By: George Adams, MA (Cantab)
Reprinted by permission from the British Homeopathic Journal Vol.78 No.2 April 1989
doi: 10.1016/S0007-0785(89)80051-1

INTRODUCTION

Projective Geometry and Amnesia
One of the minor pleasures of studying homeopathy is its sense of his­tory, which contrasts so sharply with the ahistoricity of mainstream medicine. Most doctors feel that there was no real medicine before the dis­covery of Penicillin (but this is little more than a feeling, for there is virtually no teaching of medical history in medical schools). Before Penicillin all seems to have been darkness, pierced only by an occasional brilliant shaft of light associated with a great name—a Harvey, Virchow or Pasteur—but since 1940 all is clarity and reason. This is, of course, a highly distorted image.

In homeopathy, we have a much greater sense of continuity, indeed we rest too much on our laurels, accepting far too readily the opinions of famous teachers of the past. Yet while every word of Hahnemann or Kent is treated with exaggerated reverence, other important historic discoveries originating in homeopathy are almost forgotten. Hering it was who intro­duced nitrates into medicine (Glonoine)—a fact which was recalled recently in the journal Circulation, but almost forgotten by his heirs in homeopathy. Reilly, in researching his recent work on hayfever, discovered that hayfever was first correctly attributed to pollen allergy by Blackleg, a British homeopath.

Many other episodes of intellectual amnesia among homeopaths could be cited. This seems to be mainly a short-term memory loss; more recent contributions are less likely to be remembered than older ones! It is for this reason that I make no apology for reprinting, from time to time, classical but neglected pieces of work. The paper which follows, "Potentization and the peripheral forces of nature" by George Adams, is based on a lecture given at the 1961 British Homeopathic Congress. To judge from the congress report, and the recollections of those who were present, it aroused great excitement at the time. Certainly it has important implications for the nature of extreme dilutions, implications which are not widely recognized, and have not been developed, but instead have fallen victim to our collective short-term memory loss.

—Peter Fisher, MB, Hon. editor of the British Homeopathic Journal.

 

Dr. Twentyman, Ladies and Gentlemen:

May I begin by saying that I feel it a great privilege and satisfaction to be invited as a layman to address this Congress. My theme will be to tell of new ideas and discoveries—well founded, though still in their initial stages—which, among other things, should contribute to the long desired scientific explanation of the effectiveness of high potencies in medicine. Let me remind you to begin with where the difficulty lies. For generations past the effectiveness of high potencies has been a fact of experience for the physician and of untold benefit to countless patients. Also in recent decades, in the work of L. Kolisko.1 Boyd 2 and others, it has been experimentally es­tablished by biological as well as purely physical and chemical reactions. Yet it is difficult to account for, both in the light of rough and ready com­mon sense and of prevailing scientific notions. The chemist who surmises that a particular component present in small quantities in a solution or mix­ture, is responsible for some physical or physiological effect, will contrive by distillation, crystallization or the like to concentrate it. His theory is con­firmed if the effect increases; thus with Madame Curie, when with endless pains she extracted a few grams of radium from tons of pitchblende. Why, in the preparation of homeopathic remedies, do we dilute instead of con­centrating? I am, of course, aware that potencies are no mere dilutions. "Dilution alone," says Hahnemann, "say when a grain of common salt is dissolved, produces the merest water. Diluted with a vast amount of water, the salt simply disappears. This never makes it into a medicine. Yet by our well-prepared dynamizations the medicinal virtue of common salt is wondrously revealed and enhanced."3 Nevertheless, there is no denying that among other things the potentizing or dynamizing process does dilute the substance and in so doing brings forth its virtue. To quote Hahnemann again: "The homeopathic dilution of medicaments brings about no reduc­tion, but on the contrary a true enhancement of their medicinal virtues; thus our dilutions represent a truly wonderful unveiling, nay more, a calling-to-­life of the medicinal and healing spirit of the substance."

The down-to-earth, common sense difficulty of understanding how this can be, is reinforced by the prevailing molecular theories of matter, accord­ing to which the number of molecules in a gram-molecule of any substance is of the order of 10 to the 23rd power. The exact figure, variously known as Avogadro's or Loschmidt's number, has been found consistently by several methods. In terms of molecular theory, therefore, starting with a normal solution and with the normal technique of potentization, by the 23rd or 24th decimal potency only a single molecule would be left, and from then onward it is ever more unlikely that even this will be there in the medicine bottle or am­pule bearing the name of the substance! Ways of escape from this theoretical dilemma have indeed been suggested by the more recent theories of physics. The Nineteenth Century conceived the molecules or their con­stituent atoms more or less naively as ultimate and self-contained pieces of matter. The atoms and subatomic 'particles'—protons, electrons, and so on, in terms of which even the chemical affinities and biological effects of sub­stance are today explained—have become purely ideal entities figuring in recondite mathematical equations. Thinking of the mysterious duality of particle and wave, the philosophically minded physicist can even aver with scientific reason that with its sphere of influence each single atom is co-ex­tensive with the entire universe. Some people therefore pin their hopes on a future science of biophysics in which the subtle influences of life will be illumined by the idealized conceptions of atomic physics. Yet it should not be forgotten that the experiments and discoveries on which the latter are based have been increasingly remote from the realm of living things, depending as they do on the deliberate enhancement of conditions—high values, high-tension electric fields and the resulting radiations and 'bombardments'—downright inimical to life. It is therefore better to regard the apparent gulf between the experience of homeopathic medicine and the conventional scientific outlook in a wider historic setting, not only in terms of the ever-changing theories of twentieth-century physics. .

The growth of physical science from the times of Galileo and Torricelli, Newton, Boyle and Huyghens, Dalton, Lavoisier and Faraday down to the present day is a wonderful chapter in the intellectual and spiritual history of mankind. Hahnemann's long life (1755-1843) spans an important period in this development, leading from the celestial mechanics of the Eighteenth to the electro-magnetic theories and growing chemical discoveries of the Nineteenth Century. Still in his youth when hydrogen and the composition of water are discovered, he is in his prime when Dalton enunciates the atomic theory, Cavendish in 1772 confirms the inverse-square law in electrostatics, Oersted and Ohm make their discoveries on the electric cur­rent in the 1820s, Faraday's electro-magnetic researches culminate in 1831. In 1828 Wohler's synthesis of urea undermines the old vitalist ideas of or­ganic chemistry which Hahnemann—himself a creative chemist—still enter­tained in common with his contemporaries. 

It is well to remember this when reading Hahnemann's forms of expres­sion, which as I shall hope to show are scientifically important to this day. For the vitalism, inevitably abandoned in its old philosophic form, the vagueness of which stood in the way of true research, can now be reborn on a clear and scientific basis. Hahnemann's vitalism underlies his use of the word 'dynamic' and the noun 'dynamis' which he adopts, or coins for him­self. "From the beginning,” says Tischner, “his notion of the vital force prevailing in the living body was essentially spiritual."4 He attributes illnesses to immaterial, dynamic causes, and in his essay of 1801 describes the medicinal effects of high dilutions as 'dynamic' rather than 'atomic'—a con­trast the literal significance of which will, I hope, emerge in the course of this lecture. We also have to remember that the clear distinction of energy and matter and the law of conservation of energy were not yet current in Hahnemann's day. The 'mechanical equivalent of heat' was discovered by Mayer and Joule almost exactly at the time of his death (1842-45). Heat, light and other energies—bio- and psycho- logical as well as physical, even in­cluding 'animal magnetism,' for example—were until then still being thought of as tenuous if not imponderable substances. The supposed sub­stance of warmth was called 'caloric.' Lavoisier in 1789 still included heat and light among the chemical elements. Rumford's experiment was widely supposed to have released the 'caloric' from the iron made hot by friction. Even in 1824, when in his Puissance motrice du feu Carrot in effect discovered the second law of thermodynamics, soon to become a cornerstone of physics, he still interpreted it in terms of 'caloric.' Perhaps this idea of im­ponderable essences is in the light of present-day ideas no longer quite so wide of the mark as it might have seemed sixty years ago. It should at any rate be borne in mind when reading Hahnemann's expressions, when for example he describes as feinstofflich, 'delicately substantial,' or as 'virtual' or 'well-nigh spiritual' the medicinal effects set free from the material during the rhythmic processes of dilution, trituration and succussion.

I have deliberately drawn attention to these aspects. The history of science is not the unidirectional process which neatly finished textbooks lead one to suppose. Many streams run side by side; the most essential dis­coveries, experimental or theoretical, may lie unnoticed for decades till a fresh aspect emerges to reveal their importance.

Let us consider for a moment in a human and historic spirit what it was that gave the orthodox scientific outlook its strength, accounting too for the intolerance with which the claims of homeopathy have only too often been met. It was the combination of an instinctive and robust materialism with the mathematical clarity and cogency of theories supported by experiment and observation. The instinctive materialism is well illustrated by the story of Dr. Johnson's angry reaction after listening to a sermon in which Bishop Berkeley put forward his idealistic theory of the world. 'I refute it thus,' the learned doctor exclaims, kicking his foot against a stone. In scientific atomism until the close of the Nineteenth Century, Johnson's stone—vastly reduced in spatial but proportionately grown in spiritual dimensions—be­came the highly satisfying football, better perhaps the baseball, of science. For it is this intuitive feeling of the ultimate reality of tangible material things which underlies the older forms of scientific atomism. It is a very genuine element in the consciousness of Western man throughout the Seventeenth to Nineteenth Centuries, inseparable from the age of explora­tion, the growth of natural history and of artistic naturalism, the dawn of in­dustrialism. Nor is it out of harmony with the patriarchal, simply believing, strongly Old Testament forms of religion then prevailing.

Yet the instinctive materialism is reinforced by another, more ideal fac­tor—and this alone accounts for the spiritual tenacity of a materialistic science—namely, the confidence born of the intellectual clarity and probity of mathematical thinking. It is too apt to be forgotten how many purely ideal, in other words spiritual, elements are built into the resulting scientific system. Mathematics is an activity of pure thought, and in the past (if not in the extreme formalism and empty nominalism which is now the fashion) was never quite remote from philosophical and even religious thinking. Certainly Isaac Newton, whom we may justly think of as the founder of modem physics, was in his own dominant interests a philosopher, even a theologian, as for example his correspondence with Henry More and the Cambridge Platonists reveals. For all the scientific care and skepticism sin­cerely voiced in his 'Hypotheses non fingo'  he—who was afterwards to describe his Universal Space as 'the sensorium of God'—built into his Prin­cipia, in formal quality if not in intention, an almost theological masonry of thought. The implications of it were but inverted by the French atheists and rationalists! Over a century later, other Englishmen of philosophic and religious disposition brought a like clarity of geometrical imagination and mathematical analysis into the rising science of electric and magnetic forces. I refer, of course, to Faraday and Clerk Maxwell. It is this mathematical ele­ment in physics which gives it strength and power—power for technical uses, strength in its influence upon our mental outlook. There is an element of tragedy in this, for the resulting system becomes a rigid framework bar­ring access to the more spiritual aspects of reality, of which the truths of homeopathic medicine are an example. But the spiritual power of geometri­cal and mathematical thinking which has helped build this framework can also help in the much needed release. Of this I am about to tell.

Till about half a century ago—the time of Einstein and Minkowski—the space in which the real events of the universe were supposed to be taking place was that of Euclid, the geometry of which we learn at school. It is the space measured in finite and rigid lengths, or areas and volumes based on the measurement of length. It is determined by the well-known laws of parallelism and of the right angle, as in the theorem of Pythagoras or in the statement that opposite sides of a parallelogram are equal. The same type of space was held to prevail down to the smallest and up to the largest dimen­sions. Inward and outward, the identical scale of length leads to the mil­limicrons of atomic science and to the parsecs and light-years of astronomical speculation. What happens when a straight line is extended to the infinite, was held to be an idle question, of philosophic interest perhaps, but beyond the effective range of science.

Occasionally, scientists of the Nineteenth Century—W.K. Clifford, for example—reflected that cosmic space might after all be 'non-Euclidean,' its structure differing from the Euclidean to so slight an extent as to escape our instruments of measurement. But neither this nor Einstein's four-dimen­sional space-time did more than modify the profoundly Euclidean—I might call it earthly—way of thinking about space and the realities it contains. This is so taken for granted as to be difficult to describe; few people realize that there is any other way. Space is conceived as a vast empty container—the Irishman's box without sides, top or bottom—populated (in some regions more and in others less densely) by point-centered bodies sending their forces and radiations to one another. It becomes a field of manifold potential forces, but the real sources of activity are, once again, point-centered— material or at least quasi-material—bodies. Apart from these, there would be emptiness, mere nothing. That, surely, is a fair description, both of the popular idea and of the mathematical analysis.

As against this, I now have to tell of what opens out quite new pos­sibilities, both of pure thought and of insight into the realities of nature. For in the Seventeenth to Nineteenth Centuries, while physicists and astronomers were busily applying to their problems the ancient geometry of Euclid—rendered more handy and more elegant but in no way altered by the new analytical methods of Descartes, Leibniz and Newton—among pure mathematicians a new form of geometry was arising. It is a form which, while including the Euclidean among other aspects, is far more comprehen­sive, also more beautiful and more profound. I refer to the school of geometry variously known as projective geometry, modern synthetic geometry, or the geometry of position. In the Seventeenth Century its truths began to be apprehended by the astronomer Kepler and the mystical philosopher Pascal, also by Pascal's teacher, Girard Desargues, a less known but historically important figure. It was, however, in the early Nineteenth Century, about the last twenty years of Hahnemann's own life, that the new geometry really began to blossom forth. Once again, French mathe­maticians—among them Poncelet, Gergonne and Michel Charles—were the pioneers, soon to be followed by a few brilliant thinkers in Switzerland and Germany, England, Italy and other countries. Largely unnoticed save among pure mathematicians, upon whose thought it was to have a deep and lasting influence, it grew into an ever wider insight, which by the end of the century was seen to embrace most if not all of the known forms of geometry, Euclidean and non-Euclidean alike. Today, as I shall presently contend, it opens out new ways of understanding nature—above all, living nature and the subtler, more spiritual forces which the intuitive genius of Hahnemann was perceiving.

Like that of Euclid, projective geometry is not only a discipline of pure thought, resting securelyon its own ideal premises or axioms; it is also re­lated to practical experience, though to begin with in a rather different direction. Our experience of the spatial world is above all visual and tactile. There are indeed other and less conscious senses—senses more 'proprioceptive' of our own spatial body both in itself and in its interaction with the world, such as the sense of movement and that of balance—to which our spatial awareness and geometrical faculty are largely due. But in our outward consciousness it is the sense of touch and that of sight which reinforce and confirm geometrical reasoning and imagination. Now the geometry of Euclid relates above all to the sense of touch; hence too its natural connection with a scientific outlook taking its start from tangible material things. The inch, the foot, the yard, derive from our own body. We measure as we touch the earth, foot by foot and step by step, or in the rhyth­mic act of measurement with finger-tip and yard-stick. By tactile experien­ces we confirm the constant distance between parallels, the symmetry laws of the right angle. We even prove the first theorem of Euclid by the im­agined tactile experiment of applying one triangle to another. But our ex­perience of space is also visual, and as such far more extensive, more manifold and satisfying. We see things we can never touch by hand or foot or tool; our vision reaches to the infinite horizon and to the stars. Now in the Fifteenth to Seventeenth Centuries the beginnings of modern science coin­cided with the increasingly naturalistic art of the Renaissance. Both were in­spired by the same love of nature and wish to penetrate her secrets. So as to give an outwardly 'true' picture of the scenes of landscape and the forms and works of men, artists such as Leonardo da Vinci and Duerer studied the science of perspective vision, which from its practical and aesthetic applica­tions presently gave birth to a new purely geometrical discipline—to wit, projective geometry. The latter therefore naturally deals not only with tan­gible and finite forms but with the infinite distance of space, represented as these are by the vanishing lines and vanishing points of perspective. Thus in the new geometry the infinitely distant is treated realistically, in a way that was foreign to the classical geometry of Euclid and the Greeks.

To include the infinitely distant, sometimes referred to as the 'ideal elements' of space, no less definitely than those at a finite distance, is a bold step in thought, and is rewarded by a twofold insight of an importance hitherto unsuspected for the science of living things.1 Attention is focused no longer on rigid forms such as the square or the circle, but on mobile types of form, changing into one another in the diverse aspects of perspec­tive, or other kinds of geometrical transformation. In Euclid, for instance, we take our start from the rigid form of the circle, sharply distinguished from the ellipse, parabola and hyperbola, as are these from one another. In projec­tive geometry it is the 'conic section' in general of which the pure idea arises in the mind and of which various constructions are envisaged. As in real life the circular opening of a lampshade will appear in many forms of ellipse while moving about the room, or as the opening of a bicycle lamp projects on to the road in sundry hyperbolic forms, so in pure thought we follow the transformations from one form of conic section to another. Strictly speaking, the 'conic section' of projective geometry is neither circle, ellipse, parabola nor hyperbola; it is a purely ideal form, out of which all of these arise, much as in Goethe's botany 5 the ‘archetypal leaf’ is not identical with any par­ticular variety or metamorphosis of leaf (foliage leaf varying in shape from node to node, petal, carpel and so on) but underlies them all. The new geometry begets a quality of spatial thinking akin to the metamorphoses of living form.

The other insight 2 is perhaps even more important. Projective geometry recognizes as the deepest law of spatial structure an underlying polarity which to begin with may be called, in simple and imaginative language, a polarity of expansion and contraction, the terms being meant in a qualitative and very mobile sense. (If I now illustrate by using, after all, some of the more rigid and symmetrical forms, the limitations of which I have just referred to, it is only to make it easier by starting with familiar pictures.) Think of a sphere—not the internal volume but the pure form of the surface. One sphere can only differ from another as to size; apart from that, the form is the same. Now the expansion and contraction of a sphere leads to two ul­timate limits. Contracted to the uttermost, the sphere turns into a point; ex­panded, into a plane. The latter transformation, though calling for more careful reflection, is no less necessary than the former. A large spherical sur­face is less intensely curved than a small one; in other words, it is flatter. So long as it can still grow flatter, a sphere has not yet been expanded to the ut­most limit, which can only be the absolute flatness of a plane.

The above experiment in thought—the ultimate contraction and expan­sion of a sphere—leads in the right direction. Point and plane prove to be the basic entities of three-dimensional space—that is, the space of our universe and of the human imagination. Speaking qualitatively, the point is the quintessence of contraction, the plane of expansion. Here comes the fun­damental difference as against both the old geometry of Euclid and the naive and rather earthly spatial notions which culminate in a onesidedly atomistic outlook. For in the light of the new geometry, three-dimensional space can equally well be formed from the plane inward as from the point outward. The one approach is no more basic than the other. In the old-fashioned explanation, we start from the point as the entity of no dimension. Moving the point, say from left to right, we obtain the straight line as the first dimension; moving the line forward and backward, we get the two dimensions of the plane; finally, moving the plane upward and downward, the full three dimensions. To modern geometry this way of thinking is still valid, but it is only half the truth—one of two polar-opposite aspects, the in­terweaving harmony of which is the real essence of spatial structure. In the other and complementary aspect we should start from the plane and work inward. To mention only the first step: just as the movement of a point into a second point evokes the straight line that joins the two, so does the movement of a plane into a second plane give rise to the straight line in which the two planes interpenetrate. We can continue moving in the same line and ob­tain a whole sheaf of planes, like the leaves of an open book or a door swinging on its hinges. We thus obtain a 'line of planes,' as in the former in­stance a 'line of points.' In the space-creating polarity of point and plane, the straight line plays an intermediate role, equally balanced in either direction. Just as two points of space always determine the unique straight line which joins them, so do two planes: we only need to recognize that parallel planes too have a straight line in common; namely, the infinitely distant line of either. At last we see that all the intuitively given relationships of points, lines and planes have this dual or polar aspect. Whatever is true of planes in relation to lines and points, is equally true of points in relation to lines and planes. Three points, for example, not in line, determine a single plane (prin­ciple of the tripod), but so do three planes, not in line (e.g. the ceiling and two adjoining walls of a room) determine a single point. The planes must again be extended to the infinite and thought of as a whole to see that this is true without exception.

All spatial forms are ultimately made of points, lines, and planes. Even a plastic surface or a curve in space consists of an infinite and continuous se­quence, not only of points, but of tangent lines and tangent or osculating planes. The mutual balance of these aspects—pointwise and planar, with the linewise aspect intermediating—gives us a deeper insight into the es­sence of plasticity than the old-fashioned, one-sidedly pointwise treatment.

The outcome is that whatever geometrical form or law we may con­ceive, there will always be a sister form, a sister law equally valid, in which the roles of point and plane are interchanged. Or else the form we thought of—as for example a tetrahedron with its equal number of points and planes—proves to be its own sister form, arising ideally out of itself by the polar interchange of point and plane. The principle just enunciated, as it were a master-key among the truths of projective geometry, is known as 'the principle of duality.' It would perhaps have been better had it been described as a 'principle of polarity' from the outset, for in its cosmic aspect it is also one of the essential keys to the manifold polarities of nature. The recognition of it leads to a form of scientific thinking calculated to transcend one-sided atomism and materialistic bias.

A simple instance is shown in Figure 1. A sphere is placed inside a cube just large enough to contain it. Touching the six planes of the cube, the sphere picks out six points of contact. Joined three by three, the latter give eight planes, forming the double pyramid of the octahedron. Octahedron and cube are sister forms, in polar relation to one another. The structure and number relations are the same, only with plane and point—the prin­ciples of expansion and contraction—interchanged. The octahedron has eight planes, each of them bearing a triangle or triad of points and of the lines that join them; so has the cube eight points, each of them bearing a triad of planes and lines. The octahedron on the other hand has six points or apices, each with a four-fold structure, answering to the cube with its six four-square planes. The number of straight lines or edges is the same in each; namely, twelve.