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The arrangement of subjects is systematic; no principle is anticipated, or used in the explanation of another, until it has itself been explained. Subjects intimately connected are grouped together in the order of their dependence.

In connection with the Notation of Integers, that of Decimals is taught to three places below units, corresponding to dimes, cents, and mills. Decimal or Metric Weights and Measures are placed next after Decimal Currency. Percentage is followed by its applications in their proper order, as Profit and Loss, Commission, etc.

General Analysis, covering the several departments of Commercial Arithmetic, has received special attention. The articles devoted to Test Questions, and to the Entrance Problems of different Colleges, will be found a valuable addition and excellent practice. Thanks are due to the College Officers who have kindly furnished copies of their Examination Papers.

In the preparation of the work the author has carefully weighed the discussions in the various journals of education respecting the present wants of our schools, and has endeavored to provide for them. He has availed himself of many valuable suggestions from business men, practical teachers, and educators, all of whom he desires to thank most cordially for the aid they have rendered.

He cheerfully submits the result of his labors to his friends, the teachers, and the public, for whose favorable verdict upon his former efforts he desires to express renewed obligations.

BROOKLYN, April, 1882.

J. B. T.

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Art. 1. A Unit is one or any single thing; as, one, one book, one chair.

2. A Number is a unit or a collection of units.

Thus, one, or one book, is a unit; five, or five books, is a collection of units.

3. The Unit of a Number is one of the collection forming that number.

Thus, the unit of four books is one book, of seven is one.

4. An Abstract Number is one that is not applied to any object.

Thus, four, five, thirteen, etc., are abstract numbers.

5. A Concrete Number is one that is applied to some object. Thus, five boys, seven apples, etc., are concrete numbers.

6. Like Numbers are those which express units of the same kind.

Thus, eight pears and five pears, four and seven, are like numbers.

7. Unlike Numbers are those which express units of different kinds.

Thus, seven, six peaches, nine days, are unlike numbers.

8. Numbers are expressed by words, by figures, or by letters.

9. Arithmetic is the science which treats of numbers and their applications.

No

OTATION AND NUMERATION.

10. Notation is expressing numbers by figures or letters.

11. Numeration is reading numbers expressed by figures or letters.

NOTE.-There are two methods of Notation, called the Arabic and the Roman. The former is the method in general use, and is so called because it was introduced into Europe in the 10th century by the Arabians.

12. The Arabic Notation expresses numbers by ten different characters, called Figures; viz.,

1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Naught. 13. The first nine are called Significant figures, because they always express some value. They are also called Digits.

14. The last one is called Naught, because when standing alone it has no value. It is also called Cipher or Zero.

15. The Value of a figure is the number it represents. 16. Nine is the largest number expressed by one figure. 17. The significant figures standing alone, express single things or ones; as, 4 apples, 5, 7.

18. To express the numbers from nine to one hundred requires two figures written side by side.

19. The first figure at the right denotes Ones, which are called Units of the First Order.

20. The figure in the second place denotes ten ones, which are called Tens, or Units of the Second Order.

Thus, the figures 35, denote 5 ones and 3 tens, and are read, "Thirtyfive."

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