EXAMPLES. 1. What will 4001. amount to in 4 years, at 6 per

cent. per annum, compound interest ?

400X1,06x1,06x1,06x1,06=£504,99+ or

[£504 19s. 9d. 2,75qrs. + Ans.
The same by Table I.
Tabular amount of £1=1,26247
Multiply by the principal 400

Whole amount=£504,98800 2. Required the amount of 425 dols. 75 cts. for 3 years, *at 6 per cent. compound interest.

Ans. $507,7 cts.-t. 3. What is the compound interest of 555 dols. for 14 years, at 5 per cent. ? By Table I. Ans. $543,86cts. +

4. What will 50 dollars amount to in 20 years, at 6 per cent, compound interest? Ans. $160 35cts. 6km.

INVOLUTION.' Is the multiplying any number with itself, and that product by the former multiplier; and so on; and the several products which arise are called powers.

The number denoting the height of the power, is called the index, or exponent of that power:

Wliat is the square of 17,1 ?
What is the square of ,085 ?
What is the cube of 25,4?
What is the biquadrate of 12 ?
What is the square of 71?

Sis. 292,41
Alls. ,007225
Ans. 16387,064

Ans. 20736
Ans. 525

EVOLUTION, OR EXTRACTION OF ROOTS. WHEN the root of any power is required, the business of finding it is called the Extraction of the Root.

T'he root is that number, which by a continual multipli. cation into itself, produces the given power.

Although there is no number but what will produce a perfect power by involution, yet there are many numbers of which precise roots can never be determined. But, by the help of decimals, we can approximate towards the root to any assigned degree of exactness.

The roots which approximate, are called surd roots, and those which are perfectly accurate are called rational roots.

1 Table of the Squares and Cubes of tlie nine digits. Roots. | 1 | 2 | 3 | 41 1 61 81 9! Squares. 1| 4| 9 | 16 | 25 36 49 | 64 | 81 Cubes. 1118 | 27 | 64 | 125 | 216 | 343 | 512 | 7291

EXTRACTION OF THE SQUARE ROOT. Any number multiplied into itself produces a square.

To extract the square root, is only to find a number, which being multiplied into itself, shåll produce the given number.

RULE. 1. Distinguish the given number into periods of two. figures eachi, by putting a point over the place of units, another over the place of hundreds, and so on; and if there are decimals, point them in the same manner, from units towards the right hand; which points show the number of figurès the root will consist of.

2. Find the greatest square number in the first, or left band period, place the root of it at the right hand of the given number, (after the manner of a quotient in division) for the first figure of the root, and the square number under the period, and subtract it therefroin, and to the Temainder bring down the next period for a dividend.

3. Place the double of the root, already found, on the left hand of the dividend for a divisor.

4. Place sach a figure at the right hand of the divisor, and also the same figure in the root, as when inultiplie into the whole (increased divisor), the product shall be equal to, or the next less than the dividend, and it will be the second figure in the root.

5. Subtract the product from the dividend, and to the rernainder join the next period for a new dividend.

6. Double the figures already found in the root, for a new divisor, and from these find the next figure in the root as last directed, and continue the operation in the same manner, till you have brought down all the periods.

Or, to facilitate the foregoing Rule, when you liave brought down a period, and formed a dividend, in order to find a new figure in the root, you may divide said dividend, (omitting the right hand figure tlicreof,) by double the root already found, and the quotient will commonly be the figures sought, or being made less one, or two, will generally give the next figure in the quotient.

a

EXAMPLES.

1. Required the square root of 141225,64. 141225,64(575,8 the root exactly without a remainder; 9

but when the periods belonging to any

given number are exhausted, and still 67,512 leave a remainder, the operation

may 469

be continued at pleasure, by annexius

periods of cyphers, &cx 745)4525

3725

7508) 60064

2. What is the squarc root of 1996 ?
S. Of

56644?.
4. Of

5499025 ? 5. Of

36372961? 6. Of

184,2 ? 7. Of

9712,693809? 8. Of

0,45309 ? 9. Of

,002916 ? 10. Of

45?

Answers.

36 23,8 2345 6031 19,57 + 98,555 ,673 +

,054 6,708 *

TO EXTRACT THE SQUARE ROOT Or
VULGAR FRAOTIONS.

RULE. Reduce the fraction to its lowest terms for this and all other roots; then

1. Extract the root of the numerator se: a new numerator, and the root of the denominator, for a new denominator.

2. If the fraction be a surd, reduce it to a decimal, and extract its root.

EXAMPLES. 1. What is the square root of

His. 2. What is the square root of op

Ans. Bei 3. What is the square root of its

Ans. 4. What is the square root of 201 ?

Ins. 41

is. 153 5. What is the square root of 24818?

SURDS. 6. What is the square root of ? Ans. 91287. What is the square root of Ins. ,7745+ 8. Required the square root ofś.+?

Ans. 6,0207+

APPLICATION AND USE OF TITE SQUARE

ROOT. PROBLEM I. A certain General nas anw my ot 5184 men; low many must he place in rank anc file, to foxrin them into a square ?

RULE.
Extract the square root of the yiven number,

✓51845-72 Ins. Prop. II. A certain square pavement contains 20736 square stones, all of the saine size ; I demand how many are contained in one of its sides 20731=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 hetween 18 and 72 ?

72*13=1246, and ✓ 1296336 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 duuble, trple, &c. and the product will be the number in rank.

EXAMPLES.

het 13122 nen be so fornells as that the number in rank inay be double the number in file.

13122+2=45561, and 71:561=Si in file, and 81 X2 =162 in rank,

PROD. V. Admit 10 hhds. of water are discharged through a leaden pipe of 2 inches in diameter, in a certain tire; i demand what the diameter of another pipe must be, to discharge four times as much water in the sane tiine,

RULE. Square the given diameter, and multiply said squaire by the given proportion, and the square root of the product is the answer: 23=2,5, and 2,5x2,5=6,25 square.

4 given proportion. ✓25,00=5 ioch. diam. Ans.