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. 1853 Excerpt: ...angles at S of the tetrahedral angle S. ABCD are together less than four right angles. Generally. The pentrahedral angle is reduced in the same manner to the tetrahedral, and its plane angles s'hown to be less; and hence less than those of the tetrahedral angle, and thence again less than four right angles. PROPOSITION IV. If from any point within a trihedral angle, perpendiculars be drawn to the faces, these perpendiculars will be the edges rf a new trihedral angle, which has the following relations to the original one: --(1.) Its edges will be perpendicular to the faces of the original; (2.) Its faces will be perpendicular to the edges of the original; (3.) Its plane angles will be the supplements of the opposite profile angles of the original; and (4.) Its profile angles will be the supplements of the plane angles of the original. (1.) This is but a repetition of the hypothesis, for the purpose of enumerating the connected properties seriatim. (2.) Let S. ABC be the original trihedral angle; and from any point s within it let perpendiculars sa, sb, sc to the faces BSC, CSA, ASB be drawn: the planes bsc, csa, asb will be perpendicular to the edges SA, SB, SC, respectively. For let the plane bsc cut the planes BSA, ASC in c A, b A. Then since sc is perpendicular to the plane BSA, the plane bsc through it is perpendicular to BSA (Prop. xvi., Chap. nX In like manner the plane bsc through bs is perpendicular to the plane ASC. Wherefore the plane bsc, being V ?f.2' Ihe angle CDA is equal to two right angles: and in Fig. S, the angle CDA is the reverse angle. perpendicular to both the planes BSA, ASC, is perpendicular to SA their common intersection Prop. xvni., Chap. n.). The two perpendiculars sb, sc determine, therefore, a plane perpendicular to the edge..