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AN

INTELLECTUAL ARITHMETIC,

UPON THE

INDUCTIVE METHOD,

WITH AN

INTRODUCTION TO WRITTEN ARITHMETIC.

BY

JAMES S. EATON, M. A.,
INSTRUCTOR IN PHILLIPS ACADEMY, ANDOVER, AND AUTHOR OT
"EASY LESSONS IN MENTAL ARITHMETIC," " THE COMXOX
SCHOOL ARITHMETIC,” AND “A TREATISE ON

WRITTEN ARITHMETIC."

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BOSTON:
THOMPSON, BROWN & COMPANY,

25 AND 29 CORNHILL.

1877.

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.

Entered, according to Act of Congress, in the year 1864, by

JAMES S. EATON, M.A., In the Clerk's Office of the District Court of the District of Massachusetts.

PREFACE.

THE Pestalozzian or Inductive Method of teaching the science of numbers is now universally approved by intelligent teachers. The first attempt in this country to apply this method to Mental Arithmetic resulted in the publication of Colburn's First Lessons, a work whose success has not exceeded its merit. It was, however, a useful experiment rather than a perfect realization of the inductive system of instruction. That the subsequent books of the same class and purpose have failed to correct its defects, and thus meet the demand it created, is due evidently to their departure from the true theory as developed and exemplified by Pestalozzi.

The Author of this work has endeavored to improve upon all his predecessors, by adhering more closely than even Colburn did to the original method of the great Swiss educator, and by presenting at the same time, in a practical and attractive form, such improvements in the application of his principles as have stood the test of enlightened experience.

In accordance with this design, the subjects are so arranged that each step of the learner prepares him for that which follows. By this suggestive and natural order of arrangement, together with copious illustrations of principles and applications by means of small concrete numbers, the pupil is led to a clear apprehension of the properties and relations of numbers, and

is enabled to understand everything as he advances, till ho acquires a thorough knowledge of the nature and use of the essential numerical operations.

While the general arrangement of the subjects and examples is strictly progressive and logical, the difficulty of the problems is occasionally varied, in order to prevent the weariness of a long, unbroken ascent, and to afford a grateful alternation of effort and relaxation, like that experienced by the traveler in crossing a country diversified by hill, valley, and plain.

The analytical process which this method requires at every step is calculated to develop and strengthen the mental powers, and to form the habit of rapid and accurate thought. Some illustrations of modes of analyzing questions have been presented merely as suggestions to the pupil; but the plan of the work does not embrace set forms of analysis for the various classes of examples, a contrivance little likely to stimulate invention or promote self-reliance. On the contrary, its distinctive feature is its special adaptation to the mode of teaching which leads the learner to ascertain for himself each step to be taken, to think and reason independently, and to rely upon his own powers and resources, thus securing a vigorous and healthful discipline of his intellectual faculties.

Though this work is intended as a connecting link between The Primary and Written Arithmetics of the Author, thus com pleting the Series on which he has been so long engaged, it is also complete in itself. It presents a mental analysis of Arithmetic adapted to the younger pupils by its easy gradations, and to advanced pupils by its scientific arrangement and its logical development of the art of computation; and yet it has been limited to the true province of Intellectual Arithmetic, which is to serve as an introduction to Written Arithmetic, and not as a substitute for it, as some authors seem to imagine.

In the spirit of the inductive method, concrete numbers — numbers applied to physical objects — have been largely employed in treating of each topic, as the only fit preparation for the exercises upon abstract numbers, which are far more difficult for the youthful mind to grasp.

A few pages of Written Arithmetic have been appended, embracing examples in the ground rules and compound numbers, which may be profitably studied in connection with the mental lessons illustrating the same principles.

Fully aware of the difficulty of the task he has undertaken, the Author has spared no pains in its execution, and he gratefully acknowledges his obligations for the numerous valuable suggestions with which he has been favored by several eminent practical teachers.

The favorable reception of the other books of his Series, encourages him to hope that this attempt to perfect and modernize the original Inductive System of Mental Arithmetic, and adapt it to the wants of schoois of the present day, will meet with the general approbation of teachers and educators.

PHILLIPS ACADEMY, ANDOVER,

MARCH 14, 1864.

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