.1 Euclid's Axioms and Common Notions
In addition to the great practical value of Euclidean geometry, the ancient Greeks also found great esthetic value in the study of geometry. Much as children assemble a few kinds blocks into many varied towers, mathematicians assemble a few definitions and assumptions into many varied theorems. The blocks are assembled with Hands, the axioms are assembled with Reason.All of Euclidean geometry (the thousands of theorems) were all put together with a few different kinds of blocks. These are called "Euclid's five axioms":A-1 Every two points lie on exactly one line.A-2 Any line segment with given endpoints may be continued in either direction.A-3 It is possible to construct a circle with any point as its center and with a radius of any length. (This implies that there is neither an upper nor lower limit to distance. In-other-words, any distance, no mater how large can always be increased, and any distance, no mater how small can always be divided.)A-4 If two lines cross such that a pair of adjacent angles are congruent, then each of these angles is also congruent to any other angle formed in the same way.A-5 (Parallel Axiom): Given a line l and a point not on l, there is one and only one line which contains the point, and is parallel to l.In addition to its axiom, Euclidean geometry is based on a number of common notions that Euclid listed in "The Elements". Unlike the axioms which deal with objects of geometry, the common notions are general rules of logic:CN-1 Things which are equal to the same thing are also equal to one another.CN-2 If equals be added to equals, the wholes are equal.CN-3 If equals be subtracted from equals, the remainders are equal.CN-4 Things which coincide with one another are equal to one another.CN-5 The whole is greater than the part.