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1. QUANTITY is anything which can be increased, diminished, and measured.
To measure a thing, is to find how many times it con tains some other thing of the same kind taken as a standard. The assumed standard is called the unit of measure. A quantity may be measured by a direct appli cation of the unit of measure, or the measurement may be made indirectly by comparing it with some other quantity whose measure is known.
2. The unit of measure being always of the same kind as the thing measured, there must be as many kinds of quantity as there are species of units.
In pure Mathematics there are but eight species of units, and consequently but eight kinds of quantity, viz.: Units of Number, Units of Currency, Units of Weight, Units of Time, Units of Length, Units of Surface, Units of Volume, and Units of Angular Measure. The first four kinds of quantity are Arithmetical, the last four are Geometrical; and these two classes form the entire subject of mathe matical inquiry. By comparing quantities with each other, we arrive at a knowledge of their properties and relations
3. MATHEMATICS is the science which treats of the pro perties and relations of quantities.
4. The conventional methods of representing quantities, and the laws for combining the symbols employed, consti tute the LANGUAGE of Mathematics. Every branch of Mathematics has its own special language, which is but a dialect of what may be called the universal language of Mathematics.
5. The Science of Mathematics is divided into three principal branches, the basis of classification being the nature of the language employed. These branches are:
1st. ARITHMETIC, in which the quantities considered are expressed by figures and combinations of figures;
2d. ANALYSIS, in which the quantities considered are represented by letters, or by combinations of letters and figures, and in which the operations to be performed are indicated by signs;
3d. GEOMETRY, in which the quantities considered are represented by pictorial symbols.
The methods of reasoning are essentially the same in each branch, the language alone being different.
By combining the three branches, Arithmetic, Analysis, and Geometry, all the subordinate branches of mathe matics are obtained.
6. ALGEBRA is a branch of Analysis; it is, indeed, the foundation of all the other branches of Analysis and its