A strange extended Orb of Joy
Proceeding from within,
Which did on ev'ry side display
Its force; and being nigh of Kin
To God, did ev'ry way
Dilate its Self ev'n instantaneously,
Yet an Indivisible Centre stay,
In it surrounding all Eternity.
'Twas not a Sphere;
Yet did appear
One infinite: 'Twas somewhat everywhere.'
Observe the distinct description of how the relation between circumference and centre is inverted by the former becoming itself an 'indivisible centre'. In a space of this kind there is no Here and There, as in Euclidean space, for the consciousness is always and immediately at one with the whole space. Motion is thus quite different from what it is in Euclidean space. Traherne himself italicized the word 'instantaneous', so important did he find this fact. (The quality of instantaneousness - equal from the physical point of view to a velocity of the value âž - will occupy us more closely as a characteristic of the realm of levity when we come to discuss the apparent velocity of light in connexion with our optical studies.)
By thus realizing the source in man of the polar-Euclidean thought-forms, we see the discovery of projective geometry in a new light. For it now assumes the significance of yet another historical symptom of the modern re-awakening of man's capacity to remember his prenatal existence.
We know from our previous studies that the concept of polarity is not exhausted by conceiving the world as being constituted by polarities of one order only. Besides primary polarities, there are secondary ones, the outcome of interaction between the primary poles. Having conceived of Point and Plane as a geometrical polarity of the first order, we have therefore to ask what formative elements there are in geometry which represent the corresponding polarity of the second order. The following considerations will show that these are the radius, which arises from the point becoming related to the plane, and the spherically bent surface (for which we have no other name than that again of the sphere), arising from the plane becoming related to the point.
In Euclidean geometry the sphere is defined as 'the locus of all points which are equidistant from a given point'. To define the sphere in this way is in accord with our post-natal, gravity-bound consciousness. For in this state our mind can do no more than envisage the surface of the sphere point by point from its centre and recognize the equal distance of all these points from the centre. Seen thus, the sphere arises as the sum-total of the end-points of all the straight lines of equal length which emerge from the centre-point in all directions. Fig. 8 indicates this schematically. Here the radius, a straight line, is clearly the determining factor.
We now move to the other pole of the primary polarity, that is to the plane, and let the sphere arise by imagining the plane approaching an infinitely distant point evenly from all sides. We view the process realistically only by imagining ourselves in the plane, so that we surround the point from all sides, with the distance between us and
the point diminishing gradually. Since we remain all the time on the surface, we have no reason to conceive any change in its original position; that is, we continue to think of it as an all-embracing plane with regard to the chosen point.
The only way of representing the sphere diagrammatically, as a unit bearing in itself the character of the plane whence it sprang, is as shown in Fig. 9, where a number of planes, functioning as tangential planes, are so related that together they form a surface which possesses everywhere the same distance from the all-relating point.
Since Point and Plane represent in the realm of geometrical concepts what in outer nature we find in the form of the gravity-levity polarity, we may expect to meet Radius and Sphere as actual formative elements in nature, wherever gravity and levity interact in one way or another. A few observations may suffice to give the necessary evidence. Further confirmation will be furnished by the ensuing chapters.