ARITHMETIC Is the science of numbers. It would be evidently impossible to express each number by a separate character; for in that case we must have an infinite number of characters.

For example, to express every number, from one up to one million, we must have a million of different signs or marks, unless we can accomplish the end by changes and combinations of a few simple characters.

(ART. 1.) Numbers were at first written separately, either in words at length, or, as among the Romans, in characters, commonly the letters of the alphabet; and for large numbers, such letters were used as would make up, by the addition of their separate values, the number required. Thus, the number four hundred and fifty-eight was written CCCCLVIII., the value of C being one hundred, L fifty, V five, and I one.

The system of notation now in use, and which originated in Arabia, is so contrived as to express all numbers with ten characters only. Nine of these are significant, and represent numbers, and the other is used to denote nothing, or the absence of quantity.

To express numbers greater than nine, recourse is had to a law which assigns different values to the figures, according to the position which they occupy. According to this law, units of the first order occupy the first place on the right of the written expression, units of the second order, the second place, and so of the others.

The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, are each, as we perceive, separate characters; but when we would write

ten, which is a unit of the second order, we place unity in the second place, and a 0 (cypher) in the first, thus, 10, which may either be considered as one unit of the second order, or ten units of the first. Proceeding with units of the second orter in the same manner as with those of the first, we shali have 10, 20, 30, 40, 50, 60, 70, 80, 90, and next a unit of the third order, which is written 100, the significant figure being written in the third place. In like manner, we pass from the third to the fourth order.

(ART. 2.) To form an adequate and correct conception of large numbers, we must have names for the different orders of places from the right hand, which will enable us to read off the numbers in words. This naming and reading is called

And we may thus go on, and repeat the tens and hundreds of trillions, and then take another name; and again and again repeat the tens and hundreds belonging to the

new name.

Hence we observe, that we can divide these orders into periods of three figures each, and give to each period its distinctive name, reading off the hundreds, tens and units, in each period.

80.652.941.600.807.362.546.278.009.650.208.

Which is read, eighty nonillions, six hundred fifty-two octillions, and so on to the last period, to which the name (units) is not added.

Divide off by points, and read the following numbers.

864321

76247523

9210043747829

100210031002478298

To write numbers from words, we begin on the left hand, taking care to fill up those periods or places that are omitted in the question.

Write the following numbers:.

Nine millions, seventy-two thousand, and two hundred.

Eight hundred millions, forty-four thousand, and fiftyfive.

Eight billions, sixty-five millions, three hundred and four thousand, and seven.

Fifty-four sextillions, three hundred trillions, sixty-seven millions, four hundred and twenty.

Seventy decillions, two hundred and thirty-one octillions, one billion, one hundred thousand, and three hundred.

Six hundred and forty thousand, four hundred and eighty-one.

Three millions, two hundred sixty thousand, one hundred and six.

From the system of notation already explained, it is evident that figures have a simple value and a local value. When a figure stands in the place of units, it has a simple value only.

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THE FOUR RULES OF ARITHMETIC.

(ART. 3.) As quantity admits of no other changes than increase and diminution, it is evident that all the operations of arithmetic must be based upon these only.

Under the first are comprehended addition and multiplication, and under the second, subtraction and division. Addition consists in finding the sum of two or more numbers.

Multiplication is the successive addition of a number to itself a given number of times.

Subtraction consists in finding the difference between two numbers, or diminishing one given number by another.

Division is the same as subtracting one number successively from another, in order to find how many times the smaller number is contained in the larger.

Addition, subtraction, multiplication and division, are called the four rules of arithmetic; all arithmetical operations whatever are combinations of these.

(ART. 4.) In arithmetical, and in all subsequent mathematical operations, characters and SIGNS are used to denote operations and conditions.

The perpendicular cross, thus +, denotes addition. The horizontal line, thus —, denotes subtraction.

The diamond cross, or a point, thus, X. (either one), indicates multiplication.

A horizontal line, with a point above and below, thus , indicates division. Points, thus, : :: or, :=: proportion. Double horizontal lines, thus =, equality. The following character represents square root, . The same character, with a small figure annexed, as 3, 4, 5, &c., thus 3, '√, 5√, indicates third, fourth and fifth roots.

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(ART. 5.) Numbers are referable to things; but when no reference is made, the number is said to be abstract; for instance, the number 3 is abstract, and can be applied

to any object whatever, as three apples, three horses, three dollars, &c.; and when it is so applied, it is no longer abstract.

Abstract numbers can be added together, as they are then understood to refer to the same thing or scale of measure, but numbers referring to different things, or different scales or standards of measure, cannot be put

into one sum.

Let the pupil early imbibe this important idea, that numbers referring to different things, cannot be added together, under any rule, nor in any part of arithmetic. For example: 3 apples and 5 apples are 8 apples; but 3 apples and 5 dollars are not 8 of the one or the other, and it is manifest that they cannot be added together.

As an objection to this, a person might say, that in a certain orchard there are 23 apple-trees, 5 pear-trees, and 7 cherry-trees; how many in all?

We certainly cannot add them together as apple-trees, because they are not all apple-trees, nor are they all peartrees or cherry-trees; but we can add them together under the general name of trees.

In the same way, in estimating a farmer's stock, we may add together horses, oxen, cows, sheep, &c., under the general name of animals.

Indeed we may go further, and say, In a certain field I saw 6 trees, 3 hay-stacks, and 2 oxen; how many objects were before my vision?

Here we may add together trees, hay-stacks, and oxen, or throw them into one sum, under the general name of objects.

Yet the first assertion holds good, that we cannot add numbers together unless they refer to one common standard. In the first observation above cited, we may put the numbers together as trees, in the second, as animals, in the third, as objects.

With these preliminary remarks, we commence with the first arithmetical operation.