Entered according to Act of Congress, in the year 1848, by

J. HOMER FRENCH,

In the Clerk's Office of the District Court of the Northern District of
New York.

Stereotyped by
GEORGE A. CURTIS;

NEW ENGLAND TYPE AND STEREOTYPE FOUNDERY,
BOSTON.

PREFACE.

THE introduction of the analytic or inductive method of instruction is ascribed to PESTALOZZI, a late distinguished teacher in Switzerland. He applied this method to Arithmetic, and to the other sciences, and thus did much in advancing the great cause of Education.

The inductive method was first successfully applied to Mental Arithmetic, in our own country, by WARREN COLBURN. Since the publication of "Colburn's First Lessons," the importance of the study of Mental Arithmetic has been steadily gaining favor, till, at this time, a knowledge of this branch is considered indispensable to a good English Education. Indeed, many popular educators maintain that, if but one can be obtained, a knowledge of Mental Arithmetic is preferable to that of Written. However true this may be, it is certain that the pupil who has a thorough knowledge of the former, invariably advances in the study of the latter with much greater facility, and with a clearer understanding of the subject, than the pupil who has had no proper mental discipline in numbers.

The following work, designed as an introduction to the study of Arithmetic, is adapted to the gradual expansion of the intellect of the young pupil, and is also designed to give him as thorough mental discipline as is obtained by studying any other elementary work upon the same subject. The work is divided into two parts. In Part First stars are placed at the head of the lessons in Addition and Subtraction. These are designed for counters to assist the pupil in preparing his lessons; they may be used in the class at the discretion of the teacher. In Multiplication and Division, that part of the multiplication or division table with reference to which the lesson is arranged, is placed at the head of the lesson.

After a few examples have been presented in any rule in Part Second, the operation is defined, and the terms and signs used in performing examples under the rule are explained. The manner of performing operations upon tens and units is distinctly brought out, and forms an important item in the fundamental rules.

Part First is designed for beginners, and should in all cases be completed before the pupil commences either Part Second, or Written Arithmetic. Part Second may be completed before the pupil takes up Written Arithmetic, or it may be used in connection with the "New Arithmetic," the arrangement of the subjects being the same as that in the latter work.

The work commences with the simplest combinations of numbers, and advances to those that are more difficult no faster than the pupil is prepared to master them.

The examples are mostly of a practical nature, and being deduced from every day business transactions, are at once interesting and instructive.

Different forms of solution are given, which, with slight modifications, may be applied to any example contained in the work. These forms require the pupil to give a reason for each step in the solution, and from the reasoning to draw a conclusion.

Notes and suggestions to teachers are placed at the end of the book, reference being made to them in all cases where they will apply. This arrangement places a valuable key in the hands of the young teacher, to which the pupil will seldom refer.

The design of the author in the preparation of the work has been, to lead the pupil on step by step, wholly in the order of discovery. This arrangement is based upon the principle, that "If the understanding is thoroughly reached, the memory will take care of itself."