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EXAMPLES. 1. What will 4001. amount to in 4 years, at 6 per cent per annum, compound interest ?

400 x 1,06 1,06x1,06 X 1,06=£504,99 +0.

[£504 19s. 9d. 2,75 qrs.+ 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. $507,74 cts. +

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

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

INVOLUTION, IS the multiplying any number with itself, and that pro. duct 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.

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What is the squarn, of 17,1 ?
What is the square of ,085 ?
What is the cube of 25,4?
What is the hiquadrate of 12?
What is the square of 71?

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

Ans. 20736
Ans. 52

51

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.

The root is that number, which by a continued 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 uy assigned degree of exactness.

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

A Table of the Squares and Cubes of the nine digits. Roots. 112|341 61 71 81 91 Squares. 11| 4| 9 | 16 | 25 | 36 | 49 | 64 81 Cubes. | 118 | 27 | 64 | 125 | 216 | 343 | 512 | 729

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 shall produce the given sumber.

RULE.-1. Distinguish the given number into periods of nyo figures each, 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 figures the root will consist of.

2. Find the greatest square number in the first, or left nand period, place the root of it at the right hand of the

an.

given number, (after the manner of a quotient in division) for the first figure of the root, and the square

number der the period, and subtract it therefrom, and to the re.. mainder 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 such a figure at the right hand of che divisor, and also the same figure in the root, as when multiplied into the whole (increased divisor) the product Sriad 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 remainder 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 havo 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 thereof) 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.

EXAMPLES 1. Required the square root of 141225,64. 141225,64(375,8 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 annexing

periods of ciphers, &c. 745) 4325

3725

7508)60064

60064

V remains.

2 What is the square root of 1296 ? 3 Of

56044? 4. Of

51990255 ? 5. Of

36:37:2901 Ž 6 Of

104,2? 7. Of

9712,693209? 8 Of

0,45369? 9. Of

,00:2916 ? 10. Of

45?

ANSWERS. 36

23,8 2345 6031 13,57+ 98,553

,673 ,055 6,708+

TO EXTRACT THE SQUARE ROOT OF VUL

GAR FRACTIONS.

RULE.

Reduce the fraction to its fuwest termis for this and all other roots; thien

1. Extract the root of tlre vunerator for a new numera's **, arid the root of the denoviralor, for a new denominator,

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

EXANPLES. 1. What is the square root of ? ANSWERS. 3. What is the square root of the

3 3. What is the square root of it?? 4. What is the square root of 20! ?..

41 5. What is the square root of 24876?

157 SURDS. 6. Wliat is the square root of 3?

9128 7. What is the square root of 4%?

,7745 8. Required the square root of 36: ?

6,0207+ APPLICATION AND USE OE TIE SQUARE

ROOT. PROBLEM I.-A certain general has an army of 5184 .nen; how many must hc place in rank and tile, to forma them into a square?

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RULE.-Extract the square root of the given number.

15184=72 Ans. ProB. SI. A certain square pavement contains 20750 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 twe. numbers.

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

EXAMPLES,

What is the mean proportional between 18 and 72?

72 * 18=1296, and 1296336 Ans. Prop. 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 rank.

EXAMPLES.

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

13122 - 226561, and V6561=81 in filc, and 81 x2 =102 in runk.

PROB. V. Admir 10 hhds. of water are discharged hrough a lenden pipe of 24 inches in diameter, in a cerusein time; I demand what the diameter of another pipe must be to discharge four times as much water in the same time.

RuLF.-Square the given diameter, and multiply said square by the given proportion, and the square root of the oroduct is ine answer. 2=25, a : 2,5 x 2,306,25 square.

4 given proportion.
500 nő intl. diam. Ans.

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