28 ? 2. Required the cube root of 99252847 ? Ans. 463. 3. Required the cube root of 259694072 ? 638. 4. Required the cube root of 7612.812161 ? 19.67 5. Required the cube root of 61218.00121 ? 39.41 6. Required the cube root of 219365327.791 ? 603.1 7. Required the cube root of 67527834239 ? 4072.18. 8. Required the cube root of 36155.027576 ? 33.0365. 9. Required the cube root of .0069761218? .19107. VULGAR FRACTIONS, RULE—The same as given in the Square Root, page 122, only observe to extract the Cube Root instead of the Square Root. EXAMPLES. 10. Required the cube root of 324? Ans. 11. Required the cube root of 15 20 ? 5130 12. Required the cube root of 4 ? .829 13 Required the cube root of s? .873. 14. Required the cube root of 1:17? 15. Required the cube root of 4:)5123 16. Required the cube root of 9? 2.092. 17. Required the cube root of ef? 2.057. QUESTIONS FOR EXERCISE. 18. If a cubical piece of timber be 47 inches long, 47 inches broad, and 47 inches deep: What is the contents in cubical inches? Ans. 103823. 19. There is a cellar dug, that is 12 feet every way in length, breadth, and depth : how many cubical feet of earth were taken out of it? Ans. 1953.125. 20. There is a stone of a cubic form, which contains 389017 solid feet : Required the superficial content of one of its sides ? Ans. 5329. 21. There are two numbers, whereof the greater is 2579890752, their difference is 1152: What is the lesser, and what the cube root of their sum ? Ans. The lesser 2579889600, and the cube root of their sum is 1728. PROBLEMS. 1. To find two mean proportionals between two given numbers. RULE-Divide the greater extreme by the less, and the cube root of the quotient multiplied by the less extreme gives the less mean; multiply the said cube root by the less mean, and the product will be the greater mean proportional, EXAMPLES 22. What are the two mean proportionals between 7 and 189? Ans. 21 and 63. 23. Required two mean proportionals between 4 and 256 ? Ans. 16 and 64. II. To find the side of a cube that shall be equal in solidity to any given solid, as globe, cylinder, prism, &c. &c. Rule-The cube root of the solid content of any solid body given, is the side of the cube of equal solidity. EXAMPLES. 24. If the solid content of a globe be 10648 : What is the side of a cube of equal solidity ? Ans. 22. 25. Required the side of a cubical vessel that shall contain 80 wine gallons, each 231 cubic inches? Ans. 26.43 in. Ill. The side of the cube being given, to find the side of the cube that shall be double, treble, &c. in quantity to the given cube. RULE-Cube the given side, and multiply it by 2, 3, &c. the cube root of the product is the side sought. EXAMPLES.. 26. There is a cubical vessel, whose side is 12 inches, it is required to find the side of another vessel, that shall contain three times as much ? Ans, 17.306 27. Suppose the length of a ships keel to be 125 feet, the breadth of the midship beam 25 feet, and the depth of the hold 15 feet; required the dimensions of another ship of the same form, that shall carry three times the burthen ? Length of the keel 180.28 feet. Depth of the hold 21,63 feet. dimensions of a similar solid of a different capacity. RuleLike solids are in triplicate proportion to their homologous sides, therefore it will be as the cube of a dimension : is to its given weight : : so is the cube of any like dimension : to the weight required. EXAMPLES. 28. A brass bullet of 5 inches diameter weighs 20 lb, required the diameter of a like bullet that weighs 160 lb. Ans. 10 inches. 29. If a ship of 300 tons burthen, bę 75 feet long in the keel : I demand the burthen of another ship, whose keel is 100 feet long ? Ans. 711.111 tons. 30. If a brass saker, whose diameter is 11.5 inches, weighs 1000 lb. what will another piece of ordnance (of the same metal and shape) weigh, whose diameter is 20.83 inches ? Ans. 5942.5697 lb. BIQUADRATE ROOT. RULE_Extract the square root of the square root. ROOT OF THE SIXTH POWER. Rule-Extract the cube root of the square root. ROOT OF THE EIGTH POWER. Rule-Extract the square root, of the biquadrate root. ROOT OF THE NINTH POWER. RULE-Extract the cube root of the cube root. TO EXTRACT ANY ROOT WHATEVER. RULE-Let P be the given power or number, n the index of the power, A the assumed power, r its root, R the required root of P. Then, as the sum of n+1 times A, and n-1 times P: is to the sum of n+1 times P and n-1 times A :: so is the assumed root r: to the required root R. Note-For greater exactness the operation may be repeated as often as we please, by using always the last found root for the assumed root, and its n’th power for the assumed power A. EXAMPLES 1. Required the sursolid, or 5th root 21035.8 ? The fifth root is found to be between 7.3 and 7.4. The 5th power 7.3=20730.71593. Hence P=21035.8, nX5, r=7.3 and A=20730,71593 n+1xA=124384.295581 +1XP=126214.8 na-1X P= 84143.2 9-1XA=82922.86372 208527.49558 209137.76372::7.3:7.32136 2. Required the 4th root of 2? Ans. 1.189207 3. Required the 5th root of 2 ? 1.148699 4. Required the 6th root of 2 ? 1.122462 5. Required the 7th root of 2 ? 1.104089 6. Required the 7th root of 21035.8 ? 4.145392 ARITHMETICAL PROGRESSION. Arithmetical Progression is a rank, or series of numbers, which increase or decrease by common difference ; in which. five particulars are to be observed, viz : Let f, represent the first term, called extremes. the last term, the number of terms.. d, the equal or common difference. the sum of all the terms. Note--In any series of numbers in Arithmetical Progres. sion, the sum of the two extremes will be equal to the sum of any two terms equally distant therefrom ; as 2, 4, 6, 8, 10, 12; where 2+12=14; so 4+10=14 ; or 3, 6, 9, 12, 15; where 3+15=18; alse 6+12=18 ; and also 9+9=18. I. Given f. I. and n. to find s.--RULE. f+x=s. EXAMPLES. 1. How many times does the hammar of a clock strike in 12 hours. Ans. 78. 2. A man buys 17 yards of Osnaburgh, and gave for te first yard 2s. and for the last 10s. Quere, the amount of the whole ? Ans. 51. 2s. 3. A butcher bought 100 head of oxen, and gave for the first ox 1 crown, for the second, 2 crowns, for the third, 3 crowns, &c. Quere, the sum paid for the cattle. Ans. 12621. 108. 4. Suppose a basket and 500 stones were placed in a straight line a yard distant from one another: Required in what time a man could bring them one by one to the basket, allowing him to walk at the rate of 3 miles an hour ? Ans. 142.329 miles.--and 47 ho. 26 m. 34.8 .. ni II. Given fi l. and n. to find d._RULE. =d. EXAMPLES. 5. A man had 8 sons, the youngest was 4 years old, and the eldest 32, they increase in arithmetical progression : Required the equal difference of their ages ? Ans. 4. 6. A man is to travel from Washington to a certain place in 12 days, and to go but 3 miles the first day, increasing ev. ery day an equal excess, so that the last day's journey may be 58 miles : Required the daily increase, and distance from Washington ? Ans. 5 daily increase, and distance 366 m. -f III. Given f. I. and d. to find n.-Rule. +1=n. EXAMPLES. 7. A person travelling into the country, went 3 miles the first day, and increased every day by 5 miles, till at last he went 58 miles in one day : How many days did he travel, Ans. 12. 8. A man being asked how many sons he had, said, that the youngest was 4 years old, and the oldest 32, and that his family increased one every 4 years : How many had he ? Ans. 8. IV. Given l. n. and d. to find f-RULE. L-Xn-l=f. EXAMPLES. 9. A man in 10 days went from New-York to a certain town in the country, every day's journey increasing the for. mer by 4, and the last he went was 46 miles. What was the first? Ans. 10 miles. 10. A man takes out of his pocket at eight several times, so many different numbers of dollars, every one exceeding the former by 6, the last at 46. What was the first ? Ans. dxn-1 V. Given, n. d. and 3. to find f.-Rule. 2 EXAMPLE. 11. A man is to receive 360 eagles, at 12 several payments, each to exceed the former by 4 eagles, and is willing to be. stow the first payment on any one that can tell him what it is. What will that person have for his pains ? Ans. 8 eagles. VI. Given, f. n. and d. to find l.--Rule. nxd_d+f=l. EXAMPLE. 12. Required the last number of an arithmetical progression, beginning at 6, and continuing by an increase of 8 to 20 places ? Ans. 158. GEOMETRICAL PROGRESSION. Geometrical Progression is a series of numbers increasing or decreasing by one continual multiplier, or divisor, called the ratio ; as, 2, 4, 8, 16, 32, &c. increase by the continual multiplication of 2 ; and 32, 16, 8, 4, 2, decrease continually by the divisor 2. I. When the first term is unity, the ratio and number of terms being known; to find the last. Rule_Raise such a power of the ratio, multiplied into the first term ; or, take a convenient number of terms (called indices) in Arithmetical Progression, beginning and increasing with an unit, under which, place the leading terms, of the given Geometrical Progression ; then the square of any term under an indice, will be the term represented by twice that indice, &c. If the first term and ratio be different, begin the indices with a cipher, and the sum which is to be made choice of, |