This example will suffice to show the method of obtaining the nth power or the _n_th root of _any_ number. OTHER METHODS OF OBTAINING POWERS AND ROOTS. A simple method of obtaining powers and roots, which may serve on occasion, is by scaling off proportional lengths on the D scale (or the A scale) of the ordinary rule. Thus, to determine the value of 1·25^{1·67} we take the actual length 1–1·25 on D scale, and increase it by any convenient means in the proportion of 1 ∶ 1·67. Then with a pair of dividers we set off this new length from 1, and obtain 1·44 as the result. One convenient method of obtaining the desired ratio is by a pair of proportional compasses. Thus to obtain 1·52^{¹⁷⁄₁₆}, the compasses would be set in the ratio of 16 to 17, and the smaller end opened out to include 1–1·52 on the D scale; the opening in the large end of the compasses will then be such that setting it off from 1 we obtain 1·56 on D as the result sought. [Illustration: FIG. 11.] The converse procedure for obtaining the _n_th root of a number N will obviously resolve itself into obtaining (1)/(_n_)th of the scale length 1-N, and need not be further considered. Simple geometrical constructions are also used for obtaining scale lengths in the required ratio. A series of parallel lines ruled on transparent celluloid or stout tracing paper may be placed in an inclined position on the face of the rule and adjusted so as to divide the scale as desired. When much work is to be done which requires values to be raised to some constant but comparatively low power, _n_, the author has found the following device of assistance:—On a piece of thin transparent celluloid a line OC is drawn (Fig. 11) and in this a point B is taken such that (OC)/(OB) is the desired ratio. It is convenient to make OB = 1–10 on the A scale, so that assuming we require a series of values of _v_^{1·35}, OB would be 12·5 cm. and OC, 16·875 cm. On these lines semi-circles are drawn as shown, both passing through the point O. Applying this cursor to the upper scales so that the point O is on 1 and the semi-circle O M B passes through _v_ on A, the larger semi-circle will give on A the value of _v^n_. Thus for _p_ _v^n_ = 39·5 × 4·9^{1·35}, set 1 on B to 39·5 on A (Fig. 12) and apply the cursor to the working edge of B, so that O agrees with 1 and O M B passes through 4·9 on B. The larger semi-circle then cuts the edge of the slide on a point, giving 337 on A as the result required. Of course any number of semi-circles may be drawn, giving different ratios. If a number of evenly-spaced divisions are used as bases, the device affords a simple means of obtaining a succession of small powers or roots, while it also finds a use in determining a number of geometric means between two values as is required in arranging the speed gears of machine tools, etc. The converse operation of finding roots will be evident as will also many other uses for which the device is of service.