23 value of the fraction then, it is obvious, is somewhere intermediate between and ; and if we neglect the small fraction , may be considered as the first approximation to the value of the fraction. The fraction is, however, too great. For the denominator is not 4, but 4 increased by an additional quantity; and consequently the true value is less than 1. 23 To obtain a nearer approximation, we may divide both numerator and denominator of the fraction by 23, and the So that the fraction result is 1 923 1 the fractions 41 be equal to 216 887° 1 41 may be represented by nearly. 4 is equal to 37, and 1 divided by 9 37, is equal to 1 multiplied by, or to 37, which may be regarded as a second approximation to the value of the fraction 1. And if the steps of the process be attentively considered, it will be easy to perceive that the approximation is too small. For the denominator 9 being employed instead of 93, the fraction added to the denominator 4 is too great, and consequently the last fraction is too small. farther, if we divide the last fraction by its numerator 9. The as a third approximation to the value of the fraction 1. 1 To bring back the expression to a simple fraction, we 41 9/ proceed thus: 9 is equal to 19, and is equal to 1 di vided by, or equal to 4 is equal to 78, and 1 divided by & is equal to . So that is the third approximation to the value of 1. is, however, too great; for 19 78. 887. 19 1 the fraction being employed instead of 25 forms when join ed to 9 a denominator too great; and the fraction added to 4 will consequently be too small, and therefore the last fraction will be too great. By continually dividing the last fraction by its numerator, the continued fraction may be extended still farther, until a fraction is found having unity for its numerator. It is unnecessary to trace each step minutely. An accurate idea of the whole process may be obtained from the mere inspection of the following statement, exhibiting the successive forms in which the fraction 19 may be represented. 216 (1.) (2.) (3.) (4.) (5.) Or, as it is sometimes written, adopting the symbols of alge bra. 4+1 9+1 2+1 1+1 1+1 4, From what has been said, it will be easy to convert each of these continued fractions into a simple fraction, and thus to obtain an approximation to the true value. 216 The fraction originally was 887 The 1st approximation is, which is too great by 154 From this example, it would appear that the fractions derived from successive combinations of the members of a continued fraction, approximate gradually to the true value of that quantity from which the continued fraction has been derived. These fractions, it appears, also, are alternately greater and less than the true value, and therefore the value of the quantity will always be found between any two consecutive fractions. 57. By means of continued fractions, we are enabled to solve the following general problem. A fraction expressed by a great number of figures being given, to find all the fractions, in less terms, which approach so near the truth, that it is impossible to approach nearer without employing greater ones. A few examples will illustrate this. Ex. 1. It is ascertained by astronomers, that the length of the tropical year is 365 days, 5 hours, 48 minutes, and 48 seconds, which is longer than the common year of 365 days, by 5 hours, 48 minutes, and 48 seconds, or 20,988 seconds. Now, one day is equal to 86,400 seconds; therefore the year exceeds 365 days by the fraction ; so that, at the end of 86,400 years, it is necessary to intercalate 20,928 days. It is required, therefore, to find a series of fractions, expressed 20928 20928 in lower terms, which approximate to the fraction 86400* 20928 86400 To reduce to a continued fraction, we have to perform an operation precisely similar to that stated, (No. 44.) 20928)86400(4 2688)20928(7 2112)2688(1 576)2112(3 384)576(1 192)384(2 Hence we obtain the continued fraction 1 4+1 7+1 1+1 3+1 1+1 And by combining the successive members into a simple fraction, we will obtain a series of fractions, each approaching more nearly than its predecessor to the true value. These 20928 86400' The first fraction indicates that we must intercalate one day every four years, in order to preserve the correspondence between the common year and the year of the seasons. This intercalation was first made by Julius Cæsar every fourth year has therefore 366 days, and is called Leap year. As the fraction is greater than the true value of the fraction the intercalation of one day in four years, renders the common year a little longer than the tropical year. A small error was thus introduced into the calendar. By continual accumulation, this error, in the year 1582, amounted to ten days, and thereby deranged the time of celebrating the church festivals. This error was rectified by Pope Gregory XIII. By his decree, he ordered the ten days, which had progressively accumulated from the time of Julius Cæsar, to be cut off from the year 1582; and, to prevent future accumulations of this kind, directed, that the last year of every century not divisible by 4, should be reckoned only a common year. So that the years 1700, 1800, 1900, &c. which, according to the Julian calendar, are leap years, are considered as common years in the Gregorian calendar. The mode of reckoning by the Gregorian calendar, or new style, was not introduced into Great Britain until the year 1752. It is now adopted by all the European states except Russia, where the old style, or Julian mode of reckoning, still prevails. The difference between the two stiles at present amounts to twelve days, and will continually increase. In the Gregorian calendar only 97 days are intercalated in 400 years. From the series of approximate fractions stated above, it is obvious that it would be more accurate to intercalate 109 days in 450 years. The error, however, is so slight, that it will not amount to a day during the lapse of many centuries. 8 33 It The third approximate fraction mentioned above is intimating that 8 days must be intercalated in 33 years. is a curious fact, that this was the correction proposed by Omar, an astronomer at the court of Persia, in the year 1079. This simple method differs from the truth only one minute in a century. Ex. 2. It is ascertained that the diameter of a circle is nearly 100000 very of its circumference. It is required to find a series of fractions expressed in lower terms that shall approximate to this value. |