## Euclid

It is possible to draw a straight line from any point to any point.

It is possible to extend a finite straight line continuously in a straight line (i.e. a line segment can be extended past either of its endpoints to form an arbitrarily large line segment).

It is possible to create a circle with any center and distance (radius).

All right angles are equal to one another (i.e. "half" of a straight angle).

If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.

Things which are equal to the same thing are equal to each other.

If equals are added to equals, the wholes (sums) are equal.

If equals are subtracted from equals, the remainders (differences) are equal.

Things that coincide with one another are equal to one another.

The whole is greater than the part. From these statements which could not be proved, such that 'straight lines do not meet', but which could be used to prove very useful theorems.

The “Elements” includes work on the 'Golden Ratio', contains formulae for calculating the volumes of cones, pyramids and cylinders; proofs about series, perfect numbers and primes; algorithms for finding the greatest common divisor and least common multiple of two numbers; a proof of Pythagoras’ Theorem, and a proof that there are an infinite set of numbers satisfying Pythagoras Theorem and a proof that there are only five possible regular Platonic Solids, work on reflection of light and optics and astronomy. There is also a proof that numbers can be uniquely factorised (The Fundamental Theorem of Algebra), a proof that there are an infinite number of primes and illustrated the use of a proof by contradiction.