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XLV. Description of a new Proportion-Table equivalent to a 

 Sliding -Ride 13 feet 4i inches long. By J. D. Everett, D.C.L., 

 Assistant Professor of Mathematics in Glasgow University*. 



A LOGARITHMIC scale is a scale of divisions with num- 

 bers written against them or otherwise indicated, charac- 

 terized by the property that equal distances between divisions 

 correspond to equal ratios between the corresponding numbers. 

 With the aid of such a scale, a fourth proportional to any three 

 numbers can be found by measuring the distance between the 

 first and second, and setting off this distance from the third. 



The common sliding-rule consists essentially of two such scales, 

 which are perfect fac-similes of each other, one of them being on 

 the solid wood of the rule, and the other on a slidingpiece which 

 moves beside it. In order to find a fourth proportional by means 

 of this instrument, it is only necessary to move the slide until the 

 first and second terms are opposite one another, and the fourth 

 term will then be found opposite the third. Multiplication and 

 division can of course be performed in this way, since they are 

 reducible to proportions in which one term is unity. 



Each of the two scales begins with 1 and ends with 100 ; and 

 the first half, which extends from 1 to 10, is precisely similar to 

 the second half, which extends from 10 to 100 ; for the numbers 

 1, 2, 3, &c. having the same ratios to one another as 10, 20, 30, 

 &c, must be placed at the same distances apart as these latter. 

 In general the division which stands for any one number may 

 also stand for the product or quotient of this number by any 

 power of 10. Each of the two scales may therefore be said to 

 contain all numbers twice over, this repetition being adopted in 

 order to avoid passing out of range when the slide is moved from 

 its initial position. 



There is another form in use, consisting of two concentric 

 circles with logarithmic scales placed round their circumferences, 

 the inner circle being constructed to turn about the common 

 centre. In this form of the slide-rule there is no necessity for 

 duplicates, as passing out of range is impossible ; and the com- 

 plete scale from 1 to 10 is made to extend once round each cir- 

 cumference, the division for 10 being identical with that for 1. 



This arrangement effects so great an economy of space as com- 

 pared with the straight sliding-rule, that it is compatible with 

 the adoption of a much larger, scale. In fact the circumferences 

 of the circles may conveniently be made as much as 2 feet, which 

 implies the same size of scale as in a straight rule 4 feet long. 

 This advantage, however, is counterbalanced by some drawbacks, 

 of which perhaps the most serious is the position in which the 



* Communicated by the Author, having been read at the Meeting of the 

 British Association at Nottingham. 



