METHOD OF PROOF.

Add the subtrahend and remainder together; if their sum is equal to the minuend, the work is right.

Take 16 10

£.

S.

d.

From 24

7

9

11

Remainder 7

16

10

Proof

24 7

9

Example.

Since the remainder is the difference between the two given numbers, it is evident, that this difference, added to the smaller number, will produce the greater.

Or, subtract the remainder from the minuend, and if this last remainder is equal to the subtrahend, the work is right.

USE OF COMPOUND SUBTRACTION.

The difference between sums and quantities of different denominations is often required in the transaction of business. By this rule the result is readily accomplished.

SECTION IX.

DECIMAL NUMBERS.

By Decimal Numbers, are meant any numbers which have a uniform mode of increasing by tens, i. e. ten of one denomination making one of the next higher. Thus, 10 ones, or units, make 1 ten; 10 tens make 1 hundred; 10 hundreds make 1 thousand, &c.

When the coin of the United States is examined, it will be found, that this is the manner of reckoning it: thus, 10 mills make one cent; 10 cents make one dime; 10 dimes make one dollar; 10 dollars make one eagle. (See table, page 172.)

If all the things, about which we have occasion to speak, were divided in this manner, it would greatly increase the facilities for the transaction of business. If, in long measure, a mile were divided into ten parts, and each of those into ten others, and each of these into ten others, &c. there would be far less liability to make errors in adding, subtracting,

&c..

When things are thus divided, we call the parts Decimal Fractions. See Sec. 15.

Decimal numbers are added in the same manner as the numbers in Simple Addition, as may be seen by the following examples.

The samé remark applies to Subtraction, &c.

The Denominations in Federal Money are Eagles, Dollars, Dimes, Cents, and Mills. The manner of division or increase is by a decimal proportion, as is seen in the last Section. An eagle divided into ten parts, each of those parts is a dollar; a dollar divided into ten parts, each of those parts is a dime, &c.

ADDITION OF Federal MONEY.

In this example we add these several sums of Eagles, Dollars, &c. as in Simple Addition. The amount is 12 Eagles, 8 Dollars, 9 Dimes, 5 Cents, and 7 Mills; or $128 95 cts. and 7 m., which is the more usual mode of reading sums in Federal Money, separating them by points, thus $128.95.7.

SUBTRACTION of Federal MONEY.

A man had 15 eagles, 6 dollars, 5 dimes, 7 cents and 5 He owed 12 eagles, 9 dollars, 8 dimes, 5 cents and 9 How much will he have left after paying his debt?

mills. mills.

RULE. Subtract as in Simple Subtraction. The remainder is 2 eagles, 6 dollars, 7 dimes, 1 cent and 6 mills, or $26, 71 cts. and 6 mills, = $26.71.6.

MULTIPLICATION of Federal Money.

Eight men had each 27 eagles, 5 dollars, 7 dimes, 8 cents How much had all?

and 5 mills.

=

4

Eight times 5 mills are 40 mills cts.; 8 times 8 cts. are 64, and the 4 reserved from the mills make 68: 6 ds. and 8 cts. 8 times 7 dimes are 56, and 6 added make 62; 8 times 5 dollars are 40, and 6 added make 46: 4 E. and

220 6 2 80 $6; 8 times 7 are 56, and 4 added make 60; 8 times 2 are 16, and 6 added make 22. All the men therefore have 220❤ eagles, 6 dollars, 2 dimes, and 8 cents= $2206.28. Here it will be seen that we multiply just as in Simple Multiplication.

DIVISION OF FEDERAL MONEY.

Eight men have 220 eagles, 6 dollars, 2 dimes, and 8 cents. How much money has each?

EXCHANGE FROM ONE CURRENCY TO ANOTHER. 213

Eagles.

→ Dollars. Dimes.

Mills.

8) 220 6 2 80

27 5 7 8 5

220 eagles divided by 8, is 27, and 4 E. to 40 dollars remain. To these add the 6 dollars; 8 in 46, 5 times, and 6 dollars to 60 dimes remain; to these add the 2 dimes, 8 in 62, 7 times, and 8 dimes = =to 80 cents remain; to these add the 8 cents; 8 in 68, 8 times, and 4 cents= = to 40 mills remain; 8 in 40, 5 times. Observe, that the quotient is of the same denomination as the dividend. Each man, then, has 27 eagles, 5 dollars, 7 dimes, 8 cents, and 5 mills, or $275.78.5.

In every sum in Federal Money which has mills, the right hand figure denotes mills, the two next cents, or cents and dimes, and the rest dollars, or dollars and eagles.

By the preceding examples in Addition, Subtraction, Multiplication and Division of Federal Money, it will be seen that all operations in this coin are performed just as in Simple Addition, and of course need no farther illustrations.

SECTION XI.

EXCHANGE

From one Currency to another.

By Currency is meant the mode of calling or counting money. In different countries these modes vary. Sterling is money reckoned by pounds, shillings, pence and farthings. Federal money is reckoned by eagles, dollars, &c. In all places it is agreed to divide a pound into 20 parts, and to call each of these parts a shilling. A shilling is divided into 12 parts, and each part is called a penny. But the worth of a pound in one place is not the same as in some other places. Hence some other standard must be used, in order to know the value of a pound in different places. The law establishes Federal Money as the currency of the United States. Still there are several currencies which are nominal in different parts. The pound is not often used; but shillings and pence are as commonly spoken of as dimes and cents. A shilling in Canada is a fifth part of a dollar; in New-England it is a sixth part of a dollar; in New-York, an eighth part; in

214 EXCHANGE FROM ONE CURRENCY TO ANOTHER.

Pennsylvania it is two fifteenth parts. In Canada 20 cents is the value of a shilling; in New-England, 16 cents and 7 mills, nearly; in New York, 12 cents and 5 mills; and in Pennsylvania, 13 cents and 5 mills, nearly. In reading the price of a ton, yard, &c. when given in shillings, it is necessary to know what is the currency of the place in order to have a knowledge of the real worth of the article. The most important principles or rules are those which direct how to bring pounds, shillings, &c. to Federal Money, and the contrary. If I wish to know how many dollars, cents, &c. are contained in 23 £. 7 s. and 9 d., I can ascertain in several ways. As 6 s. make a dollar in New-England, and there are 20 s. in 1 £., there must be one sixth as many dollars as there are shillings; one sixth of 20-3 and one third. If there are 3 dollars in a pound, there are 23 times as many in 23 £.76 and two thirds dollars=$76.66.6+. In 7 shillings there is 1 dollar, and I shilling remains - 16 cents and 7 mills, nearly. As 9 d. is three fourths of 12 d. or 1 s., it is equal to 12 cents and 5 mills. The whole sum is $76.66.6 +$1.16.7+12 cents and 5 mills $77.95.8.

=

But a much shorter way is that of multiplying the pounds by 10, which makes the value of a unit then worth one third of a dollar.

In 2 £. how many dollars?

2×10=20. Each unit is worth one third part of a dollar, because 20 is one half the number of shillings in 2 £., and of course a unit is worth 2 s., and 2 s. is equal to one third of 6 s. or $1.

If in a proposed sum there are shillings, pence and farthings, the following may be adopted as general

RULES.

1. Write down the pounds, and at the right hand put half the number of shillings.

If there is an odd shilling, reduce it to farthings, and add to these the farthings in the pence and farthings. (A unit of this number will be as near the value of of 1 cent, as 48 is to the number 50. If the odd shilling is called 50 instead of 48, and a proportionate addition made to the farthings in the given pence and farthings, and then part of the amount be taken, it will be cents.)

If the change is to be made to New-England currency,