92 
MR. DAVIES GILBERT ON THE NATURE 
to the same quantity taken in opposite directions, rather than the usual ones 
of plus and minus. 
In the next place, I believe that the law of the signs has never been stated 
according to its full generalization. 
In common language, and for ordinary purposes, multiplication is consi- 
dered as an abbreviated addition ; but it would be a superfluous waste of time 
to demonstrate before this Society that multiplication is always an affair of 
ratios. Length and breadth multiplied together give areas, because an unit of 
length by an unit of breadth has previously been established as the superficial 
unit. Length, width, and depth give solidity for the same reason, and from the 
want of such a preestablished unit, arises the utter absurdity of a question, here- 
tofore proposed in various treatises on arithmetic, for multiplying some deno- 
mination of coin by itself, and ascertaining the product. 
When a multiplication of two quantities is therefore to be made, unity must 
be understood as the antecedent ; but here an extraneous limitation insinuates 
itself unperceived. The common antecedent taken in usual practice is not 
simply an unit, but unity in the scale of (a). With this limitation the law of 
the signs is correct, namely that like signs produce (a), and unlike signs pro- 
duce ( b ). But let unity, the common antecedent, be taken in the scale of (b ) ; 
the law will then immediately change to like signs producing ( b ), and unlike 
signs producing (a). 
The general rule therefore must be, that like signs give the sign of the 
assumed universal antecedent, and unlike signs the contrary. 
Admitting, therefore, that both scales are in themselves equally affirmative, 
and that either may be taken as negative to the other, it necessarily follows 
that by using the scale of (b), and consequently by assuming the unit of that 
scale as the universal antecedent, all even roots in the scale of (a) will become 
imaginary ; and thus the apparent discrimination of the two scales is entirely 
removed: and in the same way, and by varying the signs according to the scale 
in which the universal antecedent is taken, all formulae will become equally 
applicable to both. 
For example : (See plate III.) 
The natural numbers and their logarithms will be expressed for both scales 
by the correlative curves in the following figures, where all ordinates taken 
