ALLIGATION.

ALLIGATION teaches how to compound or mix together several Simples of different qualities, so that the Composition may be of some intermediate quality. It is commonly distinguished into ALLIGATION MEDIAL, and ALLIGATION ALTERNATE.

ALLIGATION MEDIAL.

ALLIGATION MEDIAL is the method of finding the rate or quality of the Composition, from having the quantities and rates or qualities of the several simples given. It is performed as follows:

RULE.-Multiply the quantity of each ingredient by its rate or quality; add all the products together into one sum, and all the quantities into another sum; then divide the sum of the products by the sum of the quantities-and the quotient will be the required rate or quality of the composition.

Example.-If four different qualities of gunpowder be mixed together-namely, 60 lb. at 12d. per pound, 50 lb. at 10d., 36 lb. at 9d., and 30 lb. at 8d. per pound; what is the value of a pound of the composition?

Here 60, 50, 36, and 30 are the quantities, and 12, 10, 9, and 8 are the rates or qualities; hence

Therefore, the rate or price of the composition is 10d. per pound.

THE REASON of the rule is obvious from the example, for each of the products is the price in pence of the given quantity from which it arises; hence the sum of the products is the price of the composition in pence, which, being divided by the number of pounds in it, gives the price of one pound of the mixture in pence.

Exercises.

1. A grocer mixes 24 lb. of tea at 6s. a pound, with 20 lb. at 5s., and 30 lb. at 4s. 6d. a pound; what was the value of a pound of the mixture? Ans. 5s. 14d. 3.

2. A composition being made of 10 gallons at 7s., 18 gal. at Ss. 6d., and 15 gal. at 5s. 10d.; what is a gallon of the mixture worth? Ans. 7s. 2 d. 28.

3. Mixed 4 lb. of tea at 4s. 10d. per pound, with 10 lb. at 5s. 3d. per pound, and 16 lb. at 5s. 8d. per pound; what is the value of a pound of this mixture? Ans. 5s. 5d.

4. Having mixed 18 gallons of spirits at 12s. 6d. per gallon, with 24 gal. at 10s. 6d., 12 gal. at 9s., and 30 gal. at 8s. 6d.; I wish to know what a gallon of the mixture is worth, Ans. 108.

5. Having melted together 7 lb. of gold of 22 carats fine, with 10 lb. at 21 carats fine, and 19 lb. at 19 carats fine; what is the fineness of the composition? Ans. 20 carats fine.

6. Find the average price of 25 qr. wheat at 40s., 36 at 44s., 16 at 48s., 15 at 54s., and 18 at 60s. Ans. £2, 7s. 73d. 23.

ALLIGATION ALTERNATE.

ALLIGATION ALTERNATE is the method of finding what quantity of any number of Simples, whose rates or prices are given, will compose a Mixture of a given rate or price, intermediate between the rates of the simple quantities.

RULE.-Set the rates or prices of the Simples in a column under each other. Link, or connect with a line, the rate of each Simple which is less than that of the mixture, with one or more of those that are greater than the mixture; and connect each greater rate with one or more of the less.

Write the difference between the rate of the mixture and the rate of each simple, opposite the rate or rates with which each is linked. Then if only one difference stand opposite a rate, it will be the quantity belonging to that rate; but if there be several differences, their sum will be the quantity belonging to the rate.

Questions in this rule often admit of a variety of correct answers, which are obtained by linking the rates in a variety of different ways. This will be evident from the following

example:

Example.-A merchant would mix wines at 14s., 19s., 15s., and 22s. per gallon, so that the mixture may be worth 18s. What quantity of each may he take?

THE REASON of the rule may be shewn thus: Take any two quantities which result from one linking, as, for instance, 4 and 3, in the first solution, which result from the linking of 15s. with 228.; since 15s. is less than 18s. by 3s., the gain on 4 gallons is 3s. X 4 = 12s., and since 22s. is greater than 18s. by 48., the loss on 3 gallons is 4s. × 3 = 12s.; therefore the gain and loss are equal. In the same manner, it may be shewn that the gain and loss on the quantities resulting from any other linking are equal, and hence the gain and loss on the aggregate of the linkings are equal.

Note 1.-When the whole composition is limited to a certain quantity, find an answer by linking; then say, as the sum of the quantities thus determined is to the given quantity, so is the quantity of each ingredient found by linking, to the required quantity of each.

Note 2.-When one of the ingredients is limited to a certain quantity, link as before; then say, as the quantity found of that which is limited, is to the limited quantity, so is any other quantity found to the quantity of it to be taken to form the mixture.

Exercises.

7. A merchant would mix wines at 16s., at 18s., and at 23s. per gallon, so that the mixture may be worth 20s. per gallon; what quantity of each must he take?

Ans. 3 gal. at 16s., 3 gal. at 18s., and 6 gal. at 23s., or 1, 1, and 2, or any other quantities in the same proportion.

8. How much corn at 2s. 8d., 3s. 4d., 3s. 8d., and 4s. per bushel, must be taken to form a compound worth 3s. 6d. per bushel? Ans. 6, 2, 2, and 10 or 2, 8, 12, 2, or 2, 6, 10, 2. 9. How many pounds of sugar at 4d., at 6d., and at 74d. per lb. must be mixed together to form a mixture at 52d. per lb. ? Ans. 8 lb., 7 lb., and 7 lb.

10. A goldsmith has four sorts of gold-namely, of 24 carats fine, of 22, 20, and 15 carats fine, and wishes to mix of each sort together so as to have 63 oz. of 17 carats fine. How much must he take of each sort? Ans. 6 oz. of 24, 6 oz. of 22,

6 oz. of 20, and 45 oz. of 15 carats fine.

11. A distiller would mix 48 gallons of French brandy at 18s. per gallon, with British at 10s. 6d., and spirits at 6s. per gallon. What quantity of each sort must he take that the mixture may be worth 12s. per gallon? Ans. 48 gallons French brandy, 48 British, and 36 of spirits; or 48 gallons French brandy, 64 British, and 32 of spirits, &c.

12. A grocer has teas at 5s., 4s. 6d., and 3s. per pound, from which he makes up two parcels; the first contained 28 lb. at 48. per lb., and the second 56 lb. at 4s. 3d. How many pounds of each sort was taken to form the parcels? Ans. 8 lb. at 5s., 8 lb. at 4s. 6d., and 12 lb. at 3s., to form the first; 20 lb. at 5s., 20 lb. at 4s. 6d., and 16 lb. at 3s. per lb., to form the second.

13. A farmer would mix 30 bushels of wheat at 6s. per bushel with other sorts worth 5s., 4s. 6d., and 4s. per bushel, so that the mixture may be worth 4s. 9d. per bushel. How many bushels of each sort may he take? Ans. 30 bushels at 6s., 50 bushels at 4s., and any equal quantities at 5s. and 4s. 6d.

DUODECIMAL MULTIPLICATION, OR

MENSURATION.

DUODECIMAL MULTIPLICATION is that which is employed in the measurement of walls, flooring, &c.; and solid bodies, such as logs of wood, in which feet, inches, and their subdivisions are multiplied together, to ascertain the required dimensions.

It is so named from duodecim, a Latin word, signifying twelve, because, in multiplying the feet, inches, &c., the number carried from one denomination to another is 12, instead of 10, as in decimal multiplication.

In measurement, some line of a determinate length is employed, as an inch, a foot, a yard, &c. The assumed line is called the unit of measure.

The number of times that the unit of measure is contained in any line is called its length or its measure.

Surface has two dimensions-length and breadth; the unit of measure for surfaces is the Square described on the unit of length. Thus, the unit may be a square inch, a square foot, a square yard, &c.

The number of times that any surface contains the unit of measure is called its Area, or its Content. Thus, if a surface contain a square foot 30 times, its area is 30 square feet.

A solid has three dimensions-length, breadth, and thickness; the unit of measure for solids is the Cube described on the unit of length. Thus, the unit may be a cubic or solid inch, a cubic or solid foot, a cubic or solid yard, &c.

The number of times that any solid contains the unit of measure is called its Volume, or Solidity. Thus, if a solid contain a cubic or solid inch 30 times, its solidity is 30 solid inches.

Note. For further definitions, rules, and their demonstrations, the pupil is referred to the Treatise on Practical Mathematics of this Course.

RULE FOR MULTIPLICATION.

1. Place the feet of the multiplier below the lowest denomination of the multiplicand; the inches, one place further to the right; the seconds, beyond the inches; and so on.

2. Multiply each denomination of the given quantity, by the feet of the multiplier, as in Compound Multiplication; the twelves being taken out of each product, and carried to the denomination above it, and the remainder written below the denomination multiplied.

3. Multiply each denomination in the same way, by the inches of the multiplier, carrying the twelves as before, but writing the remainders of each product one place further to the right, than those in the previous line; the first remainder of the product being thus written immediately below the inches of the multiplier.

4. Multiply in the same way by the seconds, and so on; always writing the remainders in each new line of products, one place further to the right than those in the previous linethe first remainder in each line, being thus written immediately below the figure used as the multiplier.

5. Then add all the products together for the answer, which is in square or cubic measure, as the case may be.

THE PRODUCT of feet, inches, &c., multiplied by feet, inches, &c., is expressed in feet, firsts, seconds, thirds; and so on. A first is 1-12th of a square foot; a second, 1-12th of a first, &c., each denomination being 1-12th of that preceding it. Each lower denomination is written a place further to the right than the one above it.