distinguished philosopher and logician, who flourished about 400 years before Christ. The Athenians having prohibited the Megareans from entering their city on pain of death, this philosopher disguised himself in women's clothes, in order to attend the lectures of Socrates. After the death of Socrates, Plato and other philosophers went to Euclid at Megara, in order to shelter themselves from the tyrants who then governed Athens. Euclid admitted but one chief good, which he sometimes called God, sometimes Spirit, and sometimes Providence.

Euclid of Alexandria, author of the most ancient elements of geometry which have come down to us, and for this reason justly regarded as one of the fathers of science. He has been sometimes confounded with Euclid of Megara, the disciple of Socrates, and the founder of a philosophic sect who pushed to an extreme the subtleties of dialectics. The place of his birth is unknown. But Proclus Diadochus, one of his commentators, informs us that he opened a school of mathematics in Alexandria in the reign of Ptolemy the son of Lagus; and Pappus extols his kindness and affection for those who laboured to advance the study of geometry. This being all that is known of the life and character of Euclid, it only remains therefore to speak of his works, some of which have been lost. Amongst those which we possess, however, the most remarkable is that which is simply entitled Elements, as if to indicate that it contained the entire body of principles upon which the pure mathematics then rested. This work now consists of fifteen books; but the last two are attributed to Hypsicles, a mathematician of Alexandria who flourished at a later period. Euclid, however, neither nor could be the inventor of all that is contained in his work; for geometers more ancient than he, including Hippocrates of Chio, had written Elements; but, on the other hand, there can be little doubt that he added to the pre-existing stock of elementary truths, improved the demonstrations in which his predecessors had failed, and composed a whole which, by more severe forms of reasoning, and a more exact concatenation of propositions, superseded all works of the same description which had been previously written, and became the basis of instruction in the mathematics. These Elements were first commented on by Theon, and by Proclus; but whatever success such commentaries may have had in the school of Alexandria, they remained wholly unknown to the occidentals of the middle ages, who derived all their knowledge of geometry, such as it was, from the works of Boeckius, and from a production entitled De Principiis Geometriae, ascribed to St Augustin. It was only in the twelfth and thirteenth centuries that Athelard in England and Campano in Italy laboured to decipher and translate Euclid from the Arabic versions, including the commentary of the Persian geometer Nazir-Eddin, which was held in great estimation among the Saracens; for although there is reason to believe that Boeckius had made a complete Latin translation of Euclid, it has not come down to our times; and, in fact, it was not till long after the revival of letters, and when versions had been multiplied by means of the press, that a part of the Elements of Euclid was introduced into the course of instruction in the schools. In order to form an idea of the entire work, however, it may be considered as composed of four parts. The first of these parts comprehends the first six books, and may be divided into three sections, viz. the demonstration of the properties of plane figures treated in an absolute manner, as in books first, second, third, and fourth; the theory of the proportions of quantities in general, which is the object of the fifth; and the application of this theory to plane figures, as in book sixth. The second part contains the seventh, eighth, and ninth books, which are denominated arithmetical, because they treat of the general properties of numbers. The third part consists only of the tenth book, in which the author considers in detail incommensurable quantities, and which he terminates by proving that the diagonal of a square and its side cannot have a common measure; a doctrine, we may add, which is much more ancient than Euclid, since Plato, towards the close of his seventh book of Laws, pronounces those who have no idea of such incommensurability as sunk in almost brutish ignorance. The fourth part, which is composed of the last five books, treats of planes and solids. But of all this great body of geometrical doctrine, the only portions which have been considered as adapted to the purposes of instruction are the first six books, together with the eleventh and the twelfth; the propositions which they contain having formed the basis of all the elements of geometry which, under whatever form, have from time to time been given to the world. The fifth book, however, has often been omitted in such publications, because the notation of our arithmetic, and still more that of our algebra, have greatly simplified the theory of proportions; and, for the same reason, the other arithmetical books, which it is now difficult to read, are justly considered as more curious than useful. But, in borrowing their materials from the work of Euclid, modern authors have frequently altered the arrangement; and on this subject there have arisen two contradictory opinions, which have been debated with very great warmth, yet still remain as irreconcilable as ever. The concatenation established by Euclid, and even the forms which he has employed in demonstrating his propositions, are regarded by some as almost the last term of perfection in works of this kind; whilst others, again, have considered these as mere essays, which, however excellent in themselves, leave room for the introduction of a more natural order and more simple demonstrations. Ramus, who declared war on the dialectics of Aristotle, accuses Euclid of omissions and redundancies, and expresses his conviction that it was with reference to these imperfections that Ptolemy inquired if there were not a more easy method than that usually employed for learning geometry; an inquiry which, as is well known, elicited from Euclid the reply, that in the mathematics there was no road for kings. Antony Arnauld, and the author of the Logic of Port-Royal, have also blamed the order followed by the Greek geometer, and disputed some of his definitions; but if Arnauld, either from not being very profound in the mathematics, or by reason of the great difficulty of the subject, failed, as Ramus and so many others had done, in the changes which he attempted to introduce into the Elements, his reasons for making the attempt still remain in all their force. For, whatever may be said to the contrary, it is certain that they are deficient in that order which, causing the propositions, as far as possible, to arise out of one another, exhibits in full evidence the analogies which connect them, assists the memory, and prepares the mind for the investigation of truth. But whether, in the actual state of the science, it be practicable to reconcile

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1 Montucla, Histoire des Mathématiques, tom. i. p. 212 and 492. 2 See Nouveaux Éléments de la Géométrie, and the Fourth Part of the Logique Port-Royal. Euclid, this order with the rigour of demonstration, or to obviate objections the force of which, abstractedly considered, cannot be disputed, is a question which it would far exceed the limits of this notice to discuss. If it be resolved affirmatively, which appears to us possible, then no reason would exist for giving an absolute preference to the Elements of Euclid. As a precious relic of antiquity, and as one of those works of science which time has thrown least in arrear of actual knowledge, the Elements would doubtless continue to be classed in the first rank of mathematical productions; but their too arbitrary arrangement, and the style in which they are written, sometimes too prolix and at other times too concise, would no longer constitute the essential character of the geometric or synthetic method, in opposition to the analysis of the moderns. The true difference of these two methods of treating the science of quantities consists in this, that the one is founded on the immediate consideration of the properties of figures, whilst the other employs arbitrary signs, combined by the operations of calculation. The first is geometry itself, not that of Euclid more than of any other; the second is an application of algebra, which ought not to be confounded with analysis, insomuch as synthesis may be effected with algebraic signs as well as with the figures of geometry. The latter, however, which may also be treated analytically, presents operations equivalent to the resolution of certain equations. Of this some propositions contained in the book of Data are remarkable examples; propositions which, by a natural and certain path, lead to the solution of problems otherwise undiscoverable. The book of Data was particularly relished by Newton, who, persuaded that a proposition scarcely deserved to see the light unless it could be demonstrated without the assistance of calculation, conceived that a more profound study of this treatise would have enabled him to dispense with the aid of such an instrument. But it may well be doubted, to say nothing more, whether his successors, by pursuing a similar path, would have ever attained those great and striking results which they have derived from the new methods of calculation. Besides the Elements and the Data, which are the most important works of Euclid, Pappus and Proclus mention the following: Introductio Harmonica, Sectio Canonis, relating to music; Phenomena, containing an exposition of the appearances produced by the motion attributed to the celestial sphere; a work subjoined to the book De Sphaera Mobili of Autolycus; Optica, Catoptrica, concerning direct vision and mirrors; Liber de Divisionibus, treating of the divisions of polygons; and the lost works, entitled Perissatum libri, Locorum ad superficiem libri, Fullaciarum liber, and Conicorum liber. At the end of the works of Euclid is a short fragment, entitled De Levi et Ponderoso, the author of which is unknown; it is, however, of almost no value. The editions of the works of this geometer are so numerous that we cannot undertake to indicate all of them, and must therefore confine ourselves to the principal ones. Of the complete works there are, 1. Euclidis Opera, Graece, cum Theonis expositione, cura Grynaei, Bâle, 1590, in fol.; 2. Euclidis quae supersunt omnia, ex recensione Davidis Gregorii, Graece et Latine, Oxford, 1703, in fol.; 3. Les Ouvres d'Euclide, en Grec, en Latin, et en Français, d'après un Manuscrit très ancien, qui était resté inconnu jusqu'à nos jours, par Peyrard, Paris, 1814, in 8vo. The manuscript here referred to was one of several sent from Rome by Monge, and supposed to date from the end of the ninth century; in it the Data are placed immediately after the thirteenth book, and thus separate from the rest of the work the fourteenth and fifteenth books, which are ascribed to Hypsicles. In 1533, Herwage published at Bâle, in folio, a complete edition of the Elements, in the Greek text, with the exposition of Theon, and the Commentaries of Proclus on the first book. The following are the principal Latin translations: 1. Praecellentissimum Opus Elementorum Euclidis perspicillatis in artem Geometriae, the first publication of the Elements by means of printing, Venice, 1482; 2. Euclidis Elementorum libri XV, una cum scholiis antiquis, a Frederico Commandino Urbinate in Latinum conversi, Commentariis quibusdam illustrati, Pesaro, 1572, in fol.; 3. Euclidis Elementorum libri XV, demonstrationibus accuratis scholiis illustrati, auctore Christophoro Clavio, 1574, in 8vo; 4. Euclidis Elementorum libri XV, breviorem demonstrati, opera I. Barrow, London, 1678, in 8vo; 5. Elementorum Euclidis libri XV, ad Graeci contextus fidem recensiti et ad usum tyronum accommodati, edente Baermann, Leipzig, 1769, in 8vo; 6. Euclide Megarensis philosophiae solo introduttore delle scientie mathematiche diligentemente reassettato per Nicola Tortaleo Brisciano. This last, however, is rather a paraphrase than a translation. The following are editions which only contain part of the Elements: 1. Analysis Geometrica sex librorum Euclides a Christiano Herlino, una cum commentariis et scholiis brevibus in eodem libros a Cunrado Duxypodio, 1566, in fol.; 2. Euclidis Elementorum libri priores sex, item undecimus et duodecimus, Oxford, 1747, in 8vo; 3. Euclidis Elementorum libri priores sex, etc. sublatis nisi quibusdam libri a Theone aliisque vitiat sunt, et quidemdam demonstrationum restitutis a Roberto Simson, Glasgow, 1756, in 4to; 4. Éléments de la Géométrie d'Euclide, ou les premiers livres d'Euclide, avec le XI et le XII, traduction nouvelle par Frédéric Castillon, Berlin, 1775, in 8vo.

The other works of Euclid, printed separately, are, 1. Euclidis Data, Claudius Hardy Graece nunc primum edit, Latine veritatis, scholiis illustravit; adjectus est Marii philosophi Commentarius, Graece et Latine, Paris, 1625, in 4to; 2. Euclidis Rudimenta Musices, Graece et Latinae causis, J. Penai interpret., Paris, 1557, in 4to; 3. Euclidis Introductio Harmonica, Graece etc. Meliboomus veritatis explicavit in Antiqui Mathematici Auctores, VII, Amsterdam, 1652, in 4to; 4. Optica et Catoptrica, Graece et Latinae reddita, per Jo. Penain, Paris, 1557, in 4to.

For further details see Murhard, Bibliotheca Mathematica (tom. ii. pp. 1-48), and other similar works. We have thought it necessary to notice the more recent editions of the Elements, which are in the hands of every body.