Analytic Geometry (Classic Reprint)

Excerpt from Analytic GeometryIn accordance with the general plan of this series of textbooks, the authors of the present volume have had constantly In mind the needs of the student who takes his mathematics primarily with a view to its applications as well as the needs of the student who pursues mathematics as an element of his education.The processes of analytical geometry find their application, for the most part, in the scientific laboratory where it is often necessary to study the properties of a function from certain observed values. The fundamental concept is, therefore, that of functional correspondence and the methods of representing such correspondence geometrically. For this reason rather more than usual attention has been given to these subjects (Chapter III; also Chapter IX, Arts. 135 to 140).An intelligent appreciation of functional correspondence requires an intimate knowledge of the relation between an equation and the graphical representation of the functional correspondence determined by the equation. Such a knowledge is most easily obtained by a study of linear equations and equations of the second degree together with their corresponding loci. This knowledge is not only of importance to the student of applied mathematics, but it has a special disciplinary value for the general student.The standard forms of the equations of a number of important loci are developed early (Chapter IV), and the properties of these loci are discussed in detail later (Chapters VI and VII) by means of the equations already at hand. By this arrangement, it is hoped that some unnecessary repetition has been avoided.The equations of tangents to the conic sections have been derived by means of the discriminant of the quadratic equation whose roots are the 3;-coordinates of the points of intersection with a variable secant, rather than by means of the derivative.