In this brief explanation is included all that need now be said with regard to the properties of logarithms. The main facts to be borne clearly in mind are:—(1.) That to find the _product_ of two numbers, the logarithms of the numbers are to be _added_ together, the result being the logarithm of the product required, the value of which can then be determined. (2.) That in finding the _quotient_ resulting from the division of one number by another, _the difference_ of the logarithms of the numbers gives the logarithm of the quotient, from which the value of the latter can be ascertained. (3.) That to find the result of _raising a number to the nth power_, we _multiply_ the logarithm of the number by _n_, thus obtaining the logarithm, and hence the value, of the desired result. And (4.) That to find the n_th root of a number_, we _divide_ the logarithm of the number by _n_, this giving the logarithm of the result, from which its value may be determined. NOTATION BY POWERS OF 10. A convenient method of representing an arithmetical quantity is to split it up into two factors, of which the first is the original number, with the decimal point moved so as to immediately follow the first significant figure, and the second, 10^{_n_} where _n_ is the number of places the decimal point has been moved, this index being _positive_ for numbers greater than 1, and _negative_ for numbers less than 1.[1] In this system, therefore, we regard 3,610,000 as 3·61 × 1,000,000, and write it as 3·61 × 10^6. Similarly 361 = 3·61 x 10^2; 0·0361 (= (3·61)/(100)) = 3·61 × 10^{−2}; 0·0000361 = 3·61 × 10^{−5}, etc. To restore a number to its original form, we have only to move the decimal point through the number of places indicated by the index, moving to the right if the index is positive and to the left (prefixing 0’s) if negative. This method, which should be cultivated for ordinary arithmetical work, is substantially that followed in calculating by the slide rule. Thus with the slide rule the multiplication of 63,200 by 0·0035 virtually resolves itself into 6·32 × 10^4 × 3·5 × 10^{−3} or 6·32 × 3·5 × 10^{4–3} = 22·12 x 10^1 = 221·2. It will be seen later, however, that the result can be arrived at by a more direct, if less systematic, method of working. THE MECHANICAL PRINCIPLE OF THE SLIDE RULE. [Illustration: FIG. 1.]