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Euclid's great work consisted of thirteen books covering a vast body of mathematical knowledge, spanning arithmetic, geometry and number theory.

The books are organized by subjects, covering every area of mathematics developed by the Greeks: The basic structure of the elements begins with Euclid establishing axioms, the starting point from which he developed 465 propositions, progressing from his first established principles to the unknown in a series of steps, a process that he called the 'Synthetic Approach.' He looked at mathematics as a whole, but was concentrated on geometry and that particular discipline formed the basis of his work.

The reason that Euclid was so influential is that his work is more than just an explanation of geometry or even of mathematics.

The way in which he used logic and demanded proof for every theorem shaped the ideas of western philosophers right up until the present day.

A line is said to be “a breadthless length”, and a straight line to be a line “which lies evenly with the points on itself”.

This may help convince readers that they share a common conception of the straight line, but it is no use if unexpected difficulties arise in the creation of a theory—as we shall see.

There were also significant advances in the domain of abstract geometries, such as those proposed by David Hilbert.

It follows that the terms ‘geometry’ and ‘physical space’ do not have simple meanings in the 19 century.

This is a flaw in the proof of the first congruence theorem (I.4) which says that if two triangles have two pairs of sides equal and the included angle is equal then the remaining sides of the triangles are equal. Theorem I.2 carries a scrupulous, and by no means obvious, proof that a given line segment in a plane may be copied exactly with one of its end points at any prescribed point in the plane.

Theorem I.4 properly requires a proof that an angle may likewise be copied exactly at an arbitrary point, but this Euclid cannot provide at this stage (one is given in I.23, which, however, builds on these earlier results).

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