a

meters; and, though these are not so gross as the others now mentioned, they ought by no means to remain uncorrected.

Upon these accounts it appeared necessary, and I hope will prove acceptable to all lovers of accurate reasoning, and of mathematical learning, to remove such blemishes, and restore the principal books of the Elements to their original accuracy, as far as I was able ; especially since these Elements are the foundation of a science by which the investigation and discovery of useful truths, at least in mathematical learning, is promoted as far as the limited powers of the mind allow; and which likewise is of the greatest use in the arts both of peace and war, to many of which geometry is absolutely necessary. This I have endeavoured to do, by taking away the inaccurate and false reasonings which unskilful editors have put into the place of some of the genuine demonstrations of Euclid, who has ever been justly celebrated as the most accurate of geometers, and by restoring to him those things which Theon or others have suppressed, and which have these

many ages been buried in oblivion. In this edition, Ptolemy's proposition concerning a property of quadrilateral figures in a circle is added at the end of the sixth book. Also the note on the 29th

prop.

book 1st, is altered, and made more explicit, and a more general demonstration is given, instead of that which was in the note on the 10th definition of book 11th; besides, the translation is much amended by the friendly assistance of a learned gentleman.

To which are also added, the Elements of Plane and Spherical Trigonometry, which are commonly taught after the Elements of Euclid.

THE

ELEMENTS OF EUCLID.

BOOK I.

DEFINITIONS.

I.

Book I. A POINT is that which hath no parts, or which hath no mag- See nitude.

Notes. 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.

a

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 being taken, See N. the straight line between them lies wholly in that superficies.

VIII. " A plane angle is the inclination of two lines to one another See N.

“ in a plane, which meet together, but are not in the same
“ direction.”

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.

B

Book I.

Ą D

B

C

E

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 some-
where upon one of those straight lines, and the other upon
the other line: thus 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 ADB, or in
· 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 ano-

ther 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.

Book 1. 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 cen-
tre, and terminated both ways by the circumference.

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

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

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