The great contribution of Euclid was his use of a deductive system for the presentation of mathematics. Primary terms, such as point and line, are defined; unproved assumptions, or postulates, regarding these terms are stated; and a series of statements are then deduced logically from the definitions and postulates. Although Euclid's system no longer satisfies modern requirements of logical rigor, its importance in influencing the direction and method of the development of mathematics is undisputed.
One consequence of the critical examination of Euclid's system was the discovery in the early 19th cent. that his fifth postulate, equivalent to the statement that one and only one line parallel to a given line can be drawn through a point external to the line, can not be proved from the other postulates; on the contrary, by substituting a different postulate for this parallel postulate two different self-consistent forms of non-Euclidean geometry were deduced, one by Nikolai I. Lobachevsky (1826) and independently by János Bolyai (1832) and another by Bernhard Riemann (1854).
See D. Berlinski, The King of Infinite Space: Euclid and His Elements (2013).
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