FOOTNOTES:

FOOTNOTES:[1]As exemplified in the Pythagorean discovery of the relationship between the length of a vibrating string and the pitch of its note, a discovery utilised in musical instruments. Another example is represented by Archimedes’ solution of the problem of Hieron’s gold tiara.[2]The appellationGalilean motiondoes not appear to have been adopted generally. However, as it is shorter to designate “uniform translationary motion” under this name, we shall adhere to the appellation.[3]We need not discuss here the difficult problems that relate to the connectivity of the continua. For instance, a point on a closed line does not divide the line into two parts, and yet the line remains one-dimensional. Problems of this sort pertain to one of the most difficult branches of geometry, namely, Analysis Situs, with which the names of Riemann, Betti and Poincaré are associated.[4]The discoveries of du Bois Reymond have shown that we could proceed still further and interpose an indefinite number of additional points, but in the present state of mathematics this extension is viewed only as a mathematical curiosity. We may mention, however, that the rejection by Hilbert of the axiom of Archimedes (to be discussed in the following chapter), leading as it does to the strange non-Archimedean geometry, would be equivalent to considering the mathematical continuum as of this more general variety.[5]That the choice of an ordering relation is all-important for the determination of dimensionality may be gathered from the following examples. Consider a number of diapasons emitting notes of various pitch. We should have no difficulty in ranging these various diapasons in order of increasing pitch. This ordering relation would be instinctive and not further analysable, since it would issue from that mysterious human capacity which enables all men to assert that one sound is shriller than another. Human appreciation is thus responsible for this definition of order in music; and we may well conceive of men whose reactions to sonorous impressions might differ from ours and who in consequence would range the diapasons in some totally different linear order. As a result, a note which we should consider as lying “between” two given notes might, in the opinion of these other beings, lie outside them. In either case, however, if we should assume that the various notes differed in pitch from one another by insensible gradations, a sensory continuum of sound pitches would have been constructed, though what we should call a continuum of pitches would appear to the other beings as a discontinuity of notes, and vice versa. In either case the respective sensory continua would be one-dimensional, since by suppressing any given note, continuity of passage would be broken. And now let us suppose that these strange beings were still more unlike us humans. Let us assume that not only would every note of their continuum appear to them as indiscernible from the one immediately preceding it and the one immediately following it, but that it would also be impossible for them to differentiate each given note from two additional notes. This assumption is by no means as arbitrary as it might appear, for we know full well that even normal human beings experience considerable difficulty in differentiating extreme heat from extreme cold and we know, too, that people afflicted with colour blindness are unable to distinguish red from green. But then, to return to our illustration, we should realise that the sound continuum, when ordered with this novel understanding of nextness or contiguity, would no longer remain one-dimensional.In other words, in virtue of this new ordering relation, imposed by the idiosyncrasies of their perceptions, the same aggregate of notes would range itself automatically into a two-dimensional sensory continuum. This is what is meant when we claim that an aggregate of elements has of itself no particular dimensionality and that some ordering relation must be imposed from without. We must realise, therefore, that when we speak of the space of our experience as being three-dimensional in points, no intrinsic property of space can be implied by this statement. It is only when account is taken of the complex of our experiences interpreted in the light of that sensory order which appears to be imposed upon out co-ordinative faculties that the statement can acquire meaning. These conclusions are by no means vitiated by such facts as our inability to get out of a closed room or to tie a knot in a space of an even number of dimensions. Unfortunately, owing to lack of time, we cannot dwell further on these difficult questions.[6]Here again we are neglecting, from motives of simplicity, our awareness of the focussing effort and of that of convergence. If we take these into account, we are in all truth considering not merely one private perspective, but a considerable number. Even so, our data would be incomplete.[7]This difficulty of counting points might be obviated to a certain degree were space to be considered discrete or atomic; for in that case we might count the atoms of space separating points and thereby establish absolute comparisons between distances. But here again the procedure would be artificial, for it would be nullified unless we were to assume that the spatial voids separating the successive atoms were always the same in magnitude; furthermore (as Dr. Silberstein points out in his book, “The Theory of Relativity”), we should be in a quandary to know how a succession of atoms would have to be defined, since this definition would depend on a definition of order. At any rate, we need not concern ourselves with atomic or discrete manifolds, for Riemann assumed that mathematical space was a continuous manifold. In view of the quantum phenomena we may eventually be led to modify these views and to attribute a discrete nature to space, but this is a vague possibility which there is no advantage in discussing in the present state of our knowledge.[8]Were it not for this restriction, comparisons of distance in space situated in different places could never be obtained by transporting rods from one place to another; for since the measurements yielded by our standard rods when they had reached their point of destination would depend essentially on the route they had followed, they could scarcely be called rigid. When the restriction is adhered to we obtain the most general type of Riemannian spaces or geometries exemplified by the three major types, the Euclidean type, the Riemannian type and the Lobatchewskian type. Weyl, however, dispenses with Riemann’s postulate and thereby obtains a more generalised type of space or geometry.The non-mathematical reader is likely to become impatient at this rejection on the part of mathematicians of apparently self-evident postulates. But it must be remembered that a postulate which can be rejected and whose contrary leads to a perfectly consistent doctrine can certainly not be regarded as rationally self-evident; so that in a number of cases the legitimacy of so-called self-evident propositions can only be discusseda posterioriand nota priori. As a matter of fact our belief in self-evident propositions is derived in the majority of cases from crude experience and we cannot exclude the possibility that more refined observations may compel us to modify our opinions in a radical way. Einstein’s discovery that Euclid’s parallel postulate would have to be rejected in the world of reality is a case in point.So far as Weyl’s exceedingly strange geometry is concerned, it is conceivable that it also may turn out to represent reality after all, for Weyl found it possible to account for the existence of electromagnetic phenomena in nature by assuming that the space-time of relativity was of the more general Weylian variety and not, as Einstein had assumed, of the more restricted Riemannian type.[9]The association of a straight line with the shortest distance between two points only holds, however, provided the several dimensions of the space are of an identical nature. When we consider continua in which the several dimensions differ in nature, as in the space-time of relativity, the straight line may turn out to be the longest distance between two points.[10]In the chapter on Weyl’s theory we shall mention Weyl’s method briefly.[11]We are referring solely to those bodies which would yield the numerical results of Riemann’s or Lobatchewski’s geometry—both these types of geometry being compatible with the homogeneity of space.[12]Also the behaviour of light during refraction proved that a form of Least Action known as Fermat’s principle of minimum time was involved.[13]In this discussion we are always assuming that the observer is standing right above the coin; hence we are not considering the variations in apparent shape due to a slanting line of sight. This latter problem is of a totally different nature.[14]The reason why Riemann refers specifically to the infinitely small is because he took it as proved that for ordinary extensions experiment had shown space to be Euclidean, and because for exceedingly great extensions he did not consider that measurements would be feasible.[15]This metrical ether must not be confused with the classical ether of optics and electromagnetics.[16]In all fairness to Einstein, however, it should be noted that he does not appear to have been influenced directly by Riemann.[17]A fact by no means evidenta priori.[18]Callingthe dimensionality of the non-Euclidean space the number of dimensions of the generating Euclidean space would beand not,as at first sight we might be inclined to believe.[19]An equivalent form of the criticisms discussed in the text consists in assuming that as Euclidean geometry is associated with the number zero, it must be logically antecedent to the non-Euclidean varieties since these are associated with non-vanishing numbers. But here, again, it is only because we take “curvature” as fundamental that Euclidean geometry is connected with zero. Should we choose to take “radius of curvature,” Euclidean geometry would be associated with infinity, and the non-Euclidean varieties with finite numbers. Hence, the problem would now resolve itself into determining whether “curvature” or “radius of curvature” was the more fundamental; and, in point of fact, “curvature” is a more complex conception than “radius of curvature.”Then again, we might characterise the various geometries by means of the parallel postulate. In this case, Riemann’s geometry would be associated with the number zero, Euclidean geometry with the number one, and Lobatchewski’s geometry with infinity. Still another method of presentation would be to approach the problem through projective geometry. Then we should find that Lobatchewski’s geometry was associated with a circle, and Euclid’s with the intersection of the line at infinity with two imaginary lines, yielding the “circular points at infinity,” whereas Riemann’s geometry would be connected with a circle of imaginary radius. With this method of representation, zero would never enter into our discussions. And no one would maintain that the concept of imaginary points at infinity (the circular points) was logically antecedent to a real circle of finite radius. In short, we see that there are a large number of different methods of representing the various geometries; and, according to the method selected, the number zero may be associated with Euclideanism or with non-Euclideanism, or, again, with neither. All that we are justified in saying is that Euclidean geometry is the easiest of the geometries, but not necessarily the most fundamental.[20]It is to be noted that had the source of light been placed at the centre of the sphere instead of at the north pole, all the great circles on the sphere would have cast straight lines for shadows on the tangent plane. Still, the shadows on the plane would, as before, have yielded Riemann’s geometry (of the elliptical variety). We see then, once again, that absolute shape, size and straightness escape us in every case, and we cannot even say that Reimann’s straight lines are curved with respect to Euclidean ones. All that is relevant is the mutual behaviour of bodies, their laws of disposition.[21]For the sake of completeness we may mention that connectivity can be studied by the same procedure as that by which dimensionality was investigated. Thus, a space was considered to be two-dimensional when it was possible to intercept the path of continuous transfer between any two points by tracing a continuous line throughout the space. If we apply this test to the surface of a doughnut, it will be seen that a continuous line, such as a circumference enclosing part of the doughnut, like a ring, would be insufficient to divide the doughnut into two separate parts which could not be connected by a route of continuous transfer over the surface. This property would differentiate the surface of a doughnut from that of a sphere or an ellipsoid, and yet the surface of the doughnut would still be two-dimensional, but its connectivity would be different owing to the hole passing through its substance.[22]An instant of time is of course a mere abstraction, but so is a mathematical point in space.[23]We are not referring solely to the tides of the sea, but also to the solid tides generated in the earth’s substance.[24]The problem is of course much more complex than would appear from the present analysis. For the slowing down of the earth’s rate of rotation, owing to tidal action, would also result in causing a retreat of the moon, producing thereby a variation in the period of its revolution round the earth.[25]This is because in Michelson’s experiment it is not necessary to consider a sphere. The two arms of the apparatus may be of different lengths; and all that is observed is the continued coincidence of the interference-bands with markings on the instrument.[26]More precisely[27]When we consider the’s in this light, we realise that the expression formight have been anticipated directly without any regard to measurement. We might, for instance, have attempted to construct in a purely mathematical way the possible expressions containing variables such asand,together with corrective factors whose rôle it would be to ensure invariance. Riemann remarked that in addition to the classical expression, a number of such expressions could be constructed. But if we wish our value ofto be compatible with the existence of the Pythagorean theorem, namely (for a right triangle), the classical expression ofmust be adhered to. For this reason the type of space which is obtained under these conditions is called Pythagorean space; and in what is to follow, we shall have no occasion to consider any other variety.[28]In a very brief way the difficulties are as follows: Differential geometry involves continuity; hence in a discrete continuum it would lose its force. But even this is not all, for there are various kinds of continuity, and continuity must be of a special type for differential geometry to remain applicable. For instance, in the foregoing exposition of the method we assumed that the expression ofwould tend to a definite limitwhen the points were taken closer and closer together. In particular we assumed that for infinitesimal areas of the surface the Pythagorean theoremwould hold. This was equivalent to stating that infinitesimal areas of our curved surface could be regarded as flat or Euclidean, hence as identified with the plane lying tangent to them. Inasmuch as the restriction of Euclideanism in the infinitesimal is precisely one that Riemann has imposed on space, we need have no fear of applying the differential method to the types of non-Euclidean space thus far discussed.Nevertheless, we may conceive of spaces wherewould not tend to a limit, and where, however tiny the area of our surface, we should still be faced with waves within wavesad infinitum. The situation would be similar to that presented by curves with no tangents at any point. However, such cases are only of theoretical interest.[29]It should be mentioned that there exists another type of curvature, different from the Gaussian curvature. This other type of curvature is called themean curvatureat a point and is given by.In this chapter we have mentioned only the Gaussian curvature, for, as Gauss discovered, it is this type of curvature alone which characterises the geometry of the surface when explored with Euclidean measuring rods. This discovery is a direct consequence of the following considerations:Two surfaces which have the same Gaussian curvature can be superposed on each other without being torn or stretched, whereas two surfaces which have different Gaussian curvatures can never be applied on each other unless we tear them or stretch them. Obviously the metric relations of a surface are not disturbed so long as we do not stretch the surface; and this is why surfaces which have the same Gaussian curvature and which can therefore be superposed without being stretched, must necessarily possess the same geometry.A few illustrations may be helpful: A plane sheet of paper has a zero Gaussian curvature and a zero mean curvature at every point. A cylinder, a cone or a roll has a zero Gaussian curvature but a positive mean curvature. It is the differences in the mean curvatures that cause these surfaces to appear to us visually as differing from the plane. But owing to the fact that the plane, the cylinder, the cone and the roll have the same zero Gaussian curvature, we can wrap the plane on to the cylinder or cone without stretching it.For this reason all these surfaces have the same Euclidean geometry (Euclidean measuring rods being used). On the other hand, aminimal surface(such as the surface defined by a soap film, stretched from wires situated in different planes) has a zero curvature just like the plane, but in contradistinction to the plane it has a negative Gaussian curvature; hence, cannot be applied to a plane, and its geometry is therefore non-Euclidean.For the same reason a sphere, whose Gaussian curvature is a constant positive number, or that saddle-shaped surface called the pseudosphere whose Gaussian curvature is a constant negative number, can neither of them be flattened out on to a plane without being stretched. As a result their geometries differ from the Euclidean geometry of the plane. We have seen that these geometries are Riemannian and Lobatchewskian, respectively.[30]If we were to represent non-Euclidean geometry as arising from the behaviour of our measuring rods, squirming when displaced as compared with Euclidean rods, we should see that Euclideanism in the infinitesimal implies that when our rods are of infinitesimal length and are displaced over infinitesimal distances, they behave in the same way as rigid Euclidean rods.[31]These invariant types of curvature are given byand.If we consider the termsandseparately, we obtain “tensors.” In a four-dimensional space there are 10 components for thetensor and 20 for thetensor. These components represent the curvatures measured in various directions.[32]It is necessary to make this distinction for a space may be flat and yet only semi-Euclidean, as will be understood in later chapters when discussing space-time.[33]The’s were also the components of a tensor.[34]We may note that the geometry, hence the nature, of a space is fully determined only when we express it in terms of the Riemann-Christoffel tensor.Thusat every point denotes perfectly flat or homaloidal space;,on the other hand, comprises a whole series of generally non-homogeneous spaces of which(the homogeneous and homaloidal variety) is only one particular illustration. Hence,by itself does not yield us any very definite information.Likewise, a perfectly homogeneous spherical space of four dimensions,i.e., the type of space which corresponds to Riemann’s geometry of four dimensions, is given bywhereis a constant depending on the intensity of the curvature. On the other hand,represents a whole series of spaces, generally non-homogeneous, of which the above-mentioned truly spherical and homogeneous variety is only one particular case.[35]Quoted from one of the standard text-books of philosophy.[36]We are endeavouring to explain things as simply as possible, but as a matter of fact the statement we are making that acceleration remains absolute in Einstein’s special theory is not quite correct. The acceleration of a body which in Newtonian science remained the same regardless of our selection of one Galilean frame or another, varies in value in the special theory under similar circumstances. Nevertheless, inasmuch as a sharp distinction still persists between velocity and acceleration, we have felt justified for reasons of simplicity in presenting the problem as we have done.[37]Subject to the restrictions mentioned in a note in the previous chapter.[38]This formulaorcan also be written:,since.We may conclude that the velocity of the ball with respect to the embankment is equal to its velocity in the train plus the velocity of the train with respect to the embankment. This expresses the classical belief that velocities add up like numbers.[39]In the classical theory, however, aberrational observations of very high refinement should reveal our speed through the ether, but observations of this sort are beyond our present powers; and Einstein’s theory has since proved that however precise our experiments, this velocity could never be revealed.[40]Although no experimental results could be claimed to have justified any such assumption, Maxwell introduced it unhesitatingly into his theory. His celebrated equations of electromagnetics represented, therefore, the results of experiment, supplemented by this additional hypothetical assumption. The advisability of making this hypothesis was accentuated when it was found to ensure the law of the conservation of electricity.[41]If two magnetic poles of equal strength, situated in empty space at a distance of one centimetre apart, attract or repel each other with a force of one dyne, either pole is said to represent one unit of magnetic pole strength in theelectromagnetic system of units. Owing to the interconnections between magnetism and electricity, we can deduce therefrom the unit of electric charge also in the electromagnetic system. Likewise, if two electric charges of equal strength, also situated in empty space at a distance of one centimetre apart, attract or repel each other with a force of one dyne, either charge is said to represent one unit of electric charge in theelectrostatic system of units. From this we may derive the unit of magnetic pole strength in the electrostatic system.[42]It is to this pressure of light that the repulsion of comets’ tails away from the sun is due.[43]By a first-order experiment, we mean one that is refined enough to detect magnitudes of the order ofwhereis the velocity of the earth through the stagnant ether andis the velocity of light. Likewise, by a second-order experiment, we mean one capable of detecting magnitudes of the order of.Inasmuch asis certainly very much smaller than,is an extremely small quantity, andis very much smaller still. So we see that a second-order experiment is necessarily very much more precise than a first-order one. We may also mention that no experiments have yet been successful in exceeding those of the second order in precision.[44]The classical or Galilean transformations were,,,,wherewas the velocity of the frame with respect to a frame at rest in the ether, hence with respect to the ether itself; this velocity being directed along theaxis and the two frames being assumed to have coincided at the initial instant.Under the same conditions the Lorentz transformations were,,;wherebeing the velocity of light. It is easy to see that whenis small compared to,the difference between the two types of transformations becomes negligible.[45]As a matter of fact the appellation “FitzGerald contraction” should also be abandoned in Einstein’s theory, since FitzGerald and Lorentz had always regarded the contraction in the light of a real physical contraction in the stagnant ether. Nevertheless, if the reader realises the difference in the two conceptions of the contraction, it will simplify matters to retain the original name.[46]It is necessary to take both of these conditions into account. If, for instance, we limited ourselves to the invariance of light’s velocity without taking into consideration the relativity of velocity, transformations such as;would also satisfy our requirements, and these would not be the Lorentz-Einstein transformations and would not be in accord with the relativity of Galilean motion.[47]These results had already been anticipated by Lorentz, since as we have seen, he considered that his transformations were applicable to purely electromagnetic magnitudes. But Lorentz was always dealing with a fixed ether and with motion through the fixed ether; hence the interpretation of these discoveries was much less satisfactory.[48]We must remember that it is only lightin vacuowhich is propagated with the speed,hence which possesses an invariant velocity. Light passing through a transparent medium moves with a velocity which is less than;hence in this case its velocity ceases to be invariant and is affected by the velocity of the medium, as also by that of the observer.[49]The objective world of science has nothing in common with the world of things-in-themselves of the metaphysician. This metaphysical world, assuming that it has any meaning at all, is irrelevant to science.[50]The identification of mass with energy necessitated by the Einstein theory allows us to obviate the paradoxical appearance of the following example: Consider an incandescent sphere radiating light in every direction. If placed at rest in some Galilean frame, it should remain there undisturbed, since the backward pressure exerted by the light radiation is distributed symmetrically round the sphere. But suppose now we view the incandescent sphere from some other Galilean frame moving away from the first with a velocity.From this new frame the incandescent sphere will be moving with a velocity,whereas the light waves radiated by it will still possess a velocityin all directions in our frame. It will then appear to us that the light waves are speedingaway from the spherein the direction of its motion with a velocitywhile they are traveling with a speedin the opposite direction. The conditions of light pressure will no longer be symmetrical, and calculation shows that the motion of the sphere should gradually slow down under the action of this resistance. Now this is impossible; for if the sphere appeared to us to be slowing down, it could never remain attached to its evenly moving Galilean frame, and we should be faced with a contradiction. As soon, however, as we recognise that the incandescent sphere by radiating light is losing energy and hence mass, the principle of the conservation of momentum shows that this gradual loss of mass would have for effect an increase in the velocity of the sphere and would thus compensate exactly the loss of velocity generated by the unsymmetrical light pressure. In this way the paradox is explained very simply.[51]A further verification was afforded when the mechanism of Bohr’s atom was studied by Sommerfeld. In Bohr’s atom the electrons revolve round the central nucleus in certain stable orbits. Sommerfeld, by taking into consideration the variations in mass of the electrons due to their motions as necessitated by Einstein’s theory, proved that in place of the sharp spectral lines which had always been observed it was legitimate to expect that more accurate observation would prove these sharp lines to be bundles of very fine ones closely huddled together. These anticipations were confirmed by experiment both qualitatively and quantitatively.[52]Whenever we refer to “another Galilean frame,” we invariably mean one in motion with respect to the first. Were it not in a state of relative motion, it would constitute the same frame.[53]We may mention that the physical existence of a finite invariant velocity is by no means impossible. It is acceptable mathematically, and the only question that we shall have to consider is whether it corresponds to physical reality.[54]By an external event we do not necessarily mean an event occurring outside our body. We mean one that does not reduce to a mere awareness of consciousness. A sudden pain in our toe would constitute an external event in exactly the same measure as would the explosion of a barrel of gunpowder a mile distant.[55]The relativity of simultaneity is a most revolutionary concept, as will be seen from the following illustration:Consider two observers, one on a train moving uniformly along a straight line, the other on the embankment. At the precise instant these two observers pass each other at a point,a flash of light is produced at the point.The light wave produced by this instantaneous flash will present the shape of an expanding sphere. Since the invariant velocity of light holds equally for either observer, we must assume that either observer will find himself at all times situated at the centre of the expanding sphere.Our first reaction might be to say: “What nonsense! How can different people, travelling apart, all be at the centre of the same sphere?” Our objection, however, would be unjustified.The fact is that the spherical surface is constantly expanding, so that the points which fix its position must be determined at the same instant of time; they must be determined simultaneously. And this is where the indeterminateness arises. The same instant of time for all the points of the surface has not the same significance for the various observers; hence each observer is in reality talking of a different instantaneous surface.[56]“The Principles of Natural Knowledge.”

[1]As exemplified in the Pythagorean discovery of the relationship between the length of a vibrating string and the pitch of its note, a discovery utilised in musical instruments. Another example is represented by Archimedes’ solution of the problem of Hieron’s gold tiara.

[2]The appellationGalilean motiondoes not appear to have been adopted generally. However, as it is shorter to designate “uniform translationary motion” under this name, we shall adhere to the appellation.

[3]We need not discuss here the difficult problems that relate to the connectivity of the continua. For instance, a point on a closed line does not divide the line into two parts, and yet the line remains one-dimensional. Problems of this sort pertain to one of the most difficult branches of geometry, namely, Analysis Situs, with which the names of Riemann, Betti and Poincaré are associated.

[4]The discoveries of du Bois Reymond have shown that we could proceed still further and interpose an indefinite number of additional points, but in the present state of mathematics this extension is viewed only as a mathematical curiosity. We may mention, however, that the rejection by Hilbert of the axiom of Archimedes (to be discussed in the following chapter), leading as it does to the strange non-Archimedean geometry, would be equivalent to considering the mathematical continuum as of this more general variety.

[5]That the choice of an ordering relation is all-important for the determination of dimensionality may be gathered from the following examples. Consider a number of diapasons emitting notes of various pitch. We should have no difficulty in ranging these various diapasons in order of increasing pitch. This ordering relation would be instinctive and not further analysable, since it would issue from that mysterious human capacity which enables all men to assert that one sound is shriller than another. Human appreciation is thus responsible for this definition of order in music; and we may well conceive of men whose reactions to sonorous impressions might differ from ours and who in consequence would range the diapasons in some totally different linear order. As a result, a note which we should consider as lying “between” two given notes might, in the opinion of these other beings, lie outside them. In either case, however, if we should assume that the various notes differed in pitch from one another by insensible gradations, a sensory continuum of sound pitches would have been constructed, though what we should call a continuum of pitches would appear to the other beings as a discontinuity of notes, and vice versa. In either case the respective sensory continua would be one-dimensional, since by suppressing any given note, continuity of passage would be broken. And now let us suppose that these strange beings were still more unlike us humans. Let us assume that not only would every note of their continuum appear to them as indiscernible from the one immediately preceding it and the one immediately following it, but that it would also be impossible for them to differentiate each given note from two additional notes. This assumption is by no means as arbitrary as it might appear, for we know full well that even normal human beings experience considerable difficulty in differentiating extreme heat from extreme cold and we know, too, that people afflicted with colour blindness are unable to distinguish red from green. But then, to return to our illustration, we should realise that the sound continuum, when ordered with this novel understanding of nextness or contiguity, would no longer remain one-dimensional.In other words, in virtue of this new ordering relation, imposed by the idiosyncrasies of their perceptions, the same aggregate of notes would range itself automatically into a two-dimensional sensory continuum. This is what is meant when we claim that an aggregate of elements has of itself no particular dimensionality and that some ordering relation must be imposed from without. We must realise, therefore, that when we speak of the space of our experience as being three-dimensional in points, no intrinsic property of space can be implied by this statement. It is only when account is taken of the complex of our experiences interpreted in the light of that sensory order which appears to be imposed upon out co-ordinative faculties that the statement can acquire meaning. These conclusions are by no means vitiated by such facts as our inability to get out of a closed room or to tie a knot in a space of an even number of dimensions. Unfortunately, owing to lack of time, we cannot dwell further on these difficult questions.

In other words, in virtue of this new ordering relation, imposed by the idiosyncrasies of their perceptions, the same aggregate of notes would range itself automatically into a two-dimensional sensory continuum. This is what is meant when we claim that an aggregate of elements has of itself no particular dimensionality and that some ordering relation must be imposed from without. We must realise, therefore, that when we speak of the space of our experience as being three-dimensional in points, no intrinsic property of space can be implied by this statement. It is only when account is taken of the complex of our experiences interpreted in the light of that sensory order which appears to be imposed upon out co-ordinative faculties that the statement can acquire meaning. These conclusions are by no means vitiated by such facts as our inability to get out of a closed room or to tie a knot in a space of an even number of dimensions. Unfortunately, owing to lack of time, we cannot dwell further on these difficult questions.

[6]Here again we are neglecting, from motives of simplicity, our awareness of the focussing effort and of that of convergence. If we take these into account, we are in all truth considering not merely one private perspective, but a considerable number. Even so, our data would be incomplete.

[7]This difficulty of counting points might be obviated to a certain degree were space to be considered discrete or atomic; for in that case we might count the atoms of space separating points and thereby establish absolute comparisons between distances. But here again the procedure would be artificial, for it would be nullified unless we were to assume that the spatial voids separating the successive atoms were always the same in magnitude; furthermore (as Dr. Silberstein points out in his book, “The Theory of Relativity”), we should be in a quandary to know how a succession of atoms would have to be defined, since this definition would depend on a definition of order. At any rate, we need not concern ourselves with atomic or discrete manifolds, for Riemann assumed that mathematical space was a continuous manifold. In view of the quantum phenomena we may eventually be led to modify these views and to attribute a discrete nature to space, but this is a vague possibility which there is no advantage in discussing in the present state of our knowledge.

[8]Were it not for this restriction, comparisons of distance in space situated in different places could never be obtained by transporting rods from one place to another; for since the measurements yielded by our standard rods when they had reached their point of destination would depend essentially on the route they had followed, they could scarcely be called rigid. When the restriction is adhered to we obtain the most general type of Riemannian spaces or geometries exemplified by the three major types, the Euclidean type, the Riemannian type and the Lobatchewskian type. Weyl, however, dispenses with Riemann’s postulate and thereby obtains a more generalised type of space or geometry.The non-mathematical reader is likely to become impatient at this rejection on the part of mathematicians of apparently self-evident postulates. But it must be remembered that a postulate which can be rejected and whose contrary leads to a perfectly consistent doctrine can certainly not be regarded as rationally self-evident; so that in a number of cases the legitimacy of so-called self-evident propositions can only be discusseda posterioriand nota priori. As a matter of fact our belief in self-evident propositions is derived in the majority of cases from crude experience and we cannot exclude the possibility that more refined observations may compel us to modify our opinions in a radical way. Einstein’s discovery that Euclid’s parallel postulate would have to be rejected in the world of reality is a case in point.So far as Weyl’s exceedingly strange geometry is concerned, it is conceivable that it also may turn out to represent reality after all, for Weyl found it possible to account for the existence of electromagnetic phenomena in nature by assuming that the space-time of relativity was of the more general Weylian variety and not, as Einstein had assumed, of the more restricted Riemannian type.

The non-mathematical reader is likely to become impatient at this rejection on the part of mathematicians of apparently self-evident postulates. But it must be remembered that a postulate which can be rejected and whose contrary leads to a perfectly consistent doctrine can certainly not be regarded as rationally self-evident; so that in a number of cases the legitimacy of so-called self-evident propositions can only be discusseda posterioriand nota priori. As a matter of fact our belief in self-evident propositions is derived in the majority of cases from crude experience and we cannot exclude the possibility that more refined observations may compel us to modify our opinions in a radical way. Einstein’s discovery that Euclid’s parallel postulate would have to be rejected in the world of reality is a case in point.

So far as Weyl’s exceedingly strange geometry is concerned, it is conceivable that it also may turn out to represent reality after all, for Weyl found it possible to account for the existence of electromagnetic phenomena in nature by assuming that the space-time of relativity was of the more general Weylian variety and not, as Einstein had assumed, of the more restricted Riemannian type.

[9]The association of a straight line with the shortest distance between two points only holds, however, provided the several dimensions of the space are of an identical nature. When we consider continua in which the several dimensions differ in nature, as in the space-time of relativity, the straight line may turn out to be the longest distance between two points.

[10]In the chapter on Weyl’s theory we shall mention Weyl’s method briefly.

[11]We are referring solely to those bodies which would yield the numerical results of Riemann’s or Lobatchewski’s geometry—both these types of geometry being compatible with the homogeneity of space.

[12]Also the behaviour of light during refraction proved that a form of Least Action known as Fermat’s principle of minimum time was involved.

[13]In this discussion we are always assuming that the observer is standing right above the coin; hence we are not considering the variations in apparent shape due to a slanting line of sight. This latter problem is of a totally different nature.

[14]The reason why Riemann refers specifically to the infinitely small is because he took it as proved that for ordinary extensions experiment had shown space to be Euclidean, and because for exceedingly great extensions he did not consider that measurements would be feasible.

[15]This metrical ether must not be confused with the classical ether of optics and electromagnetics.

[16]In all fairness to Einstein, however, it should be noted that he does not appear to have been influenced directly by Riemann.

[17]A fact by no means evidenta priori.

[18]Callingthe dimensionality of the non-Euclidean space the number of dimensions of the generating Euclidean space would beand not,as at first sight we might be inclined to believe.

[19]An equivalent form of the criticisms discussed in the text consists in assuming that as Euclidean geometry is associated with the number zero, it must be logically antecedent to the non-Euclidean varieties since these are associated with non-vanishing numbers. But here, again, it is only because we take “curvature” as fundamental that Euclidean geometry is connected with zero. Should we choose to take “radius of curvature,” Euclidean geometry would be associated with infinity, and the non-Euclidean varieties with finite numbers. Hence, the problem would now resolve itself into determining whether “curvature” or “radius of curvature” was the more fundamental; and, in point of fact, “curvature” is a more complex conception than “radius of curvature.”Then again, we might characterise the various geometries by means of the parallel postulate. In this case, Riemann’s geometry would be associated with the number zero, Euclidean geometry with the number one, and Lobatchewski’s geometry with infinity. Still another method of presentation would be to approach the problem through projective geometry. Then we should find that Lobatchewski’s geometry was associated with a circle, and Euclid’s with the intersection of the line at infinity with two imaginary lines, yielding the “circular points at infinity,” whereas Riemann’s geometry would be connected with a circle of imaginary radius. With this method of representation, zero would never enter into our discussions. And no one would maintain that the concept of imaginary points at infinity (the circular points) was logically antecedent to a real circle of finite radius. In short, we see that there are a large number of different methods of representing the various geometries; and, according to the method selected, the number zero may be associated with Euclideanism or with non-Euclideanism, or, again, with neither. All that we are justified in saying is that Euclidean geometry is the easiest of the geometries, but not necessarily the most fundamental.

Then again, we might characterise the various geometries by means of the parallel postulate. In this case, Riemann’s geometry would be associated with the number zero, Euclidean geometry with the number one, and Lobatchewski’s geometry with infinity. Still another method of presentation would be to approach the problem through projective geometry. Then we should find that Lobatchewski’s geometry was associated with a circle, and Euclid’s with the intersection of the line at infinity with two imaginary lines, yielding the “circular points at infinity,” whereas Riemann’s geometry would be connected with a circle of imaginary radius. With this method of representation, zero would never enter into our discussions. And no one would maintain that the concept of imaginary points at infinity (the circular points) was logically antecedent to a real circle of finite radius. In short, we see that there are a large number of different methods of representing the various geometries; and, according to the method selected, the number zero may be associated with Euclideanism or with non-Euclideanism, or, again, with neither. All that we are justified in saying is that Euclidean geometry is the easiest of the geometries, but not necessarily the most fundamental.

[20]It is to be noted that had the source of light been placed at the centre of the sphere instead of at the north pole, all the great circles on the sphere would have cast straight lines for shadows on the tangent plane. Still, the shadows on the plane would, as before, have yielded Riemann’s geometry (of the elliptical variety). We see then, once again, that absolute shape, size and straightness escape us in every case, and we cannot even say that Reimann’s straight lines are curved with respect to Euclidean ones. All that is relevant is the mutual behaviour of bodies, their laws of disposition.

[21]For the sake of completeness we may mention that connectivity can be studied by the same procedure as that by which dimensionality was investigated. Thus, a space was considered to be two-dimensional when it was possible to intercept the path of continuous transfer between any two points by tracing a continuous line throughout the space. If we apply this test to the surface of a doughnut, it will be seen that a continuous line, such as a circumference enclosing part of the doughnut, like a ring, would be insufficient to divide the doughnut into two separate parts which could not be connected by a route of continuous transfer over the surface. This property would differentiate the surface of a doughnut from that of a sphere or an ellipsoid, and yet the surface of the doughnut would still be two-dimensional, but its connectivity would be different owing to the hole passing through its substance.

[22]An instant of time is of course a mere abstraction, but so is a mathematical point in space.

[23]We are not referring solely to the tides of the sea, but also to the solid tides generated in the earth’s substance.

[24]The problem is of course much more complex than would appear from the present analysis. For the slowing down of the earth’s rate of rotation, owing to tidal action, would also result in causing a retreat of the moon, producing thereby a variation in the period of its revolution round the earth.

[25]This is because in Michelson’s experiment it is not necessary to consider a sphere. The two arms of the apparatus may be of different lengths; and all that is observed is the continued coincidence of the interference-bands with markings on the instrument.

[26]More precisely

[27]When we consider the’s in this light, we realise that the expression formight have been anticipated directly without any regard to measurement. We might, for instance, have attempted to construct in a purely mathematical way the possible expressions containing variables such asand,together with corrective factors whose rôle it would be to ensure invariance. Riemann remarked that in addition to the classical expression, a number of such expressions could be constructed. But if we wish our value ofto be compatible with the existence of the Pythagorean theorem, namely (for a right triangle), the classical expression ofmust be adhered to. For this reason the type of space which is obtained under these conditions is called Pythagorean space; and in what is to follow, we shall have no occasion to consider any other variety.

[28]In a very brief way the difficulties are as follows: Differential geometry involves continuity; hence in a discrete continuum it would lose its force. But even this is not all, for there are various kinds of continuity, and continuity must be of a special type for differential geometry to remain applicable. For instance, in the foregoing exposition of the method we assumed that the expression ofwould tend to a definite limitwhen the points were taken closer and closer together. In particular we assumed that for infinitesimal areas of the surface the Pythagorean theoremwould hold. This was equivalent to stating that infinitesimal areas of our curved surface could be regarded as flat or Euclidean, hence as identified with the plane lying tangent to them. Inasmuch as the restriction of Euclideanism in the infinitesimal is precisely one that Riemann has imposed on space, we need have no fear of applying the differential method to the types of non-Euclidean space thus far discussed.Nevertheless, we may conceive of spaces wherewould not tend to a limit, and where, however tiny the area of our surface, we should still be faced with waves within wavesad infinitum. The situation would be similar to that presented by curves with no tangents at any point. However, such cases are only of theoretical interest.

Nevertheless, we may conceive of spaces wherewould not tend to a limit, and where, however tiny the area of our surface, we should still be faced with waves within wavesad infinitum. The situation would be similar to that presented by curves with no tangents at any point. However, such cases are only of theoretical interest.

[29]It should be mentioned that there exists another type of curvature, different from the Gaussian curvature. This other type of curvature is called themean curvatureat a point and is given by.In this chapter we have mentioned only the Gaussian curvature, for, as Gauss discovered, it is this type of curvature alone which characterises the geometry of the surface when explored with Euclidean measuring rods. This discovery is a direct consequence of the following considerations:Two surfaces which have the same Gaussian curvature can be superposed on each other without being torn or stretched, whereas two surfaces which have different Gaussian curvatures can never be applied on each other unless we tear them or stretch them. Obviously the metric relations of a surface are not disturbed so long as we do not stretch the surface; and this is why surfaces which have the same Gaussian curvature and which can therefore be superposed without being stretched, must necessarily possess the same geometry.A few illustrations may be helpful: A plane sheet of paper has a zero Gaussian curvature and a zero mean curvature at every point. A cylinder, a cone or a roll has a zero Gaussian curvature but a positive mean curvature. It is the differences in the mean curvatures that cause these surfaces to appear to us visually as differing from the plane. But owing to the fact that the plane, the cylinder, the cone and the roll have the same zero Gaussian curvature, we can wrap the plane on to the cylinder or cone without stretching it.For this reason all these surfaces have the same Euclidean geometry (Euclidean measuring rods being used). On the other hand, aminimal surface(such as the surface defined by a soap film, stretched from wires situated in different planes) has a zero curvature just like the plane, but in contradistinction to the plane it has a negative Gaussian curvature; hence, cannot be applied to a plane, and its geometry is therefore non-Euclidean.For the same reason a sphere, whose Gaussian curvature is a constant positive number, or that saddle-shaped surface called the pseudosphere whose Gaussian curvature is a constant negative number, can neither of them be flattened out on to a plane without being stretched. As a result their geometries differ from the Euclidean geometry of the plane. We have seen that these geometries are Riemannian and Lobatchewskian, respectively.

[29]It should be mentioned that there exists another type of curvature, different from the Gaussian curvature. This other type of curvature is called themean curvatureat a point and is given by.

In this chapter we have mentioned only the Gaussian curvature, for, as Gauss discovered, it is this type of curvature alone which characterises the geometry of the surface when explored with Euclidean measuring rods. This discovery is a direct consequence of the following considerations:

Two surfaces which have the same Gaussian curvature can be superposed on each other without being torn or stretched, whereas two surfaces which have different Gaussian curvatures can never be applied on each other unless we tear them or stretch them. Obviously the metric relations of a surface are not disturbed so long as we do not stretch the surface; and this is why surfaces which have the same Gaussian curvature and which can therefore be superposed without being stretched, must necessarily possess the same geometry.

A few illustrations may be helpful: A plane sheet of paper has a zero Gaussian curvature and a zero mean curvature at every point. A cylinder, a cone or a roll has a zero Gaussian curvature but a positive mean curvature. It is the differences in the mean curvatures that cause these surfaces to appear to us visually as differing from the plane. But owing to the fact that the plane, the cylinder, the cone and the roll have the same zero Gaussian curvature, we can wrap the plane on to the cylinder or cone without stretching it.

For this reason all these surfaces have the same Euclidean geometry (Euclidean measuring rods being used). On the other hand, aminimal surface(such as the surface defined by a soap film, stretched from wires situated in different planes) has a zero curvature just like the plane, but in contradistinction to the plane it has a negative Gaussian curvature; hence, cannot be applied to a plane, and its geometry is therefore non-Euclidean.

For the same reason a sphere, whose Gaussian curvature is a constant positive number, or that saddle-shaped surface called the pseudosphere whose Gaussian curvature is a constant negative number, can neither of them be flattened out on to a plane without being stretched. As a result their geometries differ from the Euclidean geometry of the plane. We have seen that these geometries are Riemannian and Lobatchewskian, respectively.

[30]If we were to represent non-Euclidean geometry as arising from the behaviour of our measuring rods, squirming when displaced as compared with Euclidean rods, we should see that Euclideanism in the infinitesimal implies that when our rods are of infinitesimal length and are displaced over infinitesimal distances, they behave in the same way as rigid Euclidean rods.

[31]These invariant types of curvature are given byand.If we consider the termsandseparately, we obtain “tensors.” In a four-dimensional space there are 10 components for thetensor and 20 for thetensor. These components represent the curvatures measured in various directions.

[32]It is necessary to make this distinction for a space may be flat and yet only semi-Euclidean, as will be understood in later chapters when discussing space-time.

[33]The’s were also the components of a tensor.

[34]We may note that the geometry, hence the nature, of a space is fully determined only when we express it in terms of the Riemann-Christoffel tensor.Thusat every point denotes perfectly flat or homaloidal space;,on the other hand, comprises a whole series of generally non-homogeneous spaces of which(the homogeneous and homaloidal variety) is only one particular illustration. Hence,by itself does not yield us any very definite information.Likewise, a perfectly homogeneous spherical space of four dimensions,i.e., the type of space which corresponds to Riemann’s geometry of four dimensions, is given bywhereis a constant depending on the intensity of the curvature. On the other hand,represents a whole series of spaces, generally non-homogeneous, of which the above-mentioned truly spherical and homogeneous variety is only one particular case.

Likewise, a perfectly homogeneous spherical space of four dimensions,i.e., the type of space which corresponds to Riemann’s geometry of four dimensions, is given bywhereis a constant depending on the intensity of the curvature. On the other hand,represents a whole series of spaces, generally non-homogeneous, of which the above-mentioned truly spherical and homogeneous variety is only one particular case.

[35]Quoted from one of the standard text-books of philosophy.

[36]We are endeavouring to explain things as simply as possible, but as a matter of fact the statement we are making that acceleration remains absolute in Einstein’s special theory is not quite correct. The acceleration of a body which in Newtonian science remained the same regardless of our selection of one Galilean frame or another, varies in value in the special theory under similar circumstances. Nevertheless, inasmuch as a sharp distinction still persists between velocity and acceleration, we have felt justified for reasons of simplicity in presenting the problem as we have done.

[37]Subject to the restrictions mentioned in a note in the previous chapter.

[38]This formulaorcan also be written:,since.We may conclude that the velocity of the ball with respect to the embankment is equal to its velocity in the train plus the velocity of the train with respect to the embankment. This expresses the classical belief that velocities add up like numbers.

[39]In the classical theory, however, aberrational observations of very high refinement should reveal our speed through the ether, but observations of this sort are beyond our present powers; and Einstein’s theory has since proved that however precise our experiments, this velocity could never be revealed.

[40]Although no experimental results could be claimed to have justified any such assumption, Maxwell introduced it unhesitatingly into his theory. His celebrated equations of electromagnetics represented, therefore, the results of experiment, supplemented by this additional hypothetical assumption. The advisability of making this hypothesis was accentuated when it was found to ensure the law of the conservation of electricity.

[41]If two magnetic poles of equal strength, situated in empty space at a distance of one centimetre apart, attract or repel each other with a force of one dyne, either pole is said to represent one unit of magnetic pole strength in theelectromagnetic system of units. Owing to the interconnections between magnetism and electricity, we can deduce therefrom the unit of electric charge also in the electromagnetic system. Likewise, if two electric charges of equal strength, also situated in empty space at a distance of one centimetre apart, attract or repel each other with a force of one dyne, either charge is said to represent one unit of electric charge in theelectrostatic system of units. From this we may derive the unit of magnetic pole strength in the electrostatic system.

[42]It is to this pressure of light that the repulsion of comets’ tails away from the sun is due.

[43]By a first-order experiment, we mean one that is refined enough to detect magnitudes of the order ofwhereis the velocity of the earth through the stagnant ether andis the velocity of light. Likewise, by a second-order experiment, we mean one capable of detecting magnitudes of the order of.Inasmuch asis certainly very much smaller than,is an extremely small quantity, andis very much smaller still. So we see that a second-order experiment is necessarily very much more precise than a first-order one. We may also mention that no experiments have yet been successful in exceeding those of the second order in precision.

[44]The classical or Galilean transformations were,,,,wherewas the velocity of the frame with respect to a frame at rest in the ether, hence with respect to the ether itself; this velocity being directed along theaxis and the two frames being assumed to have coincided at the initial instant.Under the same conditions the Lorentz transformations were,,;wherebeing the velocity of light. It is easy to see that whenis small compared to,the difference between the two types of transformations becomes negligible.

[45]As a matter of fact the appellation “FitzGerald contraction” should also be abandoned in Einstein’s theory, since FitzGerald and Lorentz had always regarded the contraction in the light of a real physical contraction in the stagnant ether. Nevertheless, if the reader realises the difference in the two conceptions of the contraction, it will simplify matters to retain the original name.

[46]It is necessary to take both of these conditions into account. If, for instance, we limited ourselves to the invariance of light’s velocity without taking into consideration the relativity of velocity, transformations such as;would also satisfy our requirements, and these would not be the Lorentz-Einstein transformations and would not be in accord with the relativity of Galilean motion.

[47]These results had already been anticipated by Lorentz, since as we have seen, he considered that his transformations were applicable to purely electromagnetic magnitudes. But Lorentz was always dealing with a fixed ether and with motion through the fixed ether; hence the interpretation of these discoveries was much less satisfactory.

[48]We must remember that it is only lightin vacuowhich is propagated with the speed,hence which possesses an invariant velocity. Light passing through a transparent medium moves with a velocity which is less than;hence in this case its velocity ceases to be invariant and is affected by the velocity of the medium, as also by that of the observer.

[49]The objective world of science has nothing in common with the world of things-in-themselves of the metaphysician. This metaphysical world, assuming that it has any meaning at all, is irrelevant to science.

[50]The identification of mass with energy necessitated by the Einstein theory allows us to obviate the paradoxical appearance of the following example: Consider an incandescent sphere radiating light in every direction. If placed at rest in some Galilean frame, it should remain there undisturbed, since the backward pressure exerted by the light radiation is distributed symmetrically round the sphere. But suppose now we view the incandescent sphere from some other Galilean frame moving away from the first with a velocity.From this new frame the incandescent sphere will be moving with a velocity,whereas the light waves radiated by it will still possess a velocityin all directions in our frame. It will then appear to us that the light waves are speedingaway from the spherein the direction of its motion with a velocitywhile they are traveling with a speedin the opposite direction. The conditions of light pressure will no longer be symmetrical, and calculation shows that the motion of the sphere should gradually slow down under the action of this resistance. Now this is impossible; for if the sphere appeared to us to be slowing down, it could never remain attached to its evenly moving Galilean frame, and we should be faced with a contradiction. As soon, however, as we recognise that the incandescent sphere by radiating light is losing energy and hence mass, the principle of the conservation of momentum shows that this gradual loss of mass would have for effect an increase in the velocity of the sphere and would thus compensate exactly the loss of velocity generated by the unsymmetrical light pressure. In this way the paradox is explained very simply.

[51]A further verification was afforded when the mechanism of Bohr’s atom was studied by Sommerfeld. In Bohr’s atom the electrons revolve round the central nucleus in certain stable orbits. Sommerfeld, by taking into consideration the variations in mass of the electrons due to their motions as necessitated by Einstein’s theory, proved that in place of the sharp spectral lines which had always been observed it was legitimate to expect that more accurate observation would prove these sharp lines to be bundles of very fine ones closely huddled together. These anticipations were confirmed by experiment both qualitatively and quantitatively.

[52]Whenever we refer to “another Galilean frame,” we invariably mean one in motion with respect to the first. Were it not in a state of relative motion, it would constitute the same frame.

[53]We may mention that the physical existence of a finite invariant velocity is by no means impossible. It is acceptable mathematically, and the only question that we shall have to consider is whether it corresponds to physical reality.

[54]By an external event we do not necessarily mean an event occurring outside our body. We mean one that does not reduce to a mere awareness of consciousness. A sudden pain in our toe would constitute an external event in exactly the same measure as would the explosion of a barrel of gunpowder a mile distant.

[55]The relativity of simultaneity is a most revolutionary concept, as will be seen from the following illustration:Consider two observers, one on a train moving uniformly along a straight line, the other on the embankment. At the precise instant these two observers pass each other at a point,a flash of light is produced at the point.The light wave produced by this instantaneous flash will present the shape of an expanding sphere. Since the invariant velocity of light holds equally for either observer, we must assume that either observer will find himself at all times situated at the centre of the expanding sphere.Our first reaction might be to say: “What nonsense! How can different people, travelling apart, all be at the centre of the same sphere?” Our objection, however, would be unjustified.The fact is that the spherical surface is constantly expanding, so that the points which fix its position must be determined at the same instant of time; they must be determined simultaneously. And this is where the indeterminateness arises. The same instant of time for all the points of the surface has not the same significance for the various observers; hence each observer is in reality talking of a different instantaneous surface.

[55]The relativity of simultaneity is a most revolutionary concept, as will be seen from the following illustration:

Consider two observers, one on a train moving uniformly along a straight line, the other on the embankment. At the precise instant these two observers pass each other at a point,a flash of light is produced at the point.The light wave produced by this instantaneous flash will present the shape of an expanding sphere. Since the invariant velocity of light holds equally for either observer, we must assume that either observer will find himself at all times situated at the centre of the expanding sphere.

Our first reaction might be to say: “What nonsense! How can different people, travelling apart, all be at the centre of the same sphere?” Our objection, however, would be unjustified.

The fact is that the spherical surface is constantly expanding, so that the points which fix its position must be determined at the same instant of time; they must be determined simultaneously. And this is where the indeterminateness arises. The same instant of time for all the points of the surface has not the same significance for the various observers; hence each observer is in reality talking of a different instantaneous surface.

[56]“The Principles of Natural Knowledge.”

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