IV.

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

V.

A superficies or surface is that which has length and breadth, but not thickness.

VI.

The extremities of a surface are lines.

VII.

A plane surface or plane is a surface, in which, if any two points be taken, the straight line of which they are the extremities, lies wholly in that surface, i. e. every point in the straight line is also a point in the surface.

VIII.

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

IX.

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

OBS. 1. Unless the contrary be expressly stated, whenever an angle is spoken of, a plane rectilinear angle is to be understood. OBS. 2. When there are several angles at one point, any one of them is denoted by three letters, of which the letter put between the other two denotes the point where the straight lines meet together; and one of these two denotes some point in one of the two straight lines, that include the angle, and the other denotes some point in the other, the order of the first and third being indifferent.

angle included by the straight lines AC, AD is denoted by CAD or DAC; and the angle included by the straight lines AB, AD is denoted by BAD or DAB.

OBS. 3. But when there is only one angle at a point, it may be denoted either by the single letter that denotes the point, or by three letters as above.

Ex. The angle at the point E (Fig. 2) may either be denoted by E, or by FEG, or by GEF.

that straight line, or with that straight line produced, if necessary, the adjacent angles equal to one another, each of these angles is defined to be a right angle; and each of the straight lines is defined to be perpendicular to the other.

XI.

An obtuse angle is an angle which is greater than a right angle.

XII.

An acute angle is an angle which is less than a right angle.

XIII.

The boundary is the extremity of any thing.

XIV.

A figure is a portion of space which is enclosed by one or more boundaries; and when all the points in a figure are also points in the same plane, the figure is called a plane figure.

OBS. Unless the contrary be expressly stated, whenever a figure is spoken of a plane figure is to be understood.

This one line is called the circumference of the circle; the point within the figure is called the centre of the circle; and any one of the straight lines drawn from the centre to meet the circumference is called a radius of the circle.

OBS. It is usual to denote a circle by three letters, these three letters denoting any three points in the circumference of the circle; and the same three letters may be taken to denote the circumfer

ence.

Ex. In the above figure the circle may be denoted by ABC, or by DBC, or by BEA, &c.; and we may speak of the circumference ABC, or DBC, &c. F is the centre; and FA, FB, FC, &c., are radii.

XVII.

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

XVIII.

A semicircle is the figure contained by any diameter of a circle, and by either of the two parts of the circumference, into which it is divided by the diameter.

XIX.

This definition is the same as Bk. iii. Def. 6.

XX.

When the lines which contain a plane figure are all straight lines, it is called a rectilinear figure, and the straight lines are called its sides.

XXI.

A rectilinear figure which has three sides is called a triangle.

XXII.

A rectilinear figure which has four sides is called a quadrilateral figure; one which has five, a pentagon; and one which has six, a hexagon.

XXIII.

Polygon is the general name for a rectilinear figure of any number of sides, including the triangle, quadrilateral figure, &c., as particular cases.

XXIV.

An equilateral triangle is a triangle which has its three sides all equal; an equilateral polygon is a polygon with all its sides equal; and an equiangular polygon is a polygon with all its angles equal.

XXV.

An isosceles triangle is a triangle which has two of its three sides equal.

XXVI.

A

A scalene triangle is a triangle which has its three sides all unequal.

XXVII.

A right-angled triangle is a triangle

one of the three angles of which is a

right angle.

XXVIII.

An obtuse-angled triangle is a triangle one of the three angles of which is an obtuse angle.

XXIX.

An acute-angled triangle is a triangle each of the three angles of which is an acute angle.

XXX.

A square is defined to be a four-sided figure which has all its sides equal and all its angles right angles.

XXXI.

An oblong is a four-sided figure which has all its angles right angles, but has not all its sides equal.

XXXII.

A rhombus is a four-sided figure which has all its sides equal, but its angles not right angles.

XXXIII.

A rhomboid is a four-sided figure which has its opposite sides equal, but all its sides are not equal, and its angles are not right angles.

XXXIV.

Trapeziums are such four-sided figures as are not included in the four preceding definitions.

XXXV.

Parallel straight lines are such straight lines as are in the same plane, and as, being

produced ever so far both ways, do not meet.

XXXVI.

A parallelogram is a four-sided figure which has its opposite sides parallel; and each of the two