the first yard, 3 d.; for the second yard, 9 d. ; and for the third yard, 27 d. How much did he receive for the whole, at that rate?

Case 2d,

Case 1st, 319x3=3486784401, the last term. 3486784401-3÷2=1743392199.+ 3486784401-5230176600 the sum of all the terms in pence=21792402 £. 10 s.

2. What would 12 horses cost, if 4 cents were allowed for the first, 16 cents for the second, and 64 cents for the third horse, &c. the value thus increasing in a quadruple ratio to the last or twelfth horse? Ans. $223696.20.

3. A gentleman gave his daughter, on the day of her marriage, one dollar, promising to triple it on the first day of each month in the year. What was the amount of her portion? Ans. $265720.

CASE 4th.-THE EXTREMES AND NUMBER OF TERMS GIVEN, TO

FIND THE RATIO.

RULE.-Divide the greater extreme by the less, and the quotient will be that power of the ratio, which is equal to the number of terms less 1. The corresponding root will, therefore, be

the ratio.

Ex. 1. The first term of a geometrical series is 2, and the last term 354294, and the number of terms 12. What is the ratio?

354294÷2=177147, and the eleventh root of this number is the ratio required; therefore, 177147=3, the ratio.

2. The first term of a certain series is 4, and the last 65536, and the number of terms 8. What was the ratio? Ans. 4.

QUESTIONS.-What is Geometrical Progression? What is the ratio of the series? What are the terms of the series? What terms of a series are called extremes? And what are called means? In geometrical progression, how many things are to be considered? How many of these must be given to find the others? What are the five things given? What is Case 1st? What is the rule for Case 1st ? What is Note 1st ? What is Case 2d? What is the rule? What is Case 3d? What is the rule? What is Case 4th? What is the rule?

ALLIGATION.

Alligation is the method of mixing several simples of different qualities, so as to obtain a compound of a mean or middle quality.

CASE 1st.-WHEN THE QUANTITIES AND PRICES OF SEVERAL SIMPLES ARE GIVEN, TO FIND THE MEAN PRICE OF THE MIXTURE.

RULE. Find the total value of the several kinds to be mixed, and divide the amount of this value by the whole number of articles.

Ex. 1. A farmer mixed together 8 bushels of rye worth $0.50 per bushel; 12 bushels of corn worth $0.65 per bushel; and 6 bushels of oats worth $0.30. What was the value of one bushel of the mixture?

8 bushels of rye at 50 ct.-$4.00; 12 bushels of corn at 65 ct. $7.80; and 6 bushels of oats at 30 ct. - $1.80. And 8+ 12+6=26 bushels, and $4.00+$7.80+$1.80-$13.60, and $13.30-26 bushels=$0.523,+ price of one bushel of the mixture.

2. A grocer mixed 6 lb. of tea at $1.20 per lb.; 12 lb. at $1.60; and 8 lb. at $1.80. What was the value of one lb. of the mixture? Ans. $1.569.

3. If 15 bushels of wheat worth $1.40 per bushel, be mixed with 12 bushels of rye at $0.60 per bushel, and ten bushels of oats at $0.35, what is the value of one bushel of the mixture? Ans. $0.856.+

4. If 6 lb. of gold, 20 carats fine, be mixed with 12 lb. at 18 carats fine, what is the fineness of the mixture? Ans. 18 carats fine.

5. If 6 gallons of wine at $0.67 per gallon, 7 gallons at $0.80 per gallon, and 5 gallons at $1.20 per gallon, be mixed together, what will be the value of one gallon of the mixture? Ans. $0.867.+

CASE 2d.-THE PRICES OF SEVERAL COMMODITIES BEING GIVEN,

TO DETERMINE HOW MUCH OF EACH COMMODITY MUST BE TAKEN, TO FORM A COMPOUND OF A CERTAIN PROPOSED MEDIUM VALUE.

RULE. Write down the prices of the several simples under each other, placing that price which is least in value uppermost, and the remaining prices in the order of their values.

Connect, by a line, any price less than the given mean price, with one that is greater, and continue thus to do till they are all connected; then place the mean price on the left, and separate it from the other numbers by a perpendicular line. Write the difference between the proposed price of the mixture, and the price of each simple, opposite the number or numbers with which that simple is connected; and, finally,

Notice whether more than one difference stands opposite any one price; if so, their sum will express the quantity of that price to be taken; but if only one difference stands there, that will be the quantity required.

Note.-One difference at least must stand against each price. Ex. 1. How much corn at 48 cents, barley at 36 cents, and oats at 24 cents per bushel, must be taken to make a compound worth 30 cents per bushel?

6+18=24 bushels at 24 cents.

Mean price, 30

24 36

The difference between 30, the mean, and 24, is placed opposite of both 36 and 48, as it is connected with them both; and the difference between 30, the mean, and 36, and also between 30 and 48, are both placed opposite 24, because these numbers are both linked with 24, and the sum of their differences determines the number of bushels required of that price. Of the oats, therefore, 24 bushels are required, and of the corn and barley, only 6 bushels of each.

2. I have four kinds of sugar valued at 8, 12, 15, and 18 cents per pound. How much of each kind must be taken to make a mixture worth 14 cents per pound?

3. A grocer mixed together three kinds of tea, valued at 6, 9, and 10 shillings per pound, so that the compound was worth 8 shillings per pound. How much of each sort did he take? Ans. 3 lb. at 6 s., 2 lb. at 9 s., and 2 lb. at 10s.

4. A merchant has three kinds of wine. For the one kind he charges 3 s. 4 d., for the second 5 s., and for the third 7 s. per gallon. How much of each is required to form a mixture worth 6 s. per gallon? Ans. 12 gal. at 3 s. 6 d., 12 gal. at 5 s., and 44 gal. at 7 s.

5. How much gold at 16, 19, 21, and 24 carats fine, will be required to form a compound of 20 carats fine? of 16, 1 of 19, 1 of 21, and 4 of 24 carats fine.

Ans. 4 parts

CASE 3d.-THE PRICE OF EACH OF SEVERAL SIMPLES, THE QUANTITY OF ONE AND THE PRICE OF THE COMPOUND BEING GIVEN, TO FIND HOW MUCH OF EACH OF THE OTHER SIMPLES IS REQUIRED.

RULE.-Link the several prices together, as in the last case, and find their differences; then multiply the given quantity by the differences standing severally against the other quantities, and divide the product by the difference standing against itself. Or say, as the difference opposite the given quantity is to the given quantity, so are the other differences severally to their required quantities.

Ex. 1. How much barley at 30 cents, rye at 36 cents, and corn at 48 cents per bushel, must be mixed with 12 bushels of oats at 18 cents per bushel, so that the compound may be worth 22 cents per bushel?

The re

The price of the given quantity is 48; therefore, 48: 12:: 4: 1, the quantity required at 30 cents per bushel. maining statements and answers are the same, since the differences are all the same. Therefore, one bushel at 30 cents, one at 36 cents, and one at 48 cents, would be required to be mixed with 12 bushels at 18 cents, to form a mixture worth 22

cents.

2. A grocer has three kinds of beer for sale, valued at 7 s.,

5 s., and 3 s. per gallon, which he proposes to mix with 20 gallons of a superior quality, worth 6 s. per gallon, so that the mixture may be sold at 4s. per gallon. How much of the first three kinds must he take? Ans. 120 gal. at 3 s., and 20 gal. at 5 s. and 7 s.

3. How much tea at 80, 60, and 40 cents per lb. must be mixed with 30 lb. at $1.00 per lb. so that the mixture may be sold at 70 cents per lb.? Ans. 10 lb. at 80 cents and 60 cents, and 30 lb. at 40 cents.

4. How much water of no value must be mixed with 100 gal. of wine at 7.s. 6 d. per gal., to reduce the price to 6 s. 3 d. per gallon? Ans. 20 gal.

(

CASE 4th.-THE PRICE OF THE SIMPLES BEING GIVEN, AND ALSO THE COMPOUND TO BE FORMED, TO FIND HOW MUCH OF EACH SIMPLE MUST BE TAKEN.

RULE.-Connect the prices of the simples as in the preceding cases, and find the amount of the differences; then say, as the amount of the differences is to each of the differences taken separately, so is the whole compound to the part required.

Ex. 1. A compound of 15 gallons, which shall be worth 8 shillings per gallon, is to be made of three sorts of wine, valued at 5, 7, and 12 shillings per gallon. How much of each kind will be required?

Then, 12: 4:: 15:5, Ans. 5 lb. of each kind are required.

Proof, 5 s. x5=25 s. 7 s. X5 s. 35 s.; and 12 s. x 5 s. 60 s.; and 25+35+60-120 s.; and 120÷8-15 gallons.

2. I have four sorts of tea, of which the first kind is worth 1 s. per lb.; the second kind, 3 s.; the third, 6 s.; and the fourth, 10 s. How much of each kind will be required to make a compound of 120 lb. worth 4 s. per lb? Ans. 60 lb. at 1 s.;

20 lb. at 3 s.; 10 lb. at 6 s.; and 30 lb. at 10 s.

3. How much of each of four kinds of coffee, worth 8, 12, 18, and 22 cents per lb. will be required to make a compound