rithm of the principal. The sum will be the logarithm of the amount for the given time. 2. From the amount subtract the principal, and the remainder will be the interest. EXAMPLES. 1. What is the amount of 20 dollars, at 6 pr. cent compound interest, for 100 years? Amount of 1 dollar for 1 year $1.06 log. 0.0253059 Multiply by the time 100 2.5305900 Add log. of $ 20, given principal 1.3010300 Amount required $ 6786 3.8316200 interest, for 4 years? 2. What is the amount of 425 dollars, at 5 pr. cent compound Amount of 1 dollar for 1 year = $105 log. Multiply by the time 0.0211893 0.0847572 2.6283889 2.7131461 Note 1. If the the interest becomes due semianually, or quar terly; find the amount of one dollar, for the half-year or quarter, and multiply the logarithm, by the number of half-years or quarters in the given time. Note 2. As Simple Interest is performed by a rank of numbers, arithmetically proportional, so it may be shown, that Compound Interest is performed by a rank of numbers geometrically proportional. And it is a principle in Mathematicks, that, if three numbers be in geometrical proportion, the product of the two extremes, is equal to the square of the mean. (See Euclid's Eliments, 20th prop. 7th book.) And on the contrary, if the rec. tangle contained by the extremes of any three numbers, be equal to the square of the mean, then those three numbers are in geometrical proportion. Now if 3 dollars be the compound interest of 8 100 for 1 year, or 6 months, then these three numbers 100,103, 106, should be in geometrical proportion; but it may be proved by the aforesaid proposition, they are not; for the reetangle of 100 into 106 is but 10600, and the square of the mean 103, is 10609, which is greater than the product of the two extremes. But the square root of 10600 will be found to be 102,956: so that the true proportional interest of $100, for year, is but $ 2 95 cts. 6 m.. 3. What is the amount of 1000 dollars, at 6 pr. cent com pound interest, for 10 years ? Ans. $1790 80 4. Required the amount of 100 dollars, at 6 pr. cent compound interest, for 3 years? Ans. 119 10 5. What will 1000 dollars amount to at 7 pr. cent, compound interest, in 4 years? Ans. 1310 80 6. What is the compound interest of 876 dollars 90 cts. at 6 pr. cent pr. annum, for 3 years and 6 months? Ans. $ 198 83 7. What will 100 dollars amount to in 3 years, at 6 pr. cent compound interest, allowing that it becomes due semiannually ? Ans. 127 05+ 8. What is the amount of 400 dollars, at 5 pr. cent compound interest, for 1 year, payable quarterly? Ans. 420 37 9. What is the amount of 1 cent, at 6 pr. cent compound interest, in 500 years? Amount of 1 dollar for 1 year= $ 1 06 log. 0.0253059 500 A POWER is a number produced by multiplying any given number continually by itself a certain number of times. The number denoting the power, is called the Index, or Exponent of that power. To raise a given number, we have the following RULE. Multiply the given number, or first power, continually by itself, till the number of multiplications be 1 less than the index of the power to be found, and the last product will be the power required. Note. Powers are commonly denoted by writing their indices above the first power; as follows. 2 X 2 = 4, the 2d power, or square of 2, 2 X 2 X 2 = or 22. 8, the 3d power, or cube of 2, or 23. 2 X 2 X 2 X 2 = 16, or biquadrate of 2, or 24. &c. EXAMPLES. 1. Let it be required to raise 45 to its cube, or third power? Ans. the 3d power, 2. What is the square of 3758? 3. What is the cube of 327 ? Ans. 14122564 4. What is the biquadrate, or fourth power of 376 ? 5. What is the fifth power of 029? Ans. 6. What is the sixth power of 48? Ans. 7. Required the seventh power of 7? Ans. 19987173376 000000020511149 12230590464 Ans. 823543 EVOLUTION. Is that, by which we extract the roots of numbers; or find a radical quantity, which multiplied into itself a certain number of times will produce the given power. TO EXTRACT THE SQUARE ROOT. RULE. 1. Having distinguished the given number in periods of two figures each, beginning at the place of units, 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, (in the manner of a quotient figure in division,) for the first figure of the root, the square of which subtract from the first period, and to the remainder bring down the next period for a dividend. 2. Place the double of the root, already found, on the left of the dividend for a divisor. Note. Roots are sometimes denoted by writing ✔before the power, with the index of the root against it. Thus the third root of 80 is 80, and the second root of 80 is 80, the index 2, though omitted, is always to be understood, when the radical sign is written without a numeral index. 3. Consider what figure must be annexed to the divîsor, so that if the result be multiplied by it, the product may be equal to, or next less than the dividend, and it will be the second figure of the root. 4. Find a divisor as before, by doubling the figures already in the root; and from these find the next figure of the root, as in the last article; and so on through all the periods to the last, *Note. When the given number is a surd; that is, when its root cannot be found exactly, without a remainder, the evolution may be carried on, until we obtain a root, sufficiently near the truth, by annexing cyphers to the remainder, and proceeding as in rational numbers. In the 10th example; although 12.64911, is not the exact root of 160, yet if it be multiplied by itself, the product will be 159,9999837921, which is not two parts, of which 10000 mske an unit, wide of the truth. TO EXTRACT THE CUBE ROOT. RULE. 1. Having distinguished the given number into periods of three figures, find the nearest less cube in the first period, set its root in the quotient, and subtract the said cube from the first period; to the remainder bring down the second period, and call this the RESOLVEND. 2. To three times the square of the root, just found, add three times the root itself, setting this one place farther to the right than the former, and call this sum the DIVISOR. Then divide the resolvend, excepting the right hand figure, by the divisor, for the next figure of the root, which annex to the former; calling this last figure e, and the part of the root before found, call a. 3. Add together these three products, namely, three times the square of a multiplied by e, three times a multiplied by the square of e, and the cube of e, setting each of them one place farther towards the right than the former, and call the sum the SUBTRAHEND : which must not exceed the resolvend; if it does, then make the last figure e less, and repeat the operation for finding the subtrahend. 4. Subtract the subtrahend from the resolvend, and to the remainder bring down the next period of the given number for a new resolvend; to which form a new divisor from the whole none root found; and thence another figure of the root, as before and thus continue till the whole is finished. |