Eudemus' history of Mathematics

ZA NS Do 2 13 Eudemus’ History of Mathematics’ Leonid Zhmud About the founder of the historiography of science Eudemus of Rhodes we know very little. Ancient sources depict him as a devoted student of Aristotle, who even considered him (along with Theophrastus) as a possible scholarch of the Lyceum (fr. 5). We know neither when he was born, nor when he joined the Lyceum. Eudemus | certainly was younger than Theophrastus (born c. 370) and after Aristotle’s death he returned to Rhodes, where he continued to study $ and to teach (fr. 88). While Eudemus’ Physics belongs to the second period of his activity, his works on the history of science must have been written while still in Athens. Being a part of Aristotle’s historio‘ graphical project, along with Theophrastus’ “physical” doxography and Menon’s medical doxography, they can be dated between 335/4 (foundation of the Lyceum) and 322/1 (Eudemus’ return to Rhodes). In the (ay aber OF RAGES “This paper is a part of my project Historiography of Science in Antiquity financially supported by the Russian Scientific Fund for Humanities (RGNF), grant Nr. 9703-04184. I would like to thank Sir Geoffrey Lloyd and Carl Huffman for their com‘ ments on earlier drafts of this paper and Istvan Bodnar for his generous help and patience.

list of Theophrastus’ works there are three writings on the history of science with the same titles as Eudemus’ works.' Since there are no traces of such writings by Theophrastus, the editors of his fragments believe that these were Eudemus’ works, which were later mistakenly added to Theophrastus’ list.” This misunderstanding indirectly confirms Eudemus’ History of Mathematics 265 And yet in this particular genre he appeared to have no followers even among those who displayed some interest in the history of mathematics and astronomy. Eratosthenes dealt with the problem of the duplication of the cube in his dialogue Platonicus and in the letter to king Ptolemy.* Hellenistic biographies of the famous scientists, popular “inthat all these works were written before Eudemus left Athens, otherwise troductions” to the mathematical sciences, and later philosophical and they would hardly have been in Theophrastus’ catalogue. scientific commentaries also included information on the history of the Beginning with Simplicius, the majority of those who have studied exact sciences. Each of these genres had, however, its own aims that Eudemus’ theoretical treatises agree that in this domain he was a true only partially coincided with the aims of the history of science. Thus, follower of Aristotle, clarifying his ideas and arranging them more sysin a general perspective, Eudemus’ history of exact sciences happened tematically. But although Eudemus did not succeed much in developto be not so much a promising beginning, as an exception. ing Aristotle’s system, or in creating his own philosophical system, this Does his failure to found his own school explain the abortive develdoes not mean that he was bereft of originality. Since Aristotelianism opment of the history of the exact sciences? Even if Eudemus really had by its very nature resisted any attempt at substantial modification, the Peripatetics became famous not so much in philosophy, but in the specific sciences. There is no doubt that ancient musicology. geography, botany and physics would appear incomparably more inferior without Aristoxenus, Dicaearchus, Theophrastus and Strato. Such an appraisal only two students, Theophrastus had two thousand listeners (D.L. seems all the more appropriate to the historiography of science, as Eudemus’ writings happened to be not only the first but also the last specimens of that genre in antiquity. To be sure, neither Eudemus himself, nor his works have been forgotten: they were still extant in the interests and spirit of the epoch. It is revealing that Eratosthenes, who sixth century AD? and a special biography was devoted to Eudemus.* V.37), and nonetheless his botanical research was not further developed. Meanwhile the biographical genre founded by Aristoxenus and Dicaearchus, of whose students we also know nothing, was immediately picked up by the Hellenistic writers, since it corresponded to the as a mathematician and historian could have continued Eudemus’ history of mathematics, attempted instead of this to change the framework of the genre. In accordance with Eratosthenes’ philosophical predisposition, the fictitious plot of the Platonicus combines a sober history of mathematics with the academic legends about Plato as architect of sci- ' ’AotpoAoyıkfig iotopiag a’-5', ‘AprOpuntikay iotopròv (without the number of books), ‘lotopiküv yewpetpika@v a’-8 (D.L. V.48, 50 = 137 Nr, 43, 264 Nr. 2-3 FHS&G). 2 This was suggested first by H. Usener. In the same list we find another work, Tôv nepi tó Beiov iotopias a’—c’, which contrary to Wehrli’s opinion should be identience. However, even this attempt to make this genre more popular and accessible had only limited success: although the Platonicus was widely known, we do not hear of any followers of Eratosthenes in the “belletristic” history of the exact sciences. fied with Eudemus’ History of theology, known from Damascius (251 Nr. 2 FHS&G In spite of the general decay of this genre, I would not say that = Eud. fr. 150). Cf. Damascius’ reference xatà thy Etôuov iotopiav (Wehrli p. 70.6). 3 Simplicius gives a long verbatim quotation from the History of geometry (fr. 140) and cites about a hundred passages from Eudemus’ Physics; he also gives three of the Eudemus “was virtually unknown” in the Hellenistic period,’ especially seven quotations preserved from the History of astronomy (frr. 146, 148-49); in quoting Eudemus, he makes it clear that he is dealing with his book (fr. 149). Eutocius also indicates that he has read the History of geometry (fr. 139). Damascius apparently used Eudemus’ History of theology without intermediaries (fr. 150). 4 This biography (fr. 1), written by a certain Damas, is mentioned only by Simplicius. taking into account that we possess only meager remains of the vast Hellenistic literature. Eratosthenes, and probably Archimedes, drew upon his History of geometry, Diogenes Laertius and Clement of Alex3 Knorr (1986) p. 17ff., Knorr (1989) p. 131 ff. brings convincing arguments that this letter is not a forgery. When Damas lived is unknown, but one can guess that he was Eudemus’ student rather 6 Gottschalk p. 26. This seems to me rather improbable. A fragment of Eudemus’ than a later biographer. Eudemus was not such a popular figure that a late author would Physics, written on Rhodes, depicts a typical picture of a teacher lecturing to a group be interested in writing his biography; besides, what were his possible sources? of students: xdyò nVBoAoynow 16 paBdiov Exav buiv KabnpEvois oro (fr. 88). 7 Ibid. Eudemus’ biography is mentioned in the Arabic sources (Rosenthal p. 36).

andria, known for their extensive use of the Hellenistic sources, cite his 267 ematicians as quite normal. Indeed, the mathematicians do not want to History of astronomy (frr. 143-44).® Later, Eudemus’ theoretical treaprove their dpyai not because of their failure to think things through, tises remained of interest only to Aristotle’s commentators, whereas his but rather because such a position corresponds to the general rule acworks on the history of the exact sciences were not infrequently used cording to which no one scientific discipline can prove its own &pyat. by those who took these sciences up in one way or another: Theon of Proceeding on such an understanding of the division of labor, one can Smyrna, Porphyry, Pappus, Proclus, Eutocius. Thus, Eudemus — the suppose that when Eudemus deals with subjects that are in the jurisdicexpert in the exact sciences, their first and perhaps only historian, was tion of the mathematicians, he prefers to follow their norms and rules. not less important for the classical tradition than Eudemus — the true Owing to this, his exposition of the history of exact sciences appears to follower of Aristotle. be free from the preconceived philosophical tenets — at least free to Judging by his works, Eudemus received a good mathematical edusuch an extent that they do not harm the professional approach.!? cation and was very competent in the problems of contemporary mathematics.? Eudemus’ professional training is also manifest in the fact that his histories of the exact sciences are devoted to strictly mathematical problems and methods, and not to the philosophical interpretation of mathematics which was so characteristic of Plato and all his students, including Aristotle. To be sure, a professional approach to mathematics was not the only possibility for Eudemus, as a fragment from his work On Angle shows (fr. 30). There he treats an angle as a certain quality, i.e., in the spirit of Aristotle.!° An interesting fragment from Eudemus' Physics (fr. 34) reveals one of the possibilities of combining philosophical and mathematical approaches. According to Eudemus, after the mathematicians have defined their apxai and reached an agreement about them, they carry their research further. They refuse to study and to prove mathematical apyat, saying that this is not their business («4AA' oddé paoiv abr@v elvaı tadta émokoneiv). Therefore, there is another science (étépa iAocogia) to deal with these &pyat, i.e., Aristotelian “first philosophy,” or “wisdom.”!' Eudemus’ position is thus very close to that of Plato and Aristotle, but in contrast to Plato who reproaches the mathematicians for the lack of interest in proving their &pyat (Resp. 510C-E), Eudemus seems to consider the division of labor between philosophers and math- 2 If we confine ourselves to those of Wehrli's fragments where Eudemus’ name is mentioned, leaving aside for the moment the socalled Catalogue of geometers.!? it turns out that there is not much material preserved from the History of geometry. Frr. 134-35 concern theorems of Thales, frr. 136-37 Pythagorean mathematics, fr. 138 Oenopides, frr. 139-40 the quadrature of a circle by Antiphon and the quadrature of lunes by Hippocrates of Chios, fr. 141 Archytas’ solution to the problem of the duplication of the cube, and fr. 141.1 the theory of irrationals developed by Theaetetus. The origins of these fragments vary and even taken together they are far from giving us an idea of what was the original scope of Eudemus’ book.' Of the nine fragments preserved under his name, five deal with the propositions of Euclid's book I (they come from Proclus’ commentary on this book), one with those of book X (on the theory of irrationals, from Pappus), and the other three concern problems outside the scope of the Elements (two from Eutocius and one very long quotation from Simplicius). Thus, in their entirety Eudemus’ fragments do not cover even one tenth of the material that — judging by the Catalogue in Proclus — was collected and examined in the History of geometry. Of the twenty mathematicians 12 The case of Aristoxenus is instructive as to the influence of Aristotelian philoso- 8 See also: Eud. fr. 89 ap. D.L. 1.9; fr. 145, Dercyllides ap. Theon of Smyrna. 9 Contrary to Aristotle, who believed that Hippocrates of Chios committed a logical phy on the professional approach. In his youth Aristoxenus studied with the fallacy in his quadrature of lunes (Soph. El. 171b12f.), Eudemus did not find any Pythagoreans, but joining the Lyceum he abandoned the mathematical theory of music mistakes in Hippocrates’ proofs (fr. 140). See: Lloyd. Aristotle was quite competent and developed his own harmonics, based on the qualitative approach to musical sound. in elementary mathematics, but did not reveal a particular interest in the higher math- 13 Procl. In Eucl. p. 64.16-68.23 Friedlein = fr. 133. See below, section 3. The only fragment from his History of arithmetic (fr. 142) also deals not so much ematics of the period: Heath (1949) p. 1f. 10 Arist. Cat. 10a11-24, Phys. 188a25, Mer. 1020a35-b8. See Heath (1927) p. 177f. with arithmetic, as with Pythagorean harmonics, i.e., with the mathematical theory ll Cf. Met. 995b4f.; 996b26f., 1005a19ff. of music.

269 mentioned in the Catalogue, !$ we find only five in the fragments, plus authors who made Pythagoras the originator of all five regular solids.?! Antiphon, who is omitted from the Catalogue. The anonymous scholia to book V of the Elements twice call Eudoxus In reconstructing the original scope of the History of geometry we the author ofthis book, which also may have Eudemus as the source.?? can rely on these fragments as solid ground, yet we cannot confine our- The last proposition of the fourth book, on the fifteen-angled figure selves only to them. It is well known that for the late authors Eudemus was one of the main sources, if not the main source of information on pre-Euclidean geometry. This does not mean, of course, that any anonymous evidence concerning early Greek mathematics goes back to Eudemus. Nevertheless, there are many cases where his authorship inscribed in the circle (IV.16), deserves special attention. Proclus, rethere are not a few theorems and problems useful for astronomy, e.g.. seems to be firmly established, Proclus, for example, informs us about (p. 283.7f.). Since the last reference, as we noted, goes back to two of Thales’ theorems with a reference to Eudemus (frr. 134-35), and Eudemus, one can suspect that this remark too has the same origin. about two others without mentioning his sources.! It seems rather ob- Proclus explains further that by inscribing the side of the fifteen-angled lying on the earlier commentaries on Euclid, notes that in the Elements the problem IV.16 (p. 269.8ff.). The same is said here about the problem 1.12 that belongs to Oenopides: “he thought it useful in astronomy” vious that the latter two pieces of evidence are likewise based on figure in the circle we get the angle between the celestial equator and Eudemus’ authority.!” The same conclusion can be reached about two the zodiacal circle, i.e. 24°. Here we should remember that Oenopides of Oenopides’ discoveries, one of which Proclus mentions with referwas that very astronomer who first measured the angle of the obliquity ence to Eudemus (fr. 138) and the other without it (p. 283.7f.)."? of ecliptic (41 A7). In a short but regrettably error ridden résumé of Here is another example: who was the authority for Proclus’ information (p. 304.11f.) that the Pythagoreans had knowledge of the theorem that only the following polygons can fill up the space about a point: the equilateral triangle, the square, and the regular hexagon? There is Eudemus’ History of astronomy we find decisive evidence: “Oenopides celestial equator is equal to the side of the fifteen-angled figure, or 24°” no such theorem in Euclid, but his older contemporary Eudemus could (fr. 145). K. von Fritz demonstrated long ago that originally both statewas the first who found the obliquity of the zodiacal circle,”2 whereas “the others found that the angle between the zodiacal circle and the have referred to this, as well as to other Pythagorean discoveries.!? It ments pertaining to the zodiac referred to Oenopides.?* Therefore, seems very likely that the two pieces of evidence from the scholia to Proclus’ evidence on the astronomical significance of the problem Euclid also go back to Eudemus: first, that the whole fourth book of Euclid belongs to the Pythagoreans, and second, that they constructed IV.16 also goes back to Eudemus, although Proclus mentions neither his, nor Oenopides’ name.?* three regular solids (pyramid, cube and dodecahedron), whereas the In fr. 141.1 from the Arabic translation of Pappus, overlooked in the octahedron and icosahedron were discovered by Theaetetus.”? first Wehrli edition, Eudemus speaks about Theaetetus’ contribution to Eudemus, as we remember, wrote both on the Pythagoreans and on the theory of irrationals. This fragment was added to the second edition, Theaetetus and definitively knew the subject better than those later but Wehrli, following Burkert, does not include in it the preceding note on the Pythagoreans. It is clear from the text, however, that Eudemus 15 | include in this number Philip of Opus, but not Plato. Hippias of Elis is mentioned here only as a source, and not as a mathematician. 16 fm Eucl. p. 157.10f., 250.208. 17 van Pesch p. 78f.; Heath (1927) p. 36. 18 Ibid. It is very possible also that Eutocius, who derived Archytas' solution to the problem of doubling the cube from Eudemus (fr. 141), owes his information about the solutions of Eudoxus and Menaechmus to the same source (In Archim. De sphaer. III, p. 56.1ff., 78.13-80.24 Heiberg). See: Wehrli p. 118; Knorr (1986) p. 21; Knorr (1989) p. 77ff., 94ff. 19 van Pesch p. 79; Heath (1927) p. 36. 20 Schol. in Eucl. p. 273.3-13, 654.3f. Heiberg. See Burkert (1972) p. 450. 21 This tradition is preserved in Proclus’ Catalogue (In Eucl. p. 65.15f.). See below, p. 286. 2 Schol. in Eucl. p. 280.7f., 282.12f, Heiberg. 23 Diels’ correction, A6Eworg instead of manuscript 81áLwo1s (Theon Smyrn. Exp. p. 198.15 Hiller = 41A7), is fully justified by the parallels in Aétius, Diodorus and Macrobius (41 A7). 24 von Fritz (1937) col. 2260f. 25 According to Neuenschwander p. 374 the style of this problem differs from the other propositions of book IV, so it seems to be a later addition.

considered Theaetetus to continue and develop already existing theory. Hence, the preceding note (omitted by Wehrli), “This science had its origin in the school of Pythagoras, but underwent important development at the hands of Theaetetus,”? perfectly fits in such a context.”” Among the predecessors of Theaetetus at least one person had to be mentioned: his Pythagorean teacher Theodorus of Cyrene. Theodorus is named as well in the Catalogue (p. 66.6), and Plato (Theaet. 147D) ascribes to him a proof of irrationality of the magnitudes between /3 and ‚A 7. Further, it seems very improbable that Eudemus, who left a detailed account on the authorship of many elementary propositions, would neglect to say something about the discovery of irrationality, which was made before Theodorus. In any case, it would be contrary to Eudemus’ usual manner to mention Theaetetus (and Theodorus) without saying a word about the np@rog evpetie of irrationality, Hippasus of Metapontum.2 Although Hippasus’ name is not attested either in the Catalogue, or in the fragments, Eudemus in all likelihood did write 271 ter which the modems discover four others as well.? Thus, it comes out that the first three proportions were discovered by Pythagoras, the second three by contemporaries of Plato and Aristotle, and the last four by later mathematicians. This information is correct, but it lacks details that would allow us to connect it with Eudemus. We find such details. however, in Jamblichus’ commentary to Nicomachus: Of old there were but three means in the days of Pythagoras and the mathemalicians of his times, the arithmetic, the geometric, and the third in order, which once was called the subcontrary, but had its own name changed forthwith to harmonic by Archytas and Hippasus, because it seemed to embrace the ratios that govern the harmonized and tuneful. And it was formerly called subcontrary because its character was somehow subcontrary to the arithmetic ... After this name has been changed, those who came later, Eudoxus and his school, invented three more means, and called the fourth properly subcontrary because its properties were subcontrary to the harmonic ... and the other two they named simply from their order, fifth and sixth. The ancients and their successors thought that this number, i.e., six, of means could be set up; but the moderns have found four more in addition . . Y about him too. In the same fr. 141.1 it is said that Theaetetus used in his theory three proportions, the arithmetic, geometric and harmonic, whereas in the Catalogue we read that Eudoxus added to the three known proportions three others (p. 67.2f.). If the information of the Catalogue goes back to Eudemus (and there are no grounds to doubt this), we can surmise that he mentioned also who discovered the first three proportions. In this connection I would like to draw attention to the reports of Nicomachus and Iamblichus on the discovery of the proportions. According to Nicomachus, the first three of them “came down to Plato and Aristotle from Pythagoras,” the second three, subcontrary to them, which are called the fourth, fifth, and sixth, were discovered later; af5 26 Pappus, p. 63-64; cf. Burkert (1972) p. 440 n. 182. 27 Scholia to book X of Euclid also mention Pythagoreans as the originators of the It is clear that we have a fragment of the history of mathematics before us, taken by lamblichus from some reliable and well-informed source. It contains the names of Hippasus, Archytas and Eudoxus that were missing in Nicomachus, along with much additional information on the early history of proportions. This information perfectly matches the fragment of Archytas quoted by Porphyry: there are three means in music, the arithmetic, geometric, and subcontrary “that we call harmonic.”?! Since Eudemus ascribes an application of the first three proportions to Theaetetus and a discovery of the three others to Eudoxus, lamblichus’ passage likewise has to be connected with Eudemus. Specifically, he seemed to mention not only Eudoxus but also Pythagoras (in connection with the discovery of the first three means), as well as Hippasus and Archytas (as his followers).* Interestingly, lamblichus theory of irrationality (p. 415.7, 416.13, 417.12f. Heiberg). Burkert’s position in this question is inconsistent. He states that: 1) Eudemus, quoted by Pappus, does not mention the Pythagoreans; 2) scholia to book X are mostly from Pappus; 3) the same scholia, ascribing discovery of the irrationality to the Pythagoreans, are based ultimately on Eudemus. Cf. Burkert (1972) p. 450 n. 13, 457, 458 n. 57, 462 nn. 73-74. This contradiction can be easily reconciled by suggesting that Pappus’ reference to the Pythagoreans goes back to Eudemus as well. 28 Cf. his note on the discovery of the regular solids by the Pythagoreans and Theaetetus (Schol. in Eucl. p. 654.3f. Heiberg) and below, n. 49. On Hippasus see below, p. 287. 29 Inır. arith. p. 122.11f., 142.21 f. Hoche. 30 In Nicom. p. 100.19-101.9 Pistelli = 18A15, tr, M. L. D’Ooge. 31 In Prol. Harm. p. 92 Düring = 47B2. Philolaus also called this mean “harmonic” (Nicom. /ntr. arith. p. 135.10f. Hoche = 44A24), which makes clear that it has been renamed before Archytas. Tannery believed that Archytas quoted Hippasus (Tannery [1885] p. 190). Even if this was not the case, Archytas surely could have mentioned his name (cf. 47A13, Bl). 3? According to Aristoxenus (fr. 90 Wehrli), Hippasus made an experiment with four bronze discs, using the so-called “musical proportion" (6:8 = 9:12) that includes

273 turns two more times to the history of proportions, saying again that the will make the original proposition evident, he adds: so, for example, the first three come from Hippasus and Archytas, whereas the first six were problem of doubling the cube was reduced to the finding of two means used from the times of Plato until Eratosthenes.?? His immediate source here was in all probability Porphyry's commentary on the Elements, which traced the history of means from Pythagoras to Eratosthenes, relying mainly on Eudemus.** The name of the first author of Elements, Hippocrates of Chios, occurs both in the Catalogue (p. 66.4) and in two of Eudemus’ fragments (frr. 139-40). As follows from his detailed account on the quadrature of lunes (fr. 140), Eudemus was well acquainted with Hippocrates’ in continuous proportion between two given straight lines. They say that the first to effect reduction (araywyn) ofdifficult constructions (t@v Aropovpévov Siaypapuárov) was Hippocrates of Chios, who also squared the lune and made many other discoveries in geometry, being a man of genius when it came to construction, if there ever was one.?? As follows from Proclus’ aot, we have here a reference to a source, which seems to be the same as that of Eratosthenes. In both cases work and highly estimated his contribution to mathematics. In Era- Hippocrates is called ap&tog ebpetn of the reduction of the problem tosthenes’ letter to king Ptolemy, devoted to the problem of the dupliof doubling the cube, and this was a standard method of Eudemus for cation of the cube, it is said that describing mathematical and astronomical discoveries.°® In both cases Hippocrates of Chios first conceived (np@tog £revöngev) that if, for two given lines, two mean proportionals were found in continued proportion, the cube will be doubled. Whence he turned his puzzle (&ròpnpa) into another no less puzzling.* the same Gropnua, i.e., a difficult geometric puzzle, is in question, which Hippocrates reduced to the other problem. It is revealing that for the first time «rayoyh is mentioned in Aristotle (An. Pr. 69a25), and as an example of its application he brings the problem of quadrature of the circle with the help of the lunes (69a30-34), i.e., the famous prob- These words almost certainly go back to Eudemus, on whose material Eratosthenes heavily relied.% In Proclus we find a similar account of lem of Hippocrates!?? It is quite natural that Aristotle’s student also applied the term &naywyn to Hippocrates’ method, which allows us to Hippocrates. Giving a definition of the anaywyn, i.e., of the reduction of a complicated problem to another which, if known or constructed, ascribe Proclus’ note to Eudemus. To him, probably, belong as well the words of admiration of Hippocrates’ talent, much more suitable for the classical than for the later author. both the arithmetic and harmonic means. This clearly shows that the first three means were known not only to him, but also to his teacher Pythagoras, whose acoustical experiments Hippasus has followed: Zhmud (1997) p. 191ff. lamblichus connects the “musical proportion” with Pythagoras and Philolaus (/n Nicom. p. 118.23f. Pistelli = 44424). On the theory of proportions in the Pythagorean school see Heath (1921) vol. I, all the ten E In Nicom. p. 113.16f., 116:1f, Pistelli. Since Eratosthenes studied p. 850. means, it was probably him who discovered the last four: Wolfer p. 20ff.; van der Waerden (1956) p. 385. lamblichus erroneously ascribes these four means not to Eratosthenes, but to otherwise unknown Pythagoreans (ibid. p. 116.5). # See below, p. 283. Lasserre (1966) p. 175 connects lamblichus’ passage with Eudemus via Eratosthenes’ Platonicus. Wolfer, however, rightly points out (p. 24) that in this dialogue only the first three proportions known to Plato were discussed. Besides, in lamblichus’ passage the name of the Platonicus’ main hero is missing, whereas Nicomachus, Theon of Smyrna, and Pappus, who knew the material of this dialogue, did not mention Eudoxus in connection with the proportions. 35 Eutoc. In Archim. De sphaer. INI, p. 88.18-23 Heiberg = 42A4, tr. Knorr (1989) p. 147. Cf. above, n. 5. 36 Wehrli p. 118f.; Knorr (1986) p. 21. In Diogenes Laertius’ biography of Archytas there is a passage that also can be connected with Eudemus. In this passage Archytas is called TPÚTOC twice: He was the first to bring mechanics to a system by applying mathematical principles; he also first employed mechanical motion (xivnoig ópyavixñ) in a geometrical construction, namely, when he tried, by means of a section of a half-cylinder, to find two mean proportionals in order to duplicate the cube (V111.83, tr. R. Hicks). The immediate source of Diogenes is likely to be Favorinus: he was very much interested in various ebpfuataæ and mentioned elsewhere Archytas’ discoveries in mechanics.* Favorinus, in turn, could have 37 In Encl. p. 213,7-11, tr. G. Morrow. 38 See below, n. 158. 39 Soph. El. 171b12f. = 42A3; see below, n. 46 and Knorr (1986) p. 71f. 40 Fr. 66 Mensching = 47A 10a. The passages in Diogenes Laertius where he enu-

relied on Eratosthenes, who certainly used Eudemus.*! The second part of Diogenes’ passage closely matches both Eratosthenes' letter that also mentions half-cylinders*? and Archytas” solution to the problem of doubling the cube, known from Eudemus (fr. 140). Does the first part come from the same source too? An obvious symmetry between the two parts of the passage indicates that they belong to the same context: Archytas applied mathematics to mechanics and (mechanical) motion to mathematics.* Eudemus wrote on Archytas both in the History of geometry and in the Physics, where he states that Archytas considered things unequal (tó &vıcov) and uneven (to &VOpLaAov) to be causes of motion (fr. 60). Fr. Krafft has appropriately connected this idea with the main principle of motion in Aristotle’s Mechanical problems. i.e., with the principle of the unequal concentric circles, and concluded that Aristotle drew upon Archytas’ Mechanics.“ Hence, Archytas as the founder of mechanics was known to Eudemus, which confirms his authority in Diogenes Laertius” passage. Another piece of evidence is the passage from the Quadrature ofparabola, where Archimedes refers to the geometers who tried to sue a circle but, according to the opinion of most experts, failed to do so. He clearly means here the attempts of Hippocrates, Ai enr: and 275 The following example is less obvious, but no less interest ing. In the introduction to his work the Merhod, Archimedes writes: In the case of the theorems the proof of which Eudoxus was first to discover, namely, that the cone is a third part of the cylinder , and the pyramid of the prism, having the same base and equal height, we should give no small share of credit to Democritus, who was the first to make assertio n with regard to the said figure though he did not prove it.47 Archimedes refers to Democritus only once and hardly knew his works. Eudoxus, as it seems, also does not mention Democr itus. This follows from the introduction to Archimedes’ treatise On the sphere and cylinder, written before the Method, where he refers to the same discoveries of Eudoxus, adding that before him no geometer came to these ideas.** Thus, at that point, Archimedes did not know Eudoxus’ predecessors in this field. Later, in the Metho of any of d he corrects his view, but was this correction due to his acquaintance with Democritus’ books? The following prompts us rather to suppose that Archimedes took the comparison of Eudoxus and Democritus from Eudemus’ History of geometry: 1) It was characteristic of Eudemus’ style to compare the results of several scientists who worked on the same problem.*? over Bryson in silence). Since Eudemus, in contrast to Aristotle, usu- 2) ally presented both the geometrical procedures and his own judgment concerning them, he is almost surely included among the experts to The expression énbpnxev npörtoc is also typica l for Eudemus; it occurs in almost every fragment of his histori cal work50 s 3) Eudemus was concerned to specify whether a strict mathematical proof was given or not.5! 4) Inhis Physics, Eudemus mentions Democritus (fr. 54a-b) and at one place he even addresses him in the vocative (fr. 75). 5) Eratosthenes, to whom the Method was addres sed, certainly used Eudemus’ work. This reinforces the possibility, implied by the Bryson to square a circle that were briefly mentioned by Aristotle” and more extensively treated by Eudemus (frr. 139-40; he, however, passes whom Archimedes alludes. merates one eVpnua after another usually derive from Favorinus. In his Manifold history there was a special book on npwtoı edpetoi : Mensching p. 31f., 161 (index on the word TpoTOG). o 4! Cf. fr. 27 Mensching (Eratosthenes as a source). Favorinus contemporary and friend Plutarch also drew upon Eratosthenes’ Platonicus and mentioned some Opyavırai Kat pnyavixaì KatacKevat designed by Archytas, Eudoxus, and Menaechmus (Quaest. conv. 718E). | 42 Eutoc. In Archim. De sphaer. III p. 96.6 Heiberg. | 43 See Heath (1921) p. 246f.; van der Waerden (1956) p. 150f.; cf. Knorr (1989) p. 109. 44 Krafft p. 1498. | _ ee ai 45 Siônep adtoic dad TOV rAciotuv OLY ebpioxôpeva tadta roteyvooBev (II, . 262.13ff. Heiberg). passage from the Quadrature of parabola, that Archimedes too was acquainted with the History ofgeome try. i 46 Car. 7b271., An. Pr. 69a30-34, 75b37f.. Soph. El. 171b12f., b34ff.; Phys. 185al4f. 4? IL, p. 430.1 ff. Heiberg = fr. 61¢ Lasserre. Cf. 68B155 DK. 481, p. 4.5f. Heiberg = fr. 62b Lasserre. 49 Frr. 139-40, 146; Schol. in Eucl. p. 654.3f. Heiberg; Pappus, p. 63. 50 See below, n. 158. 51 Cf. his remarks on Thales: fr. 135: Procl. In Eucl. p. 157.10f., 250.20f.; Heath (1921) vol. 1, p. 130f.

If my assertion is right, we can add Democritus to the list of the mathematicians named by Eudemus. An absence of his name in Proclus’ Catalogue, which appeared rather strange to many historians of mathematics,5? could be due to the Neoplatonic editing of this text. An attempt to attribute to Eudemus mention of all the significant mathematicians before Euclid is justified not in the least by the fact that practically every mathematician that is known to us from the times of Thales to the second third of the fourth century does appear in Eudemus’ fragments and in the evidence that can be safely attributed to him (including the Catalogue). More than that: from the other sources we do not know of any mathematician of that period who would not be mentioned by Eudemus (except for Hippasus and Democritus). Some of them, e.g., Mamercus, Neoclides, Leon, Theudius, Athenaeus and Hermotimus are known only from Eudemus. This appears to be a sufficient ground for further attempts to connect with his works those names as well, which for various reasons are omitted in the Catalogue (if, indeed, we have independent and reliable evidence on them). On the other hand, this very fact prompts us to examine the chronology of the mathematicians nor mentioned by Eudemus. In Proclus we find several references to a certain Amphinomus, taken from the mathematical encyclopedia of Geminus.5 In one case, he associates Amphinomus with Speusippus, opposing them to the mathematicians of Menaechmus’ school (p. 77.16f.); in the other, on the contrary, he writes about the mathematicians from the circle of Menaechmus and Amphinomus (p. 254.3f.). In all the references to Amphinomus specific mathematical discoveries are not in question; rather methodological debates and problems of terminology. This seems to be a sufficient reason for omitting Amphinomus in the History of geometry. After all, Speusippus and Xenocrates, who wrote a lot on these subjects, do not appear in this book either. There is, however, another possibility: Amphinomus worked after Eudemus had finished his book. This would make him a younger contemporary of Menaechmus, whose generation is the last to appear in Eudemus. According to Geminus, Amphinomus held the view that mathematics does not investigate the causes, while the originator of this view was Aristotle (p. 201.11). This might imply: Eudemus’ History of Mathematics that Amphinomus lived after him. To be sure, Geminus (or was not interested in chronology and could his source) associate the persons according to the similarity of their views, regardless of when they lived.54 One more reference of Proclus, likewise taken from Geminus. concerns a certain Zenodotus, “who belonged to the success ion of Oenopides, although he was a pupil of Andron” (p. 80.15f.). The mathematicians Zenodotus and Andron are otherwise unknown, while Oenopi des of Chios was an older contemporary of Hippocrates. But did Geminus really have in mind that Oenopides? Again, he speaks here about the distinction between theorems and problems: their definiti Zenodotus are very close to those of Posidonius and ons given by his followers (p. 80.15f.). It seems, then, that Oenopides. Zenodotus, and Andron (1956) p. 150. 53 In Eucl. p. 77.16f., 202.11, 220.9, 254.3f.; cf. Tannery (1887) p. 24; van Pesch p- 112f. On Geminus see below, p. 282. belong to the Hellenistic period. 3 Moving from the evident cases to the less evident, we come to one of our central problems: Who was the author of the Catalog ue of geometers and how did this document come to Proclus?5 It is customary since the second half of the 19th century to think of the information in the Catalogue as going back, albeit through intermed iaries, to Eudemus’ History of geometry (fr. 133).57 Although Proclus does not mention Eudemus in connection with the Catalog ue, he refers to “those | who wrote the history of geometry” before Euclid (p. 68.4). Besides, Eudemus’ fragments, including those quoted by Proclus himself , coincide thematically with the Catalogue: they tell us about the ment of mathematics from the sixth to the fourth 34 Cf. Lasserre (1987) E mus analoge pAmphino developcentury (frr. 134-41). 149f., 5871. Knorr (1986) p.p. 75 was indecisiv indecisive e about ¿ Cf. Knorr (1986) p. : 374 n. . 70. 70. Since Geminus’ meth odologic i al discussi i i on of theorems and problems Un Eucl. p, 77.7-78.10 and 80.15-81 .4) is based on Posidonius (see fr. 195 Edelstein-Kidd with commentary), the other referenc es to Amphinomus Speusippus, Menaechmus, Oenopides, Zenodotus, and Andron might derive from the same source. Cf. Lasserre (1987) p. 552f., 614f. and 20F 17-23. A Stoic Oenopides is Nm in Aëtius (1.7.17). See: Zeller, p.48 n. 1. This section partly overlaps with m y recent : article, i TI 52 Tannery (1882) p. 172; van Pesch p. 82; Heath (1927) p. 36; van der Waerden 277 mn “> in more detail: Zhmud (1998) p. 219ff. where ore I hz È i have teenie hs It was suggested first in 1866 by his editor L. Spengel. See also Tannery (1882) p. 171 ffi van Pesch p. 80: “Inter omnes viros doctos constat originem id ab Eudemi historia duxisse.”

However, in the last decades this opinio communis has been challenged, especially after the publications of the History of the Academy by Philodemus (first century Bc), formerly known as the Index academicorum. In column Y of the Herculanum papyrus 1021, which contains a part of Philodemus’ book, we read the following: At this time the mathematical sciences were also greatly advanced, with Plato being the architect of this development; he set problems to the mathematicians, who in turn eagerly studied them. In this way, netpoAoyla (the theory of proportions?) and the research on definitions reached their peak, as Eudoxus of Cnidus and his students completely revised the old theory of Hippocrates (of Chios), Especially great progress was made in geometry, as the methods of analysis and of diorism were discovered. Optics and mechanics also were not (left in contempt?) . . .*8 The name of the author of this passage, taken by Philodemus from some early source, is omitted in the papyrus, and several theories have been proposed about his identity. Lasserre su ggested that the passage comes ultimately from the book Tlepı [IA&twvog by Plato’s secretary Philip of Opus,5® whereas Gaiser argued in favor of the Peripatetic Dicaearchus.© Lasserre’s suggestion was supported by Burkert and Dorandi,®! and it seems to be much more plausible than Gaiser’s. Whoever wrote this passage certainly belonged to the Academy, since only an Academician could have ascribed to Plato such an important role in the development of mathematics. On the other hand, the passage from Philodemus closely matches the description of Plato given in the 279 whereas Philip is described here as one of those “academic mathematicians” who studied mathematics under Plato’s methodological direction: Philip [. .. .] of Mende, a pupil whom Plato had encouraged to study mathematics, also carried on his investigations according to Plato's instructions and set himself to study all the problems that he thought would contribute to Plato’s philosophy.f? Against the background of Eudemus’ fragments and the other sections of the Catalogue, both these passages look rather odd. Eudemus records everywhere specific discoveries and achievements of the Greek geometers, but in these two passages there is not a single word about any discovery. It is hard to believe that Eudemus considered Philip’s foremost merit in mathematics to be that he studied problems connected as he thought with Platonic philosophy. And was it really relevant for Eudemus’ history of geometry that Plato’s writings were “thickly sprinkled with mathematical terms”? In this connection, the following question arises: Does it follow, as Lasserre thought, that we should credit Philip, instead of Eudemus, with the second part of the Catalogue — which begins with Plato and ends with Philip himself — if not with the Catalogue in its entirety? I believe that there are no grounds for such a conclusion. First, it is far from evident that Philip was the author of the papyrus passage and that it was taken from his book epi MAdtwvoc, of which we know virtually nothing. Hermodorus of Syracuse, another student of Plato, may well have been its author.© Besides Mepi po8nudtav,TM he also Catalogue: Plato [....] greatly advanced mathematics in general and geometry in particular wrote a work Tlepi TAd&twvoc.® Philodemus mentions his book on thickly sprinkled with mathematical terms and that he everywhere tries to two other Academicians who shared Philip's and Hermodorus' interests because of his zeal for these studies. It is well known that his writings are arouse admiration for mathematics among students of philosophy. 58 For the text see Gaiser p. 152; Dorandi (1991) p. 126f. It is worth mentioning that the occurrence of mechanics in the passage that praises Plato’s achievemtoents this destroys all the attempts of Plutarch to ascribe to him a contemptuous approach science (Marc. 14.9-11; Quaest. conv. 718E-F). 59 Lasserre (1987) 20F15a-15b, p. 61 ff. 60 Gaiser p. 76f., 97f., 342ff. bo 44, 71 Wehrli), s of Dicaearchus (frr. 42, 43, Burkert = > e:rom de preserved fragment (1993) VI; col. us, Philodem See also 6! Burkert (1993) p. 26f.; Dorandi (2001). he had a quite critical opinion of Plato. p. 26f. Plato (col. VI), so it must have been available to him. Further, there are in mathematics and in Plato’s biography. Xenocrates was the author of Mepi tod MAdtwvog Btov‘® and many works on mathematics.99 63 In Eucl. p. 66.8f., 67.23f., tr. G. Morrow. 64 Lasserre (1987) p. 611 ff. Earlier he shared a traditional view on the Catalogue: Lasserre (1966), fr. 22. 65 Lasserre (1987) p. 433f. considered Hermodorus only an intermediary figure between Philodemus and Philip. 66 D.L. 1.2 and 8 = fr. 6 Isnardi Parente. 67 Frr, 7-8 Isnardi Parente = FGrHist 1008F2a-b. 68 Frr, 264-66 Isnardi Parente = FGrHist 1010Fla-c. 69 Nepi tóv padquétov, Mepi yewpetpav, Mepi &piOpav, Mepi dotpoAoytag,

Speusippus wrote [Adtwvoc nepideinvovTM that was known to Philodemus,7! as well as the Mo@npotixòg and Nept TOV Mvdayopix@v Apıduav.’* Since Philodemus apparently did not have any difficulties in finding the fourth-century sources, it is very hard to decide . “ani the Catalogue contains too much detailed information that thored the passage he cited. has no connection with Plato, especially in its first part which narrates the development of geometry from Thales to Hippocrates and coincides with Eudemus' fragments. To be sure. Plato occupies a central place in the second part of the Catalogue and such a perspective brings it closer to the papyrus passage. But in spite of some similarities Caba the papyrus passage and the Catalogue,” the latter has a number of features that prevent us from suggesting a common origin for both texts. To = nothing of the first part of the Catalogue, even the perspective O | a second part is much wider than that of the Philodemus passage, whic is continued further with a polemic against certain students of Plato who used “the fruits of learning” for their own purposes. Although the end of columnY is ina very damaged state, and Gaiser’s o of what follows after “optics and mechanics” is only tentative,” nothing that is discernible here implies that this quotation comes from a work on the history of mathematics. This work clearly focuses on Plato, and the passage on the progress of poBhuato appears to be justa gras sion, intended to show that his influence has reached this sphere too. ye IN ia ( (D.L, 2-L. 1V.13-14). @ Bewpia | 'Aprdpov TP, Mepi1 yewpetpias, ram 1009T2-3, F1-3). | 10 This work might be the same as his MAdtwvos Eykapıov (fr. 1 Tarän = FGrHist 71 Col. VI and Pap. Here. 164, fr. 12. See Gaiser p. 185, 441f.; Dorandi (1991) p. 178. 72 DL. IV.5; fr. 28 Tarn. | 73 Fg. the references to Eudoxus who “applied the method of analysis to vin theory of the section, which originated with Plato” (p. 67.6), and to another geometer, Leon, who discovered the method of diorism (p. 66.20). Cf. below, n. 96. | 74 Gaiser p. 153. Third, although Plato could not have been a referen ce point in Eudemus’ history of fourth-century geometry, he could play such a role both for the early Academicians and for the Neoplatonists. Thus, it is possible to come up with the following alternatives: | a 75 In the same direction points the beginning ofcitation: [atelvevonto rai TOV pabnuctwv Enidooıg ROAAT Kat’ éxeivov tov xpóvov. The motif of the rapid develeither the Catalogue was taken from a book by one of Plato’s students and does not have any connection with Eudemus, or it was compiled on the basis of Eudemus’ writing and its Platonic features are explained by later Neoplatonic editing. The second alternative seems to me preferab le, since the traces of such editing are discernible as well in the first part of the Catalogue, e.g., in the section where a discovery of the five regular solids is ascribed to Pythagoras (p. 65.15f.) and typically Neoplatonic terms occur («vAwc, vosp@c). Also the referenc e to the late pseudo-Platonic dialogue Anterastai (p. 66.3) can belong neither to Philip, nor to Eudemus. Further, if the Catalogue goes back to the same text as Philodemus’ quotation, he must have known this text. Meanwhile, in his list of Plato’s students (col. VI) only two of the twelve mathematicians of the fourth century, listed in the Catalogue, are named: Amyclas of Heraclea (who, however, is called here Amyntas) and Archytas, whereas Philip is inexplicably missing! Both of these names also occur in the source of Diogenes Laertius’ list of Plato’s students and can thus be traced back to a common tradition has no connection with the Catalogue, where Archytas is that not called Plato’s student.” When analyzing the Catalogue many have overlooked the fact that the Platonizing tendency in this text does not end with Philip, but includes Euclid as well. Proclus, once again uniting all the previous mathematicians around Plato, says: | Euclid was younger than oi repi NAdtwvea.. sg 281 ., but belonged to his school and had an excellent knowledge of his philosophy, and he even set the final goal of the Elements as the construction of the five Platonic bodies (p. 68.17f.). For simple chronological reasons, this description could come neither from Eudemus nor from any of Plato’s students. It is, however, so similar to Philip's description that the two passages might appear to have opment of the exact sciences is known from Aristotle’s Protrepticus: ev oAlya tosadınv éxidoaw tiv tov ualnuétov Gewpiav Aoßeiv (fr. 5 Ross = fin 5 { Düring); tocobtrov Se viv rporAnA baci Ex pikpôv EHOPHÜV EV Ehaxtot@ xpove Entodvieg of te nepi thy yeoperpiav Kai toba AGyous Koi TAG dA GG rocio, öcov ovdiv Erepov yévoc ev odSep1& Tv texvav (fr. 8 Ross = fr. C 55:2 Düring). 76 Diogenes’ list (111.46) probably goes back to Plato’s biography written by Theon of Smyrna (Gaiser p. 439f., 444). Theon’s list, preserved in Arabic, contains both Amyclas and Archytas, whereas in Diogenes Archytas is omitted, since he belonged to the Pythagorean school, and not to the Academy.

283 been written by the same hand. Both mathematicians worked under Plato’s guidance (although in Euclid’s case this guidance was indirect), and for both Platonic philosophy was the final goal in mathematics. This similarity proves again that the whole historical digression in Proclus’ second introduction was subjected to Platonizing editing after foundations of the exact sciences and to their classification; his main goal was to show the logical consistency of Euclid’s mathematics and the fourth century. to refute its critique by the Sceptics and Epicureans.83 There is no evi- In a paper devoted to the Catalogue, Eggers Lan also notes many passages where Neoplatonic influence is clearly observed, but this leads him to completely different conclusions: the Catalogue was compiled by Proclus himself and besides two or three references it does not go back to Eudemus, either directly or indirectly.”” To be sure, in addition to all the aforesaid, the late composition of the Catalogue is evident from the fact that, besides Euclid, it mentions Archimedes and Eratosthenes. This, however, does not mean that we should exclude Eudemus from its main sources. Even if we had not known about his History of geometry, we would have to infer the existence of sucha work, proceeding from the detailed information of the Catalogue on mathematics of the 6th-4th centuries and precise chronological indications. The very fact that it contains many of the mathematicians’ names, practically unknown to us from the other sources, suggests its early origin. Where, if not in Eudemus, could a later author get the information that Neoclides was younger than Leodamas (p. 66.18), and that his student Leon was a little older than Eudoxus (p. 67.2), if Neoclides and Leon are not mentioned elsewhere? Considering Eudemus as the main source of the Catalogue, the scholars differently replied to the question whether his History of geometry was available to Proclus and whether Proclus himself could have been the editor of the Catalogue. Whereas Tannery denied both propositions, van Pesch and Heath were inclined to the following conclusion: although Proclus probably had access to Eudemus’ book (as later did Simplicius and Eutocius)’® he did not compile the Catalogue, but took it from an earlier source.?? Both van Pesch and Heath emphatically denied Tannery’s idea that Posidonius’ student Geminus was an editor of the Catalogue and an intermediary between Eudemus and Proclus.# In fact, nothing that we know about Geminus and his work Ma@nuéray dempia8' suggests him as a source of either Neoplatonic influence in the Catalogue or its special interest in those predecessors of Euclid who compiled Elements.8? Geminus’ work was devoted to the dence that he was particularly interested in the history of mathematics before Euclid, or that he knew Eudemus’ book.3* One gets the same impression from his Introduction to phaenomena: clearly, Eudemian History of astronomy was not used there. the Generally, one has to say that Tannery’s arguments in favor of Geminus are so weak that it makes no sense even to repeat them.85 Incidentally, apart from Geminus’ encyclopedia there were three commentaries to the Elements among Proclus’ sources: by Hero, Porphyry and Pappus. There is no point in discussing Hero as a Platonizing editor of Eudemus. Pappus, although he suits this role better than Hero,®° does not mention Eudemus in his vast Collectio and only once in the commentary on book X of Euclid. Plato’s name occurs only twice in the Collectio: first, in connection with the “nature of proportion” and second, concerning the so-called Platonic bodies.®’ Pappus was clearly too “technical” an author to be enthusiastic about Plato’s contribution to geometry. As for the Neoplatonist Porphyry, he is the most appropriate 8! This title is given in Eutocius (/n Apoll. Con. II, p. 170.25f, Heiberg), while Pappus refers to Nepi Tis tOv pabnpdtov téEews (Coll. VII, p. 1026.8f. Hultsch). Tannery (1887) p. 18f. and Tittel col. 1040f, were inclined in favor of the first title, Mansfeld (1998) p. 24 n. 21 — the second. 82 See van Pesch p. 80f.; Heath (1927) p. 37; Eggers Lan p. 140f. 83 Tittel col. 1040ff. The same was characteristic of his teacher Posidonius (frr. 46-47, 195-99 Edelstein-Kidd). 84 Tittel col. 1048 thought that Geminus knew Eudemus through Eratosthenes, but in this case Eratosthenes becomes the editor of the Catalogue. The only mathematician of the fourth century mentioned in that part of Proclus’ commentary which can be safely attributed to Geminus (see: van Pesch p. 112ff.) is Menaechmus, the older contemporary of Euclid. In two cases methodological debates and terminology are in question (In Eucl. p. 72.24f., 78.9f, 78.17f.), and only once he refers to Menaechmus’ discovery of the conic sections (p. 111.21). But even here his authority is Eratosthenes’ Fans and not Eudemus. About the earlier mathematicians he says nothing. Cf. above, 77 Eggers Lan p. 154ff. 78 Cf. above, n. 3. 79 van Pesch p. 84; Heath (1927) p. 37f. 80 Tannery (1882) p. 172ff.; Tannery (1887) p. 71ff. 85 Tannery (1887) p. 71f., cf. van Pesch p. 81f. 86 On Neoplatonic influence on Pappus sce Mansfeld (1998) p. 99ff. 87 III, p. 86.19f., V, 352. 11f. Hultsch.

option for the role of an intermediary-editor. Actually, una too 4) considered this option, but then rejected it in favor of Geminus.* Such a decision was largely affected by his own, rather mechanical division of the “spheres of influences” between Porphyry and Geminus. Tannery 65.15f.) contains. on the one hand, Neoplatonic terms and, on the there is a short mention of Theodorus and Hippocrates (p. 77.24f. Festa), missing in the parallel passage of his biography of Pythagoras (VP 89). This remark is very similar to that place in the Catalogue where both mathematicians are also mentioned in phyry.# What makes Porphyry the preferable candidate? one sentence (p. 66.4f.). It looks as though lamblichus used the same source as Proclus,?? i.e., Porphyry. To the same source must 1) Proclus in his Commentary five times refers to Porphyry’ work go back lamblichus” information on the development of the theory of means from Pythagoras to Eratosthenes.% that was, in all probability, a commentary to Euclid.” Such a commentary, written by the Neoplatonist Porphyry, could explain both the Platonism of the second part of the Catalogue (as well as of some portions of the first part) and its interest in the predecessors of Euclid. 2) Proclus twice refers to Eudemus immediately after Porphyry, the first time in the space of one page (p. 297.4-298.10 and 299.3, on prop. XIV and XV), and the second time in the very same line (p. 352.14, on prop. XXVI). This reinforces the probability that Porphyry’s commentary included references to Enger book (in both cases theorems of Thales are in question). À arithmetic (fr. 142). | | 88 Tannery (1882) p. the same second introduction 82 Er i UE Evpyurá Enrñ poro indirect references do not give {p. 56.24), where the Catalogues located. To be sure, an adequate picture. In the first introduction Proclus extensively borrows from lamblichus’ De communi mathematica scientia, but he nowhere mentions his name " "50 id 255.12-14, 297.1-298.3, 315.11-316.13, 323.5-326.5, 347.20-352.14 ire Commentary. Sec Mueller (19876). = (rr. 482-86 Smith. See Tannery (1882) p. 170f.; van Pesch p. 127f.; Heath (1921) vol. 2, p. 529; (1927) p. 24; Mansfeld (1998) p. 24, On the influence of Porphyry’s commentary on Proclus, see Mueller (1987a) p. 311f. 5) Porphyry wrote a history of philosophy that embraces practically the same period as the Catalogue: it starts from the Oriental predecessors of philosophy; covers Thales, Pythagoras and other Presocratics; and ends with Plato.% Porphyry clearly viewed Plato’s system as the relos of the whole previous philosophy. and this attitude seems to be very close to the “Platocentrism” of the second part of the Catalogue, where the construction of the five Platonic bodies is called “the final goal of the whole Elements” (p. 68.18f.). This provides additional support to the hypothesis that Porphyry was the principal intermediary between Eudemus 3) Porphyry was the author of a commentary to Ptolemy’s Harmonics that preserved the only fragment from Eudemus’ History of i The passage from the Catalogue relating to Pythagoras (p. other, partly coincides with the passage from the work by Porphyry’s student Tamblichus.?? In the same work of lamblichus believed that Geminus is mostly presented in the first part of Proclus” commentary, where the Caralogue is located, and Porphyry in the second. Yet this picture is far from being correct: in both introductions to the commentary there is material both from Geminus and from Por- 171f. 285 | 9 tt is possible that Proclus took these two citations of Eudemus not directly from the History of geometry (as was the case with the other excerpts), but from Porphyry. It is equally arguable, however, that Porphyry’s references prompted Proclus to look in Eudemus’ book. Simplicius too used Eudemus’ book both directly and indirectly, through Alexander's paraphrase (fr. 140). See Knorr (1987) p. 29ff. and Proclus.°6 To sum up, we find in Porphyry exactly what is missing in Geminus that would have enabled us to consider him the editor of the Catalogue: predisposition to Platonism, obvious knowledge of Eudemus’ works, 92 De comm. math. sc. p. 70.1f. Festa. See: Burkert (1972) p. 410. 93 Björnbo col. 1782; Burkert (1972) p. 458 n. 59. In lamblichus' passage there are two examples ofthe “progressive” terms (ër16186va and npowyayeiv) , so characteristic of the Catalogue. See below, n. 151. 94 See above, p. 271. The latter is mentioned at the very end of the Catalogue (p. 68.18-20). 95 piddcowos iotopía Ev PiBAloic è’ = frr. 193-224 Smith. Apart from the fragments, only one part of the first book is preserved, containing the biography of Pythagoras. 96 Porphyry might have derived additional information from the same work as Philodemus, which would explain certain similarities between the Catalogue and column Y. It is noteworthy that he knew, although indirectly, Hermodorus’ book On Plato (frr.'7-8 Isnardi Parente = FGrHist 1008F2a-b = fr. 146 Smith).

al and interest in the intellectual history of the 6th—4th centuries in gener ient suffic a be to seems in Euclid’s predecessors in particular. This Catathe of form ground to think that we owe to Porphyry the present of the logue, i.e.. a very abridged and tendentiously revised versiontion, he History of geometry. Following Eudemus’ chronological exposi iate immed the to s Thale compiled a short outline of geometry from precursors of Euclid and used it as a historical introduction to his comyry mentary on the Elements. It is conspicuous that as soon asthePorph le reliab lack to s begin he , leaves Eudemus' text and turns to Euclid historical data. An absence of Eudemus’ followers in the Hellenistic a rough period made it very problematic for Porphyry to figure out even date for Euclid: he had to rely on casual remarks of Archimedes and st Eratosthenes, and on historical anecdotes (p. 68.10f.). What a contra e! with Eudemus’ chronologies, which are precise almost to the ondecad the of It would be mistaken to think, however, that the final versi bes to ge that aseri Catalogue is due to Porphyry. Specifically, the passafive regular solids the of and y Pythagoras the discovery of irrationalit we do not rate, any At (p. 65.15f.) can likely be attributed to Proclus.? . (De find this information in the parallel passage in lamblichus “Panpcomm ythamath. sc., p. 70.1f.) nor in his other works. Meanwhile, us, Procl of than ichus lambl of ic terist goreanism” is much more charac elsefrom or yry Porph from so if he had known this tradition, either n rer where, he would have certainly mentioned it. While the versious, | the Procl ring all the regular solids to Pythagoras is known before idea that he was the author of the theory of irrationals (tov GAOYOV this connecKpaypateta) is not attested in any other ancient source.ngInTV GAOYOV tion it was proposed long ago to change the readi the theor y of rpoypateia into Tv dvà A0yov rpoypateto, l.e., into in one manu proportions.” This reading, however, 15 attested only from script, all the others have tv aXoyov npaypoteia. To judge three first lamblichus’ passages that connect Pythagoras with the about Tov means,!°! Porphyry and, correspondingly, Eudemus wrote and by ¿vá Aöyav npaypateta. It is confirmed both by Nicomachus the obvious fact that Pythagoras really dealt with proportions. What 97 Cf. Burkert (1972) p. 41 If. 98 Aét. 11.6.5 = 44A15; Burkert (1972) p. 70 n. 113. 99 DK 1, p. 98.23; Heath (1921) vol. 1, p. 84f. 100 | discussed this passage in Zhmud (1997) p. 158f. 101 See above, p. 271. 287 made Proclus change the “theory of proportion” into the “theory of irrationals”? I suppose he did it for the same reason he is silent about Hippasus, one of the important mathematicians of the early fifth century. Late tradition connects with Hippasus two significant discoveries: the construction of the dodecahedron inscribed in the sphere and the discovery of irrationals. One part of the evidence (Clemens of Alexandria, lamblichus) mentions Hippasus” name, the other speaks about some anonymous Pythagorean.!° At least two pieces of evidence from the last group, which concern the discovery of irrationality, are based on Eudemus: Pappus’ commentary on book X of Euclid and the scholia to the same book.!% To them we should add the first scholium to book XIII that relates to the Pythagoreans the discovery of the first three regular solids, including the dodecahedron. Does it follow from this evidence that Eudemus has referred to some anonymous Pythagoreans, and the late tradition filled in Hippasus’ name? If so, we are not in a position to determine who in fact was the author of these discoveries. Besides, it means that Hippasus totally disappears from the history of mathematics, since no other discoveries are ascribed to him. In other words, if Eudemus and his contemporaries had not known the mathematician Hippasus, he did not exist. Meanwhile, Hippasus was known as a philosopher to Aristotle and Theophrastus;!TM Aristoxenus referred to his acoustical experiment, based on mathematical proportions;!® lamblichus, relying on the tradition that most likely derives from Eudemus, likewise connected his name with the first three proportions.'% Thus, the Pythagorean Hippasus who took up philosophy, harmonics, and mathematics did not merely exist but was known at the end of the fourth century. On the other hand, Eudemus mentioned the discovery of the irrationals and the construction of the dodecahedron by the Pythagoreans. I believe, therefore, that the late tradition assigning these discoveries to Hippasus, contains a historical core and might go back to Eudemus. If, however, 102 The evidence and its analysis, see Zhmud (1997) p. 170ff. 103 Pappus, p. 63f.; Schol. in Eucl. p. 415.7, 416.13, 417.12f. Heiberg. See above, n. 27. 10% Arist. Mer. 984a7; Theophr. fr. 225 FHS&G, cf. Aët. 1.3.11, IV.3.4 = 18A9. 105 Fr. 90 Wehrli, cf. 18A13. See above, n. 32. 106 In Nicom. p. 100.23, 113.17, 116.4 Pistelli. See above, p. 271f.

Eudemus named Hippasus, why then is his name missing from the Catalogue? There is at least one important ground for such an omission.Proclus, unlike Eudemus, attributed to Pythagoras these very discov eries that the late tradition connects with Hippasus: the theory of the irrationals and the construction of all five regular solids, including the dodecahedron. Therefore, for the mathematician Hippasus there was no place left in the Catalogue. One might surmise that Proclus had decided to trust the tradition that persistently connected Hippasus with plagiarism and with divulging of the Pythagorean secrets!” and to sacrifice quirements of history, he did much more for its development than any ancient philosopher.!!? Therefore, it is difficult to agree with the views that he held history in very low regard, because it deals only with the particular,'!* or that history (including intellectual history, the history of ideas and of philosophy) interested Aristotle not per se, but only as ts ed Rise dialectical discussion in the course of developing his own | We can better understand what value the Peripatetics did assign to historical research, if we remember that Aristotle’s theory of science stresses the empirical origin of any knowledge. The question of Ott this figure by “returning” his discoveries to Pythagoras.'® (collection and description of the facts) precedes and makes possible the question of 51671 (explanation of the general or particular causes) 4 Any scientific explanation must be based on the previously arranged Eudemus’ history of the exact sciences has to be considered against the background of other historical genres that were initiated or cultivated in the Lyceum. Aristotle himself compiled lists of the Olympian relevant facts.'!5 In this way, even purely descriptive work is a legitimate and necessary part of scientific procedure, if it presupposes subsequent theoretical analysis, and the Peripatetics wrote, as we know ' and and Pythian victors, of the dramatic victors at the Dionysia, wrote a number of other writings of a historical character.'!° Besidebys cultural history and philosophical doxography, developed and Aristotle.!!! and after him by Dicaearchus (Biog "EAAAÖöOG) was phy, Theophrastus (®voıx@v 66Ga1), a new genre, the biogra founded by Aristoxenus and Dicaearchus. Being fully aware of the hisphitorical character of such human accomplishments as the state, art,their of logic inner the losophy or science, Aristotle tried to reveal development. Even if Aristotle does not entirely meet the modern re107 This tradition is presented in Iamblichus’ work that Proclus certainly used (De comm, math. sc. p. 77.18f. Festa, cf. VP 88, 246-47). concerned 108 The motives of Pappus’ silence can be more prosaic: he was not verysays that the chus Nicoma While . ticians mathema the of names the ing about mention arith. (Inır. ras” Pythago first three means “came down to Plato and Aristotle from p. 84.1f. Hultsch). Thisp. 122.11f.), Pappus quoting him omits all the names (Coll. Wl, presents three methis not the only example: in the fourth book of the Collectio Pappus , see Knorr (1989) ods of angle trisection without any attribution. On Pappus' method p. 227ff. 109 D.L. V.26, Nr. 130-31, frr. 615-17 Rose = frr. 408-14 Gigon; D.L. V.26, Nr. where 110 See, e.g., his dialogue On poets (frr. 70-77 Rose = frr. 14-22 Gigon), hishis lly especia and genres, nt differe the of s founder the much attention is paid to De Cf. Gigon). 123-34 tory of rhetoric in Texv@v cuvavyayi (frr. 136-41 Rose = frr. 133, Ste. Croix p. 29 with further references. III On Aristotle’s doxography, see Mansfeld (1990) p. 28ff. 289 hundreds of such works. It seems very probable that the entire historiographical project, conceived by Aristotle, i.e., Eudemus’ histories of the exact sciences and his history of theology, Theophrastus’ “physical” doxography, and Menon's medical doxography, set as a general task the collection, systematization and at least preliminary analysis of evidence for subsequent philosophical examination. It is conspicuous, further that the study of the different fields of knowledge was distributed among Aristotle's students according to his division of theoretical sciences into mathematics, physics, and theology.!!$ This corresponds a See von Fritz (1956). Zoepffel p. 37: “so ist Historia in ihrem Bereich, vom philosophischen Stadpunkt ' aus betrachtet, kein wirkliches Wissen.” Cf. De Ste. Croix p. 29: Aristotle “evidently saw research into historical facts as a necessary condition, though not of course a dr condition, of full scientific knowledge of human society.” Baltussen, e.g., suggests that Aristotle was not a historian sensu stricto and that we can hardly regard him and his followers as historians in the modern sense of the word (p. 335f.). This is of course correct, yet is “a historian in the modern sense of the word” the only possible kind of historian? What would remain of Jacoby’s Fragmente der griechischen Historiker, if we apply to them modern conceptions of A historical research? The Peripatetic way of doing the history of philosophy certainly was methodologically immature and subjective. Still, I do not think that the best way to study doxography is to isolate it from the other historical genres practiced in the nai and to regard it as a part of Aristotle’s dialectical procedure. ane Post. 89b29, 93a16ff. See Kullmann p. 204ff. Met. 1026a18ff., 1064b1 ff.

291 quite well to Aristotle’s common method and to his other project, the collection and the arranging of material for his Politics, in the course of which the 158 polities of the different Greek cities were written. They were also classified according to his theoretical scheme of the different types of power.!!? It hardly needs reminding, however, that Aristotle’s Athenian polity is a historical treatise of independent value, addressed to a wide audience, and not just a dossier. In fact, nothing suggests that Aristotle and his followers self-consciously limited their historical writings only to the question of facts and of particulars, trying to avoid any generalization.''® Modern historians will agree with Aristotle that history as such does not belong to the theoretical sciences, although there are some historical regularities that we can discern and importance for defining its genre. Although Thucydides’ book had no title, and he never uses the word iotopia this does not prevent us from study." referring his book to the historical genre. The sentence already quoted What was the difference between the approaches of Eudemus, Theophrastus, and Menon to the various fields of knowledge within the framework of the Peripatetic project? How were Eudemus' writings arranged: chronologically, thematically or problematically? Is it appropriate to define them as a history of developing knowledge, or were they rather “a classification of the authors according to different math- '?! ematical topics”?!?0 Let us begin from the titles of Eudemus” books. What, e.g., does the title leountpixh iotopia mean: “geometric inquiry,” “inquiry into geometry” or “history of geometry”? It is clear that this book was not a mathematical treatise and that Eudemus was inter117 See the title: Constitution of 158 cities, according to the type, democratic, oligarchic, tyrannical, aristocratic (D.L. V.26). ested not in geometrical problems themselves, but in the way they were solved by others. Thus, the subject of his study coincides with the subject of the history of science, as we understand it. That is why the titles of his works should be compared not so much with the notion nepi pvoews Lotopio, reflected in the titles of Aristotle’s and Theophrastus’ works (iotopia Chov and repi purûv iotopia), but rather with that usage of totopia which is attested already in Herodotus (VII.96) and is understood commonly as a “written account of one’s inquiries, narrative, history.”** But even if Eudemus’ book had been entitled simply repi yewnerpiag,'? or had no title at all, this would not have crucial from the Catalogue, oi Tag iotopiac dvaypawavtes péypi todrov MPOcyouai TV tig EMO TA Ns Tadıng teAeiwon (p. 68.4) definitively shows that readers took Eudemus’ book as a historical exposition of the development of geometry. To be sure, for Aristotle and for the Lyceum in general iotopia nepi pÜoeuxs was not yet so strictly separated from iotopia of human npübeıg and yevôueva as it is for us. Comparing the fragments that come from the books with the same title ‘Iotopixà drouvnuara, by Theophrastus (fr. 196A FHS&G), Aristoxenus (fr. 131) and Hieronymus of Rhodes (frr. 35-36), we can see that Theophrastus deals with “natural history,” whereas the two other Peripatetics examine the historical (biographical) material.'?* A specific feature of human history 118 Dicaearchus’ Biog ‘EAAGSo¢ deals apart from particular events and individuals with the general stages of the development of civilization (frr. 47-51). At the beginning of his History of geometry Eudemus states a general rule of the cognitive evolution from aïoBnoic to Aoyiouôç and further to voög (fr. 133), and this is clearly not the only case of 81671. Two of Theophrastus’ works, Historia plantarum and De causis plantarum,-are devoted respectively to the research of Sti and of S16t1. In Historia plantarum one finds, however, not merely assembled data but botanical classification, morphology, and taxonomy. 119 De Ste. Croix p. 24f. discusses in this connection an important Aristotelian concept, dag éxi td ROAD, “as a general rule.” According to Aristotle, this kind of regularity along with the universal and the necessary also belongs to the theoretical £rıotnun (Met. 1027a20-21, 1064b32-36, 1065a1-6). 122 LSJ, s.v. II; Hornblower p. 9. On iotopia as “history” see: Arist. Rher. | 359b30f., 1360a30-37; Poet. 1451b1-7, 1459a21-24; Anaximenes (FGrHist 72F3, 9). Ephorus, the older contemporary of Eudemus, gave the title ‘Iotoptat to his universal history. Theophrastus’ student Praxiphanes criticizes Thucydides in his Tlepi tatopias (fr. 18 Wehrli). The content of Theophrastus’ work Mepì totoptac (D.L. V.47) is unknown. '23 Cf. Aristoxenus’ epi &pBpnrueñs (fr. 23 Wehrli) that also deals with the history of mathematics. 124 It is worth noting that whereas totopia Cowv was considered by Aristotle as “natural history,” repì pvoews iotopía was traditionally understood as “natural philosophy.” This hardly justifies Zoepffel’s idea that totopía as such rules out any research into the general causes. At De caelo 268al nepi pooeux iotopia means the same as nepi pÜoeux ériotpn (298b2), and at De an. 402a4 ih TG yuyfig iotopia is cha- 120 Eggers Lan p. 130. 1?! They are given by several authors, the most exact variant being preserved by Simplicius: Fewperpich iotopia (fr. 140), "AotpoAoyıxn iotopia (fr. 148). Porphyry Ts Sopiac iv Sn Kadobat nepi poses iotopiav that investigates tac aitias Erdotov refers to "ApiBuntuch iotopia (fr. 142), at PI. Phaedo 96A). Quite consistently with this usage Theophrastus employs the notion racterized as a purely theoretical investigation (cf. Socrates’ ironical narration about

is that the chronological principle is the most natural and convenient way to systematize the facts (Arist. Poet. 1459a21-24), whereas in the totopía epi gboewg other principles of systematization are usually 293 at least partially, a historical dimension to the doxography were based on the idea, very important for Aristotle, of the gradual perfection of philosophy from its first immature ideas to the present state.'28 If the applied. idea of development and the historical approach that follows from it were not applied consistently, it is because the systematic pattern anlast mathematicians he refers to in the History ofgeometry are Eudoxus’ with. The subject of Eudemus’ works was the development of the exact sciences from their origin to the second third of the fourth century. The students. This corresponds to the chronological framework of his History of astronomy which starts from Thales (fr. 143) and ends with Eudoxus’ student Callippus (fr. 149), Eudemus’ own contemporary. The fact that in the Catalogue all the names of the geometers are given in chronological order, often with indications of who was older and younger, who was whose student, etc., also speaks to the historical character of Eudemus’ writings.!? Besides, we know that Eudemus considered the Pythagoreans in the first book of the History of arithmetic (fr. 142), the quadrature of lunes in the second book of the History of geometry (fr. 140), and the theories of Eudoxus and Callippus in the second book of the History of astronomy (frr. 148-49). In the doxographic treatises of Theophrastus and Menon the material is arranged as a rule along different lines. They classified Soñar not according to the chronology of their authors, but according to certain topics or problems, such as the nature of the moon’s light or the causes of diseases. The same pattern of organization is to a large degree characteristic of the doxographic passages in Aristotle, where he discusses the &pyai of his predecessors (Met. 1.3-7; De An. 1.2; Phys. 1.2-4). This pattern was originated by Hippias of Elis, Gorgias, Isocrates and others, who classified ideas on thematic or systematic principles, e.g., according to the character and/or number of &pxai that were held bya philosopher.!?6 We cannot say, however, that the Peripatetics blindly followed the schemes of the early doxography. Already Aristotle, and particularly Theophrastus tried to combine the systematic pattern with the chronological, by tracing the development of the ideas from teacher to student, from one school to another, etc.!?? These attempts to give, repi púceos iatopica both to Thales’ and to Plato’ philosophy (fr. 225, 230 FHS&G). See also fr. 224 on Xenophanes. 125 Heath (1927) p. 38. 126 Mansfeld (1990) p. 5Sf. Doxography aimed to describe all the (relevant) 50Ga1, not merely the true ones; putting it another way, its subject was pvoixdv d6&aı, and not voix émiotun. Furthermore, in the physical and medical theories studied by Theophrastus and Menon there were no hard proofs, and to separate the true 66Ea1 from the untrue was difficult. The ideas of the sixth-century thinkers often appeared to be more convincing than those of the more recent periods, thus they did not always fit the chronological pattern. As for Eudemus, he could employ an historical approach much more consistently than his colleagues mostly due to the character of his material. Indeed, at that time the cumulative development of the exact sciences was more manifest than that of natural philosophy or medicine. The discoveries of many mathematicians immediately depended on what was done before them: Hippocrates and Theaetetus relied on Pythagorean mathematics, Archytas and Eudoxus developed the theories of Hippocrates, etc. Progress in mathematics is all the more evident, because, according to Aristotle’s concession, “it alone among human activities knows of proofs.”!?? That is why a mathematician could base his research on a solid foundation created by his predecessors and move further in his quest for the truth. To be sure, Eudemus records some unsuccessful attempts to solve mathematical . problems, e.g., that by Antiphon to square a circle (frr. 139-40). It lies, however, in the nature of mathematics that its history, more than that of any other science, lists a much larger number of victories than of failures, And yet I would not exaggerate the dependence of Eudemus’ approach on the material he was dealing with. Both his inclination to historical (or, at least, chronological) schemes and his interest in the history of ideas are revealed outside the exact sciences. In the History of theology he also names the theologoi according to their chronol- | o 127 Ibid. p. 28ff. It is conspicuous especially in the first book of buarxav dólar (von Kienle p. 38f., 52f., 581.). swered much better the purposes of doxography and the material it dealt 128 Mer. 993a15-17; fr. 53 Rose; see below, p. 296f. 129 Arist. ap. lamb. De comm. math. sc. p. 78.8f, Festa. See Burkert (1972) p. 447f.

ogy.!* In the fragments of his Physics there are unusually many doxographical digressions, presenting and/or criticizing the views of Parmenides. Zeno, Melissus, Anaxagoras, Empedocles, the Eudemus’ History of Mathematics a = is seen in the Catalogue, and not from one problem to 295 the A few words about Eudemus' style. Simplicius twice notes the la- Pythagoreans, Archytas, and Plato.'?! The same names (except for Archytas) could be found in Aristotle’s Physics, but Eudemus, it seems, conic and clear style of his History of astronomy (fr. 149) and in quotpays much more attention to them. Eudemus’ memorandum-like style (fr. 140). It is interesting that Returning to the problem of how material was organized in Eudemus’ history, I quote Wehrli’s opinion: “Der Stoff war nach Auftreten und Entwicklung der maßgebenden Gedanken, nicht nach Autoren geordnet.”!3? To be sure, one and the same mathematician, e.g., Hippocrates, could appear in connection with the problem of doubling the cube (along with Archytas and Eudoxus) and in connection with the quadrature of the circle (with Antiphon). The same can be said about the Pythagoreans and Theaetetus, Democritus and Eudoxus, etc, But J still do not think that the thematic or problematic principle predominates in Eudemus. Such a principle is suitable for examination of how different mathematicians solved the same problem — e.g., the duplication of the cube — but it does not fit the historical study of whoma particular mathematical discovery belongs to. It is evident, at least, that such figures as Thales, Mamercus, Pythagoras and Oenopides were treated by Eudemus in connection with their discoveries. By the way, it is hard to call Oenopides’ discoveries maßgebende Gedanken; Eudemus very likely referred to them only because he knew that it was Oenopides who made them. As to Mamercus, Eudemus scarcely knew more about him than that he was famous as a mathematician; such a figure could be mentioned only in a historical context. Eudemus appeared to deviate from the chronological principle only when he treated the problems that several mathematicians of different generations had worked on. But usually he moves from generation to generation, from teachers to stu- 130 Fr. 150. With one exception: Acusilaus (ca. 500 nc) is mentioned after Hesiod, but before Epimenides and Pherecydes who are older than he. One reason for this might be that Acusilaus appears to follow Hesiod very closely in his Genealogiai (DK 9A4). This was explicitly mentioned by Plato (Symp. 178B = DK 9B2), on whom Eudemus could have relied. It could be also that Eudemus simply did not know the correct chronology of Acusilaus. 131 Frr. 31, 35-47, 49, 53-54, 60, 65, 67, 75, 78, 82, 89, 110-11, 118. 132 Wehrli p. 113; “problemgeschichtliche Anordnung,” ibid. p. 119. ing a long passage from the History of geometry he refers again to Simplicius mentions the “archaic brevity” of his account at the very place (fr. 140, p. 59.26) where Eudemus himself was going to consider the problem in detail (fr. 140, p. 60.1). It seems, therefore, that in this case Eudemus simply reproduced Hippocrates’ proofs, which were not sufficiently complete from Simplicius’ point of view. In any case, we know that Eudemus did not confine himself to the abbreviated exposition of the material. Speaking about Thales’ discoveries, he notes both his archaic terminology,'** and whether the latter gave a scientific proof or not (fr. 135). At one point Eudemus even tries to “reconstruct” the theorem that he ascribes to Thales (fr. 134). Referring to the Pythagorean application of the areas, he emphasizes that this discovery was ancient (fr. 137), and his commentary on the quadrature of the circle contains a criticism of Antiphon (fr. 140). Generally, one can say that Eudemus paid attention not only to the results and the form of their exposition but also to the method of the proof and whether it corresponded to what was customary in his own time. In the Catalogue, the perfecting of the geometrical methods is discussed in much detail. It is said here that Thales’ method was in some cases more empirical, and in others more general, i.e., scientific (In Eucl., p. 65.10), whereas Pythagoras is credited with the transformation of geometry into abstract science and its inclusion in the canon of higher education.!% Owing to the efforts of Leodamas, Archytas, and Theaetetus geometry became “more scientific and systematic,” while Leon discovered the method of diorism, Eudoxus used the method of 133 Heath (1927) p. 38; Edelstein p. 95. Quite a different approach was characteristic of the later authors like Pappus or Eutocius. They could describe various solutions of the same problem ignoring either the chronology of their authors (Eutoc. /n Archim De sphaer. Ill, p. 57.13ff. Heiberg), or even their names (Papp. Coll. IV, p. 270 if. Kulisch): see Knorr (1989) p. 77ff., 213ff. In Eucl. p. 250.20f. See the similar note on Oenopides 135 Cf. Arist. Met. 985b23f., Protr. frr. 18, 20 bains er

297 analysis, and his students Menaechmus and Dinostratus “made the expressed by the poets of the fourth century.'“° Although these paralwhole of geometry still more perfect” (¿ti teAewtepav Exoinoay nv Anv yempuetpiav). It is significant that the Catalogue starts with the words about the transition and tod atedodg eig TO TEAELOV and ends lels do not cast doubt upon Eudemus’ Aristotelianism, they show that he did not slavishly follow his teacher’s ideas, but rather that both shared views common for that period.!'*! The same common attitude is with the notice that “those who compiled histories of geometry bring seen in the thesis that all the sciences appeared due to their practical the perfecting (teAciwotc) of this science up to this point.” The idea of necessity, the Egyptians being the first to discover geometry (/n Eucl., p. 64.17). The author of the Ancient medicine also believed that medithe “completion” of geometry at the end of the fourth century, i.e., before Euclid, Archimedes, and Apollonius, could not belong to any author later than Eudemus. This confirms once again that the beginning of the Catalogue also goes back to him. cine was discovered due to necessity and need.'#? We find analogous notions on the origin of téxvn in many classical writers.'43 The idea that geometry originated with the Egyptians has even more predecessors.!44 I have suggested above that in his histories of the exact sciences But whereas Eudemus follows Herodotus’ view concerning the practi- Eudemus employed the professional approach, to the extent, of course, that it was possible for a Peripatetic philosopher. This does not contracal origin of Egyptian geometry and shares Philip’s conviction that the Greeks bring to perfection all the knowledge they receive from the dict the fact that his views on development in general, and on developbarbarians, '*° Aristotle points out that paOnpatixai téyvo were first ment of science in particular, rely on the Aristotelian doctrine which can discovered in Egypt because the local priests were allowed to be at leibe tentatively called “teleological progressivism.” According to this doctrine, everything in nature, society and culture develops from a sure (Met. 981b23). This remark clarifies the preceding passage, where he states that in every civilization practical téyvoa. are born first, then primitive state to a perfect one,'? and for many things this state of perappear the fine arts, and after them theoretical &mıotfjuan, for which. fection was thought to have already been achieved or almost achieved. adds Aristotle, leisure time is needed. So, tragedy has already “attained its very nature” (Poet. 1449a15), so- It is noteworthy that the same historical scheme was known before Aristotle; some of his predecessors did notice the role of oyoAn, ciety has found its best and final state in the polis (Pol. 12522261253a9), and philosophy, whose early ideas were as immature as childish speech (Met. 993a15-17), will be completed soon (fr. 53 Rose).!*7 Against this background, Eudemus’ idea that geometry had whereas others did not.'* It is possible therefore that in this case Eudemus followed the more simple version that occurs also in Aristotle.!* The general line of the cognitive development from achieved or almost achieved perfection seems quite natural. !*8 To be sure, the epistemological optimism that Eudemus shared with his teacher was not exceptional in this period of classical rationalism. The author of the Ancient medicine was convinced: “what still remains undiscovered in medicine will be discovered” (VM 2), if the inquirer uses the right method. The other Hippocratics believed the whole of medicine to be already discovered,!* and no less optimistic ideas were 140 Chairemon: oùk Éotiv obSév tav Ev dvOpdmnorg Gti oùk gv xpóve Entodorv é€evpioxetat (TGF 71F21); Alexis: &xavta tà Entobueva eEevptoxertar (fr. 31 Kassel-Austin). 141 Cf. Isoc. Paneg. 10, Nic. 32, Euag. 7, Antid. 82, 185. 142 On the possible influence of Democritus or Protagoras, sec Jouanna p. 34f., 45f. 143 Democritus (68B 144); Isocrates (Bus. 12-15, 21-23); Philip (Epin. 974D8f., 975C9f.); Aristotle (fr. 53 Rose = Protr. fr. 8 Ross; Pol. 1329b25f.; Mer. 981b12-22, 136 “There is evolution towards a state of excellence all over in the design of nature; the goal is the end, telos,” Burkert (1997) p. 31. 137 “Since in few years great progress has been achieved, philosophy will be finished and perfected in a short time.” Cf, his similar remarks on the arts (EN 1098a22f.) and political systems (Pol. 1264a3). 138 Theophrastus expressed similar views: were men to live longer, the arts and sciences might be brought to perfection (fr. 34A FHS&G). See Edelstein p. 148 n. 31. 139 De locis in hom. 46; De arte 1. See Jouanna p. 43f. 982b22f.). 144 Herodotus (11.109); Isocrates (Bus. 28, cf. 23); Plato (Phaedr. 274D1f.; Leg. 747A-C); Philip (Epin. 986D8ff., 987D9); Aristoxenus (fr. 23). 145 K&AALOV todto eig tédos anepyalovraı (987E1) clearly corresponds to teAciwois in Eudemus. 146 Democritus (68B144), Plato (Leg. 677A-683B), and Philip (Epin. 974D3977B8) do not mention oxoAn. Cf.: Isocrates (Bus. 21-23), Plato (Crit. 147 Fr. 53 Rose = Protr. fr. 8 Ross: Pol. 1329b25ff.

299 aíc8nors to Aoyıoyög and further to vodc, which is known from the Catalogue (p. 65.1f.) has quite obvious parallels in Aristotle.'*® In ac- TPÒTOG Ebperng, so popular in the sixth and the fifth centuries,'*4 maincordance with this scheme Eudemus stressed the practical origin of Some of them contributed to heurematography, having written books with the standard title nepi edpnudtov.!% Again, ebpñuata as a stan- Egyptian geometry (but not its practical character), since even Thales, tains its significance for many genres practiced by the Peripatetics. who borrowed this science (Gewpia) from Egypt, still proved some things aio@ntiKdtepov and others xaBoA1x@tepov (p. 65.10). Thus, genre, Aristoxenus (frr. 23-24). The same topic is widely presented in for Eudemus the real founder of theoretical geometry was not Thales, Dicaearchus’ but Pythagoras (p. 65.15f.). In this opinion he hardly contradicts Aristotle, who writes in the same book of the Metaphysics: the Pythagoreans were the first to advance the mathematical sciences dard topic of philosophical biography goes back to the originator of this Bios "EAAadog as well as in Aristoxenus’ and Dicaearchus’ works on the history of poetry and music.'® The principle of the np@tog ebpeths was no less relevant for Theophrastus’ doxography than for Aristotle’s.!57 It is quite natural, therefore, that the question tig ti ebpev underlines (985b23f.). The idea of the progressive growth of mathematical knowledge prethe entire work of Eudemus on the history of science, the more so as in dates Eudemus, of course. In the Protrepticus Aristotle refers to signifithis field the idea of priority was particularly important. Indeed, exprescant progress being achieved in mathematics in a very short time.!% In Philodemus’ passage the term éxidooic, designating progress, is used sions connected with the idea of np@tog evpetic occur in almost every fragment of Eudemus’ historical writings,!% so that these writings twice: first, with regard to all mathematical sciences (tOv poOnpatov can be conceived as a detailed answer to the question “who discovered eridoois noAAN Kat’ éxelvov tov xpóvov), and second, in connection with geometry (EAaBe Sè Kai N yemuetpia noAAnv Extdoorv). Alwhat?” Yet, in spite of their closeness, the history of science was rather a by-product of the early heurematographic tradition than its direct heir. though éxidoo1c appears only once in Eudemus!* the verbs from the same semantic group as émdiddvar (avEavetv, npoayayeiv, rpoépxeoda1) occur in the Catalogue very frequently.!*! Incidentally, the term éxtdootc and its synonyms were not the only means to express In itself, the question about np@tog ebperng does not entail a historical approach. Depending on the genre and context, every element of this question — “who,” “discovered,” and “what” — could receive primary significance or retreat into the background. For the history of the idea of the progress of knowledge. From the time of Xenophanes poetry and music the personality of the np@tog ebperng was more im- (21A18) this idea was connected with two key notions, Gino and eV peorc.' It is with these words that the author of the Hippocratic treaportant than for heurematography, which was interested not so much in the actual discoverers of the past, whose names were of course untise On ancient medicine describes the origin and development of his téyvn. In his view, the history of medicine is the history of the inquiries and discoveries that increase our knowledge about human nature and the causes of diseases (VM 2-4). This history begins with the raptor: evpetat, who discovered medicine itself.153 The quest for 148 Mer, 981a121f., NE 1139al7f. See: Wehrli p. 114. 149 See above, n. 75. On the progress of knowledge, see Soph. El. 183b20f.; De an. 417b5f.; EN 1098a22f. 150 In Eucl. p. 66.9. This passage of the Catalogue seems to me rather suspicious, since it describes Plato’s role in the development of geometry. 151 éxavEdver (p. 66.16), ab&dverv (p. 67.5), mpoayayeiv (p. 67.7, 67.22) rpogpyeoda: (p. 66.17). See: Edelstein p. 92; Thraede (1965) col. 141f., 154. 154 Kjeingünther; Thraede (1962) col. 1191ff. 155 Theophrastus (frr. 728-34 FHS&G), Heraclides of Pontus (fr. 152 Wehrli), Strato (frr. 144-47 Wehrli). Aristotle also wrote on this subject (Irr. 382, 479, 501, 600, 602 Rose = tr. 924 Gigon). 156 Aristox. frr. 78-81, 83, Dicaear. frr. 57. 75-76, 85. 157 Theophr. Phys. op. frr. 1, 2, 4, 6a, 17 Diels = frr. 225, 226A, 227D-E, 228A FHS&G. Aristotle called Thales rpòtog ebperng of natural philosophy (Mer. 983b20), Socrates — of ethics (Mer. 1078b17), Empedocles — of rhetoric, Zeno — of dialectic (fr. 65 Rose), the Pythagoreans — the founders of mathematics and number philosophy (Met. 985b23f.); see also fr. 72 Rose; Mansfeld (1990) p. 44f. 158 Frr. 135-41, 144-48. See also Procl. In Eucl. p. 64.18, 65.7f., 65.21, 66.4f., 67.2f., 67.20f., 213.7f., 250.20f., 283.7f.; Eutoc. In Archim. De sphaer. 111, p. 88.1823, Schol. in Eucl. p. 273.3-13, and my discussion above, p. 272-75. Cf. the subtitle 152 Cf. Archytas (47B3). for the abridged version of his History of astronomy: tig ti ebpev év aPnnatikoîc: 153 of SE Cntnoavtés te kai ebpóvtes intpiKAy (5), ot rpator ebpovtes (14). (Ps.-Heron. Deff. IV, p. 167.9 Heiberg = Eud. fr. 145).

301 known, as in the cultural innovations ascribed to them. As a result, more relying mainly on the evpjpata of the musicians. Glaucus’ book was than half of the rp@to1 evpetai mentioned in the ancient “catalogues used by Aristoxenus (fr. 91), so it must have been known to Eudemus. of discoveries” are gods and heroes, and the catalogues themselves re- Itis hardly by chance that his histories reveal the same pattern of orgasemble dry inventories, lacking even systematic, to say nothing of chronizing material according to the chronology of the mp@to1 edpetai, as nological order. In doxography, the references to mp@toc edpetng are in Glaucus. We know that Glaucus has not limited his book to the mubuilt into the chronological-systematic scheme and are used only as sicians: he mentioned, e.g., Empedocles and the Pythagorean teachers supplementary classifying signs. As for Eudemus, in attempting to exof Democritus (frr. 5-6 Lanata), whereas Aristoxenus referred to press the dynamics of development of the exact sciences, he bases his Glaucus in his description of Hippasus’ acoustical experiment, based on inquiry not on systematics of discoveries but on the chronology of their mathematical proportions (fr. 90). Thus, Glaucus’ book could have been authors, which gives his works a historical perspective. a source of information not only on Hippasus, but also on the early theory of proportions, which from its very beginning was closely con5 The most difficult question concerns the origin of Eudemus’ information on the mathematicians before the middle of the fifth century. All that I have to say here has a preliminary character. Eudemus evidently used the works of Oenopides, Hippocrates, Archytas, Theaetetus, Eudoxus, and his students, but it is hard to say definitively whether they contained any historical information. In principle, this does not seem impossible: we know that Archytas approvingly refers to the views of his Pythagorean predecessors (oi nepi po0npora, 47B1) and mentions Philolaus’ student Eurytus (A13). One could guess that Hippocrates too referred to the Pythagorean mathematical compendium, preceding his Elements and comprising the basis of the first four books of Euclid.!% Such a guess is indirectly confirmed by the notice, going back to Eudemus, that the whole book IV of the Elements belongs to the Pythagoreans.!$ Eudemus obviously could not refer to the fourth book of Euclid’s Elements, but it is very likely that this book, devoted to the relations between regular polygons and a circle, took up the same place in the Elements prior to Euclid.'9! Eudemus’ other possible source was the book On the ancient poets and musicians by Glaucus of Rhegium, a contemporary of Democritus. Glaucus can be viewed as a predecessor of Eudemus in the history of Greek music and poetry: he gives a chronological sketch of these arts, 159 See: Neuenschwander p. 357ff., 378; van der Waerden (1978) p. 354ff. 160 Sce above, n. 20. 161 According to Neuenschwander (p. 374f.), the fourth book was compiled by the Pythagoreans before Hippocrates, who used it in his quadrature of lunes, and was subjected later to very minor changes. nected with harmonics.!® Hippias of Elis, to whom Eudemus’ reference to Mamercus (In Eucl., p. 64.11f.) can be traced, is considered to be one of his main sources, Judging by the context of Hippias’ book on “related ideas,” recently investigated by A. Patzer, this reference was hardly isolated. Since Hippias played a very important role as a transmitter of Thales’ d0G a. it is more probable that he named Mamercus along with the other famous geometers of the sixth century, Thales and Pythagoras. But while Patzer believes that geometry also belongs to the Themenkreisen examined by Hippias,'® his own reconstruction of Hippias’ book leaves barely any room to suppose that it contained particular mathematical material. Now, it would be worthwhile to make more precise what is meant here by “geometry.” Hippias quite likely wrote that Thales, Mamercus, and Pythagoras became famous for their studies of geometry, and that among the barbarians the Egyptians are well known for their geometrical knowledge. I would not exclude the possibility that he even made references to some particular problems studied by the early mathematicians. What seems to me highly tmprobable is that Hippias quoted geometrical propositions at length, including their proofs, e.g., the Pythagorean proof of the theorem that the sum of the interior angles of a triangle is equal to two right angles.!% Eudemus clearly knew substantially more about the proofs offered by Thales and 162 See above, p. 271f., nn. 31-32, cf. 47B2 (Archytas) and Eud. fr. 142. Glaucus’ origin in Rhegium might point to his Pythagorean connections. 163 Patzer p. 106f. 164 Eud. fr. 136. Cf. the other ancient proof that probably goes back to Thales (Arist. An. Pr. 41b13-22).

303 by the other early mathematicians than could be included in Hippias’ thing else, they seem to be unconcerned with the actual mathematical Zvvayayn.!6 problems of their time. Although it would be unfair to say that the Aca- In the passage excerpted by Philodemus it is easy to discern several ideas familiar to us from Eudemus’ works: the notion of scientific demicians were interested in the history of science only to the extent progress, the idea that it was especially rapid in the last decades, the attributed to him the role of the “architect of the exact sciences.”!?0 replacement of old theories by new ones (with a reference to the authors Therefore, we can suppose that the academic writings were not so much of these theories), the appearance of new methods (analysis and diorism), and even the idea of approaching perfection (tà mept a basis for Eudemus’ work, but rather a stimulus to him in elaborating uetpodoyiav AABev érì kopuepñv). It is very possible that Eudemus in his history of the exact sciences there was no place left for Plato. that it was connected with Plato’s activity, they nevertheless wrongly his own conception of scientific development. It is thus no surprise that knew the work quoted by Philodemus, but what could he borrow from it? This “biography” of Plato touched the exact sciences only in passBibliography ing, in connection with his influence on the progress of geometry and the other sciences. Earlier, Aristotle also mentioned rapid progress in mathematics, indeed, without naming Plato,'9° and Eudemus certainly shared his attitude. As for the specific facts mentioned in the papyrus passages, Eudemus had to know of them at first hand, and not from the book on Plato. Hermodorus’ book Mepi ua@nudtwv was probably a more widely Baltussen, H. “A ‘Dialectical’ Argument in De anima A 4,” in Polyhistor. Studies in the History and Historiography of Ancient Philosophy; presented to J. Mansfeld, ed. K. Algra et al., 333-44. Leiden: Brill 1996. Björnbo, A. “Hippokrates von Chios,” Paulys Realencyclopädie 8.2 (1913) col. 1787-1801. oriented work. Regrettably, one can only guess about its subject matter: in the two short citations from it, it is not mathematics but Zoroaster Burkert (1972) = Burkert, W. Lore and Science in Ancient Pythagoreand the Persian Magi who are mentioned.!6’ Not much more is known about the other academic treatises, which, judging from their titles, Burkert (1993) = Burkert, W. Platon in Nahaufnahme: ein Buch aus could contain some information on the history of the exact sciences.!% anism. Cambridge, Mass.: Harvard University Press 1972. Herculaneum. Leipzig: Teubner 1993. There is too little evidence to consider Speusippus and Xenocrates to Burkert (1997) = Burkert, W. “Impact and Limits of the Idea of be the sources and/or predecessors of Eudemus;!% apart from every- Progress in Antiquity,” in The Idea of Progress, ed. A.Burgen et al., 19-46. Berlin: de Gruyter 1997. De Ste. Croix, G. E. M. “Aristotle on History and Poetry,” in Essays on 165 We can more confidently consider the Evvaywyh as one of the sources for Eudemus’ History of theology. Aristotle's Poetics, ed. A. O. Rorty, 23-32. Princeton: University 166 See above, n. 75. 167 D.L. 1.2 and 8. And yet, it is worth noting that Diogenes mentions Hermodorus Press 1992, | Dorandi (1991) = Dorandi, T. Filodemo: Platone e l’Academia: in the context of discussion of the origin of philosophy, and that next to him he quotes the opinions of Aristotle and Eudemus about the Magi (1.8-9 = Arist. fr. 6 Rose, Eud. Dorandi (2001) = Dorandi, T. “La tradizione papirologica da Dicearco fr. 89). Aristotle’s opinion comes from his dialogue De philosophia that traced the barbaric origins and further development of philosophy; that of Eudemus comes most probably from the History of theology (fr. 150), and not from the Physics, as Wehrli thought. Relying on the analogies (for we have nothing else in this case), One can guess (PHerc. 1021 e 164). Napoli: Bibliopolis 1991. a Demetrio del Falero,” in Dicaearchus of Messana, ed. W. W. Fortenbaugh and E. Schütrumpf = Rutgers University Studies in that Hermodorus’ book too was devoted to the history of scientific, especially mathematical knowledge, beginning from its origin in the Orient. 168 See above, p. 279, nn. 69, 72. 169 A fragment from Speusippus’ book On the Pythagorean numbers (fr. 28 Taran) contains some information on the Pythagorean arithmetic, which is, however, intermingled with his own arithmological interpretations. The only thing that can be found in Xenocrates is his reference 10 Pythagoras’ discovery of harmonic intervals (fr. 87 Isnardi Parente). 170 For further instances, see Zhmud (1998) p. 21511.

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