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ches at the bung diameter, is ordered to make another cask of the same shape, but to hold just twice as much; wha will be the bung diameter and length of the new cusk ? 10 x 40 x 40x2=128000 then 128000=50,3+-length. 32 x 32 x 32 x 2=65536 and 7655:36–40,3+ bung diam.

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A General Rule for extracting the Roots of all Powers.

RULE.

1. Prepare the given number for extraction, by pointing sit from the unit's place, is the required root directs.

2. Find the first figure of the root by trial, and stract vis power from the left hand period of the given number.

3. To the remainder bring down the first figure in the #ext period, and call it the dividend.

4. Involve the root to the next inferior power to that which is given, and multiply it by the number denoting the iiven 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 it (always) from as many periods of the given number as you have found figures in the root.

7. Bring dowu the first figure of the next period to the remainder for a new dividend, to which find a new divisor as before, and in like marner proceed till the whole be finished.

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

EXAMPLES.

1. Required the cube root of 135796,744 by the above general method.

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5X5X3=75 first divisor.
51 x 51 x 51=132651 second subtrahend
51 x 51 x3=7803 second divisor.

514 X 514 X 514=135796744 3d subtrahent 2. Required the sursolid or 5th root of 6436343.

6436343(23 root.
32

2x2x2x2x5=80)323 dividend.
23 x 23 x 23 x 23 x 23=6436343 subtrahend.

Note.—The roots of most powers may be found by the square and cube roots only; therefore, when any ever power is given, the easiest method will be (especially in e very high power) to extract the square root of it, which re duces it to half the given power, then the square rool o that power reduces it to half the same power; and so on till you come to a square or a cribe.

For example: suppose a 12th power be given; the square root of that reduces it to a 6th power: and the square rooi of a 6th power to a cube.

EXAMPLES,

3. What is the biquadrate, or 4th root of 199871733761

Ans. 376. 4. Extract the square, cubed, or 6th root of 12230590 464.

Ans. 48. 5. Extract the square quadrate, or Sth root of 72138 95789338836.

Ans 96.

ALLIGATION, is the method of mixing several simples of different quaijties, so that the composition may be of a mean or middle quality : It consists of two kinds, viz.. 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 whole composition : is to the whole value : : 80 18 any part of the composition : to its mean price.

EXAMPLES. 1. A farmer mixed 15 bushels of rye, at 64 cents a bushe, 18 bushels of ludian corn, at 55 cts. a bushel, and 21 bushels of oats, at 28 cts. a bushel; I demand what a hushel of this mixture is worth?

bu. cts. $cts. bu. $ cts. bu.
15 at 64=9,60 As 54 : 25,38 : :1
18 559,90

1
21 28=5,88

-cts.

54)25,38(,47 Ans. 54 25,38 2. If 20 bushels of wheat at 1 dol. 35 cts. per bushel be mixed with 10 bushels of rye at 90 cents per bushel, what will a bushel of this mixture be worth ?

Ans. $1,20 cts. 3. A tobacconist mixed 36 lb. of tobacco, at 1s. 6d. per lb. 12 lb. at 2s. a pound, with 12 lb. at 1s. 10d. per ib. ; what is the price of a pound of this mixture ?

Ans. ls. 8d. 4. A grocer mixed 2 C. of sugar at 50s. per C. and I C. at 43s. per C. and 2 C. at 50s. per C. together; I demand the price of 3 cwt. of this mixture ? Ans. £7 13s.

5. A wine merchant mixes 15 gallons of wine at 48. Bd. per gallon, with 24 gallons at 6s. 8d. and 20 gallons a Gs. 3d i what is a gallon of this composition worth ?

Ins. 5s. Ind. 240 grs.

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178

ALLIGATION ALTERNATE,

6. A grocer hath several sorts of sugar, viz. one sort & 8 dols. per cwt. another sort at 9 dols. per cwt. a third sold at 10 dols. per cwt. and a fourth sort at 12 dols. per cwt. and he would mix an equal quantity of each together; ) demand the price of 31 cwt. of this mixture ?

Ans. $34 12 cts. 5 m. 7. A goldsmith melted together 5 lb. of silver bullion, of 8 oz. fine, 10 lb. of 7 oz. fine, and 15 lb. of 6 oz. fine; pray what is the quality or fineness of this composition ?

Ans. 6 oz. 13 put. 8 gr. fine. 8. Suppose 5 lb. of gold of 22 carats fine, 2 lb. of 21 carats fine, and 1 lb. of alloy be melted together; what is the quality or fineness of this mass ?

Ans. 19 carats fine.

ALLIGATION ALTERNATE, IS the method of finding what quantity of each of the ingredients whose rates are given, will compose a mixture of a given rate; so that it is the reverse of Alligation Me. dial, and may be proved by it.

CASE I. When the mean rate of the whole mixture, and the ratos of all the ingredients are given, without any limited quas tity.

RULE. 1. Place the several rates, or prices of the simples, be. ing reduced to one denomination, in a column under each other, and the mean price in the like name, at the left hand

2. Connect, or link the price of each simple or ingredi. ent, which is less than that of the mean rate, with one of any number of those, which are greater than the mean rate, and each greater rate, or price, with one, or any num. ber of the less.

3. Place the difference, between the mean price (or mir. ture rate) and that of each of the simples, opposite to the rates with which they are connected.

4. Then, if only one difference stands against any rate, It will be the quantity belonging to that rate, but if there be several, their sum will be the quantity.

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Answer

EXAMPLES. 1. A merchant has spices, some at 9d. per lb. some at Is. come at 2s. and some at 2s. 6d. per lb. how much of each jort must he mix, that he may sell the mixture at ls. 8d. per pound? d. 16. d.

d. 16. 9 10 at 9

9 4 2. 12 4 12 Gives the

d. 12?

10 20 ) 24 8 24 | Answer ; or

20 245

11 30- 11 30

30 8 2. A grocer would mix the following qualities of sugar; riz, at 10 cents, 13 cents, and 16 cents per lb. ; what quanity of each sort must be taken to make a mixture worth ? cents per pound? ains. 5 lb. at 10 cts. 2 lb. at 13 cts, and 2 lb. at 16 cts. per lb.

3. A grocer has two sorts of tea, viz. at 9s. and at 158. Mcr lb. how must he mix them so as to afford the composiiwn for 12s.

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Ib. 3 Ans. He must mir an equal quantity of each sort. 4. A goldsmith woul.l mix gold of 17 carats fine, with some of 19, 21, and 24 carats fine, so that the compound way be 22 carata fine; whát quantity of each must he takc?

Ans. 2 of each of the first three sorts, and 9 of the last. 5. It is required to mix several sorts of rum, viz. at 5s. is. and 9s. per gallon, with water at O per gallon, togeTher, so that the mixture may be worth 6s. per gallon; how Touch of each sort must the mixture consist of? Ans. I gal. of rum at 5s., I do. at 7s., 6 do, at 9s. and 3 gals. water. Or, 3 gals. rum at 5s., 6 do. at 7s., I do. at 9s. ană

| gal. water. 0. A grocer hath several sorts of sugar, viz. one sort at 12 ets. per lb. another at 11 cts. a third at 9 cts. and a fourth at 8 cts. per lb. ; I demand how much of each sort he must mix together, that the whole quantity may be afforded at 10 cents per pound?

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