The Axioms of Projective Geometry (Classic Reprint)

Excerpt from The Axioms of Projective Geometry

In this tract only the outlines of the subject are dealt with. Accordingly I have endeavoured to avoid reasoning dependent upon the mere wording and on the exact forms of the axioms (which can be indefinitely varied), and have concentrated attention upon certain questions which demand consideration however the axioms are phrased.

Every group of the axioms is designed to secure the deduction of a certain group of properties. For the most part I have stated without proof the leading immediate consequences of the various groups. Also I have ignored most of the independence theorems, as being dependent upon mere questions of phrasing, and have only investigated those which appear to me to embody the essence of the subject; though, as far as I know, no formal line can be drawn between these two classes of theorems.

But there is one group of deductions which cannot be ignored in any consideration of the principles of Projective Geometry. I refer to the theorems, by which it is proved that numerical coordinates, with the usual properties, can be defined without the introduction of distance as a fundamental idea. The establishment of this result is one of the triumphs of modem mathematical thought. It has been achieved by the development of one of the many brilliant geometrical conceptions which we owe to the genius of von Staudt.

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Alfred North Whitehead (1861 1947) was a prominent logician, mathematician and philosopher in the early twentieth century and helped pioneer the 'process philosophy' approach to metaphysics. He was Professor of Philosophy at Harvard University until his retirement in 1937.