For natural, hyperbolic, or Napierian logarithms, the base is 2·718. A special line marked ε or _e_ serves to locate the exact position of this value on the E scale, and placing this to 1 on D we read log._{_e_} 4·35 = 1·47; log._{_e_} 7·4 = 2·0; antilog._{_e_} × 2·89 = 18, etc. The other parts of the scale are read as already described for common logs. Calculations involving powers of _e_ are frequently met with, and these are facilitated by using the special graduation line referred to, as will be readily understood. If it is required to determine the power or root of a number which does not appear on either of the log.-log. scales, we may break up the number into factors. Usually it is convenient to make one of the factors a power of 10. EX.—3950^{1·97} = 3·95^{1·97} × 10^{3 × 1·97} = 3·95^{1·97} × 10^{5·91}. Then 3·95^{1·97} = 15, and 10^{5·91} (or antilog.) 5·91 = 812,000. Hence, 15 × 812,000 = 12,180,000 is the result sought. Numbers which are to be found in the higher part of the log.-log. scale may often be factorised in this way, and greater accuracy obtained than by direct reading. The form of log.-log. rule which has been mainly dealt with in the foregoing gives a scale of comparatively long range, and the only objection to the arrangement adopted is the use of a separate slide. _The Jackson-Davis Double Slide Rule._—In this instrument a pair of aluminium clips enable the log.-log. slide to be temporarily attached to the lower edge of the ordinary rule, and used, by means of a special cursor, in conjunction with the C scale of the ordinary slide. In this way both the log.-log. and ordinary scales are available without the trouble of replacing one slide by the other. Since the scale of exponents is now on the slide, the value of _x^n_ will be obtained by setting 1 on C to _x_ on E and reading the result on E under _n_ on C. By using a pair of log.-log. slides, one in the rule and one clamped to the edge by the clips, we have an arrangement which is very useful in deducing empirical formulæ of the type _y_ = _x^n_. _The Yokota Slide Rule._—In this instrument the log.-log. scales are placed on the face of the rule, each set comprising three lines. These, for numbers greater than 1, are found above the A scale while the three reciprocal log.-log. lines are below the D scale. Both sets are used in conjunction with the C scale on the slide. Other features of this rule are:—The ordinary scales are 10 in. long instead of 25 cm. as hitherto usual; hence the logarithms of numbers can be read on the ordinary scale of inches on the edge of the rule. There is a scale of cubes in the centre of the slide and on the back of the slide there is a scale of secants in addition to the sine and tangent scales. [Illustration: FIG. 15.]