In the first of these examples I divide the minutes by 60, to bring them into hours, by cutting off the o from the 6, and dividing by 6 only, as taught in Divifion, and the quotient is 410828 hours, and there remains 4 minutes, which, placed before the 1 cut off from the dividend, makes 41 minutes for a remainder. This quotient I again divide by 2, and that quotient by 12, which is equal to dividing by 24, the number of hours in 1 day, and the quotient is 17117 days, and there remains 10; but to find the true remainder, I multiply the 10, the last remainder, by 2, the first divifor of the hours, and the product is 20 for the true remainder: thus the answer to the question is, 17117 days, 20 hours, 41 minutes.

In the fecond example, I divide the grains by 2 and 12, to bring them into pennyweights, as 24 grains make 1 pennyweight; the quotient is 1974 pennyweights, and 22 grains remain. The pennyweights I divide by 20, to bring them into ounces, and the quotient is 98 ounces, and 1 pennyweight remains, to which I bring down the 4 I cut off from the dividend, and the laft remainder is 14 pennyweights: thus the answer is 98 ounces, 14 pennyweights, and 22 grains.

In the laft example, I divide the perches by 40, as 40, perches make i rood, and the quotient is 736824 roods, and 26 perches; the roods I divide by 4, to bring them into acres; and the answer is found to be 184,206 acres, and 26 perches.

Reduction, both afcending and defcending, may be per-1 formed by one divifor or mul iplier: thus, to bring farthings into pounds, the pounds may be divided by 960, the number of farthings in a pound, and the quotient will give the number of pounds, and the remainder (if any) must be refolved into the inferior denominations. And to reduce hundredweights into fingle pounds, they may be multiplied by 112, the pounds in an hundred-weight, and the product is the anfwer. But the method before laid down is the more regular, and at the fame time the more expeditious way of performing this rule.

Examples

Examples of both Kinds for Practice.

Example 8. If 420 pieces of cloth contain 8420 ells Flemish, it is required to know how many ells English they contain? -Anf. 5052 ells English.

Example 9. In 220 puncheons of rum how many hogsheads?-Anf. 293 hogfheads, and 21 gallons remain.

Example 10. A filversmith hath 1000 ounces of filver to be made into spoons, falts, and tankards; each spoon to weigh 2 oz. 12 pwt. each falt 3 oz. and each tankard 30 07. and to make an equal quantity of each, it is defired to know how many he can make of each-Anf. 28, and 64 pwt. remain.

In the first of thefe examples the 8420 ells Flemish are reduced into quarters of a yard, by multiplying by 3 (as there are 3 quarters in an ell Flemish), and then brought into English ells, by dividing thefe quarters by 5, the number of quarters in an ell English.

The 9th example is wrought in the fame manner: viz. by reducing the 220 puncheons into gallons, by multiplying by 84, and bringing these gallons into hogfheads, and by dividing them by 7 and 9, which is equal to 63, the gallons in a hogfhead.

This method of reduction always takes place when the lefs denomination

Y 2

denomination is not contained any certain number of times exactly in the greater.

In the roth example I reduce the weight of 1 spoon, 1 falt, and I tankard, into pennyweights, by multiplying by 20, and then add them together; and by the total 712, I divide the 1000 ounces of filver, which is also reduced into pwts. and it quotes 28 of each, and 64 pwts. remain.

By reduction we are enabled to reduce the coin of one country into that of another, without having recourfe to the rule of three, or exchange.

Example 11. What is the value, in English coin, of 350 ducats, at 4s. 2d. per ducat?

Example 12. In 2461, 18s. 6d. Flemish money, how much English, the course of exchange being 30s. 6d. per pound fterling?

Example 13. How much money English is there in 4420 pieces of eight, the course of exchange being at 49 d. fterling?

In the 11th example, I multiply the 350 ducats by 50, to bring them into pence, and then divide by 12, and it quotes 1458 fhillings and 4 pence: I then divide the fhillings by 20, and the quotient is 727. 18s. 4d. for the answer.

In the 12th example the 2467. 18s. 6d. Flemish money is reduced into pence, and then divided by 366, the course of exchange; and the answer is 1617. 18s. 44d. and 114 farthing's remain.

The 13th example is wrought in the fame manner as the 11th: viz. by reducing the pieces of eight into pence; and for the fraction of a penny, I multiply the given number of pieces of eight by 5, the numerator, or upper figure of the fraction, and divide the product by 8, the denominator, or lower figure of the fraction, and to the quotient I add the pence contained in the pieces of eight, and then reduce the whole into fhillings and pounds, by dividing by 12 and 20, as before, and the answer is 9137. 18s. 4d.

This method of reducing foreign coin into English may serve for those who are unacquainted with the rule of practice; for practice performs this much more expeditiously, as will be fhewn in its proper place.

Thofe fums in reduction, in which both divifion and multiplication are used, muft be proved by multiplication and divifion; as, for example, that part which is performed by multiplication must be proved by divifion, and that part performed by divifion must be proved by multiplication.

The foregoing examples, perfectly understood, will be fufficient to give the learner a complete knowledge of this rule, and the various uses to which it may be applied.

It is abfolutely neceffary that the learner be perfectly acquainted with what has been delivered in the foregoing part of this chapter, as all the following rules in arithmetic are performed by one or more of the foregoing rules; I have therefore been more explicit in the former part; being the bafis of the whole.

SECT.

SECT. VII.

THE GOLDEN RULE; OR, SINGLE Rule of thrEE DIRECT.

THIS rule, which, for its universal ufe in most parts of the mathematics, is called the golden rule, is alfo called a rule of proportion, because the number fought bears a certain proportion to one of the numbers given.

It is called the rule of three, because it confifts of three given numbers, from which a fourth number is to be found, which, in the direct rule, bears the fame proportion to the fecond number as the third does to the first.

Rule. Multiply the fecond and third numbers together, and divide the product by the first number, and the quotient is the answer fought, or the fourth number.

ift number.

2d number.

3d number. Example 1. If 3 yds. of muslin coft 125. what will 9 yds.

coft at that rate?

9

3)108 Answer 36s.

Here the fourth number or answer, 36, bears the fame proportion to the second number 12, as 9 the third number, bears to the first number 3, that is, it contains it three times; or it bears the fame proportion to the third number 9 as the fecond number 12 does to the first number 3, viz. contains it four times. This proportion is called direct proportion; from whence this rule is called the rule of three direct, and is always performed as above.

But when the fourth number bears the fame proportion to

the