death is referred to by Horace in the following lines: "Te, maris et terræ numeroque carentis arenæ Pulveris exigui prope litus parva Matinum IV VERY similar to the relation between the schools of Miletus and Crotona (or Tarentum) was that between the latter and the school of Athens. Eudoxus (409355 B.C.) was a pupil of Archytas at Tarentum, but set up on his own account at Cyzicus, and ere long removed to Athens. Plato also, if not a pupil of Archytas, was an intimate friend. It is recorded that when Plato was in Sicily, and got into a difficulty" with Dionysius, Archytas saved his life by intervention with the tyrant. It seems hopeless to attempt a reconciliation between various statements as to the relations between Eudoxus and Plato, their friendship and their enmity. The following is, upon the whole, perhaps the likeliest account of the matter; but it must be admitted that it is in direct opposition to statements which may possibly be true. Our supposition is that the two were friends in their youth, and were fellow-travellers in Egypt; that, on their return, Eudoxus went to Cyzicus and Plato to Athens; that their friendship lasted as long as Eudoxus remained in Cyzicus; but that Plato resented his removal to Athens, and his opening of a rival school to his own Academy. Tantæne animis cœlestibus ira? EUDOXUS was a mathematician of a high order, though it ought in fairness to be stated that Pro fessor De Morgan, than whom there are few higher authorities, speaks most disparagingly of him as an astronomer; and it is difficult to believe that so utterly incompetent an astronomer as De Morgan represents him to have been, could have been an accomplished geometer. Yet such he certainly was. He founded some beautiful theorems on the division of a line in Medial Section (Euc. II. 11, VI. 30). These are now known as Euc. XII. 4, 5, 6, and are given in many modern editions of Euclid as corollaries from II. 12, or as additional propositions in Book II. He gave special attention to the subject of proportion, and is supposed by some to have been virtually the author of the Fifth Book of Euclid. This supposition rests on a statement in an anonymous fragment of a commentary on Euclid which has been ascribed to Proclus, but which was probably later than his time. The conjecture seems to have grown, after the manner of the classical Three Black Crows, and to have been consolidated into positive assertion. Thus we have now before us a folio edition of Euclid, by a Jesuit of the seventeenth century, which in its title-page' professes to be a commentary on the thirteen books of Euclid's Elements, and on certain treatises by Isidorus, Hypsicles, and Proclus, 1 Euclidis Elementorum Geometricorum Libros tredecim; Isidorum et Hypsiclen et recentiores de corporibus regularibus; et Procli propositiones Geometricas immissionemque duarum rectarum linearum continue proportionalium inter duas rectas, tam secundum antiquos, quam secundum recentiores, Geometras (qu. Geometros ?) novis ubique fere demonstrationibus illustravit, et multis definitionibus, axiomatibus, propositionibus, et animadversionibus ad Geometriam recte intelligendam necessariis, locupletavit Claudius Ricardus, e Societate Jesu sacerdos, patria Ornacensis, in libero Comitatu Burgundiæ, et Regius Mathematicarum Professor.-Antverpiæ, ex officina Hieronymi Verdussii, 1645. but which gives the titles and headlines of Books V. and VI. respectively as Commentarius in Librum Quintum (Sextum) Elementorum Geometricorum Eudoxi et Euclidis. The idea of a joint-authorship of Book V., here assigned to the two geometers, is at once set aside by the simple fact that Eudoxus was born 408 B.C. and Euclid about 330 B.C.—an interval of some eighty years. Eudoxus died in 355 B.C.—a quarter of a century before Euclid was born. But, apart from joint-authorship, which is impossible, there is no reason to question that Euclid made the same use of the labours of Eudoxus in the construction of his Fifth Book that, in the construction of his other books, he made of those of Pythagoras and Archytas and other predecessors. It is interesting to note that the geometers of this period exercised themselves greatly in attempts to solve the problem of the duplication of the cube, the rectification and quadrature of the circle. These attempts were unsuccessful, yet it is not to be supposed that they were made in vain. The old fable of the treasure pretended to be buried in the vineyard by the old man, in order that his sons might trench deeply in order to find it, has its application here as elsewhere. The fabulist tells us that the diggers found no treasure, for the very good reason that there was none to find, but that they were rewarded for their toil by a series of abundant vintages. But he omits to mention the additional gain that they derived from the labour, in their own brawny arms and more firmly knit loins, and habits of patient perseverance. It has been thus with the diggers in several departments of the scientific vineyard. Astronomy and chemistry might have been elevated to their actual high position by other and more satisfactory means; but as a matter of actual fact it was in astrology and alchemy respectively that they had their beginning. Honest labour is never wholly useless. But the attempts of these early geometers to solve these problems were not altogether unproductive even of direct results. Hippocrates1 of Chios, in attempting to square the circle, succeeded in squaring certain figures bounded by circular arcs. This is of so much interest that we must endeavour to make it understanded" by the non- mathematical reader. The simplest case is this. If a semicircle be described on the chord of a quadrant in any circle as diameter, there will be formed a figure called a lunula or lune, bounded by a quadrantal arc of the one circle and a semicircular arc of the other. Now, with the assumption that the areas of circles are proportionate to the squares of their diameters, it is very easily proved that this lune is equal to a certain rectilineal figure. It may be presumed that Hippocrates so proved it. He probably believed that he had made a near approach to the squaring of the circle; and some who ought to have known better asserted that he had. He had made no approach at all, but had left that problem just where he found it. But it is his proud fame to have been the first to square a figure with circular boundaries, though he necessarily failed to square the circle itself, that is, the figure with one circular 1 He was a contemporary with his namesake of Cos, the celebrated physician, with whom he has been unhappily confounded, even as we have seen that our Euclid has been confounded with his namesake of Magaera, with whom he was not even a contemporary. |