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examples prepared and arranged with special reference to their practical utility, and their adaptation to the real business of active life. The answers to a part of the examples have been omitted, that the learner may acquire the discipline resulting from verifying the operations.

Particular attention is invited to improvements in the subjects of Common Divisors, Multiples, Fractions, Percentage, Interest, Proportion, Analysis, Alligation, and the Roots, as it is believed these articles contain some practical features not common to other authors upon these subjects.

The improvements in Percentage made necessary by the financial changes of the last few years are especially noticable. The different kinds of United States' Securities, Bonds, and Treasury Notes are described, and their comparative value in commercial transactions illustrated by practical examples. The difference between Gold and Currency, and the corresponding difference in prices, exhibited in trade, are taught and illustrated, and many other things that every commercial student and business man ought to know and understand.

There has also been added a full and practical presentation of the Metric System of Weights and Measures, containing many new and original improvements in the arrangement, notation, and applications, not before presented to the public, and which greatly simplify and adapt the system to general use.

It is not claimed that this is a perfect work, for perfection is impossible; but no effort has been spared to present a clear, scientific, comprehensive, and complete system, sufficiently full for the business man and the scholar; not encumbered with unnecessary theories, and yet combining and systematizing real improvements of a practical and useful nature. How nearly this end has been attained the intelligent and experienced teacher and educator must determine.

THE AUTHOR.

CONTENTS.

Decimal Notation and Numeration,. 116 | Subtraction of Decimals, ...........
Reduction of Decimals, ............. 121 | Multiplication of Decimals,.........
Addition of Decimals, .............. 124 | Division of Decimals,...............

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PRACTICAL ARITHMETIC.

DEFINITIONS.

1. Quantity is any thing that can be increased, diminished, or measured.

2. Mathematics is the science of quantity.
3. A Unit is one, or a single thing.
4. A Number is a unit, or a collection of units.
5. An Integer is a whole number.

6. The Unit of a Number is one of the same kind or name as the number. Thus, the unit of 23 is 1; of 23 dollars, 1 dollar; of 23 feet, 1 foot.

ng. Like Numbers have the same kind of unit. Thus, 74, 16, and 250; 7 dollars and 62 dollars ; 19 pounds, 320 pounds, and 86 pounds; 4 feet 6 inches, and 17 feet 9 inches.

8. An Abstract Number is a number used without reference to any particular thing or quantity. Thus, 17; 365 ; 8540.

9. A Concrete Number is a number used with reference to some particular thing or quantity. Thus, 17 dollars ; 365 days; 8540 men. Notes. 1. The unit of an abstract number is 1, and is called Unity.

2. Concrete numbers are, by some, called Denominate Numbers, Denomination means the name of the unit of a concrete number.

10. Arithmetic is the Science of numbers, and the Art of computation.

11. A Sign is a character indicating an operation to be performed.

12. A Rule is a prescribed method of performing an operation.

Define quantity. Mathematics. A unit. A number. An integer. The unit of a number. Like numbers. An abstract number. A concrete number. The unit of an abstract number. Denominate numbers. Arithmetic. A sign, or symbol. A rule.

NOTATION AND NUMERATION.

13. Notation is a method of writing or expressing numbers by characters; and,

14. Numeration is a method of reading numbers expressed by characters.

15. Two systems of notation are in general use — the Roman and the Arabic.

Note. The Roman Notation is supposed to have been first used by the Romans ; hence its name. The Arabic Notation was introduced into Europe by the Arabs, by whom it was supposed to have been invented. But investigations have shown that it was adopted by them about 600 years ago, and that it has been in use among the Hindoos more than 2000 years. From this latter fact it is sometimes called the Indian Notation.

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THE ROMAN NOTATION 16. Employs seven capital letters to express numbers, thus : Letters, I V X L C D M Values, one, five, ten, fifty, hundred, hundred, thousand.

17. The Roman notation is founded upon five principles, as follows:

1st. Repeating a letter repeats its value. Thus, II represents two, XX twenty, CCC three hundred.

2d. If a letter of any value be placed after one of greater value, its value is to be united to that of the greater. Thus, XI represents eleven, LX sixty, DC six hundred.

3d. If a letter of any value be placed before one of greater value, its value is to be taken from that of the greater. Thus, IX represents nine, XL forty, CD four hundred.

Define notation. Numeration. What systems of notation are now in general use? From what are their names derived? What are used to express numbers in the Roman notation? What is the value of each? What is the first principle of combination: Second: Third :

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