation of new colonies. As the passages from the Alexandra and the Alexander Romance show, an oracle – a directive from a god – is a particularly irrefutable way to justify the Greek claims to land across the Mediterranean. A fragment from Hesiod’s Melampodia, a hexameter poem tracing the lives of mythical seers, further demonstrates that archaic poets were aware that calculated claims to land in oracular contexts could involve contestation. Here is the fragment and Strabo’s introduction to it:

Colophon was founded when the seer Manto arrived there, having left Thebes in the aftermath of the war of the Seven against Thebes (cf. e.g. Epigonoi fr. 3 EGF). The famous seer Calchas, in the aftermath of the Trojan War, arrived at Colophon and challenged Manto’s son, Mopsus, to a contest of their oracular abilities. Numerous versions of the meeting between Mopsus and Calchas have survived (Strabo 14.1.27; Apollod. Epit. 6.2–4).Footnote ⁷⁹ Across the range of retellings, as Naoíse Mac Sweeney has shown, there is variability in the agency ascribed to Manto and to her son, Mopsus, regarding which of the two founded Colophon and the Oracle at Clarus.Footnote ⁸⁰ Whichever narrative one follows, though, a notable constant in the accounts is that Mopsus prevails in the contest with Calchas. In its broadest outline, the contest constitutes an aition for the continued Theban and Mantid control of that oracular site following the Trojan War and the challenge of Calchas. The second constant is that the oracular challenge always has a numerical element.

Archimedes may have had this story, or a version of it, in mind. As befits a contest, ‘calculating’ the figs on the tree has a question-and-answer format. This ‘tell me’ formula is recognisable from the Contest of Homer and Hesiod passage (above, introduction to Part II) and is similar to the opening of the CP (ὦ ξεῖνε, μέτρησον, 1). More notably, Mopsus’ answer has two stages: he gives Calchas the exact number of figs, but then goes on to explain how that number might be expressed as a volume measurement by introducing the medimnus. The Alexandra preserves a variant account which places Calchas in southern Italy by the banks of the river Siris: ‘there lies unhappy Calchas, a Sisyphus of uncountable figs’ (ἔνθα δύσμορος | Κάλχας ὀλύνθων Σισυφεὺς ἀνηρίθμων | κεῖται, 979–81). The scholium to Lycophron’s elliptical reference describes how Calchas met not Mopsus, but Heracles after he had carried off the oxen of Geryon, and how he successfully responded to Heracles’ challenge to enumerate the figs on a tree. Calchas numbered them as ten medimni and one fig and mocked Heracles when ‘having measured them and greatly forcing the one left-over fig into the measure [i.e. medimnus], he was unable to’ (τοῦ δὲ Ἡρακλέος ἀναμετρήσαντος καὶ πολλὰ βιαζομένου τὸν ἕνα ὅλυνθον περισσὸν ἐπιτιθέναι τῷ μέτρῳ καὶ μὴ δυναμένου, Schol. on Alexandra 980a). In response Heracles kills Calchas for mocking him. Both narratives of Calchas’ death focus on the fact that certain numerical totals cannot be expressed in a geometric form, such as the volume of a medimnus. Archimedes similarly structures the CP.Footnote ⁸¹ The CP first asks for the number of the Cattle of the Sun from the given ratios and then second provides the parameters that the white bulls together with the black bulls are a square number and that the brown bulls and dappled bulls are a triangular number. Given the different objects of calculation, Archimedes substitutes volume for area. As I outlined above, the first half of the problem (5–26) yields infinitely many solutions, with the smallest positive integer solutions yielding cattle in their millions.Footnote ⁸² It is the second half (33–40) and the requirement to fit the cattle into a rectangular and triangular arrangement which makes the sum astronomically large and ultimately incalculable for a Hellenistic mathematician. Arguably, the CP’s structural echo of the contest in the Melampodia and the similar retellings constitutes a hint that the further parameters lead inevitably to failure. Elsewhere Archimedes employs literary allusions to suggest to the astute reader the (im)possibility of their success, and here too they will know from earlier poetry such as the Melampodia that you cannot force and fudge a calculation when sensible and indivisible bodies are involved – that is, when doing λογιστική. Just like the lone fig, for the ancient reader, these cattle could not be forced simply into any old measure.

Archimedes’ use of this structure also geographically frames the stakes of solving the mathematics of the CP: failure in a numerical challenge leads to a failure to gain possession of land. In the Melampodia, Mopsus succeeds in the competition and so retains control over Clarus. The CP similarly offers up a numerical challenge but also sets the challenger up to fail in that task. Since these are Sicilian cattle and since such counts as those discussed above connect censuses of the land with possession over the same land, it would be logical to suppose that Archimedes presents the calculation in the CP as offering the potential for possessing Sicily. Archimedes, just like Mopsus at Clarus, retains dominion over Sicily, whereas Eratosthenes would have failed in his attempt to calculate the number of the cattle, as did Calchas. Unlike the passages from Theocritus’ Idyll, the Alexandra or the Odyssey, where those counting seem only to have to assert their possession over the land, I would suggest that the Melampodia (or something like it) provided Archimedes with a model of an arithmetical challenge between two famed intellectuals who have competing claims to a location. Given that, as I discussed above, this is a poem about Sicily sent to an intellectual who denied its Homeric pedigree, the importance of this model helps clarify the purpose of the CP and the nature of the challenge Archimedes sent Eratosthenes: if you can calculate the number of the Cattle of the Sun, you can then claim possession of (knowledge about) Sicily.

The focus on the number of Sicilian livestock finds a contemporary parallel in Theocritus’ Idyll 16, as Marco Fantuzzi notes and Reviel Netz develops. That ‘patriotic’ Idyll, addressed to Hieron II of Sicily, looks towards the island’s reinvigoration with ἀνάριθμοι | μήλων χιλιάδες (‘countless thousands of sheep’, Theocritus Idyll 16.90–1). Netz pushes this numerical aspect, suggesting that Theocritus’ emphasis on ‘those who wished to slaughter its [Sicily’s] cattle’ refers to contemporary events, perhaps Marcellus’ attacks and siege of the city.Footnote ⁸³ Thus, in two political poems, Theocritus’ poetry preserves two contrasting political connotations of enumeration. For the Ptolemies in Idyll 17 (above), fertility is something which can be emphatically brought under control and measured; for Sicily, conversely, its fecundity is immeasurable as the island teems with cattle. In the CP, admittedly, it is the number of the legendary Cattle of the Sun and the Thrinakia of Homeric poetry that is to be calculated and so controlled rather than the contemporary livestock of Sicily. Nevertheless, many such political interactions between Hellenistic states and poleis were effected through appeals to their (fictive and recently fashioned) epic past.Footnote ⁸⁴ Whereas Theocritus states the immeasurability of Sicily’s cattle, Archimedes offers the expectation of grasping the quantity of cattle, which the arithmetical complexity duly thwarts; Sicily’s cows are innumerable and Sicily unlimited in its resources. His language and mathematics equally contrive an uncontrollable, incalculable situation in the same vein as the teeming livestock of Idyll 16, and it is directed against a Ptolemaic intellectual who might well have been in a position to calculate the number of cities in the vein of Idyll 17. Unlike the earlier counting contests over land, Archimedes’ CP resists simple scientific judgements being made about Sicily. It cannot be counted by – and so potentially ruled by – the Ptolemaic Empire.

* * *

This chapter set out to demonstrate that the CP engages with its readers on literary, intellectual and cultural levels as well as on the arithmetical level: evident by now, I hope, is the sophistication of Archimedes’ agonistic arithmetic aesthetics aimed at Eratosthenes. The CP works because it problematises scientific and mathematical descriptions of cultural and literary artefacts, especially for Eratosthenes, whose rationalising geography sees him strip Sicily of its Homeric past. Archimedes beats Eratosthenes at his own game, pairing poetry and mathematics, and offers a scientific expression of the Greek cultural idea of the Cattle of the Sun (not to mention the dimensions of Sicily itself). The irresolvable ratios of cattle underscore the sheer fecundity of the Sicilian land and its inability to be fully encompassed, an immeasurability that might even be seen to stand for the boundlessness of the Homeric tradition. This is an aesthetics of arithmetic, in other words, that points up the very tension of setting arithmetic in verse as well as the contested capabilities of mathematics as a means of describing the world.

Archimedes’ Cattle Problem is an early, extended and complex case of a poem seeking to interlace arithmetic and aesthetics, but it is not the only case. The focus of analysis in this chapter are the so-called arithmetical poems preserved in Book 14 of the Palatine Anthology (henceforth AP). They similarly challenge their readers to solve the outlined simultaneous equations, and this time, all the arithmetic is solvable. The poems constitute an odd collection: their authorship, date and purpose are all contested. AP 14.116–46 in the modern numbering are a collection of arithmetical poems, which are preceded by a collection of riddles (AP 14.14–47, 52–64, 101–11) and oracles (14.65–100, 112–15, 148–50). The arithmetical poems are attributed to one Metrodorus, whose identity is difficult to ascertain.Footnote ¹ There seems to be no consensus as to whether Metrodorus should be thought the author or the compiler of the collection.Footnote ² Poems 14.1–4, 6, 7, 11–13 and 48–51 are also arithmetical in nature, and there is evidence that some of them are part of the Metrodoran collection.Footnote ³

An explanation of the purpose of AP 14 is given in a prefatory statement preceding the first poem: γυμνασίας χάριν καὶ ταῦτα τοῖς φιλοπόνοις προτίθημι, ἵνα γνῷς τί παλαίων παῖδες, τί δὲ νέων (‘for the sake of mental exercise I also provide the following for the industrious, so that you might know what both the children of former times [did] and those of recent times’).Footnote ⁴ It is unclear whether this preface goes back to Constantinus Cephalas, the Byzantine schoolmaster who compiled and edited together earlier epigram anthologies into a vast collection, which serves as the basis of the codex Palatinus (the modern day AP) and its some 3,700 poems.Footnote ⁵ In any case, the preface can be no later than the codex, formed in the middle of the tenth century ce, which contains AP in its current shape. The preface could equally apply to the oracles and riddles as well as the arithmetical poems, as examples of mental exercises. A contrast between the genres may then be implied in the contrast between children in the present and those of earlier generations. A reference to the arithmetical poems, though, is prima facie probable, given Plato’s description in the Laws of calculating with real objects, that is, λογιστική. There the mixing and the dividing of tangible objects is a game employed by teachers in order to ‘connect practices in elementary numbers to play’ (εἰς παιδιὰν ἐναρμόττοντες τὰς τῶν ἀναγκαίων ἀριθμῶν χρήσεις, 819b–c).Footnote ⁶ The practice of λογιστική is a particularly apposite referent of the preface’s comment, in other words, since it is the kind of mental training, on Plato’s authority, that was engaged in by children.Footnote ⁷

Given the place of λογιστική at the lower end of the educational ladder and the comments of the preface, scholarship has tended to approach the arithmetical poems within the context of the history of mathematics and of mathematical education.Footnote ⁸ As will become clear with discussion of specific poems, there is an awareness of the poetry’s potential pedagogical function, and this chapter will show that the dialogue between number and poetry was one operating in an educational frame at least from the time of the Metrodoran collection. Equally, the literary influences on the arithmetical subjects of individual epigrams are various, and their form cannot be explained only as the result of a schoolroom context. The CP demonstrates that the intersection of arithmetic and poetry occurred already in the Hellenistic period, anticipating the poems in the Metrodoran collection by at least three centuries (for the estimated dates of the poems and the collections see below). It supports the assertion that these later arithmetical poems need not be aimed solely at educating readers and that poems containing arithmetic could be refined literary products. Indeed, a recent flurry of interest in studies by Simonetta Grandolini, Jenny Teichmann and Jan Kwapisz has elucidated the literariness of the arithmetical poems.Footnote ⁹ These largely philological studies have examined the constitution of the poems and their scholia and highlighted the sophisticated – even allusive – imagery and language that they contain. Building on that trend, this chapter seeks to analyse the poems more fully – individually and as a collection – and to provide a clearer cultural context for their intertwining of arithmetic and aesthetics.

I proceed in four sections. In the first section, I offer an overview of the types of poems found in the Metrodoran collection and provide detailed study of select compositions. I pay close attention to the strategies for placing arithmetic information in poetry and the extent to which they rely on recognisable verse forms. That is, the first section outlines a literary archaeology for the arithmetical poems. I then consider a series of novel compositions by Ausonius and Optatian Porphyry in order to situate the poems’ workings within the wider late antique literary landscape and to identify a shared practice of involving the reader in the construction of the poems’ meaning and of setting numbers in a literary form as means of displaying one’s cultural capital. My claim will be that they circulate in a context where arithmetical ability could be flaunted effectively by converting numbers into numbered aspects of the cultural and literary past. In Section 3, I turn to the arithmetical poems as a collection and propose that their arrangement and framing aims to present the poems as handed down the generations and central to the educational process. If the second section underscores the notably late antique nature of the arithmetical poems, then the third section shows that the editor of the collection figured the intertwining of literary and arithmetical learning as a highly conservative operation within the Graeco-Roman tradition. Section 4 concludes the chapter by looking to the later Byzantine incorporation of the collection into AP 14. Even at the ‘end’ of the tradition, it will be seen, there remains an awareness of the literary potential of arithmetic in verse.

4.1 An Archaeology of Arithmetical Poetry

This section examines the literary genres which the composers of arithmetical poems develop. My aim is to show how the arithmetical poets read these earlier works and genres as already containing the seeds of arithmetical operations in poetry and built on these models in versifying their own arithmetical challenges.

I begin with an epigram that not only poses a mathematical challenge: it is about a mathematician.

This is a neat composition employing a number of epigrammatic motifs. A deictic identifying the tomb in front of the reader is common in funerary epigrams, as is the emphasis on finality (cf. τέρμ’ … βίου) placed in the final position in the epigram. The exclamatory ἆ μέγα θαῦμα has an equally strong pedigree in the epigrammatic tradition.Footnote ¹¹ The use of τηλύγετον brings an epic colour to the poem, although it is a term which is often considered to be ambiguous in meaning.Footnote ¹² However, the description of Diophantus’ son as τηλύγετον and the fact that something seizes the ‘measure of his life’ (μέτρον … βιότου, 14.126.8) recall the description of the Eleusinian Demophoon in the Homeric Hymn to Demeter. He is a ‘late-born’, τηλύγετος child of Metaneira (Hymn. Hom. Cer. 164) whom his sisters hope the disguised Demeter will raise in their house so that he might reach ‘the measure of youth’ (ἥβης μέτρον, Hymn. Hom. Cer. 166). Demeter attempts to deify the child in her care until Metaneira spies her and halts the attempt, after which, in some versions of the myth, the child dies.Footnote ¹³ This background is certainly not necessary to an appreciation of the poem, although being aware of the echo would elevate the status of Diophantus’ child and make his death a matter of divine and epic significance, while at the same time marking a grim contrast between Demophoon, who is spared by Demeter, and Diophantus’ child, who is not. But the hymn was also an important model for funerary epigrams and especially for young women, who are often likened to Persephone snatched in her prime.Footnote ¹⁴ The author of this arithmetical poem follows in that tradition but draws poetic language instead from the characterisation of the male child in the hymn.Footnote ¹⁵

The poet also makes a play with language. He provides an etymological interpretation of Diophantus as ‘conspicuous (cf. φαίνω) because of Zeus (cf. Διά, Διός, etc.)’: θεός governs both ὤπασε and ἐπένευσεν, actions that are associated with Zeus, and the providing of a marriage ‘light’ or ‘torch’ could imply that the god is making Diophantus manifest in some respect. He may also be offering a further pun on the fact that λογιστική traditionally dealt with the division of apples or sheep, both μῆλα in Greek; here the poet uses the same word with another meaning: ‘cheek’ (LSJ s.v. μῆλον II.2).

Thematically, this is not the first epigram to consider mathematicians in connection with their mathematics, but all others that are extant have a geometrical focus.Footnote ¹⁶ Unlike many Greek mathematicians, however, Diophantus’ focus in his Arithmetica was on arithmetic and in particular on determinate and indeterminate linear and quadratic equations of the kind also employed by Archimedes in the Cattle Problem. In this poem, though, the author has provided a sufficient number of equations to be able to identify the unknowns. The poem thus embodies the intertwined nature of Diophantus’ life and arithmetical interests, following a tradition that can already be seen, for example, in two epigrams on the scholars Philetas and Eratosthenes, where their deaths are closely connected to their intellectual activities.Footnote ¹⁷ The combination of epigrammatic style and Diophantine equations allows his life and learning to be exemplified in just five couplets, where the μέτρα and τέρμα of his life converge.Footnote ¹⁸

In terms of the deeper literary history reflected in the epitaph on Diophantus, and others with a funerary subject matter (14.123, 128 and 143), the poet has exploited a connection that underlies countless compositions. Number and enumeration relating to age are, unsurprisingly, generically determined in funerary epigrams. For example, an epitaph on a fourth-century marble Attic lekythos describes the deceased Kerkope as ‘numbering nine decades of years in old age’ (γήραι ἀριθμ[ή]σασ’ ἐννέα ἐτῶν δεκά<δ>ας, CEG 592.4). In a second-century bce inscription from Smyrna the length of one Dionysius’ life is a particular focus: ‘You will find the length of my life by counting seven decades from the years and a small bit in addition’ ἑπτά που ἐξ ἐτέων δεκάδας καὶ βαιὸν ἐπόν τι | εὑρήσεις ἀριθμέων μῆκος ἐμῆς βιοτῆς· (SGO 05/01/38 1–2). All the key terms used to enumerate the deceased’s age can be seen in the epigram.Footnote ¹⁹ What is more, it both provides the deceased’s age and figures the reader as enquiring after and calculating his lifespan: epigrammatic enumerations were as much an interest for the reader encountering a grave site as those commemorating a loved one.

Certainly, enumeration of objects occurred elsewhere in the epigrammatic tradition: victories were counted and dedications inventoried.Footnote ²⁰ Yet the idea that sepulchral epigrams were particularly oriented to provide a reckoning was at least well-known enough in mid-first-century ce Rome for Philip of Thessalonica to develop it: ‘everyone once counted Aristodice a proud mother since six times she had thrust away the pain of labours …’ (ἠρίθμουν ποτὲ πάντες Ἀριστοδίκην κλυτόπαιδα | ἑξάκις ὠδίνων ἄχθος ἀπωσαμένην, 29.1–2 GP = AP 9.262.1–2). The interrelation of tombs and tallies can be seen most clearly in the Milan Posidippus.Footnote ²¹ The section of the collection (provisionally) entitled the ἐπιτύμβια (lit. ‘things upon tombs’) variously explores the notion of keeping count. The section may well open with the fantastic age of one hundred: ἡ ἑκατ[ (42.1 AB), just as Onasagoratis is at 47 AB: ‘at the age of one hundred, the people of Paphos deposited here the blessed offspring of On[asas] in the [fire-devoured] dust’ (ἣν ἑκατονταέτιν Πάφιοι μακαριστὸν Ὀν[ασᾶ] | θρέμμα πυ[ριβρώτ]ωι τῆιδ’ ἐπέθεντο κόνει, 5–6).Footnote ²² Similarly, the woman praised in 45 AB ‘was eighty years old, but still capable of weaving the [delicate] warp with her shrill shuttle’ (ὀγδωκοντ[αέτις μέ]ν̣, ἔτι κρέξαι δὲ λιγε[ίαι] | κερκίδι λε[πταλέον] στήμονα δυναμ̣έ̣[νη, 3–4), as is Menestrate at 59.1–2 AB. Such successful aging is poignantly contrasted with the youths who do not survive: Hegedike who was only eighteen (ὀκ[τωκαιδε]κέτιν, 49.3 AB) and Myrtis who was ten (τὴν δεκέτιν Μυρτίδα, 54.2 AB). The deceased’s lives are also measured by the children they produce: the anonymous mother at 45 AB ‘saw the fifth crop of daughters’ (θυγατέρων πέμπτον ἐπεῖδε θέρος, 6) and Menestrate (59.3 AB) and Aristippus (61.6 AB) are both blessed with numerous grandchildren. Onasagoratis is a wonder of fecundity, and the rhythmically dactylic third line of the epigram tots up her tots: ‘the group is four times twenty; [she], in the hands of her eighty children …’ (τετράκις εἴκοσι πλῆθος· ἐν ὀγδώκοντ’ ἄ̣[ρα] παίδω[ν] | χερσὶ, 47.3–4 AB ). Already in the Hellenistic era there is a keen awareness that enumeration is a mode of accounting for life particularly suited to funerary epigram.

The numbered nature of time and its progression, as opposed to a lifespan, also finds a place in Posidippus. Poem 56 AB describes an unnamed Asiatic woman who gives birth five times (πέντε, 1), who ‘died during the sixth labour’ (ἕ]κτης δ’ ἐξ ὠδῖνος ἀπώλεο, 3) and whose infant died ‘on the seventh day’ (ἐν ἑβδομάτωι … ἠελίωι 5). The question of causality hangs uneasily over the sequence and the extent to which it means anything: there is an unclear connection between the sixth labour and the infant’s death on the seventh day. In different numerological contexts the number seven was connected with significant changes within the body and was known as an unproductive number.Footnote ²³ Enumeration underscores a dread sense of the natural, arithmetical inevitability of things.

The passage of time is a recurrent interest in the Metrodoran collection as well beyond the epitaph for Diophantus: how long it takes women (14.134 and 142) or bricklayers (14.136) to complete tasks and how much time has passed according to astrological phaenomena (14.140–1). The most basic form of time calculation also finds a place.

The tradition must be early since Posidippus composes an epigram that describes, and is represented as accompanying (see the deictic τοῦθ’, 52.1 AB), a sundial which the deceased father Timon has set up for his daughter Aste.Footnote ²⁴ The closing makes the father’s intention clear and touching: ‘so that she might measure the beautiful sun through many a year’ (σωρὸν ἐτέων μέτρει τὸν καλὸν ἠέλιον, 52.6 AB). Following those lives spanning a century mentioned earlier on in Posidippus’ collection, the reader is asked here, together with the youthful addressee (cf. κούρη at 5), to reflect on the much shorter and perhaps more precious measures of a human life. A keen focus on not only the age of the deceased, but also the day and hour at which they died, is evidenced by numerous Latin inscriptions that detail specific horae.Footnote ²⁵ A further Greek example focuses on life, instead of death.

The epigram is preserved in the Palatine Anthology. There is a probable reference to the epigram on a sundial at Herculaneum (cf. IG 5862), which suggests that the epigram or something like it was known already by the mid-first century ce. It interweaves literary and numerical thinking by employing the same numerical-cum-literary reading practice explored in Chapter 2. The epigram explains that the seventh through tenth hours, when written in Greek numerals (ζ, η, θ, ι), can be interpreted as the imperative ζῆθι. The two poems in the collection above are building on the long tradition of epigrams on sundials toying with epigrammatic and time-keeping conventions, but they innovate by taking the accounting seriously.Footnote ²⁶

A further genre that employs enumeration is sympotic epigram, encapsulated by another Posidippean epigram representing the arithmetical Realien of the symposium.

Posidippus presents the situation numerically: the amount of wine, the number of guests, the time of the party. Time, as I have already noted, was a theme turned to the advantage of arithmetical exercises, and the same is true of the other factors. The amount of wine at a symposium was understood early on to be regulated by number. For Posidippus, the proportions of wine mixed with other ingredients elsewhere served as an image for his range of literary influences: ἕβδομον Ἡσιόδου, τὸν δ’ ὄγδοον εἶπον Ὁμήρου | τὸν δ’ ἔνατον Μουσῶν, Μνημοσύνης δέκατον (‘The seventh [measure] of Hesiod, the eighth I say is of Homer, the ninth of the Muses and Mnemosyne the tenth’, Posidippus 140.5–6 AB = AP 12.168.5–6). This undoubtedly had a programmatic function within his own collection, given that other poems draw on sympotic themes in introducing epigram collections.Footnote ²⁷ Closer to the time of the arithmetical epigrams, Ausonius’ Riddle of the Number Three (Griphus tenarii numeri; more on which below) underscores the orderliness that numbers gave to sympotic proceedings and the arithmetical extremes to which that might be taken: ‘drink thrice, or three times three … [or] nine times uneven three to complete the cube!’ (ter bibe uel totiens ternos … imparibus nouies ternis contexere coebum, Auson. Griph. 1 and 3).Footnote ²⁸ If three is the numerical rule to follow, why stop at nine: ‘three cubed’ drinks also works! Beyond the world of poetry, arithmetic at the symposium does not escape the interest of Athenaeus. In Book 15 of his Dinner Sophists (Ath. 15.670f–671a), he discusses the division of apples and wreaths at symposia not only in language that suggests he has mêlitês and phialitês numbers in mind, but with specific reference to Plato’s discussion of arithmetical games in education (Laws 819b–c) discussed at the beginning of this chapter. In addition to the influence of Posidippus’ sympotic epigrams, that is, ‘sympotic calculation’ remained an interest for the intellectual figures at – and readers of – Athenaeus’ literary dinner.

Sympotic calculations are found among the arithmetical poems. A notable development of Posidippus’ calculation of guests is observable in the following epigram.

The epigram draws on a pre-existing dialogue between funerary and sympotic themes, making the connection explicit in verses 3–4 and by exploiting the bivalency of κείμεθα (5; cf. Simonides el. 102 Sider). In terms of content, the identity of Antiochus is unknown, but the scene is familiar. It recalls the story of Simonides’ presence at a feast hosted by his patrons the Scopadae and his surviving the collapse of the banquet-hall when the Dioscouri appear and request his presence outside the building.Footnote ²⁹ According to Cicero and Quintilian, that story was used to explain Simonides’ ‘invention’ of mnemonics, since he was subsequently asked to remember who had been at the banquet and where they were sitting, although in all likelihood it is a biographical fiction.Footnote ³⁰ This epigram rehearses the basic idea of the story, although in order to exemplify a different sort of mental dexterity. The epigram does not ask the reader to remember who was at the banquet, but to do the kind of sympotic summing seen in Posidippus’ epigram and calculate how many dined and died at the dinner. Accounting for the dead is itself an aspect of Simonidean poetry, such as in his epitaph for all those who died at Thermopylae: ‘once, four thousand from the Peloponnese fought against 3 million’ (μυριάσιν ποτὲ τῇδε τριηκοσίαις ἐμάχοντο | ἐκ Πελοποννάσου χιλιάδες τέτορες, 9 Sider = Hdt. 7.228). The overall effect is thus to present counting as an activity important for memory. The epigram presents enumeration as connected to the memorialisation of the war dead as seen in funerary inscriptions, but it also offers an explanation of that activity’s origin by drawing on the recognisably Simonidean narrative that provided the origin of mnemonics. It employs the sympotic context to reposition the aetiology of commemoration, as well as its recognising and identifying of the fatalities, closer to the practice of arithmetic.

Beyond the Palatine Anthology, there survives another arithmetic poem with a sympotic setting, and it is to be found in Diophantus’ Arithmetica. There are six books of the Arithmetica extant in Greek and a further four in Arabic; the order is thought to be the first three Greek books, then the four in Arabic, followed by the final three Greek books.Footnote ³¹ At the end of the fifth Greek book there is an epigram that versifies an arithmetic problem.

It is not the work of Diophantus himself: the Arithmetica otherwise exhibits little in the way of literary flourishes besides the introductory address to Dionysius, an orientation for the reader not uncommon in mathematical treatises.Footnote ³³ I think it is safest to consider it a later composition interpolated into the text which reworks the prose arithmetical problem into verse, and for my present purposes a poetic response to his arithmetic inserted into the Arithmetica only adds to the picture of Diophantus’ poetic reception.Footnote ³⁴ It is clear from the scholia to the Palatine Anthology that Diophantus was an important source for resolving the poems in Book 14.Footnote ³⁵ So too, whether it was composed specifically for its place in the Arithmetica or taken from elsewhere, the interpolation of this epigram likewise shows an arithmetical poem being read together with Diophantine mathematics.

The epigram draws on a range of sympotic themes. The reference to the wine-mixer being ordered by his fellow sailors (ὁμοπλοῖσι, 2), if this is the correct reading,Footnote ³⁶ leans on a well-trodden equation of symposiasts as sailing together in a ship.Footnote ³⁷ The central problem is working backwards from the mixing of two wine measures (χόες) that were bought for different prices. The mixing of wine is a common theme in sympotic epigram, as Posidippus attests; mention of the units consumed also occurs (cf. Hedylus 3.2 HE = Ath. 11.486b2 and 6.2 HE = Ath. 11.473a5). Likewise, the commercial aspect of buying the wine recalls shopping-list epigrams recounting transactions (Asclepiades 25.9 HE = AP 5.181: λογιόυμεθα, ‘we will reckon’; 26.3 HE = AP 5.185.3: ἀριθμήσει δέ σοι αὐτός, ‘he [the fishmonger] will count them himself’). A further important sympotic resonance is the speaker’s concluding address to a παῖς to carry out the calculation. The request brings to mind sympotic addresses to a youth functioning as wine-pourer, for example Anacreon’s command ‘come now, bring to us the bowl, o youth’ (ἄγε δή, φέρ’ ἡμίν, ὦ παῖ | κελέβην, fr. 33 Gentili). Given the nature of the request, though, the epigram is also characterising the symposium as a site of intellectual competition and education. Challenges were set to test one’s cultural prowess: in the game of skolia symposiasts could each be required to contribute a verse to a song;Footnote ³⁸ they could probe each other’s knowledge of, say, Homer;Footnote ³⁹ or they could be interrogated about which fish is best in which season.Footnote ⁴⁰ Equally, the symposium in Archaic and Classical Greek society was where younger elite males were expected to absorb Greek culture as well as to learn how to behave, and that idea lasted well after education became more formalised outside of the dining room.Footnote ⁴¹ In this respect, the speaker offers a sympotic challenge to a younger participant as a test of his educational progress in arithmetic. The epigram stages a youth being put on the spot and asked to calculate the number of wine measures in total just before they would be serving up the wine to the attendants: even complex arithmetic is part of one’s sympotic acculturation.

As well as integrating numbers into various generic forms, there are poems in the collection that take a playful approach to the content of tradition. The single couplet of AP 14.12 looks back to a figure more well known from Herodotus’ Histories: ‘Croesus the king dedicated six bowls weighing six minae, each one heavier than the next by one drachma’ (ἓξ μνῶν ἓξ φιάλας Κροῖσος βασιλεὺς ἀνέθηκεν | δραχμῇ τὴν ἑτέρην μείζονα τῆς ἑτέρης: the first bowl weighs 97.5 drachmas). Herodotus had surveyed Croesus’ dedications to Apollo at Delphi, which included two large bowls – one of gold, one of silver – weighing many talents and minae (Hdt. 1.51.1–2). He gives both the geometric form of solid gold ingots and their total – ‘he made the longer sides six palm-lengths, the shorter sides three palm-lengths and the height one palm. Their number was one hundred and seventeen’ (ἐπὶ μὲν τὰ μακρότερα ποιέων ἑξαπάλαστα, ἐπὶ δὲ τὰ βραχύτερα τριπάλαστα, ὕψος δὲ παλαστιαῖα, ἀριθμὸν δὲ ἑπτακαίδεκα καὶ ἑκατόν, Hdt. 1.50.2) – and their weight in talents – ‘four of them were refined gold, each weighing two and a half talents, the others ingots were of white gold, with a weight of two talents’ (ἀπέφθου χρυσοῦ τέσσερα, τρίτον ἡμιτάλαντον ἕκαστον ἕλκοντα, τὰ δὲ ἄλλα ἡμιπλίνθια λευκοῦ χρυσοῦ, σταθμὸν διτάλαντα, Hdt. 1.50.2). Moreover, he also provides a little calculation of his own when accounting for the solid gold lion which weighed ten talents that Croesus dedicated but which was burnt in a fire at Delphi: ‘and now it lies in the treasury of the Corinthians, but weighs only six and a half talents, for the fire melted away three and a half talents’ (καὶ νῦν κεῖται ἐν τῷ Κορινθίων θησαυρῷ, ἕλκων σταθμὸν ἕβδομον ἡμιτάλαντον· ἀπετάκη γὰρ αὐτοῦ τέταρτον ἡμιτάλαντον, Hdt. 1.50.3). Herodotus had already demonstrated that one needs arithmetical acumen to count up Croesus’ gifts, and this poem develops that numerically exacting survey to offer a more challenging account of Croesus’ ‘ever growing’ (cf. τὴν ἑτέρην μείζονα τῆς ἑτέρης) riches.

Two further poems revolve around the number of Muses, who divide apples among themselves. In one, the Graces share apples with the Muses (AP 14.48). It asserts the intrinsic numerical nature of the goddesses even though, as Bonnie MacLachlan’s study on the Graces and Tomasz Mojsik’s on the Muses have shown, their number varies depending on the ancient tradition and on the choices of each cult.Footnote ⁴² In the other, the setting and language bring to mind two parallel literary themes, with Eros complaining to his mother Aphrodite that the Muses have stolen his apples (Πιερίδες μοι μῆλα διήρπασαν, AP 14.6.3). Although late in the tradition, this recalls the use of apples in contexts of declaring one’s love and more specifically of the apple of discord that ultimately precipitated the Trojan War, which according to Colluthus Aphrodite wanted for her Erotes (De rapt. 67). Two others in the collection have apples apportioned not by the Muses or Graces, but by the Bacchants Agave, Ino, Autonoe and Semele (14.117–18). There is humour in replacing their famous sparagmos of Pentheus with a different, less lethal kind of ‘dividing up’, and this replacement is thematised through the similar-sounding μῆλα (‘apples’) and μέλη (‘limbs’). The fact that Semele is included in both poems – while dead during the events of Euripides’ Bacchae (cf. 1–63) – places this particular apportioning prior to the fatal events that conclude the play: even before Dionysus’ arrival, that is, Theban women knew well how to divide things between themselves.

As the Cattle Problem so clearly demonstrates, composing arithmetic in verse went hand in hand with searching the literary past for a suitable image or images through which to express the manipulation of numbers. The arithmetical poems in AP 14, it should now be clear, enact the same sort of excavation of traditional genres and content, in order to furnish their poems with the sensible bodies – the ‘stuff’ – of logistic that must be calculated. To put it another way, this section has shown that producing arithmetical poetry involved a specifically numerical reception and (re)reading of the earlier tradition.

4.2 The Cultural Capital of Calculation

The preceding section has demonstrated that, at the level of individual poem, the result of packaging arithmetical content in poetic form is a trend of reading pre-existing genres and motifs as containing the seeds of arithmetic. That is, the intent to cultivate mathematical dexterity through poetry pushed authors of the poems to reinterpret and reuse traditional literary forms and to reify their numerical aspects. Late antique poetry is now a burgeoning area of scholarship, with numerous studies seeking to reappraise its poetry as creative reactions to changing literary and cultural contexts and not as belated and derivative show pieces palely imitating earlier models.Footnote ⁴³ This section thus aims to provide a wider intellectual context for the arithmetical poems and how they might have functioned within a late antique literary culture. I look to the Latin poetry of Late Antiquity and its reflections on number in poetry; the analysis is offered as comparative material informing a reading of the Greek arithmetical poems: I am not claiming that they were composed with knowledge of the following Latin works. In terms of the level of mathematics, too, there is nothing comparable, but the poems should nevertheless be understood as constructing a recognisably late antique mode of engagement for their readers as well as being representative of a wider trend of incorporating arithmetic within displays of poetic novelty and learning. Arithmetic finds a place within poetry, I propose, as an additional means for gauging the cultural capital of both educated elite composers and readers.

First, however, it is worth locating the arithmetical poems’ context of production. Their common thread is the numericalisation both of pre-existing literary forms and of figures or objects from the literary past. This is a strategy of fitting calculations into verse that arose, inter alia, with Archimedes’ Cattle Problem. The difference, though, is both the lack of surrounding cultural historical context, as there is in the case of the Cattle Problem and its exchange between two famous intellectuals, and the lack of a broader poetic project into which the poems fit, as there is, for example, in the passages from Lycophron’s Alexandra or Hesiod’s Melampodia discussed in Chapter 3. A parallel for the arithmetical poems’ reworking of earlier genres and topics as well as their self-contained nature can be identified in the wider educational curriculum. The preserved rhetorical handbooks or progymnasmata detail the literary education of the imperial student, providing a series of different exercises in the art of speaking and writing; these included how to deploy anecdotes, recount mythical narratives and fables, offer arguments for and against a proposition and deliver encomia and invective. One of the later exercises to be completed is prosopopoeia, the personification of an object or a person from history or myth.Footnote ⁴⁴ The student would have to compose a response in verse or prose to such questions as ‘what words would Cyrus say as he attacks the Massagetae?’ (τίνας ἂν εἴποι λόγους Κῦρος ἐλαύνων ἐπὶ Μασσαγέτας, Aelius Theon 115.17–18 Spengel) or ‘what words would Andromache say to Hector?’ (τίνας ἂν εἴποι λόγους Ἀνδρομάχη ἐπὶ Ἕκτορι, Hermogenes 15.7 Spengel). The exercises not only asked students to dwell on the material of the inherited literary tradition, they asked them to recompose it, to produce compositions matching the style and metre of the original but with new things to say. This educational background is part of the impetus for the return to Homeric subject matter and the Homeric voice or, say, to rhetorical performances in the style of the Attic orators, while at the same time offering something novel.Footnote ⁴⁵ Yet many short verse compositions survive in the Palatine Anthology, exemplifying what such exercises might produce, and they may have once formed a collection (for example, AP 9.457–80). In their reliance on the forms and models of the past as well as in the revivification of mythical or historical figures, the arithmetical poems echo the strategies of these progymnasmata. Their rehearsing and reconfiguring of the literary past not only produces mythical ‘what would X say to Y’ scenarios, it reaches across disciplines to incorporate aspects of mathematical education too.

The parallel of the progymnasmata proposes a post-Hellenistic context of production for the arithmetical poems. In terms of their context of reading and of reception, I think that it makes most sense to view them as a late antique development. To exemplify what is particular to engagements with the reader in the poetry of Late Antiquity, I want to consider two Latin works that underscore the importance of arithmetic for conceptualising the form and interpretability of a poem. Ausonius’ preface to his Cento nuptialis (Wedding Cento) is a key passage of late antique literary theory, and it rests on an explicitly arithmetical comparison. A cento is a poem stitched together from lines of existing poetry, and in this case the poem is a bricolage of Vergilian half-lines reassembled in order to describe a night of nuptial consummation. In introducing the poem to his correspondent Paulus, he outlines the practice of composing centos.Footnote ⁴⁶ His explanation exemplifies the composing of centos with the Greek game called στομάχιον (‘Belly-teaser’), a tangram in which a square cut into fourteen polygons can be rearranged to create many other figures (such as a ship or a gladiator).Footnote ⁴⁷ It was also explicitly theorised by Archimedes, who dedicated a treatise to the topic, a single fragment of which has been recovered from a palimpsest.Footnote ⁴⁸ Whatever the precise focus of Archimedes’ treatise, it undoubtedly influenced the later use of the image for underscoring the possibilities of combination.Footnote ⁴⁹ Given the close relationship I argued for in Chapter 3 between mathematics and Homeric epic in a work by Archimedes, Ausonius’ choice of the στομάχιον to describe his own use of epic may have been informed by a now lost literary or cultural implication of the calculations mentioned in the treatise.Footnote ⁵⁰ As Fabio Acerbi has shown, moreover, combinatorics was certainly a matter of theory by the time of Hipparchus (second half of the second century bce), who criticised the Stoic thinking of Chrysippus and his calculation of the possible claims that could be made given ten ‘assertables’ connected by a conjunction such as ‘and’.Footnote ⁵¹ For my purposes, it is sufficient to note that combinatorics was applied in the domain of language and the construction of sentences from the Hellenistic period. Ausonius’ example of the στομάχιον, while not requiring the application of arithmetic, attests to an arithmetic understanding of compositional possibilities and of the construction of new meanings out of canonical forms.

An additional example that again functions with the idea of compositional combination will set in high relief the role of arithmetic in conceiving of a poem’s interpretability. The early fourth-century poet Publilius Optatianus Porfyrius – commonly known as Optatian – is a late antique poet who is now gaining his fair share of scholarly interest. He was a composer of carmina cancellata (‘latticed poems’), poems in grid-like patterns that preserve hidden sentences and verses at their edges and in variegated patterns across the gridded page.Footnote ⁵² For example, poem 16 presents 38 hexameters comprising a panegyric to Constantine with an acrostic that likewise extols Constantine as ruler and inheritor of Augustus’ mantle. Three further mesostichs also run vertically from the top to the bottom of the grid starting from the tenth, nineteenth and twenty-eighth letter of each verse. They produce a string of letters that, when converted into Greek, announce instead Christ’s bestowing of power on Constantine.

Rather different from these carmina cancellata is poem 25.

The poem develops a Hellenistic model of composition first attempted by Castorion of Soli (SH 310 = Ath. 10.454f), in which the feet of his Hymn to Pan can be arranged in any order, and where the content of the words also advertises the fact. With Optatian’s poem, the reader is freer since the words rather than the feet can be reorganised. Thus, these four verses of five words each can be combined in a truly staggering array of combinations.Footnote ⁵⁴ The poem sets an arithmetic challenge to the reader: in just how many ways can the words be rejigged (confusis … uerbis, 4)? There were attempts – possibly dating back to the fourth century ce – to calculate the poem’s potential permutations, and the question is equally alive in modern scholarship (just over 39 billion variations, according to one commentator).Footnote ⁵⁵ Optatian’s four-line poem is a textual Rubik’s cube that outdoes Leonides’ isopsephic epigram by concealing not a numerical account, but an innumerable amount of further poetry. Poem 25 also outdoes the Cento nuptialis: it is the poetic instantiation of the στομάχιον game, since it provides the ‘square’ of words that – unlike the Cento – can be rearranged any way the reader likes. Fascinating, in this respect, is that in some manuscripts the combinatory challenge has led the copyist to try out the permutations, scaling the poem up as far as 84 verses.Footnote ⁵⁶ A later reader has attempted to answer the implicit question exhaustively. This evidences the imbrication of numerical and literary appreciation that confronts readers of the poem; the copyist – and indeed the scholiast – makes a claim about the numerical extent of the poem’s reconfigurability.

Ausonius and Optatian’s explorations of poetic form establish that arithmetic had a role in conceptualising the possibilities and the limits of literary innovation. Their importance for understanding the arithmetical poems lies in the connection between the arithmetic and the deep involvement of the reader in the construction of meaning. Ausonius explains this through a mathematical image, and the readers of Optatian’s poem 25 clearly aimed to calculate the number of meanings possible. The arithmetical poems are neither as self-conscious nor as theoretical in their comments. Nevertheless, they demand the work of the reader to make sense of the poem and get beyond the surface of the expressed ratios, just as a reader must work to configure the many meanings of Optatian’s chequerboard carmina cancellata and to appreciate the Vergilian undercurrent of Ausonius’ Cento. In a seminal study of Hellenistic epigram, Peter Bing argued that they were often written in such a way as to require the reader to supply further information about context, addressee or imagined location not made explicit, an effect which he called ‘the game of supplementation’ (Ergänzungsspiel).Footnote ⁵⁷ Given that many arithmetical poems are influenced by Hellenistic epigram, this readerly demand may be part of the genre’s adaptation to arithmetical content. A similar process is at work: in both cases the reader must take the epigram’s contents and out of that construct a plausible scenario beyond what the poem describes on the surface. While number and epigrammatic poetry thus have a long interrelation, the notably late antique development of the arithmetical epigrams is the extent to which the experiment with form is taken. The presence of numbers on epitaphs has become a full series of calculations that require computing, just as Optatian’s poem outdoes earlier ‘reconfigurable’ poems in its possible permutations.Footnote ⁵⁸ The arithmetical poems belong to Late Antiquity, simply put, in their increased reliance on the role of the reader in uniting the individual components of a text into a meaningful whole.

A further operative aspect of poetry in Late Antiquity is the construction of innovative poetry and the display of virtuosic skill using the material building blocks of the literary tradition: Vergilian lines are cut and pasted to form Ausonius’ Cento, while Optatian’s poems draw on numerous canonical works which disintegrate and reform in front of the reader’s eyes.Footnote ⁵⁹ The arithmetical poems, by contrast, do not work at the level of the material text but with the constituent objects described within it. However, as I noted in the previous section, these topics themselves draw heavily on the heritage of various literary forms. The matter of the tradition itself becomes the objects with which the poets demand the reader grapples and engages. Fortunately, the poetry of Late Antiquity also furnishes an example of where tradition and its constituent objects and tropes are treated numerically. A poem that is almost entirely composed out of numbers and enumeration is Ausonius’ Riddle of the Number Three. The poem rings the changes on things existing in threes or nines, under the influence of three cups at the symposium. There has been much discussion about the possible ‘answer’ to the riddle, although I am most persuaded by the proposal that, since the Greek γρῖφος means riddle but also a woven fishing-basket or net (LSJ s.v. γρῖφος A.1; cf. Ath. 10.457c–458a), the title Griphus indexes ‘the dense texture of its literary allusions’.Footnote ⁶⁰ Ausonius is steeped in the numbered-ness of tradition. His prose preface contains multiple references to no less than four canonical Latin authors.Footnote ⁶¹ The Griphus’ composition is figured as the result of drinking, following the style of Horace’s poem (3.19) ‘in which, on account of midnight, the new moon and Muraena’s augurship the inspired bard calls for three times three cups’ (in qua propter mediam noctem et nouam lunam et Murenae auguratum ternos ter cyathos attonitus petit uates, Praef. ad Griph. 16–17). Two further references to Horace, in significant third positions (Satires 1.3 and Odes 3.1), make for a three-pronged allusion to the Augustan lyricist, supported also by an opening reference to Catullus c. 1, which plays with the idea of a three-book collection (see Chapter 2, Section 3).

While the preface establishes that talking in threes is a habit inherited from canonical authors, the verses aim to affirm the three-ness of various cultural institutions. As in the arithmetical poem on the Muses and Graces, Dunstan Lowe has noted that Ausonius in the Griphus asserts the numerical nature of the Muses as nine (22), despite elsewhere thinking of them as either three or eight (Epist. 13.64), and that he numbers the Sibyls at three, although that number is nowhere else attested.Footnote ⁶² Ausonius’ strategy amounts to an attempt to collect and order cultural data from the past and to regulate it so as to make it manageable. Indeed, Ausonius’ regulatory mode is a key part of the preface. He sent the letter to Symmachus with the expectation that the enclosed poem may be either approved or destroyed (Praef. ad Griph. 11–13), but this is paired with the concern that his original composition has been ‘mutilated for a long time by secret yet popular readings’ (diu secreta quidem, sed uulgi lectione laceratus, Praef. ad Griph. 10–11). Regulation is part of the impetus for preserving the text; literary and cultural artefacts are associated with specific numbers, which Ausonius must protect against the distortions of time and populism: indeed his list of threes does not extend to anything related to the profanum uulgus (‘unitiated crowds’).Footnote ⁶³ He is aiming, not exhaustively but symbolically, to impress the idea of literature and culture’s numerical nature within elite circles and their shared late antique paideia. The Griphus too is an argument for the cultural capital of numbers.

Reading the Griphus in this way makes Ausonius’ allusiveness in the preface particularly piquant. He characterises the original composition of the Griphus as nothing more than ‘a frivolous piece worth less than Sicilian baskets’ and ‘a trifling booklet’ (haec friuola gerris Siculis uaniora … nugator libellus, Praef. ad Griph. 9–10). Yet his claims to mere playfulness belie his referentiality. The second phrase refers back to the Catullan allusion at the start of the preface and Catullus’ opening poem responding to a three-volume history (c. 1.6). The first phrase refers to gerrae, another form of wickerwork that had a metaphorical meaning of nonsense, but it is also modified by Siculus, which makes it a product of the three-cornered Sicily.Footnote ⁶⁴ Ausonius intimates that the composition is certainly playful and may be nothing more than an experiment; but his allusiveness suggests that even in trifling works, reading a little deeper uncovers a whole world of numbers and numberedness.

Of course, the arithmetical poems are more challenging than the Griphus in that its answer – if that is the right word – is not hidden to the reader. Nonetheless, further works do reveal Ausonius’ cultural capital of numbers in action.Footnote ⁶⁵ Epistle 14 addressed to Theon – an otherwise unknown friend – records a gift of thirty oysters and, noting the lack of literary accompaniment, reworks an old letter in return, the poem that follows the prose introduction. The poem is divided into four metrical schemes: hexameter (1–18), iambic (19–23),Footnote ⁶⁶ hendecasyllable (24–35) and asclepiads (36–56). The hexameters introduce the theme of the oysters and list a series of single verses (monosticha, 4) characterising the number: for example ‘as many as the Geryones, if they were multiplied by ten’ (Geryones quot errant, decies si multiplicentur, Epist. 14b.6). In the following iambics, Ausonius characterises the number arithmetically, for example ‘three times ten, I think, or five times six’ (ter denas puto quinquiesue senas, 24). The hendecasyllables describe the sourcing and cooking of the oysters. The focus is on lexical dexterity and ‘a general luxuriance of expression’.Footnote ⁶⁷ In the concluding asclepiads, he notes the excessive length of his writing and commands his pen to stop writing (or the composition be erased), in case the parchment costs more than the oysters. The concluding reflection that the papyrus may cost more than the (presumably free) thirty oysters invokes the relationship – by now recognisable from Part I – between poetic content and the extent required to express it. The humour here is that Ausonius may have overdone his attempt to supply a composition in lieu of one from Theon. This virtuosic piece displays the mythological, mathematical and lexical skills required to be a learned writer. Significant for my purposes is the arithmetical section’s introduction.

These lines mark the transition between the literary and arithmetical characterisation of thirty and mark out the stakes attached to the different types of learning. Unlocking the number through literature requires a knowledge of narratives and an ability to decipher obscurities or riddles, whereas arithmetical calculation alone belongs to the uulgus. Theon is mocked for his size elsewhere (Epist. 16.31), but the imagery in these lines makes a more general point about mental exercise: a mind wrapped in fat (out of disuse) will not be able to deal with the already obscure and wrapped-up descriptions of numbers. It is the expression of numbers through poetry that makes the exercise intellectual and not accessible to the masses, that is, what makes it elite.

Ausonius’ use of ambages – an obscurity or enigma – to characterise his descriptions of the number thirty through literary references provides one explanation for the designation of the Griphus and its three-counting as a riddle. More importantly, however, Ausonius’ distinction provides a parallel for the nature of the arithmetical epigrams. It is my contention that their form is a result of the same sense of the cultural capital of numbers. Their exercising of the reader’s knowledge of, and control over, the numerical aspects of the cultural and literary past is part and parcel of the wider habit of deploying learning competitively. Around a third of the arithmetical epigrams directly invoke mythological topics, while others take on topics such as the constellations (e.g. AP 14.124). But it is not solely about content. As I have demonstrated, almost all wrap their arithmetic in the ambages of a pre-existing poetic form. The possibility of solving the series of simultaneous equations encoded in the arithmetical poems offers readers the opportunity to cash in their own cultural capital, and it is a capital derived from knowing literary tropes, traditions and clichés as much as it is knowing enumerable ‘objects’ or ‘stuff’ of the mythical past. More than an awareness of the numbered nature of the cultural and literary past, though, the poems provide real and serious arithmetic problems to be solved that go beyond Ausonius’ display of factorisation: they are both an arithmetical and literary exercise. Ausonius’ poem explicitly notes, furthermore, that in erudite exchanges, arithmetic retains its currency best when expressed in obscure and circumlocutory language, framed in stories (cf. fabulis). By versifying numerical aspects of antiquity, the poems not only provide a literary and arithmetic challenge for the reader simultaneously, they supply it in a form that also increases the distinction of the author (and solver) within an elite group.

The arithmetical poems are a quintessentially late antique product in that sense, since their insertion of arithmetic into poetry demands the reader’s participation in the construction of meaning, displays the authors’ education and skill in transforming the literary tradition and results in an innovative and experimental poetic form. Accounting for the potential educational context, moreover, does not mitigate this claim. Rather, if they are the product of rhetorical training, then they evidence a practice taking seriously Ausonius’ emphasis on the value of computing in poetry, not to mention providing exempla for the combination of literate and arithmetical learning. Ultimately, though, attempting to distinguish definitively between the function of the poems as either educational exempla or virtuosic show pieces is unhelpful; the two are not mutually exclusive and the poems beyond the Metrodoran collection show that they circulated in multiple contexts. The wider significance of the cultural capital of calculating for which I have argued in this section, then, is that it modifies the literary historical trajectory of numbers in poetry. Whereas the poems of Archimedes and Leonides have often been imagined to be esoteric, peculiar experiments of form that survive only out of curiosity, the view from Late Antiquity is rather different. The arithmetical poems make clear that composing calculations was not only the preserve of mathematicians grappling with the inherent difficulty of incorporating their disciplinary content into verse, but an expectation for educated late antique authors as well as readers.

4.3 Arithmetic Anthologised

At some point after the appearance of the arithmetical poems, they were brought into a collection by the shadowy figure Metrodorus. In this section, I want to trace out the poems’ reception and interpretation as they were anthologised by Metrodorus. I illuminate the dialogue between poems encouraged by their editorial organisation and selection within the Metrodoran collection. I then identify an overarching theme that comments on the nature of the collection and the purpose of the compositions. Following the conclusion of the previous section, I argue that the nature of the Metrodoran collection foregrounds the same sophisticated balance of arithmetic novelty expressed through traditional poetic forms that was operative in individual poems.

First, though, it is necessary to set out the evidence for Metrodorus’ collection. The organisation of the poems within Book 14 may date back to Constantinus Cephalas in the early tenth century ce and is no later than the formation of the codex Palatinus in the middle of that century, the basis for the modern AP.Footnote ⁶⁹ His collection is reconstructed on the basis of a comment in the scholia to Book 14 in the codex Palatinus. At poem AP 14.116 the scholiast introduces the following poems as ‘the arithmetic epigrams of Metrodorus’ (Μητροδώρου ἐπιγράμματα ἀριθμητικά), and this probably extends all the way to 14.146.Footnote ⁷⁰ The collection is accompanied by an intermittent marginal numbering that in all likelihood represents the poems’ order in the Metrodoran collection: AP 14.116 is designated β (2), 117 as γ (3), etc. This coherence is supported by the wording of arithmetical solutions given in a number of scholia which implies that the poems are drawn from a single collection.Footnote ⁷¹ Outside of this section, poems have been found with a marginal numbering that is missing in the core sequence: AP 14.6 and 14.7 are 19 (ιθʹ) and 28 (κηʹ) respectively. They are thus also added to the Metrodorus collection, as are AP 14.2–5 and 14.48–51, which are thematically and stylistically of a piece with the securely Metrodoran compositions. AP 14.1 is not thought to be from the Metrodoran collection.

The arithmetical epigrams share with the wider literary context an immersive participation in the production of significance on the part of the reader. The ‘game of supplementation’, however, could also operate across a poetic collection, in which the arrangement invites the reader to make connections between poems as they navigate through the work. An early example of this editorial ordering is the Milan papyrus of Posidippus, the individual poems of which echo and cap each other, both within and across its thematic sections.Footnote ⁷² Posidippus’ ἐπιτύμβια is again particularly important here, since the theme of accounting operates across the section, contrasting different forms of enumerating and valuing life. Admittedly, the Metrodoran collection and its bounds cannot be identified with the same precision as Posidippus’, recovered from a mummy cartonnage (more or less) intact, but its integration into Book 14 is sufficiently contained to allow for analysis and cautious conclusions.

Preliminarily, poems in the Metrodoran collection evince an order suggestive of editorial placement, with similar poems set in thematic dialogue and close proximity, creating a cohesive anthology playing variations on a theme.Footnote ⁷³ The epigram on Diophantus appears within a sequence of epigrams counting up life and death (AP 14.124–7) and other funerary-themed epigrams ask instead for the inheritance to be calculated from its respective proportions (14.123, 128 and 143). Poems were also connected on the lexical and stylistic level. For example, 14.125 is a funerary epigram for Philinna that asks for the number of her children to be calculated. Philinna is a common enough name to encounter in an epigram, but it is noteworthy that it appears in two earlier epigrams in the collection.Footnote ⁷⁴ Philinna is the name of one of the maidens who divide up the walnuts in 14.116 and 14.120. The shape of the collection parallels a reader’s progress with the mathematically themed events in an imagined Philinna’s life: as a youth she plays with her age-mates and later is buried by her remaining family. Given the epigrams on dividing up walnuts and apples (14.116–20), the description of her offspring as the ‘fruit of her womb’ (καρπὸν … λαγόνων, AP 14.125.2) frames the calculation as following the same rubric as those that began the collection: the topic is new, but the arithmetical process will be the same as before. In a similar vein, 14.120 begins as a poem on dividing walnuts following 14.116, but by the end of the poem it has resumed a focus on the Graces and the Muses, echoing 14.3. Likewise, the dual focus of 14.124 on astronomy and the lifespan of an unnamed man echoes the language of the Diophantus epigram. The child of the unnamed man is also ‘late-born’ (τηλύγετον, AP 14.124.6), he sees his child (and wife) perish (7–8), and then he attains the end of life (βίου … τέρμα περήσεις, 9; cf. 14.126.10 above). Represented as a prediction, though, its future tenses invert the funerary finality with which Diophantus’ life is laid out. There is little to determine which poem has priority. Important rather is the shared language echoing across the collection. It points to an editorial arrangement that expects readers to move through the collection, make connections and read the compositions in a similar manner to other literary anthologies, in addition to possibly extracting a poem for educative or socially competitive purposes.

One further theme in the collection that has (to the best of my knowledge) received no attention is the focus on family relations. It is not just the inheritance for children, the number of offspring or family members that must be calculated. Three extant poems have a more marked sense of familial connection.

The mother asks an initial question which prompts the delineation, and then the child outlines the proportions but does not offer the calculation. Embedded alongside the intertwining of arithmetical and literary and generic allusion is a frame that presents the arithmetical challenge as one exchanged between mother and child, in which the child requires help with resolving the ratios. Since this occurs in both hexameter and elegiac compositions it is reasonable to think that there is an underlying explanation for the shared frame (whether they are the product of multiple authors, or the concerted variation of a single author).

An anonymous poem preserved in an appendix to the Planudean Anthology helps to shed light on this framing and its connection to arithmetical problems.

The poem is recorded as being addressed to Euclid and, although he is not mentioned, this is supported by the final words of the poem: Euclid would be a likely candidate for the title of best geometer in the ancient mind. While the poem is ostensibly a conversation between a mule and an ass, a further operative frame emerges in verse four, which forms a hinge between the set-up and the arithmetic. It heightens the language of lament from the previous two lines and employs the simile used by Achilles to address the petulant Patroclus in the Iliad (16.7), in order to imply a parental association between the mule and ass. The arithmetic, however, is confined to verses 5–6, where there is no language to distinguish it as particularly poetic or to locate it within the framework of a mother-and-child relationship, to say nothing of implicating it as the words of a talking mule. Those two verses are reminiscent of two poems from Book 14.Footnote ⁷⁵

The fact that this sort of arithmetical challenge circulated freely suggests that the poet of the verses addressed to Euclid has surrounded a core arithmetic challenge with lines that imbue (or at least seek to imbue) the arithmetic with a literary quality and contextualise it as an exchange between mother and child.Footnote ⁷⁶ It is external, supporting evidence for an author embedding within the poems their context of use as well as for an author setting arithmetic within the frame of a maternal exchange.

These three arithmetical poems are placed as the second (AP 14.116), third (14.117) and fifth poems (14.3) of the Metrodoran collection. The fifth poem in Metrodorus’ collection thus pointedly varies the theme of the first two: AP 14.116 described walnuts divided by a group of maidens in hexameter, AP 14.117 addressed the division of apples, but this time by Corinthian women in elegiacs, and then AP 14.3 combines the use of hexameter with a return to the topic of apples. Accordingly, three of the five opening poems of the original Metrodoran collection frame the exchange of arithmetical problems as a maternal matter, the third even doing so archetypally in the use of gods, as well as of the eternal child, Eros. The fact that the maternal framing is intentional is supported by the identity of the author or editor Metrodorus. Francesco Grillo has recently shown that it is difficult to identify the Metrodorus mentioned in AP; through a combination of scholarly mistakes and wishful thinking a range of figures have been suggested, but none can be proposed with any degree of certainty.Footnote ⁷⁷ Buffière raised the possibility that the name may be a pseudonym, although he is not explicit why ‘for an author of problems in verse, it would not be unwelcome’.Footnote ⁷⁸ I assume him to have had in mind a *Μετρόδωρος, which characterise the collection as a gift (δῶρον) of measures (μέτρα). That meaning may have been intended on the aural level, but the spelling Μητρόδωρος speaks against it. Nevertheless, Μητρόδωρος already makes sense as a pseudonym playing an etymological name game: just like the arithmetical education framed in the opening poems, the collection itself is a ‘mother’s gift’ (μητρο-, δῶρον).

The focus on the maternal, to my mind, encapsulates the unique nature of the arithmetical poems for which I have been arguing. On a pragmatic reading, since mothers would have been expected to care for infants, the framework of mother–child interactions mirrors the probable reality of early education, and it may imply the pedagogical function of the poems. According to Plato in the Laws, λογιστική is part of their early education, while in the second book of the Republic (377c) he charges mothers with teaching children through (the approved) myths. Arithmetic cloaked in mythical dress would seem to be a particular maternal form of early education. Yet their function only partially explains the maternal framing; many of the poems do not frame their problem as an exchange between two people. As I argued in the preceding section, the repertoire of the elite as perceived by Ausonius included displays of arithmetic (preferably in verse) but also a proficient grip on the numerical nature of the late antique literary and cultural inheritance, and the reader brings these to bear in approaching, solving and appreciating the arithmetical epigrams. Ausonius’ conservative cultural outlook is metaphorised in the maternal framing of the arithmetical poems. The connection between mother and child not only provides a continually valid context in which to place the poems, it also implies an unbroken lineage transmitting the traditions of antiquity to the subsequent generations. Indeed, individual poems in the collection pay close attention to the literary past, looking for legitimation within pre-existing literary forms for their use of arithmetic in verse. That it is a maternal as opposed to a paternal relationship arguably further emphasises the conserving of the tradition unchanged.Footnote ⁷⁹ In each case, moreover, either the child is expected to answer, or they provide only the series of equations before the poem concludes, without the maternal voice resuming. As the reader identifies the proportions embedded in the verses, they take on the role of the child, aiming both to solve arithmetic problems and to discern and construct the underlying meaning of the poem from its constituents.

Thus, the pseudonym Metrodorus figures the cultural exchange across generations in the ambiguity of the μητρο- stem, since it could index either a subjective genitive (the gift the mother gives) or an object genitive (the gift the mother receives). As educational poems they would be given from parents to children, but equally that education can be repaid and reproduced, as demonstrated by AP 14.3, in which Eros gives apples (read perhaps: new compositions) to Aphrodite as a gift. The thematic shape of the Metrodoran collection, in short, associates the interpreting and deciphering of the arithmetic problems with the core pattern of generational cultural transmission and preservation at large. Although the poems are innovative in their reworking of the literary past and integration of arithmetic, the collection presents them as deeply traditional.

4.4 Arithmetical Poetry beyond Late Antiquity

The precise date of the Metrodoran collection is difficult to ascertain, although I have suggested that the compositions’ investment in the past and the involvement they demand from the reader make most sense within a late antique literary context and that this conservative literary approach is also indexed by the form of the collection. In the concluding section of this chapter I want to emphasise that the appreciation of these poems and their editorial engagement does not end with the late antique collection of Metrodorus. Rather, it can be observed in the final stages of the Palatine Anthology’s formation. It, too, exhibits a conscious arrangement of the poems aware of their literary and arithmetical significance.

On the broadest level, it is clear that the arithmetic poems were purposefully set in a book alongside both riddles and oracles. While it might be thought that arithmetical poems fit uneasily with riddles and oracles, there is a deep literary logic to the combination. Consider again Herodotus’ first oracle: it exhibits generic aspects of riddles and of arithmetic. On the one hand, the Pythia’s claim to know the number of the sands – οἶδα δ᾿ ἐγὼ ψάμμου τ᾿ ἀριθμόν (Hdt. 1.47) – ascribes to her numerical abilities, whereas the further adynaton of hearing the dumb (κωφοῦ συνίημι) and the subsequent mixing of expected categories (tortoise and lamb: κραταιρίνοιο χελώνης | … ἅμ᾿ ἀρνείοισι κρέεσσιν) is reminiscent of the paradoxes that riddles offer to their audiences to (re)solve. Numbers are also part of the riddle genre, which can be seen in those collected in Book 14.Footnote ⁸⁰ The similarity of these poems is aided by the shared metrical form: oracles are invariably in hexameter, while many riddles and arithmetic problems are as well.Footnote ⁸¹ An earlier parallel for the mixing of riddles and arithmetic poetry in AP 14 can be found in the collection of Latin riddles by Symphosius, which exhibits the influence of Ausonius’ Griphus in its prefatory material and the many three-line poems.Footnote ⁸² The author of that collection evidently saw a link between Ausonius’ reflection on the numerical aspects of culture and the nature of his riddles. As well as a formal dialogue between oracles, riddles and arithmetic problems, there is also a shared intellectual challenge in that they all require reader interpretation. With riddles and oracles, this usually requires lateral thinking with regards to the description of an object and the unravelling of the poem’s use of, inter alia, metonymy and double meaning, whereas the arithmetical poems require the objects to be treated ‘laterally’ as numbers or ratios.Footnote ⁸³ What binds these generic forms is the involvement of the reader in the construction of meaning and exercising of their intellectual grasp of Graeco-Roman culture. In this sense, they all fall under the category of ‘how children in the past learnt’, as described by the book’s introductory lemma.

In addition to being combined with riddles and oracles, arithmetical poems also take pride of place as the first four compositions in the book. The transmission history of the opening poems of AP 14 and its relation to the opening of Metrodorus’ collection require discussion, since scholarship on this point contains much supposition. The opening poem of AP 14 appears to be attributed to one Socrates by the scholiast, since it is preceded by the lemma Σωκράτους, but the scholiast says nothing more about him. Paul Tannery, in his edition of the works of Diophantus, which included the arithmetical poems, identified the Socrates as an epigrammatist mentioned by Diogenes Laertius (2.47), but again nothing more is known about this figure.Footnote ⁸⁴ Whether the two figures are the same person is a moot point. What there is certainly no external evidence for is that AP 14.1 is the first poem of a collection by this Socrates, as Tannery suggests, nor that his collection shared poems with Metrodorus’.Footnote ⁸⁵ Tannery’s reasoning is as follows. The marginal account that accompanies the core arithmetical poems is lacking for those that open AP 14. Therefore, those poems must have been shared by Socrates’ collection and the copyist must have not wanted to add the further Metrodoran numbering and instead stuck with Socrates’ ordering. The Socratean numbering relies on reading the ordinal designation at the beginning of AP 14 (α at 14.1, β at 14.2, γ at 14.3) as coterminous with the order in Socrates’ collection. Tannery’s proposal was developed by Félix Buffière, who identified AP 14.2 as Metrodorus’ inaugural poem (followed by AP 14.116–18), with AP 14.3 occupying the fifth place in the collection and AP 14.4, like 14.1, belonging to Socrates’ collection only.Footnote ⁸⁶

However, in addition to there simply being no evidence for this collection, nor of an independent Socratean numbering, the marginal numbering of AP 14.6 (κηʹ = 28) and 14.7 (ιθʹ = 19) show that the arithmetical poems outside the preserved core were still identified by their number in the Metrodoran collection. The argument that the prime position of 14.1, and the second position of 14.2 (etc.), reflects the position also in a Socratean collection cannot be proved or disproved given that the numbering in each case is the same. The arithmetical scholia certainly help to determine inclusion in Metrodorus’ collection, but this is not a watertight rule.Footnote ⁸⁷ Nor, importantly, is the inverse – that those without scholia are from the Socratean collection – a necessary consequence. The existence of a Socratean collection ultimately relies solely on the lemma Σωκράτους immediately following the preface to the book: a particularly precarious castle of sand.Footnote ⁸⁸

When the spectre of the Socratean collection is removed, it can be said that the first poem offers no clear signs of belonging to Metrodorus’ collection, but neither does it exhibit anything alien to the collection. Nonetheless, it is my working assumption that it is not part of the Metrodoran collection. AP 14.2–4, however, bear all the hallmarks of being from Metrodorus’ collection, since they are closely related in form and theme and 14.2–3 are even accompanied by arithmetical scholia. Before considering the introductory poem of AP 14 and its programmatic aspects, then, I want to consider AP 14.2–4 and suggest that they have been moved from the Metrodoran collection to the beginning of the book in order also to have a programmatic function. In other words, I am arguing that the later compiler is reading these poems and actively arranging them into a poetic-cum-arithmetical programme.

First is AP 14.2.

The speaking statue explains the ratios of gold given for the construction of the statue which was (presumably) made by Aristodicus. As Jan Kwapisz has brilliantly elucidated, the epigram can be read programmatically, since the contributors are designated as poets and Solon and Thespis are even recognisable figures. Since this was most probably Metrodorus’ opening poem, it self-referentially indexes the collection as formed by the contribution of numerous poets and at the same time represents that act of (editorial) combination as an arithmetical operation.Footnote ⁸⁹ The entire material form of the statue is a gift from numerous poets, and in opening Metrodorus’ collection it would likewise have signalled the collection as a gift. If my argument about the maternal framing of the collection is correct, then Athena as a female goddess who is renowned for her wisdom and knowledge of many crafts makes this a collection that has hypostasised female intellectual prowess as its frontispiece, so to speak. As a virgin goddess she is not a mother of children; her progeny is rather the mathematical abilities transmitted in the collection. In any case, in moving the poem out of the Metrodoran collection to the opening of the book, the editor of AP 14 retains the epigram’s programmatic force and its use of numerical combination to imbue literary significance.

In the case of AP 14.3 there is also recontextualisation at work from the Metrodoran collection into the wider book, but it occurs in tandem with AP 14.4. It is not assigned to the Metrodoran collection, because it lacks accompanying scholia, but is placed in the supposed Socratean collection by Buffière.Footnote ⁹⁰

The poems bear strong similarities. They are framed as a dialogue and conclude with an amount of fifty left for the addressee. As I noted above, the application of λογιστική involved mêlitês numbers (μηλίτας … ἀριθμούς). This could be interpreted as referring to either apples or herds. By setting these two poems side by side, the compiler knowingly alludes to and rings the changes on the debate about what mêlitês numbers in poetry might look like.Footnote ⁹¹

Moreover, the two poems resonate on the metapoetic level when read at the opening of Book 14. In Meleager’s Garland, the epigrams that he weaves into a collection are described in his introductory poem mostly by comparisons to flowers, but also by comparison to fruits such as ‘intoxicating grapes’ (μαινάδα βότρυν, 1.25 HE = AP 4.25; of Hegesippus), ‘sweet apple’ (γλυκύμηλον, 27; of Diotimus) and ‘wild pear’ (ἀχράδα, 1.30; of Simias). Likewise in Philip’s Garland, modelled on Meleager’s, he states that he has formed the collection for the reader by ‘plucking the flowers [for you] from Helicon, having cut the first-growing buds of famous-wooded Pieria’ (ἄνθεά σοι δρέψας Ἑλικώνια καὶ κλυτοδένδρου | Πιερίης κείρας πρωτοφύτους κάλυκας, 1.1–2 GP = AP 4.2.1–2). The apples that Eros takes from Helicon – and which survive the division of the Pierian Muses – stand in for the arithmetical problems that the editor has sourced and that the poem introduces.Footnote ⁹² A similar reading works for AP 14.4, too.Footnote ⁹³ Multiple epigrammatic variations composed on Myron’s cow were subsequently conceptualised as a ‘herd of poems’ in need of rounding up.Footnote ⁹⁴ A further epigram by Artemidorus imagines a collection of bucolic poetry in a similar manner: βουκολικαὶ Μοῖσαι σποράδες ποκά, νῦν δ’ ἅμα πᾶσαι | ἐντὶ μιᾶς μάνδρας, ἐντὶ μιᾶς ἀγέλας (‘Once the Bucolic Muses were scattered, but now they are all together in one fold, in one herd’, 1 FGE).Footnote ⁹⁵ The poem can be read as a competing programmatic introduction to an arithmetical poetry collection that instead conceptualises the poems as a herd, by drawing on a pre-existing motif for editorial activity and on a Homeric motif deeply connected to counting. The position of both poems in juxtaposition at the opening of the book is a (further) programmatic placement by the later compiler.Footnote ⁹⁶

Furthermore, AP 14.4 displays an approach to arithmetic in poetry observable in Archimedes’ Cattle Problem. As I demonstrated in Chapter 3, Archimedes is a keen reader of Homer’s poetics of enumeration, since he combines Homer’s reflection on whether he has the capacity to recall the entire πληθύς at Troy with the imagery preceding the Invocation and Catalogue that likens accounting for the troops to the counting and controlling of herds. Regardless of whether in the second verse the poet is cognizant of, and refers back to, Archimedes’ πληθύν Ἠελίοιο βοῶν, ὦ ξεῖνε, μέτρησον (‘the multitude of the Cattle of the Sun calculate, O stranger’, 1), the verse-initial πληθύν with the genitive βουκολίων undoubtedly shows their awareness of Homer’s archetypal exploration of handling numbers in poetry. Not insignificantly, then, the programmatic allusion to the Invocation prior to the Catalogue of Ships also informs the final arithmetical poem of the book. As I have noted, the poet’s invocation in Iliad 2 was adapted into a pointedly numerical challenge in the Contest of Homer and Hesiod. Remarkably, those same verses are appended immediately after the Metrodoran section.

This poem is contextualised in the manuscripts with the following comments: Ὅμηρος Ἡσιόδῳ ἐρωτήσαντι πόσον τὸ τῶν Ἑλλήνων πλῆθος τὸ κατὰ τῆς Ἰλίου στρατεῦσαν (‘Homer, to Hesiod after he asked how great was the number of Greeks that campaigned against Ilium’). The emphasis on the πλῆθος and the suggestion of the question-and-answer format of Homer and Hesiod’s exchanges make it plausible that the verses were drawn directly from the Contest.Footnote ⁹⁷ So too, they make the cultural value of the arithmetic poems clear in that they give their combination of numerical calculation and poetry the greatest possible pedigree. Important for my argument, however, is its concluding position following the arithmetic epigrams; the lines take on new meaning when placed in Book 14. Again, the reworking plays on the two possibilities raised in the Invocation to the Muses, namely the recalling and naming of the πληθύς and the counting of it. It takes advantage of multiplication’s ability to avoid the linear relationship between poetic content and poetic extension and reduces the much-prized 285 hexameters of the Catalogue of Ships to three verses. What is more, it looks back to the ‘πληθύς of poems’ programmatically introduced in AP 14.4 with a further allusion to Homer’s counting in poetry. In terms of the arrangement of the collection in AP 14, placing lines that collapse the Catalogue of Ships into three verses at the end of a catalogue of arithmetical poems provides fitting metatextual closure: lines that end the need for a catalogue through calculation signal the end of a catalogue of calculations.Footnote ⁹⁸

Thus, there are signs that the compiler of Book 14 appreciated the significance of arithmetical poems as products of a simultaneously arithmetical and literary education and sought to reflect that in their arrangement of the book. This approach is nowhere more evident than in the opening poem of the book, which I take to be attributed to one Socrates and which is most probably not from the Metrodoran collection.

Polycrates was the tyrant of Samos, and Pythagoras one of its most famous inhabitants. They were contemporaries – Pythagoras left Samos because of Polycrates’ rule – and this poem imagines a dialogue between them. The opening verse’s address to Pythagoras as an offspring of the Muses connects a foundational figure of mathematics and numerology to poetry, which the Muses inspire. The term ἔρνος – literally, a ‘sprout’ or ‘offshoot’ of a plant – subordinates Pythagoras to the Muses. The collection which intertwines poetry and arithmetic contains in its opening gambit the claim that mathematical interests are dependent on, and develop out of, the traditional cultural practices which the Muses represent (that is, poetry, but also music, history and astronomy). In terms of form, the dialogue also makes clear the question-and-answer format that is implicit in many of the subsequent poems, in that they are to be posed by one person to another. Most notable about the poem, however, is its meta-pedagogical stance. As commentators have observed, the groupings in the poem seem to reflect the division of Pythagoreans found in some sources into the ἀκουστικοί, who meditate in silence, the μαθηματικοί, studying sciences, and the φυσικοί, contemplating the nature of the universe.Footnote ⁹⁹ Pythagoras enumerates those in his circle, and the different forms of enquiry that they make, in a poem that introduces a collection which contains arithmetical poetry (as well as riddles and oracles) gathered together for educational purposes.Footnote ¹⁰⁰ The mix of numerical and poetic learning is thematised as well in that μαθήματα could refer to ‘lessons’ or ‘knowledge’ broadly conceived but also had the specific sense of ‘mathematical sciences’ (LSJ s.v. μάθημα A.3). These ‘Pythagorean’ students within the poem have a range of interests that encompass the cultural and the mathematical, but they are nevertheless all interpreters of the plural Muses (Πιερίδων). These students reflect the aim of the collection. The effect of reading through it is that one is initiated into the house of Pythagoras, a teacher of mathematics home-grown on Helicon, and that one is endowed not just with mathematical knowledge, but is in commune with all the Muses.

The Byzantine compiler’s ordering shows that they too appreciated AP 14.1’s self-reflexive comment on the dialogue between poetic and arithmetical learning; their positioning of the work at once emblematises and instigates the educational process of dealing with mathematics alongside the Muses. In other words, although probably placed in that location well after Graeco-Roman antiquity as commonly conceived, the poem nevertheless was seen to comment on and justify the significance of arithmetic poetry. Far from these poems being thought of as marginal literary experiments, the Byzantine compiler actively engaged with the significance of arithmetic in poetry. Similarly, I have argued in this chapter that the arithmetic poems themselves encapsulate a broader conversation between poetry and arithmetic in Late Antiquity. Individually and as a (Metrodoran) collection, the poems demonstrate how well arithmetic could not only be versified, but also presented and framed in a way that provides an additional means of enhancing poetry as an object of cultural value and social exchange. Whether it was arithmetical skill or cultural prestige, there was something to be gained by producing and appreciating the arithmetical aesthetics of these poems. The collection testifies that over the course of a millennium the practice of composing calculations in verse really did count for something.