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.

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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.