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 Excerpt: ...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 x 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 M/ /a in both planes. / A J 7 V 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 some point of CD, the common intersection of the two planes; M but AB cannot meet CD, since it is parallel to it; hence it will not meet the plane MN. Therefore (Def. 4) it is parallel to that plane. Cor. 1. A straight line, AB, and a plane, MN, perpendicular to the same straight line, OP, are parallel. o For the plane BAP intersects the plane MN in a line, PC, perpendicu lar to OP, and therefore parallel to AB. Hence, AB is parallel to MN. N. w Cor. 2. Through a point, A, without a plane, MN, any number of lines may be drawn parallel to that plane. Cor. 3. If two intersecting planes, AF and CF, contain two parallels, AB and CD, their common intersection, EF, will be parallel to these lines. For, the straight line AB, being parallel to CD, must be parallel to the plane CF, and being parallel to this- B plane it must be parallel to ...
