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8d payment April 1, 1801,

12,00 4th payment May 1, 1801,

30,00 How much remains due on said note the 16th of Sep tember, 1801 ?

$ cts. 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,

467,50 Interest to May 1, 1801, (1 year,)

28,05

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

8,00 l'aid 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 Sept. 16, 1801, (4; mo.)

10,92 Balance due on the note, Sept. 16, 1801, $455,57

The payments being applied according to this 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 pound, or one dollar, for one year, at the rate per cent. given, until the number of multiplications are equal to the given number of years, and the product will be the amount required.

Or, In Table 1, 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. amount to in 4 years, at 6 per cent per annum, compound interest ?

400 x 1,06 X 1,06 x 1,06 X 1,06=£504,99+04 W
[£504 19s. 9d. 2,75 qrs. + Ans.

W
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,

TI at 6 per cent. compound interest ? Ans. $507,74 cts. +

catio! 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 dollárs amount to in 20 years, at 6 per whit cent, compound interest ? Ans. $160, 35 cts. 62m.

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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.

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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
Ans. ,007225
Ans. 16387,064

Ans. 20736
Ans. 524

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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 iny 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. | 12 3 / 4 51 61 7 l

81 91
Squares. 11/4 | 9 | 16 25 36 | 49 | 64 | 81
Cubes. 1118 | 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
number.

Rule.-1. Distinguish the given number into 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 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

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 subtraet 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 dow 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.

F be 1

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

th

O remains.

I die 2 What is the square root of 1296 ?

ANSWERS. 36 umber a 3 Of

56644?

23,8 4. Of 5499025?

2345 end. 5. Of 36372961 ?

6031 184,2?

13,57+ 7. Of 9712,693809?

98,553 8. Of 0,45369?

,673+ 9. Of ,002916?

,054 10. Of

45?

6,708+

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anaTO EXTRACT THE SQUARE ROOT OF VULnd.

GAR FRACTIONS.

RULE,

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

1. Extract the root of the numerntor for a new numera. Ei tör, and the root of the denominatór, for a new denominator.

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

EXAMPLES.
1. What is the square root of ? ANSWERS - 3
2. What is the square root of ?

1 / 1 / 2
3. What is the square root of 113?
4. What is the square root of 201?

41 5. What is the square root of 24816?

153
SURDS.
6. What is the square root of 31 ?

9128* 7. What is the square root of 43

?

,77454 8. Required the square root of 361 ?

6,0207+ APPLICATION AND USE OF THE SQUARE

ROOT. PROBLEM I. --A certain general has an army of 5184 men; how many must le place in rank and file, to forget them into a square ?

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