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performing the operation; as the proof may render simple and clear some part or the whole of an operation that was before complicated and obscure.
If answers are placed in the back part of the book, the pupil will at once refer to them whenever he is in any doubt or difficulty in performing an operation. Hence the object aimed at is not accomplished by placing the answers together in this manner.
Again, if all the answers are omitted, the pupil may become involved in doubt and uncertainty, and acquire a distaste for the study; and from this discouragement, subsequently make but limited advancement in Mathematical Science.
In order, therefore, that pupils may receive the advantages of both methods, the answers to nearly one half of the examples in this book are omitted. They will be found, together with full and clear solutions of all the examples, in a Key to this work, which has been prepared for the use of teachers and private learners.
Many valuable hints and suggestions which have been received from teachers and friends of education, have been incorporated into this work. The author desires to make especial acknowledgment of the valuable services. rendered in the preparation of this work by Mr. D. W. Fish, of Rochester, N. Y., and J. H. French, LL. D., of Syracuse, N. Y. These gentlemen have had long and successful experience as teachers, and an intimate acquaintance with the plans and operations of some of the best schools in the country. It is due to them to state that they are in fact the authors of the Practical and Higher Arithmetics of this series.
August 1, 1860.