cially meets the wants of Graded Schools. But sixty pages cover the same ground as the Intermediate Arithmetic, and of these not more than twenty pages are in any sense a repetition. The repeated matter consists of definitions, principles, and rules, all the problems being new. The subjects before treated are not only concisely reviewed, but from a higher stand-point. Of the twenty-four pages devoted to the fundamental rules, eight present new abbreviated methods; and of twenty-eight pages devoted to Denominate Numbers, simple and compound, more than sixteen discuss new topics. A similar difference is observable in the treatment of Common Fractions and United States Money. Among the added articles worthy of special mention are those on Denominate Fractions, the Metric System, and Longitude and Time.

In the number of problems, the author has aimed to hit the golden mean between a paucity and an excess, and the greatest pains have been taken to make them sufficiently progressive, varied, and difficult, to afford the requisite drill and practice. Instead of rehashing old problems, with their incorrect data and obsolete terms, the author has gone to science and history for statistical information of practical value, and he has aimed to present the current values, terms, forms, and usages of American business. The mental problems will be found as difficult and comprehensive as those which constitute the latter half of the standard Mental Arithmetics, and are sufficiently numerous to afford thorough drills in analysis.

The explanations of the written processes are not designed to serve as models for the pupil to memorize and repeat. They are intended to supplement the analysis. In some cases, a formal analysis is given; in others, a principle is deduced or demonstrated; and in others, the process is described or its principles stated. Neither teacher nor pupil is denied the privilege of determining his own explanations.

Another characteristic feature of this work is the prominence

given to PRINCIPLES. A clear comprehension of the principles of arithmetic is essential to its thorough mastery, and their induction, proof, and illustration are mental exercises of great value. Until the pupil can step inductively from processes to principles, he has not a thorough knowledge of numbers. In this work the principles are concisely and formally stated in connection with the rules which are based upon them.

The author invites special attention to the sections treating of Percentage, Ratio and Proportion, and Involution and Evolution. Over eighty pages are devoted to Percentage and its applications, and it is believed that the treatment will be found not only full and thorough, but of great practical value. The student who masters these pages will certainly have a fair knowledge of the nature, laws, and usages of the business of the country. The introduction of Formulas, it is hoped, will prove a useful feature.

The thorough treatment of Ratio before Proportion, and of the latter before its application to the solution of problems, will make the mastery of this subject easy. The treatment of Involution and Evolution will not escape notice. The geometrical explanations of Square Root and Cube Root are the reverse of those usually given, and are believed to be new. They will be found both simple and natural.

All useless and obsolete subjects have been omitted; and those of interest only to more advanced students and teachers, are presented in an appendix. The typography and illustrations are beautiful and appropriate.

The COMPLETE ARITHMETIC is submitted to American teachers in the hope that it may not only be found new in its general plans and in many of its methods and details, but that it may prove eminently adapted to the present wants and condition of GRADED SYSTEMS OF INSTRUCTION.

COLUMBUS, OHIO, July, 1870.

SUGGESTIONS TO TEACHERS.

1. The Mental Problems should be made a thorough drill in analysis; but, since the reasoning faculty is not trained by mere logical verbiage, the solution should be concise and simple. They should also be made introductory to the written processes of which they are often a complete elucidation. The corresponding problems in the two classes of exercises should be recited in connection, as well as separately.

2. All Written Problems should be solved by the pupils on slate or paper, and the solutions should be brought to the recitation for the teacher's inspection and criticism. From three to five minutes at the beginning of the recitation will suffice to compare the answers of the class, and ascertain the accuracy and neatness of each pupil's work. Time thus taken from the class-drill is more than made good by the increased interest, self-reliance, and accuracy, which the absence of answers secures. The explanations of the written processes, given by the pupil, should be both analytic and inductive.

3. The Definitions should be deduced and stated by the pupils under the guidance of the teacher, and this can usually be done in connection with the solution of the problems. See Int. Arith., p. 5, Sug. 3. When the definitions are placed before the problems, as in the applications of Percentage, they should be studied by the pupils, but their recitation may be deferred until the problems are solved, and the processes mastered.

4. The Principles should be taught inductively, when this is possible, and each should be proven or illustrated, or both, by the pupil. A thorough mastery of every principle should be made an essential condition of the pupil's progress. The recitation should secure a constant application of known principles, and a clear comprehension of all new ones.

5. The Rules should also be deduced and stated by the pupils. The true order is this: 1. A mastery of the process. 2. Recognition of the successive steps in order, and a statement of each. 3. Combination of these several statements into a general statement. 4. Comparison of this generalization with the author's rule. 5. Memorizing of the rule approved. See Int. Arith., p. 6.

6. When two or more methods or solutions are given, the one preferred should be thoroughly taught. It is well for pupils to understand different processes and explanations, but they should be made familiar with one of them.

7. Before a subject is left, the pupils should be required to make a topical analysis of the definitions, principles, and rules, and the same should be recited with accuracy and dispatch. Their knowledge of the subject should finally be tested by a series of questions and problems. See General Review, pp. 268-284.

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