Elements of Geometry, Tr. by J. Kaines

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: ...which it is more useful to consider than any of the others. If we conceive that a square or rectangle Ahcii is moved parallel to itself in such a way that its four angles A, B, G, and H, traverse each of the four lines A D, B C, G F, and He, perpendiculars to the plane of the rectangle Abgh, that rectangle by the. movement now described, will form the parallelopiped aBcdEFgH. It is hardly necessary to state that by a line perpendicular to a plane we understand a line which does not incline towards any side of that plane; and, in like manner, a plane which does not incline more to one side than to another upon a second plane is said to be perpendicular to that second plane. These two definitions are analogous to that given of a line perpendicular to another line. It follows from this that the line a B, which is perpendicular to the plane x, must be perpendicular to every line a c, a D, A E, kc., in that plane drawn from the foot a of the perpendicular; for it is obvious that if it inclined towards any one of these lines, it would be inclined towards some part of the plane, therefore it would not be perpendicular to it. In order to represent clearly to the mind how the line A B can be perpendicular to all the lines which extend from its extremity A, a figure may be made in relief in the following manner. Making in some flat flexible substance, like cardboard, a rectangle F G D E divided into two equal parts by the line A B, perpen Fig. 125. dicular to the sides Ed and Fg, we can fold that rectangle along the line a B, and place it so folded upon the plane x. It is obvious that, however much or little the two parts F B a E and G B A D of the folded rectangle E A D G B F may I opened apart, those two parts will always be in contact with the plane x, without the...