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Both author and teacher must yield to the demands of the age, arid 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 thuB 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 conscioua of augmented strength and certain of reward, he will enter new fields cf 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 sucli a task; and to him the public are indebted for much that is valuable, both in the matter and arrangemen'1 of this treatise,

October. 18G0.

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