Geometry, Plane, Solid, and Spherical. to Which Is Added, in an Appendix, the Theory of Projection [&C. by P. Morton]. (Paperback)

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. 1830 Excerpt: ...If one plane be perpendicular to another, any straight line which is drawn in the first plane at right angles to their common section shall be perpendicular to the other plane: and, conversely, if a straight line be perpendicular to aplane, any plane which passes through it shall be perpendicular to the same plane. It being: provided also, that Die parallel planes lie, each pair, upon the same side, or each pair upon opposite sides of the plane which passes through the common sections (A B, C D in the figure of 12. Cor.); for, if one pair lie towards the same parts, and the other pair towards opposite parts of that plane, the dihedral angles will be supplementary, not equal, to one another. See tie note at Prop. 15. Let the plane ABC be perpendicular to DBC, and let any straight line A B be drawn in the plane ABCperpendicular to the common section B C: the straight line AB shall be perpendicular to the plane DBC. From the point B, in the plane DB C, let BD be drawn at right angles to BC. Then, because the planes are at right angles to one another, the angle A B D is a right angle (17. Cor.): but ABC is likewise a right angle; therefore (3.) A B is perpendicular to the plane DBC. Next, let the straight line AB be perpendicular to the plane BCD; and let A B C be any plane passing through AB: the plane ABC shall be perpendicular to B C D. Let BC be the common section of the two planes; and, from the point B, in the plane BCD, draw BD at right angles to BC. Then, because AB meets the line BD drawn in the plane to which A B is perpendicular, the angle ABD is a right angle (def. 1.). But the angle AB D is contained by straight lines drawn in the two planes perpendicular to their common section B C. Therefore (17. Cor.) the dihedral angle A B C D is likewise a rig...