Our space conceptions are so intimately connected with those which we derive from the existence of gravitation that it is difficult to realise the condition of a plane being, without picturing him as in material surroundings with a definite direction of up and down. Hence the necessity of our somewhat elaborate scheme of representation, which, when its import has been grasped, can be dispensed with for the simpler one of a thin object slipping over a smooth surface, which lies in front of us. It is obvious that we must suppose some means by which the plane being is kept in contact with the surface on which he slips. The simplest supposition to make is that there is a transverse gravity, which keeps him to the plane. This gravity must be thought of as different to the attraction exercised by his matter, and as unperceived by him. At this stage of our enquiry I do not wish to enter into the question of how a plane being could arrive at a knowledge of the third dimension, but simply to investigate his plane consciousness. It is obvious that the existence of a plane being must be very limited. A straight line standing up from the surface of his earth affords a bar to his progress. An object like a wheel which rotates round an axis would be unknown to him, for there is no conceivable way in which he can get to the centre without going through the circumference. He would have spinning disks, but could not get to the centre of them. The plane being can represent the motion from any one point of his space to any other, by means of two straight lines drawn at right angles to each other. Let AX and AY be two such axes. He can accomplish the translation from A to B by going along AX to C, and then from C along CB parallel to AY. The same result can of course be obtained by moving to D along AY and then parallel to AX from D to B, or of course by any diagonal movement compounded by these axial movements. [Illustration: Fig. 5.] By means of movements parallel to these two axes he can proceed (except for material obstacles) from any one point of his space to any other. If now we suppose a third line drawn out from A at right angles to the plane it is evident that no motion in either of the two dimensions he knows will carry him in the least degree in the direction represented by AZ. [Illustration: Fig. 6.] The lines AZ and AX determine a plane. If he could be taken off his plane, and transferred to the plane AXZ, he would be in a world exactly like his own. From every line in his world there goes off a space world exactly like his own. [Illustration: Fig. 7.]