About the Book
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. 1881 edition. Excerpt: ...it met any Air J-b one of these lines lying in the plane Mr--WV J4--v MN, it would meet that plane, which is o' i impossible by the hypothesis (Def. 4). B ----H Also, the lines CD, EF, GH, etc., being parallel to AB, are parallel to' s each other (Prop. XI., Cor. 2). Cor. 1. If a line, AB, is parallel to a plane, MN, through any point, C, of this plane, a line may be drawn parallel to AB in the plane. For, if through the point C and the line AB we pass a plane, the intersection, CD, of this plane with MN will be parallel to AB. Conversely, when a line, AB, is parallel to a plane, MN, a parallel to AB through any point, C, of MN lies in that plane; otherwise there would be two parallels through the same point, C, to the same straight line, AB. Cor. 2. A straight line, AB, parallel to two planes, MN and PQ, which intersect each other, is parallel to their line of intersection. For, the parallel to AB through any point, C, of their intersection, must lie j--y- in both planes. PROPOSITION XIII. Theorem. If a straight line, AB, is parallel to a straight line, CD, drawn in the plane MN, it will be parallel to that plane. For, if the line AB, which is in the plane ABCD, could meet the plane MN, this could only be in A B some point of CD, the common / 7 intersection of the two planes; M I. I but AB cannot meet CD, since it V 7 7 is parallel to it; hence it will not ft J meet the plane MN. Therefore (Def. 4) it is parallel to that x plane. Cor. 1. A straight line, AB, and a plane, MN, perpendicular to the same straight line, OP, are parallel. For the plane BAP intersects the plane MN in a line, PC, perpendicular to OP, and therefore parallel to AB. Hence, AB is parallel to MN. Cor. 2. Through a point, A, without a plane, MN, any number of lines...
