The Foundations of Geometry

Cover The Foundations of Geometry
The Foundations of Geometry
Edward T Edward Travers Dixon
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If not AC intersects the r. F. In a single point Q, and the s. 1. PQ is in both p. And r. F. And they cannot intersect in any point outside PQ (HI. 2).
VI. Two parallel r. F s. Cannot intersect; and any two which are not parallel intersect in a p.
For if ABCD, EFGH be two r. F s. Which are not parallel, and A be any point in the first which is not in the second, then EFGH is not parallel to all of AB, AC, AD. Therefore it intersects at least one of them 96 PART III.
in a single point, P (III. 4
...). And EFGH is not parallel to either of the p s. ABC, ABD. Therefore it intersects each of them in a s. 1. PQ, PR (III. 5). Therefore also the p. PQR is in both r. F s. And they cannot intersect in any point outside this p. (III. 2).
VII. Any two p s. Extending in four i. D s. Intersect in a single point.
For if A BC, DEF be two p s. Which extend in four i. D s. , and if A be any point in the first which is not in the second, then the direction AD is not independent of the four i. D s. AB, AC, DE, DF (P).


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