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No'rE.-The foregoing example gives a perfect root; and if, when all the periods are exhausted, there happens to be a remainder, you may annex periods of cyphers, and continue the operation as far as you think it necessary. ·

Answers. 2. What is the cube root of 205379 ?

59 3. Of

614125? 4, Of


346 5. Of

146363,183 ?

52,7 6. Of

29,503629 ?

3,0980,763 ?

4,324 8. Of

,162771336 ?

9. Of
,000684134 ?

10. Of

1. Find ly trial, a cube near to the given number, and
call it the supposed cube. ::

2: Then, as twice the supposed cube, added to the given
number, is to twice the given number added to the sup.
posed cube, so is the root of the supposed cube, to the
true root, or an approximation to it.

3. By taking the cube of the root thus found, for the
supposed cube, and repeating the operation, the root will
be had to a greater degree of exactness.

Let it be required to extract the cube root of 2.

Assume 1,3 as the root of the nearest cube ; then
1,3x1,3 X1,3=2,197 =supposed cube.
:. Then, 2,197 2 ,000 given number...

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- As 6,394 ; 6,197 :: 1,3 : 1,2599 root,
which is true to the last place of decimals; but might by.
repeating the operation, be brought to a greater exactness.
2. What is the cube root of 584,277056 ?

Ans. 8,36.

3. Required the cube root of 729001101?

Ans. 900,0004

Shewing the use of the Cube Root. 1. The statute bush-i contains 2150,495 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 sinilar sides or diameters,

2. If a bullet S inches diameter, weigh 41b. what will a bullet of the same metal weigh, whose diameter is 6 inches?

3X3X3=27 6x6x6=216 As 27 : 41b. : : 216: 321b. Ans.

3. If a solid globe of silver, of 3 inches (liameter, be worth 150 dollars; what is the value of another globe of silver, whose diameter is sis inches ?. S

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 tube 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 inuch ?

2x2x2=8 and 8X864 3/64=4 inches. Ans. 5. There is a cubical vessel, whose side is 4 feet;I: 1 demand the side of another cubical vessel, which shall contain 4 times as much ?

4X4X4=64 and 64x4=2563 256=6,349+ft. Ans.
6. A cooper having a cask 40 inches long, and 32 iz-

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 ?
40X40X40X2=128000 then 3128000=50,3+ length.
32X32X32X2=65536 and 3765536=40,3+bung diam.

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General Rule for Extracting the Roots of all Powers,

RULE. 1. Prepare the given number for extraction, by points ing 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.

Ś. To the remainder bring down the first figure in the pext 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 a3 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.


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1. Required the cube root. of 1357963744 by the above general method.

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2x2x2x2x5=80)323 dividend.

23x23x23x23x23 m6436343 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 biquadrate, or 4th root of 19987173376?

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

Ans. 48. 5. Extract the square, biquadrate, or 8th root of 72158 95789338336.

Ans. 96.

ALLIGATION, Is the method of mixing several siinples of different ities, so that the composition may be of a mean or in quality : It consists of two kinds, viz. Alligation M and Alligation Alternate.

ALLIGATION MEDIAL, Is when the quantities and prices of several thing given, to find the mean price of the mixture compos those materials.

· RULE. As the whole composition : is to the whole value : is any part of the composition : to its mean price.

1. A farmer mixed 15 bushels of rve, at 64 ce
bushel, 18 bushels of Indian corn, at 55 cts. a bushel,
21 bushels of dats, at 28 cts. a bushel ; I demand w
bushel of this mixture is worth?

bul. cts. Scts. bu. S cts. bu.
15 at 64=9,60 As 54 : 25,38 : :1
18 .5559,90
21 28=5,88 ;


54)95,58(,47 Answer 54 25,58 2. If 20 bushels of wheat at 1 dol. 35 cts. per bus be mixed with 10 bushels of rye at 90 cents per bus what will a bushel of this mixture be worth?

...ans. $1, 2001 3. A Tobacconist mixed 36 lb. of Tobacco, at is. per lb, 12 lb. at 2s, a pound, with 12 lb. at is. 10d. lb.; what is the price of a pound of this mixture ?

ens.' Is. 8 4. A Grocer mixed 2 C. of surar, at 56s. per C. ar C. at 433 per C. and 2 C. at 50s. per C. together; I mand the price of 3 cwt. of this mixture ? Ans. 47 13

5. A Wine merchant mixes 15 gallons of wine at

2d. per gallon, with 24 gallons at 6s. 8d. and 20 gall il ... at 6s. 3d., what is a gallon of this composition wort!

· Ans. 5$. 10d. 2ar

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