EXAMPLES. 1. What will 400l. amount to in 4 years, at 6 per cent. per annum, compound interest ? 400×1,06×1,06×1,06×1,06=£504,99+ or [504 19s. 9d. 2,75grs. + 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. $507,7cts. 3. What is the compound interest of 555 dols. for 14 years, at 5 per cent. By Table I. Ans. $543,86cts.+ 4. What will 50 dollars amount to in 20 years, at 6 per cent, compound interest ? Ans. $160 35cts. 64m. 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, What is the square of 17,1? Ans. 292,41 Ans. ,007225 Ans. 16387,064 Ans. 20736 Ans. 52 9 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 continual multiplication 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 any assigned degree of exactness. The roots which approximate, are called surd roots, and those which are perfectly accurate are called rational rects. A Table of the Squares and Cubes of the nine digits. Roots. | 1 | 2 | 3 | 41 4 9 16 5 | 671 81 91 512729 Squares. Cubes. 1|8|27| 64 | 125 | 216 | 343 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 and 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 inultiplied 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 Foot 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. 9 EXAMPLES. 1. Required the square root of 141225,64. 141225,64(375,8 the root exactly without a remainder; but when the periods belonging to any given number are exhausted, and stifl leave a remainder, the operation may be continued at pleasure, by annexing periods of cyphers, &c. 67,512 469 745)4325 3725 7508)60064 60064 O remaine Reduce the fraction to its lowest terms for this and all other roots; then 1. Extract the root of the numerator fer a new numerator, and the root of the denominator, for a new denomi nator. 2. If the fraction be a surd, reduce it to a decimal, and extract its root. EXAMPLES. 7023 1. What is the square root of 98 6. What is the square root of 15 Hus. Ans. 9128 Ans. ,7745+ S. Required the square root of 66+? Ans. 6,0207+ APPLICATION AND USE OF TIÏE SQUARE ROOT. PROBLEM I. A certain General nas an a my ot 5184 men; how many must he place in rank and file, to form them into a square? 16 RULE. Extract the square root of the given number, ✓5184=72 Ans. PROB. II. A certain square pavement contains 20736 square stones, all of the same size; I demand how many are contained, in one of its sides? 20730-144 Ans. PROB. III. To find a mean proportional between two RULE. numbers. Multiply the given numbers together, and extract the square root of the product. EXAMPLES. What is the mean proportional between 18 and 72 ? 72X13=1296, and 1296-36 Ans. PROB. IV. To form any body of soldiers so that they may be double, triple, &c. as many ingank 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+2=6561, and ✓6561–81 in file, and 81x2 =162 in rank. PROB. V. Admit 10 hds. of water are discharged through a leaden pipe of 2 inches in diameter, in a certain time; i demand what the diameter of another pipe must be, to discharge four times as much water in the same time. RULE. Square the given diameter, and multiply said square by the given proportion, and the square root of the product is the answer. 23=2,5, and 2,5×2,5=6,25 square. 4 given proportion. ✔25,005 inch. diam. Ans. |