The following is a high-level overview of the Greek mathematician Euclid and his five famous axioms.
- Euclid (c. 325-265 BC), of Alexandria, probably a student of one of Plato’s students, wrote a treatise in 13 books (chapters), titled The Elements of Geometry, in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry.
- The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates), and general quantitative principles (called common notions) from which all the rest of geometry could be logically deduced.
- The axioms, according to Plato, should be simple and self-evident principles, so clearly true that they need no proof.
- Euclid’s first four axioms meet this criterion, but the fifth, even if replaced by Playfair’s Axiom, is not simple, and most would say not self-evident like the first four.
- Following are his five axioms, somewhat paraphrased to make the English easier to read. - (1) points can be joined by a straight line.; (2) Any finite straight line can be extended in a straight line.; (3)A circle can be drawn with any center and any radius.; (4)All right angles are equal to each other.;(5)If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate).