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RULE.
Extract the square root of the given number.

75184=72 Ans. Prob. II. A certain square pavement contains 20736 square stones, all of the same size; I demand how many are contained in one of its sides ? 20736=144 ins.

Prob. III. To find a mean proportional between two numbers.

RULE. Multiply the given numbers together, and extract the square root of the product. What is the mean proportional between 18 and 72 ?

72x18=1296, and ✓ 1296=36 Ans. PROB. IV. To form any body of soldiers so that they may be double, triple, &c. as many in ranik as in file.

RULE. Extract the square root of 1-2, 1-3, &c. of the given number of men, and that will be the number of men ic file, which double, triple, &c. and the product will be the number in rank.

EXAMPLES.

EXAMPLES.

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Let 13122 men be so formed, as that the number in rank may

be double the number in file.. 131222=6561, and 6561=81 in file, and 81 x2 162 in rank.

PROB. V. Admit 10 hhds. of water are discharged through a leaden pipe of 24 inches in diameter, in a cer. tain time; I demand what the diameter of another pipe must be, to discharge four times as much water in the same time.

RULE. Square the given diameter, and multiply said square by the given proportion, and the square root of the product is the answer. 2}=2,5, and 2,5 x2,5=6,25 square.

4 given proportion.

✓ 25,00=5 inch. diam. Ans.

PROB. VI. The sum of any two numbers, and their products being given, to find each number.

RULE. From the square of their sum, subtract 4 times their product, and extract the square root of the remainder, which will be the difference of the two numbers; then half the said difference added to half the sum, gives the greater of the two numbers, and the said half difference subtracted from the half sum, gives the lesser number. The sum of two numbers is 43, and their product is

The sum of the numb. 43x43=1849 square of do. The product of do. 442 x 4=1768 4 times the pro. Then to the sum of 21,5

[numb. tand

4,5

V81=9 dift. of the

EXAMPLES.

The sum of those two numbers ?

442; whata

are

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EXTRACTION OF THE CUBE ROOT. A cube is any number multiplied by its square.

To extract the cube root, is to find a number, which, being multiplied into its square, shall produce the given number.

RULE. 1. Separate the given number into periods of three figures each, by putting a point over the unit figure, and every third figure from the place of units to the left, and if there be decimals, to the right.

2. Find the greatest cube in the left hand period, and place its root in the quotient.

3. Subtract the cube thus found, from the said period, and to the remainder bring down the next period, calling this the dividend.

4. Multiply the square of the quotient by 300, calling it the divisor.

5. Seek how often the divisor may be had in the divi. dend, and place the result in the quotient; then multiply the divisor by this last quotient figure, placing the product under the dividend.

6. Multiply the former quotient figure, or figures by the square of the last quotient figure, and that product by 30, and place the product under the last; tnen under these two products place the cube of the last quotient figure, and add them together, calling their sum the subtrahend.

7. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend;

with which proceed in the same manner, till the whole be finished.

Nore.--If the subtrahend (found by the foregoing rule) happens to be greater than the dividend, and consequenta ly cannot be subtracted therefrom, you must make the last quotient figure one less ; with which find a new subtrahend, (by the rule foregoing) and so on until you can subtract the subtrahend from the dividend.

EXAMPLES.

1. Required the cube root of 18399,744.

18399,744(26,4 Root. Ans.

8

2x2=4x300=1200)10399 first dividend.

1.)

7200 6.46=36x2=72x30=2160

6x6x6= 216

9576 1st subtrahend. 26 x 26=676x300=203800)823744 ad dividend.

811200 4X4=16x26=416x30= 12480

4X4X4= 64

825744 2d subtrahend.

3,09

NOTE.-The foregoing example gives a perfect root 3 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

i 614125 ?

85 4. Of

41421736 ?

346 5. Of

146363,183 ?

52,7 6. Of

29,503629 ? 7. Of

80,763 ?

4,32+ 8. Of

,1627713367

,546 9. Of

,000684134 ?

,088+ 10. Of 122615327232 ?

4968 RULE II. 1. Find by 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 supposed 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.

EXAMPLES.
Let it be required to extract the eube root of 2.

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

2

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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

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

3X3X3=27 6x6x6=216 As 27 : 41b. : : 216 : S2lb. Ans.

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.

RULE.

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 dolars; I demand the side of a cube of like silvei, 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 64 x4=256 /256=6,349+ft. Ans. 11. A cooper having a cask 40 inches long, and 32 in.

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