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Methods that are new and progressive have been adopted in the Elementary and in the Practical, so far as they have been generally accepted by teachers. In the more advanced part of the Practical Arithmetic several ways of treating a subject are sometimes presented, so that teachers may compare different methods as a useful exercise in training pupils to think for themselves.

The author has endeavored to associate, in the mind of the pupil, the rules and forms in the more advanced subjects with those in the less advanced, reducing the reasoning in each subject, as nearly as possible, to first principles. Percentage, Interest, and their applications are thus treated under four general problems which have been developed inductively in the study of Common and Decimal Fractions.

The two books of the series, for unity of method, have the definitions and drill exercises of the one corresponding with those of the other. Each, however, is provided with business forms, etc., and is complete in itself.

In presenting this new revised series of arithmetics to the attention of the public, the author begs leave to acknowledge his great obligations to his accomplished collaborator, Mr. S. T. Stewart of Brooklyn, New York, for valuable services rendered in the work of its preparation, and also to other skilled and able teachers for helpful suggestions.

C. S. V.

UNIVERSITY OF VIRGINIA,

August, 1888,

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PRACTICAL ARITHMETIC.

1. A unit is a single thing, or one.

One man, one dollar, one regiment, one, are units.

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

Three men, five dollars, nine regiments, are numbers.

3. An abstract number is a number the name of whose unit is not given.

Seven, fifteen, forty-five, one hundred six, are abstract numbers.

4. A concrete number is a number the name of whose unit is given.

Six horses, nine pounds, ten dollars, are concrete numbers.

5. A simple number is a number which expresses one or more units of the same name or denomination; as, 5 yards; 3 dollars; 7.

6. A compound (or denominate) number is a number containing units of two or more different names or denominations, but the same in kind; as, nine pounds twelve ounces; and five bushels three pecks two quarts.

7. Arithmetic teaches the use of numbers.

NOTATION AND NUMERATION.

8. Notation is the writing of numbers with figures, or with letters.

9. Numeration is the reading of numbers expressed by figures, or by letters.

10. Figures are not numbers, but characters or marks that represent numbers.

[blocks in formation]

12. The first nine figures are called digits, or significant figures. They represent, in their order, the first nine numbers.

13. The last figure is called naught, zero, or cipher. It is used with the significant figures to give them their proper place and value.

14. A figure standing alone is said to occupy the first place, and it then represents the corresponding number of ones, or units of the first order.

15. A figure standing with another figure at its right is said to occupy the second place, and it then represents the corresponding number of tens, or units of the second order.

16. A figure standing with two figures at its right is said to occupy the third place, and it then represents the corresponding number of hundreds, or units of the third order. 17. Numbers containing units of the first, second, or third place or order are read as follows:

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