Euclidean Geometry Essays

To those who decided to read the carefully and see how the crucial terms are used, it became apparent that the account is both remarkably scrupulous in some ways and flawed in others.Straight lines arise almost always as finite segments that can be indefinitely extended, but, as many commentators noted, although Euclid stated that there is a segment joining any two points he did not explicitly say that this segment is unique.

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(See the many comments collected in Heath’s edition of Euclid’s are about straight lines and circles, but it is well known that the concept of a straight line receives only a most unsatisfactory definition.

The nature of the relation between the abstract geometry and its practical expression has also to be considered.

This essay considers various theories of geometry, their grounds for intelligibility, for validity, and for physical interpretability in the period largely before the advent of the theories of special and general relativity in the 20 century itself saw a profusion of new geometries, of which the most important were projective geometry and non-Euclidean or hyperbolic geometry.

There were also significant advances in the domain of abstract geometries, such as those proposed by David Hilbert.

It follows that the terms ‘geometry’ and ‘physical space’ do not have simple meanings in the 19 century.


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