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SECT. XVIII.

OF THE Rule of false, SINGLE AND DOUBLE (GENERALLY CALLED POSITION).

THIS rule teacheth to answer fuch queftions as cannot be refolved by any direct rule in vulgar arithmetic, and must be performed by falfe or feigned numbers.

Pofition is either fingle or double.

Single pofition is when the queftion can be refolved by one falfe pofition or fet of feigned numbers, and one operation in the rule of three direct.

Rule. Take any number, and proceed exactly the fame as if it were the true number through all the proportions mentioned in the question.

Then say (by the rule of three) as the refult of this falfe operation is to any of its parts, fo is the true refult in the queftion to the correfponding part required.

Example 1. A fon asking his father his age, the father aufwered, I am double the age of your eldest brother John, and he is three times the age of your youngest brother Henry, and the fum of all our ages is So years; what is each person's age?

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Here I fuppofe Henry's age to be 6, at which supposition John's age must be 18, the father's 36, and the fum of these three is 60, whereas it fhould be 80; then I fay, as 60 the fum of the falfe fuppofition, is to 80 the fum of the true one,

1

Gg 2

fo

fo is 6 the fuppofed age of Henry to 8 his true age; therefore John's age is 24, and the father's 48, as in the example.

Qu. 2. A afked B how much money he had in his pocket; B answered, If you give me 4 guineas of the money in your pocket, I fhall have five times as much as you will then have; but if, instead of that, I fhould give you 2 guineas of the money in my pocket, you will then have twice as much as I fhall then have: how much money had each ?—Anf. 6 guineas.

Qu. 3. A perfon hired a horfe for 9 days, on the following terms for the firft 3 days he was to pay of the hire for the next 3 days, and for each of the laft 3 days as much as the hire for the firfl 6 days; the whole was 27. 8s.; what was it per day?-Anf. 15. per day the first 3 days, 35. per day the next 3 days, and 125. per day the 3 last days.

Double pofition, or the double rule of falfe, is when two falfe pofitions are requifite to give an answer to the question. Rule 1. Take any two convenient numbers, and work with each of them according to the queftion, as in fingle pofition. 2. Find the difference between the refult of each of these falfe pofitions and the refult of the question; these differences are called the errors. 3. Multiply each error by the contrary pofition, that is, the firft error by the fecond pofition, and the second error by the firft pofition; then find the fum and difference of the products. 4. If the errors are both alike, that is, if the refult of the two pofitions be both greater or both less than the refult of the question, divide the difference of the products by the difference of the errors, and the quotient will be the answer. 5. But if the errois be unlike, that is, if the refult of one pofition be greater and the other lefs than the truth, then the fum of the products must be divided by the fum of the errors, and the quotient will be the answer.

Example 1. Three perfons, A, B, C, built a houfe, which coft 500l. of which B paid half as mnch again as A, and C paid as much as A and B together;,what did each pay?

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Difference of errors 3,0)300,0 difference of the products.

£100 for A's fhare, wherefore

B must have paid 150 and D must have paid 250

being half as much again as A, being as much as both A and B,

which added togethergives£500 the original fum.

From this example nfay be feen the method of working this rule, which is always the same, except when the errors are unlike; then the fum of the products is to be the dividend, and the fum of the errors the divifor as above directed.

24.2. A falefman bought a number of oxen, sheep, and lambs, for which he paid 1157.; for the oxen he paid 10l. each, for the sheep 20s. each, and for the lambs 10s. each; how many of each fort did he buy?-Anf. 10 of each.

Qu. 3. Three perfons, A, B, C, have equal incomes; A faves of his income every year; B fpends 10l. per annum more than A, and C fpends 1cl. per annum more than B. At the expiration of five years, C finds himself in debt 50l. what is each perfon's income, and what has each faved or fpent?-Anf. the income of each is 100l. per annum; A has faved 50l.; B has faved nothing; and C has fpent 50%. more than his income.

Qu. 4. A labourer was hired for 40 days: for every day he wrought he was to receive 2s. and for every day he was idle he was to forfeit is.; at the end of the time he had to receive 445.; how many days did he work, and how many was he idle?—Ans, he wrought 28 days and was idle 12 days.

SECT.

SECT. XIX.

OF EXCHANGE.

EXCHANGE is that rule which teacheth to find what fum of the money of one country is equal to a given sum of the money of another country, the courfe of exchange being

known.

The course of exchange is that fum of the money of one country which is proposed to be given for a certain conftant fum of that of another country: thus, when we fay the courfe of exchange between England and Holland is 34s. Flemish per pound fterling, it fignifies that 1 pound sterling is equal to the value of 345. in Flemish money. This courfe of exchange varies on the part of the foreign coins, according to the ftate of public affairs.

The par of exchange is that quantity of the coin of one country which is intrinfically equal to a certain quantity of the coin of another country, according to the value of the metal.

Moft foreign countries have two forts of coins, called current money and banco money; the first is that in general use throughout the country; the latter is that kept in the banks of thofe places, and is finer than the other; the difference between any fum as it is valued in current money and banco money is called the agio.

The money used in exchange is generally imaginary, and different from that in which the accompts are kept in most places: the money ufed in exchange alfo differs from current money in its value.

Before

Before the learner can refolve any questions in this rule, it is neceffary that he know how the country with which the exchange is to be made keep their accompts.

Holland, Flanders, and Germany.

In these countries accompts are kept in gilders, ftivers, and pennings, fimilar to the English pounds, fhillings, and pence.

But the different denominations of their money are contained in the following table:

8 pennings

2 grotes

6 ftivers 20 ftivers 2florins 6 florins

grote or penny
ftiver

fchilling

make one florin or gilder

rix-dollar

pound Flemish

Note. The money of Holland and Flanders is called Flemish money, and they exchange by the pound sterling.

The courfe of exchange with these countries has mostly been (except during the troubles on the Continent) from 33s. 4d, to 36s. 6d. Flemish per pound sterling; and the agio from 3 to 6 per cent.

As the exchange with all countries is fuppofed to be made in banco money, the current money must be turned into banco before the exchange can be made.

Rule. By the rule of three, or practice.

Proof. By reverfing the queftion.

To change current money into "banco, and banco money

into current.

Say, by the rule of three, as rool, with the agio added to if, is to rool. fo is the given fum current money to its value in banco.

A nd

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