EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY

the study of points, lines, angles, surfaces, and solids, on the basis of either the 10 axioms and postulates selected by the Greek mathematician Euclid (c. 300 BC) or a modification of Euclid's system (commonly the replacement of the so-called parallel postulate). The importance of Euclidean geometry lies not so much in the actual mathematics that it contains as in the systematic method used by Euclid to present and develop that mathematics. This method, called the axiomatic-deductive method, has served as the model for the development of many other mathematical subject areas for more than 2,000 years. Euclid's Elements, a set of 13 books, dealt with various aspects of plane and solid geometrical figures, their measurement, and their relationships to one another. It appears that very little of the mathematics contained in the Elements was due to Euclid himself, although some of the results and proofs undoubtedly were. From the 10 axioms and postulates, Euclid deduced 465 theorems, or propositions. This was the first demonstration of the power of the axiomatic method, in which the truth of the derived theorems follows from the truth of the axioms and postulates. Because the latter were offered as self-evidently true, Euclid's contemporaries felt that the derived theorems constituted accurate descriptions of the world and valid tools for studying it. Euclid's parallel postulate attracted interest almost as soon as the Elements appeared, because it seemed less self-evident than the others. Its most popular equivalent is: Through a given point P not on a line l, there is only one line in the plane of P and l that does not meet l. Attempts to derive the parallel postulate from the others, thereby transforming it into a theorem, involved replacing it with its two alternativesthat there is no such line or that there are more than oneand then showing that contradictions ensue. Unexpectedly, no contradictions resulted from either substitution: the outcome was, instead, two new, non-Euclidean geometries that were found to be just as valid and consistent as Euclidean geometry. It soon became clear that it is impossible to tell which, if any, of the three geometries is the most accurate as a mathematical representation of the real world. Thus, mathematicians were forced to abandon the cherished concept of a single correct geometry and to replace it with the concept of equally consistent and valid alternative geometries. They were also forced to realize that mathematical systems are not merely natural phenomena waiting to be discovered; instead, mathematicians create such systems by selecting consistent axioms and postulates and studying the theorems that can be derived from them. It is this change of viewpoint that may prove to be the most important and far-reaching part of Euclid's intellectual legacy.