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merous illustrations are given showing the exact size of the metric weights and measures and the relation of their units to each other.

In the operations the process of simplifying by cancellation, whereby much time and labor is saved, is kept constantly in view. In the processes of Commission, Interest, Banking, Equation of Payments, etc., business men and bankers have been consulted, and those methods adopted that experience has proved to be the simplest and the best. Only such tables of compound numbers have been given in the body of the work as are used in ordinary transactions.

Circulating Decimals, Annual Interest, Foreign Exchange, Custom House Business, Arithmetical and Geometrical Progression, and Alligation, not being required in the grammar school grade, have been placed in the Appendix. Additional tables, monetary equivalents of different countries, rates of interest in the several States, and other matter of this nature used mainly for reference, are also given in the Appendix.

The author desires to return his thanks to those teachers who have so freely aided him with suggestions and examples. His obligations are specially due to Mr. L. A. Wheelock, Master of the Rice School, Boston, Mr. James A. Page, Master of the Dwight School, Boston, and Mr. L. M. Chase, Master of the Dudley School, Boston, for valuable assistance.

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ARITHMETIC..

1. A Unit is a single thing of any kind; as, one day, one book.

2. A Number is a unit, or a collection of units; as, six days, ten books.

3. A Concrete Number is a number that is applied to a particular object; as, six books, ten men, four days.

4. An Abstract Number is a number that is not applied to any particular object; as, six, ten, seventeen.

5. Arithmetic is the science of numbers, and the art of computation.

NOTATION AND NUMERATION.

6. Notation is the writing of numbers. 7. Numeration is the reading of numbers. 8. To express numbers ten figures are used, viz. : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Zero, One, Two, Three, Four, Five, Six, Seven, Eight, Nine.

9. The first figure is called zero, a cipher, or naught ; standing alone, it signifies nothing. The remaining nine figures represent in order the numbers one, two, three, four, five, six, seven, eight, nine.

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

11. 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.

12. The numbers from 10 to 20 are expressed by placing the figure 1 at the left of each of the ten figures except zero; 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, 36, 66, 98, etc. Twenty-one, Thirty-six, Sixty-six, Ninety-eight, etc. 13. One hundred is expressed by placing the figure 1 at the left of two ciphers; thus, 100. In like manner two hundred, three hundred, etc., are written; thus,

200, 300, 600, 800, etc. Two hundred, Three hundred, Six hundred, Eight hundred, etc.

14. The other numbers up to one thousand may be expressed by putting another figure in the place of one, or in the place of each, of the two ciphers; thus, Two hundred three, expressed in figures, is

203, Six hundred eighty, expressed in figures, is Nine hundred ninety-eight, expressed in figures, is 998.

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