This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1916 Excerpt: ...angles y; and it can be constructed without that assumption. The associated triangle gives us the second side b of the required triangle. This argument depends upon the theorem proved in 41-3, that we can always find H(p) when p is given, and that proved in 45, that given IT(p), we can always find p. 37. Proper and ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1916 Excerpt: ...angles y; and it can be constructed without that assumption. The associated triangle gives us the second side b of the required triangle. This argument depends upon the theorem proved in 41-3, that we can always find H(p) when p is given, and that proved in 45, that given IT(p), we can always find p. 37. Proper and Improper Points. In the Euclidean Plane two lines either intersect or are parallel. If we speak of two parallels as intersecting at "a point at infinity and assign to every straight line "a point at infinity," so that the plane is completed by the introduction of these fictitious or improper points, we can assert that any two given straight lines in the plane intersect each other. On this understanding we have two kinds of pencils of straight lines in the Euclidean Plane: the ordinary pencil whose vertex is a proper point, and the set of parallels to any given straight line, a pencil of lines whose vertex is an improper point. Also, in this Non-Euclidean Geometry, there are advantages to be gained by introducing fictitious points in the plane. If two coplanar straight lines are given they belong to one of three classes. They may intersect in the ordinary sense; they may be parallel; or they may be not-intersecting lines with a common perpendicular. Corresponding to the second and third classes we introduce two kinds of fictitious or improper points. Two parallel lines are said to intersect at a point at infinity. And every straight line will have two points at infinity, one corresponding to each direction of parallelism. All the lines parallel to a given line in the same sense will thus have a common point--a point at infinity on the line. Two not-intersecting lines have a common perpendicular. The lines ar...
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