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3d payment April 1, 1801.

12,00 4th payment May 1, 1801.

30,00 How much remains due on said nie the irtisanep tember, 1801 ?

3.ts. Principal dated February 1, 1800,

500,00 Interest to May 1, 1800, (3 mo.)

7,50

Amount, 507,50 Paid May 1, 1800, a sum exceeding the interest, 40,00

New principal, May 1, 1800,
Interest to May 1, 1801, (1 year.)

467,50 28,05

Amount, 495,55 Paid Nov. 4, 1800, a sum less than the interest then due,

8,00 Paid April 1, 1801,

do. do. 12,00 Paid May 1, 1801, a sum greater, 30,00

50,00

New principal May 1, 1801,

445,55 Interest to Sep. 16, 1801, (4) mo.)

10,02 Balance due on the note, Sept. 16, 1801, 8455,57

The payments being applied according to the Rule, keep down the interest, and no part of the interest ever forms a part of the principal carrying interest.

COMPOUND INTEREST BY DECIMALS.

RULE. MULTIPLY the given principal continually by the amount of one powd, or one dolar, for one year, at the rate per cent. given, until the nunber of multiplications are equal to the given aumber of years, and the product will be the amount required.

OR, In Table I. Appendix, find the amount of one dollar, or one pound, for the given number of years, which multiply by the given principal, and it will give the amount as before.

EXAMPLES.

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

400X1,00 x 1,00 x 1,06x1,06==£504,99+ or

[6,304 19s. 9d. 2,75grs.+ Ans.
The same by Table I.
Tabular amount of fl=1,26247
Multiply by the principal 400

i

$

Whole amount=4504,98800 2. Required the amount of 495 dots. 75 cts. for 3 years, at 6 per cent. compound interest. Ans. 9507,7 cts.

3. What is the compound interest of 555 dols. for 13 years, at 5 per cent. ? By Table I. Ans. $543,86cts. +

4. What wili 50 dollars annount to in 20 years, at 6 per cent, compound interest ? Ans. $160 35cts. 61m.

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INVOLUTION. Is the multiplying any namber 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.

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32768 == 5th power, or surselid. Airis.

What 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 ?

Ans. 292,41 Aizse afl225 ains, latina

ins. je

a

EVOLUTION, OR EXTRAC'DON 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 continual multipilcation into itself, produces the given power.

Although there is no number but what will produce a perfect power by involution, yet there are niany 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.

A Table of the Squares and Cubes of the nine digits. Roots. 1112 | 3 | 41 671 81 9

Squares. 11/41 9116! 23 55 49 | 64 81 Cubes. 1118 1 27 64 165 1716 1 343 1 519 729

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

To extract the square rout. inuniy to ind a number, which being multiplied into itself, shall produce the given Qumber.

RULE. 1. Distinguish the given number ito periods of two 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 liand; which points biow the number of figures the rnot, will consist ol.

2. Find the greatest satare number in the list, or test

51

hand 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 therefrom, and to the remainder 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 the divisor, and also the same figure in the root, as when multiplied 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 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 have 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 remaindes; 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 cyphers, &c. 745)4325

3725

7508)60064

60064

emaine .

Answers

36

2. What is the square root of 1296 ?
3. Of

56644
4. Of

5499025. 5. Of

36372961 ? 6. Of

184,2 ? 7. Of

9712,693809? 8. Of

0,45369 ? 9. Of

,002916 ? 10. Of

45 ?

99,553

,678+ ,054 6,708+

TO EXTRACT THE SQUARE ROOT OB
VULGAR FRACTIONS.

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

1. Extract the root of the numerator for a new nume. pator, and the root of the denominater, for a new denomi. nator.

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

EXAMPLES.
1. What is the square root of 128? Ans.
2. What ix the square foot of

Ans. The
3. What is the square root of 111?
4. What the square root of 204 ?

Ans. 41 5. What is the square root of 24816?

Ans. 153 SURDS. 6. What is the square root of ? Ans. 9120+ 7. What is the square root of 4 Ans. ,7745+ 8. Required the square root of 561? Ans. 6,0207+

Ans.

APPLICATION AND USE OF THE SQUARE

ROOT, PROBLEM 1. A certain General mas an any ot 5184 men; how many must he place in rank ano de, to form 3 them into a square ?

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