Elements of Non-Euclidean Geometry - Tapa blanda

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. 1919 edition. Excerpt: ...ordinary spherical trigonometry. The measurement of angle, plane or dihedral, is the same in all three kinds of space, and spherical trigonometry involves only angular measurement. This explains why spherical trigonometry is the same in all three geome 23. The trirectangular quadrilateral. As in hyperbolic geometry, there is a correspondence between a right-angled triangle and a trirectangular quadrilateral. In fact tries. we see that, by producing two opposite sides of the quadrilateral to meet, we get, corresponding to the trirectangular quadrilateral Camlb, a right-angled triangle with hypotenuse 7r&-6=6, sides o and irk m = m. and the opposite angles l and It-C. If we write the parts 7T a ir m i! 6 2+ ' I' 2k' 2'k' h in cyclio order, then we have the rules: sine (middle part) = product of cosines of opposite parts = product of tangents of adjacent parts. EXAMPLES m. 1. Prove that the bisectors of-the vertical angle of a triangle divide the base into segments whose sines are in the ratio of the sines of the sides. 2. Prove that the arc of an equidistant-curve of distance a, corresponding to a segment x on its axis, is x cos a/k. 3. Prove that the area of a circle of radius r is 4rfsin2--2k 4. Prove that the area included between an arc of an equidistantcurve of distance a, its axis, and two ordinates at distance x, is kx sin-. k 5. Prove that the area of the whole plane is 2jt 2, and the volume of the whole of space is 7r2&3. 6. Prove that the volume of a sphere of radius r is "K?-sin?) (In the following examples k is unity.) 7. If R is the radius of the circurosphere of a regular tetrahedron whose side is a, show that sin a = /ft sin R. 8. If 2d is the distance between opposite edges of a cube of edge 2a, 2h the...