Dr, Wollaston's Synoptic Scale 
the mode in which the scale of equivalents is constructed, will, 
I presume, be acceptable. 
They will observe, that the series of natural numbers are 
not placed at equal intervals on the scale; but that at all equal 
intervals are found numbers which bear the same proportion 
to each other. In fig. 3, some of the larger intervals alone 
are represented on a line similarly divided. The succession 
of intervals, marked A, B, C, D, E, are all equal, and at these 
points of division are placed numbers 1, 2, 4, 8, ib, which 
increase progressively by the same ratio. And since the series 
3 : 6 : 12 : 24, increase in the same ratio of 1 to 2, these in- 
tervals dy by Cy dy 6 y tile samc as the former. At another 
succession of different yet equal intervals, marked F, G, H, I, 
are placed numbers 1,3, 9, 27, which increase regularly by 
an equal ratio of 1 to 3 ; and by means of a pair of compasses 
it would be found that the interval from 2 to b, or from 6 to 
i8 (which are in the same ratio of 1 to 3,) is exactly equal 
to FG, the interval between 1 and 3. As any single space 
represents any one ratio, so the sum of any 2 or 3 equal spaces 
represent a double or triple ratio. If 1 be increased three 
« 
times by the ratio of 1 to 2, it becomes 8, which bears to 1 
triple the ratio of 2 to 1. This ratio is therefore rightly repre- 
sented bv AD, which is the triple of AB. 
The distances of the intermediate numbers 5, 7, 10, 11, 13, 
&c. from 1 are likewise made proportional to the ratios which 
they bear to 1, and are easily laid down by means of a table 
of logarithms ; for as these are arithmetic measures of the 
ratios which all numbers bear to unity, the spaces propor- 
tional to them become linear representations of the same 
quantities. 
