obvious – « most Greek abaci seem to have been set up to handle monetary calculations, etc. » (Wyatt 1964, 269)<fnref n="91" /> – is always the best answer to a question that deals with the most ordinary tools of everyday life. And – make no mistake about it – the ancient abacus was just another run-of-the-mill gizmo common people used one moment and forgot all about it afterwards – unless, of course, something strange happened right next to it <fnref n="92" />. Besides, it is only reasonable to assume that Ancient Greeks gave up finger counting and set up the abacus when they needed to, that is when they had to go through lengthy calculations or work out figures based on both decimal and non-decimal ratios. Needless to say, this is precisely what happened each and every time they reckoned to any degree of precision how many coppers make up how many obols and how many of these you need to have such and such amount of drachmas, minas or talents.

IF YOU PAY BEANS, YOU GET JURORS. A cautionary tale, which Aristophanes has one of his most level-headed and likeable characters tell, might just spell it out for us. How do you rip off your opinionated and gullible senior citizens ? Easy busy jurors squeezy – you set them on your political foes in court and you keep the whole lot both happy and hungry, feeding them scantily the leftovers from the pie you and your cronies have lavishly helped yourselves to :

[T17] *Aristophanis Vespae*, 655-664 : « <Bδελυκλέων :> ἀκρόασαί νυν, ὦ παπίδιον, χαλάσας ὀλίγον τὸ μέτωπον. καὶ πρῶτον μὲν λόγισαι φαύλως, μὴ ψήφοις ἀλλ' ἀπὸ χειρός, τὸν φόρον ἡμῖν ἀπὸ τῶν πόλεων συλλήβδην τὸν προσιόντα, κᾶξω τοῦτου τὰ τέλη χωρὶς καὶ τὰς πολλὰς ἑκατοστάς, πρυτανεῖα, μέταλλ', ἀγοράς, λιμένας, μισθώσεις, δημιόπρατα τούτων πλήρωμα τάλαντ' ἐγγυς δισχύλια γίγνεται ἡμῖν. ἀπὸ τούτου νυν κατάθες μισθὸν τοῖσι δικασταῖς ἐνιαυτοδ, ἔξ χιλιάσιν – “κούπω πλείους ἐν τῇ χώρᾳ κατένασθεν”. γίγνεται ἡμῖν ἑκατὸν δῆρκου καὶ πεντήκοντα τάλαντα. <Φιλοκλέων :> οὐδ' ἡ δεκάτη τὸν προσιόντων ἡμῖν ἄρ' ἐγίγνεθ' ὁ μισθός [Henderson 1998, 305 : <Loathecleon :> then listen, pop, and relax your frown a bit. First of all, calculate roughly, not with your counters but on your fingers, how much tribute we receive altogether from the allied cities. Then make a separate count of the taxes and the many one percents, court dues, mines, markets, harbours, rents, proceeds from confiscations. Our total income from all this is nearly two thousand talents. Now set aside the annual payment to the jurors, all six thousand of them, “for never yet have more dwelt in this land”. We get, I reckon, a sum of one hundred and fifty talents. <Lovecleon :> so the pay we’ve been getting doesn’t even amount to a tenth of the revenue] ».

Whilst it is just possible that the dutiful son character wishes to keep the pebbles out of his father's reach and sight, lest he gets too excited all over again and relapses even before his sobering up could begin, no one – in the last two hundred years – has missed the fact that Aristophanes set apart rough off-hand reckoning (λογίζομαι φαύλως, ἀπὸ χειρός) from accurate pebble computation (λογίζομαι ψήφοις) <fnref n="93" />. Few, on the other hand, seem to

<footnote n="91">⁹¹. Based on Letronne's archaeological data and analysis alone (the Salamis abacus file, for short), Moritz Cantor, who could still read French, drew a similar conclusion according to which all signs (monetary numerals, huge dimensions and sturdily built) supported the inference that the Salamis table was a « Zahltisch eines Wechslers », that is a money-changer's counter (Cantor 1863, 133).</footnote>

<footnote n="92">⁹². There's nothing particularly inspiring about ancient abacuses and one has no problem understanding why people did not fancy the kind of chores they were supposed to help with. Some things never change and computational assignments have always been a pain in the neck (ἐνέργεια λυπράπω : « ἡ μὲν οἰκεία ἡδονὶ ἐξακριβοῖο τὰς ἐνεργείας καὶ χρονιωτέρας καὶ βελτίους ποιεῖ [...] . φθείρουσι γὰρ τὰς ἐνεργείας αι οἰκεῖα λῦπαι, ὀιον ἐ τῷ τὸ γράφειν ἀήδες καὶ ἐπίλυπον ἦ τὸ λογίζεσθαι· ὃ μὲν γὰρ οὐ γράφει, ὃ δ' ὦ λογίζεται, λυπράης οὐσῆς τῆς ἐνεργείας [the proper pleasure of an activity makes it accurate, last longer and improves it. ...]. Pain that belongs by itself to an activity, on the other hand, destroys it. For example, someone loathes and can't stand writing or doing sums - well, he'll neither write nor will he do sums, because he finds it annoying] » (Ethica nicomachea X 5, 1175b 13-15 et 17-20). For the sake of decorum, we won't dwell upon the secret life of ancient abaci. That being said, if one were to dig for unsavoury details, he would unearth the usual amount of dirt and then some. One always does, especially when bankers are involved and money changes hands faster than you can count. A short fragment from Lysias will suffice to remind us of the close proximity - if not intimate kinship - between whoring and banking, two of the oldest and most lucrative trades of the civilized world : « ἐφ' ἐτέρου μὲν γὰρ εἶρηται ὑπὸ Λυσίου ἐν τῷ ὑπέρ Καλλαίσχρου, “μετ' ἀβακίου δὲ καὶ τραπεζίου πωλῶν ἑαυτόν” [the word “abacus” is used in still another sense by Lysias in his *On behalf of Callaeueschrus* : “selling himself between an abacus and a counter”] » (Pollucis onomasticon X 105, 221.12-14). Already Johann Georg Baiter and Hermann Sauppe suggested - p. 191 of their 1850 edition of the Attic orators - that the word ἀβακίου does not mean here « gaming table (tabula lusoria) » but « counting table ». They went even further and suspected without much proof, as Carey - p. 418 of his 2007 edition of Lysias orations and fragments - rightly pointed out, that the *servus argentarii* was the employee servicing both the *mensula* and the *mentula* (the syntagma πολεῖν ἑαυτόν, as it occurs in the Lysias' fragment possibly for the first time, has been discussed with references to Lysias and later sources by Colla 2012, 50-51). True enough, it is immaterial to ascertain here whether the hired hand worked both jobs or not, and I may have made the point a bit flippantly, but, folks, there's a serious issue here : the moral of the story is that wherever banking counters were to be found [a], abaci were not far away. Not to mention the fact that *τράπεξα* and *ἄβαξ* are occasional synonyms and therefore may refer at times to the same thing, as the epigraphic evidence from one of the Corinthian surviving specimens (SEG XI 188) shows : « ΔΑΜΟΣΙΑ ΚΟΡΙΝΘΙΩΝ » is inscribed on the lower right corner of the abacus, that is to say : *δαμοσία <τράπεξα>* – as Donati 2010, 10a-b took good notice : « the *δαμοσία* Κορινθίων identifies the counting table as the property of the Corinthian state [10b] with the feminine singular gender of *δαμοσία* alluding to *τράπεξα* (table) and not the masculine *ἄβαξ* (abacus) ». [a] As a matter of fact, we know where the Athenian counters were traditionally located, somewhere in the northwest corner of the Agora (cf. Thompson & Wycherley 1972, 171 note 12) – a corner Socrates and Hippias were pretty familiar with, as evidenced by Plato's *Apology* (17c 7-9 : ἐν ἀγορᾳ ἐπὶ τῶν τραπεζῶν) and *Hippias minor* (368b 2-5 : ἐν ἀγορᾳ ἐπὶ ταῖς τραπεζαῖς*.</footnote>

<footnote n="93">⁹³. By contemporary standards the « Dean Ireland Scholarship for the promotion of classical learning and taste »'s test is definitely elite philologists' stuff - how many people, apart from Sten Ebbesen, Philippe Hoffmann and a chosen few, do you know who would be comfortable with translating off-hand, either in Latin hexameters or in Greek iambics, stanzas from Spenser's *The Faery Queene* ? When it was established back in 1825 (cf. *Parecholae*, 1846, 203-207), it was meant for undergraduate students (who, by the way, were no longer eligible to take it after their sixteenth term, that is beyond their fourth year). As it happened, [T17] caught the examiners' imagination around 1844, for they required that year's candidates to translate Aristophanes verses and comment, albeit shortly, on their content - technically-wise if we are to judge from their other requirements... for instance, that same year, Fufidius' scam (cf. *Horatii saturae*, I, 2.14 : « quinas hic capit mercedes exsecat [Rushton Fairclough 1926, 19 : five times the interest he slices away from the principal] ») was to be assessed</footnote>
have noticed that the digital calculations Bdelukleôn is running by his old man stick to the same monetary denomination: as a matter of fact, however conspicuous, the approximate sums (ἐγγύς δισχίλια, ἑκατὸν δήπου καὶ πεντήκοντα) are all expressed in talents (τάλαντα). As a result, although the domestic whistle-blower is keeping track of a whole lot of coin, no fancy conversion is called for and even his intoxicated, delusional jury-duty fiend of a father has no problem following the money and figuring out that he's been seriously bamboozled. Just the same, few have taken notice of the fact that when Philokleôn finally catches up and realises he and his fellow minions have been feasting on crumbs ⁹⁴, he takes the figures of the racket he's been involved in and rounds them up to the nearest decimal, a tithe precisely – give or take fifty talents, that is (which is, by the way, more than he would earn in several lifetimes as a juror).

WHEN DID YOU GET TO PULL THE PEBBLES OUT OF THE BAG THEN? The answer to that question should be clear by now: you pick up the counting board when you cannot trust your fingers to do the job, either because you run out of digits before the calculation is over or because the ongoing computation involves more variables than your hands can handle on their own ⁹⁵. Albeit in short supply, literary evidence points precisely in this direction (and in this direction only): the abacus main strength and, as a result, its primary utility and overall interest laid in its reliability in carrying on long-drawn-out reckonings, especially when they involved back and forth permutations between decimal and non-decimal operands. Alexis' carousers – whom we've already met ([T11] and [T15]) – and the bull artist in Theophrastus' portrait gallery ⁹⁶ offer a glimpse into the abacus' workings:

[T18] *Theophrasti Characteres* XXIII 6, 130.20 - 132.26: « καὶ ἀγνώτων δὲ παρακαθημένων κελεῦσαι θεῖναι τὰς ψήφους ἑνα αὐτῶν καὶ ποσῶν κατὰ χιλίας [a] καὶ κατὰ μίαν καὶ προστιθεὶς πιθανὰ ἑκάστοις τοῦτων ὀνόματα ποιῆσαι καὶ δέκα τάλαντα: καὶ τοῦτο φῆσαι εἰσενηνοχέναι εἰς ἑράνους αὐτῶν· καὶ τὰς τριηραρχίας εἰπεῖν, ὅτι οὐ τίθησιν, οὐδὲ τὰς λειτουργίας, ὅσα λελειτούργηκε [Diggle 2004, 131: when he finds himself sitting next to complete strangers he will ask one of them to work the calculator, and then he does an addition counting from the thousand-drachma to the one-drachma column, and putting a plausible name to each item, and reaches as much as ten talents, and says that these are the sums he has contributed towards loans for friends – and he has not included the trierarchies and all his other compulsory public services]». [a] κατὰ χιλίας is Wilamowitz 1898's, 522 conjecture. It is widely accepted on account of the fact that, on the one hand, ancient abaci lacked a 600 drachmas column (whereas they actually had one for the 1000 drachmas) and, on the other hand, the figure 600 (καθ' ἐξακοσίας) may be explained as a confusion between the alphabetic and the acrophonic values of X (it being understood that abaci's markings are usually consistent with the acrophonic system).

[T18] and [T15] deal with similar situations: Theophrastus' braggart and Alexis' partygoers – ἑρανισταί both, as it happened – were in for more than a few rounds of additions and conversions.

Ο ἀλαζών. On top of the five talents worth of charities he handed out during the famine (5) as well as the civic contributions he's burdened with as the wealthy citizen he pretends to be (6), Theophrastus' fraud boasts about the ten talents he allegedly spent helping out friends in need. And our friendly neighbour certainly has been busy comforting indigent pals, for he's making up stories about liberalities whose figures are supposed to

according to Roman moneylending customs and laws: « what was the usual rate of interest at Rome? Mention some of the laws by which it was regulated ».

⁹⁴. As suggested in a scholium (*Scholia in Aristophanis Vespas*, ad 663), Bdelukleôn worked out the figure on the basis of jurors daily pay (τριώβολον τῆς ἡμέρας), times the number of jurors (ἔξ χιλιάσιν), times the number of available months in a year (δέκα μήνας). While the reasoning is sound and the τριώβολον as well as the number of jurors are solid enough figures (MacDowell 1971, 222; Sommerstein 1983, 198; Biles & Olson 2015, 293), three hundred court days – year in, year out – is undoubtedly more often than the Athenian calendar actually allowed and the jurors – all six thousand of them – could actually stand if they were to attend every day (Hansen 1979 reduced these figures significantly, whether he went too far or not, he was definitely headed in the right direction, as pointed out by Harris 1986).

⁹⁵. That much should be uncontroversial – but it isn't. Who disagrees? Franco Lo Piparo, for one, is of a different mind altogether. Admittedly, there's subtle and there's too subtle – and some at least of Lo Piparo's distinction are so subtle they're lost on me – for instance, the distinction between an Aristotelian notion of « symbol » and its opposite un-Aristotelian number: « our text does not claim that words are symbols of facts. Rather, it says that – when discussing – we use words-that-are-symbols » (Lo Piparo 2003, 184). His examples, on the other hand, are delightful – even when they prove exactly the opposite of what they are supposed to show. In this particular instance, let's follow Lo Piparo to the market and meddle in his salesman's business. Hermogenes buys and sells sheep and uses counters to keep track of his transactions. Does he really need them? Better safe than sorry... but let Lo Piparo tell us more about it: « this is how our salesman keeps accounts: he matches sheep and pebbles so that he puts one of these in his bag each and every time he buys one of those and does the opposite when he sells instead of buying » [Lo Piparo 2003, 184]. Let me ask again: does one need an abacus or even a bunch of pebbles to count up to ten (add four, subtract two, add eight... equals ten – attaboy !)? Whatever the answer, unless one can't be bothered to properly match one pebble and one sheep as need be while keeping track of both at one and the same time (in Lo Piparo's terse scientific prose: « se non ha fatto errori nell'operazione della messa in corrispondenza uno-a-uno di pecore e sassolini, alla fine dei suoi affari avrà tante pecore quanti sono i sassolini che si trovano nella sua bisaccia »), then he has no business counting them at all, with or without an abacus !

⁹⁶. It is worth noticing that Theophrastus mentioned the abacus on no less than three different occasions. As a matter of fact, in addition to the boastful man ([T18]), the abacus reveals peculiar features of two other characters: the moron (XIV 2, 106.3-5) and the arrogant man (XXIV 12, 134.15-17). While the former's absentmindedness is farcical and heartening, the latter's high-handedness is more informative, i.e. more supportive of the monetary and commercial agenda I've been pushing all along – see [T23] below. Millett 2007 (in particular 69-70) and Pertsinidis 2018 are two short, student-friendly introductions to Theophrastus work. Cf. Lane Fox 1996 for a more detail-oriented, almost book-length study (in particular, 134-135).
add up as high as ten talents, that is as much as sixty thousand drachmas. True enough, [T18] doesn't say much about the average amount of such loans<fnref n="97" />, but – as Diggle 2004, 439 observed – the « κατὰ χιλίας καὶ κατὰ μίαν » suggests that loans covered the full range of columns. If this is true or even half true, it does not matter how clever with their hands Theophrastus' mythomaniac and his audience were supposed to be; only an abacus would have allowed them to navigate through the fairly long string of operations involved in [T18]'s reckoning divagations.

Οἱ ἑρανισταί. Even under the best of circumstances, dinner arrangements are a sensitive matter to say the least, and you'd better discuss them beforehand, lest you get into an argument as soon as the party's over and party animals start turning on each other. This is precisely what makes [T15] an awkward and potentially hilarious situation : instead of sleeping off the booze or having it off with the flute girl<fnref n="98" />, as any decent bloke would have done instead, A and B picked up a fight over the price of mussels, cabbage and sea-urchins – what's wrong with you people ? One thing they got right though : whether they went at each other intoxicated or not, there's no way they got to the bottom of the matter relying only on their fingers for adding seven coppers of this, one drachma of that, three of those at ten obols each. etc. We ignore whether eventually A and B found some sort of closure (for all we know, they might still be quibbling and tossing the pebbles around). If they ever did, they had to thank the non-decimal notations on the abacus they called for and put to good use to add and convert – as needs be – non-decimal monetary denominations like coppers, obols and drachmas.

***

WHAT DO [POSITIONALITY] AND [HYBRIDITY] TELL US ABOUT ARISTOTLE'S PEBBLE ANALOGY? For the sake of brevity, we have left aside a few additional allusions to the abacus and a number of passing mentions of the counters in ancient Greek literature – they sing pretty much the same tune anyway <fnref n="99" />. All in all, if I'm right or even half right, then the best way to make sense of Aristotle's analogy is also the most natural, insofar as it is consistent with most of the epigraphic and literary evidence available. Specifically, everything we've gathered so far warrants two related claims. The first is that – contrary to what [ARITHMETICAL BIAS] would have us believe – there's more to the abacus comparison than just plain arithmetic. Insofar as abacus assisted calculations were first and foremost pecuniary transactions, they routinely involved operations and conversions related to monetary and weight standards. More to the point, if plain numbers and plain arithmetical rules entered the Aristotelian picture at all, they didn't do so for their own sake (wherefore the [PROXY] label our first assumption will henceforth go by). Our second claim is that Aristotle was not so much interested in comparing calculation and argumentation as such (let alone language at large), as he was in comparing why (and how) they both fail. As a matter of fact, the whole point of the pebble analogy is failure ; in this particular instance, failure to detect and prevent abusive value shifts affecting words and counters (wherefore the [FAILURE] label etc.) <fnref n="100" />.

<footnote n="97">⁹⁷. For what it is worth, Demosthenes (or, perhaps, Apollodoros himself, which is somewhat ironic considering there was no love lost between the two) recorded two such loans granted to Nicostratos, a friend turned foe, for an amount of 300 (which the former eventually condoned) and 1000 drachmas (an ἔρανος contribution for the latter's ransom) : « τάς τε τριακοσίας, ἂς τῷ ἀδελφῳ αὐτοῦ ἔδωκα ἑφόδιον ὅτε ἑπορεύετο ἐπὶ τοῦτον, ἀφιείην αὐτῷ, χιλίας τε δραχμάς ἔρανον αὐτῷ εἰς τὰ λύτρα εἰσοίσομι [Bers 2009, 59-60 : I forgave the loan of three hundred drachmas that I gave his brother when he travelled to get him and said I would contribute a thousand drachmas towards his ransom] » (Contra Nicostratum 8, 204.20-23). Demosthenes again – in an early speech against his guardians over his father's squandered estate – listed amongst the assets that should have been bequeathed to him a number of loans : « ναυτικὰ δ' ἐβδομήκοντα μνᾶς, ἐκδοσιν παρὰ Ξούθω, τετρακοσίας δὲ καὶ δισχυλίας ἐπὶ τῇ τραπέζῃ τῇ Πασίωνος, ἐξακοσίας δ' ἐπὶ τῇ Πυλάδου, παρὰ Δημομέλει δὲ τῷ Δήμωνος νιεῖτι χιλίας καὶ ἐξακοσίας, κατὰ διακοσίας δὲ καὶ τριακοσίας ὁμοῦ τι τάλαντον διακεχρημένον. καὶ τούτων αὐ τῶν χρημάτων τὸ κεφάλαιον πλέον ἢ ὀκτώ τάλαντα καὶ πεντήκοντα μναῖ γίγνονται [MacDowell 2004, 24 : in maritime assets he left 70 minas on loan to Xuthus, 2.400 drachmas at Pasion's bank, 600 at Pylades', 1.600 with Demomeles son of Demon, and various loans of 200 or 300 amounting to about a talent. The total sum of this money comes to more than 8 talents 50 minas] » (Prima in Aphobum oratio 11, 45.11-18). Korver 1941, 14-15, Thompson 1979, 227 and Millett 1991, 157 note 38 have suggested that the twenty odd loans Demosthenes mentions amongst his non-earning assets did not yield interests and are to be considered ἔρανος-like credits (Bogaert 1986, 22 disagrees). In which case, the amount of operations Theophrastus' schmoozer has his occasional acquaintance lay down on the abacus might be ridiculously high – hardly out of character, ain't it ? Be that as it may, sums may well be imaginary, the computation is not – Theophrastus' fraud may be fabricating names and contriving figures, but he calculates as if the amounts were all too real, on the abacus that is.</footnote>

<footnote n="98">⁹⁸. Admittedly, there is more about ancient musician women than meets the classicist eye (cf. e.g. Burton 1998, Harmon 2005, Goldman 2015, etc.), starting with the label itself – « flute girl » – which may well be an anachronistic fabrication (cf. West 1992a, 1). That being said, Old Comedy clichés apart (cf. e.g. Gianvittorio 2018), Alexis' characters – especially A (a man after my own heart) – strike me as they would not think twice before going for Philokleôn's bold manoeuvre and snatch the αὐλητρίς for their personal comfort... *Vespea*, 134 : « ὁρᾶς ἐγώ σ' ᾗς δεξιῶς ὑφειλόμηνμέλλουσαν ἦδη λεσβιεῖν τοῦς ζυμμότας ὑν ἐίνεκ' ἀπόδος τῷ πέει τρεῖ δχάριν [Henderson 1998, 391 : did you see how handily I sneaked you away just when you were supposed to start sucking the guests ? for that you owe my cock here a favor] » (you can quote me on that).</footnote>

<footnote n="99">⁹⁹. For instance, Pindar's tenth *Olympian* opening strophe relies heavily on ancient accounting jargon: indebtedness (*χρέος*) and repayment with interests (*τόκος*), etc. Several scholars have thus come to the conclusion that the poet chose the ψᾶφος metaphor accordingly, that is in reference to the pebbles used in money-calculations (Norwood 1974, 111 ; Kromer 1976, 426-428 and Faraguna 2008, 36-37). Others have been more nuanced (Verdenius 1987, 60). All in all, the poet seems to have conflated two images when he mentions the flow of his song washing away his debt : on the one hand, the clearing of the counters off the counting table after the reckoning has been successfully carried out and, on the other hand, the washing away of the pebbles swept by the ever-rolling wave.</footnote>

<footnote n="100">¹⁰⁰. For we lack conclusive evidence concerning how calculations were actually performed on the abacus, we haven't indulged in a thorough, albeit tentative, reconstruction of what could have possibly gone wrong on the counting board when chips were pushed around. If I were to single out the one line of speculation that – in another life – I'd pursue, I would say that, for all practical purposes, tracking pebbles</footnote>
Before we expound [PROXY] and [FAILURE] in more detail, let us first clear a technical hurdle involved in shifting the focus of Aristotle’s analogy away from the arithmetical bias that has traditionally plagued its interpretation: is Aristotle’s choice of words consistent with the idea that merchant arithmetic and bean counting were the kind of calculations he had in mind when comparing poor reckoning and poor debating skills? In so many words, yes.

Λογίζομαι (ἐπὶ τῶν ψήφων). If one were to ask what exactly Aristotle's « λογιζόμενοι » ([URTEXT], 165a 9-10) were counting, the answer would be as vague as the verb is rich in nuances – most likely a jest (« the counters, you silly ») or a shrug (« just about anything and everything the counters can stand for, I guess »). So late in the game, an attempt at narrowing down the polysemy of the expression by virtue of its association with the pebbles would look like cheating or begging the question, to an extent. That being said, the fact remains that, whether the counters are explicitly mentioned or not, λογίζομαι was used to refer to all sorts of practical computations, for the most part involving money. To stay in character, supportive fathers do not fare much better than abusive ones in Aristophanes' family sagas, especially when their offspring develop expensive addictions; their financial problems, however, were referred to and assessed in the same terms, as Strepsides – the onanist opsimath who got in deep with the sharks and thought philosophy was the easy way out (think again !) – put it when prompting the houseboy to bring him the ledger on a sleepless, anguish-fuelled night<fnref n="101" />:

[T19] *Aristophanis Nubes*, 16-20 : « ὁ δὲ κόμην ἔχων ἰπάξεται τε καὶ ξυνωρικεύεται ὀνειροπολεῖ θ huroues. ἐγὼ δ ἀπόλλυμαι ὁρῶν ἄγουσαν τὴν σελήνην εἰκάδας. οἱ γὰρ τόκοι χωροῦσιν. ἄπτε, παῖ, λύχνον κᾶκφερε τὸ γραμματεῖον, ἵν ἀναγνῶ λαβών ὀπόσοις ὁρεῖλω καὶ λογίσωμαι τοὺς τόκους [Halliwell 2015, 21 : he lets his hair grow long and his life's an obsession with horses and chariot-racing — he even dreams of horses. Meanwhile I'm distraught as I watch the moon reach the twentieth day of the month. All that interest mounting up! Hoy, slave, a lamp! And bring me out my accounts. I want to read how many my creditors are and work out the interest] ».

People being people, they hold grudges over money more than over anything else : now and then, family members fritter away their next of kin's heritage, trade partners turn on each other, bankers rob their clients blind – business as usual. It is hardly surprising then that ancient legal courts offer a wealth of lexical evidence ; and λογίζομαι figures prominently in all kinds of financial litigations : embezzlement of funds and goods, misappropriation of estates and revenues, miscalculation of profits and costs, concealment of property, creative accounting – you name it <fnref n="102" />.

[T20] *Lysiae De bonis Aristophanis ad aerarium* 9-10, 184.23 - 185.3 : « συκοφαντούμεθα καὶ κινδυνεύομεν περὶ ὃν οἱ πρόγονοι ἡμῖν κατέλιπον κτησάμενοι ἐκ τοῦ δικαίου. καίτοι, ὁ ἄνδρες δικασταί, ὁ ἐμός πατὴρ ἐν ἄπαντι τῷ βίῳ πλείω εἰς τὴν πόλιν ἀνήλωσεν ἢ εἰς αὐτὸν καὶ τοὺς οἰκείους, διπλάσια δὲ ἤ νῦν ἔστιν ἡμῖν, ὡς ἐγὼ [10] λογιζομένω αὐτῷ πολλάκις παρεγενόμην. ἡμ ἐν ἀν ποκαταγιγνώσκετε ἀδικίαν τοῦ εἰς αὐτὸν μέν μικρὰ δαπανῶντος, ὑμῖν δὲ πολλὰ καθ' ἐκαστον τὸν ἐνιαυτὸν, κτλ. [Todd 2000, 203-204 : we are being attacked by sycomants and are on trial for the property which our ancestors justly possessed and handed down to us. And yet throughout his life, gentlemen of the jury, my father spent more on the city than on himself and the members of his family : twice what we now possess, as I often heard him calculate. Do not convict prematurely of wrongdoing the person who spends little on himself but a great deal every year on you, etc.] ».

Being under suspicion as an accessory in a scheme involving a transfer of seizable assets, Lysias' client may or may not be trusted implicitly – all the more so since he seems to have been the only witness of his father's reckoning. That being said, we have no reason to think that the jurors understood the λογιζομένω αὐτῷ as referring to anything else but the process of calculating the expenses the defendant's old man incurred on behalf of the city.

More to the point, when both words (*λογίζομαι* and *ψῆφοι*) occurred in the same sentence, before you know it, you are counting money or someone is counting money for you. Demosthenes – referring back to Aeschines – and Theophrastus said it all:

[T21] *Aeschinis Contra Ctesiphontem*, 59.3-9 : « ὅσπερ ὅταν περὶ χρημάτων ἀνήλωμένων διὰ πολλοῦ χρόνου καθεξώμεθα ἐπὶ τοὺς λογισμούς, ἐρχόμεθα ἤ Που ψευδεῖς οἰκοθεν ἐνίοτε δόξας ἔχοντες· ἀλλ' ὅμως ἐπειδῶν ὁ λογισμὸς συγκεφαλαιωθῆ, οὐδεὶς ἔστιν οὖτω δύσκολος τὴν φύσιν ὄστις οὐκ ἀπέρχεται τοῦτο ὁμολογήσας καὶ ἐπινεύσας ἀληθές εἶναι, ὁ τὶ ἀν αὐτὸς ὁ λογισμὸς αἰρῇ [Carey 2000, 185 : when we take our seats at an audit session for expenditure over a long time, we may sometimes come from home with false impressions, but still when the account is reckoned up there is

on the abacus must have been nearly impossible to begin with. As far as we know, the abacus simply did not allow one to display anything but the outcome of the reckoning. As [T18] and [T15] are to suggest, we can safely assume that most calculations run on the abacus went through more than just one step – why bother otherwise to get out the counters and set up the reckoning board in the first place ? So many steps, so many manipulations resulting over and over in a different configuration of the counters on the abacus. Each successive arrangement on the pebble-board modified and replaced the one it resulted from and was superseded by the one it led to. Since we are not aware that the abacus would record any previous stage of a calculation, short of working them backwards and comparing (mental) notes along the way, it must have been extremely difficult to nail down exactly what went south. And, to be sure, a number of things could have gone wrong : a displaced counter, or a shortcut replacement between non-adjacent columns, etc.

<footnote n="101">¹⁰¹. On Strepsides' financial troubles as an « outstanding Athenian example of a "consumption loan» », cf. Millett 1991, 66. A representative selection of material evidence about money circulation and loans, is gathered in Bogaert 1976, who previously studied the world of Greek credit in Bogaert 1968 (cf. in particular 37-60 for a study of ancient banking vocabulary). For a more recent survey – building on Bogaert – cf. Shipton 2008.</footnote>

<footnote n="102">¹⁰². I defer to Cuomo 2001, 20-24 who has already reviewed and discussed the evidence I hint at here, and refer the reader to Cuomo 2013 for a few sound suggestions about ancient numeracy, accounting and accountability (cf. already Davies 1994).</footnote>
none of you of so grudging a disposition that he leaves without admitting and agreeing that the figure proved by the reckoning is true] ».

[T22] *Demosthenis De corona oratio*, 227.1-5 : « ἐιτα σοφίζεται καὶ φησὶ προσήκειν ἦς μὲν οἶκοθεν ἤκετ’ ἔχοντες δόξης περὶ ἤμων ἀμελῆσαι, ὥσπερ δ’, ὅταν οἰόμενοι περιείναι χρήματα τω λογιζησθε, ἀν καθαραὶ ὥσιν αἰ ψήφοι καὶ μηδὲν περιῆ, συγχωρεῖτε, οὕτω καὶ νῦν τοῖς ἐκ τοῦ λόγου φαινομένοις προσθέσθαι [Yunis 2005, 87 : next, he <Aeschines> made a very clever suggestion : you are to disregard the opinion that you had of us when you came here from home, and, just as when you audit people for supposedly retaining surplus funds but acquit them if the figures balance and there is no surplus, so in this case too you are to concur with the evident force of the argument] ».

[T23] *Theophrasti Characteres* XXIV 12, 134.15-17 : « ἀμέλει δὲ καὶ λογιζόμενος πρός τινα τῷ παιδὶ συντάξαι τὰς ψήφους διαθεῖναι καὶ κεφάλαιον ποιήσαντι γράγαι αὐτῷ εἰς λόγον [Diggle 2004, 135 : and you may be sure that when the arrogant man is reckoning someone’s account he instructs his slave to do the calculations, work out a total, and write him out an invoice for that amount] ».

As is well known, Aeschines and Demosthenes did not get along very well<fnref n="103" />. Still, they would have agreed between them – and with Theophrastus – on one thing : whomever the finger of blame should be pointed at, λογισμοί, λογίζομαι and ψήφοι definitely belong together and have a distinct reek of money about them.

Παρακρούω. If we are to believe ancient lexicographers <fnref n="104" />, a similar case might be argued for the other expression associated with the counters in [Urtext], namely the verb παρακρούω :

[T24] *Harpocrationis Lexicon in decem oratores*, Π 28 : « παρακρούεται· ἀντὶ τοῦ ἐξαπατᾶ. πολὺ δ’ ἐστὶ παρά τε τοῖς ἀλλοις Ἀττικοῖς καὶ παρὰ Δημοσθένει ἐν τοῖς Φιλιπποῖς. μετήκται δὲ τοῦνομα ἀπὸ τοῦ τοὺς ἱστάντας τι ἡ μετροῦντας κρούειν τὰ μέτρα καὶ διασείειν ἔνεκα τοῦ πλεονεκτεῖν, ὡς καὶ Σοφοκλῆς που “ώς μήτε κρούσῃς μήθ’ ὑπέρ χειλὸς βάλης” [παρακρούεται (strike aside, mislead) for ἐξαπατᾶ (deceive). It occurs often both in the other Attic <orators> and in Demosthenes’ *Philippics*. The word is a metaphor derived from how people who weigh or measure something flick the measures and shake them to obtain a profit – as Sophocles says somewhere : “that you neither flick nor exceed the rim”] ».

Did Aristotle actually use παρακρούω in [URTEXT], 165a 15 to convey the idea that smart pebble-movers take advantage of less experienced ones by fixing the counters ? Tempting though this is – after all, meddling with the counters for profit is not so different from tipping the scales – we’ll leave it at that and will only allow that nothing in [URTEXT] rules out the possibility that παρακρούω means cheating unwary people out of their money through a wicked sleight of hand.

[PROXY]. Despite the overwhelming epigraphic and literary evidence suggesting the opposite and against a solid consensus amongst some of the best archaeologists, numismatists and historians of Greek mathematics – [ARITHMETICAL BIASED] interpreters have long been labouring under the wrong assumption that the purpose of Aristotle’s pebble analogy was to draw a parallel between computation and speech tout court – as if the way we work out numbers in general could shed any light on how we misuse words. This is, of course, misleading on several counts. First if not foremost, nowhere does Aristotle compare numbers and linguistic expressions as such, their features or their relations to the things we talk and make calculations about. In fact, [URTEXT] offers little support to the idea that, when Aristotle referred to counters, he was leaning on a kinship of sorts – or any kinship, for that matter – between calculation and speech themselves. He wasn’t. As [COMPLICATION BIAS] and [HYBRIDITY] discussions have made it abundantly clear, leisure calculation or counting for the sake of crunching numbers – not to speak of more abstract forms of ancient logistic <fnref n="105" /> – were anything but a priority for those who

<footnote n="103">¹⁰³. On character assassination and Aeschines and Demosthenes rivalry, cf. Worman 2004, 2008, 213-274, 2018 and Kamen 2020, 60-86. Since the winner takes it all, on Demosthenes portrait of his foe as a Theophrastean character – a comic one of course – cf. Rowe 1966; stylistic and linguistic issues of the crown speech have been addressed in Yunis 2001 and, more recently, in Murphy 2016.</footnote>

<footnote n="104">¹⁰⁴. On Harpocration’s glossary, cf. Dickey 2007, 94, both concise and much to the point. Same entry in Photius (Π 253), Suda (Π 373), Lopadiota (Π 18), etc.</footnote>

<footnote n="105">¹⁰⁵. The kind of higher, more speculative disciplines investigating the true nature of numbers, their many properties and relations, which Plato had already set apart as a matter of course while separating the theoretical requirements of philosophers interested in numbers theory from the all too practical needs of ordinary people busy measuring and counting off everyday things (*Philebus*, 56d 4 - 57a 4). It is not always easy to determine whether Plato thought of philosophical logistic as a science all onto itself and to what extent exactly it was germane to other branches of human knowledge and overlapped with them – most notably arithmetic (cf. e.g. *Gorgias*, 451a 8 - 451c 5 and *Respublica* VII, 525a 10 - 527c 10). Insofar as neither is to be mistaken with counting and measuring crafts – the only maths vulgar calculators were supposed to know and arguably cared about anyway – we won’t try to address the issue here. Klein 1934-1936 brilliantly raised the problem and went a long way toward solving it; half the story though it is, Majolino 2012 may be considered the final word on this as well as on a number of related matters, most notably ancient dislike for fractions – also addressed most competently in Knorr 1982, Vitrac 1992, Mendell 2008 and Acerbi 2019. It is a little out of our jurisdiction and we probably should trust our layman’s instincts and leave it out, but Boyer 1968, 66 may have something there : « it is likely that the widespread use of the abacus accounts at least in part for the amazingly late development of a consistent positional system of notation for integers and fractions ». As a matter of fact, as pointed out by Carl Boyer himself, insofar as « the abacus can be readily adapted to any system of numeration or to any combination of systems » (Boyer 1968, 66), it made it perfectly natural to treat fractions as multiple subunits : on the counting board, a chalkous does not look anything like an eighth of an obol... rather, it takes eight coppers coins to make an obol. Likewise, on the abacus, an obol is not a sixth of a drachma, but six obols make a drachma, and so on and so forth. For it stands out as the most astute description of how abacus computations were likely to be performed, let’s hear it from Henry Mendell : « I may need to divide 2 drachmas equally among 5 people. Well, I multiply 2 drachmas by 6 obols per drachma to get 12 obols, which, in division, gives me 2 obols per payee with 2 remainder. But I multiply these by 8 coppers per obol to get 16 coppers, so that I can disperse 2 obols 3 coppers. The remaining copper is not worth much, so I will just give it to anyone » (Mendell 2018, 205-206).</footnote>
conceived and built the counting tables which have survived to this day <fnref n="106" />. In fact, if these are any indication of what ancient designers and users looked for in their abacus, then it is safe to assume that all they cared about was the comfort of merchants, retail-traders, accountants and other money handlers who dealt with numbers for no other reason than to buy and sell goods, charge interest rates or exchange currencies. Counting coin is where pebble boards really shone and proved most useful, so it definitely stands to reason if we assume that arithmetical operations by themselves hardly entered the picture for Aristotle. If they came into play at all, it was by proxy : while there ain't no such thing as two arithmetics, if Aristotle's pebbles were to be meaningful in any way, knowing one's numbers properly was not the same as moving counters around on the reckoning board. Provided that we understand Aristotle's abacus simile along the lines of the epigraphic and literary evidence available – as we should – it become obvious then that it presupposed numeracy all right, but it was not about numeracy itself. To begin with, granted that coin and weight calculations follow now and then the same arithmetical rules through and through, the fact remains that they do not reflect arithmetical procedures alone. Monetary and ponderal conventions are at least as important and they have their own set of rules concerning conversions between different denominations : it is not because one and one is two and three times four equals twelve that, say, an obol was worth eight coppers in Athens and twelve in Aegina or that it took seventy drachmas here and one hundred there to make a mina – this is simply the way monetary standards work, to the fishmongers' delight if we are to believe ancient humour <fnref n="107" />. Moreover, just as Aristotle took for granted that dialectical patsies had basic language proficiency and at least minimal argumentational awareness, it is only fair to assume that he also presupposed that inept calculators had at least crude numerical understanding and elementary computational training. However inexperienced and little acquainted with semantic subtleties, inferior debaters had to know enough Greek and questions and answers routine to follow a discussion, indeed to be involved in one ([URTEXT], 165a 15-17). Likewise, incapable though they were of carrying out digital feats with the counters on their own and poorly equipped to spot them on the abacus, incompetent calculators must nonetheless have known enough maths to sit at a counting table to start with and toss the occasional pebble around ([URTEXT], 165a 14-15). That being said, Aristotle's simile did not dwell on either, that is to say : it is neither primarily nor specifically about numeracy and computational articulateness as such, any more than it is about literacy and discursive fluency per se <fnref n="108" />. What is Aristotle's pebble analogy all about then ? Pebbles... what else ? And this is precisely the feature [ARITHMETICAL BIAS] has traditionally taken out of the equation, namely the fact that Aristotle compared logistical and linguistic symbols insofar as they are useful tools but require a degree of savoir-faire and must be handled with care. As a matter of fact, there can be little doubt that Aristotle's turn of phrase lays stress on the counters and those who used (and misused) them rather than on computation as an art or on reckoning at large. In other words, the emphasis of the analogy is definitely on the pebbles, the handling of which is the area of expertise – or, rather, the lack thereof – around which the whole simile revolves. Why else, of all calculators, would Aristotle have singled out those who are good – and not so good – at moving the stones ? One might object that we're taking a liberty with the text when we claim that

<footnote n="106">¹⁰⁶. Instead of skimming through the exhibits all over again, let all be reminded that even the most [ARITHMETICAL BIAS] abacus specialist – in a moment of great insight – acknowledged that « the Salamis abacus <IG II², 2777> is inscribed with three sequences of numerals, monetary numerals as it is *always* the case with abaci's numerals » (Schärlig 2001, 66 – his emphasis).</footnote>

<footnote n="107">¹⁰⁷. For a most succinct introduction to ancient Greek standards and the long-standing dissensions amongst scholars, see Duyrat 2014 and De Catallataý 2017. Marcellesi 2000 tackles a few practical problems Hellenistic monetary standards confronted ancient traders and accountants with on a daily basis. On the divergence between Aeginetan and Attic standards in particular, cf. *Pollucis Onomasticon* IX 76, 168.17-19 : « τὴν μὲν Αἰγιναίαν δραχμὴν μεῖζω τῆς Αὐτικῆς οὐσαν – δέκα γὰρ ὀβολοὺς Ἀττικοὺς Ἰσχυεν – Ἀθηναῖοι παχεῖαν δραχμὴν ἑκάλουν, μίσει τῶν Αἰγινητῶν Αἰγιναίαν καλεῖν μὴ θέλοντες [since the Aeginetan was larger than the Attic drachma (in fact, its worth was ten Athenian obols), Athenians preferred to call it the “big drachma” rather than the “Aeginetan drachma”, for they loathed Aeginetans] ». Atheneus (VI 224c - 227b) relays several comic tirades against fishmongers, most notably a fragment from Diphilus' *Busybody* : « ὁμν ἐγὼ τοὺς ἰχθυοπώλας τὸ πρότερον εἶναι πονηροῦς τοὺς Αἰγήνστιν μόνους. τὸδε δ', ὡς ξοίκε, τὸ γένος ὥσπερ θηρίων ἐπιβουλόν ἐστι τῇ φύσει καὶ πανταχοῦ. ἐνταθα γοῦν ἔστιν τις ὑπερηκοντικώς, κόμην τρέφων μὲν πρῶτον ιερᾶν τοῦ θεοῦ, ὡς φησίν· οὐ διὰ τοῦτό γ', ἀλλ' ἐστιγμένος πρὸ τοῦ μετώπου παραπέτασι' αὐτὴν ἔχει. οὐτος ἀποκρίνε', ἀν ἐρωτήσῃ "πόσου ὁ λάβραξ", "δέκ' ὀβολών", οὐχὶ προσθεῖς ὁποδαπῶν. ἔπειτ' ἐὰν τὰργύριον αὐτῷ καταβάλης, ἐπράξατ' Αἰγιναίον· ἀν δ' αὐτὸν δῆμ κέρματ' ἀποδοῦναι, προσπέδωκεν Αὐτικά. κατ' ἀμφότερα δὲ τὴν καταλλαγὴν ἔχει [Douglas Olson 2006, 17 : I used to think it was only the fish-sellers in Athens who were no good. But apparently this breed is like wild animals : their very nature makes them treacherous everywhere. Here, at any rate, there's one who's outdone them all ; he's growing his hair long, first of all, as an act of piety – so he says. That's not the reason ; he's been tattooed, and he uses his hair as a screen to cover his forehead. If you ask him “how much for the sea-bass ?”, he answers “ten obols”, without specifying the currency. Then if you pay him the money, he charges you on the Aeginetan standard ; and if he has to give change, he offers Attic coins ! Either way, he makes money on the deal] » (*Deipnosophistae* VI, 225a 6 - 225b 10). On fishmongers' bad reputation, see Davidson 1993 and Paulas 2010.</footnote>

<footnote n="108">¹⁰⁸. It is perfectly possible to have a decent grasp of arithmetic calculations and still get into trouble with the pebbles for exactly the same reason average people – that it is to say people who have no problem at all grasping the general principles of verbal communication and dialectical disputation – are tricked on a regular basis by those who know better. Following a different line of argument and without cluttering up his minds (or the readers') with mentions of exotic historical evidence, McCready-Flora 2019, 55-56 has arrived to this very same conclusion, which I endorse without reservation : « a person could be great at doing sums, but baffled by moving stones around ... verbal naifs go wrong in the same way that leads to bad stone-movers getting cheated. Mathematical error, though, is not what separates marks from their money. What the hustlers understand (epistēmenōn <no point in messing with the Smurf – if you get it wrong, mate>) and weaponize is how to move stones (psēphous therein) ... all this entails that what lets the hustlers cheat is an instrumental failure distinct from the cognitive capacity to do sums. If the inept stone-mover suffers instrumental failure and the same goes for word-novices, then the errant word-novice also suffers instrumental failure » – my point exactly !</footnote>
Aristotle's experts are not so much accomplished arithmeticians as they are individuals skilled at pushing the counters around. Granted, but let's turn the question around : what precisely do Aristotle's « ἐπιστήμονες » ([URTEXT], 165a 14) know that « οἱ μὴ δεινοὶ τὰς ψήφους φέρειν » (165a 15) don't ? Precisely. In fact, while anyone who picks up the counters shares, at least to a degree, the belief that we can depend on them, it is how deftly or clumsily we manipulate them that makes all the difference in [URTEXT]. Skilled and unskilled calculators alike put at least a measure of trust in their pebbles (otherwise, why use them in the first place ?), but only the former could trust themselves to come out on top of every transaction, especially the unfair ones.

[FAILURE]. Once we relinquish the idea that calculation as such took centre stage in Aristotle's abacus simile, it becomes easier to pinpoint what its terms were and why Aristotle brought pebbles and words together in the first place. More to the point, it is possible to turn the analogy on its head and set it back upon its feet by shifting its focus from trying to explain why computation and language succeed to trying to explain why pebble reckoning and dialectical argumentation fail – which, by the way, is so much more in character with the subject matter [URTEXT] is supposed to introduce us to, that is fallacies, paradoxes, falsities, improprieties and babbling. In fact, while [URTEXT] does not provide much in the way of comparing linguistic and computational habits per se (after all, we don't calculate with words any more than we speak in numbers, etc.), it definitely tells us that they both rely on symbols and – for this very reason – share the same liability : linguistic and computational substitutes alike are prone to inconspicuous and yet momentous variations, which we will fail to prevent as long as we do not come to terms with the fact that both linguistic expressions and counters may have different values. For this is the core of Aristotle's analogy : linguistic expressions are to argumentation as counters are to computation insofar as their worth may change without us always being able to keep up or keep track. Hence, linguistic symbols (οὐνόματα, λόγοι) and computational ones (ψῆφοι) play similar roles and, more to the point, have the same shortcomings. The problem with words is the same as the problem with counters – not because there's a however intimate or loose relation between argumentation and calculation, let alone between the way we talk and the way we reckon, but because words and counters fail us the same exact way when their value or their meaning as symbols shifts at the hands of unscrupulous debaters and malicious calculators without us taking duly notice or having the proper understanding of how it happens.

***

[EPILEGOMENA]. How well do verbal and computational prestidigitation compare and, more importantly, what do they teach us about Aristotle's views on language and its workings ? Provided that we understand Aristotle's pebble analogy on its own terms as the kind of heavy-duty comparison people were expected to figure out without racking their brains, it fares well enough to drive home an important, albeit unsophisticated, truth about language – and what it tells us about language is that it is, by and large, a matter of savoir-faire : after all is said and done, the answer to the question « what do we ask of words ? » is not so different from the answer to the question « what do we ask of counters ? ». In a nutshell, we ask them both to be worth something and to allow us to go about our conversational and computational business on the assumption that this is going to be the case as long as we don't change our mind and agree to use either words or counters with a different value altogether. All that is required for it to work then is that we play by the rules, keep an eye out for those who don't and pay as much attention when we speak as we do when we give the change or check our balance. Where's the excitement in all that ? Beats me, but to quote again Aristotle's tribesmen of old ([URTEXT], 164a 27) : « it is better to be bored and right than to get robbed and outsmarted at every turn » – Amen to that.

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