Acerbi ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 2013 Fabio Acerbi Ones Fabio Acerbi IOPHANTUS’ ARITHMETICA is a collection of arith- metical1 problems: to find numbers satisfying specific conditions that are stated in the enunciation.2 For instance, Ar. 1.27 requires to find two numbers such that their sum and their product are assigned numbers. Each problem of the Arithmetica is solved by concretely setting out the assigned numbers (in the case of 1.27, they are given as 20 and 96), by positing one unknown and by solving the equality (‘equation’ in our language) resulting from the constraints stipulated in the enunciation. Apart from this very general approach, there is no standard method to solve any specific set of Diophantine problems: the Arithmetica presents a host of clever tricks and specific manipulations, which a student gets acquainted with by simply doing Diophantine problems. Problem 5.30 begins with an epigram:3 1 ‘Arithmetic’ is the ancient denomination of our ‘number theory’. The discipline explaining how to calculate with specific numbers was called ‘logistic’. 2 The Diophantine writings were edited by P. Tannery, Diophanti Alexan- drini opera omnia cum graecis commentariis I–II (Leipzig 1893 text and transl., 1895 pseudepigrapha, testimonia, scholia, index graecitatis). A new edition of the Arithmetica has been provided in A. Allard, Diophante d’Alexandrie, Les Arith- métiques I–II (diss. Louvain 1980); it lies unpublished and, even in the University Library of Louvain-la-Neuve, only the volume containing text and translation is available. As we shall see, Allard’s edition is no im- provement on Tannery’s. The Arithmetica was paraphrased in English and commented on extensively in T. L. Heath, Diophantus of Alexandria. A Study in the History of Greek Algebra (Cambridge 1910). 3 I 384.6–21 Tannery; in the manuscripts, the numeral letters are iden- tified by macrons. The epigram, whose text needs emendation, was studied D FABIO ACERBI 709 ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 ὀκταδράχµους καὶ πενταδράχµους χοέας τις ἔµιξε τοῖς ὁµοπλοῖσι ποιεῖν χρήστ’ ἐπιταττόµενος, καὶ τιµὴν ἀπέδωκεν ὑπὲρ πάντων τετράγωνον, τὰς ἐπιταχθείσας δεξάµενον µονάδας καὶ ποιοῦντα πάλιν ἕτερόν σε φέρειν τετράγωνον κτησάµενον πλευρὰν σύνθεµα τῶν χοέων· ὥστε διάστειλον τοὺς ὀκταδράχµους πόσοι ἦσαν, καὶ πάλι τοὺς ἑτέρους, παῖ, λέγε πενταδράχµους. The meaning of the epigram is explained as follows: ἠγόρασέν τις βʹ ἑνῆ οἴνου, ἐκ µὲν τοῦ ἑνὸς τὸν χοέα δραχµῶν ηʹ, ἐκ δὲ τοῦ ἑνὸς τὸν χοέα δραχµῶν ε, καὶ ἀπέδωκεν ὑπὲρ πάν- των τιµὴν τετράγωνον ἀριθµόν, ὃς πρὸς µο ξʹ ἐποίει τετράγωνον πλευρὰν ἔχοντα τὸ πλῆθος τῶν χοέων· διάστειλον τοὺς ὀκτα- δράχµους καὶ πενταδράχµους. Someone bought 2 ἑνῆ of wine—a congius of the one being worth 8 drachms, a congius of the other being worth 5 drachms —and paid for all of them a square number, which, added to 60 units, made a square having the multiplicity of congii as its side: tell apart the eight-drachms and the five-drachms.4 The underlined pericope contains, untranslated, the term I shall focus on: ἑνῆ, apparently a hapax legomenon in the whole ancient Greek corpus, including papyri and inscriptions. The passage was paid only marginal attention in the available critical editions of the Diophantine writings, and has never been discussed in the literature. The aim of the present note is to offer such a discussion. Before addressing grammatical prob- lems, however, it is necessary to briefly present the Arithmetica and say something about its text. The Arithmetica was originally redacted in thirteen books.5 A ___ in P. Tannery, “Sur une épigramme attribuée à Diophante,” REG 4 (1891) 377–382, repr. Mémoires scientifiques II (Toulouse/Paris 1912) 433–439. 4 An algebraic transcription of this enunciation would read: mx + ny + a = (x + y)2, with mx + ny being a square number. In our text, x and y are the sought numbers, and the parameters m, n, a are the assigned values 8, 5, 60, respectively. 5 So Diophantus himself in his introduction: I 16.6–7. 710 ONES ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 large collection of problems arranged in six books is preserved in Greek.6 It is not clear whether such a division exactly reflects the original Diophantine partition, or rearrangements of the material have occurred at some stages of the transmission. In fact, the criterion by which the material of the Greek Arithmetica is organized into books is not always obvious, if indeed any general criterion is at work.7 As a consequence of this, not every manuscript of the Greek Arithmetica has it in six books; three of them divide Book 4 into two books, at least two other manuscripts divide Book 1 into two books.8 Even if the man- uscripts in question are late apographs, it is by no means assured that a similar phenomenon could not have occurred in very early copies. The loose deductive structure of the Arith- metica has also the consequence of easily allowing unauthentic material to creep into the text: Tannery deemed problems 2.1– 7 and 3.1–4 spurious, on the grounds that they contain mater- ial more properly pertaining to the final parts of Books 1 and 2, respectively, or that they repeat other propositions. Tannery suggested that those problems were marginalia, coming from a commentary on Books 1 and 2, that got inserted into the text.9 After comparison with the Arabic text, where problems 6.1–11 are likely interpolations, problems 4.1–2 of the Greek text 6 Four books of problems are transmitted in Arabic translation, referred to in the titles and the subscriptions as Books 4 to 7 of Diophantus’ treatise. The sets of Greek and of Arabic problems are almost completely non- overlapping. 7 Local criteria may apply; for instance, the sixth book contains problems of quite a distinctive kind. 8 See the description of the manuscripts in A. Allard, “La tradition du texte grec des Arithmétiques de Diophante d’Alexandrie,” RHT 12–13 (1982– 1983) 57–137, at 58–72. 9 P. Tannery, “La perte de sept livres de Diophante,” Bulletin des Sciences mathématiques 4 (1884) 192–206, repr. Mémoires scientifiques II 73–90, at 80–82, and Diophanti opera I 83 and 139 in app. Tannery suspected also problems 2.17–18 and 3.20–21, since they are identical with 1.22–23 and 2.15 and 14, respectively. FABIO ACERBI 711 ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 should also be regarded as spurious.10 There are at least two reasons to suspect the authenticity of Ar. 5.30.11 First, even if an epigram was a way of enunciating arithmetical problems or riddles that is well-established in the ancient corpus,12 5.30 is the only problem of the Arithmetica that is formulated in this way. Second, 5.30 is in fact the last prob- lem of Book 5, and we have seen that some problems near the end of other books have been considered later additions by the editors. On the other hand, the authenticity of 5.30 seems confirmed by a series of facts: the problem is a truly difficult one, its contents are homogeneous with those of the problems preceding it, the style in which the proof is redacted is defin- itely Diophantine. To reconcile this contradictory evidence, we might entertain the hypothesis that a late revisor modified the beginning and the end of the original problem in order to set it in its present form:13 it is an easy exercise to do it—to be fair, the same exercise could successfully be applied to any problem 10 See J. Sesiano, Books IV to VII of Diophantus’ Arithmetica, in the Arabic Translation Attributed to Qustā ibn Lūqā (New York 1982) 53. 11 Allard, Diophante 935, deems the problem spurious but I could not have access to the volume of his thesis where he discusses the issue. 12 Epigrams 14.116–146 of the Anthologia Palatina are arithmetical prob- lems; 14.126 deals exactly with the life-span of Diophantus. On the structure of this collection of arithmetical epigrams see P. Tannery, “Sur les épigrammes arithmétiques de l’Anthologie palatine,” REG 7 (1894) 59–62, repr. Mémoires scientifiques II 442–446. 13 The bulk of the problem (I 386.5–390.2) bears no trace of the (fictionally) practical context of the enunciation. On the other hand, the epigram and the explanatory enunciation contain the only two occurrences of the verb διαστέλλω “to tell apart” in the whole Arithmetica. This is a tech- nical term of the metrological domain and it is not used appropriately in our passage: it usually features in ‘problems of separation’, where what is added and has to be “told apart” are unhomogeneous magnitudes like line seg- ments and surfaces. We find a number of such problems in the pseudo- Heronian Geometrica, in fact a philological patchwork of disparate collections of metrological problems, assembled by their editor J. L. Heiberg and printed in Heronis opera omnia IV: see Geom. 21.9–10, 24.3, 24.10–13, 24.43, 24.46–47. 712 ONES ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 of the Arithmetica. Be that as it may: even if 5.30 was authored by some later scholiast and not by Diophantus himself (who probably lived about 150 CE and surely writes in a late Greek), we may safely submit that the hapax ἑνῆ was written on purpose by some Greek native speaker. But was it? The apparatus criticus of Tannery’s edition is notoriously unreliable, and one must check his readings against the man- uscripts. Now, the rich tradition of the Arithmetica (31 witnesses) can readily be reduced to no more than four independent manuscripts: Matrit. 4678, Vat.gr. 191 and 304, and Marc.gr. 308, the latter in fact containing a recension made by the renowned scholar Maximus Planudes (†1305).14 The readings are Matrit. 4678, f. 120r ἑνῆ; Vat.gr. 191, f. 386v ἑνῆ; Vat.gr. 304, f. 112r ἐνῆ; Marc.gr. 308, f. 244r ἐνῆ: the only variant is in the breathing. Tannery, who collated only the first and the last manuscripts, correctly puts ἑνῆ in the main text and relegates the vox nihili ἐνῆ to the apparatus.15 I say “correctly” since the immediately subsequent correlative ἐκ µὲν τοῦ ἑνὸς … ἐκ δὲ τοῦ ἑνὸς … leaves no doubt as to the meaning of ἑνῆ: it is a plurality of ‘ones’. The interesting point is how this plural was formed. The only scholar who has addressed this problem is Tannery, who translated the beginning of the clause containing ἑνῆ “Quidam vinum emit duarum qualitatum,” and who, in his index graecitatis, recorded the term s.v. ἑνῆ, adding “(ut plurale vocis ἕν?).”16 This suggestion is wrong, as we shall see by turning our attention to the attested plural forms of ἕν. 14 The most recent analysis of the manuscript tradition of the Arithmetica was provided by Allard, RHT 12–13 (1982–1983) 57–137, who uncon- vincingly argues that the first three manuscripts are independent copies of a lost archetype. A closer look, in fact, shows that the whole non-Planudean tradition depends on the Matritensis. 15 To be fair, also ἑνῆ with rough breathing was a vox nihili until now. 16 I 385 and II 270, respectively. Heath, Diophantus 224, translates “a certain number” and does not add any comment. Allard, Diophante 933, translates “deux sortes.” His apparatus at 896 does not record any vari- ants(!) for the ἑνῆ he writes in his text. FABIO ACERBI 713 ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 How to inflect ‘one’ in the plural: a survey of the relevant texts Admittedly, forming the plural of an adjective whose mean- ing is ‘one’ looks like doing something self-contradictory, and we shall see that some ancient grammarians agreed with this prima facie impression. Yet, there are contexts in which dealing with pluralities of ‘ones’ is perfectly natural, and this fact makes in turn necessary to create adequate lexical tools. The context most conducive to introducing a plurality of ‘ones’ is obviously the science of arithmetic, especially in view of the fact that numbers were conceived as collections of units (see below). Since nothing has survived of pre-Euclidean arithmetic, the first author where we read ‘one’ in the plural is Aristotle. This happens in three passages; in two of them the immediate context is strictly arithmetical (a number is “many ones”),17 even if they are embedded in more philosophically- coloured discussions about the potential infinity and about the dialectic between one and many, respectively. A third passage introduces a polemical, and more markedly philosophical, dimension: the targets are unidentified Platonizing philoso- phers, who claim that numbers have a primary ontological status, and that the ‘one’ is their principle—yet, Aristotle replies, it is absurd that there is a ‘one’ which is first of ‘ones’, 17 Ph. 4.7, 207b5–10: αἴτιον δ’ ὅτι τὸ ἕν ἐστιν ἀδιαίρετον, ὅ τι περ ἂν ἓν ᾖ (οἷον ἄνθρωπος εἷς ἄνθρωπος καὶ οὐ πολλοί), ὁ δ’ ἀριθµός ἐστιν ἕνα πλείω καὶ πόσ’ ἄττα, ὥστ’ ἀνάγκη στῆναι ἐπὶ τὸ ἀδιαίρετον (τὸ γὰρ τρία καὶ δύο παρώνυµα ὀνόµατά ἐστιν, ὁµοίως δὲ καὶ τῶν ἄλλων ἀριθµῶν ἕκαστος), and Metaph. I 6, 1056b20–25: οὕτως γὰρ λέγοµεν ἓν ἢ πολλά, ὥσπερ εἴ τις εἴποι ἓν καὶ ἕνα ἢ λευκὸν καὶ λευκά, καὶ τὰ µεµετρηµένα πρὸς τὸ µέτρον [καὶ τὸ µετρητόν]· οὕτως καὶ τὰ πολλαπλάσια λέγεται· πολλὰ γὰρ ἕκαστος ὁ ἀριθµὸς ὅτι ἕνα καὶ ὅτι µετρητὸς ἑνὶ ἕκαστος, καὶ ὡς τὸ ἀντικείµενον τῷ ἑνί, οὐ τῷ ὀλίγῳ. Here and henceforth, the relevant syntagmas are under- lined. In the first passage, ὄνοµα means “nominative singular” (Cat. 1, 1a12– 15). Since τρία and δύο are necessarily in the nominative plural and nominative dual, respectively, when they are used as a nominative singular this can be only done by paronymy (Cat. 8, 10a27–b11): see M. Ugaglia, Aristotele, Fisica. Libro III (Rome 2012) 170. 714 ONES ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 as they say, but not a dyad which is first of dyads, and so on.18 In this text, Aristotle does not resort to the same lexical material to denote the ‘one’ on the one hand and the numbers greater than it on the other: he writes τὸ ἓν µὲν εἶναί τι πρῶτον τῶν ἑνῶν […] δυάδα δὲ τῶν δυάδων µή, µηδὲ τριάδα τῶν τριάδων: ‘one’ is the substantivized numeral adjective τὸ ἕν, whereas δυάς and τριάς are numeral substantives. Aristotle might well have written19 ἡ ἑνὰς µὲν εἶναί τι πρῶτον τῶν ἑνά- δων or, better still, ἡ µονὰς µὲν εἶναί τι πρῶτον τῶν µονάδων, the noun µονάς being repeatedly used by Aristotle himself in the arguments immediately preceding this very passage. Quite likely, then, the nexus τὸ ἓν … τῶν ἑνῶν is intended to streng- then the connection with the criticized source. If ἕν and µονάς were used interchangeably by Aristotle20 and, as we shall see, in the early philosophical tradition, math- ematical exigencies induced a differentiation of their semantic range, a move that in its turn triggered further philosophical speculations about the difference between the two terms. Before seeing this, one must recall that numeral substantives, not substantivized numeral adjectives, were normally used to designate numbers as abstract entities: ἡ δυάς instead of τὰ δύο, ἡ τριάς instead of τὰ τρία, etc.21 This happens most 18 Metaph. M 8, 1083a21–27: εἰσὶ δ’ οὗτοι ὅσοι ἰδέας µὲν οὐκ οἴονται εἶναι οὔτε ἁπλῶς οὔτε ὡς ἀριθµούς τινας οὔσας, τὰ δὲ µαθηµατικὰ εἶναι καὶ τοὺς ἀριθµοὺς πρώτους τῶν ὄντων, καὶ ἀρχὴν αὐτῶν εἶναι αὐτὸ τὸ ἕν. ἄτοπον γὰρ τὸ ἓν µὲν εἶναί τι πρῶτον τῶν ἑνῶν, ὥσπερ ἐκεῖνοί φασι, δυάδα δὲ τῶν δυάδων µή, µηδὲ τριάδα τῶν τριάδων· τοῦ γὰρ αὐτοῦ λόγου πάντα ἐστίν. 19 The noun ἑνάς (in the plural form ἑνάδων) was firmly attested in the philosophical lexicon thanks to Plato Phlb. 15A4–7, where it refers to a list of four items, each qualified by a suitable form of the numeral adjective εἷς: ὅταν δέ τις ἕνα ἄνθρωπον ἐπιχειρῇ τίθεσθαι καὶ βοῦν ἕνα καὶ τὸ καλὸν ἓν καὶ τὸ ἀγαθὸν ἕν, περὶ τούτων τῶν ἑνάδων καὶ τῶν τοιούτων ἡ πολλὴ σπουδὴ µετὰ διαιρέσεως ἀµφισβήτησις γίγνεται. 20 See e.g. the discussion of the mathematical number at Metaph. M 6–7. 21 On all these forms see R. Kühner and F. Blass, Ausführliche Grammatik der griechischen Sprache I.13 (Hannover 1890) 621–624, §181. FABIO ACERBI 715 ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 naturally in number-theoretical contexts, and we read texts ac- cording to which a criterion that sets the theoretical science of arithmetic apart from lowly logistic is the use of the numeral substantives instead of the substantivized numeral adjectives.22 If, as is only natural, we find both in strictly arithmetical texts, their ranges of application were strictly non-overlapping. It suffices to look at the enunciation of Euclid Elem. 9.8:23 ἐὰν ἀπὸ µονάδος ὁποσοιοῦν ἀριθµοὶ ἑξῆς ἀνάλογον ὦσιν, ὁ µὲν τρίτος ἀπὸ τῆς µονάδος τετράγωνος ἔσται καὶ οἱ ἕνα διαλείπον- τες, ὁ δὲ τέταρτος κύβος καὶ οἱ δύο διαλείποντες πάντες, ὁ δὲ ἕβδοµος κύβος ἅµα καὶ τετράγωνος καὶ οἱ πέντε διαλείποντες. If as many numbers as we please beginning from a unit be in continued proportion, the third from the unit will be square, as will also those which leave out one; the fourth will be cube, as will also all those which leave out two; the seventh will be at once cube and square, as will also those which leave out five. In this sentence, µονάς is a number-theoretic entity and is hence designed by a numeral substantive, while the numeral adjectives count the tokens of a certain object (here the ἀριθµοί left out of the series). An enunciation where a numeral sub- stantive is the direct object of the verb λείπειν as in Elem. 9.8 is that of Diophantus Ar. 5.6 (I 322.2–5): 22 See the extract from Geminus in Def. 135.5, at Heronis opera IV 98.13– 18: λογιστική ἐστι θεωρία ἡ τῶν ἀριθµητῶν, οὐχὶ δὲ τῶν ἀριθµῶν, µετα- χειριστική, οὐ τὸν ὄντως ἀριθµὸν λαµβάνουσα, ὑποτιθεµένη δὲ τὸ µὲν ἓν ὡς µονάδα, τὸ δὲ ἀριθµητὸν ὡς ἀριθµόν, οἷον τὰ τρία τριάδα εἶναι καὶ τὰ δέκα δεκάδα, ἐφ’ ὧν ἐπάγει τὰ κατὰ ἀριθµητικὴν θεωρήµατα = beginning of scholium to Chrm. 165E6, in D. Cufalo (ed.), Scholia Graeca in Platonem I (Rome 2007) 173, no. 27. As for late Neoplatonism, see the whole discus- sion about subordination in sciences at Proclus In Euc. 38.1–41.2 Friedlein, in particular 40.2–9 (the entire argument relies heavily on Geminus). 23 Euclidis Elementa II 194.5–9 Heiberg-Stamatis. Note also the presence of ordinal numerals and the absence of the article in the first occurrence of the syntagma ἀπὸ (τῆς) µονάδος; Greek arithmetic avails itself of an unlimited supply of µονάδες: in its first occurrence in each proposition, “a” unit is taken. 716 ONES ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 εὑρεῖν τρεῖς ἀριθµοὺς ὅπως ἕκαστος µὲν αὐτῶν λείψας δυάδα ποιῇ τετράγωνον, ὁ δὲ ὑπὸ δύο ὁποιωνοῦν, ἐάν τε λείψῃ συναµφότερον, ἐάν τε τὸν λοιπόν, ποιῇ τετράγωνον. To find three numbers such that each of them minus a dyad24 makes a square, and the ‹rectangle contained› by two as we please, either minus their sum or minus their difference, makes a square. When numbers themselves are counted, the number desig- nating the plurality is a numeral substantive, while the number giving the amount is a numeral adjective.25 In more relaxed number-theoretic contexts (as in the extract from Domninus quoted in n.25), and invariably in logistic contexts, the names of numbers are expressed by numeral letters, preceded either by a substantivizing τά or by a ὁ, which reminds us that we are still speaking about an ἀριθµός. Occasionally, as e.g. in the Heronian Metrica, the numeral letters are preceded by a plural feminine article, intimating that one is really counting µονά- δες.26 This brings us back to the mathematical lexicon. Defining ‘number’ as a multiplicity composed of units27 requires one to define what a unit is. Elem. 7.def.1 does exactly this job: µονάς ἐστιν καθ’ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται.28 This Euclidean 24 The numeral substantive δυάς occurs 32 times in the Arithmetica and 8 in the Elements, prop. 9.32 (ter), 34 (quater), 36. 25 E.g. Domninus Ench. 8, 108.24–26 Riedlberger: τῶν δὲ περιττῶν οἱ µὲν τέµνονται εἰς ἴσους τινὰς ἀριθµούς, ὡς ὁ θʹ εἰς τρεῖς τριάδας, ὡς ‹ὁ› ιεʹ εἴς τε τρεῖς πεντάδας καὶ εἰς πέντε τριάδας, ὡς ὁ λεʹ εἰς εʹ ἑβδοµάδας καὶ εἰς ἑπτὰ πεντάδας. 26 Hero made his choice on purpose, as he himself declares at Metr. 1 praef. (Heronis opera III 6.4–7). 27 At Elem. 7.def.2: ἀριθµὸς δὲ τὸ ἐκ µονάδων συγκείµενον πλῆθος. The two definitions are at Euclidis Elementa II 103.2–4. 28 Among the several definitions attested in ancient writings, Chrysippus’ deserves a mention (Iambl. In Nic. 11.8–9 Pistelli = 2.10, 76.19 Vinel). It reads µονάς ἐστι πλῆθος ἕν, a paradoxical statement where envisaging a “unit multiplicity” is a semantic incongruity of the same kind as (and exactly the inverse of) Aristotle’s resorting to the plural of ‘one’ in his description of FABIO ACERBI 717 ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 definition makes clear the difference between the noun-token and the predicate (ἓν λέγεται), gets rid of the grammatical problem of pluralizing the adjective ‘one’, and, finally and most importantly, by resorting to the stem µον- avoids the defini- tional vicious circle that would have arisen from presenting the stem ἑν- (as in ἕν itself or in ἑνάς) both in the definiendum and in the definiens. Such a lexical convention is adhered to without exceptions in the whole of the Elements and in all the sub- sequent number-theoretical tradition. This fact neutralizes the possibility that what we read as Elem. 7.def.1 can be dismissed as irrelevant in that this definition might well have been for- mulated, and put at the very beginning of the arithmetic books of the Elements, by forcing a preexisting definition to fit the strict denotative practice of the treatise.29 It may also be that the philosophical connotations with which the term ἑνάς was charged already in early times (not to speak of τὸ ἕν, “the One” with which every Greek philosopher after Parmenides had to come to grips) might have suggested turning one’s attention to µονάς.30 The latter term almost surely belonged in the notional ___ ἀριθµός as ἕνα πλείω. 29 A circular characterization of this kind was around well before Euclid, if we are to believe Sextus Empiricus Math. 4.11: τὴν τοῦ ἑνὸς τοίνυν νόησιν διατυπῶν ἡµῖν πυθαγορικώτερον ὁ Πλάτων φησὶν “ἕν ἐστιν οὗ µηδὲν χωρὶς λέγεται ἕν”, the syntagma οὗ µηδὲν χωρίς being exactly the double negation of the Euclidean καθ’ ἣν ἕκαστον τῶν ὄντων. Note however Sextus’ for- mulation τὴν τοῦ ἑνὸς νόησιν διατυπῶν “outlining the notion of the one,” which does not seem to entail a technical, definitional context: the aim of the reported characterization is only to establish a link between the Basic Entity (noun ἕν) and our categorizations (predicate ἕν)—it remains that one would have liked to read a τὸ before the first ἕν, and that the whole clause is dangerously near to a wordplay. Doubts about the authenticity of Elem. 7.def.1 are a priori legitimate, since this definition is totally ineffective from the mathematical point of view and (as a consequence) is never applied in the Elements. 30 Recall the occurrence at Plato Phlb. 15A6 (n.19 above) and the fragment (apud Alexander apud Simplicius In Ph. 99.12–16 Diels and apud Philoponus In Ph. 42.9–17 Vitelli [Philoponus, however, does not mention any source]), where Eudemus lends Zeno the term ἑνάς for a description of 718 ONES ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 and terminological heritage of the early Pythagoreans, but in this case they did not show themselves very eager to fix a sharply defined lexicon.31 This rather plethoric terminological supply needed to be structured and hierarchized. The task was taken up by philosophically-oriented writers on technical matters, mainly of Middle- and Neoplatonic bent. A technical differentiation be- tween ἕν and µονάς, amounting to that between logistic and arithmetic and putting due emphasis on the dichotomies ‘intelligible’/‘sensible’ and ‘indivisible’/‘divisible’, can be read in Theon of Smyrna.32 Philosophically-marked speculations about the difference between ἕν and µονάς are attested in one of the pseudo-Heronian Definitiones, in a section that is a cento of extracts from Proclus and maybe other Neoplatonists. The author of this short text regards the ἕν as an entity which is “higher” than the µονάς, and in fact a principle of it and of the dyad. The text, however, suffers from a disappointing confu- ___ τὰ πολλὰ as πλῆθος ἑνάδων (Philoponus interchanges the two genera and writes τὸ πλῆθος ἐκ πλειόνων ἑνάδων σύγκειται). 31 This much is expressly asserted by Theon of Smyrna, Exp. 20.19–20 Hiller: Ἀρχύτας δὲ καὶ Φιλόλαος ἀδιαφόρως τὸ ἓν καὶ µονάδα καλοῦσι καὶ τὴν µονάδα ἕν, and is implicit in the quotation in Iamblichus In Nic. 77.9–11 Pistelli (= 4.86, 146.7–8 Vinel), making up fr.44 B 8 D.-K. of Philolaus: ἡ µὲν µονὰς ὡς ἂν ἀρχὴ οὖσα πάντων κατὰ τὸν Φιλόλαον (“οὐ γὰρ ἕν” φησιν “ἀρχὰ πάντων”). But note that the doctrine assigning to the Unity the role of a first principle is quite definitely Neoplatonic, not early Pythagorean: for them, the numbers themselves have the status of principle and matter of everything, their elements being the even and the odd; the elements of the latter are in their turn the Bounded and the Unbounded (the corres- pondence is chiastic), while the ‘one’ (which is both even and odd) proceeds from both of these (Arist. Metaph. Α 5, 986a15–21). On this fragment see C. A. Huffman, Philolaus of Croton, Pythagorean and Presocratic (Cambridge 1993) 345–346. 32 Exp. 19.18–20.4: καὶ µονὰς τοίνυν ἐστὶν ἡ τοῦ ἑνὸς ἰδέα ἡ νοητή, ἥ ἐστιν ἄτοµος· ἓν δὲ τὸ ἐν αἰσθητοῖς καθ’ ἑαυτὸ λεγόµενον, οἷον εἷς ἵππος, εἷς ἄνθρωπος. ὥστ’ εἴη ἂν ἀρχὴ τῶν µὲν ἀριθµῶν ἡ µονάς, τῶν δὲ ἀριθµη- τῶν τὸ ἕν· καὶ τὸ ἓν ὡς ἐν αἰσθητοῖς τέµνεσθαί φασιν εἰς ἄπειρον, οὐχ ὡς ἀριθµὸν οὐδὲ ὡς ἀρχὴν ἀριθµοῦ, ἀλλ’ ὡς αἰσθητόν. ὥστε ἡ µὲν µονὰς νοητὴ οὖσα ἀδιαίρετος, τὸ δὲ ἓν ὡς αἰσθητὸν εἰς ἄπειρον τµητόν. FABIO ACERBI 719 ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 sion between ἕν and ἑνάς.33 More refined distinctions between ἕν and µονάς are presented, just after the ‘technical’ one we have mentioned, again by Theon of Smyrna,34 who explains among other things why Plato in the Philebus employed ἑνάς: it is a third species of ‘one’, namely, “a unit participating of the one” (µονὰς µετοχῇ τοῦ ἑνός). Neo-Pythagorean authors such as Nicomachus (who does not even define the µονάς), Iamblichus, Anatolius, and in general all the arithmological literature, were apparently not interested in drawing such subtle distinctions, since their philosophical reference systems did not envisage hierarchies among objects as 33 Def. 136.28, in Heronis opera IV 132.22–134.3: ἔστι διαφορὰ µονάδος καὶ ἑνάδος οὕτως· ἐπειδὴ ἔστιν ἐν τοῖς οὖσιν εἰδοποιία καὶ ταυτότης, καλεῖται µονάς. ἔστι δὲ ἑτερότης· καλεῖται δυάς. ἔστιν ἑτέρα ὑπερτέρα δύναµις, ἀρχὴ κοινὴ τῶν δύο τούτων, ἥτις πάντα ἐπίσταται· αὕτη ἓν καλεῖται. ὥστε τὸ ἓν ὑπέρτερόν ἐστι τῆς µονάδος. ἰστέον δέ, ὅτι, ἐπειδὴ ἔστι δυὰς καὶ µονὰς καὶ τὸ ἕν, δυὰς µὲν αὐτὰ τὰ σώµατα, µονὰς δὲ τὸ εἶδος τὸ ἐν αὐτοῖς, ἓν δὲ ἡ φύσις. The passage (which very likely conflates two char- acterizations) has no parallels in other authors. 34 Exp. 20.12–19: οἱ δὲ καὶ αὐτῶν τούτων ἀρχὴν τὴν µονάδα φασὶ καὶ τὸ ἓν πάσης ἀπηλλαγµένον διαφορᾶς ὡς ἐν ἀριθµοῖς, µόνον αὐτὸ ἕν, οὐ τὸ ἕν, τουτέστιν οὐ τόδε τὸ ποιὸν καὶ διαφοράν τινα πρὸς ἕτερον ἓν προσειληφός, ἀλλ’ αὐτὸ καθ’ αὑτὸ ἕν. οὕτω γὰρ ἂν ἀρχή τε καὶ µέτρον εἴη τῶν ὑφ’ ἑαυτὸ ὄντων, καθὸ ἕκαστον τῶν ὄντων ἓν λέγεται, µετασχὸν τῆς πρώτης τοῦ ἑνὸς οὐσίας τε καὶ ἰδέας, which the sentence quoted above (n.31) immediately follows. Another point of view on the difference between ἕν and µονάς is related at Exp. 21.7–19: ἔνιοι δὲ ἑτέραν διαφορὰν τῆς µονάδος καὶ τοῦ ἑνὸς παρέδοσαν. τὸ µὲν γὰρ ἓν οὔτε κατ’ οὐσίαν ἀλλοιοῦται, οὔτε τῇ µονάδι καὶ τοῖς περιττοῖς αἴτιόν ἐστι τοῦ µὴ ἀλλοιοῦσθαι κατ’ οὐσίαν, οὔτε κατὰ ποιό- τητα (αὐτὸ γὰρ µονάς ἐστι καὶ οὐχ ὥσπερ αἱ µονάδες πολλαί) οὔτε κατὰ τὸ ποσόν (οὐδὲ γὰρ συντίθεται ὥσπερ αἱ µονάδες ἄλλῃ µονάδι)· ἓν γάρ ἐστι καὶ οὐ πολλά, διὸ καὶ ἑνικῶς καλεῖται ἕν. καὶ γὰρ εἰ παρὰ Πλάτωνι ἑνάδες εἴρηνται ἐν Φιλήβῳ, οὐ παρὰ τὸ ἓν ἐλέχθησαν, ἀλλὰ παρὰ τὴν ἑνάδα, ἥτις ἐστὶ µονὰς µετοχῇ τοῦ ἑνός. κατὰ πάντα δὴ ἀµετάβλητον τὸ ἓν τὸ ὡρισµέ- νον τοῦτο ἐν τῇ µονάδι. ὥστε διαφέροι ἂν τὸ ἓν τῆς µονάδος, ὅτι τὸ µέν ἐστιν ὡρισµένον καὶ πέρας, αἱ δὲ µονάδες ἄπειροι καὶ ἀόριστοι. Note the adverb ἑνικῶς, which scores 554 occurrences in the ancient Greek corpus; the adjective ἑνικός denotes the grammatical “singular,” and so does the ad- verb in this clause. 720 ONES ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 far as their ontological status is concerned. For this reason, and since the early Pythagorean tradition apparently did not pay attention to the term, the noun ἑνάς has zero occurrences in these writings.35 On the other extreme of the spectrum, a vertiginous lexical proliferation took place in the writings of the late Neoplatonic philosophers. Their ontology included com- plex hierarchies of entities, each of which had its share of uniqueness and priority, let us say of ‘oneness’: it was “a system that, after the One-Good and at the summit of multiple levels of the intelligible, preserves a place for the ‘henads’.”36 From this came the exigency of mobilising the widest possible ter- minological apparatus: ἕν, ἑνάς,37 µονάς—and, what is im- portant, of inflecting these terms in the plural. Even taking into account the bias due to the fact that Neoplatonic writings make up most of the philosophical record coming from antiquity,38 it remains the case that a remarkable number of the TLG oc- currences of such and related terms come from Neoplatonic authors. To give some figures, of the 8774 occurrences of the noun µονάς, 1011 are in Proclus, 467 in Damascius.39 As is to be expected, the substantive ἑνάς has the best score among the 35 The only exception, at Theol.ar. 76.14–15 De Falco, is a calembour be- tween ἐννεάς and ἑνάς just at the beginning of the section on the ennead. 36 Ph. Hoffmann, “What was Commentary in Late Antiquity? The Example of the Neoplatonic Commentators,” in M. L. Gill and P. Pellegrin (eds.), A Companion to Ancient Philosophy (Malden 2006) 597–622, at 598. 37 As seen above, there is a reason for the Neoplatonists being fond of this term: the occurrence in Plato Phlb. 15A6, already an object of exegesis in the Middle-platonic Theon of Smyrna, Exp. 21.14–16 (it is the underlined sen- tence in n.34). Note that Syrianus, In Metaph. 183.24 and 194.29 Kroll, had no problems in recognizing ἑνάς and µονάς as synonyms. 38 See R. Goulet, “La conservation et la transmission des textes philo- sophiques grecs,” in C. D’Ancona (ed.), The Libraries of the Neoplatonists (Leiden 2007) 29–61. 39 The adjective µοναδικός scores 838 occurrences; only 10 of them are in Damascius, 175 in Proclus. Of the adjective µοναδιαῖος there are only 5, all in technical contexts: Metr. 2 praef. (bis: Heronis opera III 94.3.6) and Pro- legomena ad Almagestum (ter: Diophanti opera II 7.19.23, 8.26). FABIO ACERBI 721 ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 Neoplatonists: of 854 occurrences, 398 of them in the plural(!), 420 are in Proclus, 164 in Damascius.40 The fact that both in Proclus and in Damascius41 we find plural forms of ἕν brings us back to the primary object of this note.42 The plural of ἕν: grammatical remarks As to the formal correctness of the plural of ‘one’, we can start by considering the compound οὐδ(µηδ)-είς. In the corpus of authors writing in Attic, the plural forms οὐδ-ένες, µηδ-ένες (implying a plural ἕνες for εἷς: οὐδ’, µηδ’ ἕνες), οὐδένων, µηδένων (οὐδ’, µηδ’ ἑνῶν), οὐδέσι, µηδέσι (οὐδ’, µηδ’ ἑσί), οὐδένας, µηδένας (οὐδ’, µηδ’ ἕνας) were currently used.43 To be sure, the parallel between εἷς and οὐδείς is only a partial one, as recognized by the late grammarians and lexicographers even if on the basis of a limited lexicographical record: they pretended that one of the main differences is that εἷς does not admit of plural forms.44 We have seen that this is not the case; 40 On the other hand, one finds only 9 occurrences of the adjective ἑναδικός, scattered among several authors. 41 Forms of the plural genitive can be found at Proclus In R. I 258.16–18 Kroll (bis), In Ti. II 143.1 Diehl, and Damascius Pr. I 55.12, 71.11, II 44.1 Westerink, In Prm. IV 83.24, 90.22, 134.4 Westerink. The several occur- rences in [Alexander] (immo Michael of Ephesus) In Metaph. 765.14–17 Hayduck, and Simplicius In Ph. 504.34–35, 505.15, are all included in com- ments on the first and third Aristotelian passages given at nn.17–18 above. 42 Note also the 861 occurrences of ἑνότης, mainly in writings on theological subject. A corresponding adjective *ἑνοτικός is not attested. 43 See Photius Lex. O 612 (III 119 Theodoridis) s.v. οὐδένες: τὸ πληθυντι- κὸν τοῦτο σύνηθες τοῖς παλαιοῖς· καὶ οὐδένων καὶ οὐδέσι καὶ οὐδένας. In the corpus, I did not find occurrences of plural neuter. 44 See Choeroboscus In Theod. IV.1 205.13–25 Hilgard (differences be- tween εἷς and the δείς of οὐ-δείς), in particular 205.20–23: τρίτον δὲ ὅτι τὸ µὲν εἷς οὐκ ἐπιδέχεται πληθυντικά· οὐδὲ γὰρ λέγοµεν οἱ ἕνες (ἄτοπον γὰρ τὸν ἕνα πληθύνεσθαι)· τὸ δὲ δείς ἐπιδέχεται πληθυντικά· λέγοµεν γὰρ οἱ οὐδένες τῶν οὐδένων τοῖς οὐδέσι. We find the same in Etym.Magn. 305.9–17 Kallierges, in particular 305.13–15: τὸ µὲν εἷς οὔτε δυϊκὰ ἔχει, οὔτε πλη- θυντικά· τὸ δὲ οὐδεὶς καὶ δυϊκὰ ἔχει καὶ πληθυντικά, οἱ οὐδένες, τῶν οὐ- δένων. The same opinion is given by Theon of Smyrna, Exp. 21.13–14 (cf. n.34 above): ἓν γάρ ἐστι καὶ οὐ πολλά, διὸ καὶ ἑνικῶς καλεῖται ἕν. 722 ONES ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 our information concerning the neuter ἕν can be summarized in the following terms: 1) Morphology: the plural of ἕν, ἑνός is ἕνα, ἑνῶν, as expressly said by [Alexander] when commenting on Aristotle Metaph. 1083a21–27 (n.18 above).45 Also everyday Attic, for expressive- ness’ sake, employed the resource of the plural of ‘one’ in specific phrases.46 2) Accentuation: ἕν is monosyllabic, hence plural ἕνα, ἑνῶν (dative *ἑσί, unattested).47 45 [Alexander] (Michael of Ephesus) In Metaph. 765.14–16, ad τῶν ἑνῶν of Arist. Metaph. 1083a25: τουτέστι τῶν µονάδων (τὸ γὰρ ‹ἑνῶν› ἐκ τοῦ ἕν, ἑνός, ἡ εὐθεῖα τῶν πληθυντικῶν τὰ ἕνα καὶ ἡ γενικὴ τῶν πληθυντικῶν τῶν ἑνῶν εἴληπται). 46 Suda Ο 830 (III 581.11–12 Adler = Aelius Dionysius Ἀττικὰ ὀνόµατα O 38 Erbse = Photius Lex. O 624 [III 121] s.v. οὐδ’ ὑφ’ ἕνων): Ἀττικοὶ ἀντὶ τοῦ ὑπ’ οὐδένων λέγουσιν ὑπερβιβάζοντες. The hyperbaton refers to the deplacement of the preposition ὑπό. It is tempting to correct the accent of ἕνων to ἑνῶν, since in this way the witz is more effective (see also the following note): as it were, a hyperbaton from “by no-one” to “not by ones” and not to “not by one.” 47 But the manuscript tradition hands down ἕνων as a plural genitive at Proclus In Ti. II 147.1.6 and Damascius Pr. I 71.11 (the latter is corrected to ἑνῶν by the editor). J. Combès, in a note (at 152) ad Damascius Pr. I 55.12, somewhat rashly asserts: “Inutile de dire que le choix est arbitraire, puisque Damascius n’écrivait pas les accents, et que la forme n’existait pas dans la langue vivante.” Of course, the choice is not arbitrary: there were accen- tuation rules in Greek antiquity—it suffices to think of Herodian’s (late second century) Καθολικὴ προσῳδία—and the change from pitch accent to stress accent that was about to be accomplished in Diophantus’ time made even more urgent the need to fix precise accentuation rules. Another matter is the evidence about written accents contained in late-antique papyri or in early (majuscule) Byzantine manuscripts—the earliest dated MSS. displaying a complete system of accentuation are Vat.Barb.gr. 336 and Vat.gr. 1666, both ca. 800. Now, given the highly technical character of our text, it is quite pointless to refer to the ‘practice’ (in fact, quite a variable one) of ac- centuation in literary papyri, a practice almost exclusively limited to poetic texts—see however C. M. Mazzucchi, “Sul sistema di accentazione dei testi greci in età romana e bizantina,” Aegyptus 59 (1979) 145–167—what matters is what we find in technical texts. Important mathematical papyri such as the first-century P.Vindob.gr. inv. 19996 (PM3 2322: metrological problems) FABIO ACERBI 723 ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 It seems thus clear that, from the point of view of everyday grammatical and linguistic competence, the neuter plurals ἕνα, ἑνῶν were considered unproblematic and in fact the most natural forms. Also for this reason, a form such as ἑνῆ can hardly be viewed as a plural of ἕν, as Tannery tentatively had it. Yet, the form ἑνῆ in Diophantus is striking both in its mor- phology and in its accent. One might be tempted to correct the text, and there are at least two possibilities: correcting to ἕνα or postulating a different nominal form. As for the first possibility, in the majuscule grapheme ΕΝΗ, -Η can hardly be a misreading of -Α (so ΕΝΗ ← ΕΝΑ, i.e. ἕνα): a technical text like the Arithmetica had every chance of being transcribed in an informal hand such as the one we see in the fragmentum mathematicum bobiense,48 and there Η is quite dissimilar from Α. The grapheme -ΝΗΟΙΝ- might instead be a palindromic quibble arising from an aural mistake of a late Greek speaker. The transliterator, who very likely knew the form ἕνα, could well have introduced it, thus correcting to ΕΝΑ (i.e. ἕνα) the presumed mistake. He could have done this but he did not. There must have been some hard thought on his part about this stretch of text: the ___ and the early second-century P.Mich. III 144 (arithmetical problems very much in the style of Diophantus) are not accented, nor are the astronomical papyri P.Oxy. 4133–4300a edited in A. Jones, Astronomical Papyri from Oxy- rhynchus (Philadelphia 1999). The 5th–6th century fragmentum mathematicum bobiense (see n.48) is replete of abbreviations but bears no accents; even in the early ninth-century majuscule Par.gr. 2389 (Ptolemy Almagestum), where no abbreviations are used, the accents were (selectively) added by a later hand. My choice in the next paragraph of assigning the accentuation of Diophantus’ Arithmetica to the medieval transliterator from a majuscule to a minuscule script is arbitrary but is likely to strike near the truth: the accents, in Proclus’ and Damascius’ passages mentioned at the beginning of this note as well as in the case of Diophantus’ ἑνῆ, are unlikely to have ancient authority. 48 This is the scriptio inferior, dated to the 5th–6th century, of the palimpsest Ambros. L 99 sup. (Isidore of Seville Etymologiae), whose pages 113–114 are reproduced in C. Belger, “Ein neues Fragmentum mathematicum Bobi- ense,” Hermes 16 (1881) 261–284. 724 ONES ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 subsequent ἑνός and the immediately preceding δύο (note the [deliberate?] oxymoron δύο [immo βʹ] ἑνῆ) suggested to him to isolate the grapheme ΕΝΗ so as to make it a word on its own and to provide it with a rough breathing, but only his gram- matical skills made him put a circumflex above the Η instead of resorting to the lectio facilior ἕνα. As a second possibility we might postulate, as an analogical formation of sorts based on its opposite πλῆθος, a singular like *ἕνος, ἕνους, whose plural should be *ἕνη, ἑνῶν, ἕνεσι, and correct the text accordingly by simply changing the accent. Of course, this amounts to replacing one unattested form with another unattested form. We have also seen that the reading ἑνῆ has a sound manu- script tradition: it would therefore be unmethodical to correct the text, in view also of the fact that the tradition is a highly technical one and that some questioning should have arisen about a form that does not coincide with those attested in the philosophical tradition (the plural ἕνα of ἕν and the plural forms of ἑνάς). The most economical solution, in fact, is to assign ἑνῆ to an adjective ἑνής ἑνές, plural ἑνεῖς ἑνῆ, having the genitive plural ἑνῶν as an entry in common with the attested plural of ἕν and with the plausible plural of an unattested *ἕνος. The redactor of our text preferred this form to ἑνικός, of which he could have seen a model in ἑνάς > ἑναδικός, but that, in its turn, was charged with grammatical overtones. An adjective such as ἑνής ἑνές, instead, widens the semantic range of the numeral ἕν in the direction of an attributive, while keeping, in the very clause in which it is introduced, an exclusive link with the originating stem (ἑνῆ … ἐκ µὲν τοῦ ἑνὸς … ἐκ δὲ τοῦ ἑνὸς …): what is at issue are not simply “two units of wine,” but “two singularities/ peculiarities of wine,” that is, two qualities of wine (“qualitates” in the pen of Tannery). This discussion should not make us forget that a dual of ‘one’ is required in our passage, not a plural. The dual of ἑνής ἑνές FABIO ACERBI 725 ————— Greek, Roman, and Byzantine Studies 53 (2013) 708–725 should be ἑνεῖ.49 But then, we might well be entitled to correct the text, postulating a mistake of quality (most likely an itacism) on the part of some copyist. Against this hypothesis stands the simple fact that forms of dual are banned from the lexicon of Greek mathematics; in particular, no dual is attested in the Arithmetica, where the substantives or adjectives denoting or qualifying pairs of numbers are always in the plural.50 September, 2013 CNRS, UMR8560 ‘Centre Alexandre Koyré’, Paris fabacerbi@gmail.com 49 On the impossibility of forming such a dual (see n.44 above on the analogous impossibility, alleged by Choeroboscus, of forming the plural), see Choeroboscus In Theod. IV.1 132.34–38: πῶς γὰρ τὸ εἷς καὶ τὸ ἕν εἰς δυϊκὰ κλίνεσθαι δύνανται δυϊκῷ ἀριθµῷ θεωρηθῆναι µὴ δυνάµενα; ἐὰν γὰρ παραληφθῶσιν ἐν δυϊκῷ ἀριθµῷ, καὶ τοῦ σηµαινοµένου ἐκπίπτουσι καὶ οὐκ ἔτι σηµαίνουσι τὸν ἕνα ἀριθµόν, ὥσπερ καὶ ἐκ τοῦ ἐναντίου ὁ δύο ἀριθµὸς οὐ δύναται ἑνικῷ ἀριθµῷ θεωρηθῆναι. On the “nominative singular” of δύο and τρία see n.17 above. 50 This note owes very much to discussions with Alessandro Lami.