Synopsis
This historic book may have numerous typos and missing text. Purchasers can usually download a * scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1922 edition. Excerpt: ...would make Pappus' theorem inevitable. Though we regard this w as, finally, inadequate, we follow the usual procedure, of allowing the suggestion of a simpler case to become a starting point of the theory of the more general case. We proceed, then, to examine the fundamental conceptions in a more concrete way than has hitherto been done. Logically, this examination should precede the foregoing formulation; but it will readily be seen that the examination, in proportion as it is complete, is prolix. Moreover, it will appear that the real geometry, which is built up on the basis of this examination, may be regarded as included in the geometry so far dealt with. Frankly recognising, now, the possibility of inaccessible points, we can assume no general proposition that every two lines in a plane intersect one another, or that any two planes intersect one another; and, requiring, now, that all constructions made shall utilise only accessible points, we need assumptions as to what points are accessible when others are known to be so. These we obtain, for points of a line, by recognising the separation effected by the assignment of a point of the line; this leads to the notion of the points of the line which are between two given points, all of which are regarded as being equally accessible. A similar notion dominates our recognition of the points of a plane which are to be regarded as accessible when three points are given as being so; and similarly for the points of (three dimensioned) space when four points are given. Betweenness for points of a line. Segments. We assume that when two points, A and are given, there exists a Q B B' point, C, which is between A and B; and then, also, a point, B', such that B is between A and B'; and also a...
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