the proportional quantities, by the above rule, we may say, As the PROPORTIONAL quantity is to the GIVEN quantity: so is each of the other PROPORTIONAL quantities: to the REQUIRED quantities of each.

4. If a man wishes to mix 1 gallon of brandy worth 16 s. with rum at 9 s. per gallon, so that the mixture may be worth 11 s. per gallon, how much rum must he

use?

Taking the differences as above, we find the proportions to be 2 of brandy to 5 of rum; consequently, 1 gallon of brandy will require 24 gallons of rum. Ans. 2 gallons.

5. A grocer has sugars worth 7 cents, 9 cents, and 12 cents per pound, which he would mix so as to form a compound worth 10 cents per pound; what must be the proportions of each kind?

Ans. 2 lbs. of the first and second to 4 lbs. of the third kind. 6. If he use 1 lb. of the first kind, how much must he take of the others? -if 4 lbs., what? if 6 lbs., what? if 10 lbs., what? if 20 lbs., what?

Ans. to the last, 20 lbs. of the second, and 40 of the third. 7. A merchant has spices at 16 d. 20 d. and 32 d. per pound; he would mix 5 pounds of the first sort with the others, so as to form a compound worth 24 d. per pound; how much of each sort must he use?

Ans. 5 lbs. of the second, and 74 lbs. of the third. 8. How many gallons of water, of no value, must be mixed with 60 gallons of rum, worth 80 cents per gallon, to reduce its value to 70 cents per gallon? Ans. 84 gallons.

9. A man would mix 4 bushels of wheat, at $1'50 per bushel, rye at $1'16, corn at $75, and barley at $50, so as to sell the mixture at $ '84 per bushel; how much of each may he use?

10. A goldsmith would mix gold 17 carats fine with some 19, 21, and 24 carats fine, so that the compound may be 22 carats fine; what proportions of each must he use? Ans. 2 of the 3 first sorts to 9 of the last. 11. If he use 1 oz. of the first kind, how much must he use of the others? What would be the quantity of the compound? Ans. to last, 74 ounces. 12. If he would have the whole compound consist of 15 oz., how much must he use of each kind? - if of 30 oz., how much of each kind? if of 374 oz., how much? Ans. to the last, 5 oz. of the 3 first, and 224 oz. of the last

Hence, when the quantity of the compound is given, we may say, As the sum of the PROPORTIONAL quantities, found by the ABOVE RULE, is to the quantity PEQUIRED, so is each PROPORTIONAL quantity, found by the rule, to the REQUIRED quantity of EACH.

13. A man would mix 100 pounds of sugar, some at 8 cents, some at 10 cents, and some at 14 cents per pound, so that the compound may be worth 12 cents per pound; how much of each kind must he use?

We find the proportions to be, 2, 2, and 6. Then, 2 + 2 +6 = 10, and 20 lbs. at 8 cts.

10 100 ::

{

2

2: 20 lbs. at 10 cts.

6 60 lbs. at 14 ets.

Ans.

14. How many gallons of water, of no value, must be mixed with brandy at $1'20 per gallon, so as to fill a vessel of 75 gallons, which may be worth 92 cents per gallon? Ans. 17 gallons of water to 574 gallons of brandy.

15. A grocer has currants at 4 d., 6 d., 9d. and 11 d. per lb.; and he would make a mixture of 240 bls., so that the mixture may be sold at 8 d. per lb. ; how many pounds of each sort may he take?

Ans. 72, 24, 48, and 96 lbs., or 48, 48, 72, 72, &c. Note. This question may have five different answers.

QUESTIONS.

medial? 3.

the

1. What is alligation? 2. rule for operating? 4. What is alligation alternate? 5. When the price of the mixture, and the price of the several simples, are given, how do you find the proportional quantities of each simple? 6. When the quantity of one simple is giver, how do you find the others? 7. When the quantity of the whole compound is given, how do you find the quantity of each simple?

DUODECIMALS.

T103. Duodecimals are fractions of a foot. The word is derived from the Latin word duodecim, which signifies twelve. A foot, instead of being divided decimally into ten equal parts, is divided duodecimally into twelve equal parts,

2

called inches, or primes, marked thus, ('). Again, each of these parts is conceived to be divided into twelve other equal parts, called seconds, ("). In like manner, each second is conceived to be divided into twelve equal parts, called thirds, (""); each third irto twelve equal parts, called fourths, ("); and so on to any extent.

In this way of dividing a foot, it is obvious, that

Duodecimals are added and subtracted in the same manner as compound numbers, 12 of a less denomination making 1 of a greater, as in the following

Note. The marks,,", "", "", &c., which distinguish the different parts, are called the indices of the parts or denominations.

MULTIPLICATION OF DUODECIMALS.

Duodecimals are chiefly used in measuring surfaces and

solids.

1. How many square feet in a board 16 feet 7 inches long, and 1 foot 3 inches wide?

=

is to take of, that is, 48'; and the 1' which we reserved makes 49', 4 feet 1'; we therefore set down the 1', and carry forward the 4 feet to its proper place. Then, multiplying the multiplicand by the 1 foot in the multiplier, and adding the two products together, we obtain the Answer, 20 feet, 8, and 9".

The only difficulty that can arise in the multiplication of duodecimals is, in finding of what denomination is the product of any two denominations. This may be ascertained as above, and in all cases it will be found to hold true, that the product of any two denominations will always be of the denomi nation denoted by the sum of their INDICES. Thus, in the above example, the sum of the indices of 7' X 3' is "; consequently, the product is 21"; and thus primes multiplied by primes will produce seconds; primes multiplied by seconds produce thirds; fourths multiplied by fifths produce ninths, &c. It is generally most convenient, in practice, to multiply the multiplicand first by the feet of the multiplier, then by the inches, &c., thus:

2. How many solid feet in a block 15 ft. 8' long, 1 ft. 5' wide, and 1 ft. 4' thick?

From these examples we derive the following RULE: Write down the denominations as compound numbers, and in multiplying remember, that the product of any two denominations will always be of that denomination denoted by the sum of their indices.

EXAMPLES FOR PRACTICE.

3. How many square feet in a stock of 15 boards, 12 ft, 8' in length, and 13' wide?

Ans. 205 ft. 10'. 4. What is the product of 371 ft. 2′ 6′′ multiplied by 181 ft. 1' 9" ? Ans. 67242 ft. 10' 1" 4" 6.

Note. Painting, plastering, paving, and some other kinds of work, are done by the square yard. If the contents in square feet be divided by 9, the quotient, it is evident, will be square yards.

5. A man painted the walls of a room 8 ft. 2' in height, and 72 ft. 4' in compass; (that is, the measure of all its sides;) how many square yards did he paint?

Ans. 65 yds. 5 ft. 8' 8". 6. There is a room plastered, the compass of which is 47 ft. 3', and the height 7 ft. 6'; what are the contents? Ans. 39 yds. 3 ft. 4' 6". 7. How many cord feet of wood in a load 8 feet long, 4 feet wide, and 3 feet 6 inches high?

Note. It will be recollected, that 16 solid feet make a cord foot. Ans. 7 cord feet. 8. In a pile of wood 176 ft. in length, 3 ft. 9' wide, and 4 ft. 3' high, how many cords?

Ans. 21 cords, and 7 cord feet over. 9. How many feet of cord wood in a load 7 feet long, 3 feet wide, and 3 feet 4 inches high? and what will it come to at $40 per cord foot?

Ans. 48 cord feet, and it will come to $175. 10. How much wood in a load 10 ft. in length, 3 ft. 9' in width, and 4 ft. 8' in height? and what will it cost at $1'92 per cord?

Ans. 1 cord and 21% cord feet, and it will come to $2'621.

¶ 104. Remark. By some surveyors of wood, dimensions are taken in feet and decimals of a foot. For this purpose, make a rule or scale 4 feet long, and divide it into feet, and each foot into ten equal parts. On one end of the rule,