no use to understand them. At the end of it he says, that Euclid has not used the synthetical but the analytical method in delivering them: in which he is quite mistaken; for in the analysis of a theorem, the thing to be demonstrated is assumed in the analysis; but in the demonstrations of the Data, the thing to be demonstrated, which is, that something or other is given, is never once assumed in the demonstration, from which it is manifest, that every one of them is demonstrated synthetically; though indeed, if a proposition of the Data be turned into a problem, (for example, the 84th or 85th in the former editions, which here are the 85th and 86th,) the demonstration of the proposition becomes the analysis of the problem.

WHEREIN this edition differs from the Greek, and the reasons of the alterations from it, will be shewn in the Notes at the end of the Data.

OF

THE LIFE AND WRITINGS

OF

EUCLID,

AND OF HIS TRANSLATOR AND COMMENTATOR,

DR. ROBERT SIMSON.

OF EUC
EUCLID.

EUCLID, the celebrated mathematician, according to the account of Pappus and Proclus, was born at Alexandria, in Egypt, where he flourished and taught mathematics with great applause, under the reign of Ptolemy Lagos, about 280 years before Christ. Some Arabian historians, however, inform us, that he was born at Tyre, that his father's name was Naucrates, an inhabitant of Damas. The particular place of his nativity appears, therefore, to be uncertain: but whether or not Alexandria had the honour of giving birth to this celebrated mathematician, all historians agree, that he flourished and taught mathematics there at the time above mentioned; which city, from that period to the conquest of it by the Saracens, seems to have been the residence, if not the birth-place, of all the most eminent mathematicians of that time. Euclid 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 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, Phænomena, Optics, Catoptrics, a Treatise on the Division of Superficies, Porisms, Loci ad Superficiem, Fallacies, and Four Books of

Conics. The most celebrated of these, is the first work, the Elements of Geometry; of which there have been numberless editions in all languages; and a fine edition of all his works, now extant, was printed in 1703, by David Gregory, Savilian Professor of Astronomy at Oxford.

The Elements, as commonly published, consist of fifteen books, of which the last two, it is suspected, are not Euclid's, but a comment of Hypsicles of Alexandria, who lived two hundred years after him. They are divided into three parts, viz. the Contemplation of Superficies, Numbers, and Solids: the first four Books treat of planes only; the fifth 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 11th, 12th, 13th, 14th and 15th, treat of the doctrine of solids.

There is no doubt but, before Euclid, Elements of Geometry 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 Eudoxus, 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 demon

strations.

History is silent as to the time of Euclid's death or his age. But Pappus represents him as a person of a courteous and agreeable behaviour, and in great esteem with Ptolemy Lagos, king of Egypt; who one day asking him, whether there was not any shorter way of coming at geometry than by his Elements, Euclid is said to have answered, "that there was no royal road to geometry."

OF DR. ROBERT SIMSON.

DR. ROBERT SIMSON, Professor of Mathematics in the University of Glasgow, was the eldest son of Mr. John Simson, of Kirtonhall in Ayrshire, and was born on the 14th of October 1687. Being designed by his father for the church, he was sent to the University of Glasgow about the year 1701,

where he was distinguished by his proficiency in classical learning, and in the sciences.

Having procured a copy of Euclid's Elements, with the aid only of a few preliminary explanations from some more advanced students, he entered on the study of that oldest and best introduction to mathematics. In a short time he read and understood the first six, with the 11th and 12th books, and afterwards proceeding still farther in his mathematical pursuits, by his progress in the more difficult branches he laid the foundation of his future eminence. His reputation as a mathematician in a few years became so high, and his general character so much respected, that in 1710, when he was only twenty-two years of age, the members of the college voluntarily made him an offer of the mathematical chair, in which a vacancy in a short time was expected to take place. From his natural modesty, however, he felt much reluctance, at so early an age, to advance abruptly from the state of a student to that of a professor in the same college, and therefore solicited permission to spend one year at least in London, where, besides other obvious advantages, he might have opportunities of becoming acquainted with some of the eminent mathematicians of England, who were then the most distinguished in Europe. In this request he was readily indulged; and without delay he proceeded to London, where he remained about a year, diligently employed in the improvement of his mathematical knowledge.

When the vacancy in the professorship of Mathematics at Glasgow did occur, the University, while Mr. Simson was still in London, appointed him to fill it; and the minute of election, which is dated March 11, 1711, concluded with this very proper condition, "That they will admit the said Mr. Robert Simson, providing always that he give satisfactory proof of his skill in mathematics, previous to his admission." He was duly admitted professor of mathematics on the 20th of November of that year.

His manner of teaching was uncommonly clear and successful; and among his scholars, several rose to distinction as mathematicians; among whom may be mentioned the celebrated Dr. Matthew Stewart, professor of Mathematics at Edinburgh the two rev. Drs. Williamson, one of whom succeeded Dr. Simson at Glasgow; the rev. Dr. Trail, formerly professor of Mathematics at Aberdeen; Dr. James Moor,