147·326 147326 ------ 1000 &c. &c. The figures on the left of the point by themselves make the whole number which the fraction contains. Of those on its right, the first is the numerator of the fraction whose denominator is 10, the second of that whose denominator is 100, and so on. We now come to those fractions which do not contain a whole number. 134. The first of these is ¹⁴⁷³²⁶/₁₀₀₀₀₀₀ which the number of _ciphers_ in the denominator is the same as the number of _figures_ in the numerator. If we still follow the same rule, and mark off all the figures, by placing the point before them all, thus, ·147326, the observation in (133) still holds good; for, on looking at ¹⁴⁷³²⁶/₁₀₀₀₀₀₀ in the table, we find it is 1 4 7 3 2 6 --- + --- + ---- + ----- + ------ + ------- 10 100 1000 10000 100000 1000000 The next fraction is ¹⁴⁷³²⁶/₁₀₀₀₀₀₀₀, which we find by the table to be 1 4 7 3 2 6 --- + ---- + ----- + ------ + ------- + -------- 100 1000 10000 100000 1000000 10000000 In this, 1 is not divided by 10, but by 100; if, therefore, we put a point before the whole, the rule is not true, for the first figure on the left of the point has the denominator which, according to the rule, the second ought to have, the second that which the third ought to have, and so on. In order to keep the same rule for this case, we must contrive to make 1 the second figure on the right of the point instead of the first. This may be done by placing a cipher between it and the point, thus, ·0147326. Here the rule holds good, for by that rule this fraction is 0 1 4 7 3 2 6 --- + --- + ---- + ----- + ------ + ------- + -------- 10 100 1000 10000 100000 1000000 10000000 which is the same as the preceding line, since ⁰/₁₀ is 0, and need not be reckoned. Similarly, when there are two ciphers more in the denominator than there are figures in the numerator, the rule will be true if we place two ciphers between the point and the numerator. The rule, therefore, stated fully, is this: