Publisher's Synopsis
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1914 edition. Excerpt: ... CHAPTER II PERPENDICULARITY AND SIMPLE ANGLES 36. Introductory. From this point the theorems of ordinary geometry presupposed. We shall now take up perpendicularity and various kinds of angles in very much the same way that these subjects are taken up in the text-books. We shall find the relation of the perpendicular line and hyperplane analogous to the relation of the perpendicular line and plane of three-dimensional geometry, but a new kind of perpendicularity in the case of two planes absolutely perpendicular in hyperspace, every line of one plane through their point of intersection being perpendicular to every line of the other through this point. The relation of perpendicular planes in a hyperplane now takes a very simple form, as that of a plane intersecting in a line each of two absolutely perpendicular planes. This chapter will conclude with a treatment of perpendicular planes and hyperplanes, and of hyperplane angles, which are analogous to dihedral angles. Chap. Ill will take up what may be called two-dimensional angles, and angles which correspond to polyhedral angles; and chap. IV will consider questions of order, symmetry, and motion. After finishing these chapters we shall return to the polyhedroids and other hypersurfaces already introduced. As indicated at the beginning of the preceding section (Art. 29), the three chapters which follow do not depend on that section. Indeed, only a few simple facts are needed from the entire first chapter besides what are already familiar to the student. These facts are easily stated and understood, and a study of their application to the theorems which follow will help us to realize their truth. Except for the properties of points on a line referred to in Art. 5, the development of most...
