The Elements of Analytic Geometry

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. 1900 Excerpt: ...the section APB in the line ES. Pass the plane VMN through VO perpendicular to the plane APB, meeting it in the line AB, meeting the plane HKR in HK, and the line ES in D; then the plane VMN is also perpendicular to the plane HKR, and therefore perpendicular to ES. Let P be any point on the section. After studying the straight line and the circle, the old Greek mathematicians turned their attention to the conic sections, and by investigating them as sections of a cone soon discovered many of their characteristic properties. The most important of these discoveries were probably made by Archimedes and Apollonius, as the latter wrote a treatise on conic sections about 200 B. C. These curves are worthy of careful study, not only on account of their historic interest, but also on account of their importance In the physical sciences and their frequent occurrence in the experiences of everyday life. For example, the orbit of a heavenly body Is a conic section. For this reason they were thoroughly studied by the astronomer, Kepler, about 1600 A. D. The path of a projectile is a parabola. The law of falling bodies, the pressure-volume law of gases, the law of moments in uniformly loaded beams, all give conic sections. The bounding line of a beam of uniform strength, the oblique section of a stove-pipe, the shadow of a circle, the apparent line dividing the dark and light parts of the moon, etc., are all conic sections. The reflectors in head-lights and searchlights are parabolic. Draw PF, and the element PF which will be tangent to the sphere at R. Through P draw a line perpendicular to the plane HKR, which will meet CR produced in Q; and through PQ pass a plane perpendicular to ES meeting it in S. Let /3 = Z PBQ = Z AHD, the complement of the semi-vertical angle of...