RULE. ✓ 5184=72 Ans. " À certain square pavement contains 20730 resso alici ili vi the same size; I demand how many video mis viie of its sides ? ✓20736=144 Ans: PROB. ill. To find a mean proportional between two Qumbers. RULE. Multiply the given numbers together, and extract the square root of the product. What is the mean proportional between 18 and 72 ? 72x18=1296, and ✓1296336 Ans. PROB. IV. To form any body of soldiers so that they may be double, triple, &ic. as many in rank as in file. RULE. Extract the square root of 1-2, 1-3, &c. of the given twitter of men, and that will be the number of men in 02. wrich double, triple, &c. and the product will be the *runk. EXAMPLES. EXAMPLES. with be so formed, as tnat the number in double the number in file. 2, and 6561=81 in file, and 81 X2 -- vški rünk. prop. V. Almit 10 khus. of water are discharged byh a leaden pipe of 24 inches in diameto, in a cerorname; I demand what the diameter of another pipe seibe, to discharge four times as much water in the time. RULE. quare the given diameter, and multiply said square oy the given proportion, and the square root of the product is the answer. 21-2,5, and 2,5x2,5=6,25 square. 4 given proportion. ✓ 25.005 inch. diam. Ans. PROB. VI. The sum of any two numbers, and their products being given, to find each number. RULE. Froin the square of their sun, subtract 4 times product, and extract the square root of the res of !! half the said difference added to half the waste gins the greater ef the two numbers, and the sair! imita nirere. ar subtracted from the half sum, gives the lesser iwles. EXAMPLES. The sum of two numbers is 43, and their product is 442; what are those two numbers ? The sum of the numb. 43 X45=1849 square of do. The product of do. 442x 4=1768 4 times the Then to the sum of 21,5 inuno. tand 4,5 ✓81=9 diff. of the Greatest numb. 26,0 41 the diff Answers. Least numb. 17,0 1 EXTACTION OF THE CUBE ROOT. 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 fig. ures each, by putting a point over the unit figure, and every third figure froin 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 suqare of the quotient by S00, callire it the divisor 5. Seek how often the divisormay be had in the diviJend, and place the result in the quotient; then multiply the divisor by this last quotient figure, placing the pro. duct 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 predluct under the last; then under these two products place the cube of the last quotient figure, and ada der together, calling their sum the subtrahend. 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 subtral end found by the foregoing rule) happens to be greater than the dividend, and consequently cannot be subtracted therefron, 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 foin the dividend. EXAMPLES. 1. Required the cube root of 18399,744. 18399,744( 26,4 Root. Ans. 8 2X2=4x300=1200) 10399 first dividend. 200 6x6=56X2=2X30=2160 6x6x6= 216 9576 1st subtrahend. 26x26=676x300=202800)825744 20 dividend. 811200 4X4=16x26*416*30* 12480 4X4X4 64 825744 2d subtrahend. NOTE.—The foregoing example gives a perfect root; and if, when all the periods are exhausted, there happens to be a remainder, you may annex periods of cyphers, and continue the operation as far as you think it necessary. Answers 2. What is the cube root of 205379 ? 59 3. Of 614125 ? 85 4. Or 41421736: 346 5. Of 146363,183 : 52.7 6. Of 29,503629 ? 3,09-2 7. Of 80,769 ? 8. Of ,162771336 ? 9. Of 0000241341 10 Of 129615927932? RULE II. 1. Find by trial, a cube near to the given numbe and call it the supposed cube. 2. Then as twice the supposed cube, added to the giv en number, is to twice the given number added to the supposed cube, so is the root of 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 exactness. EXAMPLES. Let it be required to extract the cube root of 2. Assume 1,3 as the root of the nearest cube ; then1,3% 1,3X1,3=2,197=supposed cube Then, 2,197 2,000 given number. 2 : : As 6,594 0,197 : 1,2599 root, which is true to the last place of decimals; but might by repeating the operation, be brought to greater exactness. 2. What is the cube root of 584,2777056 .498. 8,76 16* 3. Required the cube root of 729001101? Ans. 900,0004 QUESTIONS, Showing the use of the Cube Root. 1. The statute bushel contains 2150,425 cubic or solid mches. I demand tho side of a cubic box, which shall contain that quantity! 32150,425=12,907 inch. Ans. Note.-The solid contents of similar figures are in proportion to each other, as the cubes of their similar sides or diameters. 2. If a bullet 3 inches diameter, weigh 4lb. what will a bullet of the mme metal weigh, whose diameter is 6 inches ? 3X3X3+27 6x6x6=216 As 27 : 41b. : : 216 : 32lb. Ans. 3. If a solid globe of silver, of 5 inches diameter, be worth 150 dollars; what is the value of another globe of silver, whose diameter is six inches ? 3x3x3=27 6x6x6=216 As 27 : 150 : : 216 ; $1200. Ans. The side of a cube being given, to find the side of that cube wich shall be double, triple, &c. in quantity to the given cube. RULE. Cube your given side, and multiply by the given proportion betw en the given and required cube, and the cube root of the product will be the side sought. 4. If a cube of silver, whose-side is twoinches, be worth 20 dollars; I demand the side of a cube of like silver, whose value shall be 8 times as much ? 2x2x2.8 and 8x8=64764=4 inches. Ans. 5. There is a cubical vessel, whose side is 4 feet: I demand the side of another cubical vessel, which shall contain 4 times as much? 4X4X4=64 and 64x4=2563256=6,5494.ft. Ans. 6 A cooper leaving a cask 40 inches long, jih pois |