OF 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 demonstrations.

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.


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 further 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 Ďr. Simson at Glasgow; the rev. Dr. Trail, formerly professor of Mathematics at Aberdeen; Dr. James Moor, Greek professor at Glasgow; and professor Robison of Edin burgh, with many others of distinguished merit. In the year 1758, Dr. S. being then 71 years age,

found it neces sary to employ an assistant in teaching; and in 1761, on his recommendation, the rev. Dr. Willianison was appointed his assistant and successor.

His only publication, after resigning his office, was a new and improved edition of Euclid's Data, which in 1762 was annexed to the second edition of the Elements. But from that period, although much solicited to bring forward some of his other works on the ancient geometry, and notwithstanding he was fully apprised of the universal curiosity excited respecting his discovery of Euclid's Porisms, he resisted every importunity on the subject.

Through Dr. Jurin, then secretary of the Royal Society, Dr. Simson had some intercourse with Dr. Halley, and other distinguished members of that society. And about the same time and afterwards he had frequent correspondence with Mr. Maclaurin, Mr. James Stirling, Dr. James Moor, Dr. Matthew Stewart, Dr. William Trail, Mr. Williamson of Lisbon, and with Mr. John Nourse his bookseller and publisher in London.

Dr. S. was originally possessed of great intellectual powers, an accurate and distinguishing understanding, an inventive genius, and a retentive memory; and these powers being excited by an ardent curiosity, produced a singular capacity for investigating the truths of mathematical science. By such talents, and with a correct taste, formed by the study of the Greek geometers, he was also peculiarly qualified for communicating his knowledge, both in his lectures and in his writings, with perspicuity and elegance.



He was esteemed a good classical scholar; and though the simplicity of geometrical demonstration does not admit of much variety of style, yet in his works a good taste in that respect may be distinguished. In his Latin prefaces also, in which there is some history and discussion, the purity of language has been generally approved. It is to be regretted indeed, that he had not had an opportunity of employing in early life his Greek and mathematical learning, in giving an edition of Pappus in the original language.

Strict integrity and private worth, with corresponding purity of morals, gave the highest value to a character which, from other qualities and attainments, was much respected and esteemed. On all occasions, even in the gayest hours of social intercourse, the Doctor maintained a constant attention to propriety. He had serious and just impressions of religion ; but he was uniformly reserved in expressing particular opinions about it; and, from his sentiments of decorum, he never introduced religion as a subject of conversation in mixed society, and all attempts to do so in his clubs were, by him, checked with gravity and decision.

He was seriously indisposed only for a few weeks before his death, having through a very long life enjoyed a uniform state of good health. He died on the 1st of October, 1768, when his 81st year was almost completed.

The writings and publications of Dr. S. were almost exclusively of the pure geometrical kind, after the genuine manner of the ancients. He has only two pieces printed in the volumes of the Philosophical Transactions : viz.

1. Two general propositions of Pappus, in which many of Euclid's Porisms are included, vol. 32, ann. 1723.—These two propositions were afterwards incorporated into the author's posthumous works, printed in 1776, by Philip, earl Stanhope.

2. On the Extraction of the Approximate Roots of Num. bers by infinite Series; vol. 48, ann. 1753.

The separate publications in his life-time were: 3. Conic Sections, in 1735, 4to. 4. The Loci Plani of Apollonius, restored ; in 1749, 4to.

5. Euclid's Elements; in 1756, 4to, and since that time, many editions in 8vo, with the addition of Euclid's Data.

6. After his death, earl Stanhope was at the expense of printing in 1776, under the title of " Opera Reliqua," several of Dr. S.'s posthumous pieces: which were (1) Apollonius's Determinate Section : (2) A Treatise on Porisms*: (3) A Tract on Logarithms : (4) On the Limits of Quantities and Ratios: (5) Some Select Geometrical Problems.

* A part of this Treatise was translated by the Rev. Jolin Lawson, and is in his Mathematical Tracts.







A Point is that which hath no parts, or which hath no See Notes. magnitude.

A line is length without breadth.


The extremities of a line are points.

A straight line is that which lies evenly between its ex-
treme points.

A superficies is that which hath only length and breadth.

'The extremities of a superficies are lines.

VII. A plane superficies is that in which any two points be- See N.

ing taken, the straight line between them lies wholly in that superficies.


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