PREFACE.

THERE is a large class of pupils whose limited time renders it impossible for them to pursue an extended mathematical course. The author, in accordance with his original intention to prepare a series of text-books in Arithmetic, has now endeavored to adapt this work to the wants of this class of pupils.

With this purpose in view, the simple, elementary, practical principles of the science are more fully presented than in his larger work, while the more intricate and less important parts have been treated more briefly or entirely omitted. A corresponding change in the character of the examples has also been made.

As in the larger work, so here, constant attention has been paid to the brevity, simplicity, perspicuity, and accuracy of expression; and no effort has been spared in the endeavor to render the mechanical execution appropriate and attractive.

Definitions, tables, and explanations of signs have been distributed through the book where their aid is needed, to enable the pupil to learn them more readily than when they are presented collectively.

Nearly all the examples have been prepared for this book, and are different from those of the larger work; still, to secure uniformity of language (a matter of great importance, as every experienced teacher knows), the leading examples in the several subjects, the definitions and rules, with few exceptions, have been intentionally retained with but little modification.

Articles on United States Money, Percentage, Stocks, CustomHouse Business, and Exchange have been prepared for this book; and all the principles requisite for a practical business life have been presented in a simple, intelligible, attractive manner, and with sufficient minuteness and fullness and a due regard to logical arrangement.

Brief, suggestive questions have been placed at the bottom of the page, designed in no way to interfere with the free, original questioning which every teacher will adopt for himself, but merely to aid the young and inexperienced pupil in fixing his attention. upon the more important parts of the subject.

Here, as in the larger work, some of the answers to examples have been given to inspire confidence in the learner, and others are omitted to secure the discipline resulting from proving the operations, a discipline and a benefit which the pupil should not forego nor the teacher neglect.

Fully appreciating the favor which has been bestowed on his other works, the author sends this forth, hoping it may commend itself to the approval of committees and teachers, and that it may be found adapted to contribute in some measure to the happiness and improvement of the class of pupils for whom it is designed.

THIS EDITION of the Common School Arithmetic, by request of many eminent Teachers, is enlarged by the substitution of the subjects of Involution and Evolution, and by several pages of Examples from my larger work.

ANDOVER, May 29, 1863.

J. S. E.

This work is now presented, containing a full exposition of the Metric System of Weights and Measures, prepared by H. A. Newton, Professor of Mathematics, Yale College.

MAY, 1867.

ARITHMETIC.

ARTICLE 1. ARITHMETIC is the science of numbers, and the art of computation.

A NUMBER is a unit or a collection of units, a unit being one, i. e. a single thing of any kind; thus, in the number six the unit is one; in ten days the unit is one day.

2. All numbers are concrete or abstract.

A CONCRETE NUMBER is a number that is applied to a particular object; as six books, ten men, four days.

An ABSTRACT NUMBER is a number that is not applied to any particular object; as six, ten, seventeen.

3. Arithmetic employs six different operations, viz. Notation, Numeration, Addition, Subtraction, Multiplication, and Division.

NOTATION AND NUMERATION

4. NOTATION is the art of expressing numbers and their relations to each other by means of figures and other symbols.

5. NUMERATION is the art of reading numbers which have Leen expressed by figures.

ART. 1. What is Arithmetic? What is a Number? A Unit? 2. What is a Concrete Number? An Abstract Number? 3. How many operations in Arithmetic? What are they? 4. What is Notation? 5. Numeration?

6. Two methods of notation are in common use: the Arabic and the Roman.

7. The ARABIC NOTATION, or that brought into Europe by the Arabs, employs ten figures to express numbers, viz.:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Naught, One, Two, Three, Four, Five, Six, Seven, Eight, Nine.

These figures are called digits, from the Latin digitus, a finger; a term probably applied to figures from the custom of counting upon the fingers.

8. The first Arabic figure, 0, is called a cipher, naught, or zero, and, standing alone, it signifies nothing.

Each of the remaining nine figures represents the number placed under it, and for convenience in distinguishing them from O, they are called significant figures.

9. No number greater than nine can be expressed by a single Arabic figure, but by repeating the figures, and arranging them differently, all numbers may be represented.

Ten is expressed by writing the figure 1 at the left of the cipher; thus, 10. In like manner, twenty, thirty, forty, etc., are expressed by placing 2, 3, 4, etc., at the left of 0; thus,

20, 30, 40, 50, 60, 70, 80, 90. Twenty, Thirty, Forty, Fifty, Sixty, Seventy, Eighty, Ninety. 10. The numbers from 10 to 20 are expressed by placing the figure 1 at the left of each of the significant figures; thus,

11,

12,

13,

14, 15, 16,

17, etc. Eleven, Twelve, Thirteen, Fourteen, Fifteen, Sixteen, Seventeen, etc.

In a similar manner all the numbers, up to one hundred, may be written; thus,

21, Twenty-one,

36, Thirty-six,

66, Sixty-six,

98,
Ninety-eight, etc.

etc.

6. How many methods of Notation? What? 7. How many figures in the Arabic Notation? What called? Why? 8. What is the first figure, 0, called? The others? Why? 9. The largest number expressed by one figure? Ten, how expressed? Twenty? 10. Numbers from ten to twenty, how expressed?