When we set out earlier in this book (Chapter VIII) to discover the source of Galileo's intuition, by which he had been enabled to find the theorem of the parallelogram of forces, we were led to certain experiences through which all men go in early childhood by erecting their body and learning to walk. We were thereby led to realize that man's general capacity for thinking mathematically is the outcome of early experiences of this kind. It is evident that geometrical concepts arising in man's mind in this way must be those of Euclidean geometry. For they are acquired by the will's struggle with gravity. The dynamic law discovered in this way by Galileo was therefore bound to apply to the behaviour of mechanical forces - that is, of forces acting from points outward.

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.

We shall now indicate some of the lines of geometrical thought which follow from this.

In a similar way we can now seek to find the source of our capacity to form polar-Euclidean concepts. As we were formerly led to experiences of man's early life on earth, so we are now led to his embryonic and even pre-embryonic existence.

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.

Let us consider a straight line extending without limits in either direction. Projective geometry is able to state that a point moving along this line in one direction will eventually return from the other. To see this, we imagine two straight lines a and b intersecting at P. One of these lines is fixed (a); the other (b) rotates uniformly about C. Fig. 7 indicates the rotation of b by showing it in a number of

positions with the respective positions of its point of intersection with a (P₁, P₂. . .). We observe this point moving along a, as a result of the rotation of b, until, when both lines are parallel, it reaches infinity. As a result of the continued rotation of b, however, P does not remain in infinity, but returns along a from the other side. We find here two forms of movement linked together - the rotational movement of a line (b) on a point (C), and the progressive movement of a point (P) along a line (a). The first movement is continuous, and observable throughout within finite space. Therefore the second movement must be continuous as well, even though it partly escapes our observation. Hence, when P disappears into infinity on one side of our own point of observation, it is at the same time in infinity on the other side. In order words, an unlimited straight line has only one point at infinity.

Before man's supersensible part enters into a physical body there is no means of conveying to it experiences other than those of levity, and this condition prevails right through embryonic development. For while the body floats in the mother's foetal fluid it is virtually exempt from the influence of the earth's field of gravity.

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.

It is clear that, in order to become familiar with this aspect of geometry, one must grow together in inward activity with the happening which is contained in the above description. What we therefore intend by giving such a description is to provide an opportunity for a particular mental exercise, just as when we introduced Goethe's botany by describing a number of successive leaf-formations. Here, as much as there, it is the act of 're-creating' that matters.

History has given us a source of information from these early periods of man's existence in Traherne's recollections of the time when his soul was still in the state of cosmic consciousness. Among his descriptions we may therefore expect to find a picture of levity-space which will confirm through immediate experience what we have arrived at along the lines of realistic mathematical reasoning. Among poems quoted earlier, his The Praeparative and My Spirit do indeed convey this picture in the clearest possible way. The following are relevant passages from these two poems.