Multitude is quantity of separate things. It is that concerning which the question "How many?" can be properly asked.

Art. 5. Measuring a quantity is applying to it a unit of the same kind, to find how many times the unit is contained in that quantity. A measure is a unit used in measuring.

In measuring magnitudes a certain quantity is generally assumed as a unit, to measure other quantities of the same kind; as a pound, for weight; an hour, for time; a yard, for length; a gallon, for capacity, &c. Such units may be called artificial units.

In measuring multitude of any kind of thing, the units are the individuals of that kind, without reference to their being equal or unequal; as a tree, a cloud, a bird, a hill, a man, &c. Such units may be called natural units.

Art. 6. In reference to the wholeness of their unit, numbers are of three classes, viz.: integral, fractional, and mixed. An integer is any whole number; as, 1, 5, 7, 9, &c.

A fraction is a number expressing one or more of the equal parts of a unit, or of a quantity; as 1, 2, 4, &c.

A mixed number is a number composed of an integer and a fraction; as, 51, 10%, 7.5, &c.

Art. 7. In reference to their application, numbers are of two classes, viz.: concrete, and abstract.

A concrete number is an expression of a particular kind of quantity; as, 5 men, 8 dollars, 20 houses, &c.

An abstract number is an expression of quantity without reference to its kind; as, 5, 8, 20, &c.

Art. 8. In reference to the measuring quality of their units, concrete numbers may be denominate, or not.

A concrete number is denominate when its units are measures of the magnitude of the thing mentioned. Thus, five feet is a denominate, because a foot is a unit which measures quantity of length. For similar reasons, seven pounds is a denominate number in weight, or force; nine

hours, in time; and ten dollars, in money, or value. Each kind of such units is called a denomination.

A concrete number is not denominate when its units are not measures of the magnitude of the thing mentioned, but merely of its multitude. Thus, two birds, seven hills, ten houses, three thoughts, &c.

Art. 9. In reference to the number of denominations in them, denominate numbers are simple or compound.

A simple denominate number is expressed in one denomination; as, 5 feet.

A compound denominate number is expressed in more than one denomination; as, 5 feet 7 inches.

Art. 10. Numbers, compared in reference to the nature of their quantities, are either like or unlike.

Like numbers are those which express the same kind of quantity. Thus, 5 and 7 are like numbers, because both have abstract units; 5 dollars and 7 dollars are like numbers, because both have concrete units of the same nature.

Unlike numbers are those which express different kinds of quantity. Thus, 5 dollars and 7 miles are unlike numbers. Art. 11. In reference to the method of computing, Arithmetic may be Mental, or Written.

In Mental Arithmetic computations are performed without the aid of written characters. It is also called Intellectual Arithmetic.

In Written Arithmetic computations are performed with the aid of written characters.

Art. 12. In reference to its subjects, Arithmetic is either pure and theoretical, or applied and practical.

Pure, or theoretical Arithmetic treats of the nature and principles of numbers, and of operations with them, without regard to their application.

Applied, or practical Arithmetic treats of computations in science, arts, and business.

Art. 13. A rule, in Arithmetic, is a statement of the mode of computing.

An example is a statement of quantities to be used in computing according to a certain rule. It is often called a question.

A solution is the act of computing the unknown or required quantities in a question, from those which are known. An operation is that which is done to accomplish a solution.

A sign is a character which indicates either a relation between numbers, or an operation to be performed with them.

A problem is a question proposed for solution. To solve a problem is to compute the required quantity.

A demonstration is a process of reasoning which proves a truth.

A proposition is a statement of a truth to be demonstrated. It is also called a theorem.

A corollary is an inference from a proposition.

An analysis is a solution whose steps are accompanied with a statement of the reasons for them.

An axiom is a self-evident truth. All the processes of mathematics are founded upon axioms. The following are some of them.

AXIOMS.

1. The whole of a quantity is greater than a part of it. 2. The whole of a quantity is equal to the sum of all its parts.

3. If equal quantities are equally increased, the sums are equal.

4. If equal quantities are equally decreased, the remainders are equal.

5. If equal quantities are unequally increased, the sums are unequal.

6. If equal quantities are unequally decreased, the remainders are unequal.

7. Quantities which are equal to the same quantity, are equal to each other.

8. Like parts of equal quantities are equal.

9. Like powers and roots of equal quantities are equal. Art. 14. In Written Arithmetic there are six operations on which all others depend, viz.: Notation, Numeration, Addition, Subtraction, Multiplication, and Division. The last four are frequently called the four fundamental rules of Arithmetic, because all computations must use one or more of them.

SYNOPSES OF MATHEMATICS AND NUMBER.

Art. 15.

CHAPTER II.

NOTATION.

Notation is the art of expressing numbers and their relations by symbols.

The notation of numbers employs letters and figures. The notation of the relation of numbers, employs signs.

Art. 16.

There are two systems of notation in common use-the Roman and the Arabic.

NOTE. The ancient Greeks and Romans used the letters of their alphabet for symbols of number. Most modern nations still use the Roman symbols for some of the purposes of numbering; but for computations they use a set of symbols introduced into Europe by the Arabians. The Arabic Notation is supposed to have been introduced into Europe at the time of the Saracenic invasion, between the seventh and eleventh centuries of the Christian era. It is probable that the system originated in Hindoostan, where it appears to have been in use more than 2000 years. Hence it is sometimes called the Indian Notation. It is, also, sometimes called the Decimal Notation, (from the Latin decem, ten) because it uses ten symbols, and each order of value is ten times the next lower order.

ROMAN NOTATION.

Art. 17. The symbols of the Roman Notation are seven capital letters.

Values, one, five, ten, fifty, hundred, hundred, thousand. The Roman Notation is founded upon the following principles:

1ST. Repeating a letter repeats its value. V, L, and D are not repeated.