Of the Babylouian and of the Egyptian geometry we have no remains whatever, though each nation has been often said to have invented the acionc'. In reference to the authorities mentioned above in favour of the Egyptians, to whom we may add Diopuee Leertius, &a, wo may may that no one of the writers who tells the story in question is known an a geometer except Proclus, the latest of them all ; and as if to give the :weapon the character of an hypothesis, this last writer also adds that the Phecniciaus, un account of the wants of their commerce, became the inventor. of arithmetic. In the Jewish writings there ie no trace of any knowledge of geometry. So that, allowing the Greeks to have received the merest rudiments either from Egypt or India, or any other country, it is impossible to name any quarter from which we can with a shadow of probability imagine them to have received a deductive system, to ever so small an extent. That their geometry, or any of it, came direct from India, is a supposition of some difficulty : those who brought it could hardly have failed to bring with it the decimal notation of arithmetic. That Pythagoras travelled into India, is (according to Stanley) only the assertion of Aptileius and Clemens Alexandrinus, though rendered probable by several of his tenets.
Thales (s.c. 600) and Pythagoras (Be. 540) founded the earliest schools of geometry. The latter is said to have sacrificed a hecatomb when he discovered the property of the hypothenuse before alluded to, and this silly story is repeated whenever the early history of geometry is given. A large collection of miscellanies might easily be made from the works of writers who were not themselves acquainted with geometry ; but, rejecting such authorities, we shall content our with citing Pappus and Proclus, both geometers, who, living in the 4th and 5th centuries after Christ, had abundant opportunities of hearing the stories to which we allude, and of receiving or rejecting them.
According to Proclus (book ii. ch. 4, 'Comm. in Encl.'), Pythagoras was the first who gave geometry the form of a science, after whom came Anaxagoras, tEnopides, Hippocrates of Chios (who invented the well-known quadrature of the luoules), and Theodorus of Cyrene. Plato was the next great advancer of the science, with whom were contemporary Leodamas, Archytas, and Themtetua of Therms, Tarentum, and Athens. After Laodamas came Neoclides, whose disciple Leo made many discoveries, added to the accuracy of the elements, and gave a method of deciding upon the possibility or impossibility of a problem. After Leo came Eudoxus, the friend of Plato, who generalised various results which came from the school of the latter. Amyclas, another friend of Plato, and the brothers Menwelimus and Dinostratue, made geometry more perfect. Theudius wrote excellent elements, and generalised various theorems. Cyzicinue of Athens cultivated other parts of mathematics, but particularly geometry. Hermotimun enlarged the results of Eudoxus and Theo.
Cetus, and wrote on 'loci.' Next is mentioned Philippus, and after him Euclid, " who was not much younger than those mentioned, and who put together elements, and arranged many things of Eudoxus, and gave unanswerable demonstrations of many things which had been loosely demonstrated before him." Ile lived under the first Ptolemy, by whom ho was asked for an easy method of learning geometry ; to which he made the celebrated answer, that there was no royal road. Be was younger than the time of Plato, and older than Eratosthenes and Archimedes. He was of the Platonic sect.
Such is, very nearly entire, the account which Proclue gives of the rise of geometry in Greece.
Before the time of Euclid demonstration had been introduced, about the time, perhaps by the instrumentality, of Pythagoras; pure geometry had been restricted to the right line and circle, but by whom is not at all known : the geometrical analysis, and the study of the conic sections, as also the considerations of the problems of the duplication of tho cube, the finding of two mean proportionals, and the trisection of the angle, had been cultivated by the school of Plato ; the quadrature of a certain circular space had been attained, and the general problem suggested and attempted by Hippocrates and others; a curve of double curvature had been imagined and used by Archytas; writings existed both on the elements, and on conic sections, loci, and detached subjects. It is in this part of the present article that we have judged it best to introduce what would otherwise have formed the article EUCLID.
It is not known where EUCLID OP ALEXANDRIA was born. He opened a school of mathematics at Alexandria, in the reign of Ptolemmus the son of Lagus (323-284 n.o.), from which school came Eratosthenes, Archimedes, Apollonius, Ptolemieus, the Theona, &c., &c., so that from and after Euclid the history of the school of Alexandria is that of Greek geometry. He was, according to Pappus, of a mild and gentle temper, particularly towards those who studied the mathematical sciences : but Pappus is too late an authority for the personal demea nour of Euclid, and moreover may have been incited to praise him for the purpose of depreciating Apollonius, of whom he is then speaking, and against whom he several times expresses himself. Besides the Elements, Euclid wrote, or is supposed to have written, the following works: 1, l&rypcnina Wevaapiew, a treatise on 'Fallacies,' preparatory to geometrical reasoning. This book, mentioned by Proclus, does not now exist, and there is no Greek work of which we so much regret the loss. Ilad it survived, mathematical students would not have been thrown directly upon the Elements, without any previous exercise in reasoning.