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THEORETICAL AND PRACTICAL

ARI T H M E T I C;

DESIGN ED FOR

COMMON SCHOOLS AND ACADEMIES,

BY

DANIEL LEACH AND WILLIAM D. SWAN.

BOSTON:

JENKS, HICKLING, & SWAN.

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Entered according to act of Congress, in the year 1850,

BY DANIEL LEACH AND WILLIAM D. SWAN, in the Clerk's office of the District Court for the District of Massachusetts.

PREFACE.

It has been the aim of the authors, in preparing this work, to make it eminently both a practical and a theoretical treatise on the science of numbers. They have therefore adopted that arrangement which has appeared the most philosophical, and, at the same time, the best suited to the comprehension of the learner.

They have bestowed great labor on the rules and definitions, in order to make them lucid, concise, and accurate; and they have carefully avoided introducing any illustrations or remarks not necessary to a clear understanding of the subject.

The examples have been prepared with much discrimination. Many of them are questions which have actually occurred in ordinary business transactions.

They would call attention first to the examples in addition, some of which have been so arranged as to bring together the same combination of figures throughout the same line. The long leger columns are designed for those who wish to acquire a facility in adding long columns. This is one of the most useful exercises in arithmetic to which pupils can be accustomed.

They would also call particular attention to the rule for finding the least common multiple, the rule of alligation, and the rule for extracting the cube root. These rules are clear and concise, and can be most rigidly demonstrated; while in the processes indicated by them there is a saving of more than one half of the figures, as compared with the processes in similar works now in common use. The section on fractions, they think, will also commend itself to every experienced teacher.

Although the preface is not the proper place for discussing the best method of teaching arithmetic, yet the authors cannot refrain from urging upon all teachers not to allow their pupils to attempt to solve a question till they fully understand all its conditions, and always to require them to state the principles upon which each solution is founded. Pupils should be accustomed to write questions of their own under each rule. This is a very important exercise.

They would also suggest that, in every question in which there are both multiplication and division, the pupil should at first indicate the processes by their appropriate signs, and then cancel the factors common to the dividend and divisor.

In the preparation of this work, a large number of the recent English and American treatises on arithmetic have been consulted, and the most valuable mathematical works of the French.

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