EXAMPLES. 1. Required the second power of the number 3.874. 2. Required the third power of the number 2.768. 3. Required the third power of the number .7916. 4. Required the twelfth power of the number-1.539. EVOLUTION BY LOGARITHMS. RULE. 1. Seek the logarithm of the given number in th table. 2. Divide the logarithm, thus found, by the denominator of the index of the root proposed. 3. Find the number corresponding to this quotient, and it will be the root required. Vote. When the index of the logarithm, to be divided, is negative, and does not exactly contain the divisor, increase it by such a number as will make it exactly divisible, and carry the units borrowed, as so many tens, to the left-hand place of the decimal, and then divide as in whole numbers. EXAMPLES. 1. Required the square root of the number 225. The root 15. . . . . . . 1.1760912 2. Required the square root of the number 1501. The log. of 1501 3.1763807 Therefore 2)3.1763807 3. What is the cube root of the number .166375? Therefore 3)-1.2210881 4. What is the square root of the number .08162? 5. What is the twelfth root of the number 176.6? MISCELLANEOUS QUESTIONS. 1. A person being asked what o'clock it was, answered, that it was between 8 and 9, and that the hour and minute hands were exactly together; what was the time? h. Ans. 8: 43: 38-21. 2. Divide the number 50 into two such parts, that of one part, added to % of the other, may make 40. Ans. 20 and 30. 3. What two numbers are those, whose difference is 12, and their squares equal to each other? Ans. +6 and -- 6. 4. There is a certain number, consisting of two places, which is equal to the difference of the squares of its digits; and if 36 be added to it, the digits will be inverted; quære the number? Ans. 48. 5. Given 3+ y2=31, and y3 + x2 and y. = 17; to find x Ans x 3 and y - 2. 6. Given y3xy = 666, and x2 + xy 406; to find x and y. Ans. x7 and y 9. 7. Given the sum of three numbers, in harmonical proportion, = 26, and their continued product the numbers. = = 576; to find Ans. 12, 8, and 6. 8. What two numbers are those, whose difference, sum, and product are to each other as the numbers 2, 3, and 5, respectively? Ans. 2 and 10. 9. To find that number whose cube being subtracted from its square, shall leave the greatest remainder possi ble. shall be all equal to each other. Ans. 3. 10. It is required to find the least 3 whole numbers, so that of the first, of the second, and of the third, Ans. 280, 294, and 300. 11. Given zx3+xz3 290, and 4+ 24 = 641; to find x and z. Ans. x = 5, and z = 2. 12. Given the sum of three numbers in continued geometrical progression 39, and the sum of their squares = 819; to find the numbers. Ans. 3, 9, 27. 13. Required the least number of weights, and the weight of each, that will weigh from one pound to 29 hundred weight. Ans. 1, 3, 9, 27, 81, 243, 729, and 2187. squares. 14. Required two numbers such, that their sums shall be equal both to their product and the difference of their Ans. 2.618034 and 1.618034. 15. It is required to find the least 4 affirmative integers such, that the square of the greatest may be equal to the sum of the squares of the other three. Ans. 3, 4, 12, and 13. *This is properly a question in fluxions, but it is answered algebraically by Mr. Emerson, as well as several others of the same nature. 16. If money be lent, at three per cent. To those who choose to borrow, In what time shall I be worth a pound, If I lend a crown to-morrow? Ans. 46.90036 years, allowing comp. int. 17. Required the two least nonquadrate numbers, x and y, such, that x2 + y2 and x3 + y3 shall be both square numbers. Ans. x = 364, and y = 273. 18. There are three numbers in geometrical proportion such, that, if the mean be subtracted from the sum of the two extremes, the remainder, multiplied by the sum of the said two extremes will be 94; but, if that remainder be multiplied by the sum of all the three numbers, the product will be 133; it is required to find the three numbers by a simple equation. Ans. 4, 6, and 9. 19. To determine two numbers whose sum shall be a cube, but their product and quotients squares. Ans. 4 and 14, 100 and 25, 900 and 100. 20. Required that arithmetical progression whose number of terms is 10, sum of the terms 185, and the sum of the cubes of the terms 104525. Ans. 5, 8, 11, 14, 17, 20, 23, 26, 29, 32. 21. To divide a given number (N) into 4 such parts, that if any other number (n) be added to the first part, deducted from the second, multiplied by the third, and the fourth part divided by it, the sum, difference, product, and quotient, shall be all equal to each other. |