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V. On the Nature of Negative and of Imaginary Quantities. 
By Davies Gilbert, Esq. P.R.S. 
Read November 18, 1830. 
I AM desirous of submitting to the Royal Society some observations on the 
nature of what are termed Negative and Imaginary Quantities, tending as I 
hope to clear away the obscurity that has hitherto surrounded them. 
The subject has occupied my attention for many years, and however plain 
and simple may be the results, they have not been obtained without much 
patient investigation ; and, in the event of their being found correct, they will 
add one authority more to an observation frequently made, and confirmed by 
extensive experience, — That paradoxes and apparent solecisms, involving them- 
selves with facts or with deductions known to be true, may always be found 
near the surface, owing their existence either to ambiguities of expression or 
to the unperceived adoption of some extraneous additions or limitations into 
the compound terms used for definition, which are subsequently taken as con- 
stituent parts of their essence. 
The first misapprehension appears to consist in our considering any quan- 
tity whatever as negative per se, and without reference to another opposed to 
it, which has previously been established as positive. 
In applying our arithmetic to whatever is continuous, some neutral point or 
zero must be selected, as in time, in space with reference to its three dimen- 
sions, in forces, in velocities ; and the opposite directions from this point will 
be mutually negative in respect to each other, and must be distinguished by 
appropriate marks or signs. But space to the left is no more essentially nega- 
tive than space to the right, nor descent than ascent, nor time past than time 
that is to come. 
I would therefore adopt for the present investigation, and to avoid pre- 
viously formed association of ideas, (a) and ( b ) as marks or signs for prefixing 
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