In approximately 300BC, Euclid wrote his major treatise on geometry of the time and what would be the geometry of those for many years after, that is The Elements. Arguably, The Elements is the second most read book of the western world, falling short only to the Bible. In his book, Euclid stated five postulates of geometry which he used as the foundation for all his proofs. It is from these postulates we get the term Euclidean geometry, that is, Euclid strove to define what constituted ‘flat-surface’ geometry. These postulates were,
1. To draw a straight line from any point to any other.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any centre and distance.
4. That all right angles are equal to each other.
5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
It may be clear from the list that the fifth postulate is very different to the other four. In fact, in The Elements, the first 28 results were proved without it. As a result of this difference, many attempts were made to try and prove it using the other four postulates. One earlier attempt at this was made by Proclus (410 – 485). Despite his attempts eventually resulting in failure, Proclus discovered an equivalent statement for the fifth postulate. That is, what is now known as Playfair’s Axiom. It said the following,
Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.
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