Greek professor at Glasgow; and professor Robison of Edinburgh, with many others of distinguished merit. In the year 1758, Dr. S. being then 71 years of age, found it necessary to employ an assistant in teaching; and in 1761, on his recommendation, the rev. Dr. Williamson 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 undertanding, 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,

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SHORT ACCOUNT OF DR. ROBERT SIMSON.

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.

of

1. Two general Propositions of Pappus, in which many 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 Numbers 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 Porisans*: (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. John Lawson, and is in his Mathematical Tracts.

THE

ELEMENTS OF EUCLID.

BOOK I.

DEFINITIONS.

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

II. A line is length without breadth.

III. The extremities of a line are points.

IV. A straight line is that which lies evenly between its extreme points.

V. A superficies is that which hath only length and breadth.

VI. 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.

VIII. "A plane angle is the inclination of two lines to one See N. "another in a plane, which meet together, but are not in "the same direction."

B

IX. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.

N.B. When several angles are at one point B, any one of 'them is expressed by three letters, of which the letter that ' is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another, 'is put between the other two letters,

and one of these two is somewhere

upon one of those straight lines, and the other upon the other line: Thus, B

D

CE

the angle which is contained by the straight lines AB, CB, is named the angle ABC, or CBA; that which is contained by AB, DB, is named the angle ABD, or DBA ; and that 'which is contained by DB, CB, is called the angle DBC, or CBD; but, if there be only one angle at a point, it may be expressed by a letter placed at that point; as the angle at

'E.'

X. When a straight line, standing on another straight line, makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands. on the other, is called a perpendicular to it.

XI. An obtuse angle is that which is greater than a right angle.

XII. An acute angle is that which is less than a right angle.

XIII. "A term or boundary is the extremity of any thing."

XIV. A figure is that which is enclosed by one or more boundaries.

XV. A circle is a plane figure contained by one line, which is called the circumference, and is such, that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.

XVI. And this point is called the centre of the circle.

XVII. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

See last fig.

See N.

See last fig.

XVIII. A semicircle is the figure contained by a diameter and See last fig. the part of the circumference cut off by the diameter.

XIX. "A segment of a circle is the figure contained by a

66

Istraight line, and the circumference it cuts off."

XX. Rectilineal figures are those which are contained by straight lines.

XXI. Trilateral figures, or triangles, by three straight lines.

XXII. Quadrilateral, by four straight lines.

XXIII. Multilateral figures, or polygons, by more than four straight lines.

XXIV. Of three-sided figures, an equilateral triangle is that which has three equal sides.

XXV. An isosceles triangle is that which has only two sides equal.

XXVI: A scalene triangle is that which has three unequal sides.

XXVII. A right-angled triangle is that which has a right angle.

XXVIII. An obtuse-angled triangle is that which has an obtuse angle.