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acknowledged improvements. Some new methods and illustrations, favorable to clearness and brevity, will be noticed; rules have been reduced to the minimum number; a single rule, each, is given for Multiplication, and for Division of Fractions; only two, for all cases of Reduction of Denominate Numbers, whether integral or fractional; formulas are substituted for rules, whenever they clearly indicate the steps to be taken in the solution of examples; and when the principles, explanations, and processes make the rule obvious, it is omitted, and the pupil left to construct one in his own language.

Cancellation is made very prominent, especially in Fractions, Interest, and Proportion. Its application in working Interest, Partial Payments, and Proportion presents some new, valuable, and practical features.

Such topics, rules, and applications as are of minor importance to the majority of Grammar School pupils, who are qualifying themselves for the ordinary business of life, have been placed in an "Addenda," and may be taken up in their proper connection with kindred matter in the body of the book, or at its close, or omitted entirely, and thus the course. completed independent of them, at the option of the pupil or teacher.

The endeavor of the author has been to present a clear, systematic, and comprehensive text-book, sufficiently full for the student and ordinary business man, introducing the latest improvements, discarding obsolete terms, repetitions and needless theories, and so in quantity and quality of matter to economize time, labor, and money, and at the same time give unity, system, and practical utility to the science and art of computation. How nearly this end has been attained can be determined by the only true test of a text-book-its use in the class-room by the intelligent and experienced teacher.

D. W. F.

BROOKLYN, July, 1883.




1. Arithmetic is the science of numbers and the art of computation.

As a science, arithmetic treats of the nature and properties of numbers. As an art, it teaches how to apply a knowledge of numbers to practical and business purposes.

2. A unit is one thing, or a group of things regarded as one.

Thus, one day, one man, one ten, one dozen, are units.

3. A number is a unit, or a collection of units.

Thus, one, six; five books, eight days, are numbers.
A number answers to the question, "How many?"

4. An integral number or integer is a number representing whole things.

Thus, 5, 7; 12 books, 9 men, 20 days, are integers or whole things.

5. The unit of a number is one of the collection forming that number.

Thus, 1 is the unit of 4; 1 dollar is the unit of 5 dollars.

6. Like numbers are numbers having the same kind of unit, or that express the same kind of quantity.

Thus, 3 and 5,2 days and 6 days, 5 cents and 10 cents are like numbers.


7. Notation is a method of writing numbers. Numbers are expressed by words or characters. The characters may be either figures or letters.

8. Numeration is the method of reading numbers expressed by characters.

9. Arabic notation employs ten different characters called figures, to express numbers, viz.:

0123 4 5 6 7 8 9

Naught, one, two, three, four, five, six, seven, eight, nine.

The first character, naught, is also called cipher, or zero, and when standing alone, has no value.

The other nine are called significant figures, because each has a value of its own. They are also called digits. These ten characters, when combined according to certain principles, can be made to express any number.

10. In representing numbers, objects are supposed to be arranged in groups of ten, each group being ten times as great as the next lower group, and having a different name. Hence, we have single things, or units; next, groups containing ten units, or one ten; next, groups containing ten tens, or one hundred; and again, groups containing ten hundreds, or one thousand, etc.

11. This method of grouping is called the decimal system, from the Latin decem, which signifies ten.

The successive order of units in a number form a scale; and when the increase or decrease is uniformly ten, it is called a decimal scale,

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