8324 PREFACE. 513 In the preparation of this work, the Author's preus treatise. "Elements of Geometry, Plane and Spherical Trigonometry, and Conic Sections," has formed the ground-work of construction. But in adapting the work to the present advanced state of Mathematical ducation in our best Institutions, it was found necessary to so alter the plan, and the arrangement of subjects, as to make this essentially a new work. The demonstrations of propositions have undergone radical changes, many new propositions have been introduced, and the number of Practical Problems greatly increased, so that the work is now believed to be as full and complete as could be desired in an elementary treatise. In view of the fact that the Seventh Book is so much larger than the others, it may be asked why it is not divided into two? We answer, that classifications and divisions are based upon differences, and that the differences seized upon for this purpose must be determined by the nature of the properties and relations we wish to investigate. There is such a close resemblance between the geometrical properties of the polyedrons and the round bodies, and the demonstrations relating to the former require such slight modifications to become applicable to the latter, that there seems no sufficient reason for separating into two Books that part of Geometry which treats of them. The subject of Spherical Geometry, which has been much extended in the present edition, is placed as before, as an introduction to Spherical Trigonometry. The propriety of this arrangement may be questioned by some; but it is believed that much of the difficulty which the student meets in mastering the propositions of Spherical Trigonometry, arises from the fact that he is not sufficiently familiar with the geometry of the surface of the sphere; and that, by having the propositions of Spherical Geometry fresh in his mind when he begins the study of Spherical Trigonometry, he will be as little embarrassed with it as with Plane Trigonometry. / (iii) Both author and teacher must yield to the demands of the age, and by a judicious combination of the abstract and the concrete, the theoretical and the practical, make the student feel that what he learns with perhaps painful effort at first, may be made available in important applications In teaching Geometry and Trigonometry, questions should be asked, extra problems given, and original demonstrations required when the proper occasions arise; but care should be taken that the pupil's powers are not over-tasked. By helping him through his difficulties in such a way that he shall be scarcely conscious of having received assistance, he will be encouraged to make new and greater efforts, and will finally acquire a fondness for a study that may have been highly repugnant to him in the beginning. A demonstration that is easily followed and comprehended by one, may be obscure and difficult to another; hence the advantage that will sometimes be gained by giving two or more demonstrations of the same proposition. When the student perceives that the same results may frequently be reached by processes entirely different, he will be stimulated to independent exertion, and in no respect can the teacher better exhibit his tact than in directing and encouraging such efforts. Instances will be found throughout the work in which the more important propositions are twice and three times demonstrated; and as the methods of demonstration are in each case quite different, it is believed that extra space has not been thus occupied unprofitably. Practical rules with applications will be found throughout the work, and in addition to these, there are in both the Geometry and the Trigonometry, full collections of carefully selected Practical Problems. These are given to exercise the powers and test the proficiency of the pupil, and when he has mastered the most or all of them. it is not likely that he will rest satisfied with present acquisition, but conscious of augmented strength and certain of reward, he will enter new fields of investigation. The Author has been aided, in the preparation of the present work, by J. F. Quinby, A. M., of the University of Rochester, N. Y., late Professor of Mathematics in the United States Military Academy at West Point. The thorough Scholarship, and long and successful experience of this gentleman in the class-room, eminently qualify him for such a task; and to him the public are indebted for much that is valuable, both in the matter and arrangement of this treatise. October, 1860. Practical Geometry,- Application of Algebra to Geometry, and |