Elements of Analytical Geometry

From the PREFACE: NO branch of pure Mathematics presents more to interest and improve the mind of the mathematical student, than Analytical Geometry. Uniting the clearness of the geometrical reasoning, with the brevity and generality of the algebraic, it not only satisfies the requirements of the closest reasoner, but gives continued and increasing pleasure, by the elegance with which its varied results are deduced and interpreted. In preparing this treatise the Author has endeavoured to preserve the true spirit of Analysis, as developed by the celebrated French mathematician, Biot, in his admirable work on the same subject, while he has made such changes, both in the arrangement of the matter and the methods of demonstration, as he believed would render the whole more attractive, and easily acquired by any student possessing a knowledge of the elementary principles of Algebra and Geometry. In discussing the Conic sections he has preferred to consider the Parabola first, not only for the reason that the properties of this curve are more simple and more easily deduced than those of the others, but because, by this course, he was enabled to treat of the Ellipse and Hyperbola together, thus avoiding much of the repetition of words, which necessarily arises from their separate discussion. Although the treatise has been prepared with special reference to the wants of the Author's own classes at the Military Academy, he trusts that it will be found acceptable and useful to all, who are disposed to advance in the higher branches of Analysis. Those who desire to make the subject as practical, as may be, will find in the last article of the work a large number of examples. As to the wisdom of departing from the usual custom of omitting decimal points entirely from logarithm tables, the author believes that the retention of the point promotes clearness of comprehension of the tables by beginners, and lessens mental effort in more experienced computers, especially when associated with the notation by powers of 10, as in the explanations here given. It seems unfortunate that this simple notation, so useful in computation and so great an aid in the explanation of numerical relations, is not universally incorporated into arithmetical instruction. The rules for the employment of logarithms and of the tables have not been prepared especially to meet the need of those entirely unfamiliar with the principles of logarithms, although they would probably be intelligible to any mature beginner. It is thought, however, that the explanations and instructions given may prove an aid even to those who are already somewhat familiar with the subject.