VOL. LXXVII.] PHILOSOPHICAL TRANSACTIONS. 263 



the same ratio. 6. Suppose a logarithmic line to contain no more than a single 

 range of numbers from 1 to 10, it will not be necessary, for the purposes of 

 computation, to repeat it; for if a slider or beam have 2 fixed points at the dis- 

 tance of the interval between 1 and 10, and a moveable point be made to range 

 between these (always to indicate the antecedent); then, if the consequent fixed 

 point fall without the rule, the other fixed point will show the division it would 

 have fallen on, if the rule had been prolonged. This may be easily applied to 

 the arrangement described, N° 4. T . If the arrangement consist only of the 

 logarithms from 1 to 10, and the parallel cross lines intersect that geometrical 

 series whose successive ratios altogether, with that of the last term to 10 times 

 the first, make by composition of the ratio -jV, the contrivance, N° 6, may be 

 applied to show such consequents as fall laterally without the rule. 8, It is 

 convenient that the arrangement of the lines be disposed so as to occupy a rect- 

 angular parallelogram ; or, in other words, that the cross line, cutting the series 

 last-mentioned, may be at right angles to the length of the rule. 



The construction of an instrument on the foregoing principles admits of various 

 dispositions of the graduated lines and apparatus for measuring intervals on them. 

 Mr. N. here gives examples of several different forms: one is a rule consisting of 

 10 parallel lines, equivalent to a double line of numbers upwards of 20 feet in 

 length. Another is a beam compass for measuring intervals. 



A 3d is a Gunter's scale, equivalent to that of 28^ inches in length, published 

 by the late Mr. Robertson. It is however but \ of the length, and contains 

 only 4- of the quantity of division. In the slider gh, fig. 10, pi. 1, is a move- 

 able piece AB, across which a fine line is drawn; and there are also lines cd, ef, 

 drawn across the slider, at a distance from each other equal to the length of the 

 rule. The line cd or ef is to be placed at the consequent, and the line in the 

 piece AB at the antecedent: then, if the piece ab be placed at any other ante- 

 cedent, the same line cd or ef will indicate its consequent in the same ratio 

 taken the same way; that is, if the antecedent and the consequent lie on the 

 same side of the slider, all other antecedents and consequents in that ratio will 

 lie in the same manner, and the contrary if they do not, &c. But if the con- 

 sequent line fall without the rule, the other fixed line on the slider will show the 

 consequent; but on the contrary side of the slider to that where it would else 

 have been seen by means of the first consequent' line. 



Fig. 11 is an instrument equivalent to the same rule of 28i inches long. It 

 consists of 3 concentric circles engraved and graduated on a plate of about I-l 

 inch in diameter. From the centre proceed two legs a, b, having rigiit-lined 

 edges in the direction of radii They are moveable either singly or together. 

 To use this instrument, place one of the edges at the antecedent, and the other 

 at the consequent, and fix them to that angle. The 2 legs being then moved 



