Abstract
THE author discusses the question whether geometry is purely a rational science or whether it also possesses an experimental side. The question is dealt with in connection with (1) the concepts of geometry, (2) geometrical axioms, and (3) the propositions the estab lishment of which forms the object of deductive geo metry. In the first chapter, M. de Freycinet finds no a priori reasons for the existence of such concepts as- space, straight line, curved line, plane or curved sur face, volume, angle, parallelism, tangency. These and other concepts are all suggested to us by our per ception of the material universe. Passing on to the axioms relating to the straight line and plane, the author considers that it can in no sense be regarded as a self-evident truth that the straight line is the shortest line between two points, that a straight line can be produced indefinitely in either direction, or that two straight lines cannot have two points in common. These and other similar facts can only be regarded as results of experience and observation. In comparing ithe purely geometrical methods of the ancients with Ithe analytical methods of Descartes and Leibnitz, the ¦latter methods will be found in reality to be no less concrete in their foundations than the former. They do not discuss the geometrical truths of which they jmake use, but they accept them as evident, relying on ipure geometry to establish them.
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