must, therefore, be three decimal places in the quotient; and so I prefix a cipher to the quotient to make up that number of places, and place the decimal point before it.

2. Divide 8.6 by 2.718 2.718)8.600(3.164+ Quotient.

8 154

4460

2718

17420

16308

11120

10872

248

Here, I annex two ciphers to the given dividend, to make up as many decimal places in it as there are in the divisor. After the division of this dividend, there is a remainder, and I carry on the operation farther, according to Note 1.

3. Divide 17.1 by 8. 8)17.1 Here, after dividing 17.1 by 8, there is a remainder, and I suppose ciphers to Quot. 2.1375 be annexed to the dividend, and continue the operation till nothing remains.

4. Divide 8564.825 by 63.21 5. Divide 56.7 by .7

6. Divide 246.1 by 6.0427 7. Divide 7.25406 by 957. 8. Divide 76 by .7438 9. Divide 65.8 by 1.2 10. Divide 27 by .05

CONTRACTION I.

Ans. 135.49+
Ans. 81.

Ans. 40.72+

Ans. .00758 Ans. 102.17+ Ans. 54.833,&c. Ans. 540.

When the divisor is an integer, with any number of ciphers on the right hand; cut off those ciphers, and remove the decimal point in the dividend as many places farther to the left as there are ciphers cut off from the divisor; prefixing ciphers to the dividend, if necessary: then divide the dividend by the remaining part of the divisor, as usual.

EXAMPLES.

1. Divide 7.14 by 7400.

Note. When the divisor is 10, or 100, or 1000, &c., the quotient may be found by merely removing the decimal point in the dividend as many places farther to the left as the divisor has ciphers; prefixing ciphers, if necessary. So 21.4÷10-2.14; and .54÷100-.0054; &c.

CONTRACTION II.

To contract division when there are many figures in the divisor, and it is required to find only a certain number of figures in the quotient.

RULE 1.

1. Take as many of the left hand figures of the divisor as will be equal to the number of figures (both integers and decimals) required to be found in the quotient, and find how many times they may be had in the first figures of the dividend, as usual.

2. Let each remainder be a new dividual, and for every such dividual reject one figure more from the divisor; and in making out each subtrahend, carry from the figures cut off from the divisor, as in the 2d contraction in Multiplication of Decimals.

Note 1.-When there are not as many figures in the given divisor as are required to be in the quotient, begin the operation with all the figures, as usual, and continue it until the number of figures in the divisor and those remaining to be found in the quotient are equal; after which use the

contraction.

Note 2.-To know where to place the decimal point in the quotient; find, by the general rule for dividing decimals, the value, or place, of the first quotient figure, and you

will then know where the decimal point must be placed.

EXAMPLES.

1. Divide 2508.9324 by 92.4105, so as to have only three decimal places in the quotient, in which case the quotient will contain five figures.

Ans. 17.8345

2. Divide 4109.2351 by 230.4091, so as to have six figures in the quotient. 3. Divide 37104.36 by 57.1396, so that the quotient may contain seven figures.

Ans. 649.3633

4. Divide 913.08 by 21372, so that the quotient may contain four places of decimals.

RULE 2.

Ans. .0427

Take, for a defective divisor, one or two more of the left hand figures of the given divisor than the number of figures required to be found in the quotient, and divide the dividend by this defective divisor, as usual.

Note. When any of the figures rejected from the divisor are integers, then you must remove the decimal points in the divisor and dividend, or suppose them to be removed, as many places farther to the left as there are integral· figures rejected from the given divisor.

EXAMPLES.

1. Divide 721175.62 by 222574.12, so as to have only hree figures in the quotient.

Operation.

22257,4.12)721175.62(

22257)72117.562(3.24 Quotient.

66771

2. Divide 250.8928 by 92.41052, and find only three figures in the quotient.

Ans. 2.71

3. Divide 12.169825 by .031415926, so as to have four figures in the quotient. Ans. 387.3

FEDERAL MONEY.

Having explained the nature of decimal fractions, and given rules for adding, subtracting, multiplying, and dividing decimals, I shall now proceed to show the application of those rules to the currency of the United-States, usually denominated Federal Money.

The denominations of federal money increase, from the lowest to the highest, in a tenfold ratio, like whole numbers and decimal fractions. The denominations are as follows:

There are coins of all these denominations, excepting that of mills, which is merely nominal.

As all the denominations of federal money increase in a tenfold ratio, like whole numbers, it is very obvious that

the value of any sum of federal money consisting of several denominations, may be expressed in the lowest denomination mentioned, by writing the numbers of the several denominations one after another, in regular order, beginning with the highest, and placing the figures so that they may be read as one whole number. Thus, 17 eagles, 5 dollars, 6 dimes, 2 cents, and 8 mills, are equal to 175628 mills. In writing down sums of federal money in this manner, if any denomination between the highest and lowest, mentioned in the given sum, be wanting, a cipher must be written in its place: Thus, 6 dollars and four cents, are equal to 604 cents; and 7 dollars and 5 mills, are equal to 7005 mills.—It is also evident, that if the number of any one denomination be considered so many units, the lower denominations will be decimal parts of a unit, and may therefore be expressed like other decimal fractions. So, 8 eagles, 4 dollars, 6 dimes, 7 cents, and 5 mills, are equal to 8.4675 eagles,-84-675 dollars, $16.75 dimes, 8467.5 cents, 84675 mills.

In reckoning in federal money, dollars are considered as units or integers, and parts of a dollar, as decimal parts of a unit: So, 5 dollars, 8 dimes, 4 cents, 2 mills, are written thus, $5.842; and 7 eagles, 2 dollars, and 4 cents, thus, $72.04; &c. Hence, Addition, Subtraction, Multiplication, and Division of Eederal Money, are performed by the foregoing rules for decimal fractions.

Eagles are usually considered as tens of dollars, and dimes as tens of cents; and the names of cagles and dimes are seldom mentioned; accounts being kept in dollars, cents and mills, or more frequently in dollars and cents only, the mills being usually considered of too little value to be retained. So, $27.57, is read thus, twenty-seven dollars and fifty-seven cents, or twenty-seven dollars and fifty-seven hundredths; and $.254, thus, twenty-five cents, four mills, or two hundred fifty-four thousandths of a dollar.

In writing down parts of a dollar in the decimal form, always remember to prefix a cipher to the cents when the number is less than 10, and two ciphers to the mills, when the fractional part of a dollar consists of any number of mills less than 10. So, write 7 dollars and 4 cents thus, $7.04; and 14 dollars and 5 mills thus, $14.005.