¹ For infinity of universe, void, and matter see Epicurus, , Ad Herod. 41–2Google Scholar. The spatial infinity of matter results from the statement of Epicurus here: If the void were finite, the infinity of bodies would have no place to be. That the extension of the infinite number of atoms must be infinite, results also from Ad Herod. 57. Here Epicurus states that if any particle has size (as atoms have), an infinite number of them would be of infinite extension.

² Some notice of pertinent passages in Aristotle and in Simplicius was taken by Gassendi, P., Opera Omnia I (Lyons, 1658), 188Google Scholar. A mere reference to Gassendi's parallels was given by Reisacker, A. J., Quaestiones Lucretianae (Bonn, 1847), 25 n. 1Google Scholar. Reisacker's note was in turn referred to by Hildebrandt, F., T. Lucretii de primordiis doctrina (Magdeburg, 1864), 28 n. 1Google Scholar. The most specific use of any of this material was made by Bindseil, Th., Nonnulla ad Lucretii de omnis inftnitate doctrinam (Berlin, 1870), 7–8Google Scholar, who noticed that Lucretius'proof of infinity in 1.1008–13 was related to the fourth proof of infinity in Aristotle, , Physics 203 b 20–3Google Scholar.

³ Bruns, Ivo (ed.), ‘Alexandri Aphrodisiensis praetercommentaria scripta minora’, Quaestiones 3. 12, Supplementum Aristotelicum n. 2 (Berlin, 1892), 101–7Google Scholar. Of this passage p. 104. 20–3 is in Usener, Epicwea, no. 297; p. 104. 20–6 ibid. no. 301.

⁴ Schenkl, H. (ed.), ‘Themistii in Aristotelis Physica paraphrasis’, Commentaria in Aristotelem graeca v (Berlin, 1900), 81. 28–30, 94. 6–9, and 99. 15–100. 23. 100. 6–11 is also in Us., no. 298Google Scholar.

⁵ Diels, H. (ed.) ‘Simplicii in Aristotelis Physicorum libros quattuor priores commentaria’, Commentaria in Aristotelem graeca ix (Berlin, 1882), 466. 31–467. 4, 500. 8–11 and 516. 3–38. f Us., no. 297 has pp. 466. 31–467. 4Google Scholar.

⁶ Vitelli, G. (ed.), ‘Ioannis Philoponi in Aristotelis Physicorum libros tres priores commentaria’, Commentaria in Aristotelem graeca xvi (Berlin, 1887), 405. 7–14, 473. 24–474. 6 and 494.2–24Google Scholar.

⁷ In his extant remarks Alexander does not mention Epicurus by name. It is clear, however, from the context that his objections to those believing in the fourth proof of infinity in Aristotle are directed against the atomists and, more specifically, probably against the atomism of Epicurus. First, one of his sentences is in wording very similar to that of Epicurus in the same, context. Alexander writes with respect to finiteness: εἰ μ⋯ν οὖν тῷ πεπερα7sgr;μένῳ т⋯ εἶναι πεπερασμένῳ ⋯σт⋯ν ⋯ν тῳ θεωρε⋯σθαι παρ' ἂλλο… (above, n. 3, 104. 20–1). The Epicurean version contains, among other similar expressions, the phrase т δ⋯ ἂκρον παρ' ἂтερόν тι θεωρεῖ7tcy;7agr;ι (Ad Herod.41).

Also, Alexander states that his opponents in this section are those ἂтομά γέ тινα μεγέθη тιθέμενοι… κ7agr;⋯ ⋯μερ⋯ тινα κα⋯ ⋯σώμαтα κα⋯ κινήσεις κα⋯ χρόνους εἰσαγόμενοι (above, n. 3, 105. 13–15). Now, these opponents are clearly atomists, and, as indicated by the reference to incorporeal time and partless motions and time, more specifically Epicurus and his followers. Sextus Empiricus attributes ⋯σώμαтον… т⋯ν χρόνον to Epicurus, (Adv. Mathem. 10. 227)Google Scholar. Partless, indivisible units of motion and time also appear to have been introduced into atomist I philosophy by Epicurus (cf. Furley, David J., Two Studies in Greek Atomists [Princeton, 1967], 121)CrossRefGoogle Scholar.

⁸ It is not clear from the Greek of Simplicius whether ⋯ρχαιοттέρου should go with the Epicureans (sc. ⋯ρχαιοттέρου т⋯ν περ⋯ Ἐπίκουρον) or with Aristotle himself. A comparison with i. Themistius, however, indicates that Simplicius meant that the proof was older than Epicurus. Themistius writes: καίтοι т⋯ν λόγον тο⋯тον (sc. the proof based on the fourth proof for infinity in Aristotle) ἠγάπησεν οὖтως Ὲπίκουρος, ὢσтε πγαιόтερον ὂνтα εἰσποιήσασθαι κα⋯ ὑποβάλλεσθαι μικραῖς оισ⋯ ϕαύλαις προσθήκαις… (above, n. 4, 100. 8–10). Unfortunately, these προσθ⋯και are not pointed out.

⁹ Above, n. 5, 467. 1–4.

¹⁰ Among the more important scholars, the meaning sum-total of matter was proposed by Goebel, P. E., Observationes Lucretianae criticae et exegeticae (Bonn, 1854), 4Google Scholar. The same interpretation was supported by Munro in his edition in 1864, by Bockemuller's edition of 1873, by Stuerenberg, F., ‘De carminis lucretiani libro primo’, Ada Societatisphilologae lipsiensis 2. 2 (Leipzig, 1874), 412Google Scholar, by Neumann, F., De interpolationibus lucretianis (Halle, 1875), 13Google Scholar, by Woltjer, J., Lucretii philosophia cumfontibus comparata (Groningen, 1877), 34Google Scholar, by J. van der Valk in his edition of Bk. I of Lucretius (Kampen, 1903), by Mussehl, J., De Lucretiani libri primi condicione et retractatione (Tempelhof, 1912), 77Google Scholar, by Merrill, W. A., ‘Criticism of the Text of Lucretius with Suggestions for its Improvement’, University of California Publications in Classical Philology 3 (1916), 12Google Scholar, by Reiley, K. C., Studies in the Philosophical Terminology of Lucretius and Cicero (New York, 1909), 117Google Scholar, by C. Pascal in his edition of Bk. I (Second edition in Rome, 1928), by C. Bailey in his commentary to Lucretius, and by Müller, G., Die Darstellung der Kinetik bei Lukrez (Berlin, 1959), 97Google Scholar.

Rerum summa in the meaningomne or ‘the universe’ was advocated already in 1832 by Madvig, J. N., ‘De aliquot lacunis codicum Lucretii’, Opuscula Academica (Hauniae, 1887), 254Google Scholar. He was followed by Lachmann in the commentary of his edition in 1850, by T. Bindseil, above n. 2, p. 7, by Susemihl, F., ‘Neue Bemerkungen zum ersten Buche des Lucretius’, Philologus 44 (1885), 81CrossRefGoogle Scholar, by Giussani in his edition of Lucretius (1896), by Brieger, A., ‘Epicurs Lehre vom Raum…’, Philologus N.F. 14 (1901), 529Google Scholar, by Smith and Leonard in their edition of Lucretius (1942), by Schulz, P. R., ‘Das Verstandnis des Raumes bei Lukrez’, Tijdschrift voor Philosophie 20 (1958), 481Google Scholar, and by A. Barigazzi in his edition of selected passages of Lucretius (Turin, 1974).

¹¹ Above, n. 10, p. 50.

¹² The exact meaning of rerum summa in 1. 1008 cannot be established on the grounds of Lucretian usage alone because rerum summa in Lucretius can refer both to universe and to matter. For examples see Bailey's commentary to 1. 235.

¹³ Polle, F., ‘Zu Lucretius’, Jahrbiicherfiir classische Philologie 93 (1866), 757–60Google Scholarconsidered lines 1012–13 as an interpolation of a learned reader, partly because of his misunderstanding of the content of these lines but partly also because he argued that the elision in alterum eorum in 1012 was against Lucretian practice. This latter argument enjoyed some popularity, being accepted by Stuerenberg (above, n. 10, p. 415) and by Neumann (above, n. 10, p. 13 n. 1). Line 1012 was attacked also by Woltjer (above, n. 10) who, not familiar with this proof of infinity, suggested non quoniam for aut etiam, thus reversing the required meaning. Finally, there was the well-known attempt by Lachmann in his commentary to place the lacuna not after line 1013 but rather after line 1012.

¹⁴ Above, n. 3, 104. 21—1: εἰ μ⋯ν οὖν тῷ πεπερασμένῳ тῷ εῖναι πεπερασμένῳ ⋯σт⋯ν тῷ θεωρεῖσθαι παρ ἂλλο… Alexander seems to be using here the very language of Epicurus, who in the same context writes: т⋯ δ⋯ ἂκρον παρ' ἒтερόν тι θεωρεῖтαι Bewpfirai (Ad Herod. 41).

¹⁵ Aristotle assumed that what was whole had to be finite: (т⋯ ŏλον sc. ⋯σт⋯) οὗ μηδήν ⋯σтινἒξω (Physics 207 a 1) and understood the universe of Parmenides to be finite: т⋯ őλον (sc. of Parmenides) πεπεράνθαι (Physics 207 a 16–17).

¹⁶ Belief in the authorship of Melissus has been nearly unanimous. The only closely reasoned argument against his authorship seems to be that of Barnes, J., The Presocratic Philosophers i (London, 1979), 201Google Scholar. In his strongest objection he maintains that if the proof of the infinity of т⋯ π⋯ν found in Aristotle — т⋯ γ⋯ρ πέρας περαίνειν ἂν πρ⋯ς т⋯ κενόν — were admitted among the other proofs of Melissus, it would introduce circularity. However, circularity would appear only if for this proof to be valid it were necessary, as is assumed by Barnes, to prove first that the universe (т⋯ π⋯ν) is one rather than many. It seems to me that this latter proof is not indispensable. There is no reason to assume that the concept т⋯ π⋯ν cannot here include everything that exists. In other words, even if what exists were many and not one, the many could be thought to be subsumed under т⋯ π⋯ν т⋯ π⋯ν and be parts of its total being and extension. The proof that т⋯ π⋯ν is one would be necessary only if the many were thought not to be part of т⋯ π⋯ν, therefore, were able to function as a limit to it.

¹⁷ For these statements in Simplicius and Themistius see n. 8.

¹⁸ For instance, Simplicius states of Melissus: ⋯π⋯ δ⋯ тο⋯ άπείρου т⋯ ⋯ν συνελογίσαтο ⋯κ тο⋯ “εἰ μ⋯ ἓν εἳη περανεῖ πρ⋯ς ἂλλο” (above, n. 5, 110. 5–6 or Diels-Kranz I, 30B5).

¹⁹ D-K ii, 67 A 1, 28–30. The μέγα κενόν is also attested by Hippolytus 67 A 10, 25–6, where Roeper plausibly substituted it for the μεтάκοινον of the mss.

²⁰ Above, n. 1.

²¹ Leucippus, like Democritus and Epicurus after him, appears to have held that atoms in juxtaposition never formed a true unity, i.e. they were separated by some void. For discussion and references see Guthrie, W. K. C., A History of Greek Philosophy II (Cambridge, 1965), 390Google Scholar. If so, for Leucippus a section of infinite matter would have to be a single atom of infinite extension. The notion of such an atom may be encouraged by the much-debated cosmos-sized atoms envisaged, according to Aetius, by Democritus (DK II, 68 A 47). Epicurus, however, held that atoms were too small to be seen (Ad Herod. 55–6).

²² Above, n. 4, 100. 8–10.

²³ Ἀρχύтα δέ, ὡς ϕησιν Eῠδημος, οὂтως ἠρώтα т⋯ν λόγον ⋯ν тῷ ⋯σχάтῳ οἲον тῷ άπλανεῖ οὐρανῷ γενόμενος, πόтερον ⋯κтείναιμι ⋯ν т⋯ χεῖρα ⋯ т⋯ν ῥάβδον εἰς т⋯ Ӗξω ⋯ οἢ; κα⋯ тῖ μεν οὗν μ⋯ ⋯κтείνειν ἂтοπον εἰ δ⋯ ⋯κтείνω, ňтοι σ⋯μα ἣ т⋯πος т⋯ ⋯κт⋯ς ἒσтαι. διοίσει δ⋯ οὐδ⋯ν ὡς μαθησόμεθα. ⋯ε⋯ οὖν βαδιεῖтαι т⋯ν αὐт⋯ν тρόπον ⋯π⋯ т⋯ ⋯ε⋯ λαμβανόμενον πέρας, κα⋯ тαὐт⋯ν ⋯ρωтήσει, κα⋯ εἰ ⋯ε⋯ ἒтερον ⋯σтαι ⋯ϕ' ộ ή ῥάβδος, δ⋯λον őтι κα⋯ ᾰπειρον (above, f n. 5, 467. 26–32).

²⁴ Furley, David J., ‘AristotleandtheAtomistsonInfinity’, in During, I.(ed.), Naturphilosophie bei Aristoteles und Theophrast (Heidelberg, 1969), 92–3Google Scholar.

²⁵ Cherniss, Harold, Aristotle's Criticism of Presocratic Philosophy (New York, 1964Google Scholar, Repr. of 1935 ed.), 20–1.

²⁶ Raven, J. E., Pythagoreans and Eleatics (Cambridge, 1948), 80–1Google Scholar.