XI.

A surface is bounded by a line, or lines.

XII.

A line is bounded by points.

XIII.

The intersection of two lines is a point.

XIV.

If a line is extended, its parts will approach more nearly to the same direction, and if it is extended till its extremities are at the greatest distance which the length of the line will permit, every part will then be in the same direction, and it may be called a stretched line, or straight line.

XV.

A straight line may be supposed to be drawn from any point, in any direction, and to any distance.

XVI.

If a line that is not straight between two points be straightened, part of the line will be drawn beyond one of the points, and the remainder will extend from one to the other; therefore, a straight line is the shortest line that can be drawn between two points.

XVII.

To continue a straight line in the original direction, may be called , producing the line.

XVIII.

To draw a straight line from one point to another, may be called joining the points.

XIX.

Two magnitudes, which being compared exactly fill the same space, may be said to coincide.

XX.

If one straight line is applied to another, they will coincide, except so far as one extends beyond the other, and if they are produced they will still coincide.

XXI.

If a straight line be drawn between two points, any other straight line between the same points will coincide with the first line.

XXII.

Two straight lines, in different directions from the same point, may be said to make an angle.

XXIII.

An angle may be named by a letter at the angular point, as E; or by letters distinguishing the lines which make the angle, as A B C,

A B D, or C B D, the letter at the angular point being between the other two.

XXIV.

The angle is determined by the difference in direction between the lines, and is not varied by their length, or by producing them.

XXV.

If two straight lines which make an angle are produced, the distance between their extremities will be increased, and the more they are produced, the more will it be increased.

XXVI.

If two straight lines meet and are produced, they will either coincide, or not coincide. If they coincide, and are produced, they will still coincide, and cannot enclose a space. If they do not coincide, they will make an angle, and cannot enclose a space. Therefore, two straight lines cannot enclose a space.

XXVII.

If a straight line standing on another straight line, makes the angles on each side equal, each of them may be called a right angle, and the lines may be said to be perpendicular to each other.

XXVIII.

An angle greater than a right angle, may be called an obtuse angle.

XXIX.

An angle less than a right angle, may be called an acute angle.

XXX.

An acute or obtuse angle, may be called an oblique angle.

XXXI.

A portion of space enclosed by one or more boundaries, may be called a figure.