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PROB. 2. The product of any two or more numbers, and the proportion between them given, to find those numbers.

RULE.

Divide the given product by the product of the given terms of the proportion, and the quotient will be a power equal to the number of terms multiplied for a divisor; and the root of that power, multiplied severally by the given terms of the proportion, will produce the required numbers.

Ex. 1. The product of two numbers is 2240, and they are in proportion to each other as 5 to 7; what are those numbers ? 5×5=35)2240(64, and √/64=8; Then 8×5, (one of the terms of the proportion,) gives

40

Ans.

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And 8×7, (the other term of the proportion,)

gives

Ex. 2. The product of three numbers is 1296, and they are in proportion to each other as 1, 2 and 3; what are those numbers ?

Thus, 1×2×3=6)1296(216,

then 3/216=6

6X

1: <= 6
2=12
3=18

}

Ans

A GENERAL RULE FOR EXTRACTING ROOTS OF ALL

POWERS.

RULE.

1. Prepare the given number for extraction, by pointing it off from the units' place, as the required root directs.

2. Find the first figure of the root by trial, and subtract its power from the left hand period.

3. Bring down the first figure in the next period to the remainder and call this the dividend.

4. Involve the root to the next inferior power to that which is given, and multiply it by the number denoting the given power for a divisor.

5. Find how many times the divisor may be had in the dividend, and the quotient will be another figure of the root. 6. Involve the whole root to the given power, and subtract that power from as many periods of the given number, as you have found figures in the root,

7. Bring down the first figure of the next period to the remainder for a new dividend.

8. Involve the whole root to the next inferior power to that which is given, and multiply it by the number denoting the given power for a divisor, as before; and proceed in this manner till the whole is finished.

Note. When the number to be subtracted is greater than the periods from which it is to be subtracted, the last quotient figure must be taken less, &c.

EXAMPLES.

1. What is the cube root of 94818,816 ?

94818,816(45,6 root.

64

4x4x3=48)308 dividend.

45 x 45 x 45-91125 subtrahend. 45X45X3=6075)3693,8 dividend. 456 × 456 × 456-94818,816 subtrahend.

00000,000

2. What is the sursolid, or 5th root of 17210368?

17210368(28 root.

32

2×2×2×2=16×4=64)1401 dividend.

28 × 28 x 28 x 28 x 28=17210368 subtrahend.

00000000

Note. The roots of most powers may be found by the square and cube roots only, by the following

RULE.

1. For the bíquadrate, or 4th root, extract the square root of the square root.

2. For the 6th root, extract the cube root of the square

root.

3. For the 8th root, extract the square root, which reduces it to the 4th power; then extract the root of that power as above,

4. For the 9th root, extract the cube root of the cube root. 5. For the 12th root, extract the square root, which will reduce it to the 6th power; then find the root of the 6th power as above.

EXAMPLES.

1. What is the biquadrate, or 4th root of 20736 Thus, the square root of 20736 is 144;

then, the square root of 144 is 12, the Ans.

2. What is the square cubed, or the sixth root of 481890304 ?

Thus, 481890304 21952; and 21952=28, Ans. 3. Extract the square biquadrate, or eighth root of 1001129150390625.

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Alligation is the method of mixing several simples of different qualities, so that the composition may be a mean, or middle quality. It consists of two kinds, Alligation Medial and Alligation Alternate.

ALLIGATION MEDIAL,

Is when the quantities and prices of several things are given, to find the mean price of the mixture composed of those materials.

RULE.

As the sum of the quantities, or whole composition, is to the whole value, so is any part of the composition, to its mean price, or value.

EXAMPLES.

1. A farmer mixed together 4 bushels, of rye, worth 90 cents per bushel, 6 bushels of corn, worth 50 cents per bushel, and 8 bushels of oats, worth 30 cents per bushel; what is a bushel of this mixture worth?

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As 18 9001: 50 cents, price of 1 bushel,

2. A grocer mixed 5cwt. of sugar, worth $10 per cwt., 8cwt. worth $12 per cwt., with 3cwt, worth $9 per cwt.; what will be the cost of 4cwt. of this mixture?

Ans. $43,25.

3. A vintner mixed together 16 gallons of wine at $1,12c. per gallon, 12 gallons at $,90 per gallon, and 20 gallons at $,95 per gallon; what is the price of a gallon of the mixture? Ans. $,99, 4m. +

4. A goldsmith melted together 5 ounces of gold 21 carats fine, and 3 ounces 19 carats fine; what is the fineness of the mixture, that is, of one ounce of this mixture?

Ans. 20 carats fine, 5. A Grocer mixed 20 gallons of water with 85 gallons of rum, worth 76 cents per gallon, what is a gallon of the mixture worth? Ans. 61cts. 5m. +

6. A Refiner melted together 12lb. of silver bullion of 6oz. fine, 8lb. of 7oz. fine, and 10lb of 8oz. fine, I demand the fineness of the mixture. Ans. 6oz. 18pwt. 16gr.

7. Suppose that 3lbs. of gold of 22 carats fine, 5lbs. of 20 carats fine, and 1lb. alloy be melted together, what will be the fineness of the compound? Ans. 18 carats fine.

ALLIGATION ALTERNATE,

Is when the prices of the several simples and the mean price or rate are given, to find what proportion of each must be taken to compose a mixture of the given rate. It is therefore the reverse of Alligation medial, and may be proved by it.

CASE I.

When the mean rate and the rates of the several ingredients are given, without any limited quantity.

RULE.

1. Reduce the several prices to the same denomination. 2. Set the several prices under each other in a column, and place the mean rate on the left.

3. Connect or link each price which is less than the mean rate, with one or any number of those which are greater than the mean rate, and each price greater than the mean rate with one, or any number of the less.

4. Place the difference between the mean rate and that of each of the simples, opposite the price with which they are connected.

5. Then if only one difference stand against any rate, it will be the quantity belonging to that price, but if there be several, their sum will be the quantity.

EXAMPLES.

1. A Grocer would mix the following qualities of sugar, viz: at 8cts., 9cts., 11cts., and 12cts. per lb., what quantity must he take of each sort, that the mixture may be worth, 10cts. per lb.

cts.
8-

cts. 10

11.

12

lbs.
2

cts.

lbs.

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Here we set down the prices of the simples in order, from the least to the greatest; placing the mean rate at the left hand. And in the first way of linkng the prices, and taking the difference between the several prices and mean rate, and placing each difference at the opposite end of the link, we find that we must take in proportion of 2lbs. at 8cts., 1lb. at 9cts., 1lb. at 11cts., and 2lbs. at 12cts. ; and in the 2d operation, we have the proportion of 1lb. at 8cts., 2lb. at 9cts., 2lb. at 11cts., and 1lb. at 12cts. It will be seen that by linking any two of the prices together, and placing the

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