line is in a given ratio to its distance from a given plane is a certain cone. (b) It may have been used to prove that the locus of a point whose distance from a given point is in a given ratio to its distance from a given plane is the surface formed by the revolution of a conic about its major or conjugate axis'. Thus Chasles may have been correct in his conjecture that the Surface-loci dealt with surfaces of revolution of the second degree and sections of the same2. 6. The Conics. Pappus says of this lost work: "The four books of Euclid's Conics were completed by Apollonius, who added four more and gave us eight books of Conics" It is probable that Euclid's work was lost even by Pappus' time, for he goes on to speak of "Aristaeus, who wrote the still extant five books of Solid Loci connected with the conics." Speaking of the relation of Euclid's work to that of Aristaeus on conics regarded as loci, Pappus says in a later passage (bracketed however by Hultsch) that Euclid, regarding Aristaeus as deserving credit for the discoveries he had already made in conics, did not (try to) anticipate him or construct anew the same system. We may no doubt conclude that the book by Aristaeus on solid loci preceded Euclid's on conics and was, at least in point of originality, more important. Though both treatises dealt with the same subject-matter, the object and the point of view were different; had they been the same, Euclid could scarcely have refrained, as Pappus says he did, from attempting to improve upon the earlier treatise. No doubt Euclid wrote on the general theory of conics as Apollonius did, but confined himself to those properties which were necessary for the analysis of the Solid Loci of Aristaeus. The Conics of Euclid were evidently superseded by the treatise of Apollonius. As regards the contents of Euclid's Conics, the most important source of our information is Archimedes, who frequently refers to propositions in conics as well known and not needing proof, adding in three cases that they are proved in the "elements of conics" or in "the conics," which expressions must clearly refer to the works of Aristaeus and Euclid'. Euclid still used the old names for the conics (sections of a rightangled, acute-angled, or obtuse-angled cone), but he was aware that an ellipse could be obtained by cutting a cone in any manner by a plane parallel to the base (assuming the section to lie wholly between the apex of the cone and its base) and also by cutting a cylinder. This is expressly stated in a passage from the Phaenomena of Euclid about to be mentioned". 7. The Phaenomena. This is an astronomical work and is still extant. A much inter 1 For further details see The Works of Archimedes, pp. lxiv, lxv, and Zeuthen, 1. c. 8 For details of these propositions see my Apollonius of Perga, pp. xxxv, xxxvi. See Heiberg, Euklid-Studien, p. 88. "If a cone or a cylinder be cut by a plane not parallel to the base, the section is a section of an acute-angled cone, which is like a shield (θυρεός).” · polated version appears in Gregory's Euclid, and a much earlier and better recension is, says Heiberg1, contained in the MS. Vindobonensis philos. Gr. 103, though the end of the treatise, from the middle of prop. 16 to the last (18), is missing. The book consists of 18 propositions of spheric geometry. Euclid based it on Autolycus' work TEρÌ KIVOνμévηs σpaipas, but also, evidently, on an earlier textbook of Sphaerica of exclusively mathematical content. It has been conjectured that the latter textbook may have been due to Eudoxus2. 8. The Optics. This book needs no description, as it has been edited by Heiberg recently, both in its genuine form and in the recension by Theon. The Catoptrica published by Heiberg in the same volume is not genuine, and Heiberg suspects that in its present form it may be Theon's. It is not even certain that Euclid wrote Catoptrica at all, as Proclus may easily have had Theon's work before him and inadvertently assigned it to Euclid'. ἁρμονική Besides the above-mentioned works, Euclid is said to have written the Elements of Musics (αἱ κατὰ μουσικὴν στοιχειώσεις). Two treatises are attributed to Euclid in our MSS. of the Musici, the KATAтоμη KAνÓνos, Sectio canonis (the theory of the intervals), and the eloaywyn ápμovień (introduction to harmony). The first, resting on the Pythagorean theory of music, is mathematical and clearly and well written, the style and the form of the propositions agreeing well with what we find in the Elements. Its genuineness is confirmed not only by internal evidence but by the fact that almost the whole of the treatise (except the preface) is quoted in extenso, and Euclid is twice mentioned by name, in the commentary on Ptolemy's Harmonica published by Wallis and attributed by him to Porphyry, but probably for the most part compiled by Pappus or some other competent mathematician. (On the other hand Tannery set himself to prove that the treatise is not authentic.) The second treatise is not Euclid's, but was written by Cleonides, a pupil of Aristoxenus". Lastly, it is worth while to give the Arabians' list of Euclid's works. I take this from Suter's translation of the list of philosophers and mathematicians in the Fihrist, the oldest authority of the kind that we possess1o. "To the writings of Euclid belong further [in addition to the Elements]: the book of Phaenomena; the book of 1 Euklid-Studien, pp. 50—1. Heiberg, op. cit. p. 46; Hultsch, Autolycus, p. xii; A. A. Björnbo, Studien über Menelaos' Sphärik (Abhandlungen zur Geschichte der mathematischen Wissenschaften, XIV. 1902), p. 56 sqq. Euclidis opera omnia, vol. VII. (1895). Heiberg, Euclid's Optics, etc. p. l. 5 Proclus, p. 69, 3. Published in the Musici Scriptores Graeci, ed. Jan (Teubner, 1895), pp. 113-166. 7 Jan, Musici Scriptores Graeci, p. 116. 8 Comptes rendus de l'Acad. des inscriptions et belles-lettres, Paris, 1904, pp. 439-445. Cf. Bibliotheca Mathematica, VI,, 1905-6, p. 225, note 1. Heiberg, Euklid-Studien, pp. 52-5; Jan, Musici Scriptores Graeci, pp. 169–174. 10 H. Suter, Das Mathematiker-Verzeichniss im Fihrist in Abhandlungen zur Geschichte der Mathematik, VI., 1892, pp. 1–87 (see especially p. 17). Cf. Casiri, I. 339, 340, and Gartz, pp. 4, 5. H. E. 2 Given Magnitudes [Data]; the book of Tones, known under the name of Music, not genuine; the book of Division, emended by Thabit; the book of Utilisations or Applications [Porisms], not genuine; the book of the Canon; the book of the Heavy and Light; the book of Synthesis, not genuine; and the book of Analysis, not genuine." It is to be observed that the Arabs already regarded the book of Tones (by which must be meant the eloaywyn apμovin) as spurious. The book of Division is evidently the book on Divisions (of figures). The next book is described by Casiri as "liber de utilitate suppositus.' Suter gives reason for believing the Porisms to be meant', but does not apparently offer any explanation of why the work is supposed to be spurious. The book of the Canon is clearly the Kaтaтоun κavóvos. The book on "the Heavy and Light" is apparently the tract De levi et ponderoso, included in the Basel Latin translation of 1537, and in Gregory's edition. The fragment, however, cannot safely be attributed to Euclid, for (1) we have nowhere any mention of his having written on mechanics, (2) it contains the notion of specific gravity in a form so clear that it could hardly be attributed to anyone earlier than Archimedes. Suter thinks that the works on Analysis and Synthesis (said to be spurious in the extract) may be further developments of the Data or Porisms, or may be the interpolated proofs of Eucl. XIII. I-5, divided into analysis and synthesis, as to which see the notes on those propositions. 1 Suter, op. cit. pp. 49, 50. Wenrich translated the word as “utilia.” Suter says that the nearest meaning of the Arabic word as of "porism" is use, gain (Nutzen, Gewinn), while a further meaning is explanation, observation, addition: a gain arising out of what has preceded (cf. Proclus' definition of the porism in the sense of a corollary). 2 Heiberg, Euklid-Studien, pp. 9, 10. 3 Suter, op. cit. p. 50. CHAPTER III. GREEK COMMENTATORS ON THE ELEMENTS OTHER THAN PROCLUS. THAT there was no lack of commentaries on the Elements before the time of Proclus is evident from the terms in which Proclus refers to them; and he leaves us in equally little doubt as to the value which, in his opinion, the generality of them possessed. Thus he says in one place (at the end of his second prologue)': "Before making a beginning with the investigation of details, I warn those who may read me not to expect from me the things which have been dinned into our ears ad nauseam (diaтelpúλntai) by those who have preceded me, viz. lemmas, cases, and so forth. For I am surfeited with these things and shall give little attention to them. But I shall direct my remarks principally to the points which require deeper study and contribute to the sum of philosophy, therein emulating the Pythagoreans who even had this common phrase for what I mean 'a figure and a platform, but not a platform and sixpence?"" In another place he says: "Let us now turn to the elucidation of the things proved by the writer of the Elements, selecting the more subtle of the comments made on them by the ancient writers, while cutting down their interminable diffuseness, giving the things which are more systematic and follow scientific methods, attaching more importance to the working-out of the real subject-matter than to the variety of cases and lemmas to which we see recent writers devoting themselves for the most part." At the end of his commentary on Eucl. 1. Proclus remarks that the commentaries then in vogue were full of all sorts of confusion, and contained no account of causes, no dialectical discrimination, and no philosophic thought. These passages and two others in which Proclus refers to "the commentators' suggest that these commentators were numerous. He does not however give many names; and no doubt the only Emportant commentaries were those of Heron, Porphyry, and Pappus. 1 Proclus, p. 84, 8. 2i.e. we reach a certain height, use the platform so attained as a base on which to build nother stage, then use that as a base and so on. ibid. p. 289, 11; p. 328, 16. Proclus, p. 200, 10. • ibid. p. 432, 15. I. Heron. Proclus alludes to Heron twice as Heron mechanicus', in another place he associates him with Ctesibius, and in the three other passages where Heron is mentioned there is no reason to doubt that the same person is meant, namely Heron of Alexandria. The date of Heron is still a vexed question, though the possible limits appear to have been practically narrowed down to the 150 years. between (say) 50 B.C. and 100 A.D. Martin concluded that Heron lived till the middle of the first century B.C., Hultsch' placed him at the end of the second century B.C. Cantor in his first two editions. took a middle course and gave 100 B.C. as the date when he flourished". But it is now certain that in his Mechanics, preserved in the Arabic and recently published', Heron quotes Posidonius the Stoic (of Apamea, Cicero's teacher) by name as the author of a definition of the centre of gravity. Now Posidonius lived till about the middle of the first century B.C.; and, assuming that his writings dated from not earlier than 90 or 80 B.C., we must put Heron at all events (say) fifty years later than Hultsch placed him. Cantor now, while maintaining that he belonged to the first century B.C., admits that he may have flourished as late as the last third of it. But in the meantime an entirely different view was elaborated by W. Schmidt, the editor of the first volume of the new edition of Heron's complete works, who assigned him to the second half of the first century A.D.9 The arguments for the terminus post quem are mainly these. (1) Vitruvius gives in the preface to Book VII. of his De Architectura (brought out apparently 14 B.C.) a list of authorities on machinationes from whom he had made extracts. This list contains twelve names and has every appearance of being scrupulously complete; but, while it includes Archytas, Archimedes, Ctesibius, and Philo of Byzantium (who come second, third, fourth, and sixth in order respectively), Heron is not mentioned. Moreover the points of difference between Vitruvius and Heron seem on the whole to be more numerous and important than the resemblances. (2) Diels concluded from the use of Latinisms by Heron that the first century A.D. was the earliest possible date. (3) A definite date was derived by Carra de Vaux from the identification of a small single-screw olivepress described by Heron (Mechanics, III. 20) with one mentioned by Pliny (Nat. Hist. XVIII. 317) as having been introduced within the last twenty-two years: this gives A.D. 55 as the date before which the Mechanics could not have been written. The terminus ante quem, 100 A.D., was arrived at (1) from internal evidence suggesting that 1 Proclus, p. 305, 24; p. 346, 13, ibid. p. 41, 10. 3 ibid. p. 196, 16; p. 323, 7; P. 429, 13. Martin, Recherches sur la vie et les ouvrages d'Héron d'Alexandrie, Paris, 1854, p. 27. Cantor, Gesch. d. Math. 12, p. 347. 7 Heronis Alexandrini opera quae supersunt omnia (Teubner, Leipzig), vol. 11. edited by L. Nix and W. Schmidt, 1900. 8 Cantor, Gesch. d. Math. Ig, p. 366. • See Heronis Alexandrini opera, vol. 1., 1899, pp. ix-xxv. |