Euclid of Alexandria

A number is a multitude composed of units.

A unit is that by virtue of which each of the things that exist is called one.

A prime number is that which is measured by a unit alone.

Any composite number is measured by some prime number.

Prime numbers are more than any assigned multitude of prime numbers.

If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.

If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.

Magnitudes are said to be commensurable which are measured by the same measure, and incommensurable which cannot have any common measure.

A rational straight line is any straight line which can be expressed as a ratio of two integers.

An irrational straight line is one which cannot be expressed as a ratio of two integers.

To construct an equilateral triangle on a given finite straight line.

To place at a given point (as an extremity) a straight line equal to a given straight line.

Given two unequal straight lines, to cut off from the greater a straight line equal to the less.

If a straight line be drawn from the center of a circle to bisect a straight line not passing through the center, it will also cut it at right angles; and if it cut it at right angles, it will also bisect it.

In a circle the diameter is the greatest line.

In equal circles equal angles stand on equal circumferences, whether they stand at the centers or at the circumferences.

To construct a square on a given straight line.

A plane figure is a figure in a plane surface contained by any number of non-collinear straight lines.

A solid is that which has length, breadth, and thickness.

The extremities of a solid are surfaces.