RULE. 5184=72 Ans. PROB. II. A certain square pavement contains 20736 square stones, all of the same size ; I demand how many are cuntained 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 ? 72x18=1296, and 1290=36 Ans. Prob. IV. To form any body of soldiers so that they may be double, triple, &c. as many in rank as in lije. 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 nuinber in rank may be double the number in file. 13122+2=6561, and 6561=81 in file, and 81x2 162 in rank. Prob. V. Admit 10 hhds. of water are discharged through a leaden pipe of 24 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 2,5x2,5=6,25 square. 4 given proportion. V55,00 5 inch. diam PROB. VI. The sum of any two numbers, and the products being given, to find each number. RULE. From the square of their sum, subìact 4 times their pro duct, and extract the square root of the remainder, which will be the difference of the two numbers ; then hali 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. 442X 4=1768 4 times the pro. Then to the sum of 21,5 [numb. tand- 4,5 ✓81=9 diff. of the 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 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 cubc 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 domn the next period, calling this the dividend. 4. Multiply the square of the quotiont by 300, calling Um divisor.' 5. Seek how often the divisoi 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 wo products place the cube of the last quotient figure, and add ihem 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 whicla 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. Ant 8 2x2=4x300=1200)10399 first dividend. 7200 6x6 =36X2=72X30=2160 6x6x6= 216 9576 Ist subtrahendo 26X26676X300=202800) 823744 2d dividend. 811200 4X4=16X26=416x30= 12480 4X4X45 64 828744 2d subtrahend. 4. Of 7. Of Note. The foregoing example gives a perfect root ; and if, when all the periods are exhausted, there happens te 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 41421736 ? 346 5. Of 146363,183 ? 52,7 6. Of 29,503629 ? 3,09-fo 80,763 ? 4,32+ 8. Of ,162771336 ? ,546 9. Of ,000634134 ? ,088 to 10. Of 122615327232 ? 4968 RULE II, 1. Find by trial, a cube near to the given number, and call it the 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 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 ; there 1,3x1,3X1,3=2,197=supposed cube. Then, 2,197 2,000 given number. 2 2 As 6,394 ; 6,197 : 1,3 : 1,2599 root, which is true to the last place of decimals; but night by repeating the operation, be brought to greater exactness. a. What is the cube root of 584,277056 . Ans. 8,36 16 3 S. Required the cube root of 729001101? Ans. 900,0004. QUESTIONS, Showing the use of the Ćute Root. 1. The statute bushel contains 2150,425 cubic or solid inches. I demand the side of a cubic box, which shall contain that quantity ? ✓2150,425-12,907 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 same metal weigh, whose diameter is 6 inches ? 3X3X3=27 6x6x6=216. As 27 : 4lb. : : 216 32lb. Ans. 3. If a solid globe of silver, of 3 inches diameter, DE 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 which shall be double, triple, &c. in quantity to the given cube. RULE. Cube your given side, and multiply by the given proportion between the given and required cube, and the cube root of the product wil be the side sought. 4. If a cube of silver, whose side is two inches, be worth 20 dollars ; I demand the side of a cube of like silver, whose value shall be 8 times as much? 2x2x248 and 8x8=64764224 inches. Ans. 5. There is a cubical vessel, whose side is 4 feet ; I deRand the side of another cubical vessel, which shall contain 4 times as much. 4X4X4z64 and 64X4--256/256–6,349+ft. Ans. 6. A comper having a cask 40 inclos long, and 92 in. |