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3. Required the cube root of 729001101?

Ans. 900,0004

QUESTIONS, Shewing the use of the Cube Root. 1. The statute bushel contains 2150,425 cubic or solid inches. I demand the side of a cubic box, which shall contain that quantity ?

32150,425=12,907 inch. Ans. Note.—The solid contents of similar figures are in proportion to each other, as the cubes of their similar sides or diameters.

2. If a bullet 5 inches diameter, weigh 4lb. what will a bullet of the same metal weigh, whose diameter is 6 inches ?

3X3X3=27 6x6x6=216 As 27 : 4lb. : : 216 : 32lb. Ano.

3. If a solid globe of silver, of 3 inches diameter, be worth 150 dollars; what is the value of another globc of silver, whose diameter is six inches ? $

3X3X3=27 6x6x6=216 As 27 : 150 : : 216 : $1200. Ans.

The side of a cube being given, to find the side of that cube wich shall be double, triple, &c. in quantity to the given cube.




Cube your given side, and multiply by the given proportion between the given and required cube, and the cube root of the product will be the side sought.

4. If a cube of silver, whose side is two inches, be worth 20 dollars; I demand the side of a cube of like silves. whose value shall be 8 times as much ?

2x2x2=8 and 8x8=64 364=4 inches. Ans. 5. There is a cubical vessel, whose side is 4 feet ; I demand the side of another cubical vessel, which shall contain 4 times as much ?

4x4x4=64 and 64x4=2567 256=6,349+ft. Ans. 6. A cooper having a cask 40 inches long, and 32 in.

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 128000=50,5+ length.
32x32x32x2=65536 and 765536=40,3+bung diam.

A General Rule for Extracting the Roots of all Powers.


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.

3. To the remainder bring down the first figure in the next period, and call iť 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 divisor, 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.


1. Required the cube root of 155796,744 by the above general method.


185796744(51,4 the root.
125 = 1st subtrahend.

75)107 dividend.

132651=2d subtrahend. 7803) 31457 =20 dividend.

135796744=3d subtrahend.

5x5 x3 = 75 first divisor.
51 x51 X51=132651 second subtrahend.
51 X51X3 31803 second divisor.

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

6436345)23 root.

8x2x2x2x5=80)323 dividend.

23 x 23 x 23 x 23 x 23 m 6436343 subtrahend. Note.--The roots of most powers may be found by ths square and cube roots only; therefore, wher, 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 hail 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 cabe.


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

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

Ans. 48. 5 Extract the square, biquadrate, or 8th root of 72158 5729338336.

Ans. 96.

ALLIGATION, Is 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 : : 90 is any part of the composition : to its mean price.



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

bu. cts. Scts. bu. $ cts. bu.
15 at 6459,60 As 54 : 25,38 : :1
18 5559,90

21 28=5,88

54)25,58(,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. $1, 20cts. 3. A Tobacconist mixed 36 lb. of Tobacco, at 1s. 6d. per lb. 12 lb. at 25. a pound, with 12 lb. at 1s. 10d.

per Ib. ; what is the price of a pound of this mixture ?

Ans. 1s. 8d. 4. A Grocer mixed 2 C. of sugar, at 56s. per C. and 1 C. at 43s per C. and 2 C. at 50s. per C. together ; I de& mand the price of 3 cwt. of this mixture ? Ans. 67 135.

5. A Wine merchant mixes 15 gallons of wine at 48. & 2d.

2d. per gallon, with 24 gallons at 68. &d. and 20 gallons, at 6. sd.; what is a gallon of this composition worth?

Ans. 5s. 10d. 24kars.


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 fourth sort at 12 dols. per cwt. and he would mix an equal quantity of each togeth. er ; I demand the price of 3 cwt. of this wixture ?

Uns. 834 12cts. 5m. 7. A Goldsmith melted together 5 lb. of silver bullion, of 8 oz. fine, 10 lb. of 7 0%.. fine, and 15 lb. of 6 0% fine; pray what is the quality, or fiueness of this coinposition

Ans. 6uz. 15pwt. 8gr. fine. 8. Suppose 5 lb. of gold of 22 carats tine, 2 lb. of 21 carats tine, 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 giveu, will compuse a mix. ture 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, be. ing reduced to one denomination, ia a column under each other, and the mean price in the ylike 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 oue, or any nuinber 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.

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