SIDELIGHTS ON RELATIVITY By Albert Einstein Contents ETHER AND THE THEORY OF RELATIVITY An Address delivered on May 5th, 1920, in the University of Leyden GEOMETRY AND EXPERIENCE An expanded form of an Address to the Prussian Academy of Sciencesin Berlin on January 27th, 1921. ETHER AND THE THEORY OF RELATIVITY An Address delivered on May 5th, 1920, in the University of Leyden How does it come about that alongside of the idea of ponderablematter, which is derived by abstraction from everyday life, thephysicists set the idea of the existence of another kind of matter, the ether? The explanation is probably to be sought in those phenomenawhich have given rise to the theory of action at a distance, andin the properties of light which have led to the undulatory theory. Let us devote a little while to the consideration of these twosubjects. Outside of physics we know nothing of action at a distance. Whenwe try to connect cause and effect in the experiences which naturalobjects afford us, it seems at first as if there were no other mutualactions than those of immediate contact, e. G. The communication ofmotion by impact, push and pull, heating or inducing combustion bymeans of a flame, etc. It is true that even in everyday experienceweight, which is in a sense action at a distance, plays a veryimportant part. But since in daily experience the weight of bodiesmeets us as something constant, something not linked to any causewhich is variable in time or place, we do not in everyday lifespeculate as to the cause of gravity, and therefore do not becomeconscious of its character as action at a distance. It was Newton'stheory of gravitation that first assigned a cause for gravity byinterpreting it as action at a distance, proceeding from masses. Newton's theory is probably the greatest stride ever made inthe effort towards the causal nexus of natural phenomena. And yetthis theory evoked a lively sense of discomfort among Newton'scontemporaries, because it seemed to be in conflict with theprinciple springing from the rest of experience, that there can bereciprocal action only through contact, and not through immediateaction at a distance. It is only with reluctance that man's desirefor knowledge endures a dualism of this kind. How was unity tobe preserved in his comprehension of the forces of nature? Eitherby trying to look upon contact forces as being themselves distantforces which admittedly are observable only at a very smalldistance--and this was the road which Newton's followers, who wereentirely under the spell of his doctrine, mostly preferred totake; or by assuming that the Newtonian action at a distance isonly _apparently_ immediate action at a distance, but in truth isconveyed by a medium permeating space, whether by movements or byelastic deformation of this medium. Thus the endeavour toward aunified view of the nature of forces leads to the hypothesis of anether. This hypothesis, to be sure, did not at first bring with itany advance in the theory of gravitation or in physics generally, so that it became customary to treat Newton's law of force as anaxiom not further reducible. But the ether hypothesis was boundalways to play some part in physical science, even if at first onlya latent part. When in the first half of the nineteenth century the far-reachingsimilarity was revealed which subsists between the properties oflight and those of elastic waves in ponderable bodies, the etherhypothesis found fresh support. It appeared beyond question thatlight must be interpreted as a vibratory process in an elastic, inertmedium filling up universal space. It also seemed to be a necessaryconsequence of the fact that light is capable of polarisation thatthis medium, the ether, must be of the nature of a solid body, because transverse waves are not possible in a fluid, but only ina solid. Thus the physicists were bound to arrive at the theoryof the "quasi-rigid" luminiferous ether, the parts of which cancarry out no movements relatively to one another except the smallmovements of deformation which correspond to light-waves. This theory--also called the theory of the stationary luminiferousether--moreover found a strong support in an experiment which isalso of fundamental importance in the special theory of relativity, the experiment of Fizeau, from which one was obliged to inferthat the luminiferous ether does not take part in the movements ofbodies. The phenomenon of aberration also favoured the theory ofthe quasi-rigid ether. The development of the theory of electricity along the path openedup by Maxwell and Lorentz gave the development of our ideas concerningthe ether quite a peculiar and unexpected turn. For Maxwell himselfthe ether indeed still had properties which were purely mechanical, although of a much more complicated kind than the mechanicalproperties of tangible solid bodies. But neither Maxwell nor hisfollowers succeeded in elaborating a mechanical model for the etherwhich might furnish a satisfactory mechanical interpretation ofMaxwell's laws of the electro-magnetic field. The laws were clearand simple, the mechanical interpretations clumsy and contradictory. Almost imperceptibly the theoretical physicists adapted themselvesto a situation which, from the standpoint of their mechanicalprogramme, was very depressing. They were particularly influencedby the electro-dynamical investigations of Heinrich Hertz. Forwhereas they previously had required of a conclusive theory thatit should content itself with the fundamental concepts which belongexclusively to mechanics (e. G. Densities, velocities, deformations, stresses) they gradually accustomed themselves to admitting electric andmagnetic force as fundamental concepts side by side with those ofmechanics, without requiring a mechanical interpretation for them. Thus the purely mechanical view of nature was gradually abandoned. But this change led to a fundamental dualism which in the long-runwas insupportable. A way of escape was now sought in the reversedirection, by reducing the principles of mechanics to thoseof electricity, and this especially as confidence in the strictvalidity of the equations of Newton's mechanics was shaken by theexperiments with beta-rays and rapid kathode rays. This dualism still confronts us in unextenuated form in the theoryof Hertz, where matter appears not only as the bearer of velocities, kinetic energy, and mechanical pressures, but also as the bearer ofelectromagnetic fields. Since such fields also occur _in vacuo_--i. E. In free ether--the ether also appears as bearer of electromagneticfields. The ether appears indistinguishable in its functions fromordinary matter. Within matter it takes part in the motion of matterand in empty space it has everywhere a velocity; so that the etherhas a definitely assigned velocity throughout the whole of space. There is no fundamental difference between Hertz's ether andponderable matter (which in part subsists in the ether). The Hertz theory suffered not only from the defect of ascribingto matter and ether, on the one hand mechanical states, and on theother hand electrical states, which do not stand in any conceivablerelation to each other; it was also at variance with the result ofFizeau's important experiment on the velocity of the propagationof light in moving fluids, and with other established experimentalresults. Such was the state of things when H. A. Lorentz entered upon thescene. He brought theory into harmony with experience by means ofa wonderful simplification of theoretical principles. He achievedthis, the most important advance in the theory of electricity sinceMaxwell, by taking from ether its mechanical, and from matter itselectromagnetic qualities. As in empty space, so too in the interiorof material bodies, the ether, and not matter viewed atomistically, was exclusively the seat of electromagnetic fields. According toLorentz the elementary particles of matter alone are capable ofcarrying out movements; their electromagnetic activity is entirelyconfined to the carrying of electric charges. Thus Lorentz succeededin reducing all electromagnetic happenings to Maxwell's equationsfor free space. As to the mechanical nature of the Lorentzian ether, it may be saidof it, in a somewhat playful spirit, that immobility is the onlymechanical property of which it has not been deprived by H. A. Lorentz. It may be added that the whole change in the conceptionof the ether which the special theory of relativity brought about, consisted in taking away from the ether its last mechanical quality, namely, its immobility. How this is to be understood will forthwithbe expounded. The space-time theory and the kinematics of the special theoryof relativity were modelled on the Maxwell-Lorentz theory of theelectromagnetic field. This theory therefore satisfies the conditionsof the special theory of relativity, but when viewed from the latterit acquires a novel aspect. For if K be a system of co-ordinatesrelatively to which the Lorentzian ether is at rest, theMaxwell-Lorentz equations are valid primarily with reference to K. But by the special theory of relativity the same equations withoutany change of meaning also hold in relation to any new system ofco-ordinates K' which is moving in uniform translation relativelyto K. Now comes the anxious question:--Why must I in the theorydistinguish the K system above all K' systems, which are physicallyequivalent to it in all respects, by assuming that the etheris at rest relatively to the K system? For the theoretician suchan asymmetry in the theoretical structure, with no correspondingasymmetry in the system of experience, is intolerable. If we assumethe ether to be at rest relatively to K, but in motion relativelyto K', the physical equivalence of K and K' seems to me from thelogical standpoint, not indeed downright incorrect, but neverthelessinacceptable. The next position which it was possible to take up in face of thisstate of things appeared to be the following. The ether does notexist at all. The electromagnetic fields are not states of a medium, and are not bound down to any bearer, but they are independentrealities which are not reducible to anything else, exactly likethe atoms of ponderable matter. This conception suggests itselfthe more readily as, according to Lorentz's theory, electromagneticradiation, like ponderable matter, brings impulse and energy withit, and as, according to the special theory of relativity, bothmatter and radiation are but special forms of distributed energy, ponderable mass losing its isolation and appearing as a specialform of energy. More careful reflection teaches us, however, that the special theoryof relativity does not compel us to deny ether. We may assume theexistence of an ether; only we must give up ascribing a definitestate of motion to it, i. E. We must by abstraction take from it thelast mechanical characteristic which Lorentz had still left it. Weshall see later that this point of view, the conceivability of whichI shall at once endeavour to make more intelligible by a somewhathalting comparison, is justified by the results of the generaltheory of relativity. Think of waves on the surface of water. Here we can describe twoentirely different things. Either we may observe how the undulatorysurface forming the boundary between water and air alters in the courseof time; or else--with the help of small floats, for instance--wecan observe how the position of the separate particles of wateralters in the course of time. If the existence of such floats fortracking the motion of the particles of a fluid were a fundamentalimpossibility in physics--if, in fact, nothing else whatever wereobservable than the shape of the space occupied by the water as itvaries in time, we should have no ground for the assumption thatwater consists of movable particles. But all the same we couldcharacterise it as a medium. We have something like this in the electromagnetic field. For we maypicture the field to ourselves as consisting of lines of force. Ifwe wish to interpret these lines of force to ourselves as somethingmaterial in the ordinary sense, we are tempted to interpret thedynamic processes as motions of these lines of force, such that eachseparate line of force is tracked through the course of time. It iswell known, however, that this way of regarding the electromagneticfield leads to contradictions. Generalising we must say this:--There may be supposed to be extendedphysical objects to which the idea of motion cannot be applied. They may not be thought of as consisting of particles which allowthemselves to be separately tracked through time. In Minkowski'sidiom this is expressed as follows:--Not every extended conformationin the four-dimensional world can be regarded as composedof world-threads. The special theory of relativity forbids us toassume the ether to consist of particles observable through time, but the hypothesis of ether in itself is not in conflict with thespecial theory of relativity. Only we must be on our guard againstascribing a state of motion to the ether. Certainly, from the standpoint of the special theory of relativity, the ether hypothesis appears at first to be an empty hypothesis. Inthe equations of the electromagnetic field there occur, in additionto the densities of the electric charge, _only_ the intensitiesof the field. The career of electromagnetic processes _in vacuo_appears to be completely determined by these equations, uninfluencedby other physical quantities. The electromagnetic fields appear asultimate, irreducible realities, and at first it seems superfluousto postulate a homogeneous, isotropic ether-medium, and to envisageelectromagnetic fields as states of this medium. But on the other hand there is a weighty argument to be adducedin favour of the ether hypothesis. To deny the ether is ultimatelyto assume that empty space has no physical qualities whatever. Thefundamental facts of mechanics do not harmonize with this view. For the mechanical behaviour of a corporeal system hovering freelyin empty space depends not only on relative positions (distances)and relative velocities, but also on its state of rotation, whichphysically may be taken as a characteristic not appertaining to thesystem in itself. In order to be able to look upon the rotation ofthe system, at least formally, as something real, Newton objectivisesspace. Since he classes his absolute space together with real things, forhim rotation relative to an absolute space is also something real. Newton might no less well have called his absolute space "Ether";what is essential is merely that besides observable objects, anotherthing, which is not perceptible, must be looked upon as real, to enable acceleration or rotation to be looked upon as somethingreal. It is true that Mach tried to avoid having to accept as real somethingwhich is not observable by endeavouring to substitute in mechanicsa mean acceleration with reference to the totality of the masses inthe universe in place of an acceleration with reference to absolutespace. But inertial resistance opposed to relative acceleration ofdistant masses presupposes action at a distance; and as the modernphysicist does not believe that he may accept this action ata distance, he comes back once more, if he follows Mach, to theether, which has to serve as medium for the effects of inertia. Butthis conception of the ether to which we are led by Mach's way ofthinking differs essentially from the ether as conceived by Newton, by Fresnel, and by Lorentz. Mach's ether not only _conditions_ thebehaviour of inert masses, but _is also conditioned_ in its stateby them. Mach's idea finds its full development in the ether of the generaltheory of relativity. According to this theory the metricalqualities of the continuum of space-time differ in the environmentof different points of space-time, and are partly conditioned by thematter existing outside of the territory under consideration. Thisspace-time variability of the reciprocal relations of the standardsof space and time, or, perhaps, the recognition of the fact that"empty space" in its physical relation is neither homogeneous norisotropic, compelling us to describe its state by ten functions (thegravitation potentials g_(mn)), has, I think, finally disposed ofthe view that space is physically empty. But therewith theconception of the ether has again acquired an intelligible content, although this content differs widely from that of the ether of themechanical undulatory theory of light. The ether of the generaltheory of relativity is a medium which is itself devoid of _all_mechanical and kinematical qualities, but helps to determinemechanical (and electromagnetic) events. What is fundamentally new in the ether of the general theory ofrelativity as opposed to the ether of Lorentz consists in this, thatthe state of the former is at every place determined by connectionswith the matter and the state of the ether in neighbouring places, which are amenable to law in the form of differential equations;whereas the state of the Lorentzian ether in the absence ofelectromagnetic fields is conditioned by nothing outside itself, and is everywhere the same. The ether of the general theory ofrelativity is transmuted conceptually into the ether of Lorentz ifwe substitute constants for the functions of space which describethe former, disregarding the causes which condition its state. Thus we may also say, I think, that the ether of the general theoryof relativity is the outcome of the Lorentzian ether, throughrelativation. As to the part which the new ether is to play in the physics ofthe future we are not yet clear. We know that it determines themetrical relations in the space-time continuum, e. G. The configurativepossibilities of solid bodies as well as the gravitational fields;but we do not know whether it has an essential share in the structureof the electrical elementary particles constituting matter. Nor dowe know whether it is only in the proximity of ponderable massesthat its structure differs essentially from that of the Lorentzianether; whether the geometry of spaces of cosmic extent is approximatelyEuclidean. But we can assert by reason of the relativistic equationsof gravitation that there must be a departure from Euclideanrelations, with spaces of cosmic order of magnitude, if there existsa positive mean density, no matter how small, of the matter in theuniverse. In this case the universe must of necessity be spatiallyunbounded and of finite magnitude, its magnitude being determinedby the value of that mean density. If we consider the gravitational field and the electromagnetic fieldfrom the stand-point of the ether hypothesis, we find a remarkabledifference between the two. There can be no space nor any partof space without gravitational potentials; for these confer uponspace its metrical qualities, without which it cannot be imaginedat all. The existence of the gravitational field is inseparablybound up with the existence of space. On the other hand a part ofspace may very well be imagined without an electromagnetic field;thus in contrast with the gravitational field, the electromagneticfield seems to be only secondarily linked to the ether, the formalnature of the electromagnetic field being as yet in no way determinedby that of gravitational ether. From the present state of theoryit looks as if the electromagnetic field, as opposed to thegravitational field, rests upon an entirely new formal _motif_, as though nature might just as well have endowed the gravitationalether with fields of quite another type, for example, with fieldsof a scalar potential, instead of fields of the electromagnetictype. Since according to our present conceptions the elementary particlesof matter are also, in their essence, nothing else than condensationsof the electromagnetic field, our present view of the universepresents two realities which are completely separated from each otherconceptually, although connected causally, namely, gravitational etherand electromagnetic field, or--as they might also be called--spaceand matter. Of course it would be a great advance if we could succeed incomprehending the gravitational field and the electromagnetic fieldtogether as one unified conformation. Then for the first time theepoch of theoretical physics founded by Faraday and Maxwell wouldreach a satisfactory conclusion. The contrast between ether andmatter would fade away, and, through the general theory of relativity, the whole of physics would become a complete system of thought, like geometry, kinematics, and the theory of gravitation. Anexceedingly ingenious attempt in this direction has been made bythe mathematician H. Weyl; but I do not believe that his theory willhold its ground in relation to reality. Further, in contemplatingthe immediate future of theoretical physics we ought not unconditionallyto reject the possibility that the facts comprised in the quantumtheory may set bounds to the field theory beyond which it cannotpass. Recapitulating, we may say that according to the general theory ofrelativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theoryof relativity space without ether is unthinkable; for in such spacethere not only would be no propagation of light, but also no possibilityof existence for standards of space and time (measuring-rods andclocks), nor therefore any space-time intervals in the physicalsense. But this ether may not be thought of as endowed with thequality characteristic of ponderable media, as consisting of partswhich may be tracked through time. The idea of motion may not beapplied to it. GEOMETRY AND EXPERIENCE An expanded form of an Address to the Prussian Academy of Sciencesin Berlin on January 27th, 1921. One reason why mathematics enjoys special esteem, above all othersciences, is that its laws are absolutely certain and indisputable, while those of all other sciences are to some extent debatable andin constant danger of being overthrown by newly discovered facts. In spite of this, the investigator in another department of sciencewould not need to envy the mathematician if the laws of mathematicsreferred to objects of our mere imagination, and not to objectsof reality. For it cannot occasion surprise that different personsshould arrive at the same logical conclusions when they have alreadyagreed upon the fundamental laws (axioms), as well as the methodsby which other laws are to be deduced therefrom. But there is anotherreason for the high repute of mathematics, in that it is mathematicswhich affords the exact natural sciences a certain measure ofsecurity, to which without mathematics they could not attain. At this point an enigma presents itself which in all ages has agitatedinquiring minds. How can it be that mathematics, being after alla product of human thought which is independent of experience, isso admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathomthe properties of real things. In my opinion the answer to this question is, briefly, this:--As faras the laws of mathematics refer to reality, they are not certain;and as far as they are certain, they do not refer to reality. It seems to me that complete clearness as to this state of thingsfirst became common property through that new departure in mathematicswhich is known by the name of mathematical logic or "Axiomatics. "The progress achieved by axiomatics consists in its having neatlyseparated the logical-formal from its objective or intuitivecontent; according to axiomatics the logical-formal alone formsthe subject-matter of mathematics, which is not concerned with theintuitive or other content associated with the logical-formal. Let us for a moment consider from this point of view any axiom ofgeometry, for instance, the following:--Through two points in spacethere always passes one and only one straight line. How is thisaxiom to be interpreted in the older sense and in the more modernsense? The older interpretation:--Every one knows what a straight lineis, and what a point is. Whether this knowledge springs from anability of the human mind or from experience, from some collaborationof the two or from some other source, is not for the mathematicianto decide. He leaves the question to the philosopher. Being basedupon this knowledge, which precedes all mathematics, the axiomstated above is, like all other axioms, self-evident, that is, itis the expression of a part of this _a priori_ knowledge. The more modern interpretation:--Geometry treats of entities whichare denoted by the words straight line, point, etc. These entitiesdo not take for granted any knowledge or intuition whatever, butthey presuppose only the validity of the axioms, such as the onestated above, which are to be taken in a purely formal sense, i. E. As void of all content of intuition or experience. These axioms arefree creations of the human mind. All other propositions of geometryare logical inferences from the axioms (which are to be taken inthe nominalistic sense only). The matter of which geometry treatsis first defined by the axioms. Schlick in his book on epistemology hastherefore characterised axioms very aptly as "implicit definitions. " This view of axioms, advocated by modern axiomatics, purges mathematicsof all extraneous elements, and thus dispels the mystic obscuritywhich formerly surrounded the principles of mathematics. But a presentation of its principles thus clarified makes it alsoevident that mathematics as such cannot predicate anything aboutperceptual objects or real objects. In axiomatic geometry the words"point, " "straight line, " etc. , stand only for empty conceptualschemata. That which gives them substance is not relevant tomathematics. Yet on the other hand it is certain that mathematics generally, and particularly geometry, owes its existence to the need whichwas felt of learning something about the relations of real thingsto one another. The very word geometry, which, of course, meansearth-measuring, proves this. For earth-measuring has to do withthe possibilities of the disposition of certain natural objectswith respect to one another, namely, with parts of the earth, measuring-lines, measuring-wands, etc. It is clear that the systemof concepts of axiomatic geometry alone cannot make any assertionsas to the relations of real objects of this kind, which we willcall practically-rigid bodies. To be able to make such assertions, geometry must be stripped of its merely logical-formal characterby the co-ordination of real objects of experience with the emptyconceptual frame-work of axiomatic geometry. To accomplish this, we need only add the proposition:--Solid bodies are related, withrespect to their possible dispositions, as are bodies in Euclideangeometry of three dimensions. Then the propositions of Euclid containaffirmations as to the relations of practically-rigid bodies. Geometry thus completed is evidently a natural science; we may infact regard it as the most ancient branch of physics. Its affirmationsrest essentially on induction from experience, but not on logicalinferences only. We will call this completed geometry "practicalgeometry, " and shall distinguish it in what follows from "purelyaxiomatic geometry. " The question whether the practical geometryof the universe is Euclidean or not has a clear meaning, and itsanswer can only be furnished by experience. All linear measurementin physics is practical geometry in this sense, so too is geodeticand astronomical linear measurement, if we call to our help thelaw of experience that light is propagated in a straight line, andindeed in a straight line in the sense of practical geometry. I attach special importance to the view of geometry which Ihave just set forth, because without it I should have been unableto formulate the theory of relativity. Without it the followingreflection would have been impossible:--In a system of referencerotating relatively to an inert system, the laws of disposition ofrigid bodies do not correspond to the rules of Euclidean geometryon account of the Lorentz contraction; thus if we admit non-inertsystems we must abandon Euclidean geometry. The decisive step inthe transition to general co-variant equations would certainly nothave been taken if the above interpretation had not served as astepping-stone. If we deny the relation between the body of axiomaticEuclidean geometry and the practically-rigid body of reality, we readily arrive at the following view, which was entertained bythat acute and profound thinker, H. Poincare:--Euclidean geometryis distinguished above all other imaginable axiomatic geometriesby its simplicity. Now since axiomatic geometry by itself containsno assertions as to the reality which can be experienced, but cando so only in combination with physical laws, it should be possibleand reasonable--whatever may be the nature of reality--to retainEuclidean geometry. For if contradictions between theory andexperience manifest themselves, we should rather decide to changephysical laws than to change axiomatic Euclidean geometry. If wedeny the relation between the practically-rigid body and geometry, we shall indeed not easily free ourselves from the conventionthat Euclidean geometry is to be retained as the simplest. Whyis the equivalence of the practically-rigid body and the body ofgeometry--which suggests itself so readily--denied by Poincare andother investigators? Simply because under closer inspection thereal solid bodies in nature are not rigid, because their geometricalbehaviour, that is, their possibilities of relative disposition, depend upon temperature, external forces, etc. Thus the original, immediate relation between geometry and physical reality appearsdestroyed, and we feel impelled toward the following more generalview, which characterizes Poincare's standpoint. Geometry (G)predicates nothing about the relations of real things, but onlygeometry together with the purport (P) of physical laws can do so. Using symbols, we may say that only the sum of (G) + (P) is subjectto the control of experience. Thus (G) may be chosen arbitrarily, and also parts of (P); all these laws are conventions. All thatis necessary to avoid contradictions is to choose the remainder of(P) so that (G) and the whole of (P) are together in accord withexperience. Envisaged in this way, axiomatic geometry and the partof natural law which has been given a conventional status appearas epistemologically equivalent. _Sub specie aeterni_ Poincare, in my opinion, is right. The ideaof the measuring-rod and the idea of the clock co-ordinated with itin the theory of relativity do not find their exact correspondencein the real world. It is also clear that the solid body and theclock do not in the conceptual edifice of physics play the part ofirreducible elements, but that of composite structures, which maynot play any independent part in theoretical physics. But it is myconviction that in the present stage of development of theoreticalphysics these ideas must still be employed as independent ideas;for we are still far from possessing such certain knowledgeof theoretical principles as to be able to give exact theoreticalconstructions of solid bodies and clocks. Further, as to the objection that there are no really rigid bodiesin nature, and that therefore the properties predicated of rigidbodies do not apply to physical reality, --this objection is byno means so radical as might appear from a hasty examination. Forit is not a difficult task to determine the physical state of ameasuring-rod so accurately that its behaviour relatively to othermeasuring-bodies shall be sufficiently free from ambiguity to allowit to be substituted for the "rigid" body. It is to measuring-bodiesof this kind that statements as to rigid bodies must be referred. All practical geometry is based upon a principle which is accessibleto experience, and which we will now try to realise. We willcall that which is enclosed between two boundaries, marked upon apractically-rigid body, a tract. We imagine two practically-rigidbodies, each with a tract marked out on it. These two tracts aresaid to be "equal to one another" if the boundaries of the one tractcan be brought to coincide permanently with the boundaries of theother. We now assume that: If two tracts are found to be equal once and anywhere, they areequal always and everywhere. Not only the practical geometry of Euclid, but also its nearestgeneralisation, the practical geometry of Riemann, and therewiththe general theory of relativity, rest upon this assumption. Of theexperimental reasons which warrant this assumption I will mentiononly one. The phenomenon of the propagation of light in empty spaceassigns a tract, namely, the appropriate path of light, to eachinterval of local time, and conversely. Thence it follows thatthe above assumption for tracts must also hold good for intervalsof clock-time in the theory of relativity. Consequently it may beformulated as follows:--If two ideal clocks are going at the samerate at any time and at any place (being then in immediate proximityto each other), they will always go at the same rate, no matter whereand when they are again compared with each other at one place. --Ifthis law were not valid for real clocks, the proper frequenciesfor the separate atoms of the same chemical element would not bein such exact agreement as experience demonstrates. The existenceof sharp spectral lines is a convincing experimental proof of theabove-mentioned principle of practical geometry. This is the ultimatefoundation in fact which enables us to speak with meaning of themensuration, in Riemann's sense of the word, of the four-dimensionalcontinuum of space-time. The question whether the structure of this continuum is Euclidean, or in accordance with Riemann's general scheme, or otherwise, is, according to the view which is here being advocated, properlyspeaking a physical question which must be answered by experience, and not a question of a mere convention to be selected on practicalgrounds. Riemann's geometry will be the right thing if the lawsof disposition of practically-rigid bodies are transformable intothose of the bodies of Euclid's geometry with an exactitude whichincreases in proportion as the dimensions of the part of space-timeunder consideration are diminished. It is true that this proposed physical interpretation of geometrybreaks down when applied immediately to spaces of sub-molecularorder of magnitude. But nevertheless, even in questions asto the constitution of elementary particles, it retains part ofits importance. For even when it is a question of describing theelectrical elementary particles constituting matter, the attemptmay still be made to ascribe physical importance to those ideasof fields which have been physically defined for the purposeof describing the geometrical behaviour of bodies which are largeas compared with the molecule. Success alone can decide as to thejustification of such an attempt, which postulates physical realityfor the fundamental principles of Riemann's geometry outside of thedomain of their physical definitions. It might possibly turn outthat this extrapolation has no better warrant than the extrapolationof the idea of temperature to parts of a body of molecular orderof magnitude. It appears less problematical to extend the ideas of practicalgeometry to spaces of cosmic order of magnitude. It might, of course, be objected that a construction composed of solid rods departs moreand more from ideal rigidity in proportion as its spatial extentbecomes greater. But it will hardly be possible, I think, to assignfundamental significance to this objection. Therefore the questionwhether the universe is spatially finite or not seems to medecidedly a pregnant question in the sense of practical geometry. I do not even consider it impossible that this question will beanswered before long by astronomy. Let us call to mind what thegeneral theory of relativity teaches in this respect. It offerstwo possibilities:-- 1. The universe is spatially infinite. This can be so only if theaverage spatial density of the matter in universal space, concentratedin the stars, vanishes, i. E. If the ratio of the total mass of thestars to the magnitude of the space through which they are scatteredapproximates indefinitely to the value zero when the spaces takeninto consideration are constantly greater and greater. 2. The universe is spatially finite. This must be so, if there isa mean density of the ponderable matter in universal space differingfrom zero. The smaller that mean density, the greater is the volumeof universal space. I must not fail to mention that a theoretical argument can be adduced infavour of the hypothesis of a finite universe. The general theoryof relativity teaches that the inertia of a given body is greater asthere are more ponderable masses in proximity to it; thus it seemsvery natural to reduce the total effect of inertia of a body toaction and reaction between it and the other bodies in the universe, as indeed, ever since Newton's time, gravity has been completelyreduced to action and reaction between bodies. From the equationsof the general theory of relativity it can be deduced that thistotal reduction of inertia to reciprocal action between masses--asrequired by E. Mach, for example--is possible only if the universeis spatially finite. On many physicists and astronomers this argument makes no impression. Experience alone can finally decide which of the two possibilitiesis realised in nature. How can experience furnish an answer? At firstit might seem possible to determine the mean density of matter byobservation of that part of the universe which is accessible to ourperception. This hope is illusory. The distribution of the visiblestars is extremely irregular, so that we on no account may ventureto set down the mean density of star-matter in the universe asequal, let us say, to the mean density in the Milky Way. In anycase, however great the space examined may be, we could not feelconvinced that there were no more stars beyond that space. So itseems impossible to estimate the mean density. But there is anotherroad, which seems to me more practicable, although it also presentsgreat difficulties. For if we inquire into the deviations shownby the consequences of the general theory of relativity which areaccessible to experience, when these are compared with the consequencesof the Newtonian theory, we first of all find a deviation whichshows itself in close proximity to gravitating mass, and has beenconfirmed in the case of the planet Mercury. But if the universeis spatially finite there is a second deviation from the Newtoniantheory, which, in the language of the Newtonian theory, may beexpressed thus:--The gravitational field is in its nature such asif it were produced, not only by the ponderable masses, but also bya mass-density of negative sign, distributed uniformly throughoutspace. Since this factitious mass-density would have to be enormouslysmall, it could make its presence felt only in gravitating systemsof very great extent. Assuming that we know, let us say, the statistical distributionof the stars in the Milky Way, as well as their masses, then byNewton's law we can calculate the gravitational field and the meanvelocities which the stars must have, so that the Milky Way shouldnot collapse under the mutual attraction of its stars, but shouldmaintain its actual extent. Now if the actual velocities of the stars, which can, of course, be measured, were smaller than the calculatedvelocities, we should have a proof that the actual attractionsat great distances are smaller than by Newton's law. From such adeviation it could be proved indirectly that the universe is finite. It would even be possible to estimate its spatial magnitude. Can we picture to ourselves a three-dimensional universe which isfinite, yet unbounded? The usual answer to this question is "No, " but that is not the rightanswer. The purpose of the following remarks is to show that theanswer should be "Yes. " I want to show that without any extraordinarydifficulty we can illustrate the theory of a finite universe bymeans of a mental image to which, with some practice, we shall soongrow accustomed. First of all, an observation of epistemological nature. Ageometrical-physical theory as such is incapable of being directlypictured, being merely a system of concepts. But these conceptsserve the purpose of bringing a multiplicity of real or imaginarysensory experiences into connection in the mind. To "visualise"a theory, or bring it home to one's mind, therefore means to givea representation to that abundance of experiences for which thetheory supplies the schematic arrangement. In the present case wehave to ask ourselves how we can represent that relation of solidbodies with respect to their reciprocal disposition (contact) whichcorresponds to the theory of a finite universe. There is reallynothing new in what I have to say about this; but innumerablequestions addressed to me prove that the requirements of those whothirst for knowledge of these matters have not yet been completelysatisfied. So, will the initiated please pardon me, if part of what I shallbring forward has long been known? What do we wish to express when we say that our space is infinite?Nothing more than that we might lay any number whatever of bodiesof equal sizes side by side without ever filling space. Supposethat we are provided with a great many wooden cubes all of thesame size. In accordance with Euclidean geometry we can place themabove, beside, and behind one another so as to fill a part of spaceof any dimensions; but this construction would never be finished;we could go on adding more and more cubes without ever findingthat there was no more room. That is what we wish to express whenwe say that space is infinite. It would be better to say that spaceis infinite in relation to practically-rigid bodies, assuming thatthe laws of disposition for these bodies are given by Euclideangeometry. Another example of an infinite continuum is the plane. On a planesurface we may lay squares of cardboard so that each side of anysquare has the side of another square adjacent to it. The constructionis never finished; we can always go on laying squares--if their lawsof disposition correspond to those of plane figures of Euclideangeometry. The plane is therefore infinite in relation to thecardboard squares. Accordingly we say that the plane is an infinitecontinuum of two dimensions, and space an infinite continuum ofthree dimensions. What is here meant by the number of dimensions, I think I may assume to be known. Now we take an example of a two-dimensional continuum which isfinite, but unbounded. We imagine the surface of a large globe anda quantity of small paper discs, all of the same size. We placeone of the discs anywhere on the surface of the globe. If we movethe disc about, anywhere we like, on the surface of the globe, we do not come upon a limit or boundary anywhere on the journey. Therefore we say that the spherical surface of the globe is anunbounded continuum. Moreover, the spherical surface is a finitecontinuum. For if we stick the paper discs on the globe, so thatno disc overlaps another, the surface of the globe will finallybecome so full that there is no room for another disc. This simplymeans that the spherical surface of the globe is finite in relationto the paper discs. Further, the spherical surface is a non-Euclideancontinuum of two dimensions, that is to say, the laws of dispositionfor the rigid figures lying in it do not agree with those of theEuclidean plane. This can be shown in the following way. Placea paper disc on the spherical surface, and around it in a circleplace six more discs, each of which is to be surrounded in turnby six discs, and so on. If this construction is made on a planesurface, we have an uninterrupted disposition in which there aresix discs touching every disc except those which lie on the outside. [Figure 1: Discs maximally packed on a plane] On the spherical surface the construction also seems to promisesuccess at the outset, and the smaller the radius of the discin proportion to that of the sphere, the more promising it seems. But as the construction progresses it becomes more and more patentthat the disposition of the discs in the manner indicated, withoutinterruption, is not possible, as it should be possible by Euclideangeometry of the the plane surface. In this way creatures whichcannot leave the spherical surface, and cannot even peep out fromthe spherical surface into three-dimensional space, might discover, merely by experimenting with discs, that their two-dimensional"space" is not Euclidean, but spherical space. From the latest results of the theory of relativity it is probablethat our three-dimensional space is also approximately spherical, that is, that the laws of disposition of rigid bodies in it arenot given by Euclidean geometry, but approximately by sphericalgeometry, if only we consider parts of space which are sufficientlygreat. Now this is the place where the reader's imagination boggles. "Nobody can imagine this thing, " he cries indignantly. "It can besaid, but cannot be thought. I can represent to myself a sphericalsurface well enough, but nothing analogous to it in three dimensions. " [Figure 2: A circle projected from a sphere onto a plane] We must try to surmount this barrier in the mind, and the patientreader will see that it is by no means a particularly difficulttask. For this purpose we will first give our attention once more tothe geometry of two-dimensional spherical surfaces. In the adjoiningfigure let _K_ be the spherical surface, touched at _S_ by a plane, _E_, which, for facility of presentation, is shown in the drawing asa bounded surface. Let _L_ be a disc on the spherical surface. Nowlet us imagine that at the point _N_ of the spherical surface, diametrically opposite to _S_, there is a luminous point, throwing ashadow _L'_ of the disc _L_ upon the plane _E_. Every point on thesphere has its shadow on the plane. If the disc on the sphere _K_ ismoved, its shadow _L'_ on the plane _E_ also moves. When the disc_L_ is at _S_, it almost exactly coincides with its shadow. If itmoves on the spherical surface away from _S_ upwards, the discshadow _L'_ on the plane also moves away from _S_ on the planeoutwards, growing bigger and bigger. As the disc _L_ approaches theluminous point _N_, the shadow moves off to infinity, and becomesinfinitely great. Now we put the question, What are the laws of disposition of thedisc-shadows _L'_ on the plane _E_? Evidently they are exactly thesame as the laws of disposition of the discs _L_ on the sphericalsurface. For to each original figure on _K_ there is a correspondingshadow figure on _E_. If two discs on _K_ are touching, theirshadows on _E_ also touch. The shadow-geometry on the plane agreeswith the the disc-geometry on the sphere. If we call the disc-shadowsrigid figures, then spherical geometry holds good on the plane _E_with respect to these rigid figures. Moreover, the plane is finitewith respect to the disc-shadows, since only a finite number ofthe shadows can find room on the plane. At this point somebody will say, "That is nonsense. The disc-shadowsare _not_ rigid figures. We have only to move a two-foot rule abouton the plane _E_ to convince ourselves that the shadows constantlyincrease in size as they move away from _S_ on the plane towardsinfinity. " But what if the two-foot rule were to behave on theplane _E_ in the same way as the disc-shadows _L'_? It would thenbe impossible to show that the shadows increase in size as theymove away from _S_; such an assertion would then no longer haveany meaning whatever. In fact the only objective assertion that canbe made about the disc-shadows is just this, that they are relatedin exactly the same way as are the rigid discs on the sphericalsurface in the sense of Euclidean geometry. We must carefully bear in mind that our statement as to the growthof the disc-shadows, as they move away from _S_ towards infinity, has in itself no objective meaning, as long as we are unable toemploy Euclidean rigid bodies which can be moved about on the plane_E_ for the purpose of comparing the size of the disc-shadows. Inrespect of the laws of disposition of the shadows _L'_, the point_S_ has no special privileges on the plane any more than on thespherical surface. The representation given above of spherical geometry on theplane is important for us, because it readily allows itself to betransferred to the three-dimensional case. Let us imagine a point _S_ of our space, and a great numberof small spheres, _L'_, which can all be brought to coincide withone another. But these spheres are not to be rigid in the senseof Euclidean geometry; their radius is to increase (in the senseof Euclidean geometry) when they are moved away from _S_ towardsinfinity, and this increase is to take place in exact accordancewith the same law as applies to the increase of the radii of thedisc-shadows _L'_ on the plane. After having gained a vivid mental image of the geometricalbehaviour of our _L'_ spheres, let us assume that in our space thereare no rigid bodies at all in the sense of Euclidean geometry, butonly bodies having the behaviour of our _L'_ spheres. Then we shallhave a vivid representation of three-dimensional spherical space, or, rather of three-dimensional spherical geometry. Here our spheresmust be called "rigid" spheres. Their increase in size as theydepart from _S_ is not to be detected by measuring withmeasuring-rods, any more than in the case of the disc-shadows on_E_, because the standards of measurement behave in the same way asthe spheres. Space is homogeneous, that is to say, the samespherical configurations are possible in the environment of allpoints. * Our space is finite, because, in consequence of the"growth" of the spheres, only a finite number of them can find roomin space. * This is intelligible without calculation--but only for thetwo-dimensional case--if we revert once more to the case of the discon the surface of the sphere. In this way, by using as stepping-stones the practice in thinkingand visualisation which Euclidean geometry gives us, we have acquireda mental picture of spherical geometry. We may without difficultyimpart more depth and vigour to these ideas by carrying out specialimaginary constructions. Nor would it be difficult to represent thecase of what is called elliptical geometry in an analogous manner. My only aim to-day has been to show that the human faculty ofvisualisation is by no means bound to capitulate to non-Euclideangeometry.