In the former case the slide is to be placed in the rule so that the T scale is adjacent to the A scales, and, with the right and left indices coinciding, when opposite any angle on T will be found its tangent on A. From what has been said above, it follows that the tangents read on the L.H. scale of A have values extending from 0·01 to 0·1; while those read on the R.H. scale of A have values from 0·1 to 1·0. Otherwise expressed, to the values of any tangent read on the L.H. scale of A a cypher is to be prefixed; while if found on the R.H. scale, it is read directly as a decimal. EX.—Find tan. 3° 50′. Placing the slide as directed, the reading on A opposite 3° 50′ on T is found to be 67. As this is found on the L.H. scale of A, it is to be read as 0·067. EX.—Find tan. 17° 45′. Here the reading on A opposite 17° 45′ on T is 32, and as it is found on the R.H. scale of A it is read as 0·32. As in the case of the scale of sines, the tangents may be found without reversing the slide, when a fixed index is provided in the back of the rule for the T scale. We revert now to a consideration of those rules in which a single tangent scale is provided. It will be understood that in this case the slide is placed so that the scale T is adjacent to the D scale, and that when the indices of both are placed in agreement, the value of the tangent of any angle on T (from 5° 43′ to 45°) may be read off on D, the result so found being read as wholly decimal. Thus tan. 13° 20′ is read 0·237. If a back index is provided, the slide is used in its normal position, when, setting the angle on the tangent scale to this index, the result can be read on C over the L.H. index of D. The tangents of angles above 45° are obtained by the formula: Tan. θ = (1)/(tan. (90 − θ)). For all angles from 45° to (90° − 5° 43′) we proceed as follows:—Place (90 − θ) on T to the R.H. index of D, and read tan. θ on D under the L.H. index of T. The first figure in the value thus obtained is to be read as an integer. Thus, to find tan. 71° 20′ we place 90° − 71° 20′ = 18° 40′ on T, to the R.H. index of D, and under the L.H. index of T read 2·96, the required tangent. The tangents of angles less than 40′ are sensibly proportional to the angles themselves, and as they may therefore be considered as sines, their value is determined by the aid of the single and double accent marks on the sine scale, as previously explained. The rules for the number of integers are the same as for the sines. Multiplication and division of tangents may be quite readily effected. EX.—Tan. 21° 50′ × 15 = 6. Set L.H. index of T to 15 on D, and under 21° 50′ on T read 6 on D. EX.—Tan. 72° 40′ × 117 = 375. Set (90° − 72° 40′) = 17° 20′ on T to 117 on D, and under R.H. index of T read 375 on D.