EXAMPLES. 1. What will 4007. 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. 2,75 grs. + Ans. The same by Table I. Tabular amount of £1=1,26247 400 Whole amount=£504,98800 2. Required the amount of 425 dols. 75 cts. for 3 years, at 6 per cent. compound interest? Ans. $507,71⁄2 cts.+ 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 dollars amount to in 20 years, at 6 cent. compound interest? Ans. $160, 35 cts. 64m. INVOLUTION, pet 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? What is the square of,085? Ans. 292,41 Ans. ,007225 Ans. 20736 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 ny assigned degree of exactness. The roots which approximate are called surd roots, and hose which are perfectly accurate are called rational roots. A Table of the Squares and Cubes of the nine digits. Roots. |1|2|3| 4| 5 | 6 | 71 81 91 1|49|16| 25 36|49| 64 81 Squares. 8|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 umber. 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 an der 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 stad 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 remainder; 9 67)512 469 but when the periods belonging to any given number are exhausted, and still leave a remainder, the operation may be continued at pleasure, by annexing periods of ciphers, &c. 745)4325 3725 7508)60064 60064 O remains. TO EXTRACT THE SQUARE ROOT OF VUL GAR 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 numera , and the root of the denominator, for a new denominator. 2. If the fraction be a surd, reduce it to a decimal, and extract its root. PROBLEM I.-A certain general has an army of 5184 nen; how many must he place in rank and file, to form them into a square? RULE.-Extract the square root of the given number. √5184=72 Ans. PROB. II. A certain square pavement contains 20730 square stones, all of the same size; I demand how many are contained in one of its sides? √20736=144 Ans. PROB. III. To find a mean proportional between two numbers. RULE.-Multiply the given numbers together and extracı the square root of the product. EXAMPLES. What is the mean proportional between 18 and 72? 72 × 18=1296, and ✓1296=36 Ans. PROB. IV. To form any body of soldiers so that they may be double, triple &c. as many in rank 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 81×2 =162 in rank. PROB. V. Admit 10 hhds. of water are discharged hrough a leaden pipe of 24 inches in diameter, in a cerain 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. 24-25, and 2,5×2,5-6,25 square. 4 given proportion. √25,005 inch. diam. Ans. |