From Fig. 5, which is a representation of the face of a Gravêt or Mannheim slide rule, it will be seen that four series of logarithmic graduations or scale-lines are employed, the upper and lower being engraved on the stock or body of the rule, while the other two are engraved upon the slide. The two upper sets of graduations are exactly alike in every particular, and the lower sets are also similar. It is usual to identify the two upper scale-lines by the letters A and B, and the two lower by the letters C and D, as indicated in the figure at the left-hand extremities of the scales. Referring to the scales C and D, these will each be seen to be a development of the elementary scales of Fig. 3, but in this case each principal space is subdivided, more or less minutely. The principle, however, is exactly the same, so that by moving the slide (carrying scale C), multiplication and division can be mechanically performed in the manner described. The upper scale-line A consists of two exactly similar scales, placed end to end, the first lying between IL and IC, and the second between IC and IR. The first of these scales will be designated the _left-hand A scale_, and the second the _right-hand A scale_. Similarly the coinciding scales on the slide are the _left-hand B scale_ and the _right-hand B scale_. Each of these four scales is divided (as finely as convenient) as in the case of the C and D scales, but, of course, they are exactly one half the length of the latter. The two end graduations of both the C and D scales are known as the _left-_ and _right-hand indices_ of these scales. Sometimes they are figured 1 and 10 respectively; sometimes both are marked 1. Similarly IL and IR are the left- and right-hand indices of the A and B lines, while IC is the centre index of these scales. Other division lines usually found on the face of the rule are one on the left-hand A and B scales, indicating the ratio of the circumference of a circle to its diameter, π = 3·1416; and a line on the right-hand B scale marking the position of (π)/(4) = 0·7854, used in calculating the areas of circles. Reference will be made hereafter to the scales on the under-side of the slide, and we need now only add that one of the edges of the rule, usually bevelled, is generally graduated in millimetres, while the other edge has engraved on it a scale of inches divided into eighths or tenths. On the bottom face inside the groove of the rule either one or the other of these scales is continued in such a manner that by drawing the slide out to the right and using the scale inside the rule, in conjunction with the corresponding scale on the edge, it is possible to measure 20 inches in the one case, or nearly 500 millimetres in the other. On the back of the rule there is usually a collection of data, for which the slips given at the end of this work may often be substituted with advantage. THE NOTATION OF THE SLIDE RULE.