Euclid of Alexandria

Let the following be granted: To draw a straight line from any point to any point.

To extend a finite straight line continuously in a straight line.

What profit have I from this?

The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.

If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend.

In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles.

To bisect a given finite straight line.

To bisect a given rectilineal angle.

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

The angles at the base of isosceles triangles are equal to one another; and, if the equal straight lines be produced further, the angles under the base will be equal to one another.

If in a circle a straight line subtend the diameter, it will form a right angle.

Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

To find the greatest common measure of two given unequal numbers.

There are infinitely many prime numbers.

To construct a regular pentagon in a given circle.

To construct a regular hexagon in a given circle.

To construct a regular icosahedron and dodecahedron, and to comprehend them in a sphere, as was done in the case of the figures before mentioned.

The five regular solids are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

The method of exhaustion is a method for finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape.

The principles of geometry are eternal and immutable.