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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 ; what will be the bung diameter and length of the new cask ? 40 x 40 x 40 x2=128000 then y128000=50,3+ length. 32X32X32X2=65536 and 765536=40,3 +bung diam.

A General Rube for Extracting the Roots of all Powers.

RULE.

1. Prepare the given number for extraction, by pointing off from the unit's 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 of the given number.

5. To the remainder bring down the first figure in the next 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 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 it (always) 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, to which find a new divi. sor, as before, and in like manner 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 quotient figure must be taken less, &c.

EXAMPLES.

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

135796744(51,4 the root.
125=1st subtrahend.

75)107 dividend.

132651=2d subtrahend. 7803) 31457=2d dividend.

135796744=3d subtrahend.

5X5X3=75 first divisor.
51 X51X51=132651 second subtrahend.
51X51X3=7803 second divisor.

514x514x514=135796744 third subtrahende 3. Required the sursolid, or fifth root of 6436343.

6436345)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 even power is given, the easiest method will be (especially in a very high power) to extract the square root of it, which reduces it to half the given power, then the square root of that power

reduces it to half the same power; and so on, till

you come to a square or a cube.

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

EXAMPLES.

3. What is the biguadrate, or 4th root of 19987173376?

Ans. 376. 4. Extract the square, oubed, or 6th root of 12250590 464.

Ans. 48. 5. Extract the square, biquadrate, or 8th root of 72138 957 09888336.

Ans. 96.

1

ALLIGATION, Is

S the method of mixing several simples of different qualities, 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 : : so is any part of the composition : to its mean price.

EXAMPLES.

1. A farmer mixed 15 bushels of rye, at 64 cents a bushel, 18 bushels of Indian corn, at 55 cts. a bushel, and 21 bushels of oats, at 28 cts. a bushel ; I demand what ೩ bushel of this mixture is worth?

bu. cts. Scts. bu. S cts. bu.
15 at 64=9,60 As 54 : 25,38 : :1
18 55=9,90

1
21 28=5,88

cts.

54)25,38(,47 Answer. 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. 81, 20cts. 3. A Tobacconist mixed 56 lb. of Tobacco, at Is. 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. Is. 8d. 4. A Grocer mixed 2 C. of sugar, at 568. per C. and 1 C. at 43s per C. and 2 C. at 50s. per C. together; I demand the price of 3 cwt. of this mixture ? Ans. £7 138.

5. A Wine merchant mixes 15 gallons of wine at 48. 2d. per gallon, with 24 gallons at 6s. 8d. and 20 gallons, at 6s. 3d. ; what is a gallon of this composition worth ?

Ans. 5s. 10d. 2grs.

wansort at 12 dols.

6. A grocer hath several sorts of sugar, viz. one sort at 8 dols. per cwt. another sort at 9 dols. per cwt. a third sort at 10 dols, per cwt. and a

per

of each together; I demand the price of 33 cwt. of this mixture ! cwt. and he would mix anar "th sort

Ans. 854 12cts. 5m, 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, 6oz. 15pwt, 8gr. 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 the ingredients, whose rates are given, will compose a mix.

of a given rate; so that it is the reverse of alligation medial, and may be proved by it.

CASE. I. When the mean rate of the whole mixture, and the rates of all the ingredients are given without any limited quantity.

RULE. 1. Place the several rates, or prices of the simples, being 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 ingre. dient, which is less than that of the mean rate, with one or any number of those, which are greater than the mean rate, and each greater rate, or price with one, or any number of the less.

3. Place the difference, between the mean price (or mixture 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.

EXAMPLES.

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Answer.

Ib. ;

1. A merchant has spices, some at:9d. per lb. soine at 1s. some at 2s. and some at 2s. 6d. per lb. how much of each sort must he mix, that he may sell the mixture at 1s. 8d. per pound ? d. d. '16

d,

ib. 10 at 97

97 d. 12 4 12! Gives the d. 12- 10 20 24 8 24 Answer. or 20 24 11 30 11-30

30- 8 2. Agrocer would mix the following quantities of sugar; yiz. at 10 cents, 13 cents, and 16 cts.

per

what quantity of each sort must be taken to inake a mixture worth 12 cents per pound ?

3. A grocer has sorts viz. at 9s. and at 158. per lb. how must hé mix them so as to afford the composition for 128. per lb. ?

Ans. He must mix an equal quantity of each sort. 4. A goldsınith would mix gold of 17 carats fine, with some of 19, 21, and 24 carats fine, so that the compound may be 22 carats fine; what quantity of each must he take.

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. 78. and 9s. per gallon, with water at o per gallon together, so that the mixture may be worth 6s. per gailon ; how much of each sort 'must the mixture consist of ? Ans. 1 gal. of Rum at 5s. 1 do. at 7s. 6 do at 9s. and 3

gals. water. Or, 3 gals. 'rúin at 5s. 6 do. at 7s. 1 do. at 9s. and 1 gal, water. 6. A grocer hath several sorts of sugar, viz, one sort at 12 cts. per lb. another at 11 cts. a third at 9'cts. and a fourth at 8 cts. per ib,

I demand how much of each sort Dust he mix together, that the whole quantity may

he afturded at 10 cents per pound?

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