over the place of thousands, and so on over each third figure to the left: the left hand period will often contain less than three places of figures. II. Note the greatest perfect cube in the first period, and set its root on the right, after the manner of a quotient in division. Subtract the cube of this number from the first period, and to the remainder bring down the first figure of the next period for a dividend. III. Take three times the square of the root just found for a trial divisor, and see how often it is contained in the dividend, Then and place the quotient for a second figure of the root. cube the figures of the root thus found, and if their cube be greater than the first two periods of the given number, diminish the last figure, but if it be less, subtract it from the first two periods, and to the remainder bring down the first figure of the next period for a new dividend. IV. Take three times the square of the whole root for a Cube second trial divisor, and find a third figure of the root. the whole root thus found and subtract the result from the first three periods of the given number when it is less than that number, but if it is greater, diminish the figure of the root: proceed in a similar way for all the periods. 314. What is the rule for extracting the cube root of a number? Find the cube roots of the following numbers: 1. Cube root of 1728? 2. Cube root of 117649? 3. Cube root of 46656? 5. Cube root of 5735339? 315. To extract the cube root of a decimal fraction : Annex ciphers to the decimal, if necessary, so that it shall consist of 3, 6, 9, &c., decimal places. Then put the first point over the place of thousandths, the second over the place of millionths, and so on over every third place to the right; after which extract the root as in whole numbers. NOTES.-1. There will be as many decimal places in the root as there are periods in the given number. 2. The same rule applies when the given number is composed of a whole number and a decimal. 3. If in extracting the root of a number there is a remainder after all the periods have been brought down, periods of ciphers may be annexed by considering them as decimals. EXAMPLES. Find the cube roots of the following numbers: 1. Cube root of 8.343? 2. Cube root of 1728.729 ? 3. Cube root of .0125 ? 5. Cube root of .387420489 ? 6. Cube root of .000003375? 7. Cube root of .0066592 ? 4. Cube root of 19683.46656? 8. Value of 3/81.729 ? 316. To extract the cube root of a common fraction, I. Reduce compound fractions to simple ones, mixed numbers to improper fractions, and then reduce the fraction to its lowest terms. 314. What is the rule for extracting the cube root of a number? 315. How do you extract the cube root of many decimal places will there be in the root? a decimal fraction? How Will the same rule apply when there is a whole number and a decimal? If in extracting the root of any number you find a decimal, how do you proceed? II. Extract the cube root of the numerator and denominator separately, if they have exact roots; but if either of them has not an exact root, reduce the fraction to a decimal, and extract the root as in the last case. 1. What must be the dimensions of a cubical bin, that its volume or capacity may be 19683 feet? 2. If a cubical body contains 6859 cubic feet, what is the length of one side: what the area of its surface? 3. The volume of a globe is 46656 cubic inches: what would be the side of a cube of equal solidity? 4. A person wished to make a cubical cistern, which should hold 150 barrels of water; what must be its depth? 5. A farmer constructed a bin that would contain 1500 bushels of grain; its length and breadth were equal, and each half the height; what were its dimensions? 6. What is the difference between half a cubic yard, and a cube whose edge is half a yard? 7. A merchant paid $911,25 for some pieces of muslin. He paid as many cents a yard as there were yards in each piece, and there were as many pieces as there were yards in one piece: how many yards were there, and how much did he pay a yard? NOTES.-1. Bodies are said to be similar when they have the same form and have their like parts proportional. 2. It is proved in Geometry, that the volumes or weights of similar bodies are to each other as the cubes of their like dimensions. 3. Those bodies which are named in the same example are supposed to be similar. 8. If a sphere 3 feet in diameter contains 14.1372 cubic feet, what are the contents of a sphere 6 feet in diameter ? 33: 63 :: 14.1372 113.0976. Ans. 9. If a ball 2 inches in diameter weighs 8 pounds, how much will one of the same kind weigh, that is 5 inches in diameter ? 10. What must be the size of a cubical bin, that will contain 8 times as much as one that is 4 feet on a side? 11. How many globes, 6 inches in diameter, will it require to make one 12 inches in diameter? 12. If a ball of silver, 1 unit in diameter, be worth $8, what will be the value of one 5 units in diameter? 13. If a plate of silver, 6 inches long, 3 inches wide, and inch thick, be worth $100, what will be the dimensions of a similar plate of the same metal worth $800? 14. If one man can dig a cellar 12 feet long, 10 feet wide, and 4 feet deep, in 3 days, what will be the dimensions of a similar cellar that requires him 24 days to dig it, working at the same rate, and the ground being of the same degree of hardness? 15. If I put 2 tons of hay in a stack 10 feet high, how high must a similar stack be to contain 16 tons? 16. Four women bought a ball of yarn 6 inches in diameter, and agreed that each should take her share separately from the surface of the ball: how much of the diameter must each wind off? ARITHMETICAL PROGRESSION. 317. If we take any number, as 2, we can, by the continued addition of any other number, as 3, form a series of numbers: thus, 2, 5, 8, 11, 14, 17, 20, 23, &c., in which each number is formed by the addition of 3 to the preceding number. 317. What is an arithmetical progression? What is the number added or subtracted called? This series of numbers may also be formed by subtracting 3 continually from a larger number: thus, 23, 20, 17, 14, 11, 8, 5, 2. An ARITHMETICAL PROGRESSION is a series of numbers in which each is derived from the preceding by the addition or subtraction of the same number. The number which is added or subtracted is called the common difference. 318. When the series is formed by the continued addition of the common difference, it is called an increasing series; and when it is formed by the subtraction of the common difference, it is called a decreasing series: thus, 2, 5, 8, 11, 14, 17, 20, 23, is an increasing series. 23, 20, 17, 14, 11, 8, 5, 2, is a decreasing series. The several numbers are called terms of the progression: the first and last terms are called the extremes, and the intermediate terms are called the means. 319. In every arithmetical progression there are five parts, any three of which being given or known, the remaining two can be determined. They are, 1st: The first term; 2d: The last term; 3d: The common difference; 4th: The number of 'terms; 5th: The sum of all the terms. 318. When the common difference is added, what is the series called? What is it called when the common difference is subtracted? What are the several numbers called? What are the first and last called? What are the intermediate ones called? 319. How many parts are there in every arithmetical progression? What are they? How many parts must be given before the remaining ones can be found? |