RulC.-Extract the square root of the given number. 75184372 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 proportio::al between 18 and 72? 72 X 18=1296, and ✓ 1296336 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 rauk. EXAMPLES. Let 13122 men be so formed, as that the number in rank may be double the number in file. 13122:2–6561, and V6561=81 in file, and 81 X2 162 in rank. PROR. V. Admit 10 hhds. of water are discharged through a leaden pipe of 2} inches in diameter, in a certain time; I demand what the diameter of another pipo 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 tho product is ige answer. 2= 12,5x2,5=6,25 square. 4 given proportion. PROB. VI. The sum of any two numbers, and their pro ducts 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 froin the half sum, gives the lesser number. EXAMPLES. The sum of two numbers is 43, and their product is 442; what are thosc two numbers ? The sum of the numb. 43 X 43=1849 square of do. The product of do. 442 x 451768 4 times the pro. Then to the sum of 21,5 (numb. .tand 4,5 V819 diff. of the EXTRACTION OF THE CUBE ROOT. A cube is any number multiplied by its square. To extract the cube rout, is to find a number, which, being multiplied into its square, shall 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 cuhe thus found, from the said period, and to the remainder bring down the next period, calling this the cliviilend. 4. Multiply the squnre of the quoticni by 300, calling it the divisor. 5. Seck how often, the divisor may be had in the dini dend, and place the result in the quotient; then multiph 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 311, 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 sun 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 con "equently 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 subtraci the subtrahend from the dividend. EXAMPLES. 1. Required the cube root of 18399,744. 18399,744(26,4 Root. Ans. NOTE.—The foregoing example gives a perfect root ; and if, when all the periods are exhausted, there happens co he a remainder, you may annex periods of ciphers, and cortinue the operation as far as you think it necessary. Answers 2. What is the cube root of 205379 ? 59 3. Of 614125? 85 4. Of 41421736 ? 346 6. Of 146363,183 ? 52,7 6. Of 29,508381 ? 3,09+ 1. Of 80,763 ? 4,32 + ,162771336? ,546 9. Of ,000684134 ? ,088+ 122615327232 ? 4968 8. Of 10. Of Rule.-1. Find by trial, a cube near to the given number, and call it ihe supposed cube. 2. Then, as twice the supposed cube, added to the given number, is to twice the given number added to the supposed cube, so is the root wf the supposed cube, to the true root, or an approximation to it. 3. By taking the cube of the root thus found, for the supposed cube, and repeating the operation, the root will be had to a greater degree of uxactness. EXAMPLES. 1. Let it be required to extract the cube root of 2. Assume 1,3 as the root of the nearest cube ; then-1,3 1,3 x 1,3=2,197=supposed cube. Then, 2,197 2,000 given number. . 2 As 6,394 : 6,197 : : 1,3 : 1,2599 root, which is true to the last place of decimas; huu might by repeating the operation he brought to gTERLER (xactress. 2. What is the cube root of 584 P 3 Required the cube root of 729001101 ? Ans. 900,0004 QUESTIONS, Showing the use of the Cube Root. 1. The siatute bushel contains 2150,425 cubic or solu inches. I demand the side of a cubic box, which shall con. tain that quantity ? 2150,425=12,907 inch. Ans. Nore. The solid contents of similar figures are in proportion to each other, as the cubes of their similar sides of diameters. 2. If a bullet 3 inches diameter weigh 4 lb. wat will a bullet of the same metal weigh, whose diam. ver is 6 in ches? 3x3x3=27 6x6x6=216. As 27 : ! . : : 216 : 32 lb. Ans. 3. If a solid globe of silver, of 3 inche, diameter, bu worth 150 dollars; what is the value of another globe o siiver, whose diameter is six inches? 313x3=27 6*6*6=216, As 27 : 150 :: 216 $1200. Ans. The side of a cube being given, to find the side of tha: cube which shall be double, triple, &c. in quantity to the given cube. Rule.-Culse your given side, and multiply by the given propor tion between the given and required cube, and the cube root of th product will be the sido sought. EXAMPLES. 4. If a cube of silver, whose side is two inches, be wort) 20 dollars ; I demand the side of a cube of like silver whose value shall be 8 times as much ? 2x2x28, and 8x8=64 V64=4 inches. Ans. 5. There is a cubical vessel, whose side is 4 feet; I de mand the side of another cubical vessel, which shall con tain 4 times as much ? 4x4x4=64, and 64 x4=256 7256–6,349+ft. Ans. 6. A cuoper having a cask 46 iwcher long, and 32 in |