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Entered according to Act of Congress, in the year 1872, by A. S. BARNES & Co., in the Office of the

Librarian of Congress, at Washington.
SILICATE BOOK SLATE SURFACE. Patented February 24, 1857 ; January 15, 1867 ; and

August 25, 1868.
JOCELYN'S SLATED BOOK. Patent Applied for.
BARNES' SLATE AND WATERPROOF FLY-LEAF COMBINATION. Patent applied for.

SCHOOL

ARITHMETIC.

ANALYTICAL AND PRACTICAL.

of

BY CHARLES DAVIES, LL.D.,
IRT DAVIES' PRACTICAL ARITHMETIC, OF THE NEW SERIES, WITH FULL MODERN TREAT
XENT OF THE SUBJECT, 18 OF THE SAME GRADE, AND DESIGNED TO TAKE THE PLACE OF
THIS WORK.)

A, S. BARNES & COMPANY,

NEW YORK, CHICAGO AND NEW ORLEANS.

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The following named volumes are entirely new works, written within the past ten years, to conform to all modern improvement, and take the place of the author's older series.

NO CONFLICT OP EDITIONS is possible, if patrons will be particular to order the book they want by its exact title. Whenever any change is made so radical as to be likely to cause confusion in classes,

THE NAME OF THE BOOK IS CHANGED. Teachers using any work by DAVIES not here-in-after enumerated, are not availing themselves of the advantages offered by

THE NEW SERIES.

per Primary, Intellectual, and Practical Arithmetics constitute the Serics proper. Other volumes are optional. DAVIES' PRIMARY ARITHMETIC.

The elementary combinations, by object lessons. DAVIES' INTELLECTUAL ARITHMETIC.

Referring all processes to the Unit for analysis. DAVIES' ELEMENTS OF WRITTEN ARITH.

Prominently practical, with few rules and explanations. DAVIES' PRACTICAL ARITHMETIC.

Complete theory and practice. Substitute for this volume.

DAVIES UNIVERSITY ARITHMETIC.
A purely scientific presentation for advanced classes.

ALSO,
DAVIES NEW ELEMENTARY ALGEBRA.

A connecting link between Arithmetic and Algebra.

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COURSE OF HIGHER MATHEMATICS.

Entered according to Act of Congress, in the year 1852, by

CHARLES DAVIES, In the Clerk's Office of the District Court of the United States for the Southern

District of New York. N. S. A.

PRE FACE.

ARITHMETIC embraces the science of numbers, together with all tho rules which are employed in applying the principles of this scienco to practical purposes. It is the foundation of the exact and mixed sciences, and the first subject, in a well-arranged course of instruction, to which the reasoning powers of the mind are directed. Because of its great practical uses and applications, it has become the guide and daily companion of the mechanic and man of business. Hence, a full and accurate knowledge of Arithmetic is one of the most ime portant elements of a liberal or practical education.

Soon after the publication, in 1848, of the last edition of my School Arithmetic, it occurred to me that the interests of education might be promoted by preparing a full analysis of the science of mathematics, and explaining in connection the most improved methods of teaching. The results of that undertaking were given to the public under the title of “Logic and Utility of Mathematics, with the best methods of instruction explained and illustrated.” The reception of that work by teachers, and by the public generally, is strong proof of the deep interest which is felt in any effort, however humble, which may be made to improve our systems of public instruction.

In that work a few general principles are laid down to which it is. supposed all the operations in numbers may be referred :

1st. The unit 1 is regarded as the base of ery number, and the consideration of it as the first step in the analysis of every question relating to numbers.

2d. Every number is treated as a collection of units, or as made up of sets of such collections, each collection having its own base, which is either 1, or some number derived from 1.

3d. The numbers expressing the relation between the different units of a number are called the SCALE; and the employment of this term enables us to generalize the laws which regulate the formation of numbers.

4th. By employing the term "fractional units,” the same principles are made applicable to fractional numbers ; for, all fractions are but collections of fractional units, these units having a known relation to l.

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In the preparation of this work, two objects have been kept con. stantly in view :

1st. To make it educational; and,
2d. To make it practical.
To attain these ends, the following plan has been adopted:

1. To introduce every new idea to the mind of the pupil by a simple question, and then to express that idea in general terms under the form of a definition.

2. When a sufficient number of ideas are thus fixed in the mind, they are combined to form the basis of an analysis ; so that all the principles are developed by analysis in their proper order.

3. An entire system of Mental Arithmetic has been carried forward with the text, by means of a series of connected questions placed at the bottom of each page; and if these, or their cquivalents, are carefully put by the teacher, the pupil will understand the reasoning in every process, and at the same time cultivate the powers of analysis and abstraction.

4. The work has been divided into sections, each containing a number of connected principles; and these sections constitute a series of dependent propositions that make up the entire system of principles and rules which the work develops.

Great pains have been taken to make the work PRACTICAL in its general character, by explaining and illustrating the various applications of Arithmetic in the transactions of business, and by connecting as closely as possible, every principle or rule, with all the applications which belong to it.

I have great pleasure in acknowledging my obligations to many teachers who have favored me with valuable suggestions in regard to the definitions, rules, and methods of illustration, in the previous cditions. I hope they will find the present work free from the defects they have so kindly pointed out. A Key to this volume has been prepared for the use of Teachers only

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