IV.

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 inclose a space.

XI.

All right angles are equal to one another.

XII.

If a straight line meet 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.

PROPOSITIONS.

PROP. I. PROBLEM.

To describe an equilateral triangle upon a given finite straight line.

PROP. II. PROBLEM.

From a given point, to draw a straight line equal to a given straight line.

PROP. III. PROBLEM.

From the greater of two given straight lines, to cut off a part equal to the less.

PROP. IV. THEOREM.

If two triangles have two sides of the one equal to two sides of the other, each to each; and have likewise the angles contained by those sides equal to one another; they shall likewise have their bases, or third sides, equal; and the two triangles shall be equal; and their other angles shall be equal, each to each, viz. those to which the equal sides are opposite.

PROP. V. THEOREM.

The angles at the base of an isosceles triangle are equal to one another: and if the equal sides be produced, the angles upon the other side of the base shall be equal.

COR. Hence every equilateral triangle is also equiangular.

PROP. VI. THEOREM.

If two angles of a triangle be equal to one another, the sides also which subtend, or are opposite to, the equal angles, shall be equal to one another. COR. Hence every equiangular triangle is also equilateral.

PROP. VII. THEOREM.

Upon the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of the base, equal to one another, and likewise those which are terminated in the other extremity.

PROP. VIII. THEOREM.

If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal; the angle which is contained by the two sides of the one shall be equal to the angle contained by the two sides equal to them, of the other.

PROP. IX. PROBLEM.

To bisect a given rectilineal angle, that is, to divide it into two equal angles.

PROP. X. PROBLEM.

To bisect a given finite straight line, that is, to divide it into two equal parts.

PROP. XI. PROBLEM.

To draw a straight line at right angles to a given straight line, from a given point in the same.

COR. By help of this problem, it may be demonstrated, that two straight lines cannot have a common segment.

PROP. XII. PROBLEM.

To draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it.

PROP. XIII. THEOREM.

The angles which one straight line makes with another upon one side of it, are either two right angles, or are together equal to two right angles.

PROP. XIV. THEOREM.

If, at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line. PROP. XV. THEOREM.

If two straight lines cut one another, the vertical, or opposite angles shall be equal.

COR. 1. From this it is manifest, that, if two straight lines cut one another, the angles which they make at the point where they cut, are together equal to four right angles.

COR. 2. And consequently, that all the angles made by any number of lines meeting in one point, are together equal to four right angles.

PROP. XVI. THEOREM.

If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

PROP. XVII. THEOREM.

Any two angles of a triangle are together less than two right angles.
PROP. XVIII. THEOREM.

The greater side of every triangle is opposite to the greater angle.
PROP. XIX. THEOREM.

The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it.

PROP. XX. THEOREM.

Any two sides of a triangle are together greater than the third side.

PROP. XXI. THEOREM.

If from the ends of a side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.

PROP. XXII. PROBLEM.

To make a triangle of which the sides shall be equal to three given straight lines, but any two whatever of these must be greater than the third.

At a given point in a given straight line, to make a rectilineal angle equal to a given rectilineal angle.

PROP. XXIV. THEOREM.

If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other; the base of that which has the greater angle, shall be greater than the base of the other.

PROP. XXV. THEOREM.

If two triangles have two sides of the one equal to two sides of the other, each to each, but the base of the one greater than the base of the other; the angle contained by the sides of that which has the greater base, shall be greater than the angle contained by the sides, equal to them, of the other.

PROP. XXVI. THEOREM.

If two triangles have two angles of one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides

adjacent to the equal angles in each, or the sides opposite to them; then shall the other sides be equal, each to each, and also the third angle of the one to the third angle of the other.

PROP. XXVII. THEOREM.

If a straight line falling upon two other straight lines make the alternate angles equal to one another; these two straight lines shall be parallel.

PROP. XXVIII. THEOREM.

If a straight line falling upon two other straight lines, make the exterior angle equal to the interior and opposite upon the same side of the line; or make the interior angles upon the same side together equal to two right angles; the two straight lines shall be parallel to one another.

PROP. XXIX. THEOREM.

If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles.

PROP. XXX. THEOREM.

Straight lines which are parallel to the same straight line are parallel to each other.

PROP. XXXI. PROBLEM.

To draw a straight line through a given point parallel to a given straight line.

PROP. XXXII. THEOREM.

If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles.

COR. 1. All the interior angles of any rectilineal figure together with four right angles, are equal to twice as many right angles as the figure has sides.

COR. 2. All the exterior angles of any rectilineal figure, made by producing the sides successively in the same direction, are together equal to four right angles.

The straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel. PROP. XXXIV. THEOREM.

The opposite sides and angles of parallelograms are equal to one another, and the diameter bisects them, that is, divides them into two equal parts.

PROP. XXXV. THEOREM.

Parallelograms upon the same base, and between the same parallels, are equal to one another.

PROP. XXXVI. THEOREM.

Parallelograms upon equal bases and between the same parallels, are equal to one another.

PROP. XXXVII. THEOREM.

Triangles upon the same base and between the same parallels, are equal to one another.

PROP. XXXVIII. THEOREM.

Triangles upon equal bases and between the same parallels, are equal to one another.

PROP. XXXIX. THEOREM.

Equal triangles upon the same base and upon the same side of it, are between the same parallels.

PROP. XL. THEOREM.

Equal triangles upon equal bases, in the same straight line, and towards the same parts, are between the same parallels.

PROP. XLI. THEOREM.

If a parallelogram and a triangle be upon the same base, and between the same parallels; the parallelogram shall be double of the triangle.

PROP. XLII. PROBLEM.

To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

of

PROP. XLIII. THEOREM.

The complements of the parallelograms, which are about the diameter any parallelogram, are equal to one another.

PROP. XLIV. PROBLEM.

To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

PROP. XLV. PROBLEM.

To describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle.

COR. From this it is manifest how to a given straight line to apply a parallelogram, which shall have an angle equal to a given rectilineal angle, and shall be equal to a given rectilineal figure; viz. by applying to the given straight line a parallelogram equal to the first triangle ABD, and having an angle equal to the given angle.

PROP. XLVI. PROBLEM.

To describe a square upon a given straight line.

COR. Hence, every parallelogram that has one right angle, has all its angles right angles.

PROP. XLVII. THEOREM.

In any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle.

PROP. XLVIII. THEOREM.

If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it; the angle contained by these two sides is a right angle.