present edition, but to a greater extent, and with increased distinctness. To avoid the confused appearance produced by the lines being scattered irregularly over the page, as in previous texts of the 'Elements' on this plan, the lines have been printed so as to commence uniformly from the side of the page. Every conclusion is what printers technically term 'indented,' and the applicable part of it—that is, the part made use of or referred to in the subsequent reasoning-is, to mark its importance, printed in italics; the final conclusion, and the thing to be proved or to be done, standing out prominently from the rest of the page in bold 'Clarendon' type, so as to keep constantly before the eye the object towards which the whole process of the reasoning is directed.

The notes appended to the text are necessarily brief, but it is hoped that they will be found to touch upon most of the points of interest connected with the definitions and propositions. The notes are followed by a classified index of the propositions of each Book, in which all the theorems relating to the same subject are, for the convenience of reference and comparison, collected in one view.

In the Appendix the enunciations and corollaries, together with questions on the definitions, axioms, &c. of each Book are printed separately, to assist teachers in reviewing their classes, and to aid the student in the work of self-examination. Algebraical and arithmetical solutions of the propositions of the Second and Fifth Books have also been given, and to the whole is appended a large collection of geometrical exercises for solution. For the convenience of those using other editions of 'Euclid,' the Appendix has been published in a separate form.

A. K. ISBISTER.

LONDON: January 3, 1865.

THE

ELEMENTS OF EUCLID.

BOOK I.

DEFINITIONS.

I.

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

A line is length without breadth.

II.

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

VIII.

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

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 containing 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 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 circlé.

Ө

XVII.

A diameter of a circle is a straight line drawn through the centre, 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.

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.

XXIX.

An acute-angled triangle, is that which has three acute

angles.