that of the second by 6, and that of the third by 7, the products are all equal. What are their shares?

Ans. $1962, $2289,77, $27477. 4. The number of a gentleman's horses is two-fifths of the number of his cows and oxen, and for every 4 of the cows and oxen he has 11 sheep. Required the number of each, the number of the sheep exceeding that of the horses by 141. Aus. 24 horses, 60 cows and oxen, and 165 sheep.

Double Position.

183. RULE 1.-Assume two different numbers, and perform on them separately the operations indicated in the question: then, as the difference of the results thus* obtained is to the difference of the assumed numbers, so is the difference between the true result and either of the others to the correction to be applied, by addition or subtraction, as the case may require, to the assumed number which gave this result.*

Example 1. Required a number from which if 2 be subtracted, one-third of the remainder will be 5 less than half the required number.

Here, suppose the required number to be 8, from which take 2, and one-third of the remainder is 2. This being taken from one-half of 8, the remainder is 2, the first result. Suppose again the number to be 32, and from it take 2; one-third of the remainder is 10, which being taken from the half of 32. the remainder is 6, the second result. Then, the difference of the results being 4, the difference of the assumed numbers 24, and the difference between 5, the true result, and 6, the result nearer it, being 1; as 4: 24:16, the correction to be subtracted from 32, since the result 6 was too great. Hence, the required number is 26.

*This rule, which was first published in substance by Bonnycastle, in his Arithmetic, is, according to Professor Thomson, the simplest and easiest that has yet appeared for the resolution of questions in which the given result is a known number, independent on the required number. See Professor Thomson's observations on this rule in his Treatise on Arithmetic.

184. RULE II.-Having assumed two different numbers, perform on them separately the operations indicated in the question, and find the errors of the results, Then, as the difference of the errors, if both results be too great or both too little, or as the sum of the errors, if one result be too great and the other too small, is to the difference of the assumed numbers, so is either error to the correction to be applied to the number that produced that correction.

2. If one person's age be now only four times as great as another's, though 7 years ago it was 6 times as great, what is the age of each?

Here, suppose the age of the younger to be 12 years; then would the age of the older be 48. Take 7 from each of these, and there will remain 5 and 41, their ages 7 years ago. Now, 6 times 5 are 30, which, taken from 41, leaves an error of 11 years. By supposing the age of the younger to be 15, and proceeding in a similar manner, the error is found to be 5 years. Hence, as 6, the difference of the errors, (both results being too small,) is to 3, the difference of the assumed numbers, so is 5, the less error to 21, the correction; which being added to 15, the sum, 171, is the age of the younger, and consequently that of the older must be 70.*

3. Required a number, to which if twice its square be added, the sum will be 100. Ans. 6.8254.

* Both the rules above given depend on the principle, that the differences between the true and the assumed numbers are proportional to the differences between the result given in the question and the results arising from the assumed numbers. This principle is quite correct in relation to all questions which in Algebra would be resolved by what is usually called Simple Equations, but not in relation to any others; and hence, when applied to others, it gives only approximations to the true results. In this case, the assumed numbers should be taken as near the true answer as possible. Then, fo approximate the required number still more nearly, assume for a second operation the number found by the first, and that one of the first two assumptions which was nearer the true answer, or any other number that may appear nearer it still. In this way, by repeating the operation as often as may be necessary, the true result may be approximated to any assigned degree of accuracy. When applied in this way, Double Position is of considerable use in Algebra, affording in complicated expressions a very convenient mode of approximating the roots of equations, and of finding the values of unknown quantities.

It is easy to see that this number must be between 6 and 7. These numbers being assumed, therefore, the sum of 6 and twice its square is 78, and the sum of 7 and twice its square 105. Then, as 105-78: 7-6 :: 105-100: .18; which being taken from 7, the remainder, 6.82, is the required number nearly. To this let twice its square be added, and the result is 99.8448. Then, as 105-99.8448: 76.82: 105-100 .1746; which being taken from 7, the remainder is 6.8254, the required number still more nearly; and if the operation were repeated, with this and the former approximate answer, the required number would be found true for seven or eight figures.

Exercises.-1. Required a number, from which if 84 be taken, 3 times the remainder will exceed the required number by one-fourth of itself. Ans. 144.

2. One being asked how old he was, answered, that the product of th of the years he had lived, being multiplied by ths of the same, would be his age. What was his age? Ans. 30 years.

3. After A had lent $10 to B, he wanted $8 in order to have as much money as B; and together they had $60. 'What money had each at first? Ans. A $36, and B $24. 4. Two persons began to play with equal sums of money; the first lost $14, the other won $24, and then the second had twice as many dollars as the first, What sum had each at first? Ans. $52.

5. A farmer had two flocks of sheep, each containing the same number; from one of these he sells 39, and from the other 93; and finds twice as many remaining in one as in the other. How many did each flock originally contain? Ans. 147.

6. A sum of money was divided between two persons, A and B, so that the share of A was to that of B as 5 to 3; and exceeded five-ninths of the whole sum by 50 dollars. What was the share of each person?

Ans. A's $450, and B's $270. 7. Required a number which exceeds 3 times its square root by 10.

Ans. 25.

8. Required a number, to which if twice its square and 3 times its cube be added, the sum will be 2000.

Ans. 8.506744.

What is position?

Questions.

How many kinds of position are there?

What is single position?

What is double position?

Repeat the rule for working questions in single position. Repeat the first rule for working questions in double position.

Repeat the second rule.

CHAPTER XIII.

EXCHANGE.

185. Exchange is the method of finding what sum of the money of one country is equivalent to any given sum of the money of another.

186. By the par of exchange between two countries is meant the intrinsic value of the money of one compared with that of the other, and estimated by the weight and fineness of the coins.*

187. The course of exchange, at any particular time, is the sum of the money of one country which at that time is given for a fixed sum of the money of another country. This is seldom at par, but is continually varying according to the circumstances of trade.†

There are two kinds of money; real and nominal, or imaginary. All gold, silver, and copper coins are called

*The par continues the same, so long as the coins in each country are of the same weight, fineness, and nominal value. There is an exception to this, when the standard coins are of different metals. In this case, the par varies as the comparative values of the metals vary. Thus, silver is the standard coin, in general, on the continent of America: gold is the standard coin in Great Britain and Ireland, but in most of the continent of Europe, silver is the standard.

†The variation in the course of exchange depends on several circumstances, which, as well as several other questions connected with exchanges, do not accord with the plan of this publication to explain. Such discussions belong properly to Political Economy. See Thomson's Arithmetic.

W

real money: and the imaginary money is a denomination used to express money of which there is no real species current, precisely of the same value, as a livre in France; and a pound and a penny in the United States. In some countries they keep their accounts and calculate their payments in imaginary money.

188. Exchange in this country may be distinguished into Domestic and Foreign.

Domestic exchange consists in the reduction of the currencies of either state into that of any other. Accounts are generally kept at present in the United States in dollars and cents; hence, domestic exchange is falling into disuse, or if it be used at all, it is for the purpose of changing the nominal currencies of the States to dollars, or of reducing dollars to the nominal currencies of the States.

All the calculations in exchange may be performed by the Rale of Proportion. In foreign exchanges, one place always gives another a fixed sum or piece of money for a variable one. The former is called the certain price or rate; the latter the variable price or rate.*

DOMESTIC EXCHANGE.

PROBLEM I.- -To reduce the currencies of the United States to dollars and cents.

189. RULE.-As the value of a dollar in the given currency is to the sum whose value is to be found in dollars and cents, so is one dollar to the dollars and cents required.

The coins of the United States are gold, silver, and copper. The gold coins are the eagle, which weighs 270 grains, half eagle, and quarter eagle. The standard gold for coinage, consists of 11 parts of pure gold, and 1 of alloy; which is usually a mixture of silver and copper.

* Thus, in exchanges with Amsterdam, London receives for one pound sterling a number of skillings and pence of the money of Amsterdam, which is continually varying, being at one time perhaps 36 skillings and 8 pence, at another 37 skillings and 7 pence, &c. Here one pound is the certain price, or rate, and 36 skillings and 8 pence, &c. variable price, or rate. In exchanges with Portugal, on the contrary, London gives for 1 milrea some times 65 pence, some times 68 pence, &c. In this case, 1 milrea is the fixed price, and 65 pence, or 68 pence, &c. the variable price.