REMARKS ON APPENDIX. 61
the quotients and the Logarithms of the divisors will be reciprocally proportional.
Otherwise the ratio will not be exactly the same on both sides; nevertheless, if the divisors be very small, and the dividends sufficiently large, so that the quotients are very many, the defect from proportionality will scarcely, or not even scarcely, be perceived.
Hence it follows that the logarithm )[C]
Let two numbers be taken, 10 and 2, or any others you please. Let the Logarithm of the first, namely 100, be given; it ts required to find the Logarithm of the second. In the first place, let the second, 2. multiply itself continuously until the number of the products ts exceeded, by unity only, by the given Logarithm of the first. Then let the last product be divided as often as possible by the first number, 10. and again in like manner by the second number, 2. The number of quotients in the latter case will be 100, (for the product ts its hundredth power; and if a number be multiplied by itself a given number of times forming a certain product, then it will divide the product as many times and once more; for example, of 3 be multiplied by itself four times it makes 243, and the same 3 divides 243 five times, the quotients being 81, 27, 9, 3, 1.) In the former case, where the product is continually divided by 10, it is manifest that the number of quotients falls short of the number of places in the dividend by one only. Therefore (by the preceding proposition) since the same product is divided by two given numbers as often as possible, the numbers of the quotients and the Logarithms of the divisors will be reciprocally proportional, But, the number of quotients by the second being equal to the Logarithm of the first, the num-
