Inhaltsangabe
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 edition. Excerpt: ... MATHEMATICS AND METAGEOMETRY. DIFFERENT GEOMETRICAL SYSTEMS. STRAIGHTNESS, flatness, and rectangularity are qualities which cannot (like numbers) be determined in purely quantitative terms; but they are determined nevertheless by the conditions under which our constructions must be made. A right angle is not an arbitrary amount of ninety degrees, but a quarter of a circle, and even the nature of angles and degrees is not derivable either from arithmetic or from pure reason. They are not purely quantitative magnitudes. They contain a qualitative element which cannot be expressed in numbers alone. A plane is not zero, but a zero of curvature in a boundary between two solids; and its qualitative element is determined, as Kant would express it, by Anschauung, or as we prefer to say, by pure motility, i. e., it belongs to the domain of the o priori of doing. For Kant's term Anschauung has the disadvantage of suggesting the passive sense denoted by the word "contemplation," while it is important to bear in mind that the thinking subject by its own activities creates the conditions that determine the qualities above mentioned. Our method of creating by construction the straight line, the plane, and the right angle, does not exclude the possibility of other methods of space-measurement, the standards of which would not be even boundaries, such as straight lines, but lines possessed of either a positive curvature like the sphere or a negative curvature rendering their surface pseudo-spherical. Spheres are well known and do not stand in need of description. Their curvature which is positive is determined by the reciprocal of their radius. B Fig. 1. Pseudo-spheres are surfaces of negative curvature, and pseudo-spherical surfaces are saddleshaped....
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