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perspicuity, and precision being remarkable throughout. Notwithstanding
the late improvements in such works, I should not be surprised should
this production soon become the general favorite.
New York, May 4, 1846.

V. DREW.

Extract from Rev. Samuel J. May, late Principal of State

Normal School, Lexington, Mass. JAMES B. THOMSON, Esq. -Dear Sir :- I have given some time to the examination of your "Practical Arithmetic;" and am happy in being able to speak of it in terms of high commendation. The plan is excellent, and the execution of the plan, so far as I have examined, is thorough. It is the best offspring, I have yet seen, of the improved method of teaching Arith. metic. Wishing you an ample reward for your labors, in the preparation of books, to assist in the instruction of the young,

I remain your sincere friend, Syracuse, May 6, 1846.

SAMUEL J. MAY.

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From L. W. HALL, A. M. Author of Moral Philosophy, and

Principal of Ravenna High School, Ohio. J. B. THOM30N, ESQ.-Dear Sir, I have derived much satisfaction from an examination of your "Practical Arithmetic," and my appreciation of its merits will be best expressed by the statement of the fact that I intend to introduce it into my school,at the earliest practical moment. Its thorough. ly practical character is what gives it great value in my estimation, and in this respect it greatly excels any Arithmetic with which I am acquainted. Ravenna, Ohio, May 19, 1846.

L. W, HALL.

From Rev. Geo.W. THOM30N, Principal of Onondaga Academy,

New-York. MR. THOMSON-Dear Sir From the examination I have been able to give your " Practical Arithmetic,” I consider it a valuable addition to that class of books. It contains many improvements and items which are not found in other Arithmetics. Your edition of “Day's Algebra," I am also very much pleased with. I shall introduce both of them into my school as soon as possible.

Very respectfully your obedient servant, Onondaga Academy, May 25, 1846.

GEO, W. THOMSON.

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From E. A. Smith, Esq. Cor. Secretary of Queens Co. Teaeh

ers Association, N.Y. MR. THOMSON-Dear Sir-I recently received from your publishers a copy of the "Practical Arithmetic', of which you are the author. Permit me to express my gratification at the internal evidence of merit which it exhibits. Thelucid arrangementexceeds any thing of the kind that I have hitherto seen; the progression from simple easy examples to those which are more complex, and especially the illustrations of the different rules by examples drawn from the every day occurrences of life, are all exceedingly well adapted to impart a real knowledge of the science of Arithmetic, and are altogether preferable to the mechanical sameness, which has so long prevailed, and which will continue, until the mere abstract expositions are laid aside. The juvenile mind needs tangible objects, familiar examples, which may be grasped without difficulty, and to which the process of reasoning may be successfully applied. Finally, I hope your book may come into genral use; and I am satisfied that to be appreciated it has only to be seen and exam. ined.

Very truly Yours, Hemstead, Jan. 31, 1846.

ELIJAH A. SMITH.

MENTAL

ARITHMETIC,

OR

FIRST LESSONS IN NUMBERS.

FOR CHILDREN.

By JAMES B. THOMSON, A. M.
AUTHOR OF PRACTICAL ARITHMETIC, EDITOR OF DAY'S SCHOOL ALGEBRA,

LEGENDRE'S GEOMETRY, &C.

AUBURN:

J. C. DERBY & CO.;
GH. DERBY & CO., GENEVA, N.Y.

DURRIE & PECK, NEW-HAVEN ;
LOOMIS & PECK, PHILADELPHIA.

* 1846.

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Entered according to Act of Congress, in the year 1846,

BY DURRIE & PECK,
in the Clerk's Office of the District Court of Connecticut.

H. OLIPHANT, PRINTER,

Auburn, N.Y.

PREFACE.

At what precise age a child should begin to go to school, or commence the different studies, it is not our province to decide. Whatever may be the diversity of opinion on this point, all practical teachers seem to agree, that Mental Arithmetic is among the first exercises, which should be presented to the youthful mind. The correctness of this sentiment is corroborated by the ease with which children understand simple combinations of numbers, their fondness for these exercises, and the obvious advantages which may be derived from them. But in order to become interesting or profitable, it is manifest, this branch, as well as others, must be taught in such a manner that the pupil shall understand it. The examples therefore must, at first, be simple, containing small numbers, and have reference to sensible objects with which the learner is acquainted ; the transition from easy to more difficult questions must be very gradual; and the reason for every step in the solution distinctly seen.

It is believed that much dislike for the study of Arithmetic, and much unnecessary discouragement, have been occasioned by the abruptness of the transitions from easy to difficult questions. It is too often forgotten that the powers of the child's mind, like those of his body, are feeble ; that while familiar mental exercises which he can comprehend, afford him the highest delight, he turns from intricate questions, which he does not and cannot understand, with indifference and disgust.

It is the design of this little work to furnish a series of mental exercises in numbers, adapted to the wants and ca, pacities of children. It commences with practical examples which relate to familiar objects and require the simplest combinations. The pupil is then introduced to others involving the same principle but somewhat harder, special care being taken to make the transition very gradual, so that instead of being disheartened at the ruggedness of the way, he shall be stimulated to take the next step by the hope of victory.

From the fact that children comprehend and remember words, more easily than figures, and reason upon them with 80 much greater facility, the numbers and Tables in the first part of the book, are expressed in words.

After the pupil has become practically acquainted with the principles of a rule and is able to solve questions under it with facility, the operation is then defined, and its more prominent terms are briefly explained. This, it is bēlieved, teachers will be glad to see. There is no reason why a child should not be informed, that a certain operation upon numbers is called Addition; another, Subtraction ; &c., as well as to be told that a certain operation of his vocal organs in connection with those of his mind, is called reading; another, singing, &c.

With this brief explanation of the object and plan of the work, the author commends it to the friends of education by whom his former efforts to subserve this noble cause, have been so favorably received.

'Although designed particularly as an introduction to the “ Practical Arithmetic," it may be used as a preparatory work to any of the larger systems of Arithmetic now be. fore the public.

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