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Immanuel Kant

1. Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry the Elements. Euclid’s method consists in assuming a small set of intuitively appealing axioms and deducing many other propositions (theorems) from these. Although earlier mathematicians had stated many of Euclid’s results, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. Euclidean geometry studies plane and solid figures based on axioms and theorems employed by Euclid. The Elements begins with plane geometry, the first axiomatic system and the first examples of formal proof, which still taught in schools. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.

2. For more than two thousand years, the adjective "Euclidean" was unnecessary, geometry meant Euclidean geometry because no other sort of geometry had been conceived. It is the most typical expression of general mathematical thinking. Rather than the memorization of simple algorithms to solve equations by rote, it demands true insight into the subject, clever ideas for applying theorems in special situations, an ability to generalize from known facts and an insistence on the importance of proof. Euclid’s axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense.

3. 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"

1) "To draw a straight line from any point to any point."

2) "To produce [extend] a finite straight line continuously in a straight line."

3) "To describe a circle with any centre and distance [radius]."

4) "That all right angle are equal to one another."

5) 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."

Today many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19^th century. The Elements remained the very model of scientific exposition until the end of the 19^th century, when the German mathematician David Hilbert wrote his famous Foundations of Geometry (1899). The modern version of Euclidean geometry is the theory of Euclidean (coordinate) spaces of multiple dimensions. Thus, Euclid’s statement of the postulates are also taken to be unique.

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