The sines of very small angles, being very nearly proportional to the angles themselves, are found by direct reading. To facilitate this, some rules are provided with two marks, one of which, a single accent (′), corresponds to the logarithm of (1)/(sine 1′) and is found at the number 3438. The other mark—a double accent (″)—corresponds to the logarithm of (1)/(sine 1″) and is found at the number 206,265. In some rules these marks are found on either the A or the B scales; sometimes they are on both. In either case the angle on the one scale is placed so as to coincide with the significant mark on the other, and the result read off on the first-named scale opposite the index of the second. In sines of angles under 3″, the number of integers in the result is −5; while it is −4 for angles from 3″ to 21″; −3 from 21″ to 3′ 27″; and −2 from 3′ 27″ to 34′ 23″. EX.—Find sine 6′. Placing the significant mark for minutes coincident with 6, the value opposite the index is found to be 175, and by the rule above this is to be read 0·00175. For angles in seconds the other significant mark is used; while angles expressed in minutes and seconds are to be first reduced to seconds. Thus, 3′ 10″ = 190″. _Tangents of Angles._—There remains to be considered the third scale found on the back of the slide, and usually distinguished from the others by being lettered T. In most of the more recent forms of rule this scale is placed near the lower edge of the slide, but in some arrangements it is found to be the centre scale of the three. Again, in some rules this scale is figured in the same direction as the scale of sines—viz., from left to right,—while in others the T scale is reversed. In both cases there is now usually an aperture formed in the back of the left extremity of the rule, with an index mark similar to that already referred to in connection with the scale of sines. Considering what has been referred to as the more general arrangement, the method of determining the tangents of angles may be thus explained:— The tangent scale will be found to commence, in some rules, at about 34′, or, precisely, at the angle whose tangent is 0·01. More usually, however, the scale will be found to commence at about 5° 43′, or at the angle whose tangent is 0·1. The other extremity of the scale corresponds in all cases to 45°, or the angle whose tangent is 1. This explanation will suggest the method of using the scale, however it may be arranged. If the graduations commence with 34′, the T scale is to be used in conjunction with the right and left scales of A; while if they commence with 5° 43′ it is to be used in conjunction with the D scale.