RULE.-Extract the square root of the given number. √5184-72 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 Ans. PROB. III. To find a mean proportional between two numbers.

RULE.-Multiply the given numbers together and extract the square root of the product.

EXAMPLES.

What is the mean proportional between 18 and 72? 72 × 18=1296, and ✓1296–36 Ans. PROB. IV. To form any body of soldiers so that they may be double, triple &c. as many in rank 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 in file,which double, triple, &c. and the product will be the number in rauk.

EXAMPLES.

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

13122-2-6561, and 6561-81 in file, and 81×2 162 in rank.

PROB. V. Admit 10 hhds. of water are discharged through a leaden pipe of 24 inches in diameter, in a certain 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 me answer.

2=2fa12,5×2,5=6,25 square.

4 given proportion.

25,00-5 inch. diam. Ans.

PROB. VI. The sum of any two numbers, and their pro ducts 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.

EXAMPLES.

The sum of two numbers is 43, and their product is 442; what are those two numbers?

The sum of the numb. 43 × 43≈1849 square f do.

The product of do.

442 × 4-1768 4 times the pro.

Then to the

sum of 21,5

(numb. V81-9 diff. of the

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 num

ber.

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 quotien、 by 300, calling it the divisor.

5. Seck how often, the divisor may be had in the divi dend, and place the result in the quotient; then multiph 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 301, and place the product under the last; then 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.

NOTE. If the subtrahend (found by the foregoing rule) happens to be greater than the dividend, and consequently 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 subtrac the subtrahend from the dividend.

EXAMPLES.

1. Required the cube root of 18399,744.

18399,744(26,4 Root. Ans.

8

2×2=4×300=1200)10399 first dividend.

7200

6×6=36×2=72×30=2160

6×6×6=216

9576 1st subtrahend.

26×26=676×300—202800)823744 2d dividend.

811200

4×4=16×26=416×30= 12480

4×4×4=

64

823744 2d subtrahend

NOTE.-The foregoing example gives a perfect root; and if, when all the periods are exhausted, there happens to he a remainder, you may annex periods of ciphers, and cortinue the operation as far as you think it necessary.

RULE.-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.

1. Let it be required to extract the cube root of 2. Assume 1,3 as the root of the nearest cube; then-1,3× 1,3 × 1,3=2,197 supposed cube.

2,000 given number.

Then, 2,197

2

2

1,3 : 1,2599 root,

which is true to the last place of decima.s; but might by repeating the operation he brought to greater exactress. 2. What is the cube root of 584' KOLE

3 Required the cube root of 729001101?

QUESTIONS,

Ans. 900,0004

Showing 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 con. tain that quantity?

√2150,425=12,907 inch. Ans.

NOTE. The solid contents of similar figures are in proportion to each other, as the cubes of their similar sides of diameters.

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

3×3×3=27 6×6×6=216. As 27: 4 b. :: 216 | 32 lb. Ans.

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

3 (3×3=27 $1200. Ans.

6×6×6=216, As 27: 150 :: 216

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

RULE.-Cube your given side, and multiply by the given propor tion between the given and required cube, and the cube root of th product will be the side sought.

EXAMPLES.

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

3

2×2×2=8, and 8×8=64 √/64–4 inches. Ans.

5. There is a cubical vessel, whose side is 4 feet; I de mand the side of another cubical vessel, which shall con tain 4 times as much?

4×4×4=64, and 64×4=256 /256–6,349+ft. Ans.

6. A cooper having a cask 40 inches long, and 32 in