Here any figure occupying the first place, reckoning from right to left, denotes only its simple value or number of units. But the figure standing in the second place denotes ten times its simple value; that occupying the third place, a hundred times its simple value, and so on to any required number of places; the value of any figure being always increased tenfold by its removal one place to the left.

Thus, in the number 1834, the 4 in the first place denotes only four units, or simply 4; the 3 in the second place signifies three tens, or thirty; the 8 in the third place signifies eighty tens, or eight hundred; and the 1 in the fourth place, one thousand; so that the whole number is read thus,—one thousand eight hundred thirty-four.

Although the cipher has no value of itself, when standing alone, yet, being joined to the right hand of significant figures, it increases their value in a tenfold proportion; thus, 5 signifies simply five, while 50 denotes five tens, or fifty; 500, five hundred, and so on.

NOTE. The idea of number is the latest and most difficult to form. Before the mind can arrive at such an abstract conception, it must be familiar with that process of classification, by which we successively remount from individuals to species, from species to genera, from genera to orders. The savage is lost in his attempts at numeration, and significantly expresses his inability to proceed by holding up his expanded fingers or pointing to the hair of his head. See Lacroix

NUMERATION TABLE.

The following is the French method of enumeration, and is in general use in the United States and on the continent of Europe.

In order to enumerate any number of figures by this method, they should be separated by commas into divisions of three figures each, as in the annexed table. Each division will be known by a different name. The first three figures, reckoning from right to left, will be so many units, tens, and hundreds, and the next three so many thousands, and the next three so many millions, &c.

The value of the numbers in the annexed table, expressed in words, is One hundred twenty-three vigintillions, four hundred fifty-six novemdecillions, seven hundred eighty-nine octodecillions, one hundred twenty-three septendecillions, four hundred fifty-six sexdecillions, seven hundred eighty-nine quindecillions, one hundred twenty-three quatuordecillions, four hundred fifty-six tredecillions, seven hundred eighty-nine duodecillions, one hundred twenty-three undecillions, four hundred fifty-six decillions, seven hundred eighty-nine nonillions, one hundred twenty-three octillions, four hundred fifty-six septillions, seven hundred eighty-nine sextillions, one hundred twenty-three quintillions, four hundred fifty-six quadrillions, seven hundred eighty-nine trillions, one hundred twenty-three billions, four hundred fifty-six millions, seven hundred eightynine thousands, one hundred twentythree units.

317,897;431,032;639,864;361,316;461,315;123,675;816, 131;123,456;123,614;315,18

NUMERATION TABLE.

Thousands.
Tredecillions.

Thousands.
Duodecillions.

Thousands.
Undecillions.

Thousands.
Decillions.
Thousands.
Nonillions.
Thousands.

Octillions.
Thousands.

The following is the old English method of enumeration, but it has become almost obsolete in this country. In order to enumerate any number of figures by this method, they should be separated by semicolons into divisions of six figures each, and each division separated in the middle by a comma, as in the annexed table. Each division will be known by a different name. The first three figures, in each division, reckoning from right to left, will be so so many units, tens, and hundreds of the name belonging to the division, and the three on the left will be so many thousands of the same name. The value of the numbers in the annexed table, expressed in words, is Three hundred and seventeen thousand, eight hundred and ninety-seven tredecillions; four hundred and thirty-one thousand, thirty-two duodecillions; six hundred thirty-nine thousand, eight hundred sixty-four undecillions; three hundred sixty-one thousand, three hundred sixteen decillions; four hundred sixty-one thousand, three hundred fifteen nonillions; one hundred twenty-three thousand, six hundred seventy-five octillions; eight hundred sixteen thousand, one hundred thirty-one septillions; one hundred twenty-three thousand, four hundred fifty-six sextillions; one hundred twenty-three thousand, six hundred fourteen quintillions; three hundred fifteen thousand, one hundred thirty-one quadrillions; three hundred ninetyeight thousand, eight hundred thirty-two trillions; five hundred sixty-three thousand, eight hundred seventy-one billions; three hundred fifty-one thousand, six hundred fifteen millions; one hundred twenty-three thousand five hundred sixty-one.

2;563,871;351,615;123,561.

Septillions.
Thousands.
Sextillions.
Thousands.
Quintillions.
Thousands.
Quadrillions.
Thousands.
Trillions.

Thousands.
Billions.

Thousands.
Millions.

Thousands.
Units.

NOTE. The student must be familiar with the names, from units to tredecillions, and from tredecillions to units, so that he may repeat them with facility either way.

Let the following numbers be written in figures:*

1. Twenty-nine.

2. Four hundred and seven.

3. Twenty-three thousand and seven.

4. Five millions and twenty-seven.

5. Seven millions, two hundred five thousand and five.

6. Two billions, two hundred seven millions, six hundred four thousand and nine.

7. One hundred five billions, nine hundred nine millions, three hundred eight thousand two hundred and one.

8. Nine quintillions, eight billions and forty-six.

9. Fifteen quintillions, thirty-one millions and seventeen. 10. Five hundred seven septillions, two hundred three trillions, fifty-seven millions and eighteen.

11. Nine nonillions, forty-seven trillions, seven billions, two millions, three hundred ninety-two.

12. Fifteen duodecillions, ten trillions, one hundred twentyseven billions, twenty-six millions, three hundred twenty thou sand four hundred twenty-six.

*To express numbers by figures, begin at the left hand with the highest order mentioned, and, proceeding to units, write in each successive order the figure which denotes the given number in that order. If any of the intervening orders are not mentioned in the given number, supply their places with ciphers.

SECTION II.

ADDITION.

ADDITION is the collecting of numbers to find their sum.

1. A man has three farms; the first contains 378 acres, the second 586 acres, and the third 168 acres. How many acres are there in the three farms?

OPERATION.

Acres.

378

586

168

In this question, the units are first added, and their sum is found to be 22; in 22 units there are two tens and two units. The two units are written under the column of units, and the 2 (tens) are carried to be added with the tens, which are found to amount to 23 tens 2 hundreds and 3 tens. The 3 is written under the column of tens, and the 2 (hundreds) is carried to the column of the hundreds, which amount to 11=1 thousand 1 hundred. The whole of which is set down. Hence the propriety of the following

1132

RULE.

Write units under units, tens under tens, &c. Then begin at the bottom and add the units upwards, and, if the amount be less than ten set it down under the column of units; but if the amount be ten or more write down the unit figure, and add the figure denoting the number of tens to the column of tens. Thus proceed, till every column of figures is added, writing down on the left the sum total of the left. hand column, and the result will be the sum of the whole as required.

PROOF.

Begin at the top and add all the columns downwards, in the same manner as they were before added upwards; then, if the two sums agree, the work may be presumed to be correct.