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3

ERRATA.
In printing a work. on a scientific subject, entirely from manu.
script, perfect accuracy is not to be expected. Accordingly, some
errors will undoubtedly be discovered in this book; but it is believed
that the following are all, which will be likely to occasion perplexi-
ty. Page 60. Ans. ex. 10, for 328, read 378.-.p. 71, in ex. 4, for 4
men read 1 man. p. 151, line 13, for 9 read (im-line 15, for multiplier,
read multiple.- p. 229, in ex. 5, insert at 5 per cent.-P. 236, ex, 2,
Ans. should be $446,985---p. 237. Ans. by Mass. method should
be $11,370.47175.-p. 244, ex. 43, for 65, read 650.-- p. 249, line
6, for miles, read hours.--p. 250 1. 30, for 3, read 315.-p. 284, in
Ans. to ex. 1, for 143 read it.

In a small part of the impression, p. 82. 3d line in ex. 87, for 65
read 75.-p. 261, line 3, for products, read quotients.-p. 283, line
38, for 3 read 7, three times.

In & Lxxxix, the rule called the Massachusetts rule is erroneously
stated to have been established by law in that state. It has been
made the rule of the state of New York, by the decision of Chancel.
lor Kent. The statement was made on the authority of a number
of writers. For this correction, as well as for other important
suggestions, the author is indebted to Professor Dewey, now Prin.
cipal of the BERKSHIRE GYMNASIUM,

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It is not to be expected that any book, however perfect, should supersede the necessity of patient and laborious effort on the part of the teacher. The nar. row compass into which it is necessary to compress all the important particulars, embraced in a whole science, in order to form a text book, suited to the wants of schools, renders it impossible for the author to go into that minuteness of detail, or to avail himself of that fulness of illustration, which it is the province of the living teacher to supply. On this account, many of the explanations and rea. sonings, contained in the following treatise, though more full than those on the same subjects in the common books, can only be regarded as furnishing hints, which the teacher may follow out at greater length.

Pupils should, by all means, be arranged for instruction in CLASSES. By this means, not only is a spirit of emulation excited, which very soon interests the learner in his study, but the teacher is enabled to devote more time to the instruction of each individual, and to the illustration of every subject. Classes may consist of as many as twelve or sixteen pupils with advantage. In clearing up any difficulties which may arise, the instructor should always avoid direct explanations. The method which the author has pursued, and which he most strongly recommends to all teachers, is to lead the pupil to make the desired discovery himself, by means of questions adapted to the case. These should commence with the utmost simplicity, and approach the difficulty by degrees. By a skilful use of this method, the author has seen many a Gordian knot unravelled, and many a sorrowful countenance brightened into smiles. It will often be well, to ask questions to the whole class at once.

Some may, perhaps, be here led to inquire, why the interrogative plan is not adopted in the book. To this it is replied, that the advantage of the method consists in rendering questions so gradually progressive that the pupil may answer hem with ease and without asststance. A body of questions of this kind, em. bracing the whole science of arithmetic, would be too voluminous for use; and on the other hand, if the questions are made fewer, and answers furnished, no plan can be more calculated to produce pernicions effects on the mind of the learner. The pupil should never be allowed to know the forms of the questions to be used in examining him ; much less should he be furnished with the language of his answer.

With respect to the method of using this book, the author recommends that all young pupils, when taking up the subject for the first time, should study with attention the introductory exercises : afterwards the mental arithmetic alone, as far at least as fractions; then that they should perform some of the simplest examples in written arithmetic, under each of the ground rules : and finally, that they should begin anew and take the whole in order. Even in this perusal, however, they should omit the observations for advanced pupils, and the rules every where. These should be reserved for the final review, the latter being only intended to supply the pupil with accurate language, for the expression of his own ideas.

When examples are recited, the whole operation should be brought in on the slate, and parts of the process should be read by each pupil in turn. many examples, it will be sufficient to read a small part of each. By this means it will be rendered certain that every pupil understands the subject, and at the same time a great temptation to deceit in copying answers, will be removed. This plan renders a separate key unnecessary, and might even dispense with answers altogether. For when a difference is perceived between two calculations, a moment's comparison, at the point where the variation commenced, will determine which is right.

The author makes the above suggestions as the result of his own experience. It is impossible, in the narrow limits allowed him, to exhibit, in full, his views on the best mode of teaching the science, or of using the following treatise; but for experienced instructors this will be unnecessary, and for those, to whom the employment is new, the above observations will not be entirely useless.

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From D. OLMSTED, Professor of Mathematics and Natural Philosophy in

YALE COLLEGE. I have examined an Elementary Treatise on Arithmetic, by Mr. FREDERICK A. P. BARNARD, and can cheerfully recommend it to the attention of the public, particularly of Instructors, as a work of much merit, exhibiting an able and luminous view of the science of numbers.

DENISON OLMSTED. YALE COLLEGE, July 22, 1830.

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From C. DEWEY, Principal of the BERKSHIRE GYMNASIUM.

BERKSHIRE GYMNASIUM,

Pittsfield, July 23, 1830. S Mr. F. A. P. BARNARD.

Dear Sir-I have examined your " Arithmetic” with some attention, and cheerfully say, that it is, in my opinion, worthy of a high place among the improved systems of Arithmetic in our country. Though it cannot be, in many respects, an original work, it happily combines the improvements that have been made, and exhibits them in a clear and easy point of view: It developes the principles of the rules generally with much simplicity; the definitions are clear and concise, and the rules are stated, after the principles are explained, with brevity; many important practical abbreviations of calculations are introduced; mental exercises, as well as those for the slate, are employed in great abundance; and it is distinguished by a fulness in the subjects treated of, and in many interesting and important related ones, which belong to few works of the kind. The first part of the work, which is designed for the use of the quite young pupil, might well be published, after some enlargement, by itself. Without expressing un. qualified approbation of every part of the work, there can be no doubt that it is fitted to be extensively useful. With much esteem, &c.

C. DEWEY

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How many anchors do you see here? Ans. ONE. How many ropes are there, fastened to the anchor ?

Now there is a single anchor, and a single rope ; and each of these single things, you tell

me,

is ONE. Then, what is a single thing called ? Hence,

A SINGLE THING OF ANY KIND IS CALLED ONE THING, OR ONE. Also A SINGLE THING OF ANY KIND IS OFTEN CALLED A UNIT, OR UNITY.

Thus, the anchor in the picture above is a unit.

If there were another anchor in the picture, how many anchors would there be ? Ans. two.

How many sheaves of wheat are there here?

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