TO EXTRACT THE CUBE ROOT OF A VULGAR FRACTION. RULE. Extract the root of its numerator for a new numerator, and the root of its denominator for a new denominator; if this cannot be done exactly, reduce the fraction to a decimal, and then extract the root. In extracting the cube root of a decimal, care must be taken that the decimal places be three, or some multiple of three, before the operation is begun; because there are three times as many decimal places in the cube, as there are in the root. See note, page 127. EXERCISES. 2. What is the cube root of 373248? 3. What is the cube root of 54872? 4. What is the cube root of 970299? 5. What is the cube root of 64964808? 6. What is the cube root of 2? 10. What is the cube root of .0278180275? 12. What is the cube root of? 19. What is the cube root of ? 14. What is the cube root of 15. What is the cube root of 3016; ? 16. What is the cube root of 134 ? MISCELLANEOUS EXERCISES. 1. Required the side of a cubical box, which will contain 2774 cubical inches of brandy? 2. Required two mean proportionals between 2 and 54*? 3. Required two mean proportionals between 6 and 1296? 4. There is a stone, of a cubic form, containing 21952 solid feet; what is the area of one of its sides? 5. An iron ball of 4 inches diameter weighs 9lbs. what is the weight of an iron ball of 8 inches diameter ? 6. What is the diameter of a 42 lb. iron ball? 7. The side of a cubical altar being 1 cubit; what is the side of one of the same form of three times that size? 8. If 20 grains of gold gild a wooden ball which weighs 512 * Two mean proportionals between two extremes may be found thus: multiply each extreme by the square of the other, and theu extract the cube root of each product for the mean proportionals sought. 2. Similar solids are to one another as the cubes of their like linear sides. 3. Spheres are to one another as the cubes of their diameters; and their surfaces as the squares of their diameters. 4. The squares of the periodic times of the planets are to each other as the cubes of their mean distances from the sun. ounces, how many grains will gild a ball of the same kind that weighs 1331 ounces? 9. The solid content of a globe is 15625 cubic inches; required the side of a cube of equal solidity? 10. The length of a ship's keel is 72 feet, the breadth of the midship beam 25 feet, and the depth of the hold 13 feet; required the dimensions of two other ships of the same form, the one to carry twice as much, and the other only half as much? 11. If a ship of 250 tons be 72 feet long in the keel, required the tonnage of another ship of the same form, whose keel is 81 feet long? 12. The proportion between Jupiter's mean distance from the sun and the earth's is, according to Dr. Maskelyne, as 5.20279 to 1; and a tropical year is 365 d. 5 h. 48 m. 48 sec. many days, &c. are there in Jupiter's year? How 13. The Georgium Sidus being 19.08352 times more remote from the sun than the earth is; required the year of the Georgium Sidus ? 14. If the earth be 95 millions of miles from the sun, how many miles are the planets Ceres and Saturn from the sun, their years being 1681 days, 12 hours, 9 min. and 10746 days, 19 hours, 16 min. 15 sec. respectively? 15. The first of Jupiter's Satellites is at the distance of 25 diameters of Jupiter from his centre, and revolves around that centre in 42 hrs. 27 m. 34 sec., and the fourth of Jupiter's Satellites revolves in 16 days, 16 hrs. 32 min. 9 sec. required the distance of the outermost Satellite from the centre of Jupiter, in diameters of Jupiter, and in English miles. Jupiter's mean diameter being 89170 English miles? The operations, both in the square and cube root, may be proved various ways. 1. By involving the root to the given power, and adding in the remainder, if any; then, if the work be right, the result will be equal to the given power. 2. By casting out the nines, as follows; cast the nines out of the root, and multiply the remainder by itself; cast the nines out of the product, reserving the excess; cast the nines also out of the remainder, subtract the excess from the dividend, then cast the nines out of what remains: if this excess be equal to the former, it may be presumed the work is right. 3. By adding the remainder, and all the lower lines, as directed for proving division. See page 28, Art. 4. DUODECIMALS. DUODECIMALS have received their name from the division of unity into 12 equal parts. By this species of arithmetic, calculations are performed as if the arithmetical scale were regulated by 12, instead of 10, and characters for 10 and 11 added to the number of the digits. Duodecimals, or cross multiplication, is a method of finding the content of any rectangular surface, the length and breadth being given in feet inches, and duodecimal parts. It is a rule used by workmen and artificers in calculating the content of their work. As several kinds of artificers work are computed by different measures, and this subject more properly belonging to practical mathematics, than arithmetic, it will be sufficient, in this place, barely to state the general rule for computing the content of a rectangle, and give one or two examples. RULE. Multiply each denomination of the length, by the feet in breadth, beginning at the lowest place, and setting each product under that denomination of the multiplicand from which it arises, observing to carry 1, for every 12, to the next higher place. Multiply, in the same manner, by the inches in breadth, if any, setting each product one place to the right hand, and always carrying by 12 when the product exceeds that number. Multiply, in a similar manner, by the next lower parts, if any, setting down each product one place farther to the right, than those of the next higher, and so on; the sum of the different products is the answer*. When the number of feet in the multiplicand is great, multiply by the feet of the multiplier, and then perform the rest of the work, by taking parts of the multiplicand for the other parts of the multiplier or questions of this kind may be often more easily performed, decimally. : If the measure be required in yards, or any other denomination greater than feet, we may first find it in feet and then bring it to the denomination required. * Inches are sometimes called primes; the next lower denomination, parts or seconds; the next, thirds, &c; and marked thus: 4 ft. 7′ 9′′ 8′′". N.B. Feet, multiplied by feet, produce feet. Feet, multiplied into inches, produce inches. Inches, multiplied into inches, produce seconds. |