band 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 Anainder 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 thé dividend, and it will be the second figure in the root. 5. Subtract the product from the dividend, and to the temainder 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 divi. dond, (omitting the right hand figure thereof,) bs double the root already found, and the quotient will comiconly 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. i 141225,64(375,8 the root exactly without a remainder; but when the periods belonging to any given number are exhausted, and still $) 512 leave a remainder, the operation may be continued at pleasure, by annexing periods of cyphers, &c. 745)4525 3725 7508) 60064 80064 TO EXTRACT THE SQUARE ROQT OF RULE. Reduce the fraction to its lowest terms for this and all other roots; then 1. Extract the root of the numerator for the new nume. rator, 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. I 1. What is the square root of 28 Answers 2. What is the square root of 12025? 3. What is the square root of 11? 4. What is the square root of 2017 5. What is the square root of 248, 4? 15 SURDS. 6. What is the square root of ? 91284 7. What is the square root of ? ,7745+ 8. Required the square root of 361 ? 6,0207 + Her er det noen some APPLICATION AND USE OF THE SQUARE ROOT PROBLEM I. A certain General has an army of 5184 men; how many must be place in rank and file, to form them into a square ? RULE. 5184=279 Anet 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 ? 720756144 Ans. PROB. III. To find a mean proportional between two numbers. RULE. Multiply the given numbers together, and extract the square root of the product. EXAMPLES 72x18== 1296, and ✓ 1296x236 Ans. PROB. IV. To form any body of soldiers so that they may be double, triple, &c. as many in rank as in file. RULK. Estract the square root of 1-2, 1-3, &c. of the given number of men, and that will be the number of men ir file, which double, triple, &c, and the product will be the number in rank. EXAMPLES. Let 13122 men be 80 formed, as that the number in mnk may be double the number in fila. 131224236561, and 656181 in file, and 81 X2 162 in rari. PROB. V. Admit 10 bhds. of water are discharged through a leaden pipe of 24 inches in diameter, in a certain time; I deniand what the diameter of another pipe must be, to discharge fuur tinies as much water is the daine time. RULL. Square the giver diameter, and multiply mid square by thie given proportion, and the square root of the pro duct is the answer. 21%, agd 2,5X2%3-6,£5 square. * giveu proportion PROB. VI. The sum of any two numbers, and their products being given, to find each number. · RULE. From the square of their sum, subtract 4 times their product, and extract the square root of the remainder, which will be the difference of the two numbers; then half the said difference added to half the sum, gives the greater of the two numbers, and the said half difference subtracted from the half 'suin, gives the lesser number. EXAMPLES. The sum of two numbers is 48, and their product is 448; what are those two numbers ? The sum of the numb. 43x43=1849 square of do. The product of do. 442x 4-1768 4 times the pro. Then to the sum of 21,5 . [numb. tand 4,5 ✓8129 dit. of the Greatest aumber, 26,0) 41 the diff. - Answers. Least number, 17,0) EXTRACTION OF THE CUBE ROOT. A cube is any number multiplied by its square. ; To extract the cube root, is to find a number, which, being multiplied into its square, shall produce the given number. RULE. 1. Separate the given number into periods of three figures each, by putting a point over the linit figure, and every third figure from the place of units to the left, and if there we decimals, to the right. 2. Find the greatest cube in the left hand period, and place its root in the quotient. 3. Subtract the cube thus found, from the sail period, and to the remainder bring down the next period, calling this the dividend.' 4. Multiply the square of the quotient by 300, calling $the divisir. 5. Seek how often the divisor may be had in the divi. dend, and place the result in the quotient; then multiply the divisor by this last quotient figure, placing the product under the dividend. 6. Multiply the former quotient figure, or figures by the square of the last quotient figure, and that product by 30, and place the product under the last; tnen under these two products place the cube of the last quotient figure, and add them together, calling their sum the subtrahend.. 7. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new divi. dend ; with which proceed in the same manner, till the whole be finished. Note. If the subtrahend (found by the foregoing rule) happens to be greater than the dividend, and consequently cannot be subtracted therefrom, you must make the last quotient figure one less ; with which find a new subtrahend, (by the rule foregoing) and so on until you can saltract the subtrahend from the dividend. EXAMPLES. 1. Required the cube root of 18399,744. 18399,744(26,4 Root. Ans. |