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It was simpler in the 15th century, when this 'Allegory of Geometry' was painted. Geometry was equated with the 'Elements' of Euclid, published around 300BC: 465 propositions that linked plane and solid geometry, number theory and geometrical algebra in a chain of deductive reasoning. For 2,000 years geometry lessons meant Euclid.

Today, although school geometry still uses ruler (radius geometricus in the picture) and compass, at higher levels alternative geometries have multiplied. Renaisssance artists developed projective geometry in the 15th centur, Descartes co-ordinate geometry in the 17th. Then Carl Gauss (1777-1855) and Bernard Riemann (1826-1866) created the first non-Euclidean geometry. Since then, the range of geometries has wildly expanded; space is seen no more as a collection of points but as a locus in which to move and compare objects, while geometry is a powerful toolkit for creative reasoning rather than an unalterable set of axioms. web: www.mnsfld.edurwalkergeometry.html Victoria Neumark m.c escher's "sky amp; water"(c) 2001 cordon art-baarn-holland all rights reserved

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