BRITISH ASSOCIATION, SEPTEMBER 1883. 
437 
784] 
beings living in a one-dimensional space (a line) or in a two-dimensional space (a 
surface), and conceiving of space accordingly, and to whom, therefore, a two-dimensional 
space, or (as the case may be) a three-dimensional space would be as inconceivable 
as a four-dimensional space is to us. And very curious speculative questions arise. 
Suppose the one-dimensional space a right line, and that it afterwards becomes a 
curved line: would there be any indication of the change? Or, if originally a curved 
line, would there be anything to suggest to them that it was not a right line? 
Probably not, for a one-dimensional geometry hardly exists. But let the space be 
two-dimensional, and imagine it originally a plane, and afterwards bent or converted 
into a curved surface (converted, that is, into some form of developable surface): 
or imagine it originally a developable or curved surface. In the former case there 
should be an indication of the change, for the geometry originally applicable to the 
space of their experience (our own Euclidian geometry) would cease to be applicable; 
but the change could not be apprehended by them as a bending or deformation of 
the plane, for this would imply the notion of a three-dimensional space in which 
this bending or deformation could take place. In the latter case their geometry 
would be that appropriate to the developable or curved surface which is their space: 
viz. this would be their Euclidian geometry: would they ever have arrived at our 
own more simple system ? But take the case where the two-dimensional space is a 
plane, and imagine the beings of such a space familiar with our own Euclidian plane 
geometry; if, a third dimension being still inconceivable by them, they were by their 
geometry or otherwise led to the notion of it, there would be nothing to prevent 
them from forming a science such as our own science of three-dimensional geometry. 
Evidently all the foregoing questions present themselves in regard to ourselves, 
and to three-dimensional space as we conceive of it, and as the physical space of 
our experience. And I need hardly say that the first step is the difficulty, and that 
granting a fourth dimension we may assume as many more dimensions as we please. 
But whatever answer be given to them, we have, as a branch of mathematics, 
potentially, if not actually, an analytical geometry of w-dimensional space. I shall have 
to speak again upon this. 
Coming now to the fundamental notion already referred to, that of imaginary 
magnitude in analysis and imaginary space in geometry : I connect this with two 
great discoveries in mathematics made in the first half of the seventeenth century, 
Harriot’s representation of an equation in the form f (x) = 0, and the consequent 
notion of the roots of an equation as derived from the linear factors of f (x), 
(Harriot, 1560—1621 : his Algebra, published after his death, has the date 1631), and 
Descartes’ method of coordinates, as given in the Géométrie, forming a short supplement 
to his Traité de la Méthode, etc., (Leyden, 1637). 
Taking the coefficients of an equation to be real magnitudes, it at once follows 
from Harriot’s form of an equation that an equation of the order n ought to have 
7i roots. But it is by no means true that there are always n real roots. In particular, 
an equation of the second order, or quadric equation, may have no real root; but 
if we assume the existence of a root i of the quadric equation x 2 + 1 = 0, then the