The Teachings of Goethe and Newton on Colour in the Light of Modern Physics.

Werner Heisenberg

To advance science, in cooperation or in competition with others, it is sufficient to concentrate one’s whole power on the small circle of an intended project. To survey progress as a whole, however, it is useful to make repeated comparisons with the scientific tasks of a previous era and to explore that peculiar change which a great problem undergoes in the course of decades or perhaps centuries. Such a problem, posed in a creative way, can appear again and again in a new light, even though it may have been satisfactorily answered at the time.

The continuous movement of modern science towards an abstract control of nature, removed from an appreciation based on common experience, recalls immediately the memory of the great writer who, over a hundred years ago, dared to fight for a more “living” science in the field of colour theory. That battle is over. The decision on “right” and “wrong” in all questions of detail has long since been taken. Goethe’s colour theory has in many ways borne fruit in art, physiology and aesthetics. But victory, and hence influence on the research of the following century, has been Newton’s. The extraordinary development of Newton’s physics since that time has stressed the consequences of this trend in research more than ever. The cold abstract concepts which enable us to control nature, as for example in modern nuclear physics, illuminate more clearly to-day the background of that famous dispute. It is primarily this background that I wish to discuss.

We know that Goethe felt impelled to concern himself with Nature during his Italian journey. The geological structure of the country, the variety of plants flourishing under a southern sky, the vivid colours of the Italian landscape, captured his interest again and again and are brought to life for us in the vivid description in his diary. We also see from the notes how these impressions, as if by themselves, assume some scientific order and how there arise from an immediate experience of nature, concepts later destined to become the foundations of Goethe’s Contemplations of Nature. After his return to Weimar Goethe began to make use of his freshly gained experiences. The first result of this work, The Metamorphosis of Plants, was published in 1790. Work on the theory of colour which Goethe had started in Italy remained for the time in the background. On his admission, it had been stimulated by the impact of Italian colouring. After his return Goethe had borrowed a prism from Hofrat Buttner in Jena to study the colour effects of refraction. It remained wrapped on his table. Probably in spring 1791 the owner asked for the return of the prism and sent a servant to collect it, and only then did Goethe make use of the opportunity to observe the well-known colour effects. He discovered to his great surprise that large white surfaces do not, as he had assumed from his studies of Newton’s theories, appear coloured, but that they are white, and that a corresponding observation applies also to large dark surfaces. Coloured borders appear only at the edges of light and dark surfaces. From this Goethe realized “that a boundary is necessary to produce colours.” This discovery, which Goethe believed to be in contradiction to Newton’s theory, provided the incentive for intensive work on the origin of colour in the process of refraction. Goethe concluded that colour is created by the combination of dark and light and not by light alone as Newton thought. This conclusion he finds confirmed by many other phenomena. The sun, which appears radiantly white during the day, looks yellow and red when obscured by an intervening layer of mist. Smoke rising from a chimney assumes in sunlight a bluish haze. Further convinced by a variety of other experiences Goethe finally came to believe in the origin of colour out of light plus dark and to have found the “underlying phenomena” (“ Urphanomen “) in the admixture of dimness ( Trube ) with light. This concept brings together into a unified, orderly whole the many effects of colours in our world of the senses rather by way of a guiding idea, based not on reason but on experience. The harmonious arrangement laid before us by Goethe’s colour theory gives even the smallest details a living content and comprises the whole range of objective and subjective colour phenomena. Just those colours which are conditioned by processes in the eye itself and are therefore really based on an “illusion” of our senses, are treated with particular care. And when Goethe speaks of the Urphanomen of the origin of colour in one of the most beautiful poems in the Westostlicher Diwan then we can sense the importance this discovery had assumed for him.

Goethe had believed that the contradiction between his and Newton’s theories could not be resolved, so for that reason we must now deal with Newton’s theory too. This theory forms to this day the basis of all Physical Optics. White light is considered to be composed of light of different colours, similar in a way to the sound of distant breakers which is composed of the rush of individual waves though appearing to our sensations as one undivided whole. By means of exernal influences individual colours can be singled out. In this process of separation some matter is always required which will remove light, and this can be compared with what Goethe calls dimness or darkness. Thus Newton’s theory also explains that colours are created from white light only as a result of reciprocal action with dimness. But the order of phenomena is completely different in the two theories. The simplest phenomenon in the Newtonian theory is the narrow monochromatic ray purified by complicated mechanisms from light of other colours and directions. Goethe’s simplest concept is the bright all-embracing daylight. Newton’s basic effect, so remote from our daily experience, makes optical phenomena accessible to measurement and mathematical treatment. Radiation and propagation of light can be determined by measurement and fixed in mathematical form. Every colour can be associated with a number —in modern notation the wave length. This makes optics into what is commonly called an exact science, in that it enables us to construct accurate optical instruments which open up parts of the universe not normally accessible to our senses. Newton’s theory makes possible a certain control over the phenomena of light and their practical use but it is plainly of no assistance to a better appreciation of the world of colour surrounding us.

This comparison shows that mutual criticism was bound to arise between Goethe’s and Newton’s theories. Newton’s starting point appeared strange and unnatural to Goethe. White light, that is, light in its purest form, is to be downgraded to a composite. Instead the physicist is to accept as the basic form a light tormented and forced through narrow slits, lenses, prisms and all sorts of complicated devices. We can well understand when Goethe gives vent to his disappointment in these words: “The physicist also wins mastery over natural phenomena, he accumulates experiences, fits and strings them together by artificial experiments . . . but we must meet the bold claim^ that this is Nature, with at least a good-humoured smile and some measure of doubt. No architect has yet had the notion of passing off his palaces as mountains and woods.” In general he deprecates the desire of physicists to penetrate through the world of phenomena, as they appear, to the causes of those phenomena. “Even if an original phenomenon ( Urphanomen) were found, there remains the dilemma that it will not be recognized as such, that we shall look behind and beyond for something further. Otherwise we should admit a limit to seeing. Let the scientist leave original phenomena in their eternal peace and splendour.” On the other hand, the physicist can legitimately reproach Goethe in that his theory cannot be regarded as scientific since it cannot lead to a real control of optical phenomena. Particular colour phenomena, not yet observed, cannot be predicted with any accuracy. This, however, is precisely what Newton’s theory can lay claim to do. Goethe’s theory also deliberately links certain elements whose careful separation is the constant preoccupation of the physicist. The first presupposition of all research is the separation of subjective and objective. Thus Goethe’s colour theory can enrich the physicist’s knowledge in particular fields. He can learn something about the reaction of the eye to the impact of colour, about the colours of chemical compounds, or about refraction phenomena. But it is just the very unity of Goethe’s theory which he cannot accept. For the reactions of the eye must find their explanation in the finer biological structure of the retina and optical nerves, which conduct the colour impressions to the brain. The colours of chemical compounds must be capable of calculation from their atomic structure, and refraction phenomena are derived mathematically from the properties of a propa-

gating wave. … On this basis, an immediate connection of the three phenomena appears unintelligible. We see here a general characteristic of nature. Processes appearing to our senses to be closely related often lose this relation when their causes are investigated.

It is clear to all who have worked more recently on Goethe’s and Newton’s theories, that nothing can be gained from an investigation of their separate rights and wrongs. It is true that a decision can be taken on all points of detail and that in the few instances, where a real contradiction exists, Newton’s scientific method is superior to Goethe’s intuitive power, but basically the two theories simply deal with different things. It is much more to the point to ask how it is possible to link the idea of colour with such different subjects.

It has been said that Goethe’s and Newton’s methods proceeded in two entirely different directions. While Newton obviously endeavoured to open the world of colour to exact measurement and thus to create order in that world by mathematical methods similar to those he had so successfully employed in mechanics, such mathematical considerations do not figure in Goethe’s work. On the contrary, Goethe explicitly dispensed with all relations of his theory to mathematics though he stressed that the assistance of exact measurement may have been desirable in some instances. On closer scrutiny, however, this difference assumes much less importance than appears at first sight. Goethe does not renounce mathematics itself but rather mathematical manipulation. When we consider mathematics in its present form, as revealed, say, in the theory of symmetry and number, it is easily seen that Goethe’s theory contains no small amount of mathematics. In the section “The Sensual- Moral Effect of Colour” he deals for example with symmetrical arrangement of colours according to polar relations. He presents an arrangement of the six primary colours in a regular hexagon or a circle divided into six equal parts: red, blue-red, blue, green, yellow and orange, in this order. Each colour in this circle lies opposite its complementary, thus red opposite green and blue opposite orange. This symmetrical arrangement of the colours led him to a study of the varied relations between them. Colours lying opposite result in “pure, self-motivated, harmonic combinations which always carry totality with them.” The combination of two colours which are separated by only one intervening colour, Goethe calls characteristic because, as he says, “they all have something significant which forces a certain impression upon us but does not satisfy us. This is because each characteristic only originates by a separation of a part from a whole to which it is related without dissolving itself in it.” Finally, the combination of neighbouring colours he calls a “characterless combination.” This treatment of colour relations on a colour disc immediately recalls mathematical symmetries such as are found in an artistic ornament or demonstrated in the simplest form in a kaleidoscope. Similar symmetrical arrangements can be found throughout the whole work.

A somewhat clearer picture of the differences between the two theories can be gained by enquiring into the purpose they are to serve. This should not be misunderstood to mean that a scientific theory is always related to a definite purpose and that its only aim is to achieve this purpose. But every scientific theory arises in a certain mental climate which implies some idea as to how the projected theory might later be applied. This background is often conditioned by the historical development of the science concerned, and the author of the theory may be only vaguely conscious of it. If we then speak of the purpose of a theory, in this sense, there can be no doubt that Goethe’s theory of colour was designed to serve the artist, particularly the painter. Goethe himself describes at length how much he has missed a theory of colour in art and how it struck him that “artists acted merely on the basis of vague tradition and a certain impulse and that dark and light, colouring and the harmonies of colours move curiously and without rhyme or reason.” It is certain that Goethe’s initial desire was to create such a theory of colour. Beyond this desire, as a more general background, there was a goal, first mentioned during his Italian journey in connection with his plans for a theory of colour. “I can see that with some effort and persistent thought I shall be able to experience further enjoyment of this world.”

The background out of which Newton’s theory grew was entirely different. The experiences of science since Galileo and Kepler have taught us that mechanics can be summarized in, and understood by, mathematical laws. Newton was the first scientist to realize to what extent such a penetration of nature was possible. In optics, too, there existed a series of investigations, showing that large parts of this subject could be mastered with the aid of laws capable of mathematical formulation. It is quite obvious that Newton’s efforts were directed precisely towards such progress in the mathematical explanation of colour. It is difficult to estimate to what extent, at that time, this desire was linked with the realization that an accurate knowledge of the physical laws can lead to the technical mastery of nature. But the fact that Newton made long and detailed investigations in the improvement of telescopes tends to show that he was quite familiar with this side of science too.

Later developments have shown how well the two theories had achieved their stated objectives. Without a mathematical theory of light there would never have been a telescope or a microscope. On the other hand many painters gathered knowledge and enrichment from Goethe’s theory.

It has also frequently been said that behind this diversity of purpose there lies a deeper difference of mental approach and that the fundamentally different attitudes of the poet and the mathematician to the world have led to such different theories. This certainly expresses an important reason for the dispute, but it would be unjust to conclude that this other poetic side of the world need necessarily be alien to the scientist. We need only mention Kepler who, after all, helped to create the most important foundations of this mathematical science. Kepler always sensed in all his varied and intricate speculations on number the harmony of spheres. Listening to the enthusiasm with which he celebrated new discoveries about the harmony of planetary orbits it would be ungenerous not to credit him with definite poetic sensibility. Newton devoted a large part of his life to philosophical and religious investigations and it is probably correct to say that the world of poetry has been familiar to all really great scientists. The physicist, at any rate, also seeks to discover the harmonies of natural events. On the other hand, it would be an equal mistake to believe that the poet Goethe had more interest in arousing a vivid impression of the world than in acquiring a real understanding of it. Every genuinely great work of creative writing transmits real understanding of all aspects of life otherwise difficult to grasp. This is especially true of a work like the theory of colour which must transmit new understanding and is written with full claims to scientific accuracy.

Perhaps the difference between the two theories is most accurately defined by saying that they deal with two entirely different levels of reality. We must remember that every word of our language can refer to different aspects of reality. The real meaning of words often emerges only in their context or is deter- mined by tradition and habit. Modem science soon made a division of reality into objective and subjective. While the latter is not necessarily common to different people, objective reality is forced on us from the outside world always in the same way and for that reason early science made it the subject of its investigations. In a way, science represents the attempt to describe the world to the extent that it is independent of our thought and action. Our senses rank only as more or less imperfect aids enabling us to acquire knowledge about the objective world. It is only natural and consistent for the physicist to try and improve on the senses through artificial means of observation until we penetrate to the most remote fields of objective reality which are entirely beyond the range of our immediate perception. At this point arises the deceptive hope that further refinement of our methods of observation may eventually enable us to get to know the whole world.

To this objective reality, proceeding according to definite laws and binding even when appearing accidental and without purpose, there stands opposed that other reality, important and full of meaning for us. In that reality events are not counted but weighed, and past events not explained but interpreted. Useful ( sinnvoll ) interrelations here mean a “belonging together” within the human mind. True this reality is subjective but it is no less powerful for all that This is the reality of Goethe’s theory of colour. Every type of art is concerned with this reality and every important work of art enriches us with a fresh understanding of its scope.

It appears at first sight as though an unbridgeable gulf will for ever divide these two realities. Goethe’s struggle against Newton’s theory would appear simply an expression of an irreconcilable conflict. The development of science in the last few decades, however, has shown that a division of the world into two sec- tions creates a very crude image of reality. To understand this we shall have to consider more recent developments in the realm of physics.

The idea that our senses are only imperfect aids in the appreciation of the objective world has guided science further and further away from our immediate world of the senses. A more refined technique of observation has brought to light new aspects of nature previously concealed from us, while parallel to this development the concepts of science have become more abstract and remote from common experience. A basic concept of Newtonian optics, the monochromatic ray of light, is already an idea to which we are accustomed in everyday life. The movement of science away from our world of the senses becomes quite plain when considering electrical phenomena. During the first half of the last century attempts had been made to link electrical theory with mechanics through the concept of force. However, the discoveries of Faraday and Maxwell have shown that electric and magnetic phenomena can best be understood by basing them on the idea of the electric field. True, the field concept can be made plainer by comparison with the oscillations of elastic bodies but this is obviously a simile for showing mathematical interrelations, and has no connection with our immediate sense-impression of electricity. For even when we talked of an ether whose elastic oscillations had an electric effect, this ether was outside the range of our sense-impressions. At the same time, however, this science, in becoming more and more abstract, reveals a new power. It can recognize the interconnection between the most diverse phenomena and relate them back to a common root. It is the finest justification of our enquiry into the objective world that it has led to unexpectedly wide interconnections, and that, in spite of all the complexity of detail, it has, more and more, simplified our ideas of nature. Through Maxwell’s discovery, light was recognized as an electromagnetic phenomenon. This led in turn to the recognition that electric and magnetic effects, light, invisible ultra violet and infra red rays and heat radiation are but different aspects of the same physical effect in spite of the fact that they belong to entirely different parts of our world of the senses. This development is carried to its logical conclusion in modem atomic physics. Atomic physics undertakes to explain all properties of matter accessible to our senses of our experiments, by tracing them back to properties of the atom. These latter can be laid down in simple mathematical laws. Thus the infinite variety of phenomena is reflected in the infinite number of deductions from a simple system of mathematical axioms. In fact modern atomic physics can explain, from the properties of atoms, the properties of solids, chemical regularities, the effects of heat and anything else arising from an observation of matter. It is true that up to the present this explanation has been carried out, with the precision ultimately required, only in relatively few cases, but in all these cases our theory has stood up to the most rigorous tests in a wonderful way. But from an explanation of the sense properties of matter, from their atoms, it becomes clear that no such sense properties can be attributed to the ultimate “brick” of matter in a simple way. While the atom can be observed, in its effects, by an extraordinary refinement of experimental techniques, it is no longer subject to our immediate sense perception. The scientist must reconcile himself to the idea of directly linking the fundamental concepts on which his science rests with the world of the senses. They justify themselves as fundamental concepts because they penetrate the infinite variety of phenomena of our world of the senses, introduce regularity and order and thus make it comprehensible. This is proved by the technical developments they have made possible and which enable man to harness the forces of nature to his purpose.

This development has been responsible for a peculiar change in our views concerning the objective world of science. Our intention to eliminate those errors which may have been introduced by the deceptions and inaccuracies of our perception, caused us to describe the world in a manner entirely independent of our own thoughts and actions. The idea was to sketch as accurate a picture of nature as possible. Now it has turned out that this picture becomes, with increasing accuracy, further and further removed from “living” nature. Science no longer deals with the world of direct experience but with a dark background of this world brought to light by our experiments. But this means that, in a way, this objective world is a product of our active intervention, and improved technique of observation. Here, too, then, we are brought face to face with the limitations of human understanding which we cannot overcome.

Goethe’s struggle against the physical theory of colour, then, will have to be continued today on an extended front. Helmholtz said of Goethe “that his theory of colour must be regarded as an attempt to save the immediate truth of the ‘sense-impression’ from the attacks of science.” To-day, this task is more urgent than ever. The whole world is being transformed by enormous extensions of our scientific knowledge and by the wealth of its technical applications but like all wealth, this can be a blessing or a curse. Hence many warning voices have been raised during recent years counselling us to turn back. Already, they say, a great scattering of intellectual effort has resulted from our negation of the world of direct sense-impressions and the division of nature into different sectors. Further withdrawal from “living” nature will, so to speak, drive us into a vacuum where life will no longer be possible. When we are not advised simply to throw over all science, pure and applied, we are exhorted to develop science in close connection with daily experience. We are told that it is not sufficient to understand the laws governing all processes of the objective world but that it is essential to visualize at any given moment all the consequences of these laws in our world of the senses. In his constant dealing with nature in his own experiments, the scientist should become so familiar with observed phenomena that laws would appear merely a useful summary of his experiences. Thus the danger of completely separating the two kinds of realities is to be avoided by making the world of experiments as direct and “living” as surrounding nature. But it is obvious from the start that the interrelations of nature can only be understood by a man who is thoroughly familiar with the manifestations of nature in the field concerned. There has never been progress and discovery without detailed knowledge based on experimental results. But the dangers of modem science are not surmounted in this way. For our experiments are not nature itself, but a nature changed and transformed by our activity in the course of research. To effect a real change would undoubtedly entail a complete abandonment of the whole of modem technology and science, which is linked with it Nobody is in a position to say whether such a break would mean happiness or disaster for mankind. But however we may feel about this, one thing is certain. Such a break is impossible. We have to reconcile ourselves to the fact that it is the destiny of our time to follow to the end of the road along which we have started.

At the beginning of our modem era navigation flourishing and the daring feats of the circumnavigators of the earth opened up the possibility of the conquest of distant lands and of the return with immense treasures to their homelands. There may have been some doubt as to whether the new wealth would weight the scales equally with happiness and distress. Perhaps there were warning voices then who advocated a return to the more peaceful and less pretentious conditions of life of a previous epoch. But at such times warning voices resound unheard. The attraction of foreign lands and treasures can only come to its natural conclusion when these countries have been explored and their treasures have been distributed. Only then shall we have the vision to see more closely defined tasks, though they may be more important, and it is thus that science and technology will continue to develop in our time. Just as frontiers could not prevent the attraction of foreign countries, so no external obstacles will be able to prevent the progress of technology. Only nature herself can call a halt to our endeavours by showing us that the field to be conquered is not infinite. It is perhaps the most important trend of modern physics that it shows us the limits of our active attitude to nature.

Atomic physics took as a starting point the apparently natural supposition that our knowledge of the atom will, with increasing accuracy of observation, perfect itself more and more. Though atoms represented the final indivisible “brick” of matter, they nevertheless appeared to be miniature parts of ordinary matter. The atom then, at least in our imagination, was endowed with all the macroscopic properties of matter. Only in the course of time was it recognized that the smallest particles, for instance electrons, could not themselves possess the “sense-properties” of matter if they were to explain these properties on a larger scale. Otherwise the question of the reason for those properties would not have been solved but only moved one step further away. For example, if we say that a stronger movement of the atoms within differentiates a hot from a cold body, then an individual atom can be neither hot nor cold. Thus the atom was progressively divested of all its “sense properties.” The only properties which appeared for a long time to be retained were geometrical ones—the atom took up space and position, and had a definite movement. The development of modern atomic physics, however, has removed even these properties by showing that the degree to which such geometrical concepts can be applied to the smallest particles depends directly on the experiment in which they are involved. True, with a comparatively moderate demand for accuracy, we can speak of the position and velocity of an electron: true also that, compared with our daily experience, this accuracy is quite considerable. But measured by an atomic scale it is insufficient, and a law characteristic for this miniature world prevents us from determining position and velocity with the desired accuracy. Experiments can be done enabling us to determine, say, the position of a particle with great accuracy, but in the course of this measurement the particle has to be exposed to strong external influences which are responsible for a considerable uncertainty as to its velocity. Nature thus escapes accurate determination, in terms of our commonsense ideas, by an unavoidable disturbance which is part of every observation. It was originally the aim of all science to describe nature as far as possible as it is, i.e. without our interference and our observation. We now realize that this is an unattainable goal. In atomic physics it is impossible to neglect the changes produced on the observed object by observation. We decide, by our selection of the type of observation employed, which aspects of nature are to be determined and which are to be blurred in the course of the observation. This is the property which separates the smallest particles of matter from the range of our commonsense concepts. The supposition that electrons, protons and neutrons, according to modem physics the basic particles of matter, are really the final, indivisible particles of matter, is only justified by this fact. It would no longer make sense to visualize a three dimensional structure of these particles.

From what has been said we can conclude, along two different lines of thought, that the range of science and technology as we know it, is finite. On the one hand, our arrival, in atomic physics, at the final indivisible particles of matter should, in the not too distant future, lead to a complete survey of all the forces of nature yet to be exploited and hence of all possible technical possibilities. On the other hand, the way in which atomic phenomena are divorced from those of our everyday experience serves as an important example that in science the way in which a question is put and the method of research employed already singles out a finite and limited field from the abundance of physical phenomena. Previously, it appeared to be the task of science to describe the motion of bodies in space and to understand their regularity. Now we recognize that the range of atomic phenomena cannot be tackled in this way. When we ask of nature position and motion within an atomic system we destroy, through the impact of essential experimental measures, certain interconnections characteristic for a world of atomic size. It is tempting to generalize these ideas and to recall Goethe’s criticism of Newtonian physics. Goethe said that what the physicist observes with his apparatus is no longer nature. He probably meant to imply that there are further and more “living” aspects of nature which are not accessible to this particular method of science. We are, of course, ready to believe that science, where it turns from inanimate to living matter, will have to be more and more careful in its interference in the course of an experiment. As our desire for knowledge also reaches out to higher, spiritual aspects of life, so we shall have to be content with a passive, contemplative kind of investigation. From this point of view, the division of nature into a subjective and an objective sector would appear an over-simplification of reality. It would be more to the point to imagine a division into many overlapping sectors, divided by the type of question we ask of nature and by the amount of interference which we allow during observation. In attempting such a classification in simple terms we are reminded of the classification of “related aspects” as it appeared in the appendixes to Goethe’s theory of colour. Goethe stressed that all the effects which we observe by experience are connected and continuous, yet the separation of one from the other is unavoidable. He classified them from low to high: accidental, mechanical, physical, chemical, organic, psychic, ethical, religious and of genius (genial). Seen in the light of modem science we might perhaps change some of the first delineations. For mechanical we might substitute all those phenomena accessible to classical physics, where a strictly causal space-time description can be given. The sphere of chemistry would include atomic processes and its scientific structure would be made plain by modem atomic physics. Besides these two categories we should not need a specific category physics as, in a sense, the previous two would be part of it. Neither should we allot a special category to accidental as accident plays a role precisely prescribed by natural laws. Thus the four lowest categories of Goethe’s arrangement can be plainly understood in their scientific structure, their interrelations and their respective delineations. As far as the next category organic is concerned, modem biology believes it can recognize its limitations though indistinctly, and understand its inner structure. It is unlikely that anybody would, at this time, dare to define the remaining higher categories. Dividing reality in this way into different aspects immediately resolves the contradictions between Goethe’s and Newton’s theories of colour. In the great structure of science, the two theories take up different positions. It is certain that an acceptance of modem physics cannot prevent the scientist from following Goethe’s way of contemplating nature too. It would of course be premature to hope, on this basis, for an early return to a more direct and unified attitude to nature. It appears to be the task of our time to grasp, by experiment, the “lower reaches” of nature and, through technology, to make them our own. In our advance in the field of exact science we shall, for the time being, have to forgo in many instances a more direct contact with nature such as appeared to Goethe the precondition for any deeper understanding of it We accept this because we can, in compensation, obtain an understanding of a wide range of interrelations, seen with complete mathematical clarity. This must, undoubtedly, also be the basis and precondition for a proper understanding of the “higher reaches.” Those who regard this as too great a sacrifice will, for the time being, be unable to devote themselves to science. They will only grasp that sense of science where, at the outer limits of present-day methods of research, science discovers its relations to life itself.

Perhaps we can liken the scientist who leaves the field of direct sense-impression in order to see nature as a whole, to a climber who wants to master the highest peak of a mighty mountain in order to survey the country below him in all its variety. The climber too must leave fertile inhabited valleys. As he ascends, so more and more of the country unfolds below him, but also life around him becomes more and more sparse. Eventually he reaches a dazzling, clear region of ice and snow in which all life has died and where he can only breathe with great difficulty, and only by traversing this region can he reach the top. But once he has reached it, in the few moments in which the whole country below him is visible with absolute clarity, he may not be so distant from life. We can appreciate that previous eras felt those lifeless regions to be only frightening wastes and an intrusion seemed an injury to some higher power which was bound to take bitter revenge against those who dared to approach them. Goethe, too, sensed an injury in the advance of science. But we may be certain that that final and purest clarity, which is the aim of science, was entirely familiar to Goethe the poet.