, the celebrated mathematician, according to the account of Pappus and Proclus, was born at Alexandria, in W. shore of the Red Sea, has a northern coast-line on the Mediterranean, and stretches S. as far as Wady…">Egypt, where he flourished and taught mathematics, with great applause, under the reign of Ptolemy | Lagos, about 280 years before Christ. And here, from his time till the conquest of Alexandria by the Saracens, all the eminent mathematicians were either born, or studied; and it is to Euclid, and his scholars, we are beholden for Eratosthenes, Archimedes, Apollonius, Ptolemy, Theon, &c. &c. He reduced into regularity and order all the fundamental principles of pure mathematics, which had been delivered down by Thales, Pythagoras, Eudoxus, and other mathematicians before him, and added many others of his own discovering: on which account it is said he was the first who reduced arithmetic and geometry into the form of a science. He likewise applied himself to the study of mixed mathematics, particularly to astronomy and optics. His works, as we learn from Pappus and Proclus, are the Elements, Data, Introduction to Harmony, Phenomena, Optics, Catoptrics, a Treatise of the Division of Superficies, Porisms, Loci ad Superficiem, Fallacies, and four books of Conies. The most celebrated of these, is the Elements of Geometry, first published at Basil, 1533, by Simon Grynaeus, of which there have been numberless editions, in all languages; and a fine edition of all his works was printed in 1703, by Dr. David Gregory, SaTilian professor of astronomy at W. of London; it is a city of…">Oxford, which is the most complete, and is illustrated by the notes of sir Henry Savile, and dissertations and discussions on the authenticity of the several pieces attributed to Euclid.

The Elements, as commonly published, consist of 15 books, of which the two last it is suspected are not Euclid’s, but a comment of Hypsicles of Alexandria, who lived 20Q years after Euclid. They are divided into three parts, viz. the Contemplation of Superficies, Numbers, and Solids the first 4 books treat of planes only the 5th of the proportions of magnitudes in general the 6th of the proportion of plane figures the 7th, 8th, and 9th give us the fundamental properties of numbers; the 10th contains the theory of commensurable and incommensurable lines and spaces; the llth, 12th, 13th, 14th, and 15th, treat of the doctrine of solids. There can be no doubt that, before Euclid, Elements of Qeometry were compiled by Hippocrates of Chios, Eudoxus, Leon, and many others, mentioned by Proclus in the beginning of his second book; for he affirms that Euclid new ordered many things in the Elements of Ludoxus, completed many things in those of Theatetus, and besides strengthened such propositions as before were | too slightly, or but superficially established, with the most firm and convincing demonstrations.

Euclid, as a writer on music, has ever been held in the highest estimation by all men of science who have treated of harmonics, or the philosophy of sound. As Pythagoras was allowed by the Greeks to have been the first who found out musical ratios, by the division of a monochord, or single string, a discovery which tradition only had preserved, Euclid was the first who wrote upon the subject, and reduced these divisions to mathematical demonstration. His “Introduction to Harmonics,” which in some Mss. was attributed to Cleonidas, is in the Vatican copy given to Pappus; Meibomius, however, accounts for this, by supposing those copies to have been only two different ms editions of Euclid’s work, which had been revised, corrected, and restored from the corruptions incident to frequent transcription by Cleonidas and Pappus, whose names were, on that account, prefixed. It first appeared in print with a Latin version, in 1498, at Venice, under the title of “Cleonidae Harmonicum Introductorium:” who Cleonidas was, neither the editor, George Valla, nor any one else pretends to know. It was John Pena, a mathematician in the service of the king of France, who first published this work at Paris, under the name of Euclid, 1557. After this, it went through several editions with his other works.

His “Section of the Canon,” follows his “Introduction;” it went through the same hands and the same editions, and is me-ntioned by Porphyry, in his Commentary on Ptolemy, as the work of Euclid. This tract chiefly contains short and clear definitions of the several parts of Greek music, in which it is easy to see that mere melody was concerned; as he begins by telling us, that the science of harmonics considers the nature and use of melody, and consists of seven parts: sounds, intervals, genera, systems, keys, mutations, and melopceia; all which have been severally considered in the dissertation. Of all the writings upon ancient music, that are come down to us, this seems to be the most correct and compressed the rest are generally loose and diffused the authors either twisting and distorting every thing to a favourite system, or filling their books with metaphysical jargon, with Pythagoric dreams, and Platonic fancies, wholly foreign to music. But Euclid, in this little treatise, is like himself, close and clear; | yet so mathematically short and dry, that he bestows not a syllable more upon the subject than is absolutely necessary. His object seems to have been the compressing into a scientific and elementary abridgment, the more diffused and speculative treatises of Aristoxenus.

History is silent as to the time of Euclid’s death, or his age. He is represented as a person of a courteous and agreeable behaviour, and in great esteem and familiarity with king Ptolemy; who once asking him, whether there was any shorter way of coming at geometry than by his Elements, Euclid, as Proclus testifies, made answer, that there was no royal way or path to geometry. 1