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effected by a man who was thoroughly versed in the old system.55

It is not without significance, too, that Eudoxus selected the retired and pleasant shores of the Propontis as the situation of the school which he founded for the transmission of his method. Among the first who arose in this school was Menaechmus, whose work I have next to consider.

IT is pleasing to see that the number of students of the history of mathematics is ever increasing; and that the centres in which the subject is cultivated are becoming more numerous; it is particularly gratifying to observe that the subject has at last attracted attention in England. Dr. Heiberg, of Copenhagen, has completed his edition of Archimedes: Archimedis Opera omnia cum commentariis Eutocii: e codice Florentino recensuit, Latine vertit notisque illustravit J. L. Heiberg, Dr. Phil., vols. II. et III.: Lipsiae, 1881. Dr. Heiberg has been since engaged in bringing out, in conjunction with Professor H. Menge, a complete edition of the works of Euclid, of which two volumes have been published: Euclidis Elementa, edidit et Latine interpretatus est J. L. Heiberg, Dr. Phil., vol. I., Libros I-IV. continens, vol. II., Libros V.-IX continens, Lipsiae, 1883, 1884. As Heiberg's edition of Archimedes was preceded by his Quaestiones Archimedeae, Hauniae, 1879; so, in anticipation of his edition of Euclid he has published: Litterargeschichtliche Studien über Euklid, Leipzig, 1882, a valuable work, to which I have referred in the fourth chapter. Dr Hultsch, of Dresden, informs me that his edition of Autolycus is finished, and that he hopes it will appear at the end of this month (June, 1885). The publication of this work-in itself so important, inasmuch as the Greek text of the propositions only of Autolycus has been hitherto published-will have, moreover, an especial interest with regard to the subject of the pre-Euclidian geometry. The Cambridge Press announce a work by Mr. T. L. Heath (author of the Articles on 'Pappus' and 'Porisms' in the Encyclopædia Britannica) on Diophantus; a subject on which M. Paul Tannery also has been occupied for some time.

55 Eudoxus may even be regarded as in a peculiar manner uniting in himself and representing the previous philosophic and scientific movement; for—though not an Ionian-he was a native of one of the neighbouring Dorian cities; he studied under the Pythagoreans in Italy; and, subsequently, he went to Athens, being attracted by the reputation of the Socratic school.

The following works on the history of Mathematics have been recently published :

Marie, Maximilien, Histoire des Sciences Mathématiques et Physiques, tomes I.-V., Paris, 1883, 1884. The first volume alone-de Thalès à Diophantetreats of the subject of these Papers. It is, in my judgment, inferior to the Histoire des Mathématiques of M. Hoefer, notwithstanding the errors of the latter, to which I called attention in p. 2, note. For the historical part of this volume M. Marie has followed Montucla without making use, or even seeming to suspect the existence, of the copious and valuable materials which have of late years accumulated on this subject. Referring to this, Heiberg (Philologus XLIII. Jahresberichte, p. 324) says: The author has been engaged with his book for forty years: one would have thought rather that the book was written forty years ago.' M. Marie commences his Preface by saying: 'The history that I have desired to write is that of the filiation of ideas and of scientific methods;' as if that was not the aim of all recent enlightened inquiries. Hear what Hankel, in Bullettino Boncompagni, V. p. 300, says: la Storia della matematica non deve semplicemente enumerare gli scienzati e i loro lavori, ma essa deve altresi esporre lo sviluppo interno delle idee che vegnano nella scienza (Quoted by Heiberg in Philologus, 1. c.).

Gow, James, A Short History of Greek Mathematics, Cambridge, 1884. This history, as far at least as geometry is concerned, is not, nor indeed does it pretend to be, a work of independent research. Unlike M. Marie, however, Mr. Gow has to some extent studied the recent works on the subject, and the reader will see that he has made much use of the early chapters of this work (published in HERMATHENA, No. v., 1877, and No. vII., 1881). On the other hand, he has left unnoticed many important publications. In particular, the numerous and valuable essays of M. Paul Tannery, which leave scarcely any department of ancient mathematics untouched, and which throw light on all, seem to be altogether unknown to him. Essays and monographs like those of M. Tannery and others are in fact, with the single exception of Cantor's Vorlesungen über Geschichte der Mathematik, the only works in which progress in the history of ancient mathematics has of late years been made: Bretschneider's Geometrie vor Euklides and Hankel's Geschichte der Mathematik are no exceptions; for the former work is a monograph, and the latter, which was interrupted by the death of the author, contains only some fragments of a history of mathematics, and consists in reality of a collection of essays. Should the reader look at Heiberg's Paper in the Philologus, XLIII., 1884, pp. 321-346, and pp. 467– 522, which has been referred to above, he will see how numerous and how important are the publications on Greek mathematics which have appeared since the opening of a new period of mathematico-historical research with the works of Chasles and Nesselmann more than forty years ago.

A glance at the subjoined list of the Papers of a single writer—M. Paul

Tannery-relating to the period from Thales to Euclid, will enable the reader to form an opinion on the extent of the literature treated of by Dr. Heiberg.

Mémoires de la Société des Sciences physiques et naturelles de Bordeaux (2o Série).—Tome 1, 1876, Note sur le système astronomique d'Eudoxe. Tome II., 1878, Hippocrate de Chio et la quadrature des lunules; Sur les solutions du problème de Délos par Archytas et par Eudoxe. Tome IV., 1882, De la solution géométrique des Problèmes du second degré avant Eudoxe. Tome V., 1883, Seconde note sur le système astronomique d'Eudoxe; Le fragment d'Eudème sur la quadrature des lunules.

Bulletin des Sciences Mathématiques et Astronomiques.—Tome VII., 1883, Notes pour l'histoire des lignes et surfaces courbes dans l'antiquité. Tome Ix., 1885, Sur l'Arithmétique Pythagorienne. Le vrai problème de l'histoire des Mathématiques anciennes.

Annales de la faculté des lettres de Bordeaux.-Tome IV., 1882, Sur les fragments d'Eudème de Rhodes relatifs à l'histoire des mathématiques. Tome V., 1883, Un fragment de Speusippe.

Revue philosophique de France et de l'étranger, dirigée par M. Ribot.—Mars, 1880, Thalès et ses emprunts á Égypte.

Novembre, 1880, Mars, Août et Décembre, 1881, L'éducation Platonicienne.

CHAPTER VI.

THE SUCCESSORS OF EUDOXUS.

I. MENAECHMUS.

Notices of Menaechmus and of his work.-His Solution of the Problem of the Duplication of the Cube.-He discovered the three Conic Sections.Passage from the 'Review of Mathematics' of Geminus quoted. --Hypothesis of Bretschneider as to the way in which Menaechmus was led to the discovery of the Conic Sections.-Comparison of these Investigations with the Solution of Archytas. Various inferences from the notices of the work of Menaechmus considered.—Successors of Eudoxus in the School of Cyzicus.-Solution of the Problem of the Duplication of the Cube attributed to Plato.—Strong presumption against its being genuine.-Plato's Solution.-The Geometrical Theorems used in it were known to Archytas.-Recapitulation.

MENAECHMUS-pupil of Eudoxus, associate of Plato, and the discoverer of the conic sections-is rightly considered by Th. H. Martin' to be the same as the Manaechmus of Suidas and Eudocia, a Platonic philosopher of Alopeconnesus; but, according to some, of Proconnesus, who wrote philosophic works and three books on Plato's

1 Theonis Smyrnaei Platonici Liber de Astronomia, Paris, 1849, p. 59. A. Boeckh (ueber die vierjährigen Sonnenkreise der Alten, Berlin, 1863, p. 152), Schiaparelli (le Sfere Omocentriche di Eudosso, di Callippo e di Aristotele, Milano, 1875, p. 7), and Zeller, (Plato and the Older Academy, p. 554, note (28), E. T.), hold the same opinion as Martin: Bretschneider (Geom. vor Eukl., p. 162), however, though thinking it probable that they were the same, says that the question of their identity cannot be determined with certainty. Both Martin and Bretschneider identify Menaechmus Alopeconnesius with the one referred to by Theon in the fragment (k) given below. Max C. P. Schmidt (Die Fragmente des Mathematikers Menaechmus, Philologus, Band XLII., p. 77, 1884), on the other hand, holds that they were distinct persons, but says that it is certainly more probable that the Menaechmus referred to by Theon was the discoverer of the conic sections, than that he was the Alopeconnesian, inasmuch as Theon connects him with Callippus, and calls them both μalnμarırol. Schmidt, however, does not give any reason in support of his opinion that the Alopeconnesian was a distinct person. But when we consider that Alopeconnesus was in the Thracian Chersonese, and not far from Cyzicus, and that Proconnesus, an island in the

Republic.' From the following anecdote, taken from the writings of the grammarian Serenus and handed down by Stobaeus, he appears to have been the mathematical teacher of Alexander the Great :-Alexander requested the geometer Menaechmus to teach him geometry concisely; but he replied: O king, through the country there are private and royal roads, but in geometry there is only one road for all." We have seen that a similar story is told of Euclid and Ptolemy I. (p. 5).

What we know further of Menaechmus is contained in the following eleven fragments :3—

(a). Eudemus informs us in the passage quoted from Proclus in the Introduction (p. 4), that Amyclas of Heraclea, one of Plato's companions, and Menaechmus, a pupil of Eudoxus and also an associate of Plato, and his brother, Deinostratus, made the whole of geometry more perfect.*

(b.) Proclus mentions Menaechmus as having pointed out the two different senses in which the word element (στοιχεῖον) is used.

(c.) In another passage Proclus, having shown that many so called conversions are false and are not properly Propontis, was still nearer to Cyzicus, and that, further, the Menaechmus referred to in the extract (k) modified the system of concentric spheres of Eudoxus, the supposition of Th. M. Martin (/. c.) that this extract occurred in the work of the Alopeconnesian on Plato's Republic in connection with the distaff of the Fates in the tenth book becomes probable.

Bretschneider (Geom.

2 Stobaeus, Floril., ed. A. Meineke, vol. IV., p. 205. vor Eukl., p. 162) doubts the authenticity of this anecdote, and thinks that it may be only an imitation of the similar one concerning Euclid and Ptolemy. He does so on the ground that it is nowhere reported that Alexander had, besides Aristotle, Menaechmus as a special teacher in geometry. This is an insufficient reason for rejecting the anecdote, and, indeed, it seems to me that the probability lies in the other direction, for we shall see that Aristotle had direct relations with the school of Cyzicus.

3 The fragments of Menaechmus have been collected and given in Greek by Max C. P. Schmidt (l. c.).

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