Setting the R.H. index of the slide to 5 on A, it is seen that 1·71 on D coincides with 1·71 on ᗺ. Then setting the L.H. index to 5 on A, further coincidences are found at 3·68 and at 7·93, the three values thus found being the required roots. Note that the first root was found on that portion of the D scale lying under 1 to 5 on A; the second root on that portion lying under 5 to 50 on A; and the third root on that portion of D lying under 50 to 100 on A. In this connection, therefore, scale A may always be considered to be divided into three sections—viz., 1 to _n_, _n_ to 10_n_, and 10_n_ to 100. For all numbers consisting of 1, 1 + 3, 1 + 6, 1 + 9—_i.e._, of 1, 4, 7, 10, or −2, −5, etc., figures—the coincidence under the first section is the one required. If the number has 2, 5, 8, or −1, −4, −7, etc., figures, the coincidence under the second section is correct, while if the number has 3, 6, 9, or 0, −3, etc., figures, the coincidence under the last section is that required. The number of digits in the root is determined by marking off the number into sections, as already explained. _Cube Root (Pickworth’s Method)._—One of the principal objections to the two methods described is the difficulty of recollecting which scales are to be employed and with which index of the slide they are to be used. With the direct method another objection is that the readings to be compared are often some distance apart, the maximum distance intervening being _two-thirds_ of the length of the rule. To carry the eye from one to another is troublesome and time-taking. With the inverted scale method the reading of a scale reversed in direction and with the figures inverted is also objectionable. With the author’s method these objections are entirely obviated. The _same scales and index are always used_, and are read in their normal position. The three roots of _n_, 10_n_ and 100_n_ (_n_ being less than 10 and not less than 1) are given with one setting and appear in their natural sequence, no traversing of the slide being needed. The readings to be compared are always close together, the maximum distance between them being _one-sixth_ of the length of the rule. The setting is always made in the earlier part of the scales where closer readings can be obtained, and finally, if desired, the result may be readily verified on the lower scales by successive multiplication. For this method two gauge points are required on C. To conveniently locate these, set 53 on C to 246 on D; join 1 on D to 1 on A with a straight-edge and with a needle point draw a short fine line on C. Set 246 on C to 53 on D, and repeat the process at the other end of the rule. The gauge points thus obtained (dividing C into three equal parts) will be at 2·154 and 4·641, and should be marked ∛(10) and ∛(100) respectively.[6] EX.—Find ∛(2·86,) ∛(28·6) and ∛(286).