The mechanical principle involved in the slide rule is of a very simple character. In Fig. 1, A and B represent two rules divided into 10 equal parts, the division lines being numbered consecutively as shown. If the rule B is moved to the right until 0 on B is opposite 3 on A, it is seen that any number on A is equal to the coinciding number on B, plus 3. Thus opposite 4 on B is 7 on A. The reason is obvious. By moving B to the right, we add to a length 0·3, another length 0·4, the result read off on A being 7. Evidently, the same result would have been obtained if a length 0·4 had been added, by means of a pair of dividers, to the length 0·3 on the scale A. By means of the slide B, however, the addition is more readily effected, and, what is of much greater importance, the result of adding 3 to _any one of the numbers_ within range, on the lower scale, is _immediately_ seen by reading the adjacent number on A. Of course, subtraction can be quite as readily performed. Thus, to subtract 4 from 7, we require to deduct from 0·7 on the A scale, a length 0·4 on B. We do this by placing 4 on B under 7 on A, when over 0 on B we find 3, on A. It is here evident that the _difference_ of any pair of coinciding numbers on the scales is constantly equal to 3. [Illustration: FIG. 2.] An important modification results if the slide-scale B is inverted as in Fig. 2. In this case, to find the sum of 4 and 3 we require to place the 4 of the A scale to 3 on the B scale, and the result is read on A over 0 on B. Here it will be noted, the _sum_ of any pair of coinciding numbers on the scales is constant and equal to 7. This case, therefore, resembles that of the immediately preceding one, except that the _sum_, instead of the _difference_, of any pair of coinciding numbers is constant. To find the difference of two factors, the converse operation is necessary. Thus, to subtract 4 from 7, 0 on B is placed opposite 7 on A, and over 4 on B is found 3 on A. From these examples it will be seen that with the slide _inverted_ the methods of operation are the reverse of those used when the slide is in its normal position. It will be understood that although we have only considered the primary divisions of the scales, the remarks apply equally to any subdivisions into which the primary spaces of the scales might be divided. Further, we note that the length of scale taken to represent a unit is quite arbitrary. THE PRIMITIVE SLIDE RULE.