01694915254237288135593220338, &c. 016393442622950819672131147540, &c. Here, then, are two numbers, the first of which multiplied by any number under 59, and the second by any number under 61, can have the products formed by carrying certain of the figures from one end to the other. But, _b_ being still prime, it may happen that remainder 1 may occur before _b_-1 figures are obtained; in which case, as shewn, the number of figures must be a measure of _b_-1. For example, take ¹/₄₁. The repeating quotient, written as above, has only 5 figures, and 5 measures 41-1. 0₁₀2₁₈4₁₆3₃₇9₁ Now, this period, it will be found, has its figures merely transposed, if we multiply by 10, 18, 16, or 37. But if we multiply by any other number under 41, we convert this period into the period of another fraction whose denominator is 41. The following are 8 periods which may be found. 0₁₀2₁₈4₁₆3₃₇9₁ | 1₉2₈1₃₉9₂₁5₅ 0₂₀4₃₆8₃₂7₃₃8₂ | 1₁₉4₂₆6₁₄3₁₇4₆ 0₃₀7₁₃3₇1₂₉7₃ | 2₂₈6₃₄8₁₂2₃₈9₁₁ 0₄₀9₈₁7₂₃5₂₅6₄ | 3₂₇6₂₄5₃₅8₂₂5₁₅ To find _m_/41, look out for _m_ among the remainders, and take the period in which it is, beginning after the remainder. Thus, ³⁴/₄₁ is ·8292682926, &c., and ¹⁵/₄₁ is ·3658536585, &c. These periods are complemental, four and four, as 02439 and 97560, 07317 and 92682, &c. And if the first number, 02439, be multiplied by any number under 41, look for that number among the remainders, and the product is found in the period of that remainder by beginning after the remainder. Thus, 02439 multiplied by 23 gives 56097, and by 6 gives 14634. The reader may try to decipher for himself how it is that, with no more figures than the following, we can extend the result of our division. The fraction of which the period is to be found is ¹/₈₇. 87)100(01149425 130 430 820 01149425 × 25 370 28735625 × 25 220 718390625 × 25 460 17959765625 × 25 25 448994140625 0114942528735625 718390625 1795976 5625 448994 ----------------------------+------ 0114942528735632183908045977|011494 | APPENDIX X. ON COMBINATIONS. There are some things connected with combinations which I place in an appendix, because I intend to demonstrate them more briefly than the matters in the text.