N.B. 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: but when several angles are at one point B, either 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 these 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.

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

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


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

A term or boundary is the extremity of any thing.

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.

And this point is called the center of the circle.

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

The center of a semicircle is the same with that of the circle.

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

Trilateral figures, or triangles, by three straight lines.

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 two sides equal.

A scalene triangle is that which has three unequal sides.

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

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

An acute-angled triangle is that which has three acute angles.

XXX. Of quadrilateral or four-sided figures, a square has all its sides equal and all its angles right angles.

XXXI. An oblong is that which has all its angles right angles, but has not all its sides equal.

XXXII. A rhombus has all its sides equal, but its angles are not right angles.

XXXIII. A rhomboid has its opposite sides equal to each other, but all its sides are not equal, por its angles right angles.

XXXIV. All other four-sided figures besides these, are called Trapeziums.

XXXV. Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways, do not meet.

A. A parallelogram is a four-sided figure, of which the opposite sides are parallel: and the diameter, or the diagonal is the straight line joining two of its opposite angles.


LET it be granted that a straight line may be drawn from any one point to any other point.

II. That a terminated straight line may be produced to any length in a straight line.


And that a circle may be described from any center, at any distance from that center.


THINGS which are equal to the same thing are equal to one another.

If equals be added to equals, the wholes are equal.

If equals be taken from equals, the remainders are equal.


If equals be added to unequals, the wholes are unequal.

V. If equals be taken from unequals, the remainders are unequal.

VI. Things which are double of the same, are equal to one another.

VII. Things which are halves of the same, are equal to one another.

VIII. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.

IX. The whole is greater than its part.

x. Two straight lines cannot enclose a space.


All right angles are equal to one another.

XII. If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles ; these straight lines being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles.

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