When, by practice with logarithms, we become familiar with the correspondence between additions of length on the logarithmic scale (on a slide-rule) and multiplication of numbers in the natural scale (including **fractional** numbers), A /5 acquires a definite meaning as the number corresponding to the extremity of a length x, on the logarithmic scale, such that 5 corresponds to the extremity of 2X.

Of especial importance is Sidney Young, **Fractional** Distillation (1903).

A rigorous demonstration was wanting for many years, Leonhard Euler's proof for negative and **fractional** values being faulty, and was finally given by Niels Heinrik Abel.

The logarithms to base io of the first twelve numbers to 7 places of decimals are log 1 =0.0000000 log 5 log 2 = 0.3010300 log 6 log 3 =0.477 121 3 log 7 log 4 =0.6020600 log 8 The meaning of these results is that The integral part of a logarithm is called the index or characteristic, and the **fractional** part the mantissa.

**Fractional** silver coin is not legal tender above 20 pesos, and bronze and nickel coins not above 1 peso, but the government maintains conversion offices where such coins can be converted into silver pesos without loss.