Euclidean geometry the version of geometry that includes among its axioms the parallel axiom, which asserts that, given a line L in a plane, there exists just one line in the plane that passes through a point not on L but never meets L. The phrase ‘Euclidean geometry’ refers both to the doctrine of geometry to be found in Euclid’s Elements (fourth century B.C.) and to the mathematical discipline that was built on this basis afterward. In order to present properties of rectilinear and curvilinear curves in the plane and solids in space, Euclid sought definitions, axioms, and postulates to ground the reasoning. Some of his assumptions belonged more to the underlying logic than to the geometry itself. Of the specifically geometrical axioms, the least self-evident stated that only one line passes through a point in a plane parallel to a non-coincident line within it, and many efforts were made to prove it from the other axioms. Notable forays were made by G. Saccheri, J. Playfair, and A. M. Legendre, among others, to put forward results logically contradictory to the parallel axiom (e.g., that the sum of the angles between the sides of a triangle is greater than 180°) and thus standing as candidates for falsehood; however, none of them led to paradox. Nor did logically equivalent axioms (such as that the angle sum equals 180°) seem to be more or less evident than the axiom itself. The next stages of this line of reasoning led to non-Euclidean geometry.
From the point of view of logic and rigor, Euclid was thought to be an apotheosis of certainty in human knowledge; indeed, ‘Euclidean’ was also used to suggest certainty, without any particular concern with geometry. Ironically, investigations undertaken in the late nineteenth century showed that, quite apart from the question of the parallel axiom, Euclid’s system actually depended on more axioms than he had realized, and that filling all the gaps would be a formidable task. Pioneering work done especially by M. Pasch and G. Peano was brought to a climax in 1899 by Hilbert, who produced what was hoped to be a complete axiom system. (Even then the axiom of continuity had to wait for the second edition!) The endeavor had consequences beyond the Euclidean remit; it was an important example of the growth of axiomatization in mathematics as a whole, and it led Hilbert himself to see that questions like the consistency and completeness of a mathematical theory must be asked at another level, which he called metamathematics. It also gave his work a formalist character; he said that his axiomatic talk of points, lines, and planes could be of other objects.
Within the Euclidean realm, attention has fallen in recent decades upon ‘neo-Euclidean’ geometries, in which the parallel axiom is upheld but a different metric is proposed. For example, given a planar triangle ABC, the Euclidean distance between A and B is the hypotenuse AB; but the ‘rectangular distance’ AC ! CB also satisfies the properties of a metric, and a geometry working with it is very useful in, e.g., economic geography, as anyone who drives around a city will readily understand. See also NON -EUCLIDEAN GEOMETRY , PHI – LOSOPHY OF MATHEMATIC. I.G.-G.