.22-85, the truth being 22'S. Again, we find 628 (fixed) just over 275 (slide), the S being estimated ; hence, by the smiles we learn that the truth being 2.2836. Thirdly, wo estimate that 1725 (fixed) falls over 757 (slide), and that 276 (fixed) falls over 121 (slide). That is, the ruler informs us that 1725 : 757 : 276 : 121 ; the fourth term should be 121:12, as found by com putation. We take a larger scale, having n inches of radius, and setting 1 on the slide to 228.5, we find 1725 appears to fall over what we should judge to be 756 rather than 757. Now 1725+757= and Both give on the scales 2.285, so that the advantage is slightly in favour of the larger scale, but not so much as we should have expected. We now try one of 24 inches radius, and setting 1 on the slide to 2.285 on the fixed ruler, we find that 1725 (fixed) falls over 7541 (slide), the last 7 being estimation. Now which gives the advantage again (but not so decidedly as might have been expected) to the large rile. The fact is, that it is rather ease than proportionate accuracy which is gained by the large rules : the preceding results required care and close attention on the 54-inch nape ; were obtained with moderate care on the 71-inch ; and taken off instantly from the 24-inch rule. More. over, divisions on wood, made in the usual way, do not allow accu racy to increase with the size : if these rulers were divided on braas, and with the precautions taken in astronomical instruments, it wool.] be a very different thing but after all, the wonder is that the common wooden rules should be so accurate as they are.
The next step in the description is as follows :-It matters nothing whether the second scale be really made consecutive with the first, or occupy any other part of space : provided that when 1 and 1 are brought together on the first scale, 1 and 1 also come together on the second, and that the first slide and its continuations slide equally. We see this in the diagram before us : a b is one slide, and A 11 are two rulers on opposite sides of the groove. 'When a b is pushed home, A and a present coinciding scales, as don and b : we should rather say. that the last is not one scale, but the and of one and the beginning of another ; the 1 of B and b being in the middle. The consequence is, that so long as 1 of the scale b is not pushed out so far as to fall out of the groove (which is never necessary, since there is a whole scale on a), there is always the power of reading esry result of the multipli cation in hand. In the diagram, 1 on b is pushed out to 2 on B, and on the upper scales (A and a) we see 2 x 2=4, 2 x 3=6, 2 x 4 = 8, 2 x 5=10 ; on the lower (B and 6), 2 x 4=8, 2 x 5=10, 2 x 6=12, 2 x 7=14, 2 x S=16, 2 x 9=18, 2 x 10=20. This modification was invented by Mr. Silvanus Bet-an (Nicholson 's 'Journal,' vol. xlix. ,p. 187) ; but thirty years before this Mr. Nicholson (' Phil. Trans.', 1787, p. 216)
had pointed out how to divide the whole radius into four parts, two on each face.
A simple plan, and in some respects the .best, is to make a revolving circle turn upon a fixed one, in which ease the scale is its own continua tion, as in the following diagram. The two circles have a common In the cut before mitre have the 1 of the slide placed at 2 of tho fixed viler; consequently 6 on the slide comes under what would be 12 of the fixed ruler if the secondary graduations were inserted. Again, 4 comes over 2, and 9 over 45, giving 4 : 2 : 9 : the decimal point being inserted by intuition. To show the sort of results which we obtain from such a slide of 51-inches radius (or from 1 to 1), we take one of this sort, and throw 1 of the slide at hazard between 225 and 230 on the fixed nuler, a little farther to the right than it is on the preceding diagram; guessing at the interval, it Hems) 228.5. We detect it more exaotly by looking at 5 on the slide, which is hardly visible in advance of 114 on the reale. As far then as the divisions, aided by our jtulgment of this interval, inform as, we lmve 114+5 pivot, and the upper one turns round on the lower; the rim of the inner circle being bevelled down to the plane of the lower. A com plete logarithmic scale is marked on each circumference, and it will readily be seen that the scales are placed so as to point out multiplier). tions by 2, as in the former instances, and also thrit the recommence ment of the scale begins its continuation. Instead of two circles there might ho two thin cylinders, turning on a common axis, the graduations being made on the rim. Thirty years ago, an instrument-maker at Paris laid down logarithmic scales on the rims of the box and lid of a common circular snuff-box : one of two inches diameter would be as good an aid to calculation as the common engineer's rule. But either calculators disliked snuff, or mnuffitakers calculation, for the scheme was not found to answer, and the apparatus was broken up.
The form first proposed by Oughtred (presently to be mentioned was a modification of the preceding. Instead of two circles, tw( pointing radii were attached to the centre of one circle, on which number of concentric circles were drawn, each charged with logarithmic scale. These pointers would either move round together united by friction, or open and shut by the application of pressure they were in fact a pair of compasses, laid flat on the circle, with thei] pivot at its centre. Calling these pointers antecedent and consequent to multiply A and n the consequent arm must be brought to point t( 1, and the antecedent arm then made to point to A. If the pointers be then moved together until the consequent arm points to is, the ante cedent arm will point to the product of A and n.