units are called concrete numbers: for example, one dollar, two dollars. But when no particular thing is indicated by the unit, it is an abstract unit; and hence arise abstract numbers: for example, one and one make two.

Without the use of numbers, we cannot know precisely how much any quantity is, nor make any exact comparison of quantities. And it is by comparison only, that we value all quantities; since an object, viewed by itself, cannot be considered either great or small, much or lit tle; it can be so only in its relation to some other object, that is smaller or greater.

ARITHMETIC treats of numbers: it demonstrates their various properties and relations; and hence it is called the Science of numbers. It also teaches the methods of computing by numbers; and hence it is called the Art of numbering.

II

NOTATION AND NUMERATION.

NOTATION is the writing of numbers in numerical characters, and NUMERATION is the reading of them.

The method of denoting numbers first practised, was undoubtedly that of representing each unit by a separate mark. Various abbreviations of this method succeeded; such as the use of a single character to represent five, another to represent ten, &c.; but no method was found perfectly convenient, until the Arabic FIGURES OF DIGITS, and DECIMAL System now in use, were adopted. These figures are, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; denoting respectively, nothing, one unit, two units, three units, &c.

To denote numbers higher than 9, recourse is had to a law that assigns superior values to figures, according to the order in which they are placed. viz. Any figure placed to the left of another figure, expresses ten times the quantity that it would express if it occupied the place of the latter. Hence arise a succession of higher orders of units.

As an illustration of the above law, observe the dif ferent quantities which are expressed by the figure 1

When standing alone, or to the right of other figures, 1 represents 1 unit of the first degree or order; when standing in the second place towards the left, thus, 10, it represents 1 ten, which is 1 unit of the second degree. when standing in the third place, thus, 100, it represents 1 hundred, which is 1 unit of the third degree; and so on. The zero or cipher (0) expresses nothing of itself, being employed only to occupy a place.

The units of the second degree, that is, the tens, are denoted and named in succession, 10 ten, 20 twenty, 30 thirty, 40 forty, 50 fifty, 60 sixty, 70 seventy, 80 eighty, 90 ninety. The units of the third degree, that is, the hundreds, are denoted and named, 100 one hundred, 200 two hundred, 300 three hundred, and so on to 900 nine hundred. The numbers between 10 and 20 are denoted and named, 11 eleven, 12 twelve, 13 thirteen, 14 fourteen, 15 fifteen, 16 sixteen, 17 seventeen, 18 eighteen, 19 nineteen. Numbers between all other, tens are de

noted in like manner, but their names are compounded of the names of their respective units; thus, 21 twenty-one, 22 twenty-two, 23 twenty-three, &c.; 31 thirty-one, 32 thirty-two, &c. &c. This nomenclature, although not very imperfect, might be rendered more consistent, by substituting regular compound names for those now applied to the numbers between 10 and 20. This alteration would give the names, 11 ten-one, 12 ten-two, 13 ten-three, &c.

As the first three places of figures are appropriated to simple units, tens, and hundreds, so every succeeding three places are appropriated to the units, tens, and hundreds of succeeding higher denominations. For illustration, see the following table.

Duodecillions.

460 725 206 194 007 185 039 000 164 396 205 013 008 741 By continuing to adopt a new name for every three degrees of units, the above table may be extended indef

initely. Formerly, the denominations higher than thousands were each made to embrace six degrees of units; taking in, thousands, tens of thousands, and hundreds of thousands. The mode of applying a name to every three degrees, however, is now universal on the continent of Europe, and is becoming so in England and America.

The learner may denote in figures, the following numbers, which are written in words.

Example 1. Four hundred seventy-eight million, two hundred forty-one thousand, and one hundred.

2. Seven million, six hundred ninety-two thousand, and eighty-nine.

3. Nineteen million, twenty thousand, and five. 4. Eight hundred billion.

5. One billion, six hundred forty-four thousand, five hundred and thirteen.

6. One trillion, five hundred thirty-four billion, three million, eighteen thousand, and four.

7. Two hundred billion, sixteen thousand and one. 8. Eleven billion, one million, and sixty.

9. Five trillion, eight billion, four million, nine thousand, and seven.

10. One hundred trillion, twenty billion, three hundred million, two thousand, and four.

11. Thirty-one trillion, five hundred, and sixty.

12. Six quadrillion, two hundred and fourteen trillion. 13. Two hundred forty-nine quadrillion, seventy-five thousand, and twenty-two.

14. Forty-six quintillion, one quadrillion, nineteen billion, seven hundred and eight.

15. Nine hundred sextillion, three hundred twentyfive trillion, two thousand, and fourteen.

INDICATIVE CHARACTERS OR SIGNS.

The sign+(plus) between numbers, indicates that they are to be added together; thus, 3+2 is 5.

The sign-(minus) indicates, that the number placed after it, is to be subtracted from the number placed be fore it; thus, 5-2 is 3

The sign

(into) indicates that one number is to be multiplied into another; thus, 4×3 is 12.

The sign

(by) indicates that the number on the left hand is to be divided by the number on the right hand; thus, 12÷3 is 4.

The sign=(equal to) indicates that the number before it, is equal to the number after it; for example, 4+2=6. 6-24. 5 X 3=15.

15÷3=5.

III.

ADDITION.

ADDITION is the operation by which two or more numbers are united in one number, called their sum. It is the first and most simple operation in arithmetic, effecting the first and most simple combination of quantities.

The primary mode of forming numbers, by joining one unit to another, and, this sum to another, and so on, exhibits the principle of addition. When numbers, which are to be added, consist of units of several degrees, such as tens, hundreds, &c., it is found convenient to add together the units of each degree by themselves; and since ten units of any degree make one unit of the next higher degree, the number of tens in the sum of each degree of units is carried to the next higher degree, and added thereto.

RULE FOR ADDITION.

Write the numbers, units under units, tens under tens, &c. Add each column separately, beginning with the column of units. When the sum of any column is not more than 9, write it under the column: when the sum is more than 9, write only the units' figure under the column, and carry the tens to the next column. Finally, write down the whole sum of the left hand column.

1. What is the sum of 370+ 90264+1470+40060? 2. What is the sum of 4000+570+99+54+273+ 69073+4000+61998+752?

3. What is the sum of 243+5021+7628+927+64 +5823+742+796+5009+ 325 +-7426 +31186 +

987+6954+2748?

4. What is the sum of two thousand and seven, forty four million five hundred and sixty-one, one hundred mil lion, six billion twenty-eight thousand and eleven?

IV.

SUBTRACTION.

SUBTRACTION is the operation by which one number is taken from another.

The number from which another is to be taken is called the minuend, and the number to be taken is called the subtrahend. The number resulting from the operation shows the remainder of the minuend, after the subtrahend has been taken out; it also shows the difference between the minuend and subtrahend, or the excess of the former above the latter. The subtrahend and remainder may be considered the two parts into which the minuend is separated by the operation; and in this view, subtraction is the opposite of addition, in as much as addition unites several quantities in one sum, and subtraction separates a quantity into two parts.

Subtraction is performed by taking the units of each degree in the subtrahend, from those of corresponding degree in the minuend, and severally denoting the remainders. When the units of any degree in the subtra hend exceed those of the same degree in the minuend, we mentally join one unit of the next higher degree to the deficient place in the minuend, and consider the units of the higher degree to be one less than they are denoted: this process is the reverse of carrying in addition. One other method may be adopted in this case; viz. Increase both the minuend and subtrahend, by mentally adding ten to the deficient place in the former, and, one to the next higher degree of units in the latter. This method is justified by the self-evident