There can be no question, in a short, elementary statement such as this, of doing justice to all the work which has been done on the basis of this new mathematical development. Published work has been primarily in plant morphology and metamorphosis, related also to the work of Goethe. There is abundant evidence to suggest that the form-giving life of Nature is determined not only in the Euclidean universal space, in which matter qua matter (as it is usually understood) is at home, but also by the polar opposite type of space formation. In respect of time-duration, and in respect of “one” and “many,” this other kind of space plays quite a different role. It is a type of space-formation, not a single universal space given once and for all. Spaces of this kind come into being and pass away again with the life-cycles of living creatures or of their several organs. Wherever, in effect, there is a living seed, a germinating point, a special focus of life or growth —whether within the watery substance of a living body or hovering just outside it as at the growing point of the higher plant—there we may look for the “infinitude within” of such a “time-space.” We find evidence of the planar formative activity around the point in question, or in the gesture of the leaf-like organs that envelop it and thence unfold. The higher plant creates its living, counter-Euclidean spaces as it develops, and grows from thence outward into three-dimensional Earth space.

The concept of the two types of space also requires another approach to the concept of forces. Here, too, the primary polarity of space has been the guide in George Adams’ work on the planar counter-Euclidean aspect of force-fields. The idea of force in classical Newtonian mechanics (no longer, indeed, the only idea of what a force can be) is, if we reflect, in harmony with the pure geometry and kinematics of the space of Euclid. The primary characteristic of this kind of force is that in its spatial activity it is directed along a line, from point to point. We may describe the typical forces of the inorganic world as “centric forces”—forces working from center to center, that is, from point to point along the line that joins them. The archetypal instance of such a force is gravity; allied thereto are all the characteristic forces of pressure and contraction.

What kind of “force,” then, will be at work in the negative-Euclidean realm? The clear conclusion is that the primary force of such a space will be levitational, suctional, planar. The balanced duality of spatial theory will express itself also in the organic balance of a living form.

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In our times, it is of paramount importance to transcend the idea of a one-fold, point-centered, material world, in which man (one of many similar units) lives his life automatically accumulating substance and losing it again. The material world of Euclidean space is, after all, not the only aspect of the universe, though perhaps of necessity man has had to become so deeply immersed in this aspect that he has not been able to see clearly beyond its limitations. As we have seen here, in the perspective of projective geometry, the universe may more adequately be seen as three-fold, material and spiritual, with life sustained between these polar realms. A truer picture of such a cosmos reveals polarities which interplay and interweave to create the diverse forms in nature.

When man in his conscious activity of thinking has taken a more profound step in the understanding of polarity, as distinct from mere contrast, he will come to a creative and fundamental use of the imagination in many fields. Contrasts such as expansion and contraction in physical space take on quite another aspect in the realm of the living. To the extent to which we can learn to understand the laws of the interweaving polarities and how to put them into practice, we shall perhaps be enabled to sort out the complicated tangle of modern life.

Current methods of investigation into substances and the forces to which they are subject do not yet gain access to the whole of the living process; biochemistry and biophysics lean heavily on concepts which rest on quantitative mathematics and apply in the physical-mechanical realm. The impetus given by a quantitative mathematics has led to the development of a quantitatively minded world; it is an essential task for the future to develop the qualitative aspect of mathematics, so that the generations to come may in time achieve a true science of the living, conscious aspect of the world.

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Olive Whicher, who is a member of the faculty of Emerson College in Sussex, England, where she lectures and gives practical courses in Projective Geometry and Plant Morphology, worked for twenty-eight years with George Adams. She is a co-author with him of several books on plant morphology, and has written a book on Projective Geometry which has been published in German and English. Ms. Whicher offers the following Selected Bibliography for those who would like to pursue further the study of projective geometry introduced in her article:

  • Adams, George, Physical and Ethereal Spaces, London 1965; Von dem Aetherischen Raume, Stuttgart 1964; Universalkrafte in der Mechanik, Dornach, 1973.
  • Adams and Whicher, Die Pfianze in Raum und Gegen-raum, Stuttgart 1960.
  • Whicher, Olive, Projective Geometry, London 1971; Pro-jektive Geometric Stuttgart 1970.
  • Locher-Ernst, Louis, Projektive Geometric Zurich 1940; Raum und Gegenraum, Dornach 1957.
  • Steiner, Rudolf. The references are to many books and lectures, but three fundamental works are: — Philosophy of Freedom, London 1964; The Theory of Knowledge Implicit in Goethe’s World-Conception, New York, 1968; Riddles of Philosophy,Spring Valley, N.Y., 1973.
  • Unger, Georg, Grundbegriffe der modernen Physik (Teil III), Stuttgart 1967.
  • Lehrs, Ernst, Man and Matter, London 1958; M.ensch und Materie, Frankfurt 1966.