Geometry Course/Eucler's Axiom
Axioms[edit | edit source]
Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are derived from a small number of axioms. Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath):
"Let the following be postulated":
- "To draw a straight line from any point to any point."
- "To produce [extend] a finite straight line continuously in a straight line."
- "To describe a circle with any centre and distance [radius]."
- "That all right angles are equal to one another."
- The parallel postulate: "That, if a straight line falling on two straight lines make 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 are the angles less than the two right angles."
Although Euclid's statement of the postulates only explicitly asserts the existence of the constructions, they are also taken to be unique.
The Elements also include the following five "common notions":
- Things that equal the same thing also equal one another.
- If equals are added to equals, then the wholes are equal.
- If equals are subtracted from equals, then the remainders are equal.
- Things that coincide with one another equal one another.
- The whole is greater than the part.
- The assumptions of Euclid are discussed from a modern perspective in Harold E. Wolfe (2007). Introduction to Non-Euclidean Geometry. Mill Press. p. 9. ISBN 1406718521. http://books.google.com/books?id=VPHn3MutWhQC&pg=PA9.
- tr. Heath, pp. 195-202.