12 REPORT—1883. 
experience (our own Euclidian geometry) would cease to be applicable; 
but the change could not be apprehended by them as a bending or deform- 
ation 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 n-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 equa- 
tion in the form f(#)=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 ‘ Géometrie,’ 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 ought to have » roots. But it is by no means true that 
there are always » 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 7 of the quadric equation a? + 1 = 0, then the other 
root is = —7; and it is easily seen that every quadric equation (with 
real coefficients as before) has two roots, a + bi, where a and D are real 
magnitudes. We are thus led to the conception of an imaginary magni- 
tude, a + bi, where a and b are real magnitudes, each susceptible of any 
positive or negative value, zero included. The general theorem is that, 
taking the coefficients of the equation to be imaginary magnitudes, then 
an equation of the order m has always n roots, each of them an imaginary 
magnitude, and it thus appears that the foregoing form a + bi of imagi- 
nary magnitude is the only one that presents itself. Such imaginary 
a a 
