Excerpt from The Elements of Analytic Geometry
Analytic Geometry is a broader subject than Conic Sections. It is far more important to the student that he should acquire a familiarity with the analytic method, and thoroughly grasp the generality of its processes and the comprehensiveness of its results, than that he should obtain a detailed knowledge of any particular set of curves. Furthermore, all branches of mathematics are fundamentally and inseparably related. Any subject, therefore, should be presented in such a way as to keep it in touch with all that has preceded, and at the same time reach forward toward that which is immediately to follow, to the end that there may be no sudden transition in passing from one branch to another. Algebra and Geometry, Analytics and Calculus are mutually helpful, and should not be studied entirely apart. No one of these subjects can be finished before the others are begun.
The general plan and scope of this book is due to a firm conviction of the soundness of these statements. For this reason a fuller treatment than usual is given of the general analytic method before taking up the study of the conic sections, and subjects have been introduced not ordinarily treated in text books on Analytic Geometry. The method of the differential calculus is the only way of studying the slope of curves, and furnishes the best means of finding the equation of the tangent and the normal. The graphical method of illustration and the derivative are indispensable in the discussion of the Theory of Equations. The use of the derivative curve in the theory of equal roots, together with the fact that the ordinate of the "derivative curve" is the slope of the "integral curve," naturally suggests a possible converse relation, and leads easily and logically to the study of Quadrature, and Maxima and Minima.
It is believed that the elementary discussion of these subjects here given will tend to meet the needs of scientific and engineering students, who now require a knowledge of the graphic method and the simple elements of the calculus at the earliest possible moment; and that it will also be helpful to the general student who pursues the study of the subject no further.
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