price or value of each kind, and the price or total value of each mixture. Questions of the second sort are usually solved by Algebra, although Arithmetic may serve for the purpose, as will be shown hereafter. The following is the rule for questions of the first kind:

Multiply the value of each kind of things by the number of things of that sort; add all the products, and divide their sum by the total number of things of all the kinds.

EXAMPLE.

A man employed 200 labourers; 50 of whom he paid at the rate of 40d. a day; 70 at the rate of 30d.; 50 at the rate of 25d.; and 30 at the rate of 20d. What is the average price each labourer receives a day?

50 labourers, at 40d. a day, receive 2000d.

70 at 30d.

50 at 25d.

30 at 20d.

2100d.

1250d.

600d.

5950d.

The cost of 200 labourers is then 5950d. a day; and dividing this sum by 200, the quotient, 29d. 316qrs. shows the average price each labourer receives.

EXAMPLES FOR PRACTICE.

1. A refiner has 5lb. of silver bullion, of 8oz. fine; 10lb. of 7oz. fine; and 15lb. 6oz. fine; on melting them all together, what would be the fineness of 1lb. of this mass? Ans. 6oz. 13dwts. 8grs. fine. 2. A farmer mixed 19 bushels of wheat, at 6s. per bushel; 40 bushels of rye, at 4s. per bushel; and 12 bushels of barley, at 3s. per bushel; what was a bushel of this mixture worth? Ans. 4s. 4d. 14qrs.

3. If 16 gallons of brandy at $1,25 per gallon, and 4 gallons of water be mixed with 40 gallons of wine, at $3 per gallon; what will the mixture be worth per gallon ? Ans. $-,35.

R

4. A grocer mixed 10 gallons of water with a hhd. of rum, worth 80,87 per gallon; what was the mixture worth per gallon? Ans. $0,75-3 5. A grocer mixed 43 gallons of wine at $1,25 per gallon, with 87 gallons worth $1,60 per gallon. What was the mixture worth per gallon? Ans. $1,48,43.

6. A composition is made of 5lb. of tea, at 7s. per pound; 9lbs. at 8s. 6d. per pound; and 144lbs. at 6s. 104d. per pound; what is a pound of the mixture worth? Ans. 7s. 42d &c,

ALLIGATION ALTERNATE.

This rule is the reverse of Alligation, and may be proved by it. The method of operation is as follows:

Write the rates of the simples in a column under each other. Connect with a continued line the rate of each simple, which is less than that of the compound, with one or any number of those, that are greater than the compound; and each greater rate with one or any number less.

Write the difference between the mixture rate and that of each of the simples opposite the rates, with which they are respectively linked. Then if only one difference stand against any rate, it will be the quantity belonging to that rate; but if there be several, their sum will be the quantity."

EXAMPLES.

A merchant would mix wines at 14s. 19s. 15s. and 22s. per gallon, and have the mixture worth 18s. per gal. What quantity of each must be taken ?

The quantities are written as follows ;

2 It is required to mix brandy at 80 cents, wine at 70 cents, cider at 10 cents, and water together, so that the mixture may be worth 50 cents per gallon. Ans. 9 gallons of brandy, 9 of wine, 5 of cider, and 5 of water.

CASE II.

When the whole composition is limited to a certain quantity, "find an answer as before by linking; then say, as the sum of the quantities, or differences thus determined, is to the given quantity, so is each ingredient, found by linking, to the required quantity of each.”

EXAMPLES.

1. How much water, considered as of no value, must be mixed with wine worth 3s. per gallon, so as to fill a vessel of 100 gallons, and that a gallon may be afforded at 2s. 6d. ?

36: 100 :: 6 : 8312 galls. of wine, 36: 100 :: 30: 163 galls. of water. Ans.

2. How much gold of 15, of 17, of 18, and of 22 carats fine, must be mixed together to form a composition of 40 ounces, 20 carats fine?

Ans. 5oz. of 15, 17, and 18, and 25oz. of 22.

CASE III.

When one of the ingredients is limited to a certain quantity, find the difference between each price and the mean rate as before; then, as the difference of that simple whose quantity is given, is to the other differences severally, so is the quantity given to the several quantities required.

EXAMPLES.

1. How much wine at 5s., 5s. 6d., and 6s. the gallon, must be mixed with 3 gallons, at 4s. per gallon, so that the mixture may be worth 56. 4d. per gallon?

10 10 3 3 gallons at 5s.

10:20:36 gallons at 5s. 6d. Ans.

10 20 3 6 gallons at 6s.

2. How much gold of 15, of 17, and of 22 carats fine, must be mixed with 5oz. of 18 carats fine, so that the composition may be 20 carats fine?

Ans. 5oz. of 15 carats fine, 5oz. of 17, and 25 of 22.

EXAMPLES FOR PRACTICE.

I. A grocer would mix sugars at 10, 13, and 16 cents per pound. What quantity of each must he take to make a mixture worth 12 cents per Ib. ?

Ans. 5lb. at 10 cents, 2lb. at 13cts. and 2lb. at 16cts. 2. A farmer had 10 bushels of wheat, worth 8s. per bushel, which he wished to mix with corn, at 3s. per bushel. so that the mixture might be worth 5s. per bushel. How many bushels of corn must he use?

Ans. 1 of wheat to 1 of corn, or 10 of wheat to 15 of corn.

3. A grocer has rum worth 75 cents per gallon; how many parts water must he put in, that he may afford to sell the mixture at 25 cents per gallon?

Ans. 2 parts water to 13 rum. 4. A grocer has two sorts of tea, one at 75 cents per pound, and the other at $1,10 per lb? How must they be mixed, that the mixture may be afforded at $1,00 per lb. ? Ans. 2 parts of the cheaper to 5 of the dearer.

ARITHMETICAL PROGRESSION.

204. Arithmetical progression is a series of terms, in which each exceeds the term that precedes it, or is exceeded by it, by the same quantity.

31. 35.

The following series is an arithmetical progression ; 1.4 7 10 13 16 19 22 25. 28 &c.:-because each term exceeds that which precedes it, by the same quantity 3.

The three points separated by a bar, which stand at the beginning of the series, denote that, in enunciating the progression, each term except the first and last, must be repeated in this manner, 1 is to 4 as 4 is to 7, as 7 is to 10, &c.

The progression is called increasing or decreasing according as the terms of the series go on augmenting or diminishing. But as the properties of both are the same, (by changing the words plus, or addition, into minus or subtraction), we shall consider the progression only as increasing.

205. It is plain, after the definition of arithmetical progression, that, with the first term and common difference or ratio of the progression, all other terms of the series may be formed by successively adding this ratio. Hence, the second term is composed of the first plus the ratio. The third is composed of the second, plus the ratio; consequently, of the first plus twice the ratio. The fourth is composed of the third, plus the ratio; and consequently, of the first, plus three times the ratio; and

so on.

206. Hence, in general, any term whatever of an arithmetical progression is composed of the first, plus as many times the ratio as there are terms in the series preceding

it.

207. Then, the first term being cipher, every other term of the progression will be equal to as many times the ratio, as there are preceding terms in the series.

208. This principle may have the two following applications.

1st. It serves for finding any term of a progression, without requiring a calculation of those, that precede it.