Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.Any straight line segment can be extended indefinitely in a straight line.Any two points can be joined by a straight line.Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms): An implication of Einstein's theory of general relativity is that Euclidean geometry is only a good approximation to the properties of physical space if the gravitational field is not too strong.įollowing a precedent set in the Elements, Euclidean geometry has been exposited as an axiomatic system, in which all theorems ("true statements") are derived from a finite number of axioms. It also is no longer taken for granted that Euclidean geometry describes physical space. Many other consistent formal geometries are now known, the first ones being discovered in the early 19th century. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute sense. Much of the Elements states results of what is now called number theory, proved using geometrical methods.įor over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. The Elements goes on to the solid geometry of three dimensions, and Euclidean geometry was subsequently extended to any finite number of dimensions. The Elements begin with plane geometry, still often taught in secondary school as the first axiomatic system and the first examples of formal proof.
Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fitted together into a comprehensive deductive and logical system. The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions ( theorems) from those axioms. It has been one of the most influential books in history, as much for its method as for its mathematical content. Euclid's text Elements was the first systematic discussion of geometry. Euclidean geometry is a mathematical well-known system attributed to the Greek mathematician Euclid of Alexandria.
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