apeiron 2023; aop Leonid Zhmud* The Menaechmi https://doi.org/10.1515/apeiron-2022-0101 Published online March 24, 2023 Abstract: In the mid-first century BC Geminus of Rhodes, a scientist and philosopher close to Posidonius, composed a comprehensive Theory of Mathematical Sciences, in the surviving fragments of which the numerous characters are referred to plainly by name, with some of them being namesakes of other, more well-known mathematicians and philosophers. This paper tries to set apart the namesakes of Geminus, of which there are four in his fragments: Theodorus, Hippias, Oenopides, and Menaechmus. Keywords: Greek mathematics, namesakes, methodology of mathematics, Geminus of Rhodes Belying its title, this article will not examine the famous play by Plautus, but the problem posed by namesakes in the history of ancient thought. Greek sources often mention writers, philosophers, and scientists only by their names, with the patronymic, origin, and field of activity either left out or not preserved, which makes it difficult for us to choose between the carriers of a certain name in each particular case. With the significant rise in written output during the Hellenistic age this grew to be seen as a problem by the Greeks themselves. Demetrius of Magnesia, a grammarian and a librarian living in Rome in the mid-first century BC, compiled a special treatise On Poets and Authors of the Same Name, complete with a catalogue of their works.1 In the majority of cases when the person in question is of some renown, the problem yields a more or less satisfactory answer, since we usually possess a great number of additional details making it possible to understand who was meant. But whenever a lesser-known author is the case, distinguishing him from his namesakes becomes very difficult. Thus, three Stoic philosophers are mentioned in Diogenes Laertius’ list of 20 Theodori (II, 103–104), about whom we know next to nothing. The problem is further aggravated by a certain penchant to allocate credit to the eminent at the expense of their minors for which Robert Merton, the founder of the sociology 1 For preserved named fragments, see Mejer (1981). *Corresponding author: Leonid Zhmud, Institute for the History of Science, RAS, St. Petersburg, Russian Federation, E-mail: l.zhmud@spbu.ru. https://orcid.org/0000-0002-4873-6442 2 L. Zhmud of science, has coined the term the Matthew effect: “for unto one that hath shall be given, and he shall have abundance”.2 Demetrius’ close contemporary Geminus of Rhodes, a scientist and philosopher from the circle of Posidonius, composed ca. 70–50 BC a comprehensive Theory of Mathematical Sciences (Μαθημάτων θεωρία) in no fewer than eight books. The scope of his encyclopedia corresponded well with its title: it established the external and internal boundaries of science (understood as mathēmata), offered a classification of individual disciplines, notions and objects, discussed the principles (ἀρχαί) of mathematics, and, in general, dealt with what is now current under the methodology and philosophy of science.3 Excerpts of this work survived mainly in the commentary of Proclus on the first book of Euclid’s Elements and in Ps.-Hero Definitions 135 and 137.4 The classification of sciences drawn by Geminus, the most detailed and elaborate in antiquity, gained momentum after the commentary of Proclus was published in 1533, translated into Latin by Francesco Barozzi in 1560, and underpinned all the subsequent classifications well into the early 19th century. Although Geminus’ fragments are of considerable value for the history of science and philosophy, they have not yet enjoyed a separate edition; the last special works on their attribution were published at the turn of the 19th century.5 In Geminus’ fragments, the numerous characters are referred to plainly by name, with some of them being namesakes of other mathematicians and philosophers. If we split all the persons mentioned into two groups – those 1) who are wellknown, for instance, Archimedes, Apollonius and others, or at least who are securely identified, for example, Nicomedes, and 2) those who are unknown to us, or allow only for insecure identification, such as Amphinomus, Andron, Zenodotus, Oenopides and others – a certain conclusion suggests itself. With just one exception, all the mathematicians in the first group belong to the Hellenistic period. There is an obvious reason for this: the works of Euclid, Archimedes, Apollonius first eclipsed and then supplanted the writings of their predecessors. It is only natural that Geminus drew on the Hellenistic mathematicians, while keeping abreast of the current methodological discussions of his contemporaries. Hence, we have excellent grounds to believe that when an obscure or insecurely identified person comes to light, he is more likely to belong to the Hellenistic rather than the classical period. These considerations may help set apart the namesakes of Geminus, of which there are four in his fragments: Theodorus, Hippias, Oenopides, and Menaechmus. 2 Merton (1968). 3 Kouremenos (1994), Vitrac (2005, 2009), and Acerbi (2010). 4 For material on optics, see: Damian. Opt., 22.10–30.11, on logistics, i.e., applied arithmetic: Schol. in Plat. Chrm. 165e7. 5 Tittel (1895, 1912) and van Pesch (1900). The Menaechmi 3 Theodorus (more than 200 namesakes in the Realencyclopädie) is mentioned in connection with the so-called mixed lines (γραμμαὶ μικταί) – curved lines, composed of straight lines and circular lines, e.g., helixes. Geminus held that the mixture (μίξις) in such lines did not come about through blending (κρᾶσις), a view to which Theodorus the mathematician erroneously adhered (Procl. In Eucl., 118.7–8). This opinion about the mixed lines, which are absent from Euclid, could not belong to Theodorus of Cyrene, a Pythagorean mathematician of the fifth century BC, the teacher of Theaetetus and Plato. We are, consequently, dealing with a later mathematician, whom Paul Tannery tentatively identified with Theodorus of Soli, an interpreter of mathematical passages in Plato’s Timaeus mentioned by Plutarch,6 but omitted in Diogenes Laertius’ list of Theodori. This identification has convinced many; I, however, remain unconvinced. Mixed lines is quite a technical subject and ὁ μαθηματικός, by implication, is a specialist in mathēmata, whereas in Plutarch Theodorus figures alongside two other interpreters of Plato – the Academic Crantor and the Peripatetic Clearchus, the author of On the Mathematical Sections in Plato’s Republic. The way Theodorus treats the five Platonic bodies in their connection with the four elements in one case, and with the numerical progression 1, 2, 3, 4, 9, 18, 27 in the other betrays a typical philosopher,7 whom Geminus, most likely, would have called φιλόσοφος. While Theodorus the philosopher lived at the end of the fourth century BC, the point made by Theodorus the mathematician concerning the properties of mixed lines definitely belongs to a later date. Hippias (18 namesakes in the Realencyclopädie) figures twice in the material derived from Geminus. First, we read in Proclus that Nicomedes trisected a rectilinear angle by means of the conchoids that he invented, while others have used quadratices of Hippias and Nicomedes to solve this problem, and still others the spirals of Archimedes (In Eucl., 272.1–12). Secondly, Proclus says that “mathematicians are accustomed to distinguish lines, giving the property of each species. Apollonius, for instance, shows for each of his conic lines what its property is, and Nicomedes likewise for the conchoids, Hippias for the quadratrices, and Perseus for the spiric curves” (In Eucl., 356.6–12, tr. G. Morrow). In both passages we encounter Geminus’ usual classificatory practice with reference to Hellenistic mathematicians, some well-known (Archimedes, Apollonius), some lesser known (Nicomedes, Perseus), and some unknown (Hippias). However, the author of the first comprehensive modern history of mathematics J.-E. Montucla did not want to leave Hippias unknown and identified him with the fifth century BC Sophist Hippias of Elis, since no other geometer of that name was known in antiquity.8 6 Plut. De defect. orac. 427A, C; De anim. procr. 1022A, 1027A; Tannery (1887, 28). 7 Dörrie and Baltes (1987, 344–348). 8 Montucla (1758, 198–199): “Je ne crois pas que l’antiquité nous fournisse aucun autre géomètre de ce nom, que celui dont je parle”. 4 L. Zhmud Although the Sophist Hippias was knowledgeable in mathēmata and taught them (Pl. Hipp. mai. 285a–d, Hipp. min. 367–368b), nothing is known about his scientific discoveries, and nobody ever called him a ‘mathematician’. To be sure, he was mentioned by Proclus in an excerpt from Eudemus of Rhodes’ History of Geometry (In Eucl. 65.12–15 = Eud. fr. 133 Wehrli), though not as a geometer, but as a source about a certain Mamercus, brother of the poet Stesichorus. Montucla’s identification was questioned as early as the 19th century,9 while Wilbur Knorr finally exposed its impossibility.10 Indeed, the quadratrix, a curve defined kinematically, cannot be attributed to the fifth century BC without doing violence to the inner logic of the development of geometry, since motion was introduced in geometry by Archytas of Tarentum, a brilliant mathematician a generation younger than Hippias, while Dinostratus, a student of Eudoxus, himself a student of Archytas, was the first to solve the problem of squaring the circle using the quadratrix (this explains its name τετραγωνίζουσα). It is worth adding that in Geminus Hippias demonstrated the properties of several quadratrices, which places him unreservedly in the context of Hellenistic mathematics. Most specialists in Greek mathematics have since then sided with Knorr,11 but are still outnumbered by those who, relying on old works, maintain a firm belief in the exceptional mathematical prowess of the Sophist Hippias. Two long excerpts from Geminus (Procl. In Eucl., 77.7–81.4, and 200.21–202.25)12 present different views of what theorems and problems are and how they differ from each other. According to Geminus himself, theorems are propositions (προτάσεις), the purpose of which is to “demonstrate the existence or nonexistence of an attribute”, while problems are propositions aimed at the construction of “what in a sense does not exist”, i.e., construction problems (In Eucl., 201.2–15, tr. G. Morrow). Earlier he quoted the definitions given by a certain Zenodotus,13 “who belonged to the succession (διαδοχή) of Oenopides, but was a student of Andron” (ibid., 80.15–17). Andron and Zenodotus are otherwise unknown, while Oenopides is usually held to be an astronomer and geometer Oenopides of Chios, active in the mid-fifth century BC (DK 41).14 This identification is, to my mind, impossible. The students of Oenopides of Chios, if any, are virtually unknown, with even his younger contemporary and 9 See discussions in Allman (1889, 92–94, 193), Björnbo (1913), and Heath (1921, 182, 225–229). 10 Knorr (1986: 80–82). 11 Cf., however, Sefrin-Weis (2010, 248–249), who erroneously states that the “sophist Hippias is mentioned as the inventor of the curve itself”. 12 See Tittel (1895, 24–26) and van Pesch (1900, 100, 102). In Eucl., 78.13–79.2 comes from Proclus himself. 13 οἱ περὶ Ζηνόδοτον means simply Zenodotus, as the use of this expression in Proclus’ commentary shows. 14 E.g. DK 41 A 12; von Fritz (1937, 2267–2271). Cf. Mueller (1991: 95–97) and Bodnár (2007: 2–3). The Menaechmi 5 compatriot Hippocrates of Chios (DK 42), who could well be one, never mentioned as such. How could this Oenopides get his own διαδοχή – a term denoting the philosophical schools of Hellenism? The only mathematical school in Greece that lasted for three generations, was Eudoxus’ school in Cyzicus, founded ca. 362 BC,15 in all other cases, we only know about teacher-student pairs. Further on, could a follower of Oenopides of Chios define a theorem and a problem in terms of post-Aristotelian philosophy, these definitions being repeated and developed by Posidonius?16 As a matter of fact, the very attempt to discuss differences between theorem and problem in geometry could not have taken place in the fifth or the first part of the fourth century BC: at that time these words had not yet become mathematical terms. In Plato, one comes across θεώρημα once in the meaning ‘spectacle’ (Leg. 953a3). Πρόβλημα, although used in the context of geometry, astronomy, and harmonics, has a broader meaning as ‘a problem or an issue in need of a theoretical solution,’ and not a specific ‘construction problem’.17 The mathematical meaning of πρόβλημα was also unknown to Aristotle as he used this word in a broader sense. Thus, Geminus clearly referred to a different Oenopides, possibly a Stoic philosopher with his own ‘succession’. The name Οἰνοπίδης, especially common on Chios, is attested there well into the second century BC,18 so that his follower Zenodotus could have been a contemporary of Geminus, to whom the Aristotelian corpus was already available. As for Proclus, the distinctness of the two Oenopideses that he mentions, three other times in the excerpts from Eudemus (In Eucl. 66.2, 283.7, 333.5), may well have escaped him. Menaechmus (5 namesakes in the Realencyclopädie) presents the most difficult case, since, if my supposition is correct, Geminus mentioned both Menaechmi. The first was a student of Eudoxus, a geometer, active in the mid-fourth century BC, the author of the third, after Archytas and Eudoxus, solution of the problem of doubling the cube using conic sections. The letter of Eratosthenes to Ptolemy III on this problem contained an epigram which refers, along with other solutions, to cutting the cone “in the triads of Menaechmus”, i.e., the parabola, the hyperbola, and the ellipsis (Eutoc. In Archim. De sphaer., 96.17). When calling Menaechmus the first discoverer of conic sections, Geminus quotes a line from the epigram (In Eucl., 111.21– 23), and this is the only case when we can be sure that Menaechmus the Elder is meant and the only reference to a pre-Hellenistic mathematician in Geminus’ fragments. Not a single trace of any circulation of Menaechmus’ mathematical works in 15 Zhmud (2006: 98–99, 284–285). 16 τὸ μὲν θεώρημα ζητεῖ, τί ἐστι τὸ σύμπτωμα τὸ κατηγορούμενον τῆς ἐν αὐτῷ ὕλης, τὸ δὲ πρόβλημα, τίνος ὄντος τί ἐστιν. ὅθεν καὶ οἱ περὶ τὸν Ποσειδώνιον κτλ (In Eucl., 80.17–20, cf. 80.20–81.4 = Posidon. F 195 E.-K. with commentary). Tannery (1887: 89 n. 1) tried to solve this problem by suggesting that Oenopides’ scientific school existed even after the time of Aristotle. 17 Theaet. 180c, Soph. 261a, Resp. 530b, 531c; Tarán (1987: 237 n. 36). 18 Lexicon of Greek Personal Names, http://clas-lgpn2.classics.ox.ac.uk. 6 L. Zhmud the Hellenistic period is left,19 and Geminus could have learnt of his discoveries from Eudemus’ History of Geometry, as well as from Eratosthenes, who used Eudemus in his turn.20 But whereas History of Geometry deals with specific discoveries, the remaining three mentions of Menaechmus in Geminus come in the context of the foundations and methodology of mathematics. Consequently, Geminus borrowed this information from another source, and it could well pertain to Menaechmus the Younger, since the mathematicians of the fourth century BC did not discuss such problems. To quote Eudemus, For the mathematicians make clear their own proper principles (τὰς οἰκείας ἀρχάς) and define what they say, each of them. And the person without any insight in the matter would appear to be ridiculous in further trying to find out what is a line and each of the other things. But as to the principles, they do not even begin to show by demonstration, of what kind they themselves say (they are), but they say that it is none of their own business to consider these matters, no, these (principles) being admitted, they demonstrate what comes after that (fr. 34 Wehrli, tr. P. Stork). The inherent anachronism of Menaechmus the methodologist is particularly evident in the terminology and examples he uses. In one case, the two meanings of the word στοιχεῖον, ‘an element’ are discussed with examples from Euclid’s Elements (In Eucl., 72.23–73.12), who lived after Menaechmus the Elder. Even if we suppose that Geminus (or Proclus) substituted his own examples for Menaechmus’, the second meaning of στοιχεῖον – “a simpler part into which a compound can be analyzed … just as postulates (αἰτήματα) are elements of theorems” (73.8–9), – contradicts the fact that in pre-Euclidean mathematics we find neither postulates, nor terminology for them.21 And if αἴτημα is also Geminus’ term, then what remains of Meneachmus in this evidence? In the second case Geminus discusses theorem conversion (In Eucl., 253.16–254.5) and remarks, with reference to the mathematicians Menaechmus and Amphinomus, that not all conversions are true, but only those involving the primary and essential attributes of a subject (ἐφ’ ὧν δὲ τὸ πρώτως ὑπάρχον καὶ τὸ ᾗ αὐτὸ λαμβάνεται, 254.2– 3). ᾗ αὐτό is a notion of Aristotelian philosophy, whereas the term for theorem conversion, ἀντιστροφή, is first attested in mathematical texts only in Apollonius (Conic. 2, 48.157). One could raise an objection that Geminus was trying to convey Menaechmus’ point (or even that Proclus was trying to convey Geminus’ point) in the scientific idiom of his time, but the mention of the otherwise unknown Amphinomus strengthens the suspicion that Hellenistic mathematicians are meant. I am unaware of any example where a pre-Hellenistic mathematician was first mentioned as late as 19 See Schmidt (1884) and Lasserre (1987, 118–124). 20 Zhmud (2006: 84–89). 21 See Mueller (1991). The Menaechmi 7 in the first century BC. Meanwhile, Geminus refers to Amphinomus on three other occasions which clearly demonstrates that he was acquainted either with his works, or with the man himself. All the references to Amphinomus also belong to the methodology and philosophy of mathematics. Following in the steps of Aristotle, he held that “geometry does not investigate the cause, that is, does not ask the question ‘why?’” (In Eucl., 202.9–12, tr. G. Morrow). In fact, Aristotle believed that geometry does discuss the causes (EE 1222 b 23–41), but a mathematician who examines geometry through the lens of issues raised by Aristotle could not have been his contemporary. Further on, Amphinomus divided all the problems into those having a unique solution, multiple solutions, and indefinite solutions (In Eucl., 220.7–12), which evidently refers to some later attempts at a classification in already fullydeveloped geometry. The most famous passage of Geminus where Menaechmus acts as a methodologist of mathematics, this time not in agreement with Amphinomus, but against him, concerns the above-mentioned division of all deductive mathematical propositions into theorems and problems. While, according to Menaechmus, all such propositions fell under problems further subdivided into two species, Speusippus (a nephew of Plato and the first scholarch of the Academy) and Amphinomus believed that all the propositions of the theoretical sciences should rather be termed theorems, especially because these sciences deal with eternal entities (In Eucl., 77.7– 78.13 = Speus. F 72 Tarán). This may leave an impression of a discussion among contemporaries, with Speusippus and Amphinomus defending an ontological perspective and Menaechmus siding with a constructivist approach to mathematics.22 This impression is, however, misleading. As I have pointed out above, in the first half of the fourth century BC πρόβλημα was not yet used as a term for the construction problem in geometry, nor did it acquire such a meaning in the second half.23 Neither in Aristotle, nor in verbatim quotes from Eudemus can one find this meaning; a fourth-century BC Platonist writing that Plato “set problems to the mathematicians, who in turn eagerly studied them”, also used this word in a broader sense.24 Even in Euclid’s Elements the term πρόβλημα, unlike θεώρημα, is absent; it is first attested in Archimedes and Apollonius.25 The imaginary discussion of the fourth-century mathematicians and philosophers about the concepts θεώρημα and πρόβλημα, therefore, has no historical basis.26 It turns out that the concepts which mathematicians participating in this discussion perceived as settled and having their own 22 Bowen (1983). 23 Mueller (1991, 92). 24 Zhmud (2006, 87–88). 25 Mugler (1958, 357). 26 Mueller (1991: 91–97) has been the most critical of this discussion, but he too did not draw chronological conclusions from his insight. 8 L. Zhmud history, so that Menaechmus split the problems into two, while Amphinomus suggested a tripartite division depending on the number of solutions, are in this form virtually unknown to the authors of the fourth century BC. In fact, Speusippus could have believed that in theoretical sciences everything requiring proof should be called ‘theorems’,27 which does not mean that Amphinomus, who shared this opinion, and Menaechmus the Younger, who challenged it, were his contemporaries. Both of them could hardly have lived before the time when Andronicus of Rhodes published the Aristotelian corpus. With this in mind, Speusippus was under no obligation to repudiate or refute the point that all such propositions are problems, this being in all likelihood a later interpretation of his thesis. On balance, Geminus did not have first-hand knowledge of Menaechmus’ solution of the problem of doubling the cube, whereas he was much better aware of his views on classification, terminology, and methodology of mathematics, and the same stands for Amphinomus’ views. How can this be explained? Let us suppose that there was one Menaechmus, yet he wrote not one treatise, but two: one offering a solution of the problem of doubling the cube, and the second on the methodology of mathematics. The first survived him, though only for a brief period, while the second was still available in the mid-first century BC, but remained virtually unknown to either those before Geminus, including Eudemus, or those after him. Let us further suppose that Menaechmus was not only the first Greek geometer to deal with the foundations and methodology of mathematics, but was well ahead of his time: after him problems of this kind have again attracted elder contemporaries of Geminus, such as the Epicurean Zeno of Sidon, who criticized Euclid, and the Stoic Posidonius, who defended him.28 Apart from Geminus, the mathematicians and/or philosophers Theodorus, Zenodotus, and Amphinomus took interest in these issues, followed later by Carpus of Antioch (the turn of first century BC – first century AD) and Hero of Alexandria (first century AD). Let us finally suppose that Geminus (or else Proclus), while repeatedly quoting Menaechmus’ views, every time refashioned them in a way that would incorporate notions and examples of later time. Could these assumptions neutralize the fact that Menaechmus and Amphinomus offered various classifications of geometric problems a century before πρόβλημα as a mathematical term was attested in Archimedes? I deem it impossible. A rather more economical explanation is, I believe, the following. In his vast encyclopedia, Geminus mentioned quite a number of persons who were namesakes of other scientists and philosophers, including Theodorus, Hippias, Oenopides, and Menaechmus. Was he at pains to distinguish between his 27 In Eucl., 181.21–23 = Speus. F 74 Tarán: θεωρήματα πάντα τὰ ἀποδείξεως δεόμενα (προσαγορεύουσιν). The mathematical meaning of θεώρημα is attested in Aristoteles and Eudemus. 28 Procl. In Eucl., 199.3–200.6, 214.15–218.11 = Posid. F 46–47 E.-K. The Menaechmi 9 near contemporaries and those who lived in the fifth and fourth centuries BC? At times, yes, judging by the way he qualifies Zenodotus as the one “who belonged to the succession of Oenopides, but was a student of Andron”. He calls Theodorus, Hippias, and Menaechmus mathematicians, which helps to dissociate them from other bearers of the same name, should they not happen to be mathematicians themselves! But it is unlikely Geminus stuck to this practice, since both for him and his contemporaries these overlaps could mean little to nothing. Hence, Proclus could have been blind to the differences between Menaechmus the Elder and Menaechmus the Younger, or else, having spotted them, he could have not stipulated them, since such was his practice with Hippias, Theodorus, Oenopides, and their namesakes. These, in fact, are the few assumptions allowing us to painlessly separate the two Menaechmi. References Acerbi, F. 2010. “Two Approaches to Foundations in Greek Mathematics: Apollonius and Geminus.” Science in Context 23: 151–86. Allman, G. J. 1889. 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