The fourth dimension

THE USE OF FOUR DIMENSIONS IN THOUGHT 89

it is worth while to bring into the full light of our attention our habitual assumptions and processes. It often happens that we find there are two of them which have a bearing on each other, which, without this dragging into the light, we should have allowed to remain without mutual influence.

There is a fact which it concerns us to take into account in diseussing the theory of the poiograph.

With respect to our knowledge of the world we are far from that condition which Laplace imagined when he asserted that an all-knowing mind could determine the future condition of every object, if he knew the co-ordinates of its particles in space, and their velocity at any particular moment.

On the contrary, in the presence of any natural object, we have a great complexity of conditions before us, which we cannot reduce to position in space and date in time.

There is mass, attraction apparently spontaneous, electrical and magnetic properties which must be superadded to spatial configuration. To cut the list short we must say that practically the phenomena of the world present us problems involving many variables, which we must take as independent.

From this it follows that in making poiographs we must be prepared to use space of more than three dimensions, If the symmetry and completeness of our representatation is to be of use to us we must be prepared to appreciate and criticise figures of a complexity greater than of those in three dimensions. It is impossible to give an example of such a poiograph which will not be merely trivial, without going into details of some kind irrelevant to our subject. I prefer to introduce the irrelevant details rather than treat this part of the subject perfunctorily,

To take an instance of a poiograph which does not lead