the additions on the sides meet, is 60X4X4 960 inches. Still there is a deficiency of a small cube in the corner, (Fig. FIG. III. 24 inches. Proof by Involution; 24X24X24-13824. Hence it appears, that a cube is a solid body, having six equal sides, and its cube root is the length of one of those sides. From the foregoing exam ple and illustration we derive the following RULE. I. Distinguish the given number into periods of three figures each, beginning at the right hand. II. Find the greatest cube in the left hand period, and place its root as a quotient in division. 24 III. Subtract the cube from said period, and to the remainder bring down the next period, for a dividend. IV. Multiply the square of the quotient by 300, calling it the triple square, and the quotient by 30, calling it the triple quotient, and the sum of these call the divisor. NOTE. The triple quotient is not indispensable in forming the divisor. V. Seek how many times the divisor is contained in the dividend, and place the result in the quotient, for the second figure of the root. VI. Multiply the triple square by the last quotient figure, and write the product under the dividend; multiply the triple quotient by the square of the last quotient figure, and place this product under the last; under these write the cube of the last quotient figure, and call their sum the sub T trahend. Subtract the subtrahend from the dividend, and to the remainder bring down. the next period, for a new dividend, and proceed as before, till the work is finished. EXAMPLES. 2. What is the cube root of 1906624 ? Operation. 1X1X300 300 1906624(124, Answer. 4. What is the cube root of 6331625? Answer, 185. 5. What is the cube root of 11543176000 ? 6. What is the cube root of 34.328125? 7. What is the cube root of .000729? 8. What is the cube root of .003375? 9. What is the cube root of 5 of 3? 10. What is the cube root of 81 11. What is the cube root of 218? 12. What is the cube root of 1728? 13. What is the cube root of? 14. What is the cube root of? 125 97 Answer, 2260. 15. A certain hill contains 11543176 Answer, 3.25. Answer, . Answer, §. Answer, 1 Answer, . cubical feet. What is the length of one side of a cubical mound, containing an equal number of feet? Answer, 226 feet. feet. 16. The contents of an oblong cellar is 9261 cubical What the length of one side of a cubical cellar, of the same capacity? Answer, 21 feet. 17. A merchant bought cloth to the amount of $393.04, but forgets the number of pieces, and also the number of yards in each piece, and what the cloth cost per yard; but remembers that he paid as many cents per yard as there were yards in each piece, and that there were as many in each piece as there were pieces. What did he pay per yard? Answer, 34 cents. 18. What is the width of a cubical vessel, containing 75 wine gallons, each 231 cubic inches? 19. Required the side of a cubic box that shall contain a bushel ? Answer, 12.9+inches. Solids of the same form are to one another as the cubes of their similar sides, or diameters. EXAMPLES. 1. If a bullet, weighing 72 lbs., be 8 inches in diameter, what is the diameter of a bullet weighing 9 lbs. ? Answer, 4 inches. 2. A bullet, 2 inches in diameter, weighs 4 lbs. What is the weight of a bullet 5 inches in diameter ? Answer, 623 lbs. 3. If a silver ball, 9 inches in diameter, be worth $400, what is the worth of another ball, 12 inches in diameter ? Answer, $948.148+. To find two mean proportionals between two numbers. RULE. Divide the greater by the less, and extract the cube root of the quotient; multiply the lesser number by this root, and the product will be the lesser mean; multiply this mean by the same root, and the product will be the great er mean. EXAMPLES. 1. What are the two mean proportionals between 4 and 256? 3 256-4-64; then /64-4 and 4X4-16, the lesser, and 16X4-64, the greater. 2. What are the two mean 625? Proof, 4:16:64 : 256. proportionals between 5 and Answer, 25 and 125. Answer, 49 and 343. 3. What are the two mean proportionals between 7 and 2401 ? EXTRACTION OF ROOTS IN GENERAL. RULE. I. Point the given number into periods of as many figures as the index of the root directs. Thus, for the square root, two figures; cube root, three; the fourth root, four, &c. II. Find, by trial, the greatest root in the left hand period, and subtract its power from that period, and to the remainder bring down the first figure of the next period, for a dividend. III. Involve the root, already found, to the next inferior power to that which is given, and multiply it by the number denoting the given power, for a divisor, by which find the second figure of the root. IV. Involve the whole root now found to the given power; subtract it from the given number, as before, and bring down the first figure of the next period to the remainder, for a new dividend, and proceed as before, till the work is finished. NOTE. The roots of most of the powers may be found by repeated extractions of the square and cube root--Thus,: For the 4th root, take the square root of the square root. take the square root of the cube root. take the square root of the 4th root. For the 6th For the 8th 66 For the 9th take the cube root of the cube root. QUESTIONS. 1. Rule tor finding a mean proportional between two numbers? 2. What is a cube? 3. What is a cube root? 4. What is it to extract the cube root? 5. What is the rule? 6. Why do you distinguish the given number into periods of three figures each? 7. Why do you multiply the square of the quotient by 300? 8. Why the quotient by 30? 9. Why the triple square by the last quotient figure 10. Why the triple quotient by the square of the last quotient figure? 11. Explain the process of illustrating this rule by blocks. What proportion have solids to one another? 13. Rule for finding two mean proportionals between two numbers? 14. Rule for extracting roots in general? 12. EXAMPLES. 1. What is the square root of 7569 ? Operation. 7569)87 8X8= 64 square, or 2d power, of the quotient. 4084101(21 2×2×2×2×2= 32 = 5th power of the quotient. 2×2×2×2×5=80)88 1st dividend. 4084101 21×21×21×21×21=4084101-5th power of the quot❜nt 3. What is the fourth root of 140283207936? Answer, 612. 4. What is the seventh root of 4586471424 ? Answer, 24. 5. What is the ninth root of 1352605460594688? Answer, 48. ARITHMETICAL PROGRESSION. ARITHMETICAL PROGRESSION is when a series of numbers increases by a common excess, or decreases by a common difference. When numbers increase by a common excess, they form the ascending series, as 2, 4, 6, 8, 10, 12, &c.. When numbers decrease by a common difference, they form the descending series, as 12, 10, 8, 6, 4, 2, &c. The numbers forming the series are called the terms; the first and last terms are called the extremes, and the other terms the means. QUESTIONS. 1. What is Arithmetical Progression ? |