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For example, the additional proofs of the fundamental operations follow the last of these operations - Division. The chapter on Scales of Notation immediately precedes Denominate Numbers and naturally leads up to them, being the steppingstone between the simple numbers of the decimal notation and the compound numbers with their varying scales found in Denominate Numbers. The Metric System has not been treated as a curiosity reserved for an Appendix, but each table of this system follows the corresponding table of the regular system. Immediately following Denominate Numbers are their practical applications — Measurements. Savings-Bank Accounts follow other banking business, Partitive Proportion precedes and leads up to Partnership, and so on with all the subjects.

The uniformity of plan in the book adds greatly to the ease with which the subjects are mastered. When once familiar with the style of the book, the pupil knows immediately where to look for and find any special point or detail in any subject. First he finds a clear and logical development of the subject, then a model example or two with the solution, and lastly the rule; but by the time he reaches the rule he knows it. He could frame it himself in his own way, and should be permitted to do so. The rule might, therefore, be omitted altogether were it not for the fact that a clear and brief summary, such as a pupil might not be able to express, aids the memory.

When two or three explanations or methods of working an example are equally good, they are all given. One method is easier for some children, another for others; and frequently one explanation elucidates another, to say nothing of the broadening influence of such a course upon the mind. The second method of working Compound Interest given on page 286 is, perhaps, the simplest (when a Compound Interest Table is not at hand) and the most easily understood and remembered; yet few, if any, of the arithmetics give it at all, or else they reserve it for treatment under Progressions.

There is so much valuable matter in this book that is not to be found in the old edition that it is hoped it will soon take the place of that book altogether; but nevertheless, for the convenience of those classes where the new book will be used side by side with the old, the old section numbering has been preserved as much as possible, and the new edition will seldom be found to vary from it more than two or three numbers.



1. Quantity is anything 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 collection of units forming the number.

Thus, the unit of 23 is 1; of 23 dollars, 1 dollar.

7. Like Numbers are numbers that have the same kind of unit.

Thus, 74, 16, and 250; 7 dollars and 62 dollars; 4 ft. and 17 ft. 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.

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


11. Notation is a method of writing or expressing numbers by characters.

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

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

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 only 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.


14. The Roman Notation employs seven capital letters to express numbers.



Ten. Fifty. One Five
hundred. hundred. thousand.


15. The Roman Notation is founded upon five principles, as follows:

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

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


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