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As the time of many pupils will not permit them to pursue this study through all of its departments, the work is issued in two parts, as well as in a single volume. This will, it is thought, be also convenient for graded schools, in supplying a separate book for classes of the higher and lower grades respectively, without requiring any unnecessary repetition or review. In this, the SECOND PART, all the higher departments of arithmetic including Mensuration are presented, commencing with Percentage, the study of which can be taken up by the pupil immediately on completing the FIRST PART. This part of the subject has been treated in a comprehensive manner, and is, in all respects, adapted to the wants of the present time, recognizing and explaining all the recent changes in Custom-house Business, Exchange, etc. An Appendix of forty-eight pages of valuable Tables and Problems has been added to this part of the work, containing much useful and practical information, fresh and important, obtained by much labor of research and inquiry, which, with many other improvements, particularly adapt this work to the wants of the student qualifying for business, and of graduating classes in High Schools and Academies, as well as of Mercantile and Commercial colleges. The Reviews interspersed throughout the book will be found to be just what is needed by the student to make his progress sure at each step, and to give him comprehensive ideas of the subject as he advances. Carefully constructed Synopses have also been inserted, with the view to afford to both teacher and pupil a ready means of drill and ea amination, as well as to present, in a clear, concise, and logical manner, the relations of all the different departments of the subject, with their respective sub-topics, definitions, principles, and rules. It is confidently believed that, on examination, the work as a whole, as well as in its separate parts, will commend itself to teachers and others, by the careful grading of its topics ; the clearness and conciseness of its definitions and rules; its improved methods of analysis and operation; the great number and variety of its examples, both oral and written, embodying and elucidating all the ordinary business transactions; and in the omission of all obsolete

terms and discarded usages, as well as in the introduction of many novel features favorable both to clearness and brevity.

Great nains have also been taken to make this work superior to all others in its typographical arrangement and finish, and in the general tastefulness of its mechanical execution.

The author takes pleasure in acknowledging his indebtedness for many valuable suggestions received from teachers of experience and others interested in the work of education ; especially to Joseph Ficklin, Ph. D., Professor of Mathematics in the University of Missouri, by whom chiefly the sections upon Involution, Evolution, Progressions, and Annuities have been prepared ; as well as to Henry Kiddle, A. M., Superintendent of Schools in the city of New York, for valuable assistance, especially in the higher departments of Percentage, and for important suggestions in relation to other parts of the work.

D. W. F. BROOKLYN, January, 1875.

IN order to teach any subject with the best success, the instructor should not only fully understand it, in all its principles and details, but should also clearly perceive what particular faculties of the mind are concerned in its acquisition and use. Arithmetic is pre-eminently a subject of practical value; that is, it is one to be constantly applied to the practical affairs of life. But this is true only in a limited sense. Very few ever need to apply to any of the purposes of business more than a small part of the principles and rules of calculation taught in the text-books. Every branch of business has its own requirements in this respect, and these are all confined within very narrow limits. The teaching of arithmetic must, therefore, to a great extent, be considered as disciplinary, as training and developing certain faculties of the mind, and thus enabling it to perform its functions with accuracy and dispatch. The following suggestions, having reference to this twofold object of arithmetical instruction are presented to the teacher, as a partial guide, not only in the use of this text-book, but in the treatment of the subject as a branch of education. Seek to cultivate in the pupil the habit of self-reliance. Avoid doing for him anything which, either with or without assistance, he should be able to do for himself. Encourage and stimulate his exertions, but do not supersede them. Never permit him to accept any statement as true which he does not understand. Let him learn not by authority but by demonstration addressed to his own intelligence. Encourage him to ask questions and to interpose objections. Thus he will acquire that most important of all mental habits, that of thinking for himself.

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Carefully discriminate, in the instruction and exercises, as to which faculty is addressed,—whether that of analysis or reasoning, or that of calculation. Each of these requires peculiar culture, and each has its appropriate period of development. In the first stage of arithmetical instruction, calculation should be chiefly addressed, and analysis or reasoning employed only after some progress has been made, and then very slowly and progressively. A young child will perform many operations in calculation which are far beyond its powers of analysis to explain thoroughly. In the exercise of the calculating faculty, the examples should be rapidly performed, without pause for explanation or analysis; and they should have very great variety, and be carefully arranged so as to advance from the simple and rudimental to the complicated and difficult. In the exercise of the analytic faculty, great care should be taken that the processes do not degenerate into the mere repetition of formula. These forms of expression should be as simple and concise as possible, and should be, as far as practicable, expressed in the pupil's own language. Certain necessary points being attended to, the precise form of expression is of no more consequence than any particular letters or diagrams in the demonstration of geometrical theorems. Of course, the teacher should carefully criticise the logic or reasoning, not so as to discourage, but still insisting upon perfect accuracy from the first. The oral or mental arithmetic should go hand in hand with the written. The pupil should be made to perceive that, except for the difficulty in retaining long processes in the mind, all arithmetic ought to be oral, and that the slate is only to be called into requisition to aid the mind in retaining intermediate processes and results. The arrangement of this textbook is particularly favorable for this purpose. Definitions and principles should be carefully committed to memory. No slovenliness in this respect should be permitted. A definition is a basis for thought and reasoning, and every word which it contains is necessary to its integrity. A child should not be expected to frame a good definition. Of course, the pupil should

be required to examine and criticise the definitions given, since this will conduce to a better understanding of their full meaning.

In conducting recitations, the teacher should use every means that will tend to awaken thought. Hence, there should be great variety in the examples, both as to their construction and phraseology, so as to prevent all mechanical ciphering according to fixed methods and rules.

The Rules and Formulce given in this book are to be regarded as summaries to enable the pupil to retain processes previously analyzed and demonstrated. They need not be committed to memory, since the pupil will have acquired a sufficient knowledge of the principles involved to be able, at any time, to construct rules, if he has properly learned what precedes them.

In the higher department of arithmetic, the chief difficulty con. sists in giving the pupil a clear idea of the nature of the business transactions involved. The teacher should, therefore, strive by careful elucidation, to impart clear ideas of these transactions before requiring any arithmetical examples involving them to be performed. When the exact nature of the transaction is understood, the pupil's knowledge of abstract arithmetic will often be sufficient to enable him to solve the problem without any special rule.

The teacher should be careful not to advance too rapidly. The mind needs time to grasp and hold firmly every new case, and then additional time to bring its new acquisition into relation with those preceding it. Hence the need of frequent reviews, in order to give the pupil a comprehensive as well as an accurate and permanent knowledge of this subject.

The Synopses for Revier interspersed through this work, are designed to afford assistance to the teacher in accomplishing this object. Each of these Synopses exhibits a brief, but definite, summary of all that is treated under the particular topic referred to, systematically and logically arranged, showing not only the different sub-topics, and their relations to each other and the general subject, but also the necessary preliminary definitions. Thus the teacher will be able readily to ask an exhaustive series of questions, without having recourse to every paragraph and page preceding.

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