The second ratio is the one given by Archimedes. The fourth is that of Adrian Metius, and is even more exact than the ratio 3.14159, from which we have derived it. 2837 42. Required the approximate values of 751 43. Find the approximate values for 4999. 5763 44. Find the approximate values for 34999. 72622 45. What are the approximate values of .785398, which is nearly the ratio of the area of a circle, to that of its circumscribing square? 46. What are the approximate values of 37729? 47. Find approximate values for 42551 13045 48. Find approximate values for 44037 31867 In the EXTRACTION OF ROOTS, we may commence with any divisor, cutting off the right-hand figure at each step, as in contracted division. At whatever place this contraction is commenced, as many additional root figures will be obtained as are equal to the number of figures in the divisor less 1, but the last figure so obtained cannot always be relied upon. To illustrate this principle, we will extract the 5th root of 69. In applying the general rule for obtaining any root, the pupil will frequently find great difficulty in determining the value of the figures in each column. To obviate this diffi culty, he should be taught to supply the zeroes, when the first root figure is in the place of tens, hundreds, &c., and to observe the place of the decimal point in each product, until he becomes familiar with the process. Oiven number, for a divisor. Divide, and reserve the 1 0 2 After obtaining the third trial divisor, we commence rejecting one figure from the trial divisor, two from the number at the foot of the preceding column, three from the third column, &c., and proceed in a similar way with each subsequent trial divisor, until the figures from the preceding columns are entirely cancelled. But in every instance, allowance must be made for the product of the figures rejected, as in simple contracted division. 49. Extract the square root of 287; of 5. 50. Extract the cube root of 11; of 25; of 693. 51. Extract the 4th root of 13; of 1.8; of 27. 52. Extract the 5th root of 797.9341. 53. Extract the 5th root of 1.0843. 54. Extract the cube root of 997641.285. The SQUARE ROOT of any number may be expressed in the form of a continued fraction, after part of the root is found,—by making each numerator equal to the remainder, and each denominator equal to twice the root found. Thus in extracting the square root of 17, the first root figure is 4, and the remainder 1. Then the true root is 4 + the continued fraction nearly, giving the first approximate root 34. Second, nearly, giving a second approximate root 317. Third, 5 6398 276 23 31 nearly, giving a third approximate root 323. This approximation is of use in affording convenient fractional expressions for those roots which are of most frequent occurrence. Thus, the diagonal of a square is to its side as √2 is to 1. By the rule just given, we obtain successively for approximate values of √2, 5 12 29, The last of these values, 128 or 98, is a very convenient one. The following is a general rule for the approximation of ANY ROOT desired. RULE. Call the first two figures of the root found in the usual way, the ASCERTAINED ROOT. Involve the ascertained root to the given power, and multiply by the index of the root for a dividend. Subtract the power of the ascertained root from the corresponding periods of the given number, for a divisor. Divide, and reserve the quotient. To 6 times the reserved quotient, add the index of the root, plus 1, for a second dividend. To 6 times the reserved quotient, add 4 times the index of the root, subtract 2 from the sum, and multiply by the reserved quotient for a second divisor. Divide, add 1 to the quotient, and multiply by the ascertained root for the true root nearly. If greater accuracy is desired, repeat the process with the root thus found. By this rule, the number of figures in surd roots, may generally be tripled at each operation. The following is the application of the rule, in extracting the 5th root of 659901. 2689120 given no. 659901 145 = 537824 1st divisor 122077 122077 = 22.02806, reserved quotient. reserved quotient 22.02806 6 132.16836 index + 1 6. second dividend 138.16836 6 x by reserved quotient = 132.16836 4 X 5 -2 = 18. 1.041768 x 14 = 14.584752, approximate root, correct to the fourth decimal place. This contraction is of use in extracting the higher roots. Any root below the 10th may be obtained in the usual way, nearly as readily, and with much greater accuracy. 58. Find convenient fractional approximations to √3. 59. What are the approximate fractional values of √5? 64. Reduce ✓13 to a continued fraction. 65. What are the approximate values of the continued fraction which is equivalent to √27? CHAPTER XXIV. ANALYSIS. ALL the operations of Arithmetic have for their object, the discovery of one or more unknown quantities; and the great difficulty in complicated questions, is to perceive the application of the simple rules which will lead to this discovery. The examination of any question, in order to determine the relation of the different quantities to each other, is called ANALYSIS. To keep the unknown terms more constantly in view, letters are frequently employed to represent them, and the work expressed in the statement of the question, is performed on these letters, as if their value was known, and we were proving the truth of the answer. EXAMPLE FOR THE BOARD. There is a fish whose head weighs 9 pounds; his tail weighs as much as his head and half his body; and his body weighs as much as his head and tail both. What is the weight of the fish? |