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Harmonices Mundi: A Look at Sir Arthur Stanley Eddington’s Cosmos

A 1610 portrait of Johannes Kepler

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But now, Urania, there is need for louder sound while I climb along the harmonic scale of the celestial movements to higher things where the true archetype of the fabric of the world is kept hidden. Follow after, ye modern musicians, and judge the things according to your arts, which were unknown to antiquity. Nature, which is never not lavish of herself, after lying-in of two thousand years, has finally brought you forth, in these last generations, the first true images of the universe. By means of your concord of various voices, and through your ears, she has whispered to the human mind, the favorite daughter of God the Creator, how she exists in the innermost bosom. — Kepler [1]

Johannes Kepler, scientist and mystic, searched for evidence of the all-encompassing harmony of Nature. Looking at the orbit of Mars, fitting Tycho Brahe’s data to his own model of the universe, Kepler found as Pythagoras said, “there is geometry in the humming of the strings, there is harmony in the spacing if the spheres.” One aspect of Kepler’s theory which must have felt alien to later scientists was the lack of a mechanical “machine.” Kepler did not formulate gravity as a force tugging on the planets, forcing them into elliptical orbits, as Newton did. Kepler, rather, based his theory on ratios, and on the assumption that the universe is orderly and in concord with itself in various allowed proportions.

Newton’s law of gravitation gave a new mechanical view of the planet’s orbits, and this theory led scientists to great discoveries and models of the known universe. After two centuries, in 1916, the world of science took another great step forward and ironically returned to an ideal closer to Kepler’s. With Einstein’s 1916 General Relativity paper [2], gravity was no longer described as force in the usual sense, but given a less intuitive, albeit simpler description. With the fall of gravity as a mechanical force, it would not surprise us to find Maxwellian electromagnetism standing in need of a less mechanical formulation, and indeed Hermann Weyl took early steps to bring electromagnetism into line [3]. But then came quantum physics.

Who could bring quantum into harmony with relativity, who could bring the universe into proportion with an atom, who indeed are our “modern musicians,” those men of science, who like Kepler, had great faith in the simplicity of Nature, belief in the consonance of the universe, those whose trust in the logical construction of the universe lead them to bold unifications? I wish to suggest that Sir Arthur Stanley Eddington stood as Kepler’s spiritual descendant. Let us now trace Eddington’s ideas from his childhood, though his scientific career, to the manuscripts found on his desk on his death in 1944. We will pay special attention to 6 of the 157 papers Eddington published, papers which clearly show the evolution of his thoughts.

English astrophysicist Sir Arthur Stanley Eddington

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Even in earliest childhood Eddington showed a fascination with numbers, especially large numbers. It is said that he memorized the 24 × 24 multiplication tables before he could read, that he attempted to count the number of words in the Bible, making it well through Genesis [4]. Eddington proved to be an extremely bright boy, and his mathematical aptitude led him to Trinity in 1902, at the age of 20. At Cambridge he was active in the Δ2(V) Club, the Cambridge Mathematical Club, the Cavendish Society and the Nonconformist Union. Eddington graduated in three years as the Senior Wrangler, the top of his class in the grueling Mathematical Tripos examinations.

Eddington’s early scientific work was characterized by careful, exact work. In his 18 months at the Greenwich Observatory he distinguished himself as an excellent observer, especially during WWI (from which he petitioned for exemption as a Conscientious Objector because of his strong Quaker beliefs) when there was a shortage of assistants. Eddington’s first great achievement was his theoretical work in stellar physics. This work on radiation pressure, degeneracy, luminosity, etc. ranks him with Lane, Emden and Chandrasekhar as the founders of theoretical stellar physics.

Along with his work on stellar physics, Eddington’s name often brings to mind his early work on General Relativity. Eddington first heard of Einstein’s 1916 GR paper from deSitter in neutral Holland. This paper had a profound impact on Eddington. As Secretary of the Royal Astronomical Society, Eddington was called on to prepare the first English language account of General Relativity. A Report on the Relativity Theory of Gravitation was published in 1918 [5]. Eddington wrote in the Preface:

Whether the theory ultimately proves to be correct or not, it claims attention as one of the most beautiful examples of the power of general mathematical reasoning. The nearest parallel to it is found in the application of the second law of thermodynamics, in which remarkable conclusions are deduced from a single principle without any inquiry into the mechanism of the phenomena.

We thus see how Eddington uses an aesthetic criterion to evaluate the Einstein’s theory. One aim of this essay shall be to look at the ways in which the development of Eddington’s own theories were ruled by this same aesthetic.

Eddington would play a major part in the confirmation of general relativity, leading an expedition to Principe in 1919 to measure the bending of light rays passing close to the eclipsed sun. Eddington’s great joy in finding the predicted deflection — he later referred to it as the greatest moment of his life [6] — can be compared to Kepler’s overflowing joy in finding harmony in the orbit of Mars:

….but now since the first light eight months ago, since broad day three months ago, and since the sun of my wonderful speculation has shone fully a very few days ago: nothing holds me back. I am free to give myself up to the divine madness, I am free to taunt mortals with the frank confession that I am stealing the golden vessels of the Egyptians, in order to build them a temple for my God. If you pardon me, I will rejoice; if you are enraged, I shall bear up. [7]

Nineteen-twenty saw the publication of Eddington’s first popular work, a non-mathematical introduction to General Relativity, entitled Space, Time and Gravitation. It is here that the first indications of Eddington’s unorthodox views come into view. Of the final chapter, “On the Nature of Things,” Eddington gives advance warning in the Preface:

As for the last chapter, containing the author’s speculations on the meaning of nature, since it touches on the rudiments of a philosophical system, it is perhaps too sanguine to hope that it can ever be other than controversial [8].

And indeed, this chapter digs to the roots of science and declares that there is no such thing as gravity or electromagnetism! According to Eddington, these laws are all truisms, not telling matter how it must behave, but rather telling how we observe it. In other words, Eddington shows that these laws of physics tell merely that things are what they are! Only atomicity, this discontinuous partition of matter and energy exists in nature, apart from the mind. He ends the chapter and the book:

All through the physical world runs that unknown content, which surely must be the stuff of our consciousness. Here is a hint of aspects deep within the world of physics, and yet unattainable by the methods of physics. And moreover, we have found that where science has progressed the farthest, the mind has but regained from nature that which the mind has put into nature. We have found a strange foot-print on the shores of the unknown. We have devised profound theories, one after another, to account for its origin. At last, we have succeeded in reconstructing the creature that made the foot-print. And Lo! It is our own! [9]

So, like Kepler, Eddington has found the mark of God in Nature, and identified it, in his Quaker philosophy, with the deity in us all.

It is also interesting to note that Eddington’s boldest steps in his philosophy always seem to take place first in his popular writings, leading to the amusing result that, in his journal papers, he often gives references to his popular books. In his professional endeavors, Eddington almost always worked with a self-assurance that he was always correct. The journals are filled with his disputes with Jeans, Milne, and most of all, Chandrasekhar, as they “hurled mathematical formulae, those most undodgable of missiles” [10] at each other. But in his early work on fundamental theory, Eddington shows uncharacteristic caution. In 1927 Eddington gave a series of lectures at the University of Edinburgh, expanding the ideas introduced in the last chapter of Space. Time and Gravitation. The lectures were later published as The Nature of the Physical World a book that was immensely popular and translated into French, German, Italian, Spanish, Dutch, Danish, Swedish, Hebrew, Finnish, Czech, Polish and Japanese. Eddington expresses his doubt in the Preface:

From the beginning I have been doubtful whether it was desirable for a scientist to venture so far into extra-scientific territory. The primary justification for such an expedition is that it may afford a better view of his own scientific domain. [11]

Even more illuminating is the book’s conclusion, where Eddington responds to anticipated criticism:

I suppose the most sweeping charge will be that I have been talking what at the back of my mind I must know is only a well-meaning kind of nonsense. I can assure you that there is a scientific part of me that has often brought that criticism during some later chapters. I will not say that I have been half-convinced, but at least I have felt a homesickness for the paths of physical science where there are more or less discernible handrails to keep us from the worst morasses of foolishness. But however much I may have felt inclined to tear up this part of the discussion and confine myself to pointer readings, I find myself holding to the main principles [12].

But one still wonders why Eddington held so firmly to these “main principles,” an all-embracing theory of electricity, gravitation and atomicity. One fact that had always been in his mind was the occurrence of certain “large number coincidences” involving the constants in physical formulae.  For example, from the seven fundamental constants (me, mp, e, h,c ,k ,λ) four pure numbers can be obtained by dimensional analysis, i.e mass ratio of electron to proton, fine structure constant, electrical to gravitational force and the number of particles in the universe. To Eddington, such pure numbers were not arbitrary, but should logically follow from the theory. Eddington wished to derive the laws of physics, along with their associated constants, from pure theory.  However, in 1929 quantum theory was not developed to a state ripe for unification with relativity, so Eddington remained frustrated, unsure of how to bring e and h into his theory, content to muse of his dream in his popular works. He often quoted Shakespeare. A favorite was:

I have had a most rare vision. I have had a dream,–past the wit of man to say what dream this was: man is but an ass, if he go about to expound this dream…Me thought I was, and methought I had,–but man is but a patched fool, if he will offer to say what methought I had…It shall be called Bottom’s dream, because it hath no bottom.[13]

A partial reconciliation between the mp,c,k and λ received a push in 1930, with the publication of Eddington’s “On the Instability of Einstein’s Spherical World ” [14]. This is a proof that Einstein’s static universe with positive λ is unstable and will, at the slightest displacement, contract to a singularity or expand asymptotically to a deSitter universe of zero average density. Hubble’s discovery of a linear relationship between the distance and redshift of spiral nebulae [15] was quickly interpreted as evidence of an expanding universe. Hubble found the nebulae to be receding on the order of 500 km/s/Mpc. From this Eddington deduced that the mass of the universe is 2.3 × 55gm.

The paper ends with what must have seemed a queer remark to other scientists. After giving the mass in solar masses and grams, Eddington writes: “It may be noted that this gives a total of 1.4 × 1079 protons in the universe.” It is reasonable to ask why he bothered to give this number, what practical value does the mass in terms of proton mass have? I believe that, at this point, Eddington had already noticed that the number of protons, N, was close to the square of the ratio of the electrical to gravitational forces between a proton and electron. This struck Eddington as significant, but without a physical explanation this was insufficient grounds to place this “numerology” before his peers. (One ironic note about this paper: It is immediately followed in the Journal by a paper by Milne, a bitter defense to Eddington’s criticism of Milne’s theory of stellar mass, temperature and luminosity. One must smile at the realization that Milne would soon start his own work in cosmology and would again lock horns again with Eddington.)

In 1928, Dirac formulated his wave equation for an electron moving in an electrostatic field [16]. Eddington immediately fell in love with Dirac’s quantum formulation and went about to finding its application to the fundamental constants. The first result was his 1930 theoretical value of exactly 137 for the fine-structure constant [17]. His more than theoretical attachment to this number is shown by reports that he always left his coat and hat on hook number 137 in a certain conference hall at Cambridge [18]! Eddington continued his work, deriving a quadratic equation whose roots were in the ratio of the masses of the proton and electron [19].

Eddington soon applied Dirac’s wave equation for the electron to an astrophysical problem, in his 1931 “On the Value of the Cosmical Constant” [20] This wave equation, as we remember, has a term of the form m2c2. Ernst Mach postulated that this mass term was due to the presence of all the other bodies in the universe — for an electron in isolation would not have any information to tell it how massive to be. After a beautiful progression of derivations, Eddington arrives at:

\lambda=(2G_M/\pi)^2(2\pi m \alpha/h)^4

Using Lemaitre’s formula for the recession constant this gives a value for the recession of 528 km/s/Mpc, and N=1.29 × 1079.

So, Eddington has progressed from determining the Cosmical Number, N, from Hubble’s redshift measurements to a derivation involving only laboratory constants. Thus we see the evolution of Eddington’s ideas. The next step would be the boldest — the determination of N from the definition of measurement without recourse to actual measurement or observation. But before we do this, we note that Eddington’s value of N hinges on his value of the recession constant, which he sees as around 500, in the usual units. As we know today, this value is absolutely wrong. It was not discovered until 1952 [21] that Hubble’s period/luminosity calibration of Cepheid variables was in error, which meant that the Cepheid variables were actually 10 times more distant than earlier thought, and H0 therefore was actually around 50-80km/s/Mpc. It is disturbing and somewhat amusing to see the contortions Eddington will under go to fit this incorrect value into his theory. This becomes an even more urgent warning when we consider that the 1931 value and following derivations were independent of Hubble’s error!

But, of course, this error was unknown in Eddington’s time, so we might wish to ask how Eddington’s early sallies into cosmology were received by the scientific community. After presentation of the 1931 “On the Value of the Cosmical Constant,” Sir James Jeans said:

I should like to offer my congratulations to Sir Arthur Eddington for his success in this line of investigation. It is remarkable how in the last few years all the knowledge necessary for the understanding of the wave equation as now modified by Eddington has come together. As time goes on, I believe that Fellows will come more and more to realize the fundamental nature of the contributions that Eddington has made in this direction. It leaves little room for doubt that the apparent velocities of the nebulae are real and not the result of some spectroscopically wrong interpretation [22].

A humorous interpretation was given by Herbert Dingle:

He thought he saw electrons swift
Their charge and mass combine.
He looked again and saw it was
The cosmic sounding line.
Their population then, said he,
Must be 1079, [23]

in the style of Lewis Carroll, for indeed, this all was Jaberwocky to many. Thus, we see that what stuck in the mind of his peers was that Eddington’s value of H0 was close to the observed value. Little attention was paid to the profundity of his derivation of it from laboratory constants.

In fact, Eddington seemed not to want his theory scrutinized too much by his peers. He put most of the “numerology” in the last page of the paper, and gives no mention of it in the abstract.

In his popular writings, however, Eddington was much more open about his findings. The 1933 book, The Expanding Universe, has as its final chapter, “The Universe and the Atom” where Eddington details the wonder and exasperation he experienced in developing his theory. He writes that he is quite sure of his results, but if he were wrong, it would be off by a factor of 2 or so, but that his theory was fundamentally correct, a diamond in the rough:

We have been going round a workshop in the basement of the building of science. The light is dim, and we stumble sometimes. About us is confusion and mess which there has not been time to sweep away. The workers and their machines are enveloped in murkiness. But I think that something is being shaped here — perhaps something big. I don’t quite know what it will be when it is completed and polished for the showroom. But we can look at the present designs and the novel tools that are being used in its manufacture; we can contemplate too the little successes which make us hopeful [24].

Another popular work, the 1935 New Pathways in Science, has a passage which quickly brings to mind Kepler and the music of his spheres: We may look on the universe as a symphony played on seven primitive constants as music is played on the seven notes of a scale [25].

These constants, of course, are the seven fundamental constants we listed earlier. Compare this to Kepler’s universe, built on the five Platonic solids. Of his own accomplishments, Eddington significantly wrote:

I have sought a harmonization, rather than a unification, of relativity and quantum theory. I do not set out to obtain an all-embracing formula, but the investigation shows in detail how to combine the conceptions of the two theories in the solution of specific problems, which would be outside the range of either theory separately [26].

Before looking at the next step in Eddington’s process of deriving N from less and less physical observation, it would be wise to look further at the role mysticism and religion played in his philosophy, for we will find that these views will pervade his science.

Eddington, unlike many scientists, did not shy away from mysticism, rather he drank fully from the cup of deepest spirituality. From his Quaker upbringing came the quiet, introspective virtues of the Meeting, the acceptance of God’s hand in creation: Be still and know that I am God.

Eddington combined this religious mysticism with a natural mysticism, an aesthetic, to create an intuitive way of thinking. As he wrote:

In science we sometimes have convictions as to the right solution of a problem which we cherish but cannot justify; we are influenced by some innate sense of the fitness of things [27].

This innate sense of the fitness of things was recalled by A.H. Wilson, who remembered a discussion with Eddington in which he become lost in his own derivation on N and declared, I can’t quite see through the proof, but I am sure the result is correct [28].

In his writings, Eddington grappled with mysticism as early as 1927, with the chapter “Science and Mysticism” in The Nature of the Physical World. Here Eddington gives four reasons why mysticism is an important part of his thought. These may be summarized as [29]:

  1. It is generally recognized that physics, and science in general, can give only a partial view of a larger experience.
  2. Strict causality is not followed, even in the material world.
  3. The physical world is abstract and therefore does not exist apart from consciousness.
  4. The connection between our physical impressions and consciousness has no greater claim than the connection between our spiritual beliefs and our consciousness.

Eddington then summons the verse of Boswell:

We are the music-makers
And we are the dreamers of dreams
Wandering by the lone sea-breakers
And sitting by desolate streams;

World-losers and world-forsakers,
On whom the pale moon gleams:
Yet we are the movers and shakers
Of the world for ever, it seems,

bringing a primal image of the dream artist Apollo and his commandment at Delphi, “know thyself” and the length to which Eddington would follow this in his epistemological evaluation of N.

Eddington’s major contribution to the philosophy of science and religion was his 1929 Swarthmore Lecture to the Society of Friends, published as Science and the Unseen World. In this lecture Eddington made clear that he has neither sought nor desired a reconciliation of religion with science. If religion were reconciled with science, what would happen to religion when science has its next revolution? Neither can be made subservient to the other.

But it is now time to end our discussion of Eddington’s mysticism and turn again to his professional writings, now to the year 1935 and the paper “The Pressure of a Degenerate Electron Gas and Related Problems.” This paper, as many of Eddington’s, has at it roots an argument, this time with Chandrasekhar. Details of the argument would take us too far astray, so it will suffice to say that Eddington was less than fair to this young man from the “Colony.” Eddington’s called Chandrasekhar’s results “absurd,” and so they indeed seemed. According to Chandrasekhar, a star above a certain mass would undergo a runaway collapse to what we today call a “black hole.” Eddington, who could accept singularities in space-time as little as he could accept an infinite universe, referred to Chandrasekhar’s work as “stellar buffoonery” and tried at all costs to restore nature to the peace he believed to be her identity. In presenting his paper to the Royal Astronomical Society, Eddington began:

I do not know whether I shall escape from this meeting alive, but the point of my paper is that there is no such thing as relativistic degeneracy!….One has to look deeper into the physical foundations, and these are not above suspicion [30].

Chandrasekhar reports Eddington once having said to him, .”..You look at it from the point of view of the star; I look at it from the point of view of nature [31].” Or in a published review, “I think there should be a law of nature to prevent a star from behaving in this absurd way ![32]” This 1935 paper, in which Eddington attempts to find this law of nature, is beyond my comprehension. By this point in his work, Eddington had strayed from the main stream of physics; he was now working in almost complete isolation. This was Eddington’s style. Out of 157 published papers, only 9 were co-written.

One of the “related problems” covered in the paper is a familiar one, the determination of N. Eddington first gives a value of N derived from laboratory constants, 1.5727×1079. Then, postulating that N should be a number proportional to 2256, he arrives at the value N=135×822256. And then the bombshell:

Since N is an integer by definition, no irrational or fractional factors can enter into its composition. The integers most directly connected with the theory are 2,3,4,10,16,136,137,256. It therefore seems rather likely that the exact value is N=136×2256[33]

This echoes Kronecker’s “Die ganzen Zahlen hat Gott gemacht, alles anderes ist Menschenwerk” and through him Kepler, Plato and Pythagoras. To the physicists of his day, however, this must have only further alienated Eddington’s theory from the mainstream of physics. Besides the mystical overtones of Eddington’s work, two other factors contributed to its lack of general acceptance. First, he made no predictions. His work was only a unification, a harmonization, or less generously, a hash of unaccepted ideas leading to inaccurate derivations of constants everyone already empirically knew. Scientists did not need a proof that space is three dimensional. To them, such was the realm of philosophy. But the philosophers did not have the mathematical sophistication to understand Eddington’s dense derivations, which relied heavily on his own formulation of Clifford algebras, his so-called “E-algebra.” Very few, only those scientists with a more philosophical bent, could fairly evaluate Eddington’s work.

Moving to 1938 we come to Eddington’s Tarner Lecture at Trinity, later published as the Philosophy of Physical Science. This book, unlike his scientific papers of this time, masterfully explains the ideas behind his derivations. After reading it, I wonder if Eddington, in his journal articles, was hiding behind the mathematics, covering some feeling of philosophical nakedness which he was embarrassed to show the Royal Society? In these lectures Eddington makes everything so clear that the mathematics seems extraneous.

Chapter XI, “The Physical Universe,” opens with what must be one of the most stunning lines ever printed in a philosophical work. Eddington writes simply:

I believe there are 15,747,734,136,275,002,577,605,653,961,181,555,468,044,717,914,527,116,709,366,231,425,076,185,631,031,296 protons in the universe, and the same number of electrons [34].

I am struck by this statement much in the same way as I am moved by Kepler’s writing out the six-voice counterpoint of the planets: a mixture of physical skepticism and artistic admiration. Eddington continues a bit later:

My belief that I know the exact number of protons and electrons in the universe does not rank among my strongest scientific convictions, but I should describe it as a fair average sort of belief. I am, however, strongly convinced that, if I have got the number wrong, it is just a silly mistake, which would speedily be corrected if there were more workers in the field. In short, to know the exact number of particles in the universe is a perfectly legitimate aspiration of the physicist [35].

1944, the last year of Eddington’s life, saw the publication of only two papers. The first, “The Recession-Constant of the Galaxies”[36], was a review of the methods he had used to determine H0, and the last paper, “The Evaluation of the Cosmical Number” details Eddington’s epistemological derivation of N. He writes:

Before calculating their number, it would not be unreasonable to demand a definition of ‘proton’ and `electron’, and adopt for this purpose the mathematical specification that has been deduced from their observational properties. But the present investigation is more ambitious. It seeks to determine N directly from the principles of measurement. The proposition is that, as soon as we become obsessed with the idea that the right way to find out about the universe is to measure things, we are committed to an analytic conception which implicitly divided the universe into 3/2 ×136×2256 particles [37].

This is the idea of Eddington’s epistemological physics: finding in nature what the mind put into nature. Eddington liked to use an analogy of an ichthyologist, taking a survey of fish in a certain lake [38]. He casts his net in, pulls up the haul and records that no fish is less than 2” long. He repeatedly cast the net and finds the same result. He generalizes to The First Law of Fish Size: no fish is less than two inches long. An observer asks, “But if there were fish less than 2”, they would slip through the net, what of them?” The scientist answers with “Anything not catchable by my net is, by definition, not a fish, and so is beyond scientific knowledge.” A third person says, “But, could you also derive the same law, not by looking at the fish, but by closely examining your net?” Eddington likes to think of himself as this third man, looking into the mind’s processes to find what others searched for in the external universe.

When Eddington died in 1944, a nearly complete manuscript was found on his desk. Published as Fundamental Theory, this work outlines the derivation of all seven fundamental constants from epistemological principles. Few have read this book, and fewer yet have understood it. The copy I examined at the Harvard College Observatory’s Wolbach Library showed little wear. It originally belonged to the American Association of Variable Star Observers who, we could guess, bought it out of interest in Eddington because of his early work in Cepheid pulsation. But this book has no mention of stellar physics. Useless to them, A.A.V.S.O. President Leon Campbell donated it to the Observatory in 1948, and there it has sat. It has no sign-out card. It doesn’t even carry the stamp of the Wolbach Library, instead carries still the old label of “Philips Library of Harvard College Observatory.” The only sign of use is on the last page, where Eddington gives his final formula for the Cosmical Number. In the margin someone has carefully carries out the expansion of (3/2)×2256 ×136, using logarithms, to give the result 2.36×1079. I wonder what this person thought when he carried out this calculation. History has recorded the remarks of famous scientists concerning Eddington’s late work, but what of the ordinary scientist, fed by the same desires, in search of the same beauty, what did he think of this great unification?

As I said at the beginning of this essay, I envision Eddington as one who, like Kepler, served to point out the harmonies of the world, but was unable to explain these harmonies by anything but aesthetic arguments. Someday, perhaps, a modern Newton will come, with a new Calculus of Thought and revolutionize physics as we know it. I would not be surprised if Eddington’s harmonies followed logically from this theory, as Kepler’s Laws follow from Newton’s.  For all the similarity between Eddington and Kepler, I am surprised to find that Eddington’s writings contain no reference to him. Only a speech, given in Weilder Stadt, Kepler’s birthplace, on the 350th anniversary of his birth, has mention of Kepler. When reading this speech, remember that it was given in 1928, three years before he went public with his harmonization:

I think it is not too fanciful to regard Kepler as in a particular degree the forerunner of the modern theoretical physicist, who is now trying to reduce the atom to order as Kepler reduced the Solar system to order. It is not merely similarity of subject matter but a similarity of outlook. We are apt to forget that in the discovery of the laws of the solar system, as well as in our laws of the atom, as essential step was the emancipation from mechanical models. Kepler did not proceed by thinking out possible mechanical devices by which the planets might be moved across the sky, -the wheels upon wheels of Ptolemy, or the whirling vortices of later speculation. I think this how most of us would have attacked the problem; we should have hunted for some concrete mechanism to yield the observed motion. But Kepler was guided by a sense of mathematical form, an aesthetic instinct for the fitness of things. In these later days it seems to us less incongruous that a planet should be guided by the condition of keeping the Action a minimum than it should be pulled and pushed by concrete agencies. In a like manner, Kepler was attracted by the thought of a planet moving so as to keep the growth of area steady – a suggestion which more orthodox minds would have rejected as too fanciful. I wonder how this abandonment of mechanical conceptions struck his contemporaries. Were there some who frowned on these rash adventures of scientific thought, and felt unable to accept the new kind of law without any explanation or model to show how it could possibly be worked? After Kepler came Newton, and gradually mechanism came into predominance again. It is only in the latest years that we have gone back to something like Kepler’s outlook, so that music of the spheres is no longer drowned by the roar of machinery [39].

A year later, Eddington wrote:

…In each revolution of scientific thought new words are set to the old music, and that which has gone before is not destroyed but refocused. Amid all our faulty attempts at expression the kernel of scientific truth steadily grows; and of this truth it might be said — The more it changes, the more it remains the same thing [40] .


[1] Kepler, J., Harmonices Mundi, trans. C.G. Wallis University of Chicago Press, Chicago, 1952. pg. 295

[2] Einstein, A., Annalen der Physik, 49 (1916)

[3] Weyl, H. Sitz. der Pruessischen Akad. d. Wiss., 1918

[4] Douglas, A.V., The Life of Arthur Stanley Eddington, Thomas Nelson and Sons Ltd. London, 1957., pg. 2

[5] Eddington, A.S., Report on the Relativity Theory of Gravitation, Physical Society of London, 1957., pg. 2

[6] Douglas, pg. 41

[7] Kepler, pg. 269

[8] Eddington, A.S., Space Time and Gravitation, University Press, Cambridge, 1953, pg. ii

[9] Ibid, pg. 200

[10] Douglas, pg. 124

[11] Eddington, A.S. The Nature of the Physical World, MacMillan Company, New York, 1929., pg vi

[12] Ibid, pg. 343

[13] A Midsummer Night’s Dream IV, i.

[14] Eddington, A.S., M.N.R.A.S., 90, 678 (1930)

[15] Hubble, E.P., Proc. Nat. Acad. Sci., 15, 168 (1927)

[16] Dirac, P.A., Proc. Roy. Soc. A., 117, 610 (1928)

[17] Eddington, A.S., Proc. Roy. Soc. A., 126, 696 (1930)

[18] Douglas, pg. 146

[19] Eddington, A.S., Proc. Camb. Phil. Soc., 27, 15 (1930)

[20] Eddington, A.S., Proc. Roy. Soc. A., 133, 605 (1931)

[21] Baade, W., Trans. Int. Astron. Union, 8, 397 (1952)

[22] Douglas, pg. 153

[23] Eddington, A.S., The Expanding Universe, University Press, Cambridge, 1933., pg. 113.

[24] Ibid., pg. 125

[25] Eddington, A.S., New Pathways in Science, University Press, Cambridge, 1935., pg. 231.

[26] Eddington, A.S., Relativity Theory of Protons and Electrons,University Press Cambrudge, 1936., pg. v.

[27] The Nature of the Physical World, pg. 337

[28] Douglas, pg. 171

[29] The Nature of the Physical World, pg. 331

[30] Douglas, pg. 161

[31] Chandrasekhar, S., Eddington: The Most Distinguished Astrophysicist of His Time, University Press, Cambridge, 1983, pg. 51.

[32] Ibid., pg. 52

[33] Eddington, A.S., Proc. Roy. Soc. A., 152, 253 (1935)

[34] Eddington, A.S., The Philosophy of Physical Science, University Press, Cambridge, 1949, pg. 170

[35] Ibid., pg. 171

[36] Eddington, A.S., M.N.R.A.S., 104, 200 (1944)

[37] Eddington, A.S., Proc. Camb. Phil. Soc., 40, 37 (1944)

[38] The Philosophy of Physical Science, pg. 16

[39] Beer, A. and D. Beer eds., Kepler: Four Hundred Years, Pergamon Press, Oxford, 1975., pg. 42

[40] The Nature of the Physical World, pg. 353.