Sir Arthur Stanley Eddington

“ The world of four dimensions, of which we are now speaking, is perfectly familiar to everybody. It is obvious to every one—even to the mathematician—that the world of solid and permanent objects has three dimensions and no more; that objects are arranged in a threefold order, which for any particular individual may be analysed into right-and-left, backwards-and-forwards, up-and-down. But it is no less obvious to every one that the world of events is of four dimensions ”

“ It is through this unification of geometry and mechanics that I should like to approach the problem of gravitation, showing that a field of force is a manifestation of the geometry of space and time. But I fear that that would be too technical; so we will approach it from a different angle.
We have shown that the contemplation of the world from the standpoint of a single observer is liable to distort its simplicity, and we have tried to obtain a juster idea by taking into account and combining other points of view. The more standpoints the better. ”

“ What has happened in the case of the earth has happened also in the case of the world, and a similar revolution of thought is needed. An observer, say at the centre of the earth, finds that there is a frame of space and time—a flat or Euclidean frame—in which he can locate things happening in his neighbourhood without distorting their natural simplicity. There is no gravitation, no tendency of bodies to fall, so long as the observer confines his observations to his immediate neighbourhood. ”

“ You see that the apple's view of things is simpler than Newton's. Newton had to invent a mysterious force dragging the apple down; the apple observes only a familiar physical agency propelling Newton up. ”

“ Einstein's special theory of relativity, which explains the indeterminateness of the frame of space and time, crowns the work of Copernicus who first led us to give up our insistence on a geocentric outlook on nature; Einstein's general theory of relativity, which reveals the curvature or non-Euclidean geometry of space and time, carries forward the rudimentary thought of those earlier astronomers who first contemplated the possibility that their existence lay on something which was not flat. These earlier revolutions are still a source of perplexity in childhood, which we soon outgrow ”

“ To free our thought from the fetters of space and time is an aspiration of the poet and the mystic, viewed somewhat coldly by the scientist who has too good reason to fear the confusion of loose ideas likely to ensue. ”

“ Our measurements of distance in space are found to be subject to certain laws—the laws of geometry. But it has now become impossible to regard the subject of space-geometry as complete in itself. Consider a triangle formed by three points (or events) in the four-dimensional world; if we happen to have drawn our instantaneous strata so that the three points lie in one stratum, then the triangle is a space-triangle and its properties fall within the scope of our classical geometry. ”

“ It is not determined by anything distinctive in the frame; it is determined by something distinctive in us—by the fact that our existence is bound to a particular planet and our motion is the motion of that planet. Nature offers an infinite choice of frames; we select the one in which we and our petty terrestrial concerns take the most distinguished position. ”

“ The sun, not the earth, is the real centre of the scheme of things—at least of those things in which astronomers at that time had interested themselves—and by transferring our view-point to the sun the simplicity of the planetary system becomes apparent. ”

“ Einstein's first great discovery was that there are many such systems of reckoning—many possible frames of space and time—exactly on all fours with one another. No one of these can be distinguished as more fundamental than the rest; no one frame rather than another can be identified as the scaffolding used in the construction of the world. ”

“ If I have succeeded in my object, you will have realized that the present revolution of scientific thought follows in natural sequence on the great revolutions at earlier epochs in the history of science. ”

“ When we give up the mediaeval geocentric standpoint, we must recognize that they belong to one geometry, of which our ordinary space-geometry is only a part or projection. But if you examine the proposition about time-triangles, you will see that it is a statement about the behaviour of clocks when they move about, a subject which obviously comes under the heading of mechanics. When we deal with the four-dimensional world we can no longer distinguish between geometry and mechanics. ”

“ We must try another plan. I do not think we can ever eliminate altogether the human element in our conception of nature; but we can eliminate a particular human element, namely, this framework of space and time. ”

“ It is for the latter reason that we emphasize the relativity of space—that lengths and distances differ according to the observer implied. Distance and duration are the most fundamental terms in physics; velocity, acceleration, force, energy, and so on, all depend on them; and we can scarcely make any statement in physics without direct or indirect reference to them. ”

“ We realize the distortion imported into the world of nature by the parochial standpoint from which we observe it, and we try to place ourselves so as to eliminate this distortion—so as to observe that which actually is. ”

“ And those relations of simplicity, which we seek when we try to obtain a comprehension of how the universe functions, must lie in the events themselves before they have been arbitrarily fitted into the frame. The most we can hope for from any frame is that it will not have distorted the simplicity which was originally present; whilst an ill-chosen frame may play havoc with the natural simplicity of things. We have seen that the simplicity of planetary motions was obscured in Ptolemy's frame, and became apparent in Copernicus's frame. ”

“ Indeed we are accustomed to extend it beyond the universe, and we even use the word 'now' with reference to the existence of those who have passed away from the material world. The investigations of the relativity theory show incontrovertibly that this supposed stratification is an illusion; there is not the slightest evidence for such a view of world-structure. The instantaneous state, which we have hitherto taken to be a natural stratum in the four-dimensional world of events, is merely an arbitrary partition created by ourselves to correspond with our geocentric outlook. ”

“ The revolution consisted in changing the view-point from which the phenomena were regarded. As presented to the earth the track of a planet is an elaborate epicycle; but Copernicus bade us transfer ourselves to the sun and look again. Instead of a path with loops and nodes, the orbit is now seen to be one of the most elementary curves—an ellipse. We have to realize that the little planet on which we stand is of no great account in the general scheme of nature ”

“ But it is a vain pursuit. Wherever we pitch our camera, the photograph is necessarily a two-dimensional picture distorted according to the laws of perspective; it is never a true semblance of the building itself. ”

“ You all know the law of the space-triangle—that if you measure with a scale from A to B and from B to C the sum of the readings is always greater than the measure from A to C. It is not so well known that there is a precisely analogous law for the time-triangle—that if you measure with a clock from A to B and from B to C the sum of the readings is always less than the reading of a clock measuring directly from A to C. In the space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together less than the third side. ”

“ If we take the standpoint of Newton the tug is applied to the apple, if the standpoint of the apple the tug is applied to Newton; so that in our synthesis of all standpoints we cannot decide which is being tugged, and the picture of gravitation as a tugging agent becomes impossible. Einstein replaces it by a different picture, which we shall perhaps better understand if we compare it with a very similar revolution of scientific thought which occurred long ago. ”

“ Surely then we can best indicate the revolutionary consequences of what we have learnt by the statement that distance and duration, and all the physical quantities derived from them, do not as hitherto supposed refer to anything absolute in the external world, but are relative quantities which alter when we pass from one observer to another with different motion. ”

“ The subject of our study is external nature, which is a world of events, common to all observers but represented by them differently in their parochial frames of space and time; it is obvious to the most commonplace experience that this absolute world contains a fourfold order. ”

“ There is a strange delusion that the fourth dimension must be something wholly beyond the conception of the ordinary man, and that only the mathematician can be initiated into its mysteries. It is true that the mathematician has the advantage of understanding the technical machinery for solving the problems which may arise in studying the world of four dimensions ”

“ I will conclude this part of the argument with an experimental application which illustrates the power of Einstein's method. Much study has of late been given to electrons moving with very high speeds; for example, the β particles shot off from radioactive substances are negative electrons which sometimes attain speeds of 100,000 miles a second. It is found by experiment that the rapid motion produces an increase of mass of these particles. I want to show that the theory of relativity gives a very simple explanation of just how this increase of mass occurs. ”

“ That is a theme which has attracted the popular novelist, and we often smile at his mistakes when sooner or later he forgets where he is supposed to be and endows his voyagers with some purely terrestrial appanage impossible on the star they are visiting. ”

“ The consequence in physics of the discovery that a yard is not an absolute chunk of space, and that what is a yard for one observer may be eighteen inches for another observer, may be compared with the consequences in economics of the discovery that a pound sterling is not an absolute quantity of wealth, and in certain circumstances may 'really' be seven and sixpence. The theorist may complain that this last statement tends to make a mystery of phenomena of currency which have really an intelligible explanation ”

“ It is so contrary to our natural prejudices to recognize that the world-wide instant now is created by ourselves and has no existence apart from our geocentric outlook, that I will spend a few moments trying to show its artificiality. ”

“ Both these cases are illustrated in Einstein's modifications of classical theory. Newton's law of gravitation explicitly refers to a space-time frame and therefore to a world stratified into instantaneous states. It proves to be impossible to free it from this reference to a particular stratification without modifying it. In fact if the crucial astronomical observations had shown that Newton's law and not Einstein's was the exact law of gravitation, this would have been evidence of a real stratification of the structure of the world—a stratification revealed by no other phenomena. ”

“ The contraction of the moving rod ought not to surprise us; it would be much more surprising if the rod were to maintain the same form in spite of the alteration of the electrical forces which determine the spacing of the molecules. But the remarkable thing is that the contraction is only apparent according to the outlook of the solar observer; and we on the earth, who travel with the rod, cannot appreciate it. The fact that the contraction happens to be very small is irrelevant. ”

“ But for ordinary terrestrial phenomena the position is reversed and Ptolemy's frame allows their natural simplicity to become apparent. In Copernicus's frame the most simple phenomena are brought about by highly complicated processes which mutually cancel one another. Ordinary objects contract and expand as they are moved about, and the changes are concealed by an elaborate conspiracy in which all the quantities of nature—electrical, optical, mechanical, gravitational—have joined. ”

“ We now recognize that the true explanation is that the earth's surface is curved; and the demoniacal complications appeared because we were forcing the earth's surface into an inappropriate flat frame which distorts the simplicity of things. ”

“ I think you will readily understand what is meant by a frame of space and time. It is the system of location to which we appeal when we state, for example, that one event is 100 miles distant from and 10 hours later than another. ”

“ We have been accustomed to regard the enduring world as composed of a continuous succession of instantaneous states, as though the world of events were stratified. Each event is supposed to lie in a definite instant or stratum, and the orderly succession of these strata makes up the whole of reality. The instant 'now' represents one such stratum running throughout the universe. ”