Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/597

Rh CONTINUED FRACTIONS,] method explained in Art. 116, the value of the unknown quantity is evidently expressed by a continued fraction. For if x be the root sought, we have # = + -, y = b + y = V + - r/, y&quot; = b&quot; + ,, &c. where a, b, b, b&quot;, &c. denote the whote numbers, which are next less than the true values of x, y, y, y&quot;, Ac. If, therefore, in the value of x we substitute b + for y, it becomes 1 X = a + - r- 6+4. y Again, if in this second value of x we substitute b + y for y, it become 1 ALGEBRA 559 Let - bea fraction nearer to u than - ; then since tho convergents are alternately too great and too small; -h 1 ? P } P. } mus t be in order of magnitude. . . If the first be the greatest, y And so on continually, 133. It is easy to see in what manner the inverse of the preceding operation is to be performed, or a continued frac tion reduced to a common fraction. The fractions which result from omitting portions of a continued fraction are termed the convergents to that frac- 1 tion. Thus, if the fraction be a, - is the first convergent, the second, ifec. 04 2 + 1 134. The principal practical application of the properties of continued fractions is to approximate to the value of a given fraction. The proposition on which this application depends is the following : - No fraction in terms equally low can give so good an ap proximation to the value of a fraction as a convergent to the continued fraction ivhich expresses it does. To demonstrate this proposition, it is requisite to estab lish three preliminary propositions, which we shall do very briefly. 77 135. (1.) If denote the nth convergent, or the re- &amp;lt;?n duced fraction which results from stopping at a n, and reduc ing, then p n+l = ff nfl ^ B + j p B _i, q n+l = +# + q n _ :. Since no denominator can be multiplied by itself, the re duced fraction must give p n = B A + B. Now p n+i is obtained by writing + for a and n+l reducing, Pn+i = +iOA + B) + A = an+l p n + A ; i.c., the multiplier of any a is the previous p, and the other term is the multiplier of a in the previous convergent, hence the proposition. 136. (2.) ^^.-^.,^ = (-1)-. This is at once obtained by eliminating a n+1 from the two equations of last article. 137. (3.) The successive convergents are alternately greater and less than the complete fraction, and each convergent approaches nearer in value to it than the preceding. If A denote the complete denominator B+1 + &c. ; u the complete fraction; then = -l!r! . an( j by su btract- A 2n + 2n-l ing successively p ~ and P from u in this form, it will be seen at once that the results have different signs, and that the latter difference is the larger. 138. We are now able to prove the proposition enun ciated. Reducing and applying Prop. 2, there results Similarly by inverting the fractions, it may be proved that p&amp;gt;p n +. Ex. 1. To determine when a transit of Venus may be expected. The relative sidereal periods of Venus and the earth are 224,700 days and 365,250 days. The continued fraction which expresses the quotient of these numbers is 29 Q The fifth convergent is ; the sixth -^ 13 382 On account of the smallness of, the former is a very close approximation, i.e., 8 years and 13 sidereal periods of Venus are very nearly equal. In consequence of this, a transit occurs after one period of 8 years, and then again not till after 235 years have been completed. The last pair of transits at the descending node occurred in 1769, 1777 ; and at the ascending node in 1639, 1647. The next pair will accordingly occur at the latter node in 1874 and 1882. The days of transit will be December 8 and December 6, respectively. Ex. 2. To find the periods of probable recurrence of eclipses of the sun. An eclipse of the sun will occur whenever the place of the new moon is within about 13 of the line of nodes. Now, the interval between two new moons is 29 5306 days ; and the mean synodic period of the earth and the line of nodes is 346-6196 days. The proportion of the latter of these numbers to the former, reduced to a con- 47 223 tinned fraction, gives as convergents, , &c. Hence, after 47 lunar months, things have come nearly to their original position, and after 223 lunar months, very nearly. This latter period, termed the saros, has been known from the remotest antiquity. It enabled the Chal dean shepherds to predict the return of eclipses. It amounts to 18 years and 10 or 11 days. Thus, there was a total eclipse on the 18th July 1860; adding 18 years 11 days, we get for an eclipse 29th July 1878. If we add 47 lunations or 1388 days, we get 6th March 1864, on which day there was an eclipse. This period of 1388 clays, multiplied by 5, makes exactly 19 years a period which is designated as the cycle of Meton, giving eclipses which occur on the same day of the month. Thus, eclipses happened 18th July 1841 and 18th July 1860, and another will happen 18th July 1879. Ex. 3. The fraction given (Art. 131) represents the ratio of the circumference of a circle to its diameter. By taking the first two terms we have 7r = 3 + ^ = y nearly: and this is the proportion which was found by Archimedes, Again, by taking the first three terms, we have _ o 1 T o ,15 333 + 7 + ^ + 106 = 106 which is nearer the truth than the former. And, by taking the first four terms, we have