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EXAMPLE.S.

1. A owes B $380 to be paid as follows, viz. $100 in 6 months, $120 in 7 months, and $160 in 10 months : What is the equated time for the payment of the whole debt 7 100X 6= 600 120X 7= 840 160X 10=1600

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2. A owes B fl04 15s. to be paid in 44 months, £161 to be paid in 34 months, and £1525s, to be paid in 5 months: What is the equated time for the payment of the whole 7 Ans. 4 months and 8 days.

3. There is owing to a merchant £698, to be paid £178 ready money, £200 at 3 months, and £320 in 8 months; I demand the indifferent time for the payment of the whole 7 Ans. 44 months.

4. The sum of $164 16c. 6m, is to be paid, 3 in 6 months, 4 in 8 months, and # in 12 months: what is the mean time for the payment of the whole 7 - Ans. 73 months.

5. A merchant has $360 due him, to be paid at 6 months, but the debtor agrees to pay £ at the present time, and 4 at 4 months; I demand the time he must have to pay the remainder, at simple interest, so that neither party may have the advantage of the other ?

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Mow as he pays 180 dollars 6 months, and 120 dollars 2 months be. fore they are respectively due, say, as the interest of 60 dollars for 1 month, is to 1 month, so is the sum of the interest of 180 dollars for 6 months, and of 120 dollars for 2 months, to a fourth number, which added to the 6 months, will give the time for which the 60 dollars ought to be retained.

Ans, 28 months,

INVOLUTION.

Involution is the method of finding the powers of numbers.

Powers of numbers are the products arising from the continual

multiplication of numbers into themselves. Any number may itself be called the root or first power. If the first power be inultiplied by its If, the product is called the second power, or the square ; if the square he multiplied by the first power, the product is called the third power, or the cube ; if the cube be multiplied by the first power, the product is called the fourth power, or the biquadrate, &c.

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The small figure points out the order of the power, and is called the Index, or Exponent.

Rule for finding the powers of numbers.

Multiply the given number, or first power, continually by itself, till the number of multiplications be one less than the index of the power to be found, and the last product will be the power required.

Note—The powers of vulgar fractions are found by raising each of their terms to the power required. If the power of a mixed number be required, either reduce it to an improper fraction, or reduce the vulgar fraction to a decimal.

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Evolution, or the extraction of roots, is the operation by which we find any root of any given number.

The root is a number whose continual multiplication into itself produces the power, and is denominated the square, cube, biquadrate, or 2d, 3d, 4th root, &c. accordingly as it is, when raised to the 2d, 3d, 4th, &c. power, equal to that power. Thus, 4 is the square root of 16, because 4×4=16. 4 also is the cube root of 64, because 4×4×4=64; and 3 is the square root of 9, and 12 is the square root of 144, and the cube root of 1728, because 12×12×12=1728, and so on.

281. What is Involution ? 282. What are powers of numbers ?–283. Howe do you find these powers ?–284. What is Evolution ?—283. What is a root ?

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There is ‘no number of which we cannot find any power ex"ctly ; but there are many numbers, of which the exact roots Jan never be obtained. Yet, by the help of decimals, we can obtain these roots to any necessary degree of exactness.

Those roots which cannot be exactly obtained, are called surd roots ; and those which can be found exactly, are called rational 7°00ts.

Roots are sometimes denoted by writing this character A/ before the power, with the index of the power over it ; thus the

cube root of 64 is expressed V 64, and the square root of 64 is expressed A/ 64, the index 2 being omitted when the square root is required.

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THE Extraction of the SQUARE Root is the method of finding a number, which, being multiplied by itself, shall produce the given number.

o 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, which points show the number of figures the root will consist of. 2. Find the greatest square number in the first, or left 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, then subtract it therefrom, and to the remainder bring down the next period for a dividend. 3. Place double of the root, already found, on the left hand of the dividend for a divisor. 4. Seek how often the divisor is contained in the dividend, (except the right hand figure,) and place the answer in the root for the second figure of it, and likewise on the right hard of the divisor ; multiply the divisor with the figure last annexed by the figure last placed in the root, and subtract the product from the dividend: to the remainder join the next period for a new dividend.

286. Can you find the power and root of any number 7 287. What is the distinction in the roots 3–288. What is the extraction of the Square Root ?—289. What is the rule 7

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5. Double the figures already found in the root, for a new divisor, (or bring down your last divisor for a new one, doubling the right hand figure of it,) and from these, find the next figure of the root as last directed, and continue the operation in the same manner, till you have brought down all the periods.”

* The rule for the extraction of the square root may be illustrated by attending to the process by which any number is raised to the square. The several products of the multiplication are to be kept separate, as in the proof of the rule for multiplication of simple numbers. Let 37 be the number to be raised to the square.

37x37–1369—37X37
37

37
*=. 49-72
+3X7 N — 210–30x7
# =33 = x3x7; # =2x30x1
9 -32 900-302

. . . (37 2x3)42 -ox, (30+7=37 49-72 Now it is evident that 9, in the place of hundredths, is the greatest square in this product; put its root, 3, in the quotient, and 900 is taken from the product. The next products are 21+21=2x3x7, for a dividend. Double the root alieady found, and it is 2x3, for a divisor, which gives 7 for the quotient, which annexed to the divisor, and the whole then multiplied by it, gives 2×3×7(=42)+7x7(=49) which, placed in their proper places, completely exhausts the remainder of the square. The same may be shown in any other case, and the rule becomes obvious. Perhaps the following method may be considered more simple and plain. Let 37–30+7, be multiplied as in the demonstration of multiplication of simple numbers, and the products kept separate. 30+7 30-4-7

900+30×7
30x7+49

900+2×30x7+49-1316, the sum and square.
# 30+7 q

2x30+7x7)2x30x7+49
2x30x7+49

- * * *

The root of 900 is 30, and leaves the two other terms, which are exhausted by a divisor formed and multiplied as directed in the rule.

290. Explain to me the nature of this rule.

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