EXAMPLES.

1. A merchant would mix wines at 16s, at 18s, and at 22s per gallon, so as that the mixture may be worth 20s the gallon what quantity of each must be taken?

16

2 at 16s

Ans. 2 gallons at 16s, 2 gallons at 188, and 6 at 228. 2. How much wine at 68 per galion, and at 4s per gallon must be mixed together, that the composition may be worth 58 per gallon? Ans 1 qt or 1 gall, &c.

3. How much sugar at 4d, at 6d. and at 11d per lb, must be mixed together, so that the composition formed by them may be worth 7d per lb ?

Ans. 1 lb, or 1 stone, or 1 cwt, or any other equal quantity of each sort.

4. How much corn at 28 6d, 38 8d, 4s, and 48 8d per bushel, must be mixed together, that the compound may be worth 38 10d per bushel ?

Ans. 2 at 28 6d, 2 at 3s 8d, 3 at 48, and 3 at 4s 8d. 5. A goldsmith has gold of 16, of 18, of 23, and of 24 caracts fine how much must he take of each, to make it 21 caracts fine? Ans. 3 of 16, 2 of 18, 3 of 23, and 5 of 24.

:

6. It is required to mix brandy at 128, wine at 10s, cyder at 18, and water at O per gallon together, so that the mixture may be worth 88 per gallon?

Ans. 8 gals of brandy, 7 of wine, 2 of cyder, and 4 of water.

RULE 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 gold of 15, 17, 18, and 22 caracts fine, must be mixed together, to form a composition of 40 oz of 20 caracts fine?

Here

Ans. 5 oz of 15, of 17, and of 18 caracts fine, and 25 oz of 22 çaracts fine*.

Ex. 2. A vintner has wine at 48, at 58, at 58 6d, and at 6s a gallon; and he would make a mixture of 18 gallons, so that it might be afforded at 5s 4d per gallon; how much of each sort must he take?

Ans. 3 gal. at 4s, 3 at 58, 6 at 5s 6d, and 6 at 6s,

* A great number of questions might be here given relating to the specific gravities of metals, &c. but one of the most curious may here suffice.

Hiero, king of Syracuse, gave orders for a crown to be made entirely of pure gold; but suspecting the workman had debased it by mixing it with silver or copper, he recommended the discovery of the fraud to the famous Archimedes, and desired to know the exact quantity of alloy in the crown.

Archimedes, in order to detect the imposition, procured two other masses, the one of pure gold, the other of silver or copper, and each of the same weight with the former; and by putting each separately into a vessel full of water, the quantity of water expelled by them determined their specific gravities; from which, and their given weights, the exact quantities of gold and alloy in the crown may be determined.

Suppose the weight of each crown to be 101b, and that the water expelled by the copper or silver was 921b, by the gold 52lb, and by the compound crown 64lb; what will be the quantities of gold and alloy in the crown?

The rates of the simples are 92 and 52, and of the compound 64; therefore

|

64 32 28 of gold

92- 12 of copper

52

=

And the sum of these is 12+28 40, which should have been but 10; therefore by the Rule,

40:10::28:31b

40: 10 28 71b of gold

the answer.

RULE

RULE III.

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

EXAMPLES.

1. How much wine at 58, at 58 6d. and 68 the gallon, must be mixed with 3 gallons at 4s per gallon, so that the mixture may be worth 58 4d per gallon?

Ans. 3 gallons at 5, 6 at 58 6d, and 6 at 68.

2. A grocer would mix teas at 12s, 108, and 68 per lb, with 20lb at 48 per lb how much of each sort must he take to make the composition worth 8s per lb?

Ans. 20lb at 48, 10lb at 68, 10lb at 10s and 20lb at 12s. 3. How much gold of 15, of 17, and of 22 caracts fine, must be mixed with 5 oz of 8 caracts fine, so that the com position may be 20 caracts fine?

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

* In the very same manner questions may be wrought when several of the ingredients are limited to certain quantities, by finding first for one limit, and then for another. The two last Rules can need no demonstration, as they evidently result from the first, the reason of whick has been already explained.

POSITION,

POSITION.

POSITION is a method of performing certain questions, which cannot be resolved by the common direct rules. It is sometimes called False Position, or False Supposition, because it makes a supposition of false numbers, to work with the same as if they were the true ones, and by their means dis covers the true numbers sought. It is sometimes also called Trial and-Error, because it proceeds by tria's of false numbers, and thence finds out the true ones by a comparison of the errors.—Position is either Single or Double.

SINGLE POSITION.

SINGLE POSITION is that by which a question is resolved by means of one supposition only. Questions which have their result proportional to their suppositions, belong to Single Position such as those which require the multiplication or division of the number sought by any proposed number; or when it is to be increased or diminished by itself, or any parts of itself, a certain proposed number of times. The rule is as follows:

TAKE or assume any number for that which is required, and perform the same operations with it, as are described or performed in the question. Then say, As the result of the said operation, is to the position, or number assumed; so is the result in the question, to a fourth term, which will be the number sought*.

The reason of this Rule is evident, because it is supposed that the results are proportional to the suppositions.

EXAMPLES,

1. A person after spending and of his money, has yet remaining 602; what had he at first?

2. What number is that, which being multiplied by 7, and the product divided by 6, the quotient may be 21? Ans. 18.

3. What number is that, which being increased and of itself, the sum shall be 75 ?

by 1, 1, Ans. 36.

4. A general, after sending out a foraging and of his men, had yet remaining 1000; what number had he in command? Ans. 6000.

5. A gentleman distributed 52 pence among a number of poor people, consisting of men, women, and children; to each man he gave 6d, to each woman 4d, and to each child 2d moreover there were twice as many women as men, and thrice as many children as women. How many were there of each ? Ans. 2 men, 4 women, and 12 children?

have

6. One being asked his age, said, if of the years lived, be multiplied by 7, and of them be added to the product, the sum will be 219. What was his age?

Ans. 45 years.

DOUBLE