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, 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 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 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. 9 67)512 469 745)4325 3725 7508)60064 60064 O renfains. 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 cyphers, &c. Reduce the fraction to its lowest terms for this and all other roots; then 1. Extract the root of the numerator for a new numerator, and the root of the denominator, for a new denomi nator PROBLEM I. A certain General has an army of 5184 men; 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 20736 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 extract the square root of the product. EXAMPLES. What is the mean proportional between 18 and 72 ? 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 through a leaden pipe of 21 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. 21=2,5, and 2,5×2,5=6,25 square. 4 given proportion. ✔55,005 inch. diam. ns. 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 pro duct, and extract the square root of the remainder, which will be the difference of the two numbers; then half the saia difference added to half the sum, gives the greater of the two numbers, and the said half difference subtracted from the half sum, gives the lesser number. EXAMPLES. The sum of two numbers is 43, and their product is 442; what are those two numbers? The sum of the numb. 43x43-1849 square of do. The product of do. 442 x 4 1768 4 times the pro. Then to the sum of 21,5 [numb. ✔81-9 diff. of the +and 4,5 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, shali produce the given num ber. RULE. 1. Separate the given number into periods of three figures each, by putting a point over the unit figure, and every third figure from the place of units to the left, and if there be 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 said period, and to the remainder bring down the next period, calling this the dividend. 4. Multiply the square of the quotient by 300, calling it the divisor. 5. Seek how often the divisor may be had in the dividend, 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; then 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 dividend; 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 subtract the subtrahend from the dividend EXAMPLES. 1. Required the cube root of 18399,744. 18399,744(26,4 Root. Ans. |