1878.] 
On Space of Four Dimensions . 
231 
cord describe a line of double curvature, as shown by this 
figure : — 
We three-dimensional beings can only tie or untie such a 
knot by moving one end of the cord through 360° in a plane 
which is inclined towards that other plane containing the 
two-dimensional part of the knot. But if there were beings 
among us who were able to produce by their will four- 
dimensional movements of material substances, they could 
tie and untie such knots in a much simpler manner by an 
operation analogous to that described in relation to a two- 
dimensional knot. 
It is by no means necessary — nay, not even probable — 
that such beings should have a contemplative consciousness 
of these actions of their will. For all our conceptions in 
relation to the movements of our limbs, and to those pro- 
duced by their means in other bodies, have been acquired by 
us solely by way of experience. Having observed from child- 
hood that a voluntary movement of our limbs is always 
connected with a corresponding change in our visional im- 
pressions, accompanying the adtion of our will, it is only in 
this way that we are now able to connect the movements of 
our body or of other objects with a corresponding conception 
of such motion. 
Berkeley demonstrated this truth in the year 1709 in his 
“ Essay Towards a New Theory of Vision ” and in his 
“ Principles of Human Knowledge.” In the last-mentioned 
treatise he remarks, on the relation of our visional percep- 
tions to the sensations of touch : — - 
“ So that in strict truth the ideas of sight, when we 
apprehend by them distance, and things placed at a 
distance, do not suggest or mark out to us things 
actually existing at a distance, but only admonish us 
what ideas of touch will be imprinted in our minds at 
such and such distance of time, and in consequence 
of such or such actions.” — Berkeley, Principles of 
Human Knowledge, (Fraser’s Edition, vol. i., p. 177.) 
Lichtenberg, in 1799, expresses himself in like manner 
when he says : — 
“To perceive something outside ourselves is a contradic- 
tion ; we perceive only within us ; that which we 
perceive is merely a modification of ourselves, there- 
fore, within us. Because these modifications are inde- 
pendent of ourselves, we seek their cause mother things 
that are outside, and say, there are things beyond us . 
