By this measure, brandy, spirits, perry, cider, mead, vinegar,

and oil are measured.

231 cubic inches make a gallon, and 10 gallons make an anchor.

100. It is evident, that if the several denominations of money, weight and measure proceeded in a decimal ratio, the fundamental operations might be performed upon these, as upon abstract numbers. This may be shown by a few examples in Federal Money. If it were required to find the sum of $46,85

and $256,371, we should place the numbers of the same denomination in the same column, and add them together as in whole numbers; thus,

4685 256371

303221

and the answer may be read off in either or all the denominations; we may say 30 eagles S dollars 22 cents 1 mill, or 303 dollars 221 thousandths, or 30322 cents and 1 tenth, or 303221 mills. It is usual to consider the dollars as whole numbers, and the following denominations as decimals. The operation then becomes the same as for decimals.

101. WHEN the different denominations do not proceed in a decimal ratio, they may all be reduced to one denomination, and then the fundamental operations may be performed upon this, as upon an abstract number. If, for example, the sum to be operated upon were £4 15s. 9d. this may easily be expressed in

pence. As 1 pound is 20 shillings, 4 pounds will be 4 times 20, or 80 shillings. If to this we add the 15s. we shall have 95s. 9d. equivalent to the above. But as 1 shilling is equal to 12 pence, 95s. will be equal to 95 times 12 or 1140 pence. Adding 9 to this, we shall have 1149 pence as an equivalent expression for £4 15s. 9d. We may now make use of this number as if it had no relation to money or any thing else; and the result obtained may be converted again into the different denominations by reversing the process above pursued. If it were proposed to multiply this sum by another number, 37 for instance, we should find the product of these two numbers in the usual way; thus,

1149

37

8043

3447

42513

42513 is, therefore, equal to 37 times £4 15s. 9d. expressed in pence; to find the number of pounds and shillings contained in this, we first obtain the number of shillings by dividing it by 12, which gives 3542, and then the number of pounds by dividing this last by 20; thus,

42513 pence then is equal to 3542 shillings and 9 pence, or to 177 pounds 2 shillings and 9 pence. Whence 37 times £4 15s. 9d. is equal to £177 2s. 9d.

It may be remarked, that shillings are converted into pounds by separating the right hand figure and dividing those on the left by 2, prefixing the remainder, if there be one, to the figure separated for the entire shillings, that remain. This amounts to dividing, first, by 10 (90), and then that quotient by 2. If 10 shillings made a pound, dividing by 10 would give the number of pounds, but as 10 shillings are only half a pound, half this number will be the number of pounds.

By a method similar to that above given, we reduce other denominations of money and the different denominations of the several weights and measures to the lowest respectively. If it were required to find how many grains there are in 2lb. 4oz. 17dwt, 5grs. Troy, we should proceed thus,

By dividing 13853 by 24, and the quotient thence arising by 20, and this second quotient by 12, we shall evidently obtain the number of pounds, ounces, pennyweights and grains in 13853 grains. The operation may be seen below.

These examples will be sufficient to establish the following general rules, namely;

To reduce a compound number to the lowest denomination contained in it, multiply the highest by so many as one of this denomi nation makes of the next lower, and to the product add the number belonging to the next lower; proceed with each succeeding denomination in a similar manner, and the last sum will be the number required.

To reduce a number from a lower denomination to a higher, divide by so many as it takes of this lower denomination to make one of the higher, and the quotient will be the number of the higher ; which may be further reduced in the same manner if there are still higher denominations, and the last quotient together with the several remainders will be equivalent to the number to be reduced.

In 58099 half pence how many pounds &c. ? Ans. 121l. Os. 94d. In 48 guineas at 28s. each how many 4 pence?

Ans. 3584.

In one year of 365d. 5h. 48′ 48′′ how many seconds?

Ans. 31556928.

102. When we have occasion to make use of a number consisting of several denominations as an abstract number, instead of reducing the several parts to the lowest denomination contained in it, we may reduce all the lower denominations to a fraction of the highest. Taking the sum before used, namely, 41. 15s. 9d. we reduce the lower denominations to the higher, as in the last article by division. The number of pence 9, or, is divided by 12, by multiplying the denominator by this number (54), we have thus,s. which being added to 15s. or 180s, the whole number being reduced to the form of a fraction of the same denominator, we have 10 and, which being added, make This is further reduced to pounds by dividing it by 20,

189.

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