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5x5x3=75 first divisor.
51 x 51 x 51=132651 second subtrahend.
51 x 51 x3=7803 second divisor.

514x514x514=135796744 3d subtrahen! 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 tbt 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 rooi o that power reduces it to half the same power; and so ou 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 6th power: and the square rool 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 12230540 464.

Ans. 48. 5. Extract the square, biquadrate, or Sth root of 721:38 95789338336.

Ans 96.

ALLIGATION, 15 the method of nixing several simples of different qua. ties, 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 giyen, to find the mean price of the mixture composed ut 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 bushe., 18 bushels of Indian corn, at 55 cts. a bushel, and 21 bishels of onts, at 28 cts. a bushel; I demand what a bushel of this mixture is worth? bu. cts. $cts.

bu.
$ 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 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 25. a pound, with 12 lb. at Is. 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 56s. 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.

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

Ans. 58. 30d. 244 qos.

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6. A grocer hath several sorts of sugar, viz. onc sort at 8 dols. per cwt. another sort at 9 dols. per cwt. a third sort at 10 dols. per cwt. and a fourth sort at 12 dols. per cwt and he would mix au 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 02. 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 pwt. 8 gr. fine. 8. Suppose 5 lb. of gold of 22 carats fine, 2 lb. of 21 carats tive, and I 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 ratet of all the ingredients are given, without any limited quan 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, wbich 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 num: ber of the less.

3. Place the difference, between the mean price (or mix turi 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

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1. A merchant has spices, some at 9d. per lb. some 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 Is. 8d. per pound? d. Ib. d.

d. 9 10 at 9

9 4 2. 12 4 12 ( Gives the

d. 122

10 20 24 8 24 Answer; or

20 24 ) 11
30-
11 30

30

8 2. A grocer would mix the following qualities of sugar; 'iz. at 10 cents, 13 cents, and 16 cents per lb. ; what quanity of each sort must be taken to make a mixture worth 2 cents per pound ? Ins. 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. er lb. how nust he inix them so as to afford the composiion for 12s. per Ib. ?

Ans. "He must mir an equal quantity of cach sort. 4. A goldsmith woul. I mix gold of 17 carats fine, with wme of 19, 21, and 24 carats fine, so that the compound nay be 22 carats fine ; what quantity of each must he takc?

. Ins. 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. 155. and 9s. per gallon, with water at O per gallon, togeher, so that the mixture may be worth 6s. per gallon; how inuch of each sort must the mixture cousist of ? Ans. I gal. of rum at 5s., 1 do. at 7s., 6 do, at 9s. and 3 gals. water. Or, 3 gals. rum at 58., 6 do. at 7s., I do. at 9s. and

1 gal. water. 6. A grocer hath several sorts of sugar, viz. one sort at 12 its. 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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ALTERNATION PARTIAL, Or, when one of the ingredients is limited to a certain quantity, thence to find the several quantities of the rest, is proportion to the quantity given.

RULE. Take the differences between each price, and the mear rate, and place them alternately as in Case I. Then, as the difference standing against that simple whose quantity in given, is to that quantity : so is each of the other differ ences, severally, to the several quantities required.

EXAMPLES.

1. A farmer would mix 10 bushels of wheat, at 70 centa per hushel, with rye at 48 cts. corn at 36 ets. and barley at 30 cts. per bushel, so that a bushel of the composition mias be sold for 38 cts.; what quantity of cach must be taken ?

70 8 stands against the given quan 48 2

[tity Mean rate, 38

36 10
30- 32

2 : 2 bushels of rye. As 8 : 10 :: 10 : 12j bushels of corn.

32 : 40 bushels of barley.

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* These four answers arise from as many various ways of linking the rates of the ingredients together.

Questions in this rule admit of an infinite variety of answers: for after the quantities are found from different methods of linking; any other numbers i the same proportion between themselves, as the numbers which compose the answer, will likewise satisfy the conditions of Use question.

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