Euclid

To construct a cube and comprehend it in a sphere.

To construct an octahedron and comprehend it in a sphere.

What profit shall I get by learning these things?

Give him three pence, since he must make a profit out of what he learns.

Parallel lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Let it be granted that a circle may be described with any centre and any radius.

The angle of a semicircle is greater than any acute rectilineal angle.

The angle of a segment is greater than any acute rectilineal angle.

The angle of a segment is less than any obtuse rectilineal angle.

The angle of a semicircle is less than any obtuse rectilineal angle.

If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which it makes with the tangent shall be equal to the angles in the alternate segments of the circle.

If a straight line be drawn from the centre of a circle to the point of contact of a tangent, it will be at right angles to the tangent.

In any right-angled triangle the square on the side opposite the right angle is equal to the sum of the squares on the other two sides.

If a straight line be cut in extreme and mean ratio, the square on the greater segment added to the half of the whole line is equal to the square on the whole line.

The sum of the angles in a triangle is equal to two right angles.

If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.

If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts together with the square on the line between the points of section is equal to the square on the half.

If a straight line be divided into two equal parts and also into two unequal parts, the square on the unequal parts together with the square on the line between the points of section is equal to twice the square on the half.

If a straight line be divided into two parts at random, the square on the whole is equal to the squares on the two parts together with twice the rectangle contained by the parts.

If a straight line be divided into two parts at random, the square on the whole and that on one of the parts are equal to twice the rectangle contained by the whole and the said part and the square on the remaining part.