Thus, to find √365, 365 (19-1049 1 29)265 261 381) 400 38204) 190000 382089) 3718400 382098) 279599 If the answer be required to many decimal places, the working becomes laborious, but may be shortened as follows: Find half or one more than half the required figures by the above rules, and then annexing one figure instead of two proceed according to the contracted method of dividing decimals. To continue the √365 found above :— 382098) 2795990(73174 2674691 121299 114630 6669 3821 2848 2675 173 153 20 Hence √36519-104973174, *The square root of a number, which is nearly a perfect square, may be approximately found as follows: If a be the root of the perfect square, and a±b be the root of the given number N, N=(a+b)=a2+2ab+b2; but if a is nearly equal to N, b is so small that b2 may be neglected. .. √17 = 4·125 instead of 4·1231. Again, to find √1300, 362 = 1290, *(6) CONTINUED FRACTIONS. It sometimes happens in finding the formula from the percentage composition of a substance (Sect. 27) that it is required to reduce the fraction representing the ratio of these results to its lowest terms, and then to find another fraction in lower terms nearly representing this ratio, the difference being due to experimental error. The only form of continued fractions required in chemistry is the one in which every denominator is a whole number with a fraction annexed, which fraction has also for its denominator a whole number with a fraction and so on, every numerator being 1. To convert a fraction into a continued one, divide the greater term by the less, the less by the remainder and so on, as in finding the greatest common measure; the quotients will be the denominators of the several fractions in the continued fraction, the numerator of each being 1. Thus, to express 100000) 141421 (1 100000 1.41421 as a continued fraction, 41421)100000 (2 17158)41421 (2 7105) 17158(2 2948)7105 (2 1209) 2948 (2 The successive convergents which can be made from this continued fraction are 1, 1, 12, 1, 118, 128, which are alternately smaller and larger than the original number, to which also each successive convergent more nearly approximates. To find the series of converging fractions from the quotients obtained as above. Write the several quotients in a line, take the first quotient as the numerator and 1 as the denominator of the first fraction, which is set below the second quotient. Then for the second fraction, multiply both the terms of the first by the quotient which stands above it and add 1 to the numerator, the result is the second fraction which is set below the third quotient. To find the succeeding fractions, multiply the terms of each fraction, when found, by the quotient which stands above it, and to the products add separately the terms of the preceding fraction. Thus the convergents corresponding to the quotients The Imperial Standard yard is the distance between the centres of two gold plugs in a bronze bar, when it is at the temperature of 62° F. = 63360 inches=5280 feet 1760 yards = 1 statute mile. 73036.8 inches 6086·4 feet = 2028-8 yards = 1.152 miles =1 geographical mile or knot 1000 fathoms. = A geographical mile is the mean length of one minute of longitude at the Equator. The polar axis of the Earth is nearly 5·005 x 10 inches in length. 27878400 sq. ft. = 3097600 sq. yds. = 623·7 ac. = 1 sq. mile. 1728 cub. ins. = 49.884 pts. = 6·314 gals. = 1 cub. ft. 46656 cub. ins. = 1346.9 pts. = 168.36 gals. = 27 cub. ft. = 1 cub. yard. = 5451776000 cub. yds. = 1 cubic mile. |